Chemical evolution from cores to disks

Chemical evolution from cores to disks
Chemical evolution from cores to disks
Chemical evolution from cores to disks
PROEFSCHRIFT
ter verkrijging van
de graad van Doctor aan de Universiteit Leiden,
op gezag van de Rector Magnificus prof. mr. P. F. van der Heijden,
volgens besluit van het College voor Promoties
te verdedigen op woensdag 21 oktober 2009
klokke 15.00 uur
door
Ruud Visser
geboren te Amersfoort
in 1983
Promotiecommisie
Promotor:
Prof. dr. E. F. van Dishoeck
Overige leden:
Prof. dr. H. V. J. Linnartz
Prof. dr. K. Kuijken
Dr. M. R. Hogerheijde
Prof. dr. G. A. Blake
Prof. dr. S. D. Doty
Prof. dr. E. A. Bergin
Dr. C. P. Dullemond
California Institute of Technology
Denison University
University of Michigan
Max-Plank-Institut für Astronomie
voor papa & mama
“Welcome the task that makes you go beyond yourself.”
– Frank McGee
Inhoudsopgave
1
2
3
Introduction
1.1 From ancient astronomy to modern astrochemistry
1.2 Low-mass star formation and the role of chemistry
1.3 Planets, comets and meteorites . . . . . . . . . . .
1.4 Chemical models . . . . . . . . . . . . . . . . . .
1.4.1 Historical development and reaction types .
1.4.2 Solution methods . . . . . . . . . . . . . .
1.5 This thesis . . . . . . . . . . . . . . . . . . . . . .
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1
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4
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10
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14
The chemical history of molecules in circumstellar disks
I: Ices
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Envelope . . . . . . . . . . . . . . . . . . .
2.2.2 Disk . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Star . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Temperature . . . . . . . . . . . . . . . . .
2.2.5 Accretion shock . . . . . . . . . . . . . . .
2.2.6 Model parameters . . . . . . . . . . . . . . .
2.2.7 Adsorption and desorption . . . . . . . . . .
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Density profiles and infall trajectories . . . .
2.3.2 Temperature profiles . . . . . . . . . . . . .
2.3.3 Gas and ice abundances . . . . . . . . . . .
2.3.4 Temperature histories . . . . . . . . . . . . .
2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Model parameters . . . . . . . . . . . . . . .
2.4.2 Complex organic molecules . . . . . . . . .
2.4.3 Mixed CO-H2 O ices . . . . . . . . . . . . .
2.4.4 Implications for comets . . . . . . . . . . . .
2.4.5 Limitations of the model . . . . . . . . . . .
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . .
Appendix: Disk formation efficiency . . . . . . . . . . . .
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19
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56
The chemical history of molecules in circumstellar disks
II: Gas-phase species
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Collapse model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
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ix
Inhoudsopgave
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. 63
. 64
. 65
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. 79
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. 97
. 97
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. 102
. 102
4 Sub-Keplerian accretion onto circumstellar disks
4.1 Introduction . . . . . . . . . . . . . . . . . . . .
4.2 Equations . . . . . . . . . . . . . . . . . . . . .
4.3 Size and mass of the disk . . . . . . . . . . . . .
4.4 Gas-ice ratios . . . . . . . . . . . . . . . . . . .
4.5 Crystalline silicates . . . . . . . . . . . . . . . .
4.5.1 Observations and previous model results .
4.5.2 New model results . . . . . . . . . . . .
4.5.3 Discussion and future work . . . . . . .
4.6 Conclusions . . . . . . . . . . . . . . . . . . . .
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105
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5 The photodissociation and chemistry of CO isotopologues:
applications to interstellar clouds and circumstellar disks
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Molecular data . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Band positions and identifications . . . . . . . .
5.2.2 Rotational constants . . . . . . . . . . . . . . .
5.2.3 Oscillator strengths . . . . . . . . . . . . . . . .
5.2.4 Lifetimes and predissociation probabilities . . .
5.2.5 Atomic and molecular hydrogen . . . . . . . . .
5.3 Depth-dependent photodissociation . . . . . . . . . . . .
5.3.1 Default model parameters . . . . . . . . . . . .
5.3.2 Unshielded photodissociation rates . . . . . . . .
5.3.3 Shielding by CO, H2 and H . . . . . . . . . . .
5.3.4 Continuum shielding by dust . . . . . . . . . . .
5.3.5 Uncertainties . . . . . . . . . . . . . . . . . . .
5.4 Excitation temperature and Doppler width . . . . . . . .
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3.3
3.4
3.5
3.6
3.7
3.8
x
3.2.1 Step-wise summary . . . . . . . . . .
3.2.2 Differences with Chapter 2 . . . . . .
3.2.3 Radiation field . . . . . . . . . . . .
Chemical network . . . . . . . . . . . . . . .
3.3.1 Photodissociation and photoionisation
3.3.2 Gas-grain interactions . . . . . . . .
Results from the pre-collapse phase . . . . .
Results from the collapse phase . . . . . . . .
3.5.1 One single parcel . . . . . . . . . . .
3.5.2 Other parcels . . . . . . . . . . . . .
Chemical history versus local chemistry . . .
Discussion . . . . . . . . . . . . . . . . . . .
3.7.1 Caveats . . . . . . . . . . . . . . . .
3.7.2 Comets . . . . . . . . . . . . . . . .
3.7.3 Collapse models: 1D versus 2D . . .
Conclusions . . . . . . . . . . . . . . . . . .
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Inhoudsopgave
5.5
5.6
5.7
6
5.4.1 Increasing T ex (CO) . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Increasing b(CO) . . . . . . . . . . . . . . . . . . . . . . .
5.4.3 Increasing T ex (H2 ) or b(H2 ) . . . . . . . . . . . . . . . . .
5.4.4 Grid of T ex and b . . . . . . . . . . . . . . . . . . . . . . .
Shielding function approximations . . . . . . . . . . . . . . . . . .
5.5.1 Shielding functions on a grid of N(CO) and N(H2 ) . . . . .
5.5.2 Comparison between the full model and the approximations
Chemistry of CO: astrophysical implications . . . . . . . . . . . . .
5.6.1 Translucent clouds . . . . . . . . . . . . . . . . . . . . . .
5.6.2 Photon-dominated regions . . . . . . . . . . . . . . . . . .
5.6.3 Circumstellar disks . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PAH chemistry and IR emission from circumstellar disks
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
6.2 PAH model . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Characterisation of PAHs . . . . . . . . . . . .
6.2.2 Photoprocesses . . . . . . . . . . . . . . . . .
6.2.3 Absorption cross sections . . . . . . . . . . .
6.2.4 Electron recombination and attachment . . . .
6.2.5 Hydrogen addition . . . . . . . . . . . . . . .
6.2.6 PAH growth and destruction . . . . . . . . . .
6.2.7 Other chemical processes . . . . . . . . . . . .
6.3 Disk model . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Computational code . . . . . . . . . . . . . .
6.3.2 Template disk with PAHs . . . . . . . . . . . .
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4.1 PAH chemistry . . . . . . . . . . . . . . . . .
6.4.2 PAH emission . . . . . . . . . . . . . . . . . .
6.4.3 Other PAHs . . . . . . . . . . . . . . . . . . .
6.4.4 Spatial extent of the PAH emission . . . . . . .
6.4.5 Sensitivity analysis . . . . . . . . . . . . . . .
6.4.6 T Tauri stars . . . . . . . . . . . . . . . . . .
6.4.7 Comparison with observations . . . . . . . . .
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . .
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Nederlandse samenvatting
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Literatuur
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Publicaties
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Curriculum vitae
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Nawoord
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xi
Introduction
1
1
Chapter 1 – Introduction
1.1 From ancient astronomy to modern astrochemistry
Mankind has ever been fascinated with the stars. Dating back to the most ancient of times,
human life has been governed by the endless cycles of day and night and of winter, spring,
summer and fall. Already in the early Stone Age, people must have seen a relationship
between the rising and setting of the Sun on the one hand, and dark turning into light
turning back into dark on the other hand. Having established the Sun as the cause of light
and warmth, it is a logical next step to wonder whether other objects in the sky could
equally influence life on Earth. As George Forbes wrote in his book History of Astronomy
(1909), “this led to a search for other signs in the heavens. If the appearance of a comet
was sometimes noted simultaneously with the death of a great ruler, or an eclipse with a
scourge of plague, these might well be looked upon as causes in the same sense that the
veering or backing of the wind is regarded as a cause of fine or foul weather.”
With today’s knowledge, it is easy to say that comets, eclipses and most other astronomical phenomena do not altogether affect our lives as much as our ancestors believed.
Nonetheless, their suppositions led them to keep detailed records of anything remarkable
taking place in the sky. Clay tablets surviving from Babylonia show that people were
keeping track of solar eclipses at least as far back as 1062 B.C. (Cowell 1905). In China,
c. 2450 B.C., the emperor Zhuanxu apparently saw a conjunction of Mercury, Mars, Saturn and Jupiter on the same day that the Moon was in conjunction with the Sun (Chambers 1889, Hail & Leavens 1940). The preferential east-west and north-south alignment
of graves and bodies in burial sites from the late Stone Age suggests that mankind was
engaged in primitive astronomy as early as 4500 B.C. (Schmidt-Kaler & Schlosser 1984).
If the Babylonians laid the foundations for modern astronomy, it was the Greek who
started building in earnest. From around 600 B.C. onwards, scholars like Thales, Pythagoras, Anaxagoras, Plato and Eratosthenes performed many revolutionary measurements
and observations, and devised many equally revolutionary theories about the Earth and
the Moon, the planets, and the Sun and the stars (Lewis 1862). The most prolific of the
Greek astronomers was Hipparchus (c. 190–120 B.C.), regarded by Forbes (1909) as the
founder of observational astronomy. Amongst other contributions, he compiled the first
comprehensive star catalogue and discovered the precession of the Earth’s axis. Another
key figure from the classical period is the Roman Ptolemy, who, around 150 A.D., wrote
one of the first textbooks on astronomy. Known as the Almagest or the Great Treatise, it
contained a summary of the astronomical knowledge then available, including a detailed
model of the motions of the Sun, the planets and the stars.
Hipparchus, Ptolemy and all astronomers before them were mostly concerned with
the motions of the stars and the planets, and not so much with their physical nature. After
all, astronomy as a science was born out of a desire to be able to predict signs in the skies
that might announce a good harvest or the start of a war. In 1608, almost a millennium and
a half after Ptolemy’s Almagest, the three Dutchmen Hans Lippershey, Zacharias Janssen
and Jacob Metius invented the telescope (van Helden 2009). This allowed Galileo Galilei
(1564–1642) to embark on a whole new kind of astronomy. Suddenly, the Sun was known
to have dark spots on its surface, Venus to have phases, Jupiter to have moons, and the
Milky Way to consist of countless individual stars (Drake 1978).
2
1.1 From ancient astronomy to modern astrochemistry
Galileo’s observations supported the heliocentric theory published by Nicolaus Copernicus in his 1543 work De Revolutionibus Orbium Coelestium, and scientists were now
gradually accepting the idea of a spherical Earth orbiting the Sun in a vast expanse. This
inevitably led to questions about the nature of the space between the planets and the stars.
Sir Francis Bacon appears to have been the first to publish on this topic. In his Descriptio
Globi Intellectualis (1653), he wrote, “Another question is, what is contained in the interstellar spaces? For they are either empty, as Gilbert thought; or filled with a body which
is to the stars what air is to flame (. . . ); or filled with a body homogeneous with the stars
themselves, lucid and almost empyreal, but in a less degree (. . . ).”1 Despite Bacon’s excellent ideas – his third option is remarkably close to the truth – it was not until the early
20th century that astronomers widely started to think of interstellar space as a very dilute
gas. The first strong evidence of interstellar material came with the discovery of a very
narrow calcium absorption line towards δ Orionis, which Hartmann (1904) concluded had
to be due to a cloud of calcium gas located somewhere between Earth and this star. Independent of him, Kapteyn (1909) theorised that “interstellar space must contain, at every
moment, a considerable amount of gas,” based on the coronal gas expelled by the Sun and,
presumably, other stars. The “dark markings of the sky” observed by Barnard (1919) provided further evidence of interstellar material, as did spectroscopic observations by Heger
(1919), Beals (1936) and Dunham (1937a,b) and theoretical considerations by Birkeland
(1913) and Thorndike (1930).
Now that astronomers knew there was an interstellar medium (ISM), they set out to
identify its chemical composition. The observations so far had established the presence
of Na, K, Ca+ , Ti+ and other metal atoms and ions.2 It was soon suggested the ISM might
also contain simple molecules (Eddington 1926, Russell 1935, Swings 1937, Saha 1937),
and this hypothesis was confirmed by the detections of CH (Swings & Rosenfeld 1937),
CN (McKellar 1940) and CH+ (Douglas & Herzberg 1941). Astrochemistry had arrived.
The advent of radio and microwave astronomy in the 1960s, followed by infrared
astronomy in the late 1970s, provided a great boost for this fledgling area of science. By
the time Neil Armstrong set foot on the Moon, the detection of OH, NH3 , H2 O and H2 CO
had increased the number of confirmed interstellar molecules to seven (Weinreb 1963,
Cheung et al. 1968, 1969, Snyder et al. 1969). Seven more were known within a year and
a half, including two of the most abundant ones: H2 and CO (Carruthers 1970, Wilson
et al. 1970). The 50th and 100th space molecules were detected in 1978 (NO, by Liszt
& Turner) and 1992 (SO+ , by Turner). Meanwhile, strong evidence had arisen for the
ubiquitous presence of a class of much larger molecules known as polycyclic aromatic
hydrocarbons or PAHs (Fig. 1.1; Gillett et al. 1973, Puget & Leger 1989, Allamandola
et al. 1989, Tielens 2008), although as of yet no individual members of this class have been
1
2
The quote comes from Bacon’s collected and translated works, edited by Spedding, Ellis and Heath. The
possibility of space being empty refers to William Gilbert’s hypothesis in his De Magnete (1600; translated
by Mottelay) that “the space above the earth’s exhalations is a vacuum.”
Worth mentioning here is nebulium or nebulum, an element conjured up by Sir William Huggins in the 1860s
to designate, as Clerke (1898) put it, “the exotic world-stuff originating the chief nebular ray at λ 5007 [and]
its companion at λ 4959.” Nebulium continued to be considered a common component of the ISM until Bowen
(1928) showed that the lines at 5007 and 4959 Å were in fact due to doubly ionised oxygen, prompting Russell
(1932) to quip that “nebulium [had] thus very literally vanished into thin air”.
3
Chapter 1 – Introduction
Figure 1.1 – A selection of polycyclic aromatic hydrocarbons (PAHs): benzene (C6 H6 ), naphthalene (C10 H8 ), tetracene (C18 H12 ), coronene (C24 H12 ) and ovalene (C32 H14 ). Each corner represents
a carbon atom; hydrogen atoms are not drawn.
firmly identified. The counter of interstellar and circumstellar molecules and molecular
ions currently stands at 162 (Woon 2009), and astrochemistry has firmly established itself
as an important field of research.3
1.2 Low-mass star formation and the role of chemistry
The ISM is now understood to consist of several components (Field et al. 1969, McKee &
Ostriker 1977, Ferrière 2001). By volume, the top three are the hot ionised medium (particle density of 10−4 –10−2 cm−3 , temperature of 106 –107 K), the warm ionised medium
(0.2–0.5 cm−3 , 8000 K) and the warm neutral medium (0.2–0.5 cm−3 , 6000–10 000 K),
together accounting for 95–99%. Most of the remaining volume is taken up by the cold
neutral medium (20–50 cm−3 , 50–100 K). The densest ISM component, with a fractional
volume of less than 1%, are the molecular clouds (102 –106 cm−3 , 10–20 K). They are of
particular importance for this thesis, as they are the birthplace of new stars.
Molecular clouds range in diameter from a few to maybe 10 or 20 pc and have a
mass between 103 and 104 M⊙ (Cambrésy 1999). They tend to be irregularly shaped and
their density distribution is far from homogeneous. Embedded in molecular clouds are
so-called clumps with typical densities of 103 –104 cm−3 and typical diameters of 0.3–3
pc (Loren 1989, Williams et al. 1994). The clumps in turn harbour the so-called cores,
whose densities are another order of magnitude higher and whose diameters are another
order of magnitude smaller (Motte et al. 1998, Jijina et al. 1999, Caselli et al. 2002). The
temperature in all these substructures is about 10 K (Bergin & Tafalla 2007).
Supported by pressure, turbulence and magnetic fields, cloud cores usually survive for
a few 105 to possibly several 106 yr. They consist of a mixture of gas and tiny dust grains
(radius of about 0.1 µm) in a mass ratio of 1 to 0.01, or a number ratio of 1 to 10−12 (Spitzer
3
4
The list of 162 molecules includes various isomers, but no isotopologues. It contains no PAHs except the
tentatively detected benzene (C6 H6 ). It also contains other tentative and disputed detections such as glycine
(NH2 CH2 COOH) and 1,3-dihydroxyacetone (CO(CH2 OH)2 ).
1.2 Low-mass star formation and the role of chemistry
1954, Kimura et al. 2003).4 An active chemistry is already taking place during this stage.
First of all, the initially atomic gas – inherited from the more diffuse ISM out of which the
cloud coalesced – is transformed into simple molecules like CO, OH and N2 . Because of
the low temperature and moderately high density, most of these molecules freeze out onto
the dust. The resulting ice mantles offer the possibility for additional chemical reactions,
leading for example to CO2 , CH4 , H2 O, H2 CO and CH3 OH (Watanabe & Kouchi 2002,
Ioppolo et al. 2008, 2009, Fuchs et al. 2009).
The various physical and chemical stages involved in the formation of low-mass stars
are illustrated in Fig. 1.2. Point 0 represents the cold phase of the static cloud core. As
described in more detail in the rest of this section, the formation of a star at the centre
of the core is initially accompanied by a warm-up of the surrounding material. At a later
stage, when a dense disk is formed around the star, the temperature decreases again in
some areas. The range of physical conditions encountered throughout the star-formation
process results in a complex chemical evolution of both the gas and the dust.
The collapse of the core under its own gravity is initiated by the loss of turbulent or
magnetic support. Material starts falling in towards the centre along trajectories such as
the one drawn in Fig. 1.2. The core is gradually warmed up by gravitational contraction,
accretion shocks and, eventually, nuclear fusion in the protostar. Volatile species such
as CO, CH4 and N2 now evaporate from the grains (van Dishoeck et al. 1993, Aikawa
et al. 2001, Jørgensen et al. 2002, 2004, 2005, Lee et al. 2004) and this also affects
the gas-phase abundances of other species (Chapter 3). The warm-up to 20–40 K further drives a rich grain-surface chemistry (Garrod & Herbst 2006, Garrod et al. 2008,
Öberg et al. 2009a; see also the review by Herbst & van Dishoeck 2009). The remaining
ice molecules may not be volatile enough at these temperatures to evaporate, but they
are mobile enough to diffuse more rapidly across the surface and react with each other.
This leads to the formation of so-called first-generation complex organic molecules like
HCOOH and HCOOCH3 at point 1 in Fig. 1.2.
As shown by Benson & Myers (1989) and Goodman et al. (1993), cloud cores rotate at
a rate of 10−14 –10−13 s−1 . Angular momentum must be conserved throughout the collapse
and this results in the formation of a disk around the protostar (Cassen & Moosman 1981,
Terebey et al. 1984). Another feature that appears at roughly the same time is a bipolar
jet, launched along the core’s rotation axis from close to the protostar (Shu et al. 1991,
Bally et al. 2007). The jets are probably another mechanism to remove excess angular
momentum, but their precise origin remains poorly understood (Ray et al. 2007). They
carve out a bipolar cavity in the remnant core material, which at this stage is usually
called the envelope. Bipolar cavities have been observed for many protostars (Padgett
et al. 1999, Arce & Sargent 2006).
The interaction between the envelope and the circumstellar disk is only understood in
the most general terms: material falls in from large radii, hits the disk at some point, and is
absorbed by it. It is still an open question whether the accretion from the envelope occurs
predominantly onto the inner parts or the outer parts of the disk. The two-dimensional
4
The canonical mass ratio of 0.01 is actually the mass ratio between the dust and the hydrogen gas. The mass
ratio between the dust and the total gas is 0.007 (Zhukovska et al. 2008).
5
Chapter 1 – Introduction
Figure 1.2 – Schematic representation of low-mass star formation and some of the chemistry involved, as reviewed by Herbst & van Dishoeck (2009). CO freezes out before the onset of collapse
and is partially hydrogenated to CH3 OH (point 0). As the core warms up during the collapse, CO
and other volatiles evaporate and first-generation complex organic molecules are formed on the
grains (point 1). Conservation of angular momentum results in the formation of a circumstellar
disk, where the temperature may become low enough for CO to freeze out again. In the hot inner
region of the core and the disk, the entire ice mantle evaporates and high-temperature reactions in
the gas phase lead to a second generation of complex organics (point 2).
(2D) axisymmetric hydrodynamical simulations of Brinch et al. (2008a,b) suggest the
latter. As the disk gets thicker (i.e., more vertically extended), material from the envelope
would have to flow across the surface of the disk in order to accrete onto the inner part.
Brinch et al. showed that this does not typically happen. Instead, most material hits the
disk near its outer edge. We also see a lot of envelope material hitting the outer parts of
the disk in our semi-analytical collapse model, but there is still a fair amount (up to 50%)
that makes its way to the inner parts of the disk and accretes there (Fig. 1.3; see also Fig.
6
1.2 Low-mass star formation and the role of chemistry
Figure 1.3 – Cartoon view of the layered accretion in one quadrant of our collapse model, based on
Fig. 2.7 from Chapter 2. The outer parts of the original cloud core (left) end up as the surface layers
of the circumstellar disk (right). The white part of the core is roughly the part that disappears into
the outflow. The arrows illustrate the flow of material onto the disk.
4.2 in Chapter 4). Both the hydrodynamical simulations of Brinch et al. and our semianalytical model result in layered accretion: the inner parts of the original cloud core end
up near the midplane of the disk, and the outer parts of the core end up near the surface
of the disk (Fig. 1.3; see Chapter 2 for details). Whether this is indeed what happens in
reality remains to be confirmed by observations and 3D models.
Depending on initial conditions like the mass of the core and its rotation rate, the
disk formed around the protostar can grow as large as 1000 AU (Andrews & Williams
2007b). Its density is several orders of magnitude higher than that of the core – in the
inner parts, 1012 cm−3 is not unusual. Because of the wide range of physical conditions
present throughout the disk, there are several distinct chemical regimes (Fig. 1.4). The
midplane is shielded from direct irradiation, so the temperature may drop to 10 K or less.
The volatile species that evaporated during the initial stages of the collapse now freeze out
again onto the cold dust (see also Fig. 1.2, between points 1 and 2). Closer to the surface
of the disk, the physical conditions are very different: the density is only 106 cm−3 or less,
the temperature easily exceeds 100 K, and there is a strong UV field from the protostar.
Ices cannot survive in such an environment and the first-generation complex organics
formed at earlier times are liberated into the gas phase (point 2 in Fig. 1.2). There they
can participate in a hot-core chemistry to form a second generation of complex organics
(Herbst & van Dishoeck 2009). The same can happen with infalling material that passes
through the hot inner part of the remnant envelope.
The entire collapse phase typically lasts a few 105 yr. The envelope gradually dissipates towards the end of that period and we are left with a pre-main sequence star surrounded by a circumstellar disk. Depending on the mass of star, it is known in this stage
7
Chapter 1 – Introduction
Figure 1.4 – Schematic representation of the three chemical regimes in a circumstellar disk: the
photon-dominated layer (irradiated by the protostar and the interstellar UV field), the warm molecular layer, and the cold midplane. Typical species for each regime are indicated.
as a T Tauri (< 2 M⊙ ) or Herbig Ae/Be star (2–8 M⊙ ). The gaseous part of the disk disappears over a period of about 10 Myr by photoevaporation and ongoing accretion onto
the star (Hollenbach et al. 2000, Haisch et al. 2001). During that time, its chemistry can
roughly be divided into the three regimes shown in Fig. 1.4 (Bergin et al. 2007). It is as
yet unknown whether the chemical composition of the disk is purely a result of the physical conditions during the T Tauri or Herbig Ae/Be stage, or whether the disk contains
some kind of chemical history from the collapse phase. This is one of the key questions
addressed in this thesis.
1.3 Planets, comets and meteorites
The evolution of the dust in the disk is governed by two processes: settling towards the
midplane and coagulation into larger grains (Weidenschilling 1980, Miyake & Nakagawa
1995, Dullemond & Dominik 2004b, D’Alessio et al. 2006, Dullemond et al. 2007a, Lommen et al. 2007, 2009; see also the reviews by Natta et al. 2007 and Dominik et al. 2007).
Together this may eventually lead to the formation of comets and planets. Whether planets are formed in all circumstellar disks is still an open question. It is also not yet clear
how exactly planets are formed: two mechanisms have been proposed (core accretion and
gravitational instabilities), but neither of them can be ruled out based on current observations (Matsuo et al. 2007, Lissauer & Stevenson 2007, Durisen et al. 2007).
Regardless of the details of how planets are formed, their initial chemical composition is inherited from the disk. This is also true for comets, and it is these objects in
8
1.3 Planets, comets and meteorites
Figure 1.5 – Molecular abundances in comets (Halley, Hale-Bopp, Hyakutake, Lee, C/1999 S4 and
Ikeya-Zhang) compared to those in ISM sources (IRAS 16293–2422 (warm inner envelope), L1157,
W3(H2 O), G34.3+0.15, Orion Hot Core and Orion Compact Ridge) as provided by BockeléeMorvan et al. (2000, 2004) and Schöier et al. (2002). The error bars indicate the spread between
sources; errors from individual measurements are not included. The dashed line represents a oneto-one correspondence between cometary and ISM abundances and is not a fit to the data.
our own solar system that provide the most direct probe of the chemistry of the pre-solar
nebula (Bockelée-Morvan et al. 2000). Spectroscopic studies of comet C/1995 O1 (HaleBopp) revealed a chemical composition that is remarkably similar to that of interstellar
ices, hot cores and bipolar outflows, suggesting that the cometary ices were formed in
the ISM and underwent little processing in the solar nebula. However, observations of
a dozen other comets show abundance variations of at least an order of magnitude for
CO, H2 CO, CH3 OH, HNC, H2 S and S2 , as well as smaller variations for other species
(Bockelée-Morvan et al. 2004, Kobayashi et al. 2007). In Fig. 1.5, the abundances from
the six comets 1P/Halley, Hale-Bopp, C/1996 B2 (Hyakutake), C/1999 H1 (Lee), C/1999
S4 (LINEAR) and 153P/Ikeya-Zhang are compared against those in the embedded protostar IRAS 16293–2422 (warm inner envelope), the bipolar outflow L1157, and the four
hot cores W3(H2 O), G34.3+0.15, Orion HC and Orion CR (Bockelée-Morvan et al. 2000,
2004, Schöier et al. 2002). On the one hand, it shows the general correspondence between
cometary and ISM abundances. On the other hand, it shows that the abundances can vary
9
Chapter 1 – Introduction
greatly from one comet to the next. These different chemical compositions may be explained by assuming that the comets were formed in different parts of the solar nebula. If
that is indeed the case, there must have been some degree of chemical processing between
the ISM and the formation of the cometary nuclei. Two-dimensional collapse models such
as the one presented in this thesis may help to clarify exactly how chemically pristine the
cometary material is and where the mutual differences come from.
Meteorites provide another set of clues about the physical and chemical conditions in
the early solar system. For example, Clayton et al. (1973) measured the abundances of the
rare oxygen isotopes 17 O and 18 O in meteorites and found that they could not be explained
with normal low-temperature isotope enhancement reactions alone.5 Recently, this socalled oxygen isotope anomaly has been interpreted as evidence for isotope-selective photoprocesses in the solar nebula – more specifically, for the selective photodissociation of
C16 O, C17 O and C18 O (Clayton 2002, Lyons & Young 2005, Lee et al. 2008). If true,
this would put strong constraints on the ambient UV intensity in the solar neighbourhood
around 5 Gyr ago.
1.4 Chemical models
1.4.1 Historical development and reaction types
Chemical models play a pivotal role in interpreting abundances derived from spectroscopic observations. One of the first such models was constructed at Leiden Observatory
in 1946 by Kramers & ter Haar and was targeted at CH and CH+ , two of the first three
molecules detected in the ISM (Sect. 1.1). Other early chemical modelling studies were
undertaken by Bates & Spitzer (1951) and Solomon & Klemperer (1972). The latter used
a network consisting of 22 monatomic and diatomic species linked by 34 reactions. They
obtained good quantitative agreement with the observations of Herbig (1968) of CH and
CN along the line of sight towards the star ζ Ophiuchi, but they could not reproduce
the abundance of CH+ . As observations identified ever more complex species over the
years, so chemical networks have expanded. The latest versions of the two most popular
networks – UMIST06 and osu_03_2008 – contain about 450 species and 4500 reactions
(Woodall et al. 2007, Hersant et al. 2009).6 However, there are still many scientific questions that can be answered with only a few dozen species and a few hundred reactions
(see, for example, Chapters 5 and 6).
The reactions in an astrochemical network can be categorised into several types. Table
1.1 lists the types used in modern networks like UMIST06 and osu_03_2008. The first
14 reaction types belong exclusively to the gas phase; types 15–18 describe gas-grain
5
6
For a set of isotopologues, the zero-point vibrational energy decreases with increasing mass. Therefore, the
abundances of heavy isotopologues like H17
O and H18
O are enhanced at low temperature relative to the lighter
2
2
O.
isotopologue H16
2
These numbers do not include recent developments like grain-surface chemistry (Hasegawa et al. 1992, Garrod
et al. 2008) or hydrocarbon anions (Millar et al. 2007, Walsh et al. 2009), which would make the networks
even larger. Adding isotopes like D or 13 C would also greatly expand the size of the network.
10
1.4 Chemical models
Table 1.1 – Reaction types in current astrochemical models.a
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
a
Typeb
neutral-neutral
ion-neutral
charge exchange
mutual neutralisation
dissociative recombination
radiative recombination
associative detachment
radiative association
photodissociation
photoionisation
cosmic-ray dissociation
cosmic-ray ionisation
cosmic-ray–induced photodissociation
cosmic-ray–induced photoionisation
adsorption (or freeze-out)
desorption (or evaporation)
grain-surface hydrogenation
grain-surface radical-radical
Example
OH + O → O2 + H
CH+2 + H2 → CH+3 + H
CH + C+ → CH+ + C
C+ + H− → C + H
CH+ + e− → C + H
CH+ + e− → CH + hν
N + H− → NH + e−
C + H2 → CH2 + hν
CO + hν → C + O
C + hν → C+ + e−
H2 + ζ → H+ + H−
H2 + ζ → H+2 + e−
CO + ζ–hν → C + O
C + ζ–hν → C+ + e−
CO → COice
COice → CO
Oice + H → OHice
CHOice + OHice → HCOOHice
Based on Woodall et al. (2007). All species are in the gas phase unless indicated otherwise. The symbols
hν, ζ and ζ–hν indicate a photon, a cosmic ray and a cosmic-ray–induced photon, respectively.
b
Reaction types in italics were already included in the model by Solomon & Klemperer (1972).
interactions and grain-surface reactions. It is interesting to note that of the fourteen gasphase reaction types in Table 1.1, nine were already included by Solomon & Klemperer
(1972) in their chemical model. A reaction type of particular importance to this thesis is
photodissociation. It is one of the key processes controlling the abundances during the
collapse phase in Chapter 3 and it sets a lower limit to the size of circumstellar PAHs in
Chapter 6. It plays an even larger role in Chapter 5, which is entirely about the selective
photodissociation of CO isotopologues.
Photodissociation can take place in several ways (van Dishoeck 1987, 1988, Kirby &
van Dishoeck 1988), two of which are illustrated in Fig. 1.6. For most simple species, the
main channel is direct photodissociation. Absorption of a UV or visible photon brings
the molecule from the electronic ground state into a repulsive excited state. Spontaneous
emission back to the ground state is a slow process, so essentially all absorptions result
in photodissociation. The corresponding cross section is a continuous function peaking
at the vertical excitation energy. Another possible mechanism is predissociation: the
molecule is first excited from the ground state into a bound upper state and then crosses
over into a repulsive state. This is the dominant dissociation mechanism for CO and
its isotopologues (Chapter 5). The predissociation cross section consists of a series of
discrete peaks, corresponding to the vibrational levels of the bound upper state.
11
Chapter 1 – Introduction
Figure 1.6 – Schematic potential energy curves and cross sections for direct photodissociation (top)
and predissociation (bottom) of a diatomic molecule AB, after van Dishoeck (1988).
For molecules with a large number of atoms, such as PAHs (Fig. 1.1), the density of vibrational levels at each electronic state becomes high enough to form a quasi-continuum.
When a PAH absorbs a UV or visible photon and is excited to an upper electronic state,
it rapidly crosses over into the vibrational quasi-continuum of the electronic ground state
(Léger et al. 1988, 1989). From here, it relaxes back to a low vibrational level by radiative
decay. As long as the internal energy of the PAH exceeds a certain threshold value, there
is the possibility of losing a hydrogen atom or a carbon fragment (Le Page et al. 2001).
In the strong radiation fields in the inner parts of the disk and the envelope, the PAH
may absorb a second photon before its internal energy falls back below the dissociation
threshold. Multi-photon absorptions keep the PAH at high internal energy, thus greatly
increasing the effective photodissociation rate (Chapter 6).
1.4.2 Solution methods
Mathematically, a network of chemical reactions results in a system of ordinary differential equations (ODEs). Each species i is formed by a certain number of reactions (say,
mf ) and it is destroyed by a certain number of other reactions (say, md ). Hence, the time
derivative of the number density (ni in cm−3 ) is
d
f
X
dni X
Rd, j ,
Rf, j −
=
dt
j=1
j=1
m
m
(1.1)
with Rf, j and Rd, j the rates (in cm−3 s−1 ) of the individual formation and destruction reactions. If a reaction involves two reactants with number densities n1 and n2 , its rate is of
12
1.4 Chemical models
the form
R = kn1 n2 .
(1.2)
This is the case for reaction types 1–8, 17 and 18 from Table 1.1. For types 9–16, where
there is only one reactant, the rate is of the form
R = kn1 .
(1.3)
Because of the low densities in most astronomical objects of interest to chemical modellers, reactions with three reactants are very slow and are not included in the standard
networks. However, they start to become important in the innermost regions of circumstellar disks.
The variable k in Eqs. (1.2) and (1.3) is the so-called rate coefficient. Rate coefficients
are obtained most reliably by measuring the reaction in controlled conditions in a laboratory – ideally at a range of conditions, so that it can be established quantitatively how
the coefficient depends for example on temperature. Another good method is to derive the
coefficient from quantum chemical state-to-state calculations. However, both methods are
time-consuming, so many coefficients in current networks are based on extrapolations or
chemical kinetic theories, and some of them are no more than educated guesses.
Given a set of species, reactions and rate coefficients, there are two ways of solving
for the abundances: find an equilibrium or follow the evolution of the abundances in time.
The two methods are now illustrated with a very simple system consisting of only two
species (CO gas and CO ice, with number densities ngas and nice ) linked by two reactions
(freeze-out and evaporation, with rates Rdes and Rads ), similar to what is used in Chapter 2.
The rate coefficients depend on the temperature T and the gas density nH ; we approximate
them as
√
(1.4)
kads = (1 × 10−18 cm3 K−1/2 s−1 )nH T ,
!
855 K
,
(1.5)
kdes = (1 × 1012 s−1 ) exp −
T
after Charnley et al. (2001) and Bisschop et al. (2006).
If we are interested in the equilibrium abundances at a given T and nH , we have to
solve the equations
(
dngas/dt = kdes nice − kads ngas = 0
(1.6)
dnice /dt = kads ngas − kdes nice = 0
under the condition of conservation of mass. Setting the total CO abundance to 10−4
relative to nH , this is simply expressed as
ngas + nice = 10−4 nH .
(1.7)
The equilibrium solution then becomes
ngas =
10−4 nH kdes
,
kads + kdes
nice = 10−4 nH − ngas .
(1.8)
13
Chapter 1 – Introduction
The procedure is essentially the same for a full chemical network: set the time derivatives of all individual abundances to zero and ensure conservation of mass. However, an
analytical solution can generally not be derived for a full network. Instead, the problem
must be solved with a numerical procedure like the Newton-Raphson routine (Press et al.
1992).
Solving for the equilibrium abundances is usually fine if the physical conditions are
constant, although one always has to check that the object of interest is old enough
to reach chemical equilibrium. If that is not the case, or if the physical conditions
change on timescales shorter than the chemical timescale, one has to solve the abundances time-dependently with an integration package like VODE (Brown et al. 1989).
Starting from some initial values, the abundances are evolved in small time steps ∆t up to
a pre-determined final time. Turning back to our CO system, we could for example begin
with all CO as ice and ask how long it takes to evaporate at nH = 104 cm−3 and T = 18 K.
We could also simulate the warm-up phase during the collapse of a cloud core by starting
at 10 K and increasing the temperature at a typical rate of 0.01 K yr−1 . If we couple this
to a physical model of the collapse to get a relationship between the temperature and the
distance from the protostar, we could then say where CO evaporates. Finally, we could
add other molecules to the mix and expand the simple freeze-out/evaporation scheme to a
full chemical network. In essence, this is what we do in Chapters 2 and 3.
1.5 This thesis
The central theme of this thesis is the chemical evolution during the formation of lowmass stars and their surrounding disks. Although the initial stages of the collapsing cloud
core are well described by spherically symmetric models, this is no longer possible once
the disk is formed. So far, the chemistry during the collapse phase has only been studied
up to the point that the disk is formed or on large scales where spherical symmetry can
still be safely assumed (> 1000 AU). In this thesis we present the first physical-chemical
model that follows the entire core collapse and disk formation process in two dimensions.
In the preceding sections we introduced several open questions related to low-mass
star formation and the chemical evolution of the material involved. The most important
ones addressed in this thesis are:
• Does material from the envelope accrete predominantly on the inner or on the outer
parts of the disk?
• How does the chemical composition of the gas and dust change from the envelope to
the disk?
• Is the chemical composition of a T Tauri or Herbig Ae/Be disk purely a result of in
situ processes or does it retain some signature of the collapse phase?
• What fraction of the cometary ices is truly pristine?
• What is the origin of the chemical diversity of comets?
The first three chapters of this thesis deal with our 2D collapse model. To begin with,
Chapter 2 contains a full description of the model. We couple the analytical collapse solu14
1.5 This thesis
tions of Shu (1977), Cassen & Moosman (1981) and Terebey et al. (1984) to the model of
Lynden-Bell & Pringle (1974) for a viscously evolving circumstellar disk. The size and
luminosity of the central sources evolve according to Adams & Shu (1986) and Young &
Evans (2005). We define a standard set of initial conditions (core mass, sound speed and
rotation rate) and find that the resulting density and velocity profiles show good agreement with those from more detailed hydrodynamical simulations (Yorke & Bodenheimer
1999, Brinch et al. 2008a,b). The temperature is a very important input parameter for the
chemistry, so we compute it with a full radiative transfer method (Dullemond & Dominik
2004a) at a series of time intervals. The semi-analytical nature of our model allows us
to easily change the initial conditions and we obtain density, velocity and temperature
profiles for a small grid of parameters.
As a first illustration of the full chemistry, we look at the freeze-out and evaporation
of CO and H2 O. We calculate infall trajectories originating from several thousand points
in the cloud core to see how material flows towards the star and the disk. The density
and temperature along each trajectory provide the input for evolving the gas and ice abundances of CO and H2 O, resulting in 2D profiles of the gas-to-ice ratios in the disk and
remnant envelope.
In Chapter 4, we revisit one particular physical aspect of our collapse model: the subKeplerian velocity at which material accretes onto the disk. This problem was studied
previously by Cassen & Moosman (1981) and Hueso & Guillot (2005), but not in the
context of a 2D model. We derive a new solution for the radial velocities inside the disk
and show that it does not strongly affect the results from the preceding chapter. However,
it does offer new insights into the related question of why the dust in circumstellar disks
is more crystalline than in the ISM. We rerun the models of Dullemond et al. (2006a) and
obtain a better match with observed crystalline abundances in disks.
We couple our 2D collapse model – including the new correction for sub-Keplerian
accretion – to a full gas-phase chemical network in Chapter 3 to analyse the abundances of
several major oxygen-, carbon- and nitrogen-bearing species. We describe the evolution
of the abundances along one specific infall trajectory and show that most changes can be
traced back to key chemical processes like the evaporation of CO or the photodissociation
of H2 O. In turn, these key processes relate back to physical events like the increase in
temperature or UV flux at some point along the trajectory. Material ending up in other
parts of the disk encounters different physical conditions and we show how that affects
the abundances of certain species. Finally, we seek to answer whether the disks around
T Tauri and Herbig Ae/Be stars retain some signature of the collapse-phase chemistry, or
if their chemical composition is fully determined by in situ processing. To that end, we
evolve the abundances obtained at the end of the collapse for another 1 Myr in a static
disk model to see how much they still change.
Having discussed the gas-phase chemistry and the dust mineralogy, we turn to the
question of isotopes in Chapter 5. We derive an extensive update to the CO photodissociation model of van Dishoeck & Black (1988) using new laboratory data. The model
includes not only the regular isotopes 12 C and 16 O, but also the less abundant 13 C, 17 O
and 18 O. We discuss the effect of the temperature on the isotope-selective photodissociation rates and we couple the photodissociation model to a small chemical network to
15
Chapter 1 – Introduction
analyse the abundance of CO and its isotopologues as a function of depth into diffuse
clouds, photon-dominated regions and circumstellar disks. A long-standing puzzle in our
own solar system is the anomalous oxygen isotope ratio found in meteorites. Our results
support the recent hypothesis by Lyons & Young (2005) that the anomalous ratio is due
to CO photodissociation in the solar nebula.
The final chapter of this thesis targets the chemistry of yet another class of compounds:
polycyclic aromatic hydrocarbons or PAHs. We adapt the chemistry models of Le Page
et al. (2001) and Weingartner & Draine (2001) and the excitation model of Draine & Li
(2007) to find the dominant charge and hydrogenation states of PAHs in disks around T
Tauri and Herbig Ae/Be stars. We explicitly calculate where in the disk PAHs are photodissociated, taking into account the possibility of multi-photon absorption events. The
2D abundance profiles thus obtained are coupled to a radiative transfer package (Dullemond & Dominik 2004a) to simulate spatially resolved spectra. Finally, we compare the
predicted spatial extent of the PAH features to observations by van Boekel et al. (2004),
Habart et al. (2006) and Geers et al. (2007) to determine the size of the PAHs responsible
for the observed emission.
Each of the Chapters 2–6 closes with a summary of the conclusions that we draw from
our model results and the comparison with observations and other models. We present
here the main conclusions from this thesis.
• Our two-dimensional semi-analytical collapse model produces realistic density and
velocity profiles, allowing us to track the chemistry all the way from pre-stellar cores
to circumstellar disks. Combined with full radiative transfer to get temperatures and
UV fluxes, this makes it an excellent tool to study the chemical evolution during lowmass star formation (Chapters 2 and 3).
• Both CO and H2 O freeze out before the onset of collapse. H2 O remains frozen
throughout the collapse phase, except when it gets into the inner 5–10 AU of a disk.
From there, it may move outwards again to colder regions as the disk expands to conserve angular momentum. CO rapidly evaporates once the collapse starts, although
some of it is likely to be trapped in the H2 O ice. In the coldest parts of a disk (< 18
K), all CO freezes out again (Chapters 2 and 3).
• The chemistry during the collapse of a cloud core and the formation of a disk is
dominated by a small number of key chemical processes that are activated by changes
in the physical conditions. Examples of these key processes are the evaporation of CO,
CH4 and H2 O at approximately 18, 22 and 100 K, and the photodissociation of CH4
and H2 O in the vicinity of the outflow wall. The photodissociation of CO requires
a stellar temperature of at least 7000 K or a colder star with excess UV emission
(Chapter 3).
• Because of the short dynamical timescales, the chemistry does not reach equilibrium
at any time during the collapse. The chemical composition of the disk at the end of
the collapse phase is therefore primarily a product of the physical conditions at earlier
times. Additional work is required to determine if any chemical signatures from the
collapse phase survive into the T Tauri or Herbig Ae/Be stages (Chapter 3).
16
1.5 This thesis
• The material from which solar-system comets are formed must be of mixed origins.
Our collapse model predicts a large degree of chemical processing towards the cometforming zone in the disk. The observed fractions of crystalline silicates in comets are
also indicative of strong processing. However, strong processing cannot explain why
the chemical composition of cometary ices so closely resembles that of interstellar
ices. The detections of amorphous silicates in comets also point at the presence of
unprocessed material. Hence, it would seem that comets were formed partially from
processed material and partially from pristine material (Chapters 3 and 4).
• The chemical diversity between individual comets is likely a result of them having
formed at different locations in the solar nebula. The physical conditions in a disk
change in time, so if two comets were formed several 104 or 105 yr apart, their chemical compositions would also be different (Chapter 3).
• It is important to take the vertical structure of a disk into account when computing
the infall trajectories. The outer parts of a disk can intercept material and keep it
from accreting onto the disk at much smaller radii, as it would if the disk is treated as
completely flat (Chapters 2–4).
• Thermal annealing followed by outward radial mixing is responsible for at least part
of the crystalline silicates observed in circumstellar disks (Chapter 4).
• The CO photodissociation rate obtained with our new model is 30% higher than the
old value. The dissociation of C17 O and 13 C17 O shows almost exactly the same depth
dependence as that of C18 O and 13 C18 O, respectively, so 17 O and 18 O are equally
fractionated with respect to 16 O. The level of fractionation is higher for cold gas than
it is for warm gas (Chapter 5).
• Isotope fractionation in circumstellar disks through the photodissociation of CO in the
surface layers requires a far-UV component in the irradiating spectrum. The interstellar radiation field is sufficient for this purpose. Our model supports the hypothesis that
the photodissociation of CO is responsible for the anomalous 17 O and 18 O abundances
in meteorites (Chapter 5).
• PAH emission from circumstellar disks is extended on a scale similar to the size of
the disks. Neutral and positively ionised PAHs contribute to the emission in roughly
equal amounts. Based on the spatial extent, the observed emission originates mostly
from PAHs with a size of at least 100 carbon atoms. Smaller PAHs are efficiently
destroyed by the stellar UV field in the inner ∼30 AU of a disk (Chapter 6).
Although our two-dimensional semi-analytical collapse model is an important step
forwards in the study of the chemical evolution during low-mass star formation, many
questions still remain unanswered. With regards to the model itself, we have only undertaken some basic tests against observations. One of the first challenges now is to couple
the model to a radiative transfer package to predict spectral lines and compare them to
observational data. The start of science operations with the Herschel Space Observatory
later this year and with the Atacama Large Millimeter/submillimeter Array (ALMA) in
17
Chapter 1 – Introduction
2011 offers both exciting possibilities and additional challenges. For example, the unprecedented spatial resolution of ALMA may well require a revision of the physics in the
inner parts of the disk and envelope.
Given the semi-analytical nature of our model, some physical aspects had to be simplified. The bipolar outflow is only included in an ad-hoc fashion and we may be underestimating how much material it sweeps up and out of the system. Another simplification
involves the amount of core material that becomes involved in the collapse. We currently
let the entire core accrete onto the star and disk, but recent interferometric observations
call this into question (Jørgensen et al. 2007). In both cases, the final disk mass and
temperature would be different, and some of the chemistry would change as well.
On the chemical side, the treatment of photoprocesses currently involves several approximations in the way the shape of the irradiating spectrum is taken into account. Also,
grain-surface processes are still largely unexplored. They have to be added to our network if we want to reproduce the observed abundances of methanol and more complex
organic molecules. If amino acids and other biologically important species are formed
in circumstellar disks, it is likely that this happens on grain surfaces, or that this at least
requires precursor molecules formed on grains. In either case, grain-surface chemistry
promises to be important as astronomers worldwide continue to unravel the history of our
solar system in general and the origins of life on Earth in particular.
18
2
The chemical history of molecules in
circumstellar disks
I: Ices
R. Visser, E. F. van Dishoeck, S. D. Doty and C. P. Dullemond
Astronomy & Astrophysics, 2009, 495, 881
19
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Abstract
Context. Many chemical changes occur during the collapse of a molecular cloud core to form a
low-mass star and the surrounding disk. One-dimensional models have been used so far to analyse
these chemical processes, but they cannot properly describe the incorporation of material into disks.
Aims. The goal of this chapter is to understand how material changes chemically as it is transported
from the cloud to the star and the disk. Of special interest is the chemical history of the material in
the disk at the end of the collapse.
Methods. We present a two-dimensional, semi-analytical model that, for the first time, follows the
chemical evolution from the pre-stellar core to the protostar and circumstellar disk. The model computes infall trajectories from any point in the cloud core and tracks the radial and vertical motion of
material in the viscously evolving disk. It includes a full time-dependent radiative transfer treatment
of the dust temperature, which controls much of the chemistry. We explore a small parameter grid to
understand the effects of the sound speed and the mass and rotation rate of the core. The freeze-out
and evaporation of carbon monoxide (CO) and water (H2 O), as well as the potential for forming
complex organic molecules in ices, are considered as important first steps towards illustrating the
full chemistry.
Results. Both species freeze out towards the centre before the collapse begins. Pure CO ice evaporates during the infall phase and readsorbs in those parts of the disk that cool below the CO desorption temperature of ∼18 K. H2 O remains solid almost everywhere during the infall and disk
formation phases and evaporates within ∼10 AU of the star. Mixed CO-H2 O ices are important
in keeping some solid CO above 18 K and in explaining the presence of CO in comets. Material that ends up in the planet- and comet-forming zones of the disk (∼5–30 AU from the star) is
predicted to spend enough time in a warm zone (several 104 yr at a dust temperature of 20–40 K)
during the collapse to form first-generation complex organic species on the grains. The dynamical
timescales in the hot inner envelope (hot core or hot corino) are too short for abundant formation of
second-generation molecules by high-temperature gas-phase chemistry.
20
2.1 Introduction
2.1 Introduction
The formation of low-mass stars and their planetary systems is a complex event, spanning
several orders of magnitude in temporal and spatial scales, and involving a wide variety
of physical and chemical processes. Thanks to observations, theory and computer simulations, the general picture of low-mass star formation is now understood (see reviews by
Shu et al. 1987, di Francesco et al. 2007, Klein et al. 2007, White et al. 2007 and Dullemond et al. 2007b). An instability in a cold molecular cloud core leads to gravitational
collapse. Rotation and magnetic fields cause a flattened density structure early on, which
evolves into a circumstellar disk at later times. The protostar continues to accrete matter
from the disk and the remnant envelope, while also expelling matter in a bipolar pattern.
Grain growth in the disk eventually leads to the formation of planets, and as the remaining dust and gas disappear, a mature solar system emerges. While there has been ample
discussion in the literature on the origin and evolution of grains in disks (see reviews by
Natta et al. 2007 and Dominik et al. 2007 or the discussion in Chapter 4), little attention
has so far been paid to the chemical history of the more volatile material in a two- or
three-dimensional setting.
Chemical models are required to understand the observations and develop the simulations (see reviews by Ceccarelli et al. 2007, Bergin et al. 2007 and Bergin & Tafalla 2007).
The chemistry in pre-stellar cores is relatively easy to model, because the dynamics and
the temperature structure are simpler before the protostar is formed than afterwards. A
key result from the pre-stellar core models is the depletion of many carbon-bearing species
towards the centre of the core (Bergin & Langer 1997, Lee et al. 2004).
The next step in understanding the chemical evolution during star formation is to
model the chemistry during the collapse phase (Ceccarelli et al. 1996, Rodgers & Charnley 2003, Doty et al. 2004, Lee et al. 2004, Garrod & Herbst 2006, Aikawa et al. 2008,
Garrod et al. 2008). All of the collapse chemistry models so far are one-dimensional, and
thus necessarily ignore the circumstellar disk. As the protostar turns on and heats up the
surrounding material, all models agree that frozen-out species return to the gas phase if
the dust temperature surpasses their evaporation temperature. The higher temperatures
can further drive a hot-core–like chemistry, and complex molecules may be formed if the
infall timescales are long enough.
If the model is expanded into a second dimension and the disk is included, the system
gains a large reservoir where infalling material from the cloud can be stored for a long time
(at least several 104 yr) before accreting onto the star. This can lead to further chemical
enrichment, especially in the warmer parts of the disk (Aikawa et al. 1997, 2008, Aikawa
& Herbst 1999, Willacy & Langer 2000, van Zadelhoff et al. 2003, Rodgers & Charnley
2003). The interior of the disk is shielded from direct irradiation by the star, so it is colder
than the disk’s surface and the remnant envelope. Hence, molecules that evaporated as
they fell in towards the star may freeze out again when they enter the disk. This was first
shown quantitatively by Brinch et al. (2008b, hereafter BWH08) using a two-dimensional
hydrodynamical simulation.
In addition to observations of nearby star-forming regions, the comets in our own solar
system provide a unique probe into the chemistry that takes place during star and planet
21
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
formation. The bulk composition of the cometary nuclei is believed to be mostly pristine,
closely reflecting the composition of the pre-solar nebula (Bockelée-Morvan et al. 2004).
However, large abundance variations have been observed between individual comets and
these remain poorly understood (Kobayashi et al. 2007). Two-dimensional chemical models may shed light on the cometary chemical diversity.
Two molecules of great astrophysical interest are carbon monoxide (CO) and water
(H2 O). They are the main reservoirs of carbon and oxygen and control much of the overall chemistry. CO is an important precursor for more complex molecules; for example,
solid CO can be hydrogenated to formaldehyde (H2 CO) and methanol (CH3 OH) at low
temperatures (Watanabe & Kouchi 2002, Fuchs et al. 2009). In turn, these two molecules
form the basis of even larger organic species like methyl formate (HCOOCH3 ; Garrod &
Herbst 2006, Garrod et al. 2008). The key role of H2 O in the formation of life on Earth
and potentially elsewhere is evident. If the entire formation process of low-mass stars
and their planets is to be understood, a thorough understanding of these two molecules is
essential.
This chapter is the first in a series of publications aiming to model the chemical evolution from the pre-stellar core to the disk phase in two dimensions, using a simplified,
semi-analytical approach for the dynamics of the collapsing envelope and the disk, but
including detailed radiative transfer for the temperature structure. The model follows individual parcels of material as they fall in from the cloud core into the disk. The gaseous
and solid abundances of CO and H2 O are calculated for each infalling parcel to obtain
global gas-ice profiles. The semi-analytical nature of the model allows for an easy exploration of physical parameters like the core’s mass and rotation rate, or the effective sound
speed. Tracing the temperature history of the infalling material provides a first clue into
the formation of more complex species. The model also offers some insight into the origin
of the chemical diversity in comets.
Section 2.2 contains a full description of the model. Results are presented in Sect.
2.3 and discussed in a broader astrophysical context in Sect. 2.4. Finally, conclusions are
drawn in Sect. 2.5.
2.2 Model
The physical part of our two-dimensional axisymmetric model describes the collapse of
an initially spherical, isothermal, slowly rotating cloud core to form a star and circumstellar disk. The collapse dynamics are taken from Shu (1977, hereafter S77), including the
effects of rotation as described by Cassen & Moosman (1981, hereafter CM81) and Terebey et al. (1984, hereafter TSC84). Infalling material hits the equatorial plane inside the
centrifugal radius to form a disk, whose further evolution is constrained by conservation
of angular momentum (Lynden-Bell & Pringle 1974). Some properties of the star and the
disk are adapted from Adams & Shu (1986) and Young & Evans (2005, hereafter YE05).
Magnetic fields are not included in our model. They are unlikely to affect the chemistry
directly and their main physical effect (causing a flattened density distribution; Galli &
Shu 1993) is already accounted for by the rotation of the core.
22
2.2 Model
Our model is an extension of the one used by Dullemond et al. (2006a) to study the
crystallinity of dust in circumstellar disks (see also Chapter 4). That model was purely
one-dimensional; our model treats the disk more realistically as a two-dimensional structure.
2.2.1 Envelope
The cloud core (or envelope) is taken to be a uniformly rotating singular isothermal sphere
at the onset of collapse. It has a solid-body rotation rate Ω0 and an r−2 density profile
(S77):
c2s
ρ0 (r) =
,
(2.1)
2πGr2
where G is the gravitational constant and cs the effective sound speed. Throughout this
thesis, r is used for the spherical radius and R for the cylindrical radius. Setting the outer
radius at renv , the total mass of the core is
M0 =
2c2s renv
.
G
(2.2)
After the collapse is triggered at the centre, an expansion wave (or collapse front)
travels outwards at the sound speed (S77, TSC84). Material inside the expansion wave
falls in towards the centre to form a protostar. The infalling material is deflected towards
the gravitational midplane by the core’s rotation. It first hits the midplane inside the
centrifugal radius (where gravity balances angular momentum; CM81), resulting in the
formation of a circumstellar disk (Sect. 2.2.2).
The dynamics of a collapsing singular isothermal sphere were computed by S77 in
terms of the non-dimensional variable x = r/cs t, with t the time after the onset of collapse. In this self-similar description, the head of the expansion wave is always at x = 1.
The density and radial velocity are given by the non-dimensional variables A and v, respectively. (S77 used α for the density, but our model already uses that symbol for the
viscosity in Sect. 2.2.2.) These variables are dimensionalised through
A(x)
,
4πGt2
(2.3)
ur (r, t) = cs v(x) .
(2.4)
ρ(r, t) =
Values for A and v are tabulated in S77.
CM81 and TSC84 analysed the effects of slow uniform rotation on the S77 collapse
solution, with the former focussing on the flow onto the protostar and the disk and the latter on what happens farther out in the envelope. In the axisymmetric TSC84 description,
the density and infall velocities depend on the time, the radius and the polar angle:
A(x, θ, τ)
,
4πGt2
(2.5)
ur (r, θ, t) = cs v(x, θ, τ) ,
(2.6)
ρ(r, θ, t) =
23
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
where τ = Ω0 t is the non-dimensional time. The polar velocity is given by
uθ (r, θ, t) = cs w(x, θ, τ) .
(2.7)
We solved the differential equations from TSC84 numerically to obtain solutions for A, v
and w.
The TSC84 solution breaks down around x = τ2 , so the CM81 solution is used inside
of this point. A streamline through a point (r, θ) effectively originated at an angle θ0 in
this description:
cos θ0 − cos θ Rc
−
= 0,
(2.8)
r
sin2 θ0 cos θ0
where Rc is the centrifugal radius,
Rc (t) =
1
cs m30 t3 Ω20 ,
16
(2.9)
with m0 a numerical factor equal to 0.975. The CM81 radial and polar velocity are
r
r
GM
cos θ
ur (r, θ, t) = −
1+
,
(2.10)
r
cos θ0
r
r
cos θ cos θ0 − cos θ
GM
1+
,
(2.11)
uθ (r, θ, t) =
r
cos θ0
sin θ
and the CM81 density is
ρ(r, θ, t) = −
−1
Ṁ Rc
P
(cos
θ
)
1
+
2
,
2
0
r
4πr2 ur
(2.12)
where P2 is the second-order Legendre polynomial and Ṁ = m0 c3s /G is the total accretion
rate from the envelope onto the star and disk (S77, TSC84). The primary accretion phase
ends when the outer shell of the envelope reaches the star and disk. This point in time
(tacc = M0 / Ṁ) is essentially the beginning of the T Tauri or Herbig Ae/Be phase, but it
does not yet correspond to a typical T Tauri or Herbig Ae/Be object (see Sect. 2.3.2).
The TSC84 and CM81 solutions do not reproduce the cavities created by the star’s
bipolar outflow, so they have to be put in separately. Outflows have been observed in
two shapes: conical and curved (Padgett et al. 1999). Both can be characterised by the
outflow opening angle, γ, which grows with the age of the object. Arce & Sargent (2006)
found a linear relationship in log-log space between the age of a sample of 17 young
stellar objects and their outflow opening angles. Some explanations exist for the outflow
widening in general, but it is not yet understood how γ(t) depends on parameters like the
initial cloud core mass and the sound speed. It is likely that the angle depends on the
relative age of the object rather than on the absolute age.
The purpose of our model is not to include a detailed description of the outflow cavity.
Instead, the outflow is primarily included because of its effect on the temperature profiles
(Whitney et al. 2003). Its opening angle is based on the fit by Arce & Sargent (2006)
24
2.2 Model
to their Fig. 5, but it is taken to depend on t/tacc rather than t alone. The outflow is also
kept smaller, which brings it closer to the Whitney et al. angles. Its shape is taken to be
conical. With the resulting formula,
log
γ(t)
t
,
= 1.5 + 0.26 log
deg
tacc
(2.13)
the opening angle is always 32◦ at t = tacc . The numbers in Eq. (2.13) are poorly constrained; however, the details of the outflow (both size and shape) do not affect the temperature profiles strongly, so this introduces no major errors in the chemistry results. The
outflow cones are filled with a constant mass of 0.002M0 at a uniform density, which
decreases to 103 –104 cm−3 at tacc depending on the model parameters. The outflow effectively removes about 1% of the initial envelope mass.
2.2.2 Disk
The rotation of the envelope causes the infalling material to be deflected towards the midplane, where it forms a circumstellar disk. The disk initially forms inside the centrifugal
radius (CM81), but conservation of angular momentum quickly causes the disk to spread
beyond this point. The evolution of the disk is governed by viscosity, for which our model
uses the common α prescription (Shakura & Sunyaev 1973). This gives the viscosity coefficient ν as
ν(R, t) = αcs,d H .
(2.14)
p
The sound speed in the disk, cs,d = kT m /µmp (with k the Boltzmann constant, mp the
proton mass and µ the mean molecular mass of 2.3 nuclei per hydrogen molecule), is
different from the sound speed in the envelope, cs , because the midplane temperature of
the disk, T m , varies as described in Hueso & Guillot (2005). The other variable from Eq.
(2.14) is the scale height:
cs,d
H(R, t) =
,
(2.15)
Ωk
where Ωk is the Keplerian rotation rate:
r
GM∗
,
(2.16)
Ωk (R, t) =
R3
with M∗ the stellar mass (Eq. (2.29)). The viscosity parameter α is kept constant at 10−2
(Dullemond et al. 2007b, Andrews & Williams 2007b).
Solving the problem of advection and diffusion yields the radial velocities inside the
disk (Dullemond et al. 2006a, Lynden-Bell & Pringle 1974):
3 ∂ √ uR (R, t) = − √
Σν R .
Σ R ∂R
(2.17)
∂Σ(R, t)
1 ∂
=−
(ΣRuR ) + S ,
∂t
R ∂R
(2.18)
The surface density Σ evolves as
25
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
where the source function S accounts for the infall of material from the envelope:
S (R, t) = 2Nρuz ,
(2.19)
with uz the vertical component of the envelope velocity field (Eqs. (2.6), (2.7), (2.10) and
(2.11)). The factor 2 accounts for the envelope accreting onto both sides of the disk and
the normalisation factor N ensures that the overall accretion rate onto the star and the disk
is always equal to Ṁ. Both ρ and uz in Eq. (2.19) are to be computed at the disk-envelope
boundary, which is defined at the end of this section.
As noted by Hueso & Guillot (2005), the infalling envelope material enters the disk
with a sub-Keplerian rotation rate, so, by conservation of angular momentum, it would
tend to move a bit farther inwards. Not taking this into account would artificially generate
angular momentum, causing the disk to take longer to accrete onto the star. As a consequence the disk has, at any given point in time, too high a mass and too large a radius.
Hueso & Guillot solved this problem by modifying Eq. (2.19) to place the material directly at the correct radius. However, this causes an undesirable discontinuity in the infall
trajectories. Instead, our model adds a small extra component to Eq. (2.17) for t < tacc :
r
3 ∂ √ GM
uR (R, t) = − √
Σν R − ηr
.
(2.20)
∂R
R
Σ R
The functional form of the extra term derives from the CM81 solution. A constant value
of 0.002 for ηr is found to reproduce very well the results of Hueso & Guillot. It also
provides a good match with the disk masses from Yorke & Bodenheimer (1999), YE05
and BWH08, whose models cover a wide range of initial conditions. The problem of
sub-Keplerian accretion is addressed in more detail in Chapter 4. There, we derive a
more rigorous solution and show that it does not affect the gas-ice results from the current
chapter in a significant way.
The disk’s inner radius is determined by the evaporation of dust as it gets heated above
a critical temperature by the stellar radiation (e.g., YE05):
s
L∗
,
(2.21)
Ri (t) =
4
4πσT evap
where σ is the Stefan-Boltzmann constant. The dust evaporation temperature, T evap, is
set to 2000 K. Taking an alternative value of 1500 or even 1200 K has no effect on our
results. The stellar luminosity, L∗ , is discussed in Sect. 2.2.3. Inward transport of material
at Ri leads to accretion from the disk onto the star:
Ṁd→∗ = −2πRi uR Σ ,
(2.22)
with the radial velocity, uR , and the surface density, Σ, taken at Ri . The disk gains mass
from the envelope at a rate Ṁe→d , so the disk mass evolves as
Z t
(2.23)
Md (t) =
Ṁe→d − Ṁd→∗ dt′ .
0
26
2.2 Model
We adopt a Gaussian profile for the vertical structure of the disk (Shakura & Sunyaev
1973):
!
z2
(2.24)
ρ(R, z, t) = ρc exp − 2 ,
2H
with z the height above the midplane. The scale height comes from Eq. (2.15) and the
midplane density is
Σ
ρc (R, t) = √ .
(2.25)
H 2π
Along with the radial motion (Eq. (2.20), taken to be independent of z), material also
moves vertically in the disk, as it must maintain the Gaussian profile at all times. To see
this, consider a parcel of material that enters the disk at time t at coordinates R and z into
a column with scale height H and surface density Σ. The column of material below the
parcel is
!
Z z
z
1
ρ(R, ζ, t)dζ = Σerf
(2.26)
√ ,
2
0
H 2
where erf is the error function. At a later time t′ , the entire column has moved to R′ and
has a scale height H ′ and a surface density Σ′ . The same amount of material must still be
below the parcel:
!
!
1
1 ′
z′
z
=
(2.27)
Σ erf
Σerf
√
√ .
2
2
H′ 2
H 2
Rearranging gives the new height of the parcel, z′ :
!#
"
√
z
′ ′ ′
′
−1 Σ
erf
z (R , t ) = H 2erf
,
√
Σ′
H 2
(2.28)
where erf −1 is the inverse of the error function. In the absence of vertical mixing, our
description leads to purely laminar flow.
The location of the disk-envelope boundary (needed, e.g., in Eq. (2.19)) is determined
in two steps. First, the surface is identified where the density due to the disk (Eq. (2.24))
equals that due to the envelope (Eqs. (2.5) and (2.12)). In order for accretion to take place
at a given point P1 on the surface, it must be intersected by an infall trajectory. Due to the
geometry of the surface, such a trajectory might also intersect the disk at a larger radius
P2 (Fig. 2.1). Material flowing in along that trajectory accretes at P2 instead of P1 . Hence,
the second step in determining the disk-envelope boundary consists of raising the surface
at “obstructed points” like P1 to an altitude where accretion can take place. The source
function is then computed at that altitude. Physically, this can be understood as follows:
the region directly above the obstructed points becomes less dense than what it would be
in the absence of a disk, because the disk also prevents material from reaching there. The
lower density above the disk reduces the downward pressure, so the disk puffs up and the
disk-envelope boundary moves to a higher altitude.
The infall trajectories in the vicinity of the disk are very shallow, so the bulk of the
material accretes at the outer edge. Because the disk quickly spreads beyond the centrifugal radius, much of the accretion occurs far from the star. In contrast, accretion in
27
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Figure 2.1 – Schematic view of the disk-envelope boundary in the upper right quadrant of the (R, z)
plane. The black line indicates the surface where the density due to the disk equals that due to the
envelope. The grey line is the infall trajectory that would lead to point P1 . However, it already
intersects the disk at point P2 , so no accretion is possible at P1 . The disk-envelope boundary is
therefore raised at P1 until it can be reached freely by an infall trajectory.
one-dimensional collapse models occurs at or inside of Rc (see also Chapter 4). Our results are consistent with the hydrodynamical work of BWH08, where most of the infalling
material also hits the outer edge of a rather large disk. The large accretion radii lead to
weaker accretion shocks than commonly assumed (Sect. 2.2.5).
2.2.3 Star
The star gains material from the envelope and from the disk, so its mass evolves as
Z t
(2.29)
Ṁe→∗ + Ṁd→∗ dt′ .
M∗ (t) =
0
The protostar does not come into existence immediately at the onset of collapse; it is
preceded by the first hydrostatic core (FHC; Masunaga et al. 1998, Boss & Yorke 1995).
Our model follows YE05 and takes a lifetime of 2 × 104 yr and a size of 5 AU for the
FHC, independent of other parameters. After this stage, a rapid transition occurs from the
large FHC to a protostar of a few R⊙ :
s




t
−
20
000
yr
 + RPS
R∗ = (5 AU) 1 −
20 000 < t (yr) < 20 100 ,
(2.30)
∗
100 yr 
where RPS
∗ (ranging from 2 to 5 R⊙ ) is the protostellar radius from Palla & Stahler (1991).
For t > 2.01 × 104 yr, R∗ equals RPS
∗ . Our results are not sensitive to the exact values used
for the size and lifetime of the FHC or the duration of the FHC-protostar transition.
The star’s luminosity, L∗ , consists of two terms: the accretion luminosity, L∗,acc , dominant at early times, and the luminosity due to gravitational contraction and deuterium
burning, Lphot . The accretion luminosity comes from Adams & Shu (1986):
(
i
p
1 h
6u∗ − 2 + (2 − 5u∗ ) 1 − u∗ +
L∗,acc (t) = L0
6u∗
i)
p
h
η∗ 1 − (1 − ηd )Md 1 − (1 − ηd ) 1 − u∗ ,
(2.31)
2
28
2.2 Model
Figure 2.2 – Evolution of the mass of the envelope, star and disk (left panel) and the luminosity
(solid lines) and radius (dotted lines) of the star (right panel) for our standard model (black lines)
and our reference model (grey lines).
where L0 = GM Ṁ/R∗ (with M = M∗ + Md the total accreted mass), u∗ = R∗ /Rc , and
Z √
1 1/3 1 1 − u
Md = u ∗
du .
(2.32)
3
u4/3
u∗
Analytical solutions exist for the asymptotic cases of u∗ ≈ 0 and u∗ ≈ 1. For intermediate
values, the integral must be solved numerically. The efficiency parameters η∗ and ηd
in Eq. (2.31) have values of 0.5 and 0.75 for a 1 M⊙ envelope (YE05). The photospheric
luminosity is adopted from D’Antona & Mazzitelli (1994), using YE05’s method of fitting
and interpolating, including a time difference of 0.38 tacc (equal to the free-fall time)
between the onset of L∗,acc and L∗,phot (Myers et al. 1998). The sum of these two terms,
L∗ (t) = L∗,acc + L∗,phot , gives the total stellar luminosity.
Figure 2.2 shows the evolution of the stellar mass, luminosity and radius for our standard case of M0 = 1.0 M⊙ , cs = 0.26 km s−1 and Ω0 = 10−14 s−1 , and our reference
case of M0 = 1.0 M⊙ , cs = 0.26 km s−1 and Ω0 = 10−13 s−1 (Sect. 2.2.6). The transition
from the FHC to the protostar at t = 2 × 104 yr is clearly visible in the R∗ and L∗ profiles.
At t = tacc , there is no more accretion from the envelope onto the star, so the luminosity
decreases sharply.
The masses of the disk and the envelope are also shown in Fig. 2.2. Our disk mass of
0.43 M⊙ at t = tacc in the reference case is in excellent agreement with the value of 0.4
M⊙ found by BWH08 for the same parameters.
29
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
2.2.4 Temperature
The envelope starts out as an isothermal sphere at 10 K and it is heated up from the inside
after the onset of collapse. Using the star as the only photon source, we compute the dust
temperature in the disk and envelope with the axisymmetric three-dimensional radiative
transfer code RADMC (Dullemond & Dominik 2004a). Because of the high densities
throughout most of the system, the gas and dust are expected to be well coupled, and the
gas temperature is set equal to the dust temperature. Cosmic-ray heating of the gas is
included implicitly by setting a lower limit of 8 K in the dust radiative transfer results.
As mentioned in Sect. 2.2.1, the presence of the outflow cones has some effect on the
temperature profiles (Whitney et al. 2003). This is discussed further in Sect. 2.3.2.
2.2.5 Accretion shock
The infall of high-velocity envelope material into the low-velocity disk causes a J-type
shock. The temperature right behind the shock front can be much higher than what it
would be due to the stellar photons. Neufeld & Hollenbach (1994) calculated in detail the
relationship between the pre-shock velocities and densities (us and ns ) and the maximum
grain temperature reached after the shock (T d,s). A simple formula, valid for us < 70 km
s−1 , can be extracted from their Fig. 13:
T d,s ≈ (104 K)
p
0.21 agr
ns
us
−1
6
−3
0.1 µm
30 km s
10 cm
!−0.20
,
(2.33)
with agr the grain radius. The exponent p is 0.62 for us < 30 km s−1 and 1.0 otherwise.
The pre-shock velocities and densities are highest at early times, when accretion occurs close to the star and all ices would evaporate anyway. Important for our purposes
is the question whether the dust temperature due to the shock exceeds that due to stellar
heating. If all grains have a radius of 0.1 µm, as assumed in our model, this is not the case
for any of the material in the disk at tacc for either our standard or our reference model
(Fig. 2.3, cf. Simonelli et al. 1997).
In reality, the dust spans a range of sizes, extending down to a radius of about 0.005
µm. Small grains are heated more easily; 0.005-µm dust reaches a shock temperature
almost twice as high as does 0.1-µm dust (Eq. (2.33)). This is enough for the shock
temperature to exceed the radiative heating temperature in part of the sample in Fig. 2.3.
However, this has no effect on the CO and H2 O gas-ice ratios. In those parcels where
shock heating becomes important for small grains, the temperature from radiative heating
lies already above the CO evaporation temperature of about 18 K and the shock temperature remains below 60 K, which is not enough for H2 O to evaporate. Hence, shock heating
is not included in our model.
H2 O may also be removed from the grain surfaces in the accretion shock through
sputtering (Tielens et al. 1994, Jones et al. 1994). The material that makes up the disk at
the end of the collapse in our standard model experiences a shock of at most 8 km s−1 .
At that velocity, He+ , the most important ion for sputtering, carries an energy of 1.3 eV.
30
2.2 Model
Figure 2.3 – Dust temperature due to the accretion shock (vertical axis) and stellar radiation (horizontal axis) at the point of entry into the disk for 0.1-µm grains in a sample of several hundred
parcels in our standard (left) and reference (right) models. These parcels occupy positions from
R = 1 to 300 AU in the disk at tacc . Note the different scales between the two panels.
However, a minimum of 2.2 eV is required to remove H2 O (Bohdansky et al. 1980), so
sputtering is unimportant for our purposes.
Some of the material in our model is heated to more than 100 K during the collapse
(Fig. 2.12) or experiences a shock strong enough to induce sputtering. This material
normally ends up in the star before the end of the collapse, but mixing may keep some of
it in the disk. The possible consequences are discussed briefly in Sect. 2.4.4.
2.2.6 Model parameters
The standard set of parameters for our model corresponds to Case J from Yorke & Bodenheimer (1999), except that the solid-body rotation rate is reduced from 10−13 to 10−14
s−1 to produce a more realistic disk mass of 0.05 M⊙ , consistent with observations (e.g.,
Andrews & Williams 2007a,b). The envelope has an initial mass of 1.0 M⊙ and a radius
of 6700 AU, and the effective sound speed is 0.26 km s−1 .
The original Case J (with Ω0 = 10−13 s−1 ), which was also used in BWH08, is used
here as a reference model to enable a direct quantitative comparison of the results with an
independent method. This case results in a much higher disk mass of 0.43 M⊙ . Although
such high disk masses are not excluded by observations and theoretical arguments (Hartmann et al. 2006), they are considered less representative of typical young stellar objects
than the disks of lower mass.
The parameters M0 , cs and Ω0 are changed in one direction each to create a 23 parameter grid. The two values for Ω0 , 10−14 and 10−13 s−1 , cover the range of rotation rates
31
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Table 2.1 – Summary of the parameter grid used in our model.a
Caseb
1
2
3 (std)
4
5
6
7 (ref)
8
Ω0
(s−1 )
10−14
10−14
10−14
10−14
10−13
10−13
10−13
10−13
cs
(km s−1 )
0.19
0.19
0.26
0.26
0.19
0.19
0.26
0.26
M0
(M⊙ )
1.0
0.5
1.0
0.5
1.0
0.5
1.0
0.5
tacc
(105 yr)
6.3
3.2
2.5
1.3
6.3
3.2
2.5
1.3
τads
(105 yr)
14.4
3.6
2.3
0.6
14.4
3.6
2.3
0.6
Md
(M⊙ )
0.22
0.08
0.05
0.001
0.59
0.25
0.43
0.16
a
Ω0 : solid-body rotation rate; cs : effective sound speed; M0 : initial core mass; tacc : accretion time; τads :
adsorption timescale for H2 O at the edge of the initial core; Md : disk mass at tacc .
b
Case 3 is our standard parameter set and Case 7 is our reference set.
observed by Goodman et al. (1993). The other variations are chosen for their opposite
effect: a lower sound speed gives a more massive disk, and a lower initial core mass gives
a less massive disk. The full model is run for each set of parameters to analyse how the
chemistry can vary between different objects. The parameter grid is summarised in Table
2.1. Our standard set is Case 3 and our reference set is Case 7.
Table 2.1 also lists the accretion time and the adsorption timescale for H2 O at the edge
of the initial core. For comparison, Evans et al. (2009) found a median timescale for the
embedded phase (Class 0 and I) of 5.4 × 105 yr from observations. It should be noted
that the end point of our model (tacc ) is not yet representative of a typical T Tauri disk
(see Sect. 2.3.2). Nevertheless, it allows an exploration of how the chemistry responds to
plausible changes in the environment.
2.2.7 Adsorption and desorption
The adsorption and desorption of CO and H2 O are solved in a Lagrangian frame. When
the time-dependent density, velocity and temperature profiles have been calculated, the
envelope is populated by a number of parcels of material (typically 12 000) at t = 0.
They fall in towards the star or disk according to the velocity profiles. The density and
temperature along each parcel’s infall trajectory are used as input to solve the adsorptiondesorption balance. Both species start fully in the gas phase. The envelope is kept static
for 3 × 105 yr before the onset of collapse to simulate the pre-stellar core phase. This is
the same value as used by Rodgers & Charnley (2003) and BWH08, and it is consistent
with recent observations by Enoch et al. (2008). The amount of gaseous material left over
near the end of the pre-collapse phase is also consistent with observations, which show
that the onset of H2 O ice formation is around an AV of 3 mag (Whittet et al. 2001). In
six of our eight parameter sets, the adsorption timescales of H2 O at the edge of the cloud
core are shorter than the combined collapse and pre-collapse time (Table 2.1), so all H2 O
32
2.3 Results
is expected to freeze out before entering the disk. Because of the larger core size, the
adsorption timescales are much longer in Cases 1 and 5, and some H2 O may still be in
the gas phase when it reaches the disk.
No chemical reactions are included other than adsorption and thermal desorption, so
the total abundance of CO and H2 O in each parcel remains constant. The adsorption rate
in cm−3 s−1 is taken from Charnley et al. (2001):
s
Tg
−18
3 −1/2 −1
Rads (X) = (4.55 × 10 cm K
s )nH ng (X)
,
(2.34)
M(X)
where nH is the total hydrogen density, T g the gas temperature, ng (X) the gas-phase abundance of species X and M(X) its molecular weight. The numerical factor assumes unit
sticking efficiency, a grain radius of 0.1 µm and a grain abundance xgr of 10−12 with respect to H2 .
The thermal desorption of CO and H2 O is a zeroth-order process:
#
"
Eb (X)
,
(2.35)
Rdes (X) = (1.26 × 10−21 cm2 )nH f (X)ν(X) exp −
kT d
where T d is the dust temperature and
"
#
ns (X)
f (X) = min 1,
,
Nb ngr
(2.36)
with ns (X) the solid abundance of species X and Nb = 106 the typical number of binding
sites per grain. The numerical factor in Eq. (2.35) assumes the same grain properties as in
Eq. (2.34). The pre-exponential factor, ν(X), and the binding energy, Eb (X)/k, are set to
7 × 1026 cm−2 s−1 and 855 K for CO and to 1 × 1030 cm−2 s−1 and 5773 K for H2 O (Fraser
et al. 2001, Bisschop et al. 2006).
Using a single Eb (CO) value means that all CO evaporates at the same temperature.
This would be appropriate for a pure CO ice, but not for a mixed CO-H2 O ice as is
likely to form in reality. During the warm-up phase, part of the CO is trapped inside the
H2 O ice until the temperature becomes high enough for the H2 O to evaporate. Recent
laboratory experiments suggest that CO desorbs from a CO-H2 O ice in several discrete
steps (Collings et al. 2004; Fayolle et al. in prep.). We simulate this in some model runs
with four “flavours” of CO ice, each with a different Eb (CO) value (Viti et al. 2004). For
each flavour, the desorption is assumed to be zeroth order. The four-flavour model is
summarised in Table 2.2.
2.3 Results
Results are presented in this section for our standard and reference models (Cases 3 and
7) as described in Sect. 2.2.6. These cases are compared to the other parameter sets in
Sect. 2.4.1. The appendix at the end of this chapter describes a formula to estimate the
33
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Table 2.2 – Binding energies and desorbing fractions for the four-flavour CO evaporation model.a
Flavour
1
2
3
4
Eb (CO)/k (K)b
855
960
3260
5773
Fractionc
0.350
0.455
0.130
0.065
a
Based on Viti et al. (2004).
The rates for Flavours 1–3 are computed from Eq. (2.35) with X = CO. The rate for Flavour 4 is equal to
the H2 O desorption rate.
c
Fractions of adsorbing CO: 35% of all adsorbing CO becomes Flavour 1, and so on.
b
disk formation efficiency, defined as Md /M0 at the end of the collapse phase, based on a
fit to our model results. The results from this section are partially revisited in Chapters 3
and 4, where we use a different solution to the problem of sub-Keplerian accretion.
2.3.1 Density profiles and infall trajectories
In our standard model (Case 3), the envelope collapses in 2.5 ×105 yr to give a star of 0.94
M⊙ and a disk of 0.05 M⊙ . The remaining 0.01 M⊙ has disappeared through the bipolar
outflow. The centrifugal radius in our standard model at tacc is 4.9 AU, but the disk has
spread to 400 AU at that time due to angular momentum redistribution. The densities in
the disk are high: more than 109 cm−3 at the midplane inside of 120 AU (Fig. 2.4, top)
and more than 1014 cm−3 near 0.3 AU. The corresponding surface densities of the disk
are 2.0 g cm−2 at 120 AU and 660 g cm−2 at 0.3 AU.
Due to the higher rotation rate, our reference model (Case 7) gets a much higher disk
mass: 0.43 M⊙ . This value is consistent with the mass of 0.4 M⊙ reported by BWH08.
Overall, the reference densities from our semi-analytical model (Fig. 2.4, bottom) compare well with those from their more realistic hydrodynamical simulations; the differences
are generally less than a factor of two.
In both cases, the disk first emerges at 2 × 104 yr, when the FHC contracts to become
the protostar, but it is not until a few 104 yr later that the disk becomes visible on the scale
of Fig. 2.4. The regions of high density (nH > 105 –106 cm−3 ) are still contracting at that
time, but the growing disks eventually cause them to expand again.
Material falls in along nearly radial streamlines far out in the envelope and deflects
towards the midplane closer in. When a parcel enters the disk, it follows the radial motion
Figure 2.4 – Total density at four time steps for our standard model (Case 3; top) and our reference
model (Case 7; bottom). The notation a(b) denotes a×10b . The density contours increase by factors
of ten going inwards; the 105 -cm−3 contours are labelled in the standard panels and the 106 -cm−3
contours in the reference panels. The white curves indicate the surface of the disk as defined in
Sect. 2.2.2 (only visible in three panels). Note the different scale between the two sets of panels.
34
2.3 Results
35
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
caused by the viscous evolution and accretion of more material from the envelope. At
any time, conservation of angular momentum causes part of the disk to move inwards
and part of it to move outwards. An individual parcel entering the disk may move out
for some time before going farther in. This takes the parcel through several density and
temperature regimes, which may affect the gas-ice ratios or the chemistry in general. The
back-and-forth motion occurs especially at early times, when the entire system changes
more rapidly than at later times. The parcel motions are visualised in Figs. 2.5 and 2.6,
where infall trajectories are drawn for 24 parcels ending up at one of eight positions at
tacc : at the midplane or near the surface at radial distances of 10, 30, 100 and 300 AU.
Only parcels entering the disk before t ≈ 2 × 105 yr in our standard model or t ≈ 1 × 105
yr in our reference model undergo the back-and-forth motion. The parcels ending up near
the midplane all enter the disk earlier than the ones ending up at the surface.
Accretion from the envelope onto the disk occurs in an inside-out fashion. Because
of the geometry of the disk (Fig. 2.1), a lot of the material enters near the outer edge and
prevents the older material from moving farther out. Our flow inside the disk is purely
laminar, so some material near the midplane does move outwards underneath the newer
material at higher altitudes.
Because of the low rotation rate in our standard model, the disk does not really begin
to build up until 1.5 × 105 yr (0.6 tacc ) after the onset of collapse. In addition, most of the
early material to reach the disk makes it to the star before the end of the accretion phase,
so the disk at tacc consists only of material from the edge of the original cloud core (Fig.
2.7, top).
The disk in our reference model, however, begins to form right after the FHC-protostar
transition at 2 × 104 yr. As in the standard model, a layered structure is visible in the disk,
but it is more pronounced here. At the end of the collapse, the midplane consists mostly
of material that was originally close to the centre of the envelope (Fig. 2.7, bottom). The
surface and outer parts of the disk are made up primarily of material from the outer parts
of the envelope. This was also reported by BWH08.
2.3.2 Temperature profiles
When the star turns on at 2 × 104 yr, the envelope quickly heats up and reaches more than
100 K inside of 10 AU. As the disk grows, its interior is shielded from direct irradiation
and the midplane cools down again. At the same time, the remnant envelope material
above the disk becomes less dense and warmer. As in Whitney et al. (2003), the outflow
has some effect on the temperature profile. Photons emitted into the outflow can scatter
and illuminate the disk from the top, causing a higher disk temperature beyond R ≈ 200
AU than if there were no outflow cone. At smaller radii, the disk temperature is lower
than in a no-outflow model. Without the outflow, the radiation would be trapped in the
inner envelope and inner disk, increasing the temperature at small radii.
At t = tacc in our standard model, the 100- and 18-K isotherms (where H2 O and pure
CO evaporate) intersect the midplane at 20 and 2000 AU (Fig. 2.8, top). The disk in our
reference model is denser and therefore colder: it reaches 100 and 18 K at 5 and 580 AU
on the midplane (Fig. 2.8, bottom). Our radiative transfer method is a more rigorous way
36
2.3 Results
to obtain the dust temperature than the diffusion approximation used by BWH08, so our
temperature profiles are more realistic than theirs.
Compared to typical T Tauri disk models (e.g., D’Alessio et al. 1998, 1999, 2001),
our standard disk at tacc is warmer. It is 81 K at 30 AU on the midplane, while the closest
model from the D’Alessio catalogue is 28 K at that point. If our model is allowed to
run beyond tacc , part of the disk accretes further onto the star. At t = 4 tacc (1 Myr), the
disk mass goes down to 0.03 M⊙ . The luminosity of the star decreases during this period
(D’Antona & Mazzitelli 1994), so the disk cools down: the midplane temperature at 30
AU is now 42 K. Meanwhile, the dust is likely to grow to larger sizes, which would
further decrease the temperatures (D’Alessio et al. 2001). Hence, it is important to realise
that the normal end point of our models does not represent a “mature” T Tauri star and
disk as typically discussed in the literature.
2.3.3 Gas and ice abundances
Our two species, CO and H2 O, begin entirely in the gas phase. They freeze out during
the static pre-stellar core phase from the centre outwards due to the density dependence
of Eq. (2.34). After the pre-collapse phase of 3 × 105 yr, only a few tenths of per cent of
each species is still in the gas phase at 3000 AU. About 30% remains in the gas phase at
the edge of the envelope.
Up to the point where the disk becomes important and the system loses its spherical
symmetry, our model gives the same results as the one-dimensional collapse models: the
temperature quickly rises to a few tens of K in the collapsing region, driving most of the
CO into the gas phase, but keeping H2 O on the grains.
As the disk grows in mass, it provides an increasingly large body of material that
is shielded from the star’s radiation, and that is thus much colder than the surrounding
envelope. However, the disk in our standard model never gets below 18 K before the end
of the collapse (Sect. 2.3.2), so CO remains in the gas phase (Fig. 2.9, top). Note that
trapping of CO in the H2 O ice is not taken into account here; this possibility is discussed
in Sect. 2.4.3.
The disk in our reference model is more massive and therefore colder. After about
5 × 104 yr, the outer part drops below 18 K. CO arriving in this region readsorbs onto the
grains (Fig. 2.10, top). Another 2 × 105 yr later, at t = tacc , 19% of all CO in the disk is
in solid form. Moving out from the star, the first CO ice is found at the midplane at 400
AU. The solid fraction gradually increases to unity at 600 AU. At R = 1000 AU, nearly
all CO is solid up to an altitude of 170 AU. The solid and gaseous CO regions meet close
to the 18-K surface. The densities throughout most of the disk are high enough that once
a parcel of material goes below the CO desorption temperature, all CO rapidly disappears
from the gas. The exception to this rule occurs at the outer edge, near 1500 AU, where
the adsorption and desorption timescales are longer than the dynamical timescales of the
infalling material. Small differences between the trajectories of individual parcels then
cause some irregularities in the gas-ice profile.
The region containing gaseous H2 O is small at all times during the collapse. At t =
tacc , the snow line (the transition of H2 O from gas to ice) lies at 15 AU at the midplane
37
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Figure 2.5 – Infall trajectories for parcels in our standard model (Case 3) ending up at the midplane
(left) or near the surface (right) at radial positions of 10, 30, 100 and 300 AU (dotted lines) at
t = tacc . Each panel contains trajectories for three parcels, which are illustrative for material ending
up at the given location. Trajectories are only drawn up to t = tacc . Diamonds indicate where each
parcel enters the disk; the time of entry is given in units of 105 yr. Note the different scales between
some panels.
38
2.3 Results
Figure 2.6 – Same as Fig. 2.5, but for our reference model (Case 7).
39
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Figure 2.7 – Position of parcels of material in our standard model (Case 3; top) and our reference
model (Case 7; bottom) at the onset of collapse (t = 0) and at the end of the collapse phase (t =
tacc ). The parcels are coloured according to their initial position. The layered accretion is most
pronounced in our reference model. The grey parcels from t = 0 end up in the star or disappear in
the outflow. Note the different spatial scale between the two panels of each set; the small boxes in
the left panels indicate the scales of the right panels.
Figure 2.8 – Dust temperature, as in Fig. 2.4. Contours are drawn at 100, 60, 50, 40, 35, 30, 25, 20,
18, 16, 14 and 12 K. The 40- and 20-K contours are labelled in the standard and reference panels,
respectively. The 18- and 100-K contours are drawn in grey. The white curves indicate the surface
of the disk as defined in Sect. 2.2.2 (only visible in four panels).
40
2.3 Results
41
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Figure 2.9 – Gaseous CO as a fraction of the total CO abundance (top) and idem for H2 O (bottom)
at two time steps for our standard model (Case 3). The black curves indicate the surface of the disk
(only visible in two panels). The black area near the pole is the outflow, where no abundances are
computed. Note the different spatial scale between the two panels of each set; the small box in the
left CO panel indicates the scale of the H2 O panels.
in our standard model (Fig. 2.9, bottom). The surface of the disk holds gaseous H2 O out
to R = 41 AU, and overall 13% of all H2 O in the disk is in the gas phase. This number
is much lower in the colder disk of our reference model: only 0.4%. The snow line now
lies at 7 AU and gaseous H2 O can be found out to 17 AU in the disk’s surface layers (Fig.
2.10, bottom).
Using the adsorption-desorption history of all the individual infalling parcels, the original envelope can be divided into several chemical zones. This is trivial for our standard
model. All CO in the disk is in the gas phase and it has the same qualitative history: it
freezes out before the onset of collapse and quickly evaporates as it falls in. H2 O also
freezes out initially and only returns to the gas phase if it reaches the inner disk.
42
2.3 Results
Figure 2.10 – Same as Fig. 2.9, but for our reference model (Case 7). The CO gas fraction is plotted
on a larger scale and at two additional time steps.
Our reference model has the same general H2 O adsorption-desorption history, but it
shows more variation for CO, as illustrated in Fig. 2.11. For the red parcels in that figure,
more than half of the CO always remains on the grains after the initial freeze-out phase.
On the other hand, more than half of the CO comes off the grains during the collapse for
43
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Figure 2.11 – Same as Fig. 2.7, but only for our reference model (Case 7) and with a different colour
scheme to denote the CO adsorption-desorption behaviour. In all parcels, CO adsorbs during the
pre-collapse phase. Red parcels: CO remains adsorbed; green parcels: CO desorbs and readsorbs;
pink parcels: CO desorbs and remains desorbed; blue parcels: CO desorbs, readsorbs and desorbs
once more. The fraction of gaseous CO in each type of parcel as a function of time is indicated
schematically in the inset in the right panel. The grey parcels from t = 0 end up in the star or
disappear in the outflow. In our standard model (Case 3), all CO in the disk at the end of the
collapse phase is in the gas phase and it all has the same qualitative adsorption-desorption history,
equivalent to the pink parcels.
the green parcels, but it freezes out again inside the disk. The pink parcels, ending up in
the inner disk or in the upper layers, remain warm enough to keep CO off the grains once
it first evaporates. The blue parcels follow a more erratic temperature profile, with CO
evaporating, readsorbing and evaporating a second time. This is related to the back-andforth motion of some material in the disk (Fig. 2.6).
2.3.4 Temperature histories
The proximity of the CO and H2 O gas-ice boundaries to the 18- and 100-K surfaces
indicates that the temperature is primarily responsible for the adsorption and desorption.
At nH = 106 cm−3 , adsorption and desorption of CO are equally fast at T d = 18 K
(a timescale of 9 × 103 yr). For a density a thousand times higher or lower, the dust
temperature only has to increase or decrease by 2–3 K to maintain kads = kdes .
The exponential temperature dependence in the desorption rate (Eq. (2.35)) also holds
for other species than CO and H2 O, as well as for the rates of some chemical reactions.
Hence, it is useful to compute the temperature history for infalling parcels that occupy a
certain position at tacc . Figures 2.12 and 2.13 show these histories for the parcels from
Figs. 2.5 and 2.6. These parcels end up at the midplane or near the surface of the disk
at radial distances of 10, 30, 100 and 300 AU. Parcels ending up inside of 10 AU have
44
2.4 Discussion
a temperature history very similar to those ending up at 10 AU, except that the final
temperature of the former is higher.
Each panel in Figs. 2.12 and 2.13 contains the history of three parcels ending up
close to the desired position. The qualitative features are the same for all parcels. The
temperature is low while a parcel remains far out in the envelope. As it falls in with
an ever higher velocity, there is a temperature spike as it traverses the inner envelope,
followed by a quick drop once it enters the disk. Inward radial motion then leads to a
second temperature rise; because of the proximity to the star, this one is higher than the
first. For most parcels in Figs. 2.12 and 2.13, the second peak does not occur until long
after tacc . In all cases, the shock encountered upon entering the disk is weak enough that
it does not heat the dust to above the temperature caused by the stellar photons (Fig. 2.3).
Based on the temperature histories, the gas-ice transition at the midplane would lie
inside of 10 AU for H2 O and beyond 300 AU for CO in both our models. This is indeed
where they were found to be in Sect. 2.3.3. The transition for a species with an intermediate binding energy, such as H2 CO, is then expected to be between 10 and 100 AU, if its
abundance can be assumed constant throughout the collapse.
The dynamical timescales of the infalling material before it enters the disk are between 104 and 105 yr. The timescales decrease as it approaches the disk, due to the
rapidly increasing velocities. Once inside the disk, the material slows down again and
the dynamical timescales return to 104 –105 yr. The adsorption timescales for CO and
H2 O are initially a few 105 yr, so they exceed the dynamical timescale before entering the
disk. Depletion occurs nonetheless because of the pre-collapse phase of 3 × 105 yr. The
higher densities in the disk cause the adsorption timescales to drop to 100 yr or less. If
the temperature approaches (or crosses) the desorption temperature for CO or H2 O, the
corresponding desorption timescale becomes even shorter than the adsorption timescale.
Overall, the timescales for these specific chemical processes (adsorption and desorption)
in the disk are shorter by a factor of 1000 or more than the dynamical timescales.
At some final positions, there is a wide spread in the time that the parcels spend at a
given temperature. This is especially true for parcels ending up near the midplane inside
of 100 AU in our reference model. All of the midplane parcels ending up near 10 AU
exceed 18 K during the collapse; the first one does so at 3.5 × 104 yr after the onset of
collapse, the last one at 1.6 × 105 yr. Hence, some parcels at this final position spend
more than twice as long above 18 K than others. This does not appear to be relevant for
the gas-ice ratio, but it is important for the formation of more complex species (Garrod &
Herbst 2006, Garrod et al. 2008). This is discussed in more detail in Sect. 2.4.2.
2.4 Discussion
2.4.1 Model parameters
When the initial conditions of our model are modified (Sect. 2.2.6), the qualitative chemistry results do not change. In Cases 3, 4 and 8, the entire disk at tacc is warmer than
18 K, and it contains no solid CO. In the other cases, the disk provides a reservoir of
45
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Figure 2.12 – Temperature history for parcels in our standard model (Case 3) ending up at the
midplane (left) or near the surface (right) at radial positions of 10, 30, 100 and 300 AU at t = tacc .
The colours correspond to Fig. 2.5. The dotted lines are drawn at T d = 18 K and t = tacc . Note the
different vertical scales between some panels.
46
2.4 Discussion
Figure 2.13 – Same as Fig. 2.12, but for our reference model (Case 7). The colours correspond to
Fig. 2.6.
47
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
relatively cold material where CO, which evaporates early on in the collapse, can return
to the grains. H2 O can only desorb in the inner few AU of the disk and remnant envelope.
Figures 2.14 and 2.15 show the density and dust temperature at tacc for each parameter
set; our standard and reference models are the two panels on the second row (Case 3 and
7). Several trends are visible:
• With a lower sound speed (Cases 1, 2, 5 and 6), the overall accretion rate ( Ṁ) is
smaller so the accretion time increases (tacc ∝ c−3
s ). The disk can now grow larger
and more massive. In our standard model, the disk is 0.05 M⊙ at tacc and extends to
about 400 AU radially. Decreasing the sound speed to 0.19 km s−1 (Case 1) results
in a disk of 0.22 M⊙ and nearly 2000 AU. The lower accretion rate also reduces the
stellar luminosity. These effects combine to make the disk colder in the low-cs cases.
• With a lower rotation rate (Cases 1–4), the infall occurs in a more spherically symmetric fashion. Less material is captured in the disk, which remains smaller and less
massive. From our reference to our standard model, the disk mass goes from 0.43 to
0.05 M⊙ and the radius from 1400 to 400 AU. The stronger accretion onto the star
causes a higher luminosity. Altogether, this makes for a small, relatively warm disk
in the low-Ω0 cases.
• With a lower initial mass (Cases 2, 4, 6 and 8), there is less material to end up on the
disk. The density profile is independent of the mass in a Shu-type collapse (Eq. (2.1)),
so the initial mass is lowered by taking a smaller envelope radius. The material from
the outer parts of the envelope is the last to accrete and is therefore more likely to end
up in the disk. If the initial mass is halved relative to our standard model (as in Case
4), the resulting disk is only 0.001 M⊙ and 1 AU. Our reference disk goes from 0.43
M⊙ and 1400 AU to 0.16 M⊙ and 600 AU (Cases 7 and 8). The luminosity at tacc is
lower in the high-M0 cases and the cold part of the disk (T d < 18 K) has a somewhat
larger relative size.
Dullemond et al. (2006a) noted that accretion occurs closer to the star for a slowly
rotating core than for a fast rotating core, resulting in a larger fraction of crystalline dust
in the former case. The same effect is seen here, but overall the accretion takes place
farther from the star than in Dullemond et al. (2006a). This is due to our taking into
account the vertical structure of the disk. Our gaseous fractions in the low-Ω0 disks are
higher than in the high-Ω0 disks (consistent with a higher crystalline fraction), but not
because material enters the disk closer to the star. Rather, as mentioned above, the larger
gas content comes from the higher temperatures throughout the disk. A more detailed
discussion of the crystallinity in disks can be found in Chapter 4.
Combining the density and the temperature, the fractions of cold (T d < 18 K), warm
(T d > 18 K) and hot (T d > 100 K) material in the disk can be computed. The warm and
hot fractions are listed in Table 2.3 along with the fractions of gaseous CO and H2 O in
the disk at tacc . Across the parameter grid, 23–100% of the CO is in the gas, along with
0.3–100% of the H2 O. This includes Case 4, which only has a disk of 0.0014 M⊙ . If that
one is omitted, at most 13% of the H2 O in the disk at tacc is in the gas. The gaseous H2 O
fractions for Cases 1, 2, 6, 7 and 8 (at most a few per cent) are quite uncertain, because the
48
2.4 Discussion
Table 2.3 – Summary of properties at t = tacc for our parameter grid.a
Case
1
2
3 (std)
4
5
6
7 (ref)
8
a
Md /M b
0.22
0.15
0.05
0.003
0.59
0.50
0.43
0.33
fwarm c
0.69
0.94
1.00
1.00
0.15
0.34
0.83
1.00
fhot c
0.004
0.035
0.17
1.00
0.0001
0.0004
0.003
0.028
fgas (CO)d
0.62
0.93
1.00
1.00
0.23
0.27
0.81
1.00
fgas (H2 O)d
0.028
0.020
0.13
1.00
0.11
0.02
0.004
0.003
These results are for the one-flavour CO desorption model.
b
The fraction of the disk mass with respect to the total accreted mass (M = M∗ + Md ).
c
d
The fractions of warm (T d > 18 K) and hot (T d > 100 K) material with respect to the entire disk. The
warm fraction also includes material above 100 K.
The fractions of gaseous CO and H2 O with respect to the total amounts of CO and H2 O in the disk.
model does not have sufficient resolution in the inner disk to resolve these small amounts.
They may be lower by up to a factor of ten or higher by up to a factor of three.
There is good agreement between the fractions of warm material and gaseous CO. In
Case 5, about a third of the CO gas at tacc is gas left over from the initial conditions, due to
the long adsorption timescale for the outer part of the cloud core. This is also the case for
the majority of the gaseous H2 O in Cases 1, 5 and 6. For the other parameter sets, fhot and
fgas (H2 O) are the same within the error margins. Overall, the results from the parameter
grid show once again that the adsorption-desorption balance is primarily determined by
the temperature, and that the adsorption-desorption timescales are usually shorter than the
dynamical timescales.
By comparing the fraction of gaseous material at the end of the collapse to the fraction
of material above the desorption temperature, the history of the material is disregarded.
For example, some of the cold material was heated above 18 K during the collapse, and
CO desorbed before readsorbing inside the disk. This may affect the CO abundance if the
model is expanded to include a full chemical network. In that case, the results from Table
2.3 only remain valid if the CO abundance is mostly constant throughout the collapse.
The same caveat holds for H2 O. This point is explored in more detail in Chapter 3.
2.4.2 Complex organic molecules
A full chemical network, including grain-surface reactions, is required to analyse the gas
and ice abundances of more complex species. While this will be a topic for a future publication, the current CO and H2 O results, combined with recent other work, can already
provide some insight.
Complex organic species can be divided into two categories: first-generation species
that are formed on the grain surfaces during the initial warm-up to ∼40 K, and second49
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
Figure 2.14 – Total density at t = tacc for our parameter grid (Table 2.1). The rotation rates (log Ω0
in s−1 ), sound speeds (km s−1 ) and initial masses (M⊙ ) are indicated. The contours increase by
factors of ten going inwards; the 106 -cm−3 contour is labelled in each panel. The white curves
indicate the surfaces of the disks; the disk for Case 4 is too small to be visible.
50
2.4 Discussion
Figure 2.15 – Dust temperature, as in Fig. 2.14. The temperature contours are drawn at 100, 60,
40, 30, 25, 20, 18, 16, 14 and 12 K from the centre outwards. The 20-K contour is labelled in each
panel and the 18-K contours are drawn as grey lines.
51
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
generation species that are formed in the warm gas phase when the first-generation species
have evaporated (Herbst & van Dishoeck 2009). Additionally, CH3 OH may be considered
a zeroth-generation complex organic because it is already formed efficiently during the
pre-collapse phase (Garrod & Herbst 2006). Its gas-ice ratio should be similar to that of
H2 O, due to the similar binding energies.
Larger first-generation species such as methyl formate (HCOOCH3 ) can be formed
on the grains if material spends at least several 104 yr at 20–40 K. The radicals involved
in the surface formation of HCOOCH3 (HCO and CH3 O) are not mobile enough at lower
temperatures and are not formed efficiently enough at higher temperatures. A low surface abundance of CO (at temperatures above 18 K) does not hinder the formation of
HCOOCH3 : HCO and CH3 O are formed from reactions of OH and H with H2 CO, which
is already formed at an earlier stage and which does not evaporate until ∼40 K (Garrod &
Herbst 2006). Cosmic-ray–induced photons are available to form OH from H2 O even in
the densest parts of the disk and envelope (Shen et al. 2004).
As shown in Sect. 2.3.4, many of the parcels ending up near the midplane inside of
∼300 AU in our standard model spend sufficient time in the required temperature regime
to allow for efficient formation of HCOOCH3 and other complex organics. Once formed,
these species are likely to assume the same gas-ice ratios as H2 O and the smaller organics. They evaporate in the inner 10–20 AU, so in the absence of mixing, complex organics
would only be observable in the gas phase close to the star. The Atacama Large Millimeter/submillimeter Array (ALMA), currently under construction, will be able to test this
hypothesis.
The gas-phase route towards complex organics involves the hot inner envelope (T d >
100 K), also called the hot core or hot corino in the case of low-mass protostars (Ceccarelli
2004, Bottinelli et al. 2004, 2007). Most of the ice evaporates here and a rich chemistry
can take place if material spends at least several 103 yr in the hot core (Charnley et al.
1992). However, the material in the hot inner envelope in our model is essentially in
freefall towards the star or the inner disk, and its transit time of a few 100 yr is too short
for complex organics to be formed abundantly (see also Schöier et al. 2002). Additionally,
the total mass in this region is very low: about a per cent of the disk mass. In order to
explain the observations of second-generation complex molecules, there has to be some
mechanism to keep the material in the hot core for a longer time. Alternatively, it has
recently been suggested that molecules typically associated with hot cores may in fact
form on the grain surfaces as well (Garrod et al. 2008).
2.4.3 Mixed CO-H2O ices
In the results presented in Sect. 2.3, all CO was taken to desorb at a single temperature.
In a more realistic approach, some of it would be trapped in the H2 O ice and desorb at
higher temperatures. This was simulated with four “flavours” of CO ice, as summarised in
Table 2.2. With our four-flavour model, the global gas-ice profiles are mostly unchanged.
All CO is frozen out in the sub-18 K regions and it fully desorbs when the temperature
goes above 100 K. Some 10 to 20% remains in the solid phase in areas of intermediate
temperature. In our standard model, the four-flavour variety has 15% of all CO in the disk
52
2.4 Discussion
at tacc on the grains, compared to 0% in the one-flavour variety. In our reference model,
the solid fraction increases from 19 to 33%.
The grain-surface formation of H2 CO, CH3 OH, HCOOCH3 and other organics should
not be very sensitive to these variations. H2 CO and CH3 OH are already formed abundantly before the onset of collapse, when the one- and four-flavour models predict equal
amounts of solid CO. H2 CO is then available to form HCOOCH3 (via the intermediates
HCO and CH3 O) during the collapse. The higher abundance of solid CO at 20–40 K in
the four-flavour model could slow down the formation of HCOOCH3 somewhat, because
CO destroys the OH needed to form HCO (Garrod & Herbst 2006). H2 CO evaporates
around 40 K, so HCOOCH3 cannot be formed efficiently anymore above that temperature. On the other hand, if a multiple-flavour approach is also employed for H2 CO, some
of it remains solid above 40 K, and HCOOCH3 can continue to be produced. Overall,
then, the multiple-flavour desorption model is not expected to cause large variations in
the abundances of these organic species compared to the one-flavour model.
2.4.4 Implications for comets
Comets in our solar system are known to be abundant in CO and they are believed to
have formed between 5 and 30 AU in the circumsolar disk (Bockelée-Morvan et al. 2004,
Kobayashi et al. 2007). However, the dust temperature in this region at the end of the
collapse is much higher than 18 K for all of our parameter sets. This raises the question
of how solid CO can be present in the comet-forming zone.
One possible answer lies in the fact that even at t = tacc , our objects are still very
young. As noted in Sect. 2.3.2, the disks cool down as they continue to evolve towards
“mature” T Tauri systems. Given the right set of initial conditions, this may bring the
temperature below 18 K inside of 30 AU. However, there are many T Tauri disk models in
the literature where the temperature at those radii remains well above the CO evaporation
temperature (e.g., D’Alessio et al. 1998, 2001). Specifically, models of the minimummass solar nebula (MMSN) predict a dust temperature of ∼40 K at 30 AU (Lecar et al.
2006).
A more plausible solution is to turn to mixed ices. At the temperatures computed for
the comet-forming zone of the MMSN, 10–20% of all CO may be trapped in the H2 O
ice. Assuming typical CO-H2 O abundance ratios, this is entirely consistent with observed
cometary abundances (Bockelée-Morvan et al. 2004).
Large abundance variations are possible for more complex species, due to the different
densities and temperatures at various points in the comet-forming zone in our model, as
well as the different density and temperature histories for material ending up at those
points. This seems to be at least part of the explanation for the chemical diversity observed
in comets. Our current model is extended in Chapter 3 to include a full gas-phase chemical
network to analyse these variations and compare them against cometary abundances.
The desorption and readsorption of H2 O in the disk-envelope boundary shock has
been suggested as a method to trap noble gases in the ice and include them in comets
(Owen et al. 1992, Owen & Bar-Nun 1993). As shown in Sects. 2.2.5 and 2.3.4, a number
of parcels in our standard model are heated to more than 100 K just prior to entering
53
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
the disk. However, these parcels end up in the disk’s surface. Material that ends up at
the midplane, in the comet-forming zone, never gets heated above 50 K. Vertical mixing,
which is ignored in our model, may be able to bring the noble-gas–containing grains down
into the comet-forming zone.
Another option is episodic accretion, resulting in temporary heating of the disk (Sect.
2.4.5). In the subsequent cooling phase, noble gases may be trapped as the ices reform.
The alternative of trapping the noble gases already in the pre-collapse phase is unlikely.
This requires all the H2 O to start in the gas phase and then freeze out rapidly. However,
in reality (contrary to what is assumed in our model) it is probably formed on the grain
surfaces by hydrogenation of atomic oxygen, which would not allow for trapping of noble
gases.
2.4.5 Limitations of the model
The physical part of our model is known to be incomplete and this may affect the chemical
results. For example, our model does not include radial and vertical mixing. Semenov
et al. (2006) and Aikawa (2007) recently showed that mixing can enhance the gas-phase
CO abundance in the sub-18 K regions of the disk. Similarly, there could be more H2 O gas
if mixing is included. This would increase the fractions of CO and H2 O gas listed in Table
2.3. The gas-phase abundances can also be enhanced by allowing for photodesorption of
the ices in addition to the thermal desorption considered here (Shen et al. 2004, Öberg
et al. 2007, 2009b,c). Mixing and photodesorption can each increase the total amount of
gaseous material by up to a factor of two. The higher gas-phase fractions are mostly found
in the regions where the temperature is a few degrees below the desorption temperature
of CO or H2 O.
Accretion from the envelope onto the star and disk occurs in our model at a constant
rate Ṁ until all of the envelope mass is gone. However, the lack of widespread red-shifted
absorption seen in interferometric observations suggests that the infall may stop already
at an earlier time (Jørgensen et al. 2007). This would reduce the disk mass at tacc . The
size of the disk is determined by the viscous evolution, which would probably not change
much. Hence, if accretion stops or slows down before tacc , the disk would be less dense
and therefore warmer. It would also reduce the fraction of disk material where CO never
desorbed, because most of that material comes from the outer edge of the original cloud
core (Fig. 2.11). Both effects would increase the gas-ice ratios of CO and H2 O.
Our results are also modified by the likely occurrence of episodic accretion (Kenyon
& Hartmann 1995, Evans et al. 2009). In this scenario, material accretes from the disk
onto the star in short bursts, separated by intervals where the disk-to-star accretion rate
is a few orders of magnitude lower. The accretion bursts cause luminosity flares, briefly
heating up the disk before returning to an equilibrium temperature that is lower than in
our models. This may produce a disk with a fairly large ice content for most of the
time, which evaporates and readsorbs after each accretion episode. The consequences for
complex organics and the inclusion of various species in comets are unclear.
54
2.5 Conclusions
2.5 Conclusions
This chapter presents the first results from a two-dimensional, semi-analytical model that
simulates the collapse of a molecular cloud core to form a low-mass protostar and its surrounding disk. The model follows individual parcels of material from the core into the star
or disk and also tracks their motion inside the disk. It computes the density and temperature at each point along these trajectories. The density and temperature profiles are used
as input for a chemical code to calculate the gas and ice abundances for carbon monoxide
(CO) and water (H2 O) in each parcel, which are then transformed into global gas-ice profiles. Material ending up at different points in the disk spends a different amount of time
at certain temperatures. These temperature histories provide a first look at the chemistry
of more complex species. The main results from this chapter are as follows:
• Both CO and H2 O freeze out towards the centre of the core before the onset of collapse. As soon as the protostar turns on, a fraction of the CO rapidly evaporates, while
H2 O remains on the grains. CO returns to the solid phase when it cools below 18 K
inside the disk. Depending on the initial conditions, this may be in a small or a large
fraction of the disk (Sect. 2.3.3).
• All parcels that end up in the disk have the same qualitative temperature history (Fig.
2.12). There is one temperature peak just before entering the disk, when material
traverses the inner envelope, and a second one (higher than the first) when inward
radial motion brings the parcel closer to the star. In some cases, this results in multiple
desorption and adsorption events during the parcel’s infall history (Sect. 2.3.4).
• Material that originates near the midplane of the initial core remains at lower temperatures than does material originating from closer to the poles. As a result, the chemical
content of the material from near the midplane is less strongly modified during the
collapse than the content of material from other regions (Fig. 2.11). The outer part of
the disk contains the chemically most pristine material, where at most only a small
fraction of the CO ever desorbed (Sect. 2.3.3).
• A higher sound speed results in a smaller and warmer disk, with larger fractions of
gaseous CO and H2 O at the end of the envelope accretion. A lower rotation rate has
the same effect. A higher initial mass results in a larger and colder disk, and smaller
gaseous CO and H2 O fractions (Sect. 2.4.1).
• The infalling material generally spends enough time in a warm zone (20–40 K) for
first-generation complex organic species to be formed abundantly on the grains (Fig.
2.12). Large differences can occur in the density and temperature histories for material
ending up at various points in the disk. These differences allow for spatial abundance
variations in the complex organics across the entire disk. This appears to be at least
part of the explanation for the cometary chemical diversity (Sects. 2.4.2 and 2.4.4).
• Complex second-generation species are not formed abundantly in the warm inner
envelope (the hot core or hot corino) in our model, due to the combined effects of the
dynamical timescales and low mass fraction in that region (Sect. 2.4.2).
• The temperature in the disk’s comet-forming zone (5–30 AU from the star) lies well
above the CO desorption temperature, even if effects of grain growth and continued
55
Chapter 2 – The chemical history of molecules in circumstellar disks, part I
disk evolution are taken into account. Observed cometary CO abundances can be
explained by mixed ices: at temperatures of several tens of K, as predicted for the
comet-forming zone, CO can be trapped in the H2 O ice at a relative abundance of a
few per cent (Sect. 2.4.4).
Appendix: Disk formation efficiency
The results from our parameter grid can be used to derive the disk formation efficiency,
ηdf , as a function of the sound speed, cs , the solid-body rotation rate, Ω0 , and the initial
core mass, M0 . This efficiency can be defined as the fraction of M0 that is in the disk at
the end of the collapse phase (t = tacc ) or as the mass ratio between the disk and the star
at that time. The former is used in this appendix.
In order to cover a wider range of initial conditions, the physical part of our model
was run on a 93 grid. The sound speed was varied from 0.15 to 0.35 km s−1 , the rotation
rate from 10−14.5 to 10−12.5 s−1 and the initial core mass from 0.1 to 2.1 M⊙ . The resulting
ηdf at t = tacc were fitted to
"
#
Md
log(Ω0 /s−1 )
ηdf =
(2.37)
= g1 + g2
M0
−13
with
g 1 = k1 + k2
"
log(Ω0 /s−1 )
−13
g 2 = k5 + k6
"
#q1
+ k3
"
#q
q2
M0 3
cs
+
k
,
4
M⊙
0.2 km s−1
"
#
#
M0
cs
log(Ω0 /s−1 )
+
k
+ k7
.
8
−13
M⊙
0.2 km s−1
(2.38)
(2.39)
Equation (2.37) can give values lower than 0 or larger than 1. In those cases, it should be
interpreted as being 0 or 1.
The best-fit values for the coefficients ki and the exponents qi are listed in Table 2.4.
The absolute and relative difference between the best fit and the model data have a root
mean square (rms) of 0.04 and 5%. The largest absolute and relative difference are 0.20
and 27%. The fit is worst for a high core mass, a low sound speed and an intermediate
rotation rate, as well as for a low core mass, an intermediate to high sound speed and a
high rotation rate.
Figure 2.16 shows the disk formation efficiency as a function of the rotation rate,
including the fit from Eq. (2.37). The efficiency is roughly a quadratic function in log Ω0 ,
but due to the narrow dynamic range of this variable, the fit appears as straight lines.
Furthermore, the efficiency is roughly a linear function in cs and a square root function in
M0 .
56
2.5 Conclusions
Table 2.4 – Coefficients and exponents for the best fit for the disk formation efficiency.
Coefficient
k1
k2
k3
k4
Value
2.08
0.020
0.035
0.914
Coefficient
k5
k6
k7
k8
Value
−0.106
−1.539
−0.470
−0.344
Exponent
q1
q2
q3
Value
0.236
0.255
0.537
Figure 2.16 – Disk formation efficiency as a function of the solid-body rotation rate. The model
values are plotted as symbols and the fit from Eq. (2.37) as lines. The different values of the sound
speed are indicated by colours and the different values of the initial core mass are indicated by
symbols and line types, with the solid lines corresponding to the asterisks, the dotted lines to the
diamonds and the dashed lines to the triangles.
57
3
The chemical history of molecules in
circumstellar disks
II: Gas-phase species
R. Visser, S. D. Doty and E. F. van Dishoeck
to be submitted
59
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
Abstract
Context. The chemical composition of a molecular cloud changes dramatically as it collapses to
form a low-mass protostar and circumstellar disk. Two-dimensional (2D) chemodynamical models
are required to properly study this process.
Aims. The goal of this work is to follow, for the first time, the chemical evolution in two dimensions
all the way from a pre-stellar cloud into a circumstellar disk. Of special interest is the question
whether the chemical composition of the disk is a result of chemical processing during the collapse
phase, or whether it is determined by in situ processing after the disk has formed.
Methods. We combine a semi-analytical method to get 2D axisymmetric density and velocity structures with detailed radiative transfer calculations to get temperature profiles and UV fluxes. Material
is followed in from the cloud to the disk and a full gas-phase chemistry network – including freezeout onto and evaporation from cold dust grains – is evolved along these trajectories. The abundances
thus obtained are compared to the results from a static disk model and to cometary observations.
Results. The chemistry during the collapse phase is dominated by a few key processes, such as
the evaporation of CO or the photodissociation of H2 O. Depending on the physical conditions encountered along specific trajectories, some of these processes are absent. At the end of the collapse
phase, the disk can thus be divided into zones with different chemical histories. The disk is found
not to be in chemical equilibrium at the end of the collapse. We argue that comets must be formed
from material with different chemical histories: some of it is strongly processed, some of it remains
pristine. Variations between individual comets are possible if they formed at different positions or
times in the solar nebula. The chemical zones in the disk and the mixed origin of the cometary
material arise from the 2D nature of our model.
60
3.1 Introduction
3.1 Introduction
The formation of a low-mass protostar out of a cold molecular cloud is accompanied by
large-scale changes in the chemical composition of the constituent gas and dust. Prestellar cloud cores are cold (∼10 K), moderately dense (∼104 –106 cm−3 ), and irradiated only by the ambient interstellar radiation field (see reviews by Shu et al. 1987, di
Francesco et al. 2007 and Bergin & Tafalla 2007). As the core starts to collapse, several mechanisms act to heat up the material, such as gravitational contraction, accretion
shocks and, eventually, radiation produced by nuclear fusion in the protostar. The inner few hundred AU of the core flatten out to form a circumstellar disk, where planets
may be formed at a later stage (see review by Dullemond et al. 2007b). The density in
the interior of the disk, especially at small radii, is several orders of magnitude higher
than the density of the pre-stellar core. Meanwhile, the protostar infuses the disk with
high fluxes of ultraviolet and X-ray photons. The chemical changes arising from these
evolving physical conditions have been analysed by various groups with one-dimensional
models (see reviews by Ceccarelli et al. 2007, Bergin et al. 2007 and Bergin & Tafalla
2007). However, two-dimensional models are required to properly describe the formation
of the circumstellar disk and the chemical processes taking place inside it.
This chapter follows the preceding chapter in a series of publications aiming to model
the chemical evolution from pre-stellar cores to circumstellar disks in two dimensions.
Chapter 2 contains a detailed description of our semi-analytical model and an analysis of
the gas and ice abundances of carbon monoxide (CO) and water (H2 O). We found that
most CO evaporates during the infall phase and freezes out again in those parts of the disk
that are colder than 18 K. The much higher binding energy of H2 O keeps it in solid form
at all times, except within ∼10 AU of the star. Based on the time that the infalling material spends at dust temperatures between 20 and 40 K, first-generation complex organic
species were predicted to form abundantly on the grain surfaces according to the scenario
of Garrod & Herbst (2006) and Garrod et al. (2008).
The current chapter extends the chemical analysis to a full gas-phase network, including freeze-out onto and evaporation from dust grains, as well as basic grain-surface
hydrogenation reactions. Combining semi-analytical density and velocity structures with
detailed temperature profiles from full radiative transfer calculations, our aim is to bridge
the gap between 1D chemical models of collapsing cores and 1+1D or 2D chemical models of mature T Tauri and Herbig Ae/Be disks. One of the key questions is whether the
chemical composition of such disks is mainly a result of chemical processing during the
collapse or whether it is determined by in situ processing after the disk has formed.
As reviewed by di Francesco et al. (2007) and Bergin & Tafalla (2007), the chemistry
of pre-stellar cores is well understood. Because of the low temperatures and the moderately high densities, a lot of molecules are observed to be depleted from the gas by freezing out onto the cold dust grains. The main ice constituent is H2 O, showing abundances
of ∼10−4 relative to H2 (Tielens et al. 1991, van Dishoeck 2004). Other abundant ices are
CO2 (30–35% of H2 O; Pontoppidan et al. 2008b) and CO (5–100% of H2 O; Jørgensen
et al. 2005, Pontoppidan 2006). Correspondingly, the observed gas-phase abundances of
H2 O, CO and CO2 in pre-stellar cores are low (Snell et al. 2000, Ashby et al. 2000, Bergin
61
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
& Snell 2002, Bacmann et al. 2002). Nitrogen-bearing species like N2 and NH3 are generally less depleted than carbon- and oxygen-bearing ones (Rawlings et al. 1992, Tafalla
et al. 2002, 2004), probably because they require a longer time to be formed in the gas and
therefore have not yet had a chance to freeze out (di Francesco et al. 2007). The observed
depletion factors are well reproduced with 1D chemical models (Bergin & Langer 1997,
Lee et al. 2004).
The collapse phase is initially characterised by a gradual warm-up of the material,
resulting in the evaporation of the ice species according to their respective binding energies (van Dishoeck et al. 1993, van Dishoeck & Blake 1998, Boogert et al. 2000, van der
Tak et al. 2000a, Aikawa et al. 2001, Jørgensen et al. 2002, 2004, 2005, Jørgensen 2004).
The higher temperatures also drive a rich chemistry, especially if it gets warm enough
to evaporate H2 O and organic species like CH3 OH and HCOOCH3 (Blake et al. 1987,
Millar et al. 1991, Charnley et al. 1992). Going back to the late 1970s, the chemical evolution during the collapse phase has been studied with purely spherical models (Gerola
& Glassgold 1978, Leung et al. 1984, Ceccarelli et al. 1996, Rodgers & Charnley 2003,
Doty et al. 2004, Lee et al. 2004, Garrod & Herbst 2006, Aikawa et al. 2008, Garrod et al.
2008). They are successful at explaining the observed abundances at scales of several
thousand AU, where the envelope is still close to spherically symmetric, but they cannot
make the transition from the 1D spherically symmetric envelope to the 2D axisymmetric
circumstellar disk.
Recently, van Weeren et al. (2009) followed the chemical evolution within the framework of a 2D hydrodynamical simulation and obtained a reasonable match with observations. However, their primary focus was still on the envelope, not on the disk. Nevertheless, they showed how important it is to treat the chemical evolution during low-mass star
formation as more than a simple 1D process.
Once the phase of active accretion from the envelope comes to an end, the circumstellar disk settles into a comparatively static situation (Bergin et al. 2007, Dullemond et al.
2007a). Observationally, we are now in the T Tauri and Herbig Ae/Be stages, and some
simple molecules have been detected in these objects (Dutrey et al. 1997, Kastner et al.
1997, Qi et al. 2003, Thi et al. 2004, Lahuis et al. 2006). They have also received a lot
of attention with 2D models, showing for example that disks can be divided vertically
into three chemical layers: a cold zone near the midplane, a warm molecular layer at intermediate altitudes, and a photon-dominated region at the surface (Aikawa et al. 1996,
1997, 2002, 2008, Aikawa & Herbst 1999, 2001, Willacy & Langer 2000, van Zadelhoff
et al. 2003, Rodgers & Charnley 2003, Jonkheid et al. 2004, Semenov et al. 2006, Woitke
et al. 2009). However, as noted above, the chemical connection between the early 1D
stages of low-mass star formation and the 2D circumstellar disks at later stages remains
an unsolved puzzle.
This chapter aims to provide the first steps towards solving this puzzle by following
the chemical evolution all the way from a pre-stellar cloud core to a circumstellar disk
in two spatial dimensions. The physical and chemical models are described in Sects.
3.2 and 3.3. We briefly discuss the chemistry during the pre-collapse phase in Sect. 3.4
before turning to the collapse itself in Sect. 3.5. There, we first follow the chemistry
in detail along a trajectory terminating at one particular position in the disk, and then
62
3.2 Collapse model
generalise those results to material ending up at other positions. In Sect. 3.6, we compare
the abundances resulting from the collapse to in situ processing in a static disk. Finally,
we discuss some caveats and the implications of our results for the origin of comets in
Sect. 3.7. Conclusions are drawn in Sect. 3.8.
3.2 Collapse model
3.2.1 Step-wise summary
Our semi-analytical collapse model is described in detail in Chapter 2; it consists of several steps, summarised in Fig. 3.1. We start with a singular isothermal sphere characterised by a total mass M0 , an effective sound speed cs , and a uniform rotation rate Ω0 .
As soon as the collapse starts, at t = 0, the rotation causes the infalling material to be
deflected towards the equatorial midplane. This breaks the spherical symmetry, so we run
the entire model as a two-dimensional axisymmetric system. The 2D density and velocity
profiles follow the solutions of Shu (1977), Cassen & Moosman (1981) and Terebey et al.
(1984) for an inside-out collapse with rotation. After the disk is first formed at the midplane, it evolves by ongoing accretion from the collapsing core and by viscous spreading
to conserve angular momentum (Shakura & Sunyaev 1973, Lynden-Bell & Pringle 1974).
Taking the 2D density profiles from step 2, and adopting the appropriate size and
luminosity for the protostar (Adams & Shu 1986, Young & Evans 2005), the next step
consists of computing the dust temperature at a number of time steps. We do this with
the radiative transfer code RADMC (Dullemond & Dominik 2004a), which takes a 2D
axisymmetric density profile but follows photons in all three dimensions. The RADMC
code also computes the full radiation spectrum at each point in the axisymmetric disk and
remnant envelope, as required for the photon-driven reactions in our chemical network
(Sects. 3.2.3 and 3.3.1). The gas temperature is set equal to the dust temperature throughout the disk and the envelope. This is a poor assumption in the surface of the disk and the
Figure 3.1 – Stepwise summary of
our 2D axisymmetric
collapse
model.
Steps 2 and 4 are
semi-analytical, while
steps 3 and 5 consist
of detailed numerical
simulations.
63
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
inner parts of the envelope (Kamp & Dullemond 2004, Jonkheid et al. 2004, Woitke et al.
2009), the consequences of which are addressed in Sect. 3.7.1.
Given the dynamical nature of the collapse, it is easiest to solve the chemistry in a
Lagrangian frame. In Chapter 2, we populated the envelope with several thousand parcels
at t = 0 and followed them in towards the disk and star. We now take an alternative approach where we define a regular grid of parcels at the end of the collapse and follow the
parcels backwards in time to their position at t = 0. Since we are usually interested in the
abundance profiles at the end of the collapse, when the disk is fully formed, it has many
advantages to have a regular grid of parcels at that time rather than at the beginning. In
either case, step 4 of the model produces a set of infall trajectories with densities, temperatures and UV intensities as a function of time and position. These data are required for the
next step: solving the time-dependent chemistry for each individual parcel. Although the
parcels are followed backwards in time to get their trajectories, we compute the chemistry
in the normal forward direction. The last step of our model consists of transforming the
abundances from the individual parcels back into 2D axisymmetric profiles at whatever
time steps we are interested in.
In Chapter 2, the model was run for a grid of initial conditions. In the current chapter,
the analysis is limited to our standard set of parameters: M0 = 1.0 M⊙ , cs = 0.26 km
s−1 and Ω0 = 10−14 s−1 . Section 3.7.2 contains a brief discussion on how the results may
change for other parameter values.
3.2.2 Differences with Chapter 2
The current version of the model contains several improvements over the version used
in Chapter 2. Most importantly, it now correctly treats the problem of sub-Keplerian
accretion onto a 2D disk. It has long been known that material falling onto the disk along
an elliptic orbit has sub-Keplerian angular momentum, so it exerts a torque on the disk
that results in an inward push. Several solutions are available (e.g., Cassen & Moosman
1981, Hueso & Guillot 2005), but these are not suitable for our 2D model. The ad-hoc
solution from Chapter 2 provided the appropriate qualitative physical correction, namely
increasing the inward radial velocity of the disk material, but it did not properly conserve
angular momentum. We now use a new, fully consistent solution, derived directly from
the equations for the conservation of mass and angular momentum. It is described in
detail in Chapter 4, where it is also shown that it results in disks that are typically a factor
of a few smaller than those obtained with the original model. The new disks are a few
degrees colder in the inner part and warmer in the outer part, which may further affect the
chemistry.
Other changes to the model include the definition of the disk-envelope boundary and
the shape of the outflow cavity. In Chapter 2, the disk-envelope boundary was defined as
the surface where the density of the infalling envelope material equals that of the disk.
Instead, we now take the surface where the ram pressure of the infalling material equals
the thermal pressure of the disk (see Chapter 4). This provides a more physically correct
description of where material becomes part of the disk. The outflow cavity now has
curved walls rather than straight ones, consistent with both observations and theoretical
64
3.2 Collapse model
predictions (Velusamy & Langer 1998, Cantó et al. 2008). The outflow wall is described
by
!1.5
!−3
t
R
z = (0.191 AU)
,
(3.1)
tacc
AU
with R and z in spherical coordinates and tacc = M0 / Ṁ the time required for the entire
envelope to accrete onto the star and disk. The t−3 dependence is chosen so that the
outflow starts very narrow and becomes increasingly wide as the collapse proceeds. The
full opening angle at tacc is 33.6◦ at z = 1000 AU and 15.9◦ at z = 10 000 AU.
3.2.3 Radiation field
Photodissociation and photoionisation by ultraviolet (UV) radiation are important processes in the hot inner core and in the surface layers of the disk. The temperature and
luminosity of the protostar evolve as described in Chapter 2, so neither the strength nor
the spectral shape of the radiation it emits are constant in time. In addition, the spectral
shape changes as the radiation passes through the disk and remnant envelope. Hence, we
cannot simply take the photorates from the interstellar medium and scale them according
to the integrated UV flux at each spatial grid point.
The most accurate way to obtain the time- and location-dependent photorates is to
multiply the cross section for each reaction by the UV field at each grid point. The latter
can be computed from 2D radiative transfer at high spectral resolution. As this is too
computationally demanding, several approximations have to be made. First of all, we
assume the wavelength-dependent attenuation of the radiation field by the dust in the disk
and envelope can be represented by a single factor γ for each reaction. The rate coefficient
for a given photoreaction at spatial coordinates r and θ and at time t can then be expressed
as
!−2
r
∗
kph (r, θ, t) = kph
e−γAV ,
(3.2)
R∗
with AV the visual extinction towards that point. The unshielded rate coefficient is cal∗
culated at the stellar surface (kph
) by multiplying the cross section of the reaction by the
blackbody flux at the effective temperature of the protostar. The term (r/R∗ )−2 accounts
for the geometrical dilution of the radiation from the star across a distance r. The factor γ
is discussed in Sect. 3.3.1.
In order to apply Eq. (3.2), we need the extinction towards each point. In a 1D model,
this can simply be done by integrating the total hydrogen number density from the star to
a point r and converting the resulting column density to a visual extinction. This approach
has been extended to 2D circumstellar disk models by dividing the disk into annuli, each
irradiated only from the top and bottom (e.g., Aikawa & Herbst 1999, Jonkheid et al.
2004). Such a 1+1D method is poorly suited to our model, which always has infalling
envelope material right above and below the disk. Instead, we compute an average UV
flux for each spatial grid point at a number of time steps and compare it to the flux that
would be obtained in the case of zero attenuation. The difference gives us an effective
extinction for each point.
65
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
Table 3.1 – Elemental abundances: x(X) = n(X)/nH .
Species
H2
He
C
N
O
Ne
Abundance
5.00(-1)
9.75(-2)
7.86(-5)
2.47(-5)
1.80(-4)
1.40(-6)
Species
Na
Mg
Al
Si
P
S
Abundance
2.25(-9)
1.09(-8)
3.10(-8)
2.74(-9)
2.16(-10)
9.14(-8)
Species
Cl
Ar
Ca
Cr
Fe
Ni
Abundance
1.00(-9)
3.80(-8)
2.20(-8)
4.90(-9)
2.74(-9)
1.80(-8)
The first step in this procedure is to run the Monte Carlo radiative transfer package
RADMC (Dullemond & Dominik 2004a) at a low spectral resolution of one frequency
point per eV. The large number of photons propagated through the grid (typically 105 )
ensures that we get a statistical sampling of the possible trajectories leading to each point.
The specific UV fluxes thus obtained for a point (r, θ, t) are integrated from 6 to 13.6 eV
to get the average flux (FUV ). This flux is lower than that at the stellar surface because
of geometrical dilution and attenuation by dust. Denoting the flux at the stellar surface as
∗
FUV
, we can express the effects of dilution and attenuation as
∗
0
FUV (r, θ) = FUV
e−τUV,eff = FUV
r
R∗
!−2
e−τUV,eff .
(3.3)
The effective UV extinction, τUV,eff , is converted to the visual extinction AV through the
0
standard relationship AV = τUV,eff /3.02. The unattenuated UV flux, FUV
, can also be
expressed as a scaling factor relative to the average flux in the interstellar medium (ISM):
0
χ = FUV
/FISM , with FISM = 8 × 107 cm−2 s−1 (Draine 1978).
3.3 Chemical network
The basis of our chemical network is the UMIST06 database (Woodall et al. 2007) as
modified by Bruderer et al. (2009), except that we do not include X-ray chemistry. The
cosmic-ray ionisation rate of H2 is set to 5 × 10−17 s−1 (van der Tak et al. 2000b, Doty
et al. 2004, Dalgarno 2006). The network contains 162 neutral species, 251 cations and
six anions, built up out of 18 elements. We take a fully atomic composition as the starting
point, except that hydrogen starts as H2 . Elemental abundances are adopted from Aikawa
et al. (2008), with additional values from Bruderer et al. (2009). The latter are reduced
by a factor of 100 from the original undepleted values to account for the incorporation
of these heavy elements into the dust grains. Table 3.1 lists the elemental abundances
relative to the total hydrogen nucleus density: nH = n(H) + 2n(H2 ).
In order to set the chemical composition at the onset of collapse (t = 0), we evolve the
initially atomic gas for a period of 1 Myr at nH = 8 × 104 cm−3 and T g = T d = 10 K. The
extinction during this pre-collapse phase is set to 100 mag to disable all photoprocesses,
except for a minor contribution from cosmic-ray–induced photons. The resulting solid
66
3.3 Chemical network
and gas-phase abundances are consistent with those observed in pre-stellar cores (e.g.,
Bergin et al. 2000, di Francesco et al. 2007), and we take them as the initial condition for
the collapse phase for all infalling parcels. In the remainder of this chapter, t = 0 always
refers to the onset of collapse, following the 1 Myr pre-collapse phase here described.
3.3.1 Photodissociation and photoionisation
Photodissociation and photoionisation by UV radiation are important processes in the
inner disk and inner envelope. Their rates are given by Eq. (3.2) using the extinction
from Eq. (3.3). The extinction factor γ from Eq. (3.2) depends on the spectral shape
of the radiation field, but it is not feasible to include this dependence in detail. Instead,
we use the molecule-specific values tabulated for a 4000 K blackbody by van Dishoeck
∗
et al. (2006). The unshielded rates at the stellar surface (kph
) are calculated with the cross
sections from our freely available database, compiled from Lee (1984), van Dishoeck
(1988), Roberge et al. (1991), Huebner et al. (1992), van Dishoeck et al. (2006) and van
Hemert & van Dishoeck (2008).1 The final output of this procedure consists of a 3D array
(time and two spatial coordinates) with rate coefficients for each photoreaction. When
computing the infall trajectories for individual parcels (step 4 in Fig. 3.1), we perform a
linear interpolation to get the rate coefficients at all points along each trajectory.
The photodissociation of H2 and CO requires some special treatment. Both processes
occur exclusively through discrete absorption lines, so self-shielding plays an important
role. The amount of shielding for H2 is a function of the H2 column density; for CO,
it is a function of both the CO and the H2 column density, because some CO lines are
shielded by H2 lines. The effective UV extinction from Eq. (3.3) can be converted to a total
hydrogen column density through AV = τUV,eff /3.02 and NH = 1.59×1021 AV cm−2 (Diplas
& Savage 1994). Assuming that most hydrogen along each photon path is in molecular
form, we simply set N(H2 ) = 0.5NH to get the effective H2 column density towards each
spatial grid point. Equation (37) from Draine & Bertoldi (1996) then gives the amount
of self-shielding for H2 . The unshielded dissociation rate is computed according to the
one-line approximation from van Dishoeck (1987), scaled so that the rate is 4.5 × 10−11
s−1 in the standard Draine (1978) field. For CO, we use the new shielding functions and
cross sections from Chapter 5. Effective CO column densities are derived from the H2
column densities by assuming an average N(CO)/N(H2 ) ratio of 10−5 . Since both H2
and CO require absorption of photons shortwards of 1100 Å, their dissociation rates are
greatly reduced at T ∗ ≈ 4000 K compared with the Draine field (see also van Zadelhoff
et al. 2003). During the collapse, this results in a zone where molecules like H2 O and
CH4 are photodissociated, while H2 and CO remain intact (Sect. 3.5.1).
3.3.2 Gas-grain interactions
We allow all neutral species other than H, H2 and the three noble gases to freeze out onto
the dust according to Charnley et al. (2001). In cold, dense environments – such as our
1
http://www.strw.leidenuniv.nl/∼ewine/photo
67
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
model cores before the onset of collapse – observations show H2 O, CO and CO2 to be the
most abundant ices (Gibb et al. 2004, Boogert et al. 2008). Of these three, H2 O and CO2
are mixed together, but most CO is found to reside in a separate layer (Pontoppidan et al.
2003, 2008b). As the temperature rises during the collapse, the ices evaporate according
to their binding energy. However, the presence of non-volatile species like H2 O prevents
the more volatile species like CO and CO2 from evaporating entirely: some CO and CO2
gets trapped in the H2 O ice (Sandford & Allamandola 1988, Hasegawa & Herbst 1993,
Collings et al. 2004). In Chapter 2, we showed that this is required to explain the presence
of CO in solar-system comets.
Incorporating ice trapping in a chemical network is a non-trivial task. The approach
of Viti et al. (2004) was primarily designed to reproduce the temperature-programmed
desorption (TPD) experiments from Collings et al. (2004). It works for astrophysical
models where the temperature is increasing monotonically, but as shown in Chapter 2, the
infalling material in our collapse model goes through periods of decreasing temperature
as well. We could still apply the Viti et al. method to a “network” consisting of only CO
and H2 O (Chapter 2), but applying it to the full network currently used consistently leads
to numerical instabilities. In addition, recent experiments by Fayolle et al. (in prep.) show
that the amount of trapping depends on the ice thickness and the volatile-to-H2 O mixing
ratio. Collings et al. performed all their experiments at the same thickness and the same
mixing ratio, so the amount of trapping in the model of Viti et al. is independent of these
properties.
We ignore trapping for now and treat desorption of all species according to the zerothorder rate equation
#
"
Eb (X)
2
,
(3.4)
Rthdes (X) = 4πagr ngr f (X)ν(X) exp −
kT d
where T d is the dust temperature, agr = 0.1 µm the typical grain radius, and ngr = 10−12 nH
the grain number density. The canonical pre-exponential factor, ν, for first-order desorption is 2 × 1012 s−1 (Sandford & Allamandola 1993). We multiply this by the number of
binding sites per unit grain surface (8 × 1014 cm−2 for our 0.1 µm grains, assuming 106
binding sites per grain) to get a zeroth-order pre-exponential factor of 2 × 1027 cm−2 s−1 .
This value is used for all ice species, with the exception of the four listed in Table 3.2.
The binding energies of species other than those four are set to the values tabulated by
Sandford & Allamandola (1993) and Aikawa et al. (1997). Species for which the binding energy is unknown are assigned the binding energy and the pre-exponential factor of
H2 O. The dimensionless factor f in Eq. (3.4) ensures that each species desorbs according
to its abundance in the ice, and changes the overall desorption behaviour from zeroth to
first order when there is less than one monolayer of ice:
f (X) =
ns (X)
,
max(nice , Nb ngr )
(3.5)
with Nb = 106 the typical number of binding sites per grain and nice the total number
density (per unit volume of cloud or disk) of all ice species combined. We briefly discuss
in Sect. 3.7.1 how our results might change if we include trapping.
68
3.3 Chemical network
Table 3.2 – Pre-exponential factors and binding energies for selected species in our network.
Species
H2 O
CO
N2
O2
ν (cm−2 s−1 )
1 × 1030
7 × 1026
8 × 1025
7 × 1026
Eb /k (K)
5773
855
800
912
Reference
Fraser et al. (2001)
Bisschop et al. (2006)
Bisschop et al. (2006)
Acharyya et al. (2007)
In addition to thermal desorption, our model includes desorption induced by UV photons. Laboratory experiments on the photodesorption of H2 O, CO and CO2 all produce
a yield of Y ≈ 10−3 molecules per grain per incident UV photon (Westley et al. 1995a,b,
Öberg et al. 2007, 2009b), while the yield for N2 is an order of magnitude lower (Öberg
et al. 2009c). Classical dynamics calculations predict a somewhat lower yield of 4 × 10−4
for H2 O (Andersson et al. 2006, Andersson & van Dishoeck 2008). The yields depend to
some extent on properties like the dust temperature and the ice thickness, but this has little
effect on chemical models (Öberg et al. 2009b). Hence, we take a constant yield of 10−3
for H2 O, CO and CO2 , and of 10−4 for N2 . For all other ice species in our network, whose
photodesorption yields have not yet been determined experimentally or theoretically, we
also take a yield of 10−3 . The photodesorption rate then becomes
0
Rphdes (X) = πa2gr ngr f (X)Y(X)FUV
e−τUV,eff ,
(3.6)
0
with f the same factor as for thermal desorption. The unattenuated UV flux (FUV
) and
the effective UV extinction (τUV,eff ) follow from Eq. (3.3). Photodesorption occurs even
in strongly shielded regions because of cosmic-ray–induced photons. We incorporate this
0
effect by setting a lower limit of 104 cm−2 s−1 to FUV
(Shen et al. 2004).
The chemical reactions in our model are not entirely limited to the gas phase. As
usual, the network includes the grain-surface formation of H2 (Black & van Dishoeck
1987). Inspired by Bergin et al. (2000) and Hollenbach et al. (2009), it also includes the
hydrogenation of C to CH4 , N to NH3 , O to H2 O, and S to H2 S. The hydrogenation is
done one H atom at a time and is always in competition with thermal and photon-induced
desorption. The formation of CH4 , NH3 , H2 O and H2 S does not have to start with the
respective atom freezing out. For instance, CH freezing out from the gas is also subject
to hydrogenation on the grain surface. The rate of each hydrogenation step is taken to be
the adsorption rate of H from the gas multiplied by the probability that the H atom finds
the atom or molecule X to hydrogenate:
s
8kT g
(3.7)
Rhydro (X) = πa2gr ngr n(H) f ′ (X)
πmp
with T g the gas temperature. The factor f ′ serves a similar purpose as the factor f in
Eqs. (3.4) and (3.6). Since the hydrogenation is assumed to be near-instantaneous as soon
as the H atom meets X before X desorbs, X is assumed to reside always near the top
layer of the ice. Hence, we are not interested in the abundance of solid X relative to the
69
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
total amount of ice (as in f ), but in its abundance relative to the other species that can be
hydrogenated:
ns (X)
f ′ (X) =
,
(3.8)
max(nhydro , Nb ngr )
with nhydro the sum of the solid abundances of the eleven species X: C, CH, CH2 , CH3 , N,
NH, NH2 , O, OH, S and SH. The main effect of this hydrogenation scheme is to build up
an ice mixture of simple saturated molecules during the pre-collapse phase, as is found
observationally (Tielens et al. 1991, Gibb et al. 2004, Tafalla et al. 2004, van Dishoeck
2004, Öberg et al. 2008).
Grain-surface hydrogenation is known to occur for more species than just the eleven
included here. For example, CO can be hydrogenated to form H2 CO and CH3 OH (Watanabe & Kouchi 2002, Fuchs et al. 2009). Grains also play an important role in the formation
of more complex species (Garrod & Herbst 2006, Garrod et al. 2008, Öberg et al. 2009a).
However, none of these reactions can be implemented as easily as the hydrogenation of
C, N, O and S. In addition, the main focus of this chapter is on simple molecules whose
abundances can be well explained with conventional gas-phase chemistry. Therefore, we
are safe in ignoring the more complex grain-surface reactions.
3.4 Results from the pre-collapse phase
This section, together with the next two, contains the results from the gas-phase chemistry
in our collapse model. First we briefly discuss what happens during the pre-collapse
phase. The chemistry during the collapse is analysed in detail for one particular parcel in
Sect. 3.5.1 and then generalised to others in Sect. 3.5.2. Finally, we compare the collapse
chemistry to a static disk model in Sect. 3.6. The results in this section are all consistent
with available observational constraints on pre-stellar cores (e.g., Bergin et al. 2000, di
Francesco et al. 2007).
During the 1.0 Myr pre-collapse phase, the initially atomic oxygen gradually freezes
out and is hydrogenated to H2 O ice. Meanwhile, H2 is ionised by cosmic rays and the
resulting H+2 reacts with H2 to give H+3 . This sets off the following chain of oxygen
chemistry:
O + H+3 → OH+ + H2 ,
(3.9)
OH+ + H2 → H2 O+ + H ,
(3.10)
H2 O + H2 → H3 O + H ,
(3.11)
+
+
H3 O+ + e− → OH + H2 /2H ,
(3.12)
OH + O → O2 + H .
(3.13)
The O2 thus produced freezes out for the most part. At the onset of collapse, the four
major oxygen reservoirs are H2 O ice (44%), CO ice (34%), O2 ice (16%) and NO ice
(3%).
70
3.4 Results from the pre-collapse phase
The oxygen chemistry is tied closely to the carbon chemistry through CO. It is initially
formed in the gas phase from CH2 , which in turn is formed from atomic C:
C + H2 → CH2 ,
(3.14)
CH2 + O → CO + 2H .
(3.15)
Another early pathway from C to CO is powered by H+3 and goes through an HCO+
intermediate:
C + H+3 → CH+ + H2 ,
CH+ + H2 → CH+2 + H ,
CH+2
+ H2 →
CH+3
+ H,
(3.16)
(3.17)
(3.18)
CH+3 + O → HCO+ + H2 ,
(3.19)
HCO + C → CO + CH .
(3.20)
+
+
The formation of CO through these two pathways accounts for most of the pre-collapse
processing of carbon: at t = 0, 82% of all carbon has been converted into CO, of which
97% has frozen out onto the grains. Most of the remaining carbon is present as CH4 ice
(14% of all C), formed from the rapid grain-surface hydrogenation of atomic C.
The initial nitrogen chemistry consists mostly of converting atomic N into NH3 , N2
and NO. The first of these is formed on the grains after freeze-out of N, in the same way
that H2 O and CH4 are formed from adsorbed O and C. We find two pathways leading to
N2 . The first one starts with the cosmic-ray dissociation of H2 :
H2 + ζ → H+ + H− ,
(3.21)
→ NH + e ,
(3.22)
NH + N → N2 + H .
(3.23)
N + H
−
−
The other pathway couples the nitrogen chemistry to the carbon chemistry. It starts with
Reactions (3.16)–(3.18) to form CH+3 , followed by
CH+3 + e− → CH + H2 /2H ,
(3.24)
CH + N → CN + H ,
(3.25)
CN + N → N2 + C .
(3.26)
The nitrogen chemistry is also tied to the oxygen chemistry, forming NO out of N and
OH:
N + OH → NO + H ,
(3.27)
with OH formed by Reaction (3.12). Nearly all of the N2 and NO formed during the precollapse phase freezes out. At t = 0, solid N2 , solid NH3 and solid NO account for 41, 32
and 22% of all nitrogen.
71
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
3.5 Results from the collapse phase
3.5.1 One single parcel
The collapse-phase chemistry is run for the standard set of model parameters from Chapter 2: M0 = 1.0 M⊙ , cs = 0.26 km s−1 and Ω0 = 10−14 s−1 . We first discuss the chemistry
in detail for one particular infalling parcel of material. It starts near the edge of the cloud
core, at 6710 AU from the center and 48.8◦ degrees from the z axis. Its trajectory terminates at t = tacc at R = 6.3 AU and z = 2.4 AU, about 0.2 AU below the surface of
the disk. The physical conditions encountered along the trajectory (χ, nH , T d and AV ) are
plotted in Fig. 3.2. This figure also shows the abundances of the main oxygen-, carbonand nitrogen-bearing species. The right four panels are regular plots as function of R: the
coordinate along the midplane. The infall velocity of the parcel increases as it gets closer
to the star, so the physical conditions and chemical abundances change ever more rapidly
at later times. Hence, the left panel of each row is plotted as a function of tacc − t: the
time before the end of the collapse phase. In each individual panel, the parcel essentially
moves from right to left.
A schematic overview of the parcel’s chemical evolution is presented in Fig. 3.3. It
shows the infall trajectory of the parcel and the abundances of several species at four
points along the trajectory. The physical conditions and the key reactions controlling those
abundances are also listed. Most abundance changes for individual species are related to
one specific chemical event, such as the evaporation of CO or the photodissociation of
H2 O. The remainder of this subsection discusses the abundance profiles from Fig. 3.2 and
explains them in the context of Fig. 3.3.
3.5.1.1 Oxygen chemistry
At the onset of collapse (t = 0), the main oxygen reservoir is solid H2 O at an abundance
of 8 × 10−5 relative to nH . The abundance remains constant until the parcel gets to point C
in Fig. 3.3, where the temperature is high enough for the H2 O ice to evaporate. The parcel
is now located close to the outflow wall, so the stellar UV field is only weakly attenuated
(AV = 0.7 mag). Hence, the evaporating H2 O is immediately photodissociated into H and
OH, which in turn is dissociated into O and a second H atom. At R = 17 AU (23 AU
inside of point C), the dust temperature is 150 K and all solid H2 O is gone. Moving in
further, the parcel enters the surface layers of the disk and is quickly shielded from the
stellar radiation (AV = 10–20 mag). The temperature decreases at the same time to 114
K, allowing some H2 O ice to reform. The final abundance at tacc (point D) is 4 × 10−8 .
The dissociative recombination of H3 O+ (formed by Reactions (3.9)–(3.11)) initially
maintains H2 O in the gas at an abundance of 7 × 10−8 . Following the sharp increase in
the overall gas density at t = 2.1 × 105 yr (Fig. 3.2), the freeze-out rate increases and
the gas-phase H2 O abundance goes down to 3 × 10−10 at point A in Fig. 3.3. Moving
on towards point B, the evaporation of O2 from the grains enables a new H2 O formation
route:
O2 + C+ → CO + O+ ,
72
(3.28)
3.5 Results from the collapse phase
O+ + H2 → OH+ + H ,
(3.29)
followed by Reactions (3.10) and (3.11) to give H3 O+ , which recombines with an electron
to give H2 O. The H2 O abundance thus increases to 3 × 10−9 at R = 300 AU. Farther in, at
point C, solid H2 O comes off the grains as described above. However, photodissociation
keeps the gas-phase abundance from growing higher than ∼10−7 . Once all H2 O ice is
gone at R = 17 AU, the gas-phase abundance can no longer be sustained at 10−7 and it
drops to 3 × 10−12 . Some H2 O is eventually reformed as the parcel gets into the disk and
is shielded from the stellar radiation, producing a final abundance of 2 × 10−11 relative to
nH .
Another main oxygen reservoir at t = 0 is solid O2 , with an abundance of 1 × 10−5 .
The corresponding gas-phase abundance is 4 × 10−7 . O2 gradually continues to freeze out
until it reaches a minimum gas-phase abundance of 3 × 10−10 just inside of point A. The
temperature at that time is 19 K, enough for O2 to slowly start evaporating thermally. The
gas-phase abundance is up by a factor of ten by the time the dust temperature reaches 23
K, about halfway between points A and B. The evaporation is 99% complete as the parcel
reaches R = 460 AU, about 140 AU inside of point B. The gas-phase abundance remains
stable at 1 × 10−5 for the next few hundred years. Then, as the parcel gets closer to the
outflow wall and into a region of lower extinction, the photodissociation of O2 sets in and
its abundance decreases to 2 × 10−8 at point C. The evaporation and photodissociation
of H2 O at that point enhances the abundances of OH and O, which react with each other
to replenish some O2 . As soon as all the H2 O ice is gone, this O2 production channel
quickly disappears and the O2 abundance drops to 1 × 10−11 . Finally, when the parcel
enters the top of the disk, O2 is no longer photodissociated and its abundance goes back
up to 4 × 10−8 at point D.
The abundance of gas-phase OH starts at 3 × 10−7 . Its main formation pathway is initially the dissociative recombination of H3 O+ (Reaction (3.12)), and its main destructors
are O, N and H+3 . The increase in total density at 2.1 × 105 yr speeds up the destruction
reactions, and the OH abundance drops to 3 × 10−10 at point B. The evaporation of solid
OH then briefly increases the gas-phase abundance to 1 × 10−8 . When all of the OH has
evaporated at R = 300 AU, the gas-phase abundance goes down again to 5 × 10−10 over
the next 150 AU. As the parcel continues towards and past point C, the OH abundance is
boosted to a maximum of 1 × 10−6 by the photodissociation of H2 O. The high abundance
lasts only briefly, however. As the last of the H2 O evaporates and gets photodissociated,
OH can no longer be formed as efficiently, and it is itself photodissociated. At the end of
the collapse, the OH abundance is ∼10−14 .
The fifth main oxygen-bearing species is atomic O itself. Its abundance is 7 × 10−7
at t = 0 and 1 × 10−4 at t = tacc , accounting for respectively 0.4 and 56% of the total
amount of free oxygen. Starting from t = 0, the O abundance remains constant during the
first 2.0 × 105 yr of the collapse phase. The increasing overall density then speeds up the
reactions with OH (forming O2 ) and H+3 (forming OH+ ), as well as the desorption onto
the grains, and the O abundance decreases to a minimum of 2 × 10−8 just before point A.
The abundance goes back up thanks to the evaporation of CO and, at point B, of O2 and
73
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
Figure 3.2 – Physical conditions (χ, nH , T d and AV ) and abundances of the main oxygen-, carbonand nitrogen-bearing species for the single parcel from Sect. 3.5.1, as function of time before the
end of the collapse (left) and as function of horizontal position (right). The grey bars correspond to
the points A, B and C from Fig. 3.3. Note that in each panel, the parcel moves from right to left.
74
3.5 Results from the collapse phase
Figure 3.3 – Overview of the chemistry along the infall trajectory of the single parcel from Sect.
3.5.1. The solid and dashed grey lines denote the surface of the disk and the outflow wall, both
at t = tacc = 2.52 × 105 yr. Physical conditions, abundances (black bars: gas; grey bars: ice) and
key reactions are indicated at four points (A, B, C and D) along the trajectory. The key processes
governing the overall chemistry at each point are listed in the bottom right. The type of each reaction
is indicated by colour, as listed in the top left.
NO, with O formed from the following reactions:
CO + He+ → C+ + O + He ,
(3.30)
NO + N → N2 + O ,
(3.31)
O2 + CN → OCN + O .
(3.32)
75
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
Heading on towards point C, the photodissociation of O2 , NO and H2 O further drives up
the amount of atomic O to the aforementioned final abundance of 1 × 10−4 .
3.5.1.2 Carbon chemistry
With solid and gas-phase abundances of 6 × 10−5 and 2 × 10−6 relative to nH , CO is the
main form of free carbon at the onset of collapse. CO is a very stable molecule and its
chemistry is straightforward. The freeze-out process started during the pre-collapse phase
continues up to t = 2.4 × 105 yr, a few thousand years prior to reaching point A in Fig.
3.3, where the dust temperature of 18 K results in CO evaporating again. As the parcel
continues its inward journey and is heated up further, all solid CO rapidly disappears and
the gas-phase abundance goes up to 6 × 10−5 at point B. During the remaining part of the
infall trajectory, the other main carbon-bearing species (e.g., C, CH, CH2 , C2 and HCO+ )
are all largely converted into CO. At the end of the collapse (point D), 99.8% of the
available carbon is locked up in CO. It also contains for 44% of the available oxygen.
The protonated form of CO, HCO+ , starts the collapse phase at an abundance of 6 ×
−10
10 relative to nH , or 3 × 10−4 relative to n(CO). It is in equilibrium with CO via the
two reactions
CO + H+3 → HCO+ + H2 ,
+
HCO + e
−
→ CO + H .
(3.33)
(3.34)
It is possible to derive a simple analytical estimate of the HCO+ -CO abundance ratio.
As shown by Lepp et al. (1987), the H+3 density does not depend strongly on the total
gas density. We find n(H+3 ) ≈ 1 × 10−4 cm−3 along the entire trajectory, except for the
part outside point A, where most CO is frozen out and therefore unable to destroy H+3 .
If the cosmic-ray ionisation rate is changed from our current value of 5 × 10−17 s−1 , the
H+3 density would change proportionally. The electron abundance is roughly constant at
3 × 10−8 relative to nH . Denoting the rate coefficients for Reactions (3.33) and (3.34) as
kf and kb , and assuming HCO+ and CO to be in mutual equilibrium, we get
kf n(CO)n(H+3 ) ≈ kb n(HCO+ )n(e−) .
(3.35)
Substituting n(H+3 ) = 1 × 10−4 cm−3 , n(e− ) = 3 × 10−8 nH , kf = 1.7 × 10−9 cm3 s−1 (Kim
et al. 1975) and kb = 2.4 × 10−7 (T g /300 K)−0.69 cm3 s−1 (Mitchell 1990), this rearranges
to
n
−1 T !0.69
n(HCO+ )
g
H
−4
.
(3.36)
≈ 5 × 10
4
−3
n(CO)
30 K
10 cm
The overall density increases by six orders of magnitude along the entire trajectory, while
the temperature changes only by one, so the HCO+ -CO abundance ratio should be roughly
inversely proportional to the density. Our full chemical simulation confirms this relationship to within an order of magnitude throughout the collapse. However, the ratio comes
out about a factor of ten larger than what is predicted by Eq. (3.36). The HCO+ abundance
reaches a final value at tacc of 8 × 10−13 relative to nH , or 1 × 10−8 relative to n(CO).
76
3.5 Results from the collapse phase
The second most abundant carbon-bearing ice at the onset of collapse is CH4 , at 1 ×
10−5 with respect to nH . The gas-phase abundance of CH4 begins at 4 × 10−9 , about a
factor of 2500 lower. At point A, the evaporation of CO provides the first increase in
x(CH4 ) through a chain of reactions starting with the formation of C+ from CO. The
successive hydrogenation of C+ produces CH+5 , which reacts with another CO molecule
to form CH4 :
CO + He+ → C+ + O + He ,
+
C + H2 →
CH+2
,
(3.38)
CH+2 + H2 → CH+3 + H ,
CH+3
CH+5
+ H2 →
CH+5
(3.37)
(3.39)
,
(3.40)
+
+ CO → CH4 + HCO .
(3.41)
The CH4 ice evaporates at point B, bringing the gas-phase abundance up to 1 × 10−5 .
So far, the abundances of CH4 and CO are well coupled. The link is broken when the
parcel reaches point C, where CH4 is photodissociated, but CO is not. This difference
arises from the fact that CO can only be dissociated by photons shortwards of 1076 Å,
while CH4 can be dissociated out to 1450 Å (Chapter 5). The 5300 K blackbody spectrum
emitted by the protostar at this time is not powerful enough at short wavelengths to cause
significant photodissociation of CO. CH4 , on the other hand, is quickly destroyed. Its
final abundance at point D is 6 × 10−10 .
Neutral and ionised carbon show the same trends in their abundance profiles, with the
former always more abundant by a few per cent to a few orders of magnitude. Both start
the collapse phase at ∼10−8 relative to nH . The increase in total density at 2.1 × 105 yr
speeds up the destruction reactions (mainly by OH and O2 for C and by OH and H2 for
C+ ), so the abundances go down to x(C) = 4 × 10−10 and x(C+ ) = 5 × 10−11 just outside
point A. This is where CO begins to evaporate, and as a result, the C and C+ abundances
increase again. As the parcel continues to fall in towards point B, the evaporation of O2
and NO and the increasing total density cause a second drop in C and C+ . Once again,
though, the drop is of a temporary nature. Moving on towards point C, the parcel gets
exposed to the stellar UV field. The photodissociation of CH4 leads – via intermediate
CH, CH2 or CH3 – to neutral C, part of which is ionised to also increase the C+ abundance.
Finally, at point D, the photoprocesses no longer play a role, so the C and C+ abundances
go back down. Their final values relative to nH are 7 × 10−12 and ∼10−14 .
3.5.1.3 Nitrogen chemistry
The most common nitrogen-bearing species at t = 0 is solid N2 , with an abundance of
5 × 10−6 . The corresponding gas-phase abundance is 4 × 10−8 . The evolution of N2
parallels that of CO, because they have similar binding energies and are both very stable
molecules (Bisschop et al. 2006). N2 continues to freeze out slowly until it gets near point
A in Fig. 3.3, where the grain temperature of ∼18 K causes all N2 ice to evaporate. The
77
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
gas-phase N2 remains intact along the rest of the infall trajectory and its final abundance
is 1 × 10−5 , accounting for 77% of all nitrogen.
The parallels between CO and N2 extend to their respective protonated forms, HCO+
and N2 H+ . An approximate equilibrium exists between N2 and N2 H+ through the reactions
N2 + H+3 → N2 H+ + H2 ,
N2 H+ + e − → N2 + H .
(3.42)
(3.43)
The recombination of N2 H+ also has a product channel of NH and N (Geppert et al. 2004),
but this still results in N2 being reformed through the additional reactions
NH + O → NO + H ,
(3.44)
NO + N → N2 + O ,
(3.45)
NO + NH → N2 + O + H .
(3.46)
Assuming that the recombination of N2 H+ eventually results in the formation of N2 most
of the time, we repeat the method outlined for the HCO+ -CO abundance ratio to derive
the expected relationship
n
−1 T !0.51
n(N2 H+ )
g
H
≈ 5 × 10−4
.
(3.47)
n(N2 )
30 K
104 cm−3
The results from the full chemical model show that within an order of magnitude, the
N2 H+ -N2 abundance ratio is indeed inversely proportional to the overall density and follows the prediction from Eq. (3.47). In the previous subsection, the ratio between HCO+
and CO was also found to be roughly inversely proportional to the density (Eq. (3.36)).
An important difference between N2 H+ and HCO+ arises from the reaction
N2 H+ + CO → N2 + HCO+ ,
(3.48)
which transforms some N2 H+ into HCO+ as soon as CO evaporates. This reaction is
responsible for the drop in N2 H+ right after point A.
The second largest nitrogen reservoir at the onset of collapse is NH3 , with solid and
gas-phase abundances of 8 × 10−6 and 2 × 10−8. The gas-phase abundance receives a short
boost at point A due to the evaporation of N2 , followed by
N2 + He+ → N+ + N + He ,
(3.49)
N + H2 → NH + H ,
(3.50)
+
+
+
78
NH + H2 →
NH+2
NH+3
NH+4
+ H,
(3.51)
NH+2 + H2 → NH+3 + H ,
(3.52)
+ H2 →
+ H,
(3.53)
NH+4 + e− → NH3 + H .
(3.54)
3.5 Results from the collapse phase
The binding energy of NH3 is intermediate to that of O2 and H2 O, so it evaporates between
points B and C. Like H2 O, NH3 is photodissociated upon evaporation. As the last of the
NH3 ice leaves the grains at R = 50 AU (10 AU outside of point C), the gas-phase reservoir
is no longer replenished and x(NH3 ) drops to ∼10−14 . Some NH3 is eventually reformed
as the parcel gets into the disk, and the final abundance at point D is 1 × 10−10 relative to
nH .
With an abundance of 6 × 10−6 , solid NO is the third major initial nitrogen reservoir.
Gaseous NO is a factor of twenty less abundant at t = 0: 3×10−7 . The NO gas is gradually
destroyed prior to reaching point A by continued freeze-out and reactions with H+ and
H+3 . It experiences a brief gain at point A from the evaporation of OH and its subsequent
reaction with N to give NO and H. As the parcel continues to point B, the solid NO begins
to evaporate and the gas-phase abundance rises to 6 × 10−6 . Photodissociation reactions
then set in around R = 100 AU and the NO abundance goes back down to 6 × 10−9 .
The evaporation and photodissociation of NH3 cause a brief spike in the NO abundance
through the reactions
NH3 + hν → NH2 + H ,
(3.55)
NH2 + O → HNO + H ,
(3.56)
HNO + hν → NO + H .
(3.57)
The evaporation of the last of the NH3 ice at R = 50 AU eliminates this channel and
the NO gas abundance decreases to 3 × 10−9 at point C. NO is now mainly sustained by
the reaction between OH and N. As described above, the OH abundance drops sharply
at R = 17 AU, and the NO abundance follows suit. The final abundance at point D is
∼10−14 .
The last nitrogen-bearing species from Fig. 3.2 is atomic N itself. It starts at an abundance of 1 × 10−7 and slowly freezes out to an abundance of 2 × 10−8 just before reaching
point A. At point A, N2 evaporates and is partially converted to N2 H+ by Reaction (3.42).
The dissociative recombination of N2 H+ mostly reforms N2 , but, as noted above, there is
also a product channel of NH and N. The N abundance jumps back to 1×10−7 and remains
nearly constant at that value until the parcel reaches point B, where NO evaporates and
reacts with N to produce N2 and O (Reaction (3.45)). This reduces x(N) to a minimum of
5 × 10−10 between points B and C. Moving in further, the parcel gets exposed to the stellar
UV field, and NO and NH3 are photodissociated to bring the N abundance to a final value
of 5 × 10−6 relative to nH . As such, it accounts for 22% of all nitrogen at the end of the
collapse.
3.5.2 Other parcels
At the end of the collapse (t = tacc ), the parcel from Sect. 3.5.1 (hereafter called our reference parcel) is located at R = 6.3 AU and z = 2.4 AU, about 0.2 AU below the surface
of the disk. As shown in Fig. 3.3, its trajectory passes close to the outflow wall, through
a region of low extinction. This results in the photodissociation or photoionisation of
79
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
Figure 3.4 – Schematic view of the history of H2 O gas and ice throughout the disk. The main
oxygen reservoir at tacc is indicated for each zone; the histories are described in the text. Note the
disproportionality of the R and z axes.
many species. At the same time, the parcel experiences dust temperatures of up to 150
K (Fig. 3.2), well above the evaporation temperature of H2 O and all other non-refractory
species in our network. Material that ends up in other parts of the disk encounters different physical conditions during the collapse and therefore undergoes a different chemical
evolution. This subsection shows how the absence or presence of some key chemical
processes, related to certain physical conditions, affects the chemical history of the entire
disk. Table 3.3 lists the abundances of selected species at four points in the disk at tacc .
Two-dimensional abundance profiles representing the entire disk’s chemical composition
are presented in Sect. 3.6.
3.5.2.1 Oxygen chemistry
The main oxygen reservoir at the onset of collapse is H2 O ice (Sect. 3.4). Its abundance
remains constant at 1 × 10−4 in our reference parcel until it gets to point C in Fig. 3.3,
where it evaporates from the dust and is immediately photodissociated. When the parcel
enters the disk, some H2 O is reformed to produce final gas-phase and solid abundances
of ∼10−8 relative to nH (Sect. 3.5.1.1).
Figure 3.4 shows the disk at tacc , divided into seven zones according to different chemical evolutionary schemes for H2 O. The material in zone 1 is the only material in the disk
in which H2 O never evaporates during the collapse, because the temperature never gets
high enough. The abundance is constant throughout zone 1 at tacc at ∼1 × 10−4 (see also
80
3.5 Results from the collapse phase
Table 3.3 – Abundances of selected species (relative to nH ) at t = tacc at four positions in the disk
(two on the midplane, two at the surface).
Species
Zonesa
H2 O
H2 O ice
O2
O2 ice
O
OH
CO
CO ice
CH4
CH4 ice
HCO+
C
C+
N2
N2 ice
NH3
NH3 ice
NO
NO ice
N2 H+
N
a
R = 6 AU,
z = 0.0 AU
7, 5, 8
5(-7)
1(-4)
1(-6)
<1(-12)
1(-6)
2(-11)
7(-5)
<1(-12)
7(-8)
<1(-12)
1(-12)
<1(-12)
<1(-12)
1(-5)
<1(-12)
5(-10)
<1(-12)
1(-8)
<1(-12)
<1(-12)
5(-11)
R = 24 AU,
z = 0.0 AU
7, 5, 8
1(-12)
1(-4)
6(-8)
<1(-12)
1(-6)
<1(-12)
7(-5)
<1(-12)
1(-7)
<1(-12)
5(-11)
<1(-12)
<1(-12)
1(-5)
<1(-12)
2(-10)
5(-9)
5(-10)
<1(-12)
<1(-12)
1(-9)
R = 6 AU,
z = 2.0 AU
2, 2, 5
1(-8)
7(-8)
1(-7)
<1(-12)
1(-4)
<1(-12)
8(-5)
<1(-12)
2(-9)
<1(-12)
3(-12)
9(-12)
<1(-12)
1(-5)
<1(-12)
4(-10)
<1(-12)
<1(-12)
<1(-12)
<1(-12)
5(-6)
R = 24 AU,
z = 6.0 AU
1, 1, 2
4(-9)
8(-5)
1(-5)
<1(-12)
8(-7)
9(-10)
6(-5)
<1(-12)
1(-5)
<1(-12)
4(-12)
<1(-12)
<1(-12)
5(-6)
<1(-12)
8(-6)
5(-8)
6(-6)
<1(-12)
<1(-12)
4(-11)
The H2 O, CH4 and NH3 zones from Figs. 3.4, 3.7 and 3.9 in which each position is located.
Table 3.3). For the material ending up in the other six zones, H2 O evaporates at some
point during the collapse phase. Zone 2 contains our reference parcel, so its H2 O history
has already been described. The total H2 O abundance (gas and ice combined) at tacc is
∼10−8 at the top of zone 2 and ∼10−6 at the bottom. The gas-ice ratio goes from ∼1 at the
top to ∼10−6 at the bottom.
The H2 O history of zone 3 is the same as that of zone 2, except that it finishes with a
gas-ice ratio larger than unity. In both cases H2 O evaporates and is photodissociated prior
to entering the disk (point C in Fig. 3.3), and it partially reforms inside the disk (point D).
Parcels ending up in zone 4 also undergo the evaporation and photodissociation of H2 O.
However, the low extinction in zone 4 against the stellar UV radiation prevents H2 O from
reforming like it does in zones 2 and 3.
The material in zone 5 has a rather different history from that in zones 2–4 because it
enters the disk at an earlier time: between t = 1.3 × 105 and 2.3 × 105 yr. The material in
81
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
Figure 3.5 – Infall trajectory for a parcel ending up in zone 6 from Fig. 3.4, showing where H2 O is
predominantly present as ice (black) or gas (grey). The two diamonds mark the time in years after
the onset of collapse.
zones 2–4 all accretes after t = 2.4 × 105 yr. The infall trajectories terminating in zone 5
do not pass close enough to the outflow wall or the inner disk surface for photoprocesses
to play a role. All H2 O in zone 5 is in the gas phase at tacc (abundance: 1 × 10−4 ) because
it lies inside the disk’s snow line. The evaporation of H2 O ice does not occur until the
material actually crossed the snow line. Prior to that point, the temperature never gets
high enough for H2 O to leave the grains.
Moving to zones 6 and 7, we find material that accretes even earlier: at t = 4 × 104
yr. The disk at that time is only 2 AU large and several 100 K hot, so the ice mantles
are completely removed. Instead of carrying on towards the star, this material remains
part of the disk and is transported outwards to conserve angular momentum. Figure 3.5
shows the path followed by one particular parcel terminating in zone 6. As indicated by
the initial grey portion of the line, H2 O has already evaporated (at R = 25 AU, not shown)
before entering the young disk. The parcel cools down during the outward part of the
trajectory and H2 O returns to the solid phase. At t = 1.7 × 105 yr, the parcel starts moving
inwards again and comes close enough to the protostar for H2 O to evaporate a second
time. Other parcels ending up in zone 6 have similar trajectories and the same qualitative
H2 O history. The parcels ending up in zone 7 also have H2 O evaporating during the initial
infall and freezing out again during the outward part of the trajectory. However, they do
not terminate close enough to the protostar for H2 O to desorb a second time. Therefore,
most H2 O in zone 7 at tacc is on the grains.
Our model is not the first one in which part of the disk contains H2 O that evaporated
and readsorbed. Lunine et al. (1991), Owen et al. (1992) and Owen & Bar-Nun (1993)
argued that the accretion shock at the surface of the disk is strong enough for H2 O to
evaporate. However, based on the model of Neufeld & Hollenbach (1994), we showed in
82
3.5 Results from the collapse phase
Figure 3.6 – Qualitative evolution of some abundances towards the seven zones with different H2 O
histories from Fig. 3.4. The horizontal axes show the time (increasing from left to right) and are
non-linear. The position of points A, B, C and D from Fig. 3.3 is indicated for zone 2, which
contains our reference parcel.
Chapter 2 that most of the disk material does not pass through a shock that heats the dust
to 100 K or more. Moreover, the material that does get shock-heated to that temperature
accretes close enough to the star that the stellar radiation already heats it to more than 100
K. Hence, including the accretion shock explicitly in our model would at most result in
minor changes to the chemistry.
As discussed in Sect. 3.5.1.1, H2 O controls part of the oxygen chemistry along the
infall trajectory of our reference parcel, and it does the same thing for other parcels.
Figure 3.6 presents a schematic view of the chemical evolution of six oxygen-bearing
species towards each of the seven zones from Fig. 3.4. The abundances are indicated
qualitatively as high, intermediate or low. The horizontal axes (time, increasing from left
to right) are non-linear and only indicate the order in which various events take place.
For material ending up in zone 1, H2 O never evaporates, but O2 does. OH is inititally
relatively abundant but most of it disappears when the overall density increases and the
reactions with O, N and H+3 become faster. The abundance of atomic O experiences a
drop at the same time, but it goes back up shortly after due to the evaporation of CO,
O2 and NO followed by Reactions (3.30)–(3.32). O2 also evaporates on its way to zones
2, 3 and 4 and because it passes through an area of low extinction, it is subsequently
photodissociated. This, together with the photodissociation of OH and H2 O, causes an
increase in the abundance of atomic O. Zones 2 and 3 are sufficiently shielded against the
stellar UV field, so O2 is reformed at the end of the trajectory. This does not happen in
the less extincted zone 4.
Material ending up in zone 5 has the same qualitative history for gas-phase and solid
O2 as has material ending up in zone 1. The evolution of atomic O initially also shows
the same pattern, but it experiences a second drop as the total density becomes even
higher than it does for zone 1. The higher density also causes an additional drop in the
83
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
Figure 3.7 – As Fig. 3.4, but for CH4 gas and ice. The main carbon reservoir at tacc is CO gas
throughout the disk.
OH abundance. En route towards zones 6 and 7, O2 evaporates and is photodissociated
during the early accretion onto the small disk. It is reformed during the outward part of
the trajectory and survives until tacc . Atomic O is also relatively abundant along most of
the trajectory, although always one or two orders of magnitude below O2 . In zone 6, the
O abundance decreases at the end due to the reaction with evaporating CH3 , producing
H2 CO and H. Zone 7 does not get warm enough for CH3 to evaporate, so O remains
intact.
3.5.2.2 Carbon chemistry
The two main carbon reservoirs at the onset of collapse are CO ice and CH4 ice (Sect. 3.4).
Their binding energies are relatively low (855 and 1080 K), so they evaporate throughout
the core soon after the collapse begins (Sect. 3.5.1.2). For our reference parcel, the main
difference between the evolution of CO and CH4 is the photodissociation of the latter (near
point C in Fig. 3.3) while the former remains intact. Following our approach for H2 O, we
can divide the disk at tacc into several zones according to different chemical evolution
scenarios for CO. However, the entire disk has the same qualitative CO history: apart
from evaporating early on in the collapse phase, CO does not undergo any processing (cf.
Chapter 2). Hence, we divide the disk according to the evolution of CH4 instead (Fig.
3.7).
For material that ends up in zone 1, the only chemical process for CH4 is the evaporation during the initial warm-up of the core. It is not photodissociated at any point
84
3.5 Results from the collapse phase
Figure 3.8 – As Fig. 3.6, but for the five zones with different CH4 histories from Fig. 3.7.
during the collapse, nor does it freeze out again or react significantly with other species.
Material ending up closer to the star, in zones 2 and 3, is sufficiently irradiated by the
stellar UV/visible field for CH4 to be photodissociated. Zone 3, which contains our reference parcel, is hardly shielded from the stellar flux and the final CH4 abundance is only
a few 10−10 . The stronger extinction towards zone 2 allows CH4 to be reformed at a final
abundance of 10−9 –10−7 relative to nH (Table 3.3).
Zone 4 contains material that accretes onto the disk several 104 yr earlier than does
the material in zones 2 and 3. It always remains well shielded from the stellar radiation, so
the only processing of CH4 is the evaporation during the early parts of the collapse. The
CH4 history of zones 1 and 4 is thus qualitatively the same. The last zone, zone 5, consists
of material that accretes around t = 4 × 104 yr and is subsequently transported outwards to
conserve angular momentum (Fig. 3.5). CH4 in this material evaporates before entering
the young disk and is photodissociated as it gets within a few AU of the protostar. The
resulting atomic C is mostly converted into CO and remains in that form for the rest of
the trajectory. Hence, even though the extinction decreases again when the parcel moves
outwards, no CH4 is reformed.
The evolution of the abundances of CH4 gas and ice, CO gas and ice, HCO+ , C and
+
C towards each of the five zones is plotted schematically in Fig. 3.8. As noted in Sect.
3.5.1.2, the HCO+ abundance follows the CO abundance at a ratio that is roughly inversely
proportional to the overall density. Hence, the HCO+ evolution is qualitatively the same
towards each zone: it reaches a maximum abundance of a few 10−10 when CO evaporates
and gradually disappears as the density increases along the rest of the infall trajectories.
The most complex history amongst these seven carbon-bearing species is found in C and
C+ . Towards all five zones, they are initially destroyed by reactions with H2 , O2 and
OH. Some C and C+ is reformed when CO evaporates (at point A in Fig. 3.3), but the
subsequent evaporation of O2 and OH causes the abundances to decrease again. En route
85
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
Figure 3.9 – As Fig. 3.4, but for NH3 gas and ice. The main nitrogen reservoir at tacc is N2 gas
throughout the disk.
to zones 2, 3 and 5, the photodissociation of CH4 leads to a second increase in C and C+ ,
followed by a third and final decrease when the parcel moves into a more shielded area.
3.5.2.3 Nitrogen chemistry
Most nitrogen at the onset of collapse is in the form of solid N2 (41%), solid NH3 (32%)
and solid NO (22%). The evolution of N2 during the collapse is the same as that of CO,
except for a minor difference in the binding energy. Both species evaporate shortly after
the collapse begins and remains in the gas phase throughout the rest of the simulation.
Neither one is photodissociated because they need UV photons shortwards of 1100 Å,
and the protostar is not hot enough to provide those.
The evolution of NH3 shows a lot more variation, as illustrated in Fig. 3.9. The disk
at tacc is divided into eight zones with different NH3 histories. No processing occurs
towards zone 1: the temperature never exceeds the 73 K required for NH3 to evaporate,
so it simply remains on the grains the whole time. Material ending up in zone 2 does get
heated above 73 K, so NH3 evaporates. However, it freezes out again at the end of the
trajectory because zone 2 itself is not warm enough to sustain gaseous NH3 . The final
solid NH3 abundance in zones 1 and 2 is about 8 × 10−6 relative to nH (Table 3.3).
NH3 ending up in zone 3 also evaporates from the grains just before entering the disk.
It is then destroyed by UV photons (only for material that ends up inside of 15 AU in zone
3) and by HCO+ (for all material in zone 3). Towards the end of the trajectory, some NH3
is reformed from the dissociative recombination of NH+4 and this immediately freezes out
86
3.5 Results from the collapse phase
Figure 3.10 – As Fig. 3.6, but for the eight zones with different NH3 histories from Fig. 3.9.
to produce a final solid NH3 abundance of 10−10 –10−8. Zone 4 has the same history,
except that there is an additional adsorption-desorption cycle before the destruction by
photons and HCO+ .
Our standard parcel ends up in zone 5; its NH3 evolution is mainly characterised by
photodissociation above the disk and reformation inside it (Sect. 3.5.1.3). As is the case
for zones 2–4, NH3 evaporates when it gets to within about 200 AU of the star, halfway
between points B and C in Fig. 3.3. It is immediately photodissociated, but some NH3
is reformed once the material is shielded from the stellar radiation. The reformed NH3
remains in the gas phase. No reformation takes place in the less extincted zone 6, which
otherwise has the same NH3 history as does zone 5.
Material ending up in zone 7 does not pass close enough to the outflow wall for NH3
to be photodissociated upon evaporation. Instead, NH3 is mainly destroyed by HCO+ and
attains a final abundance of ∼10−9 . Lastly, zone 8 contains again the material that accretes
onto the disk at an early time and then moves outwards to conserve angular momentum.
Its NH3 evaporates already before reaching the disk and is subequently dissociated by
the stellar UV field. As the material moves away from the star and is shielded from its
radiation, some NH3 is reformed out of NH+4 to a final abundance of ∼10−10 .
The abundances of N2 H+ and atomic N are largely controlled by the evolution of
N2 and NH3 , as shown schematically in Fig. 3.10. In all parcels, regardless of where
they end up, N2 H+ is mainly formed out of N2 and H+3 , so its abundance goes up when N2
evaporates shortly after the onset of collapse. It gradually disappears again as the collapse
proceeds due to the inverse relationship between the overall density and the N2 H+ -N2 ratio
(Eq. (3.47)). The atomic N abundance at the onset of collapse is 1 × 10−7 . It increases
when N2 evaporates and decreases again a short while later when NO evaporates (Sect.
3.5.1.3). For material that ends up in zones 1 and 2, the N abundance is mostly constant
87
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
for the rest of the collapse at a value of 10−13 (inner part of zone 2) to 10−10 (outer part of
zone 1). Material that ends up in the other six zones is exposed to enough UV radiation
for NO and NH3 to be photodissociated, so there is a second increase in atomic N. Zones
3, 4, 7 and 8 are sufficiently shielded at tacc to reform some or all NO and NH3 , and the
N abundance finishes low. Less reformation is possible in zones 5 and 6, so they have a
relatively large amount of atomic N at the end of the collapse phase.
3.5.2.4 Mixing
Given the dynamical nature of circumstellar disks, the zonal distribution presented in the
preceding subsections may offer too simple a picture of the chemical composition. For
one thing, there are as yet no first principles calculations of the processes responsible for
the viscous transport in disks. The radial velocity equation used in our model (see Chapter
4) is suitable as a zeroth-order description, but cannot explain important observational
features like episodic accretion (Kenyon & Hartmann 1995, Evans et al. 2009). The radial
velocity profile in real disks is probably much more chaotic, so there would be more
mixing between adjacent zones. Hence, both the shapes and the locations of the zones
are likely to be different from what is shown in Figs. 3.4, 3.7 and 3.9. A larger degree
of mixing would also make the borders between the zones more diffuse than they are in
our simple schematic representation. Nevertheless, the general picture from this section
offers a plausible description of the chemical history towards different parts of the disk.
Spectroscopic observations at AU resolution, for example with the upcoming Atacama
Large Millimeter/submillimeter Array (ALMA), are required to determine to what extent
this picture holds in reality.
3.6 Chemical history versus local chemistry
Section 3.5 contains many examples of abundances increasing or decreasing on short
timescales of less than a hundred years (see, e.g., Fig. 3.2). It appears that the abundances
respond rapidly to the changing physical conditions as material falls in supersonically
through the inner envelope and accretes onto the disk. However, this does not necessarily
mean that the abundances are always in equilibrium. In this section we explore the question as to whether the disk is in chemical equilibrium at the end of the collapse, or if its
chemical composition is a non-equilibrium solution to the conditions encountered during
the collapse phase. To that end, we evolve the chemistry for an additional 1 Myr beyond
tacc . We keep the density, temperature, UV flux and extinction constant at the values they
have at tacc , and we also keep all parcels of material at the same position. Clearly, this
is a purely hypothetical scenario. In reality, the disk would change in many ways after
tacc : it spreads in size (see Chapter 4), the surface layers become more strongly irradiated,
the temperature changes, the dust coagulates into planetesimals, gas is photoevaporated
from the surface layers, and so on. All of these processes have the potential to affect the
chemical composition. However, they would also interfere with our attempt to determine
whether the disk is in chemical equilibrium at tacc . This question is most easily addressed
88
3.6 Chemical history versus local chemistry
Figure 3.11 – Abundances of H2 O ice, O2 ice and gaseous atomic O throughout the disk at the end
of the collapse phase (t = tacc ) and after an additional static post-collapse phase of 1 Myr.
by evolving the chemistry for an additional period of time at constant conditions, hence
we ignore all of the physical changes that would normally occur during the post-collapse
phase.
The abundance profiles for the oxygen-, carbon- and nitrogen-bearing species from the
previous sections are plotted in Figs. 3.11–3.17. In each case, the left panel corresponds
to the end of the collapse phase (t = tacc ) and the right panel to the end of the static 1 Myr
post-collapse phase (t = tacc + 1 Myr). Abundances of less than 10−12 are unreliable in
our chemical code, so lower values are not plotted. Hence, the purple areas in the figures
should be interpreted as upper limits.
The 21 species in Figs. 3.11–3.17 can be divided into two categories: those whose
89
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
Figure 3.12 – As Fig. 3.11, for gaseous H2 O, O2 and OH.
abundance profile changes during the post-collapse phase, and those whose abundance
profile remains practically the same. Members of the former category are O2 , OH, CH4 ,
NH3 and NO (all gaseous), NH3 ice, and atomic O and N. The thirteen species in the
“unchanged” category are H2 O gas and ice, O2 ice, CO gas and ice, CH4 ice, C, C+ ,
N2 gas and ice, NO ice, HCO+ and N2 H+ . The individual gas and ice abundances are
summed in Fig. 3.18. The total H2 O, CO and N2 abundances do not change significantly
during the post-collapse phase, while the total O2 , CH4 , NH3 and NO abundances change
by more than two orders of magnitude in a fairly large part of the disk.
There are two areas in the disk where the abundances generally change the least: near
the surface out to R ≈ 10 AU, and at the midplane between ∼5 and ∼25 AU. In the first
area, the chemistry is dominated by fast photoprocesses, allowing the abundances to reach
90
3.6 Chemical history versus local chemistry
Figure 3.13 – As Fig. 3.11, for CO ice, CH4 ice and gaseous atomic C.
equilibrium on short timescales. The second area, near the midplane, is the densest part
of the disk and therefore has high collision frequencies and short chemical timescales. In
addition, this part of the disk is populated by material that accreted at an early time (Fig.
3.5). The physical conditions it encountered during the rest of the collapse phase were
relatively constant, aiding in establishing chemical equilibrium.
Outside these two “equilibrium areas”, one of the key processes during the postcollapse phase is the conversion of gas-phase oxygen-bearing species into H2 O, which
subsequently freezes out. Because H2 O ice already is one the main oxygen reservoirs at
tacc throughout most of the disk, its abundance only increases by about 20% during the rest
of the simulation (Fig. 3.11). These 20% represent 10% of the total oxygen budget and
correspond to 2 × 10−5 oxygen nuclei per hydrogen nucleus. One of the species converted
91
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
Figure 3.14 – As Fig. 3.11, for gaseous CO, CH4 and C+ .
into H2 O ice is NO, which reacts with NH2 through
NH2 + NO → N2 + H2 O ,
(3.58)
followed by adsorption of H2 O onto the dust. This reaction is responsible for the postcollapse destruction of gas-phase NO outside the two “equilibrium areas”, as well as for
the small increase (∼20%) in gas-phase N2 (Fig. 3.16). NO is the main destructor of
atomic N via
NO + N → N2 + O ,
(3.59)
so the conversion of NO into H2 O also explains the increased N abundance from Fig.
92
3.6 Chemical history versus local chemistry
Figure 3.15 – As Fig. 3.11, for N2 ice, NH3 ice and NO ice.
3.17. The NH2 required for Reaction (3.58) is formed from NH3 :
NH3 + HCO+ → NH+4 + CO ,
(3.60)
→ NH2 + H2 /2H .
(3.61)
NH+4
+ e
−
The gas-phase reservoir of NH3 is continously fed by evaporation of NH3 ice, so Reactions 3.60, 3.61 and 3.58 effectively transform all NH3 into N2 (Figs. 3.15 and 3.16).
Two other species that are destroyed in the post-collapse phase are gaseous O2 and
CH4 (Figs. 3.12 and 3.14). O2 is ionised by cosmic rays to produce O+2 , which then reacts
with CH4 :
CH4 + O+2 → HCOOH+2 + H ,
(3.62)
93
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
Figure 3.16 – As Fig. 3.11, for gaseous N2 , NH3 and NO.
HCOOH+2 + e− → HCOOH + H .
(3.63)
The formic acid (HCOOH) freezes out, thus acting as a sink for both carbon and oxygen.
At t = tacc + 1 Myr, the solid HCOOH abundance is 3 × 10−6 , accounting for 3% of
all oxygen and 4% of all carbon. The presence of such a sink is a common feature of
disk chemistry models. Gas-phase species are processed by He+ and H+3 until they form
a species whose evaporation temperature is higher than the dust temperature. This is
HCOOH in our case, but it could also be carbon-chain molecules like C2 H2 or C3 H4
(Aikawa & Herbst 1999). Another important post-collapse sink reaction in our model is
the freeze-out of HNO. At tacc , OH is primarily formed by
HNO + O → NO + OH
94
(3.64)
3.6 Chemical history versus local chemistry
Figure 3.17 – As Fig. 3.11, for HCO+ , N2 H+ and atomic N.
throughout most of the disk. This reaction is also one of the main destruction channels
for atomic O. Hence, the gradual freeze-out of HNO after tacc leads to a decrease in OH
and an increase in O (Figs. 3.11 and 3.12). Also contributing to the higher O abundance
is the fact that OH itself is an important destructor of O through Reaction (3.13).
The post-collapse chemistry described here is merely meant as an illustration of what
might take place in a real circumstellar disk after the main accretion phase comes to an
end. Depending on how the disk continues to evolve physically, other reactions may
become more important than the ones listed above. However, it is clear that, in general,
the disk is not in chemical equilibrium at tacc . Figures 3.11–3.18 show several examples
of abundances changing by a few orders of magnitude. Even the two “equilibrium areas”
(near the surface out to 10 AU and from 5 to 25 AU at the midplane) are not truly in
95
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
Figure 3.18 – Abundances of total H2 O, O2 , CO, CH4 , N2 , NH3 and NO (gas and ice combined)
throughout the disk at the end of the collapse phase (t = tacc ) and after an additional 1 Myr postcollapse phase (t = tacc + 1 Myr).
chemical equilibrium. For example, at 20 AU on the midplane, the abundances of solid
HCOOH, C2 O, C3 O and CH3 OH increase by factors of 10 to 25 during the post-collapse
processing.
As the disk evolves from the end of the collapse phase to a mature T Tauri disk, it
spreads to larger radii. At the same time, material continues to accrete onto the protostar
and gas is evaporated from the surface layers by the stellar radiation field. Altogether, this
results in lower gas densities and longer chemical timescales. It is therefore unlikely that
the gas will ever reach chemical equilibrium, as is indeed confirmed by chemical models
96
3.7 Discussion
of T Tauri disks (Fegley 2000). Hence, some signature of the collapse-phase chemistry
may survive into later stages. This will be the topic of a future publication.
3.7 Discussion
3.7.1 Caveats
Several caveats were mentioned in the preceding sections, which we briefly discuss here.
First of all, our model does not calculate the gas temperature explicitly, but simply sets it
equal to the dust temperature. This approach is valid for the interior of the disk (optically
thick to UV radiation), but it breaks down in the surface layers and in the inner parts of
the envelope (Kamp & Dullemond 2004, Jonkheid et al. 2004, Woitke et al. 2009). The
chemistry in these optically thin regions is mostly controlled by photoprocesses and ionmolecule reactions, neither of which depend strongly on temperature. One aspect that
would be affected is the gas-phase production of H2 O through the reactions
O + H2 → OH + H ,
(3.65)
OH + H2 → H2 O + H ,
(3.66)
which have activation barriers of 3100 and 1700 K, respectively (Natarajan & Roth 1987,
Oldenborg et al. 1992). Hence, our model probably underestimates the amount of H2 O in
the surface of the disk. Other than that, increasing the gas temperature to a more realistic
value is unlikely to change the chemical results. Taking a higher gas temperature might
change some of the physical structure. It would increase the pressure of the disk, thus
pushing up the disk-envelope boundary and possibly changing the spatial distribution of
where material accretes onto the disk. However, the bulk of the accretion currently takes
place in optically thick regions, so only a small fraction of the infalling material would be
affected.
The shape of the stellar spectrum may have larger chemical consequences. Right now,
we simply take a blackbody spectrum at the effective stellar temperature. Our star never
gets hotter than 5800 K, which is not enough to produce UV photons of sufficient energy
to dissociate CO and H2 . Many T Tauri stars are known to have a UV excess, which would
allow CO and H2 to be photodissociated. The stronger UV flux would also enhance the
photoionisation of atomic C, probably resulting in C+ being the dominant form of carbon
between points C and D in Fig. 3.3. In turn, this would boost the production of carbonchain species like C2 S and HC3 N. As soon as the material enters the disk and is shielded
again from the stellar radiation, C+ is converted back into C and CO. However, some
signature of the temporary high C+ abundance may survive.
The third caveat has to do with the trapping of volatile species like CO, CH4 and N2
in the H2 O ice matrix. We argued in Chapter 2 that trapping of CO is required to explain
the observed abundances of CO in comets. In the full chemical network from the current
chapter, trapping of CO and other volatiles would reduce their gas-phase abundances,
which in turn may reduce their controlling role in the gas-phase chemistry. However, the
97
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
trapped fraction is never more than 30% (Viti et al. 2004, Fayolle et al. in prep.; see also
Gibb et al. 2004), so their gas-phase abundances remain high compared to other species.
Hence, CO, CH4 , N2 , O2 and NO are still likely to control the gas-phase chemistry even
if they are partially trapped on the dust grains.
3.7.2 Comets
The chemical composition of cometary ices shows many similarities to that of sources in
the ISM, but the abundance of a given species may vary by more than an order of magnitude from one comet to the next (A’Hearn et al. 1995, Bockelée-Morvan et al. 2000, 2004,
Schöier et al. 2002, Ehrenfreund et al. 2004, Disanti & Mumma 2008). Both points are
illustrated in Fig. 3.19, where the abundances of several species in the comets 1P/Halley,
C/1995 O1 (Hale-Bopp), C/1996 B2 (Hyakutake), C/1999 H1 (Lee), C/1999 S4 (LINEAR) and 153P/Ikeya-Zhang are plotted against the abundances in the embedded protostar IRAS 16293–2422 (warm inner envelope), the bipolar outflow L1157, and the four
hot cores W3(H2 O), G34.3+0.15, Orion HC and Orion CR. Each point is characterised
by the mean value from the available sources (the diamonds) and the total spread in measurements (the error bars). Uncertainties from individual measurements are not shown.
The dotted line marks the theoretical relationship where the ISM abundances equal the
cometary abundances. The data generally follow this line, suggesting that the material
ending up in comets underwent little chemical processing from the ISM. However, the
plot also shows variations of at least an order of magnitude in the cometary abundances for
CO, H2 CO, CH3 OH, HNC, H2 S and S2 , as well as smaller variations for other species.
These different chemical compositions may be explained by assuming that the comets
were formed in different parts of the solar nebula. If that is indeed the case, there must
have been some degree of chemical processing between the ISM and the cometary nuclei.
Another point to note is that the elemental nitrogen abundance in comets is at least a factor of three lower than that in the ISM (Wyckoff et al. 1991, Jessberger & Kissel 1991,
Bockelée-Morvan et al. 2000). Although the reason for this deficiency remains unclear, it
does also point at a certain degree of chemical processing.
Comets are thought to have formed at the gravitational midplane of the circumsolar
disk, between 5 and 30 AU from the young Sun (Bockelée-Morvan et al. 2004, Kobayashi
et al. 2007). In our model disk, the 5–30 AU range happens to be almost exactly the area
containing material that accreted at an early time (4 × 104 yr after the onset of collapse)
and stayed in the disk for the remainder of the collapse phase (Fig. 3.5). As discussed
in Sect. 3.5.2, this material undergoes a larger degree of chemical processing than any
other material in the disk, due to its accreting close to the star. If comets are entirely
formed out of such material, it is hard to reconcile the large degree of processing with
the observed similarities between cometary and ISM abundances. It requires that either
the abundances of observed cometary species actually remain constant throughout the
collapse, or that they return to ISM values after the material reaches its final position.
In this section we argue that neither scenario is likely to be true, and that the similarity
between cometary and ISM abundances in fact results from mixing unprocessed material
from other parts of the disk into the comet-forming zone.
98
3.7 Discussion
Figure 3.19 – Molecular abundances in comets (Halley, Hale-Bopp, Hyakutake, Lee, C/1999
S4 and Ikeya-Zhang) compared to those in ISM sources (IRAS 16293–2422 (warm inner envelope), L1157, W3(H2 O), G34.3+0.15, Orion Hot Core and Orion Compact Ridge) as provided by
Bockelée-Morvan et al. (2000, 2004) and Schöier et al. (2002). The error bars indicate the spread
between sources; errors from individual measurements are not included. The dashed line represents
a one-to-one correspondence between cometary and ISM abundances and is not a fit to the data.
Figures 3.20 and 3.21 compare the cometary abundances from Bockelée-Morvan et al.
(2000, 2004) to the abundances from our model at the end of the collapse phase. The
horizontal error bars show again the spread in abundances between individual comets. In
Fig. 3.20, the vertical error bars show the spread across our entire disk; in Fig. 3.21, they
show the spread in the 5–30 AU region at the midplane. For all species, the gas and ice
abundances from the model are summed and displayed as one.
The disk-wide abundances tend to cluster around the theoretical one-to-one relationship indicated by the dotted line. Two notable exceptions are CH3 OH and HCOOCH3 .
Both of them are known to require grain-surface chemistry to get the correct abundance,
and this is not included in our model. The spread in model abundances is at least six
orders of magnitude, but given the wide range of physical conditions that are sampled,
such a large spread is to be expected. The 5–30 AU abundances from Fig. 3.21 show less
variation – between one and four orders of magnitude – but still take on a wider range of
values than do the cometary abundances. More importantly, the 5–30 AU data show no
99
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
Figure 3.20 – As Fig. 3.19, but comparing the cometary abundances to those in our model at t = tacc .
The vertical black lines indicate the spread across the entire disk. For all species, the gas and ice
abundances from the model are summed and displayed as one.
correlation with the comet data. There is no indication from Sect. 3.6 that post-collapse
processing brings the abundances back to near-ISM values. Hence, the midplane material
between 5 and 30 AU in our model does not appear to be analogous to the material from
which the solar-system comets originated.
How plausible is the large degree of chemical processing for material that ends up
in the comet-forming zone? The amount of processing is a direct result of the range of
physical conditions encountered along a given infall trajectory, which in turn depends
mostly on how close to the protostar the trajectory gets. We intentionally keep the physics
in our model as simple as possible, so a trajectory like the one drawn in Fig. 3.5 may
not be fully realistic. On the other hand, the back-and-forth motion results from the
well-known concept of conservation of angular momentum. As the inner parts of the
disk accrete onto the star, the outer parts must move out to maintain the overall angular
momentum. They may be pushed inwards again at a later time if a sufficient amount of
mass is accreted from the envelope at larger radii. This happens in our simple model, but
also in the hydrodynamical simulations of Brinch et al. (2008a,b).
If we accept the possibility of back-and-forth motion, the question remains as to how
close to the star the material gets before moving out to colder parts of the disk. Our
100
3.7 Discussion
Figure 3.21 – As Fig. 3.19, but comparing the cometary abundances to those in our model at t = tacc .
The vertical black lines indicate the spread in the comet-forming region: at the midplane, between
5 and 30 AU from the star.
new solution to the problem of sub-Keplerian accretion (see Chapter 4) results in radial
velocity profiles that are different from the ones in Chapter 2. With the old profiles, all
material that accretes inside of the snow line always remains inside of the snow line. We
therefore did not get any material in the comet-forming zone in which H2 O had evaporated
and readsorbed onto the grains. As shown in Fig. 3.5, we do get this with our current
model. However, it is not a universal outcome for all initial physical conditions. If we
run the new model on the parameter grid from Chapter 2, there are three cases (out of
eight) where material accretes inside of the snow line and then moves out beyond it. The
dividing factor appears to be a ratio of 0.2 between the disk mass at t = tacc and the core
mass at t = 0. For larger ratios, material that accretes inside of the snow line always
remains there.
The presence of crystalline silicate dust in disks provides a strong argument in favour
of trajectories like the one from Fig. 3.5. Observations show crystalline fractions at R ≈ 10
AU that are significantly larger than what is found in the ISM (Bouwman et al. 2001, 2008,
van Boekel et al. 2005). Amorphous silicates can be crystallised by thermal annealing if
they get close enough to the star to be heated to at least 800 K. The disk is less than
100 K at 10 AU, so the crystalline dust is formed at smaller radii than where it is ob101
Chapter 3 – The chemical history of molecules in circumstellar disks, part II
served. We argue in Chapter 4 that the need to conserve angular momentum results in the
outward transport of enough material to explain the observed crystalline fractions at 10
AU. Moreover, crystalline dust has been detected in several comets, including 1P/Halley,
C/1995 01 (Hale-Bopp) and 81P/Wild 2, providing direct evidence that part of their constituent material has been heated to temperatures well above the evaporation temperature
of H2 O (Bregman et al. 1987, Campins & Ryan 1989, Wooden et al. 1999, Keller et al.
2006).
If Halley, Hale-Bopp, Wild 2 and other comets contain crystalline silicates, they must
also contain ices that underwent a large degree of chemical processing. Likewise, the
presence of amorphous silicates in comets is indicative of chemically unprocessed ices.
Hence, we conclude that the material from which comets are formed must be of mixed
origins: some of it accreted close to the star and was heated to high temperatures, while
another part accreted at larger radii and remained cold. Within the context of our model,
this requires that material from beyond R = 30 AU is radially mixed into the cometforming zone between 5 and 30 AU. Alternatively, vertical mixing may bring relatively
pristine material from higher altitudes down into the comet-forming zone at the midplane. As for the chemical variations between individual comets, our model shows many
examples of abundances changing with position or with time. The variations can thus be
explained by having the comets form at different positions in the circumsolar disk, or at
different times during the disk’s lifetime.
3.7.3 Collapse models: 1D versus 2D
The model presented here and in Chapter 2 is the first one to follow the entire core collapse
and disk formation process in two spatial dimensions. Some of our conclusions were
previously drawn in other studies on the basis of 1D models. For example, it was already
known that the collapse-phase chemistry is dominated by a few key chemical processes
and that the collapsing core never attains chemical equilibrium (e.g., Doty et al. 2002,
2004, Rodgers & Charnley 2003, Lee et al. 2004, Aikawa et al. 2008). The 1D models
cannot follow the infalling material all the way into the disk, so the zones with different
chemical histories from Sect. 3.6 appear for the first time in our 2D model. Another new
feature is the back-and-forth motion inside the disk, allowing material to accrete inside
of the snow line and then move out to colder regions. This offers new insights into the
chemical origin of cometary nuclei.
3.8 Conclusions
This chapter describes the two-dimensional chemical evolution during the collapse of a
molecular cloud core to form a low-mass star and a circumstellar disk. The model is the
same as used in Chapter 2, except for the improvements described in Chapter 4. The density and velocity profiles throughout the core and the disk are computed semi-analytically.
We use a full radiative transfer method for the dust temperature and the radiation field.
The chemistry is computed with a full gas-phase network, including adsorption and des102
3.8 Conclusions
orption from dust grains, as well as basic hydrogenation reactions on the grain surfaces.
Starting from realistic initial conditions, we evolve the chemistry in parcels of material
terminating at a range of positions in the disk. Special attention is paid to parcels ending
up in the comet-forming zone. The conclusions from this chapter are as follows:
• The chemistry during the collapse phase is controlled by a small number of key chemical processes, each of which is activated by changes in the physical conditions. The
evaporation of CO, CH4 and H2 O at approximately 18, 22 and 100 K is one set of such
key processes. Another set is the photodissociation of CH4 and H2 O (Sect. 3.5.1).
• At the end of the collapse phase, the disk can be divided into several zones with different chemical histories. The different histories are related to the presence or absence
of the aforementioned key processes along various infall trajectories. Spectroscopic
observations at high spatial resolution are required to determine whether this zonal
division really exists, or if it is smoothed out by mixing (Sect. 3.5.2).
• Part of the material that accretes onto the disk at early times is transported outwards to
conserve angular momentum, and may remain in the disk for the rest of the collapse
phase. It is heated to well above 100 K as it accretes close to the star, so H2 O and all
other non-refractory species evaporate from the grains. They freeze out again when
the material cools down during the subsequent outward transport (Sect. 3.5.2).
• When the chemistry is evolved for an additional 1 Myr at fixed physical conditions
after the end of the collapse phase, the abundances of most species change throughout
the disk. Hence, the disk is not in chemical equilibrium at the end of the collapse.
Instead, its chemical composition is mainly a result of the physical conditions during
the collapse phase. A robust feature of the post-collapse processing is the partial conversion of gaseous CH4 into larger species like HCOOH or C3 H4 , which subsequently
freeze out because of their higher binding energy (Sect. 3.6).
• Material that ends up in the comet-forming zone undergoes a large degree of chemical
processing, including the evaporation and readsorption of H2 O and species trapped in
the H2 O ice. This is consistent with the presence of crystalline silicates in comets.
However, it is inconsistent with the chemical similarities observed between comets
and ISM sources, which are indicative of little processing. Hence, it appears that the
cometary material is of mixed origins: part of it was strongly processed, and part of
it was not. The chemical variations observed between individual comets suggest they
were formed at different positions or times in the solar nebula. Fully pristine ices only
appear in the upper and outer parts of this particular disk model (Sect. 3.7.2).
103
4
Sub-Keplerian accretion onto
circumstellar disks
R. Visser and C. P. Dullemond
to be submitted
105
Chapter 4 – Sub-Keplerian accretion onto circumstellar disks
Abstract
Context. Models of the formation, evolution and photoevaporation of circumstellar disks are an
essential ingredient in many theories of the formation of planetary systems. The ratio of disk mass
over stellar mass in the circumstellar phase of a disk is for a large part determined by the angular
momentum of the original cloud core from which the system was formed. While full 3D or 2D
axisymmetric hydrodynamical models of accretion onto the disk will automatically treat all aspects
of angular momentum, this is not so trivial for 1D viscous disk models.
Aims. Since 1D disk models are still very useful for long-term evolutionary modelling of disks
with relatively little numerical effort, we wish to investigate how the 2D nature of accretion affects
the formation and evolution of the disk in such models. A proper treatment of this problem also
requires a correction for the sub-Keplerian velocity at which accretion takes place.
Methods. We develop an update of our 1+1D time-dependent disk formation and evolution model
that properly treats the sub-Keplerian accretion of matter onto the disk. The model also accounts
for the effects of the vertical extent of the disk on the infall trajectories.
Results. The disks produced with the new method are smaller than those obtained previously, but
their mass is mostly unchanged. The new disks are a few degrees warmer in the outer parts, so
they contain less solid CO. Otherwise, the results for ices are unaffected. The 2D treatment of the
accretion results in material accreting at larger radii, so a smaller fraction comes close enough to the
star for amorphous silicates to be thermally annealed into crystalline form. The lower crystalline
abundances thus predicted correspond more closely to the abundances found observationally. We
argue that thermal annealing followed by radial mixing must be responsible for at least part of the
observed crystalline material.
106
4.1 Introduction
4.1 Introduction
With the enormous increase in the amount of high-quality observational data of circumstellar disks in the last few years, a picture is now gradually emerging of how these objects
evolve in time (Jørgensen et al. 2007, Looney et al. 2007, Lommen et al. 2008, SiciliaAguilar et al. 2009). They form during the collapse of a pre-stellar cloud core, undergo a
number of accretion events (FU Orionis and EX Lupi outbursts), live for 3 to 10 Myr, and,
shortly before they are destroyed, open up huge gaps visible in the dust continuum and
sometimes also in gas lines (D’Alessio et al. 2005, Goto et al. 2006, Brittain et al. 2007,
Ratzka et al. 2007, Brown et al. 2008, Pontoppidan et al. 2008a). These physical changes
are echoed in the evolution of their chemical composition and dust properties. Pre-stellar
cores contain mostly simple hydrides, radicals and other small molecules, largely frozen
out onto the cold dust grains (Bergin & Langer 1997, Lee et al. 2004). A fully formed
circumstellar disk is predicted to contain a much richer chemical mixture with a wide
variety of complex organic molecules (Rodgers & Charnley 2003, Aikawa et al. 2008),
although only simple organics have been observed so far (Lahuis et al. 2006, Carr & Najita 2008, Salyk et al. 2008). The dust by this time has grown from less than a micron to
millimetres and centimetres, and part of it has evolved from an amorphous to a crystalline
structure (Bouwman et al. 2001, 2008, van Boekel et al. 2005, Natta et al. 2007, Lommen
et al. 2007, 2009, Watson et al. 2009, Olofsson et al. 2009). Crystalline silicate dust is
observed down to temperatures of 100 K, well below the threshold of 800 K required to
convert amorphous silicates into crystalline form. One of the central questions of this
chapter is how much silicate material comes close enough to the star to be crystallised.
We also investigate how the crystalline silicates end up so far outside of the hot inner disk
where they appear to be formed.
One way to answer these questions is to construct detailed models of the evolution
of circumstellar disks based on our current understanding of the physics of these objects,
and then compare to the available observational data. However, it would require extraordinarily heavy computations to run a model that does justice to all physical processes
known to be involved. A circumstellar disk ranges from a few stellar radii to hundreds
of AU, and lasts for several million years. A full model would therefore have to resolve
hundreds of millions of inner orbits, and span some five orders of magnitude on a spatial
scale. Moreover, an accurate radiative transfer method is required to properly compute the
temperatures. All this is clearly too demanding. Most multidimensional hydrodynamical
simulations therefore solve sub-problems that only capture part of the disk, or only evolve
over a limited time. Even these models, though, require days or weeks of CPU time for a
single set of parameters.
An alternative approach is to parameterise most of the physics in some form, and
treat the disk evolution as a simpler one-dimensional (1D) time-dependent problem. One
assumes axisymmetry and integrates the density vertically to obtain the surface density
Σ, which is now only a function of the radial coordinate R and the time t. These kinds
of models go back to the pioneering work by Shakura & Sunyaev (1973) and LyndenBell & Pringle (1974). In order to use these models throughout the disk’s lifetime, some
way must be found to also include the birth phase of the disk in a reasonably realistic
107
Chapter 4 – Sub-Keplerian accretion onto circumstellar disks
way. Two-dimensional axisymmetric hydrodynamical models of disk formation show
the presence of a stand-off shock that decelerates the supersonically infalling matter as
it approaches the disk from above and below (e.g., Tscharnuter 1987, Yorke et al. 1993,
Neufeld & Hollenbach 1994). The structure of this stand-off shock is clearly multidimensional in the outer regions, but may be approximated in a simpler manner in the inner
regions (Nakamoto & Nakagawa 1994). Hueso & Guillot (2005) constructed a 1D disk
evolution model with such an approximation and used it to analyse two T Tauri stars.
This showed that simple models of disk formation and evolution can be very powerful
and yield valuable insight into the evolutionary stage of young stellar objects. Similar
models have also been used to analyse the statistics of the accretion rates measured in
pre–main-sequence stars (Dullemond et al. 2006b, Vorobyov & Basu 2008).
Another problem addressed with these 1D parameterised models is the origin, evolution and transport of gas and dust in circumstellar disks. For instance, Dullemond et al.
(2006a, hereafter DAW06) suggested that the initial outward expansion of the disk during
the disk formation phase (observationally the Class 0/I phase) may be very effective in
transporting thermally processed dust to the outer parts of the disk. Based on that work,
one would expect to find a number of disks with nearly 100% crystalline dust. However,
no such extremely crystalline disks are observed (Bouwman et al. 2008, Watson et al.
2009). This is one of the issues addressed in this chapter.
In Chapter 2, we showed yet another application of 1D disk evolution models. We
followed the envelope material from thousands of AU inwards, through the accretion
shock and into the disk, to analyse when ices evaporate and recondense on the grains.
Since much of the interesting physics happens in the outer regions of the disk (several
hundred AU), where the accretion shock no longer has a simple 1D shape, some way had
to be found to include the 2D axisymmetric nature of this region without having to resort
to a full-scale multidimensional hydrodynamical simulation. Our recipe was to construct
a semi-2D disk model, i.e., to generate the 2D density structure ρ(R, z, t) (where z is the
vertical coordinate away from the midplane) out of the 1D surface density Σ(R, t) using
the disk temperature to compute the scale height at every radius. The accretion onto the
disk was then modelled by following infalling matter on ballistic supersonic trajectories
until its density equals the density of the disk. This yields a 2D axisymmetric shape of the
stand-off shock that is similar to that obtained by full 2D axisymmetric hydrodynamical
models (Yorke et al. 1993, Brinch et al. 2008a). However, it is not trivial to link this type of
multidimensional infall structure back to the 1D disk evolution model. The main obstacle
is that a substantial fraction of the matter does not really fall onto the disk’s surface but
onto the disk’s outer edge, i.e., it accretes from the side instead of from the top. Moreover,
this matter will rotate with a very sub-Keplerian velocity. If this is not treated in some
way, the total angular momentum balance of the system is violated, potentially leading
to very wrong estimates of the evolution of the disk mass with time. We devised an adhoc solution to this problem in Chapter 2 (Eq. (2.20)), which gave reasonable results but
did not properly conserve angular momentum. We derive a more rigorous solution in the
current chapter.
The problem of sub-Keplerian accretion onto a disk is not studied here for the first
time. It has long been known that material falling in an elliptic orbit onto the surface of
108
4.2 Equations
the disk has sub-Keplerian angular momentum, even in the case of infinitely flat disks.
When mixing with the disk material, which is in Keplerian rotation, a torque is exerted
that pushes the disk material in towards the star. Cassen & Moosman (1981) described this
problem and solved it elegantly. For disks without internal torque and with small vertical
extent, they found an analytical solution for the disk surface density as a function of
time, and they also presented numerical results for viscous disks. Hueso & Guillot (2005)
derived an alternative solution: they simply calculated the radius for which the specific
angular momentum of the infalling matter is Keplerian, and inserted the matter into the
disk at that radius. This is not unreasonable, because one may assume that the material,
after it hits the surface of the disk, will first adjust its own orbit before it mixes with the
disk material below. It may be argued that as long as the angular momentum budget it
not violated, the disk evolution is not much affected by the precise treatment. However,
if one wishes to follow the radial motion and mixing of certain chemical species or types
of dust, this may depend more sensitively on the way the angular momentum problem is
treated.
In this chapter we add a consistent treatment of sub-Keplerian semi-2D accretion to a
1D model for the formation and viscous evolution of a disk (Sect. 4.2). Some basic disk
properties arising from the new treatment are discussed in Sect. 4.3, and the results from
Chapter 2 are briefly revisited in Sect. 4.4. The dust crystallisation results from DAW06
are re-evaluated in Sect. 4.5, and finally conclusions are drawn in Sect. 4.6.
4.2 Equations
The model describes the formation and evolution of a circumstellar disk inside a collapsing cloud core. The disk’s surface density is governed by two processes: accretion of
material from the envelope and viscosity-driven diffusion. This requires the continuity
equation
∂(ΣR) ∂(ΣRuR )
+
= RS ,
(4.1)
∂t
∂R
with uR the radial velocity and S the source function that accounts for accretion from
the envelope. In addition to conservation of mass, ensured by Eq. (4.1), there must be
conservation of angular momentum (Lynden-Bell & Pringle 1974):
!
∂(ΣΩK R3 ) ∂(ΣΩK R3 uR )
∂ΩK
∂
ΣνR3
+ S ΩR3 ,
(4.2)
+
=
∂t
∂R
∂R
∂R
with ΩK the Keplerian rotation rate. We employ the α description (Shakura & Sunyaev
1973) for the viscosity, so the viscosity coefficient ν is given by
ν(R, t) =
αkT m
,
µmp ΩK
(4.3)
with T m the midplane temperature and µ the mean molecular mass of 2.3 nuclei per hydrogen molecule.
109
Chapter 4 – Sub-Keplerian accretion onto circumstellar disks
In the last term in Eq. (4.2), belonging to the accreting material, the rotation rate Ω is
smaller than ΩK . If accretion would occur with the Keplerian velocity, Eq. (4.2) can be
solved to give the expression for uR used by DAW06. Working from that solution, two
methods have been used in the recent literature to correct for the sub-Keplerian velocity of
the accreting material. Hueso & Guillot (2005) and DAW06 modified the source function
so that all incoming material accretes at the exact radius where its angular momentum
equals that of the disk. One might picture this as a quick redistribution of material in
the top layers of the disk after accretion initially occurred in a sub-Keplerian manner. A
disadvantage of this method is that it introduces a discontinuity in the infall trajectories:
upon accretion onto the disk, material instantaneously jumps to a smaller radius. Hence,
in Chapter 2 we chose to modify the expression for the radial velocity instead of that for
the source function, taking
3 ∂ √ uR (R, t) = − √
Σν R − ηr
Σ R ∂R
r
GM∗
,
R
(4.4)
with M∗ the stellar mass and ηr a parameter with a constant value of 0.002. While this
method has the desired effect of transporting material to smaller radii without discontinuities, it does not properly conserve angular momentum.
As an alternative to these two methods, Eq. (4.2) can also be solved more generally
for the case that Ω < ΩK , giving
3 ∂ √ 2RS ΩK − Ω
Σν R −
.
uR (R, t) = − √
Σ
ΩK
Σ R ∂R
(4.5)
Expressions for Ω (or, rather, for the azimuthal velocity) may be found in Cassen &
Moosman (1981) and Terebey et al. (1984). Equations (4.1) and (4.5), together with the
standard expressions for S (e.g., Chapter 2), now describe a fully continuous solution to
the evolution of the surface density with proper conservation of angular momentum. A
physical interpretation of Eq. (4.5) is provided in Sect. 4.3.
The boundary between the disk and the envelope was computed in Chapter 2 as the
surface where the density from both was the same. Here, we take instead the surface
where the ram pressure of the infalling gas equals the thermal gas pressure from the disk:
ρenv u2 =
kT ρdisk
µmp
(4.6)
with u the velocity of the infalling material. The use of the isobaric surface as the diskenvelope boundary instead of the isopycnic surface can have some effect on the model
results, but, as shown in the next sections, the effects of the new treatment of the subKeplerian accretion are usually much larger. In the remainder of this chapter, we exclusively use the new method presented in this section. The rest of the model is described by
DAW06 and in Chapter 2.
110
4.3 Size and mass of the disk
4.3 Size and mass of the disk
The net effect of the rightmost terms in Eqs. (4.4) and (4.5) is the same: to provide an
additional inward flux of matter. However, the radial dependence of the additional flux
is different. In the old method (Eq. (4.4)), it was largest at the inner edge of the disk
because of the R−1/2 proportionality. In the new method, on the other hand, it is strongest
at the outer edge of the disk because of the sharp decrease in surface density at that
point. Physically, the new dependence is easy to understand. Near the disk’s inner edge,
accretion occurs into a large column of material, and the torque arising from the different
azimuthal velocities only results in a weak inward push. The same torque provides a
much stronger push at the outer edge, where there is less disk material per unit surface. In
effect, the material falling onto the disk’s outer edge pushes the disk inwards and limits
its radial growth.
In order to quantify the size of the disk, we first have to define the outer edge Rd .
Since the equations allow an infinitesimal part of the disk to spread to an infinitely large
distance, we cannot simply use the full radial extent of the disk as its outer edge. Instead,
we define Rd as the radius that contains 90% of the disk’s mass.
With the new method, the disk’s outer edge initially lies beyond the centrifugal radius
(Rc ). It grows more or less linearly in time as the disk spreads out to conserve the angular
momentum of the entire system. The centrifugal radius grows as t3 , so the ratio Rd /Rc
decreases as the collapse goes on. If the rotation rate is high enough, the centrifugal radius
overtakes the outer edge of the disk before the end of the collapse. For the remainder of
the collapse phase, Rd equals Rc , because the disk cannot be smaller than the centrifugal
radius (Cassen & Moosman 1981). If the rotation rate is low enough that the centrifugal
radius does not overtake the outer edge of the disk before the end of the collapse, Rd
continues its near-linear growth until tacc . After accretion from the envelope has ceased
and the inward push from the accreting material is no longer present, the disk is free to
expand more rapidly. Its outer edge again grows linearly in time in both the high- and
low-rotation cases, and does so at a higher rate than in the initial linear-growth regime.
Figure 4.1 shows the outer edge as a function of time for the standard model from
Chapter 2, which has a low initial rotation rate Ω0 of 10−14 s−1 . The disk grows to about
40 AU at the end of the collapse phase at tacc = 2.5 × 105 yr, and it spreads to ten times
that size over the next 7.5 × 105 yr. The method from Chapter 2 (labelled “old” in Fig.
4.1) provides a more rapid growth during the collapse phase, giving an Rd of 230 AU at
tacc . The post-infall spreading occurs at the same rate as in the new method. We find the
same qualitative differences and similarities between the old and the new method for the
rest of the parameter grid from Chapter 2.
Is the smaller disk from the new method a realistic scenario? Our model necessarily assumes uR to be the same at all heights above the midplane. In reality, one might
expect the infalling material to interact mostly with the surface of the disk, exerting its
torque on only a fraction of the total surface density. This pushes the surface material
radially inwards, while material near the midplane is unaffected. Indeed, the hydrodynamical simulations of Brinch et al. (2008a) show such behaviour, with material near the
surface moving inwards and material at the midplane moving outwards. However, as this
111
Chapter 4 – Sub-Keplerian accretion onto circumstellar disks
Figure 4.1 – Outer disk radius as a function of time for the standard model from Chapter 2. Solid:
method from the current chapter; dashed: method from Chapter 2. The dash-dotted curve shows the
centrifugal radius, and the dotted line indicates the end of the envelope accretion phase. Note that
Rc is a physical quantity only up to tacc ; for larger t, we simply plot the same mathematical function.
midplane material moves out beyond the outer edge, it becomes surface material itself,
and is in turn exposed to the inward push from the envelope material. The new method
may underestimate the outer radius by a factor of two or three because of the altitudeindependence of uR , but it produces a more realistic value than does the old method.
Although the disks are smaller with the new method, they are not necessarily less
massive. In fact, we find a rather large mass increase for the standard model from Chapter
2: from 0.05 to 0.13 M⊙ at the end of the collapse phase. There are three causes for this,
each of which accounts for 0.02–0.03 M⊙ : (1) the new definition of the disk-envelope
boundary (Eq. (4.6)); (2) the new sub-Keplerian correction (Eq. (4.5)); and (3) an improved integration scheme in the computational code. The effects are smaller for more
rapidly rotating clouds. For example, the disk mass obtained for the reference model from
Chapter 2 (Ω0 = 10−13 s−1 ) is unchanged at 0.43 M⊙ .
4.4 Gas-ice ratios
During the collapse of the pre-stellar core to form a protostar and circumstellar disk, large
changes occur in both density and temperature. Many molecular species are frozen out
112
4.4 Gas-ice ratios
onto dust grains before the onset of collapse (Bergin & Langer 1997, Lee et al. 2004).
The warm-up phase during the collapse causes some of them to evaporate, and they may
freeze out again once material settles near the disk’s relatively cold midplane. In Chapter
2, we modelled these processes for carbon monoxide (CO) and water (H2 O). Here, we
investigate whether our new sub-Keplerian accretion correction affects these results.
The model consists of several steps. First, it computes the 2D axisymmetric density
and velocity structure at regular intervals from the onset of collapse (t = 0) to the point
where the entire envelope has accreted onto the star and disk (t = tacc ). Dust temperatures are computed at the same time intervals with the radiative transfer code RADMC
(Dullemond & Dominik 2004a), and the gas temperature is assumed equal to the dust
temperature. Using the velocity structure, the model then computes infall trajectories
from a grid of initial positions in the envelope. Each trajectory represents an individual
parcel of gas and dust, for which we now know the density and temperature as a function
of time and position. This allows us to compute the adsorption and desorption rates of
CO and H2 O, and solve for their gas and ice abundances in a Lagrangian frame. Finally,
the parcels’ abundances are transformed back into 2D axisymmetric abundance profiles
for the disk and remnant envelope.
As discussed in Sect. 4.3, the main difference between the disk properties from Chapter 2 and the new method is the size of the disk. Because the mass has increased or stayed
the same (depending on the model parameters), the density of the disk is now higher.
Hence, the dust temperature along the midplane decreases more rapidly. For example, at
t = tacc in the reference model from Chapter 2 (Ω0 = 10−13 s−1 ), the temperature at 30 AU
is 45 K. With the new method applied to the same initial conditions, the temperature is
down to 31 K at that point. The midplane temperature decreases further with radius until
it reaches 21 K at 160 AU with the new method. At larger radii, photons scattering off
the surface of the disk begin to reach the midplane again and the temperature gradually
increases to 25 K at the outer edge at 500 AU.
If all CO is taken to desorb at 18 K, as it would for a pure CO ice, the new temperature
profile does not allow for any solid CO to exist in this particular model at tacc . At later
times, when the disk has spread to larger sizes and the protostar has become less luminous
(Chapter 2), there appears a region around the midplane where the temperature does go
below 18 K and CO freezes out again. In reality, solid CO forms a mixture with solid
H2 O, and some of the CO remains trapped in the ice matrix at temperatures above 18 K
(Collings et al. 2004, Viti et al. 2004, Fayolle et al. in prep.). When the gas-ice ratios are
computed accordingly, the mass fraction of solid CO averaged over the entire disk at tacc
goes from 33% with the old method to 20% with the new method.
The gas-ice ratios for pure CO in the standard model from Chapter 2 (Ω0 = 10−14 s−1 )
are unchanged. With both the old and the new method, the entire disk is warmer than 18
K at tacc , so there is no solid CO. If we allow part of the CO to be trapped in the H2 O ice,
the disk-averaged solid fraction is 15% with both the old and the new methods.
In summary, the new treatment of the sub-Keplerian accretion results in disks that
are a few degrees colder in the inner parts and a few degrees warmer in the outer parts.
Overall, the new CO ice abundances are up to 50% smaller than those obtained with the
old method. In all cases, H2 O remains solid in the entire disk except for the inner few AU.
113
Chapter 4 – Sub-Keplerian accretion onto circumstellar disks
4.5 Crystalline silicates
4.5.1 Observations and previous model results
Infrared spectroscopic observations have shown that about 1–30% of the silicate dust
in the disks around Herbig Ae/Be and classical T Tauri stars occurs in crystalline form
(Bouwman et al. 2001, 2008, van Boekel et al. 2005). Even larger fractions (up to 100%)
are found in the inner 1 AU of some sources (Watson et al. 2009). The interstellar medium,
from which this dust originates, has a crystalline fraction of at most 1–2% (Kemper et al.
2004, 2005). Two mechanisms are thought to dominate the conversion of amorphous silicates into crystalline form during the formation and evolution of a circumstellar disk. At
temperatures above ∼1200 K, the original grains evaporate (Petaev & Wood 2005). When
the gas cools down again, the silicates recondense in crystalline form (Davis & Richter
2003, Gail 2004). Alternatively, amorphous dust can be thermally annealed into crystalline dust at temperatures above ∼800 K (Wooden et al. 2005). As the disk is formed
out of the parent envelope, part of the infalling material accretes close enough to the star
that it is heated to more than 800 or 1200 K and can be crystallised. However, the observations show significant fractions of crystalline dust at least out to radii corresponding
to a temperature of 100 K. Crystalline dust is also found in comets, which are formed in
regions much colder than 800 K (Wooden et al. 1999, Keller et al. 2006). This suggests an
efficient radial mixing mechanism to transport crystalline material from the hot inner disk
to the colder outer parts (Nuth 1999, Bockelée-Morvan et al. 2002, Keller & Gail 2004).
An argument against large-scale radial mixing was recently provided by spatially resolved observations with the Spitzer Space Telescope. Bouwman et al. (2008) found a
clear radial dependence in the relative abundances of forsterite and enstatite, two common specific forms of crystalline silicate. If both are formed in the hot inner disk and
then transported outwards, one would expect the same relative abundances throughout
the entire disk. Hence, the observed radial dependence argues in favour of a localised
crystallisation mechanism such as heating by shock waves triggered by gravitational instabilities (Harker & Desch 2002, Desch et al. 2005). However, the observations do not
completely rule out the possibility of crystallisation in the hot inner disk followed by
radial mixing; at best, they provide an upper limit to how much crystalline dust can be
formed that way.
In another set of Spitzer observations, crystalline spectroscopic features in the 20–30
µm region were detected three times more frequently than the crystalline feature at 11.3
µm (Olofsson et al. 2009). This is unexpected, because shorter wavelengths trace warmer
material at shorter distances from the protostar, where all models predict the crystalline
fractions to be larger. The 11.3 µm feature may be partially shielded by the amorphous
10 µm feature, but Olofsson et al. showed that this alone cannot explain the observations.
A full compositional analysis (Olofsson et al. in prep.) is required to shed more light on
this “crystallinity paradox”.
In the model of DAW06, crystallisation occurs right from the time when the disk is
first formed. Indeed, because the disk is still very small at that time, its dust is hot and
nearly fully crystalline. As the collapse proceeds and the disk’s outer radius grows, an ever
114
4.5 Crystalline silicates
larger fraction of the infalling material does not come close enough to the star anymore
to be heated above 800 K. In the absence of strong shocks, this results in amorphous dust
being mixed in with the crystalline material. Hence, the crystalline fraction averaged over
the entire disk is expected to decrease with time. There is tentative observational support
for an age-crystallinity anticorrelation (van Boekel et al. 2005, Apai et al. 2005), but this
is far from conclusive (Bouwman et al. 2008, Watson et al. 2009). One should of course
consider the fact that observations do not probe the entire disk. If the model results from
DAW06 are interpreted over a limited part of the disk, such as the 10–20 AU region, they
show only a small difference in the crystalline fractions at 1 and 3 Myr. Add to that the
uncertainties in the ages for individual objects, and it is clear that the model results cannot
be said to conflict the observational data.
The crystalline fractions obtained by DAW06 were all on the high end of the observed
range of 1–30%, unless unreasonably high initial rotation rates were adopted for the envelope or the disk temperature was lowered artificially. If we accept that only part of the
silicates in the outer disk originate in the hot inner region – so the other part, formed in
situ, can account for the observed radial abundance variations – the discrepancy between
the DAW06 model and the observations becomes even larger. In the following, we show
that we obtain more realistic crystalline fractions with our new method.
4.5.2 New model results
The two main differences between the old method of DAW06 and our new method are
(1) the treatment of the disk as a multidimensional object instead of just a flat accretion
surface and (2) the improved solution to the problem of sub-Keplerian accretion (Sect.
4.2). The former has the largest impact on the crystallisation. In case of a fully flat
disk, material falls in along ballistic trajectories until it hits the midplane at or inside the
centrifugal radius. If the vertical extent of the disk is taken into account, the infalling
material hits the disk before it can flow all the way to the midplane. Because part of
the disk often spreads beyond the centrifugal radius, especially at early times (Fig. 4.1),
accretion now occurs at much larger radii. This is visualised in Fig. 4.2, which shows the
mass loading onto the disk at 2.0 × 105 yr (0.23 tacc ) after the onset of collapse. The model
parameters are those of the default model of DAW06: an initial envelope mass of 2.5 M⊙ ,
a rotation rate of 1 × 10−14 s−1 , and a sound speed of 0.23 km s−1 . The centrifugal radius
at 2.0 × 105 yr is 2.2 AU, but the disk has already spread to 32 AU. Accretion occurs
across the entire disk, although most mass falls in at small radii.
If the vertical structure of the disk is ignored when calculating the source function, as
happened in the old method of DAW06, the infall trajectories continue along the dotted
lines. They all intersect the midplane inside of Rc . In this case, that means all accretion
takes place inside the “annealing radius” (Rann , the radius corresponding to 800 K) and all
dust is turned into crystalline form. In the new method, only 36% of the accreting material
comes inside of Rann , so the disk gains a much smaller amount of crystalline dust.
The crystalline fractions obtained with the new method are compared to the results
from DAW06 in Fig. 4.3. In both cases, the inner part of the disk, out to a few AU, is fully
crystalline. This is followed by a near-powerlaw decrease as crystalline material is mixed
115
t
(Myr)
1.0
1.6
3.1
t
(Myr)
10
M:
19.7
21.1
22.9
10
30
100
0.20, 0.012
20.1 21.5
21.3 22.2
23.0 23.4
30
100
M: 0.20, 0b
1.0
1.6
3.1
t
(Myr)
1.0
1.6
3.1
t
(Myr)
1.0
1.6
3.1
10
M:
6.7
6.1
5.9
Ω0 = 10−14 s−1 , cs = 0.19 km s−1
R (AU)
30
100
10
30
100
0.50, 0.074
M: 1.0, 0.26
6.0
5.5
7.6
3.8
1.9
5.8
5.7
5.3
3.4
2.3
5.8
5.9
3.3
2.6
2.2
10
M:
32.1
33.3
35.1
Ω0 = 10−14 s−1 , cs = 0.26 km s−1
R (AU)
30
100
10
30
100
0.50, 0.037
M: 1.0, 0.13
32.3 33.8
17.4 16.0 15.1
33.5 34.4
16.4 15.7 15.4
35.3 35.7
16.0 15.8 15.8
10
30
100
M: 0.20, 0.051
2.1
1.9
1.6
1.8
1.7
1.6
1.6
1.5
1.5
Ω0 = 10−13 s−1 , cs = 0.19 km s−1
R (AU)
10
30
100
10
30
100
M: 0.50, 0.23
M: 1.0, 0.58
2.6
1.3
0.6
4.4
1.3
0.3
1.7
1.0
0.5
2.9
1.1
0.4
1.0
0.6
0.4
1.5
0.6
0.3
10
M:
12.1
12.0
12.1
Ω0 = 10−13 s−1 , cs = 0.26 km s−1
R (AU)
10
30
100
10
30
100
M: 0.50, 0.13
M: 1.0, 0.43
5.3
4.4
3.6
6.2
3.3
1.7
4.5
4.0
3.6
4.5
2.6
1.6
3.7
3.5
3.3
2.7
1.7
1.3
30
100
0.20, 0.019
12.0 12.0
12.0 12.0
12.1 12.2
10
30
100
M: 5.0, 2.3
17.0 2.9
0.4
23.0 3.9
0.4
24.6 4.4
0.6
10
30
100
M: 5.0, 1.9
54.1 9.3
1.2
37.5 9.7
2.2
16.5 6.1
3.3
10
30
100
M: 5.0, 4.1
5.4
1.1
0.2
5.1
1.2
0.2
8.5
0.5
0.02
10
30
100
M: 5.0, 3.5
19.3 3.9
0.6
14.6 3.3
0.7
8.6 2.4
0.8
Chapter 4 – Sub-Keplerian accretion onto circumstellar disks
116
Table 4.1 – Crystalline silicate fractions for a range of model parameters.a
4.5 Crystalline silicates
Figure 4.2 – Accretion at 2.0 × 105 yr (0.23 tacc ) for the default model of DAW06. The vertical
dotted lines indicate the current values of Rc (2.2 AU) and Rann (3.6 AU). Top: mass-loading as a
function of radius. Bottom: infall trajectories (solid grey) from the envelope onto the surface of
the disk (black). In the absence of the disk, the trajectories would extend to the midplane along the
dotted lines. The inset in the bottom panel shows a blow-up of the inner 5 × 1 AU.
Table 4.1 – footnotes.
a
The fractions are given in per cent of the total silicate dust abundance at the indicated distance from the star
(R) and time after the onset of collapse (t). The two masses (M, in units of M⊙ ) listed for each combination
of parameters are the initial cloud core mass and the disk mass at the end of the accretion phase.
b
No disk is formed at all for this combination of parameters.
117
Chapter 4 – Sub-Keplerian accretion onto circumstellar disks
Figure 4.3 – Fraction of crystalline silicates for the default model of DAW06 at 1.0 (solid), 1.6
(dotted) and 3.1 Myr (dashed). The grey lines show the results from DAW06; the black lines show
our new results. The arrows on the top axis indicate the outer disk radius for the new method.
to larger radii. At a few tens of AU, the crystallinity levels off to a base value that remains
roughly constant to the outer edge of the disk, indicated by the arrows on the top axis in
Fig. 4.3. The increase in crystallinity beyond that point should not be attributed much
significance because, in reality, this material is mixed with the fully amorphous remnant
envelope. At each of the three time steps plotted, the new crystallinity outside of a few
AU is lower than the old one. For example, at 10 AU and 3.1 Myr, the fraction is down
from 15.1 to 7.4%. At the outer edge of the disk, the new method produces crystalline
fractions down to 1%. The differences between the old and the new method become even
more pronounced when we take a slightly lower initial rotation rate of 3 × 10−15 s−1 (Fig.
4.4). The crystalline fraction at 10 AU and 3.1 Myr is now down from 72.6 to 11.3%.
The amount of crystallisation that takes place during the formation and evolution of
the disk depends on the initial conditions. DAW06 already discussed the effect of the
rotation rate (Ω0 ) of the collapsing envelope. The more rapid the rotation, the larger the
radius at which the bulk of the accretion takes place. This results in less material being
heated above 800 K, so the disk becomes less crystalline. This effect also occurs in our
new model. Two other conditions that can easily be changed are the initial mass (M0 )
and the effective sound speed (cs ). In order to get a first understanding of their effects,
we computed the crystalline fractions for the parameter grid from Chapter 2, as well as
for models with initial masses of 0.2 and 5.0 M⊙ . Table 4.1 lists the relevant model
118
4.5 Crystalline silicates
Figure 4.4 – As Fig. 4.3, but for Ω0 = 3 × 10−15 s−1 .
parameters together with the fraction of crystalline silicates at three different positions
and three different times. The first of the three positions (10 AU) is representative of the
region probed by the recent Spitzer observations, while the other two positions (30 and
100 AU) contain colder material that can be studied with the Herschel Space Observatory.
The models with a low sound speed all have a lower crystallinity than the models with
a high sound speed. As noted in Chapter 2, a low sound speed results in a lower accretion
rate. The accretion time becomes longer, so the disk grows larger and more massive. A
lower accretion rate also gives a lower stellar luminosity, so the region where silicates can
be crystallised is smaller. These effects combine to give smaller fractions of crystalline
material throughout the entire disk.
Changing the initial mass of the cloud core has a more complicated effect on the
crystallinity. Table 4.1 shows several cases where, for models that differ only in the initial
mass, the crystalline fractions increase towards larger M0 , and several cases where they
decrease in that direction. Due to the inside-out nature of the Shu (1977) collapse, models
of different mass initially evolve in exactly the same way. However, the accretion phase
of the higher-mass models in our grid lasts longer (tacc ∝ M0 ) than that of the lowermass models. This affects the crystallinity in two opposing ways. First, the protostar
becomes more luminous for the higher-mass models (D’Antona & Mazzitelli 1994), so
the region in which crystallisation takes place is larger. Second, the disk grows larger,
so the bulk of the accretion occurs farther from the star. The first effect results in higher
crystalline abundances in the inner parts of the disk in the higher-mass models, while the
119
Chapter 4 – Sub-Keplerian accretion onto circumstellar disks
Figure 4.5 – As Fig. 4.3, but for two of the models from Table 4.1 with Ω0 = 10−14 s−1 and cs = 0.19
km s−1 . Grey: M0 = 0.5 M⊙ ; black: M0 = 1.0 M⊙ .
second effect eventually results in lower crystalline abundances in the outer parts (Fig.
4.5). Depending on the exact initial conditions, the transition from the inner to the outer
disk in this context may lie inwards or outwards of the 10–100 AU region given in Table
4.1, or even within that region. Example of two of these three possibilities can be found
in the series of models with Ω0 = 10−14 s−1 and cs = 0.19 km s−1 (top part of Table 4.1).
Going from M0 = 0.2 to 0.5 M⊙ , the fraction of material accreting close enough to the
star to be crystallised is reduced, so we find smaller crystalline fractions at 10, 30 and
100 AU: ∼6% for the 0.5 M⊙ model versus ∼20% for the 0.2 M⊙ model. Increasing the
initial mass to 1.0 M⊙ , the higher stellar luminosity increases the crystallinity at 10 AU to
7.6%. However, the crystallinity at larger radii suffers from the larger fraction of dust that
remained amorphous during the accretion phase because the larger disk prevented it from
getting closer to the protostar. At 30 AU, the crystalline fraction at 1.0 Myr decreases from
6.0 to 3.8% when the initial mass goes from 0.5 to 1.0 M⊙ . The decrease in crystallinity
at 100 AU is even larger: from 5.5 to 1.9%. The differences in the other three series of
models can all be explained in similar fashion.
4.5.3 Discussion and future work
The goal of this chapter is to show how treating the disk as a multidimensional object and
correctly solving the problem of sub-Keplerian accretion affect the results of DAW06. As
120
4.5 Crystalline silicates
shown in Figs. 4.3 and 4.4, we obtain smaller fractions of crystalline silicates throughout
the disk. This is an improvement over the old model, which was noted to overpredict
crystallinity compared with observations.
A detailed parameter study is required to judge how well our current model reproduces
all available observations. One complicating factor in such a procedure is the observed
lack of correlation between crystallinity and other systemic properties such as the stellar
luminosity, the accretion rate and the masses of the star and the disk (Watson et al. 2009).
The observed absence of a correlation between two observables usually translates to a
lack of a physical correlation, but this is not always the case. For example, Kessler-Silacci
et al. (2007) showed why the crystallinity is not observed to be correlated with the stellar
luminosity. The disk around a brighter protostar is warmer throughout, so the region from
which most of the silicate emission originates lies at a larger distance from the star, where
the crystallinity is lower. At the same time, though, the higher temperatures mean that
more material can be thermally annealed, so the crystallinity at all radii goes up. The
two effects cancel each other, so the observed crystalline fraction is not correlated with
the stellar luminosity. Likewise, care must be taken when interpreting other observed
non-correlations or correlations.
The best starting point for a more detailed comparison between model and observations appears to be the observed radial dependence of the relative abundances of specific
types of crystalline silicates, such as enstatite and forsterite. Our model can be expanded
to track multiple types of silicates, each with their own formation temperature and mechanism. First of all, this may help in explaining the “crystallinity paradox” identified by
Olofsson et al. (2009; see also Sect. 4.5.1). Second, it can address the question whether
crystalline silicates are predominantly formed by condensation from hot gas (∼1200 K),
by thermal annealing at slightly lower temperatures (∼800 K), or by shock waves outside
the hot inner disk. At the moment, neither the observations nor the models can rule out
any of these mechanisms. The crystalline fractions obtained with our model suggest that
thermal annealing followed by radial mixing must be taking place and must therefore be
responsible for part of the observed crystalline silicates. A scenario in which all crystalline material is formed where it is observed, according to the model of Bouwman et al.
(2008), appears unlikely.
In addition to tracking multiple types of silicates, it may be worthwhile to investigate
different collapse scenarios. The Shu (1977) collapse starts with a cloud core with an r−2
density profile, but observations of pre-stellar cores usually show an r−1.5 density profile
instead (Alves et al. 2001, Motte & André 2001, Harvey et al. 2003, André et al. 2004,
Kandori et al. 2005). Bonnor-Ebert (BE) spheres have such a density profile (Ebert 1955,
Bonnor 1956), so they have been proposed as an alternative starting point for collapse
models (Whitworth et al. 1996). The collapse of a BE sphere results in different densities,
velocities and temperatures than those obtained with the Shu collapse (Foster & Chevalier
1993, Matsumoto & Hanawa 2003, Banerjee et al. 2004, Walch et al. 2009), leading in
turn to different crystalline silicate abundances. However, no analytical solutions exist for
the collapse of a rotating BE sphere, so we are currently unable to pursue this point in any
more detail.
121
Chapter 4 – Sub-Keplerian accretion onto circumstellar disks
4.6 Conclusions
This chapter presents a new method of correcting for the sub-Keplerian velocity of envelope material accreting onto an axisymmetric two-dimensional circumstellar disk. Unlike
the previous corrections of Hueso & Guillot (2005) and from Chapter 2, this new method
properly conserves angular momentum and produces infall trajectories without discontinuities. The latter is important for tracing changes in the chemical contents and dust
properties during the evolution of the envelope and disk.
The disks produced with the new method are smaller than those produced with the
old method by up to a factor of ten. Depending on the initial conditions, the disk masses
are between 100 and 200% of previously computed values (Sect. 4.3). The new disks are
a few degrees colder in the inner regions and a few degrees warmer in the outer regions,
resulting in lower abundances of CO ice (Sect. 4.4). By the time the system reaches the
classical T Tauri stage, at about 1 Myr, the global ice abundances still agree well with
observations. Overall, there are no major changes in the gas-ice ratios compared with
Chapter 2.
The disk was treated as geometrically flat by Dullemond et al. (2006a). As in Chapter
2, we now also take into account the vertical structure when computing the infall trajectories. This results in the bulk of the accretion occuring at larger radii. A smaller fraction of
the infalling material now comes close enough to the star to be heated above 800 K, the
temperature required for thermal annealing of amorphous silicates into crystalline form.
Therefore, the new method produces crystalline abundances that are lower by a few per
cent to more than a factor of five compared to the old model. We now obtain a better
match with observations and we argue that thermal annealing followed by radial mixing
is responsible for at least part of the crystalline silicates observed in disks. An expanded
model, which tracks specific forms of crystalline silicate, is required to establish in more
detail the importance of this and other possible crystallisation mechanisms.
122
5
The photodissociation and chemistry
of CO isotopologues: applications to
interstellar clouds and circumstellar
disks
R. Visser, E. F. van Dishoeck and J. H. Black
Astronomy & Astrophysics, 2009, 503, 323
123
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
Abstract
Aims. Photodissociation by UV light is an important destruction mechanism for carbon monoxide
(CO) in many astrophysical environments, ranging from interstellar clouds to protoplanetary disks.
The aim of this work is to gain a better understanding of the depth dependence and isotope-selective
nature of this process.
Methods. We present a photodissociation model based on recent spectroscopic data from the literature, which allows us to compute depth-dependent and isotope-selective photodissociation rates
at higher accuracy than in previous work. The model includes self-shielding, mutual shielding and
shielding by atomic and molecular hydrogen, and it is the first such model to include the rare isotopologues C17 O and 13 C17 O. We couple it to a simple chemical network to analyse CO abundances
in diffuse and translucent clouds, photon-dominated regions, and circumstellar disks.
Results. The photodissociation rate in the unattenuated interstellar radiation field is 2.6 × 10−10 s−1 ,
30% higher than currently adopted values. Increasing the excitation temperature or the Doppler
width can reduce the photodissociation rates and the isotopic selectivity by as much as a factor of three for temperatures above 100 K. The model reproduces column densities observed towards diffuse clouds and PDRs, and it offers an explanation for both the enhanced and the reduced
N(12 CO)/N(13 CO) ratios seen in diffuse clouds. The photodissociation of C17 O and 13 C17 O shows
almost exactly the same depth dependence as that of C18 O and 13 C18 O, respectively, so 17 O and 18 O
are equally fractionated with respect to 16 O. This supports the recent hypothesis that CO photodissociation in the solar nebula is responsible for the anomalous 17 O and 18 O abundances in meteorites.
Grain growth in circumstellar disks can enhance the N(12 CO)/N(C17 O) and N(12 CO)/N(C18 O) ratios by a factor of ten relative to the initial isotopic abundances.
124
5.1 Introduction
5.1 Introduction
Carbon monoxide (CO) is one of the most important molecules in astronomy. It is second
in abundance only to molecular hydrogen (H2 ) and it is the main gas-phase reservoir of
interstellar carbon. Because it is readily detectable and chemically stable, CO and its less
abundant isotopologues are the main tracers of the gas properties, structure and kinematics
in a wide variety of astrophysical environments (for recent examples, see Dame et al.
2001, Najita et al. 2003, Wilson et al. 2005, Greve et al. 2005, Leroy et al. 2005, Huggins
et al. 2005, Bayet et al. 2006, Oka et al. 2007 and Narayanan et al. 2008). In particular,
the pure rotational lines at millimetre wavelengths are often used to determine the total
gas mass. This requires knowledge of the CO-H2 abundance ratio, which may differ by
several orders of magnitude from one object to the next (Lacy et al. 1994, Burgh et al.
2007, Panić et al. 2008). If isotopologue lines are used, the isotopic ratio enters as an
additional unknown.
CO also controls much of the chemistry in the gas phase and on grain surfaces, and is
a precursor to more complex molecules. In photon-dominated regions (PDRs), dark cores
and shells around evolved stars, the amount of carbon locked up in CO compared with that
in atomic C and C+ determines the abundances of small and large carbon-chain molecules
(Millar et al. 1987, Jansen et al. 1995, Aikawa & Herbst 1999, Brown & Millar 2003,
Teyssier et al. 2004, Cernicharo 2004, Morata & Herbst 2008). CO ice on the surfaces
of grains can be hydrogenated to more complex saturated molecules such as CH3 OH
(Charnley et al. 1995, Watanabe & Kouchi 2002, Fuchs et al. 2009), so the partitioning of
CO between the gas and grains is important for the overall chemical composition as well
(Caselli et al. 1993, Rodgers & Charnley 2003, Doty et al. 2004, Garrod & Herbst 2006).
A key process in controlling the gas-phase abundance of 12 CO and its isotopologues
is photodissociation by ultraviolet (UV) photons. This is governed entirely by discrete absorptions into predissociative excited states; any possible contributions from continuum
channels are negligible (Hudson 1971, Fock et al. 1980, Letzelter et al. 1987, Cooper &
Kirby 1987). Spectroscopic measurements in the laboratory at increasingly high spectral
resolution have made it possible for detailed photodissociation models to be constructed
(Solomon & Klemperer 1972, Bally & Langer 1982, Glassgold et al. 1985, van Dishoeck
& Black 1986, Viala et al. 1988, van Dishoeck & Black 1988 (hereafter vDB88), Warin
et al. 1996, Lee et al. 1996). The currently adopted photodissociation rate in the unattenuated interstellar radiation field is 2 × 10−10 s−1 .
Because the photodissociation of CO is a line process, it is subject to self-shielding:
the lines become saturated at a 12 CO column depth of about 1015 cm−2 , and the photodissociation rate strongly decreases (vDB88, Lee et al. 1996). Bally & Langer (1982)
realised this is an isotope-selective effect. Due to their lower abundance, isotopologues
other than 12 CO are not self-shielded until much deeper into a cloud or other object. This
results in a zone where the abundances of these isotopologues are reduced with respect
to 12 CO, and the abundances of atomic 13 C, 17 O and 18 O are enhanced with respect to
12
C and 16 O. For example, the C17 O-12 CO and C18 O-12 CO column density ratios towards
X Per are a factor of five lower than the elemental oxygen isotope ratios (Sheffer et al.
2002). The 13 CO-12 CO ratio along the same line of sight is unchanged from the elemental
125
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
carbon isotope ratio, indicating that 13 CO is replenished through low-temperature isotopeexchange reactions. A much larger sample of sources shows N(13 CO)/N(12 CO) column
density ratios both enhanced and reduced by up to a factor of two relative to the elemental isotopic ratio (Sonnentrucker et al. 2007, Burgh et al. 2007, Sheffer et al. 2007). The
reduced ratios have so far defied explanation, as all models predict that isotope-exchange
reactions prevail over selective photodissociation in translucent clouds.
CO self-shielding has been suggested as an explanation for the anomalous 17 O-18 O
abundance ratio found in meteorites (Clayton et al. 1973, Clayton 2002, Lyons & Young
2005, Lee et al. 2008). In cold environments, molecules such as water (H2 O) may be enhanced in heavy isotopes. This so-called isotope fractionation process is due to the differ17
18
ence in vibrational energies of H16
2 O, H2 O and H2 O, and is therefore mass-dependent. It
18
results in O being about twice as fractionated as 17 O. However, 17 O and 18 O are nearly
equally fractionated in the most refractory phases in meteorites (calcium-aluminium-rich
inclusions, or CAIs), hinting at a mass-independent fractionation mechanism. Isotopeselective photodissociation of CO in the surface of the early circumsolar disk is such a
mechanism, because it depends on the relative abundances of the isotopologues and the
mutual overlap of absorption lines, rather than on the mass of the isotopologues. The
enhanced amounts of 17 O and 18 O are subsequently transported to the planet- and cometforming zones and eventually incorporated into CAIs. Recent observations of 12 CO, C17 O
and C18 O in two young stellar objects support the hypothesis of CO photodissociation as
the cause of the anomalous oxygen isotope ratios in CAIs (Smith et al. 2009). A crucial
point in the Lyons & Young model is the assumption that the photodissociation rates of
C17 O and C18 O are equal. Our model can test this at least partially.
Detailed descriptions of the CO photodissociation process are also important in other
astronomical contexts. The circumstellar envelopes of evolved stars are widely observed
through CO emission lines. The measurable sizes of these envelopes are limited primarily
by the photodissociation of CO in the radiation field of background starlight (Mamon et al.
1988). Finally, proper treatment of the line-by-line contributions to the photodissociation
of CO may affect the analysis of CO photochemistry in the upper atmospheres of planets
(Fox & Black 1989).
In this chapter, we present an updated version of the photodissociation model from
vDB88, based on laboratory experiments performed in the past twenty years (Sect. 5.2).
We expand the model to include C17 O and 13 C17 O and we cover a broader range of CO
excitation temperatures and Doppler widths (Sects. 5.3 and 5.4). We rederive the shielding functions from vDB88 and extend these also to higher excitation temperatures and
larger Doppler widths (Sect. 5.5). Finally, we couple the model to a chemical network
and discuss the implications for translucent clouds, PDRs and circumstellar disks, with a
special focus on the meteoritic 18 O anomaly (Sect. 5.6).
5.2 Molecular data
The photodissociation of CO by interstellar radiation occurs through discrete absorptions
into predissociated bound states, as first suggested by Hudson (1971) and later confirmed
126
5.2 Molecular data
by Fock et al. (1980). Any possible contributions from continuum channels are negligible
at wavelengths longer than the Lyman limit of atomic hydrogen (Letzelter et al. 1987,
Cooper & Kirby 1987).
Ground-state CO has a dissociation energy of 11.09 eV and the general interstellar
radiation field is cut off at 13.6 eV, so knowledge of all absorption lines within that range
(911.75–1117.80 Å) is required to compute the photodissociation rate. These data were
only partially available in 1988, but ongoing laboratory work has filled in a lot of gaps.
Measurements have also been extended to include CO isotopologues, providing more
accurate values than can be obtained from theoretical isotopic relations. Table 5.1 lists
the values we adopt for 12 CO.
5.2.1 Band positions and identifications
Eidelsberg & Rostas (1990, hereafter ER90) and Eidelsberg et al. (1992) redid the experiments of Letzelter et al. (1987) at higher spectral resolution and higher accuracy, and also
for 13 CO, C18 O and 13 C18 O. They reported 46 predissociative absorption bands between
11.09 and 13.6 eV, many of which were rotationally resolved. Nine of these have a cross
section too low to contribute significantly to the overall dissociation rate. The remaining
37 bands are largely the same as the 33 bands of vDB88; bands 1 and 2 of the latter are
resolved into four and two individual bands, respectively. Throughout this work, band
numbers refer to our numbering scheme (Table 5.1), unless noted otherwise.
Thanks to the higher resolution and the isotopologue data, ER90 could identify the
electronic and vibrational character of the upper states more reliably than Stark et al. in
vDB88. The vibrational levels are required to compute the positions for those isotopologue bands that have not been measured directly. Nine of the vDB88 bands (not counting
the previously unresolved bands 1 and 2) have a revised v′ value.
The Eidelsberg et al. (1992) positions (ν0 or λ0 ) are the best available for most bands,
with an estimated accuracy of 0.1–0.5 cm−1 . Seven of their 12 CO bands were too weak
or diffuse for a reliable analysis, so their positions are accurate only to within 5 cm−1 .
Nevertheless, we adopt the Eidelsberg et al. positions for three of these: bands 2A, 6 and
14. The former was blended with band 2B in vDB88, and the other two show a better
match with the isotopologue band positions if we take the Eidelsberg et al. values. For the
other four weak or diffuse bands, Nos. 4, 15, 19 and 28, we keep the vDB88 positions.
Ubachs et al. (1994) further improved the experiments, obtaining an accuracy of about
0.01–0.1 cm−1 , so we adopt their band positions where available. Finally, we adopt the
even more accurate positions (0.003 cm−1 or better) available for the C1, E0, E1 and L0
bands (Ubachs et al. 2000, Cacciani et al. 2001, 2002, Cacciani & Ubachs 2004).1
Band positions for isotopologues other than 12 CO are still scarce, although many more
are currently known from experiments than in 1988. The C1 and E1 bands have been
measured for all six natural isotopologues, and the E0 band for all but 13 C17 O, at an
accuracy of 0.003 cm−1 (Cacciani et al. 1995, Ubachs et al. 2000, Cacciani et al. 2001,
1
All transitions in our model arise from the v′′ =0 level of the electronic ground state. We use a shorthand that
only identifies the upper state, with C1 indicating the C 1 Σ+ v′ =1 state, etc.
127
Bandb
#
1A
1B
1C
1D
2A
2B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
ER90b
#
7A
7B
7C
7D
8A
8B
9A
9B
9C
10
11
12
13
14
15A
15B
15C
16
17
18
19
20
21
22
24
25
26
27
λ0
(Å)
912.70
913.40
913.43
913.67
915.73
915.97
917.27
919.21
920.14
922.76
924.63
925.81
928.66
930.06
931.07
931.65
933.06
935.66
939.96
941.17
946.29
948.39
950.04
956.24
964.40
968.32
968.88
970.36
ν0
(cm−1 )
109564.6
109481.0
109478.0
109449.0
109203.0
109173.8
109018.9
108789.1
108679.0
108371.0
108151.3
108013.6
107682.3
107519.8
107402.8
107335.9
107174.4
106876.0
106387.8
106250.9
105676.3
105442.3
105258.4
104576.6
103691.7
103271.8
103211.8
103054.7
v′
ID
1
Π
(5pσ) 1 Σ+
(5pπ) 1 Π
1 +
Σ
(6pπ) 1 Π
(6pσ) 1 Σ+
1
Π
(6sσ) 1 Σ+
I ′ (5sσ) 1 Σ+
(5dσ) 1 Σ+
1 +
Σ
W(3sσ) 1 Π
1
Π
1
Π
1
Π
(5pπ) 1 Π
(5pσ) 1 Σ+
1 +
Σ
I ′ (5sσ) 1 Σ+
W(3sσ) 1 Π
(4dσ) 1 Σ+
L(4pπ) 1 Π
H(4pσ) 1 Σ+
W(3sσ) 1 Π
J(4sσ) 1 Σ+
L(4pπ) 1 Π
L′ (3dπ) 1 Π
K(4pσ) 1 Σ+
0
1
1
2
0
0
2
0
1
0
1
3
2
2
0
0
0
2
0
2
0
1
1
1
1
0
1
0
f v′ 0
3.4(-3)
1.7(-3)
1.7(-3)
2.7(-2)
2.0(-3)
7.9(-3)
2.3(-2)
2.8(-3)
2.8(-3)
6.3(-3)
5.2(-3)
2.0(-2)
6.7(-3)
6.3(-3)
6.0(-3)
1.2(-2)
2.2(-2)
3.8(-3)
2.1(-2)
3.1(-2)
7.6(-3)
2.8(-3)
2.2(-2)
1.6(-2)
2.8(-3)
1.4(-2)
1.2(-2)
3.4(-2)
Atot
(s−1 )
1(10)
9(10)
1(10)
9(10)
1(11)
1(11)
5(11)
1(11)
1(11)
3(11)
1(11)
4(11)
4(10)
1(11)
1(11)
3(11)
3(10)
3(11)
1(12)
1(11)
1(11)
1(10)
1(12)
7(11)
3(11)
2(9)
2(11)
2(10)
η
B′v
D′v
−1
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.99
1.00
1.00
1.00
0.96
1.00
0.99
(cm )
1.92
5.9(-5)
1.83
1.0(-5)
1.96
1.0(-4)
1.78
5.4(-5)
1.58
6.7(-6)
1.69
1.0(-4)
1.67
7.2(-5)
2.14
4.6(-5)
1.91
6.0(-6)
1.97
6.3(-6)
1.87
4.0(-5)
1.65
1.1(-4)
1.94
3.1(-5)
1.82
2.6(-5)
1.65
1.0(-5)
1.87
4.3(-5)
2.13
1.0(-5)
1.95
0.0
2.04
8.8(-5)
1.62 −1.3(-5)
1.90
1.7(-5)
1.96
1.0(-5)
1.94
4.4(-5)
1.57
5.8(-5)
1.92
9.0(-6)
1.96
7.1(-6)
1.75
1.0(-5)
1.92
6.0(-5)
ω′e
ωe x′e
−1
(cm )
2170
13
2214
15
2214
15
2170
13
1563
14
2214
15
2170
13
2214
15
2291
0
2214
15
2170
13
1745 −4
2170
13
2170
13
2170
13
2214
15
2214
15
2170
13
2291
0
1745 −4
2214
15
2171
0
2204
0
1745 −4
2236
0
2171
0
2214
15
2204
0
References
& notes
1; c
1,2; d
d
1,2,3; c
e,f
3; d
2; c
4,5; d
g
d,h
c
6; g,i
1,2,3; c
c
c
2; d,j
1,2; d,i
5; c
4; g,j
1,6; g,i,k
1,2,3; d
g
4; g,j
6; g
g
7,8,9; g,i,k
1,2,3,9; d,k
1,2,3,9; g
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
128
Table 5.1 – New molecular data for 12 CO.a
Table 5.1 – continued.
Bandb
#
25
26
27
28
29
30
31
32
33
ER90b
#
28
29
30
31
32
33
37
38
39
λ0
(Å)
972.70
977.40
982.59
985.65
989.80
1002.59
1051.71
1063.09
1076.08
ν0
(cm−1 )
102806.7
102312.3
101771.7
101456.0
101031.0
99741.7
95082.9
94065.6
92929.9
ID
v′
f v′ 0
W(3sσ) 1 Π
W ′ (3sσ) 3 Π
F(3dσ) 1 Σ+
J(4sσ) 1 Σ+
G(3dπ) 1 Π
F(3dσ) 1 Σ+
E(3pπ) 1 Π
C(3pσ) 1 Σ+
E(3pπ) 1 Π
0
2
1
0
0
0
1
1
0
1.7(-2)
1.8(-3)
4.8(-4)
1.5(-2)
4.6(-4)
7.9(-3)
3.6(-3)
3.0(-3)
6.8(-2)
Atot
(s−1 )
1(10)
4(11)
3(11)
1(12)
1(11)
3(11)
6(9)
2(9)
1(9)
η
0.97
1.00
1.00
1.00
1.00
1.00
0.96
0.56
0.80
D′v
B′v
−1
(cm )
1.57 9.7(-5)
1.54 8.0(-6)
1.85 1.4(-5)
1.92 5.1(-5)
1.96 1.1(-5)
1.81 2.2(-4)
1.93 6.6(-6)
1.92 6.3(-6)
1.95 6.3(-6)
ωe x′e
ω′e
−1
(cm )
1745 −4
1563
14
2030
0
2236
0
2214
15
2030
0
2239
43
2176
15
2239
43
References
& notes
1,3,6; g,i,k
1,2,3; f,k
g
4; g,j
d
g
10,11,12,13; k
14,15,16
13,15,17,18; k
5.2 Molecular data
129
References: λ0 , ν0 , ID and v′ from Eidelsberg et al. (1992) and fv′ 0 , Atot , B′v and D′v from ER90, except these: (1) λ0 and ν0 from Ubachs et al. (1994); (2) Atot from
Ubachs et al. (1994); (3) B′v and D′v from Ubachs et al. (1994); (4) λ0 and ν0 from vDB88; (5) B′v and D′v from vDB88; (6) fv′ 0 and Atot from Eidelsberg et al.
(2006); (7) λ0 , ν0 , B′v and D′v from Cacciani et al. (2002); (8) Atot from Drabbels et al. (1993); (9) fv′ 0 from Eidelsberg et al. (2004); (10) λ0 , ν0 , B′v and D′v
from Ubachs et al. (2000); (11) fv′ 0 from Eidelsberg et al. (2006); (12) Atot from Ubachs et al. (2000); (13) ω′e , ωe x′e from Kepa
˛ (1988); (14) λ0 , ν0 , Atot , B′v and
D′v from Cacciani et al. (2001); (15) fv′ 0 from Federman et al. (2001); (16) ω′e , ωe x′e from Tilford & Vanderslice (1968); (17) λ0 , ν0 , B′v and D′v from Cacciani
& Ubachs (2004); (18) Atot from Cacciani et al. (1998).
Notes:
a
Many values are rounded off from higher-precision values in the references. The notation a(b) in this and following tables means a × 10b .
b
The numbering follows vDB88. Their bands 1 and 2 are split into four and two components. The corresponding ER90 indices are also given.
c
ω′e and ωe x′e from the CO ground state (Guelachvili et al. 1983). ωe y′e and ωe z′e (not listed) are included in the model.
d
ω′e and ωe x′e from the CO+ X 2 Σ+ state (Haridass et al. 2000).
e
B′v and D′v from the CO+ A 2 Π state (Haridass et al. 2000).
f
ω′e and ωe x′e from the CO+ A 2 Π state (Haridass et al. 2000). ωe y′e (not listed) is included in the model.
g
Vibrational constants derived from the different ν0 in one of six vibrational series: bands 30–27, 28–21, 25–20–16–8, 24–19, 22–18 or 15–5.
h
B′v and D′v from the CO+ X 2 Σ+ state (Haridass et al. 2000).
i
Atot depends on parity and/or rotational level (see Table 5.3). Atot and η are listed here for J ′ =0 and f parity.
j
B′v and D′v computed from the C18 O values of ER90.
k
B′v and D′v depend on parity (see Table 5.2); values are listed here for f parity.
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
Cacciani & Ubachs 2004). The positions of the E0 band are especially important because
of its key role in the isotope-selective nature of the CO photodissociation (Sect. 5.3.3).
Positions are known at lower accuracy (0.003–0.5 cm−1 ) for an additional 25 C18 O, 30
13
CO and 9 13 C18 O bands (Eidelsberg et al. 1992, Ubachs et al. 1994, Cacciani et al.
2002); these are included throughout.
We compute the remaining band positions from theoretical isotopic relations. For
band b of isotopologue i, the position is
ν0 (b, i) = ν0 (b,12 CO) + Ev′ (b, i) − Ev′′ (X, i) −
h
i
Ev′ (b,12 CO) − Ev′′ (X,12 CO) ,
(5.1)
with Ev′ (b, i) and Ev′′ (X, i) the vibrational energy of the excited and ground states, respectively. Hence, we need the vibrational constants (ω′e , ωe x′e , . . . ) for all the excited states
other than the C 1 Σ+ . These have only been determined experimentally for the E 1 Π state
(Kepa
˛ 1988). For the other states, we employ this scheme:
• if it is part of a vibrational series (such as band 30, for which the corresponding v′ =1
band is No. 27), we can derive ω′e from the difference in ν0 ;
• else, if it is part of a Rydberg series converging to the X 2 Σ+ or A 2 Π state of CO+ , we
take those constants (Haridass et al. 2000);
• else, we take the constants of ground-state CO (Guelachvili et al. 1983).
The choice for each band and the values of the constants are given in Table 5.1.
5.2.2 Rotational constants
The rotational constants (B′v and D′v ) for each excited state are needed to compute the
positions of the individual absorption lines. ER90 provided B′v values for most bands,
at an estimated accuracy of better than 1%. Their D′v values are less well constrained
and may be off by more than a factor of two. However, this is of little importance for
the low-J lines typically involved in the photodissociation of CO. More accurate values
(B′v to better than 0.1%, D′v to 10% or better) are available for 12 states from higherresolution experiments (Eikema et al. 1994, Ubachs et al. 1994, 2000, Cacciani et al. 2001,
2002, Cacciani & Ubachs 2004). Again, the data for isotopologues other than 12 CO are
generally scarce, so we have to compute their constants from theoretical isotopic relations.
This increases the uncertainty in B′v to a few per cent. In case of bands 12, 15, 19 and 28,
ER90 reported constants for C18 O but not for 12 CO, so we employ the theoretical relations
for the latter. We adopt the vDB88 constants for bands 4 and 14, because they are more
accurate than those of ER90. No constants are available for Rydberg bands 2A and 6, so
we adopt the constants of the associated CO+ states (A 2 Π and X 2 Σ+ , respectively).
In seven cases, the rotational constants of the P and R branch (e parity) were found
to differ from those of the Q branch ( f parity; Ubachs et al. 1994, 2000, Cacciani et al.
2002, Cacciani & Ubachs 2004). For these bands, the f parity values are given in Table
5.1. Table 5.2 lists the difference between the f and e values, defined as q′v = B′v,e − B′v, f
and p′v = D′v,e − D′v, f . The uncertainties in q′v and p′v are on the order of 1 and 10%,
respectively.
130
5.2 Molecular data
Table 5.2 – Parity-dependent rotational constants for 12 CO.
Band
#
16
22
23
25
26
31
33
a
b
q′v
(cm−1 )
−2.7(-3)
2.212(-2)
3.0(-2)
−7.0(-4)
−1.11(-2)
1.14(-2)
1.196(-2)
p′v a
(cm−1 )
—
7.9(-6)
—
—
—
3.0(-8)
2.1(-7)
Refs.b
1
2
1
1
1
3
4
Dashes indicate that no measurement is available, so we adopt a value of zero.
(1) Ubachs et al. (1994); (2) Cacciani et al. (2002); (3) Ubachs et al. (2000); (4) Cacciani & Ubachs (2004)
5.2.3 Oscillator strengths
ER90 measured the integrated absorption cross sections (σint ) for all their bands to a typical accuracy of 20%, but they cautioned that some values, especially for mutually overlapping bands, may be off by up to a factor of two. The oscillator strengths ( fv′ 0 ) derived
from these data are different from vDB88 for most bands, sometimes by even more than a
factor of two. In addition, there are differences of up to an order of magnitude between the
cross sections of 12 CO and those of the other isotopologues for many bands shortwards
of 990 Å. The isotopic differences are likely due in part to the difficulty in determining
individual cross sections for strongly overlapping bands, but isotope-selective oscillator
strengths in general are not unexpected. For example, they were also observed recently
in high-resolution measurements of N2 (G. Stark, priv. comm.). For CO, the oscillator
strengths depend on the details of the interactions between the J ′ , v′ levels of the excited
states and other rovibronic levels. These interactions, in turn, depend on the energy levels,
which are different between the isotopologues. We adopt the isotope-selective oscillator
strengths where available. In case of transitions where no isotopic data exist, we choose to
take the value of the isotopologue nearest in mass. This gives the closest match in energy
levels and should, in general, also give the closest match in oscillator strengths, which to
first order are determined by the Franck-Condon factors.
For ten of our bands, we adopt oscillator strengths from studies that aimed specifically
at measuring this parameter (Federman et al. 2001, Eidelsberg et al. 2004, 2006). Their
estimated accuracy is 5–15%. The oscillator strength for the E0 band from Federman et al.
(2001) is almost twice as large as that of vDB88 and ER90, which appears to be due to an
inadequate treatment of saturation effects in the older work. The 2001 value corresponds
well to other values derived since 1990. Federman et al. also measured the oscillator
strength of the weaker C1 band and found it to be the same, within the error margins,
as that of vDB88 and ER90. Recent measurements of the K0, L0 and E1 transitions
and the four W transitions show larger oscillator strengths than those of vDB88 and ER90
(Eidelsberg et al. 2004, 2006). The new values correspond closely to those of Sheffer et al.
131
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
(2003), who derived oscillator strengths for eight bands by fitting a synthetic spectrum to
Far Ultraviolet Spectroscopic Explorer (FUSE) data taken towards the star HD 203374A.
Lastly, the Eidelsberg et al. (2004) value for the L′ 1 band is 33% lower than that of
vDB88, but very similar to those of ER90 and Sheffer et al. (2003), so we adopt it as well.
We compute the oscillator strengths of individual lines as the product of the appropriate Hönl-London factor and the oscillator strength of the corresponding band (Morton &
Noreau 1994). Significant departures from Hönl-London patterns have been reported for
many N2 lines, sometimes even for the lowest rotational levels (Stark et al. 2005, 2008).
The oscillator strength measured in one particular N2 band for the P(22) line was twenty
times stronger than that for the P(2) line, due to strong mixing of the upper state with a
nearby Rydberg state. For other bands where deviations from Hönl-London factors were
observed, the effect was generally less than 50% at J ′ =10. Similar deviations are likely to
occur for CO, but a lack of experimental data prevents us from including this in our model.
Note, however, that large deviations are only expected for specific levels that happen to
be strongly interacting with another state. Many hundreds of levels contribute to the photodissociation rate, so the effect of some erroneous individual line oscillator strengths is
small.
5.2.4 Lifetimes and predissociation probabilities
Upon excitation, there is competition between dissociative and radiative decay. A band’s
predissociation probability (η) can be computed if the upper state’s total and radiative
lifetimes are known: η = 1 − τtot /τrad . ER90 reported total lifetimes (1/Atot ) for all
their bands, but many of these are no more than order-of-magnitude estimates. Higherresolution experiments have since produced more accurate values for 17 of our bands
(Ubachs et al. 1994, 2000, Cacciani et al. 1998, 2001, 2002, Eidelsberg et al. 2006).
In several cases, values that differ by up to a factor of three are reported for different
isotopologues. Where available, we take isotope-specific values. Otherwise, we follow
the same procedure as for the oscillator strengths, and take the value of the isotopologue
nearest in mass.
The total lifetimes of some upper states have been shown to depend on the rotational
level (Drabbels et al. 1993, Ubachs et al. 2000, Eidelsberg et al. 2006). In case of 1 Π
states, a dependence on parity was sometimes observed as well. We include these effects
for the five bands in which they have been measured (Table 5.3).
Recent experiments by Chakraborty et al. (2008) suggest isotope-dependent photodissociation rates for the E0, E1, K0 and W2 bands. These have been interpreted to imply
different predissociation probabilities of individual lines of the various isotopologues due
to a near-resonance accidental predissociation process. Similar effects have been reported
for ClO2 and CO2 (Lim et al. 1999, Bhattacharya et al. 2000). In this process, the boundstate levels into which the UV absorption takes place do not couple directly with the
continuum of a dissociative state. Instead, they first transfer population to another bound
state, whose levels happen to lie close in energy. For the CO E1 state, this process was
rotationally resolved by Ubachs et al. (2000) for all naturally occurring isotopologues and
shown to be due to spin-orbit interaction with the k 3 Π v′ =6 state, which in turn couples
132
5.2 Molecular data
Table 5.3 – Parity- and rotation-dependent inverse lifetimes for 12 CO.
Band
#
8
13c
16
22
25
a
Atot, f a
(s−1 )
3.6(11)+4.0(9)x
3.4(10)+7.3(10)x
1.0(11)+1.8(9)x
1.83(9)
1.2(10)
Atot,e a
(s−1 )
1.6(11)+1.3(10)x
—
1.0(11)+3.4(9)x
1.91(9)+1.20(9)x
1.2(10)+2.4(9)x
Refs.b
1
2
1
3
1
x stands for J ′ (J ′ + 1).
b
(1) Eidelsberg et al. (2006); (2) Ubachs et al. (1994); (3) Drabbels et al. (1993).
c
1 +
This is a Σ upper state, so there is no distinction between e and f parity.
with a repulsive state. The predissociation rates of the E1 state are found to increase significantly due to this process, but only for specific J ′ levels that accidentally overlap. For
example, interaction occurs at J ′ =7, 9 and 12 for 12 CO, but at J ′ =1 and 6 for 13 C18 O.
Since the dissociation probabilities for the E1 state due to direct predissociation were
already high, η = 0.96, this increase in dissociation rate only has a very minor effect
(Cacciani et al. 1998). Moreover, under astrophysical conditions a range of J ′ values are
populated, so that the effect of individual levels is diluted. Since Chakraborty et al. did
not derive line-by-line molecular parameters, we cannot easily incorporate their results
into our model. In Sect. 5.6.3, we show that our results do not change significantly when
we include the proposed effects in an ad-hoc way.
The radiative lifetime of an excited state is a sum over the decay into all lower-lying
levels, including the A 1 Π and B 1 Σ+ electronic states and the v′ ,0 levels of the ground
state. The total decay rate to the ground state is obtained by summing the oscillator
strengths from Table 5.1 for each vibrational series (Morton & Noreau 1994, Cacciani
et al. 1998). Theoretical work by Kirby & Cooper (1989) shows that transitions to electronic states other than the ground state contribute about 1% of the overall radiative decay
rate for the C state and about 8% for the E state. No data are available for the excited
states at higher energies. Fortunately, the radiative decay rate from these higher states to
the ground state is small compared to the dissociation rate, so an uncertainty of ∼10%
does not affect the η values.
The predissociation probabilities thus computed are practically identical to those of
vDB88: the largest difference is a 10% decrease for the C1 band. This is due to the larger
oscillator strength now adopted.
There have been suggestions that the C0 state can also contribute to the photodissociation rate. Cacciani et al. (2001) measured upper-state lifetimes in the C0 and C1 states for
several CO isotopologues. For the v′ =0 state of 13 CO they found a total lifetime of 1770
ps, consistent with a value of 1780 ps for 12 CO, but different from the lifetime of 1500 ps
in 13 C18 O. Although the three values agree within the mutual uncertainties of 10–15% on
each measurement, Cacciani et al. suggested that the heaviest species, 13 C18 O, has a predissociation yield of η = 0.17 rather than zero if the measurements are taken at face value
133
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
and if the radiative lifetime of the C0 state is presumed to be 1780 ps for all three species.
The accurate absorption oscillator strength measured by Federman et al. (2001) for the
C0 band, 0.123 ± 0.016, implies a radiative lifetime that can be no longer than 1658 ps
at the lower bound of measurement uncertainty in fv′ 0 . Taken together, the lifetime measurements of Cacciani et al. and the absorption measurements of Federman et al. favour a
conservative conclusion that the dissociation yield is zero for each of these three isotopologues. We assume the C0 band is also non-dissociative in C17 O, C18 O and 13 C17 O; this
is consistent with earlier studies (e.g., vDB88, ER90, Morton & Noreau 1994).
5.2.5 Atomic and molecular hydrogen
Lines of atomic and molecular hydrogen (H and H2 ) form an important contribution to
the overall shielding of CO. As in vDB88, we include H Lyman lines up to n=50 and H2
Lyman and Werner lines (transitions to the B 1 Σ+u and C 1 Πu states) from the v′′ =0, J ′′ =0–
7 levels of the electronic ground state. We adopt the line positions, oscillator strengths
and lifetimes from Abgrall et al. (1993a,b), as compiled for the freely available M
PDR code (Le Bourlot et al. 1993, Le Petit et al. 2002, 2006).2 Ground-state rotational
constants, required to compute the level populations, come from Jennings et al. (1984).
5.3 Depth-dependent photodissociation
5.3.1 Default model parameters
The simplest way of modelling the depth-dependent photodissociation involves dividing
a one-dimensional model of an astrophysical object, irradiated only from one side, into
small steps, in which the photodissociation rates can be assumed constant. We compute
the abundances from the edge inwards, so that at each step we know the columns of CO,
H, H2 and dust shielding the unattenuated radiation field.
Following vDB88, Le Bourlot et al. (1993), Lee et al. (1996) and Le Petit et al. (2006),
we treat the line and continuum attenuation separately. For each of our 37 CO bands, we
include all lines originating from the first ten rotational levels (J ′′ =0–9) of the v′′ =0 level
of the electronic ground state. That results in 855 lines per isotopologue. In addition, we
have 48 H lines and 444 H2 lines, for a total of 5622. We use an adaptive wavelength
grid that resolves all lines without wasting computational time on empty regions. For
typical model parameters, the wavelength range from 911.75 to 1117.80 Å is divided into
∼47 000 steps.
We characterise the population distribution of CO over the rotational levels by a single
temperature, T ex (CO). The H2 population requires a more detailed treatment, because
UV pumping plays a large role for the J ′′ >4 levels (van Dishoeck & Black 1986). We
populate the J ′′ =0–3 levels according to a single temperature, T ex (H2 ), and adopt fixed
columns of 4 × 1015 , 1 × 1015 , 2 × 1014 and 1 × 1014 cm−2 for J ′′ =4–7. This reproduces
observed translucent cloud column densities to within a factor of two (van Dishoeck &
2
http://aristote.obspm.fr/MIS/pdr/pdr1.html
134
5.3 Depth-dependent photodissociation
Black 1986 and references therein). The J ′′ >4 population scales with the UV intensity,
so we re-evaluate this point for PDRs and circumstellar disks in Sects. 5.6.2 and 5.6.3.
The line profiles of CO, H2 and H are taken to be Voigt functions, with default Doppler
widths (b) of 0.3, 3.0 and 5.0 km s−1 , respectively. We adopt Draine (1978) as our standard
unattenuated interstellar radiation field.
5.3.2 Unshielded photodissociation rates
We obtain an unshielded CO photodissociation rate of 2.6 × 10−10 s−1 . This rate is 30%
higher than that of vDB88, due to the generally larger oscillator strengths in our data set.
The new data for bands 33 and 24 (the E0 and K0 transitions) have the largest effect: they
account for 63 and 21% of the overall increase. Clearly, the rate depends on the choice
of radiation field. If we adopt Habing (1968), Gondhalekar et al. (1980) or Mathis et al.
(1983) instead of Draine (1978), the photodissociation rate becomes 2.0, 2.0 or 2.3×10−10
s−1 , respectively. The same relative differences between these three fields were reported
by vDB88.
5.3.3 Shielding by CO, H2 and H
Self-shielding, shielding by H, H2 and the other CO isotopologues, and continuum shielding by dust all reduce the photodissociation rates inside a cloud or other environment relative to the unshielded rates. For a given combination of column densities (N) and visual
extinction (AV ), the photodissociation rate for isotopologue i is
ki = χk0,i Θi exp(−γAV ) ,
(5.2)
with χ a scaling factor for the UV intensity and k0,i the unattenuated rate in a given radiation field. The shielding function Θi accounts for self-shielding and shielding by H,
H2 and the other CO isotopologues; tabulated values for typical model parameters are
presented in Table 5.5. The dust extinction term, exp(−γAV ), is discussed in Sect. 5.3.4.
Equation (5.2) assumes the radiation is coming from all directions. If this is not the case,
such as for a cloud irradiated only from one side, k0,i should be reduced accordingly.
For now, we ignore dust shielding and compute the depth-dependent dissociation rates
due to line shielding only. Our test case is the centre of the diffuse cloud towards the star
ζ Oph. The observed column densities of H, H2 , 12 CO and 13 CO are 5.2 × 1020, 4.2 × 1020,
2.5 × 1015 and 1.5 × 1013 cm−2 (van Dishoeck & Black 1986, Lambert et al. 1994), and we
take C17 O, C18 O, 13 C17 O and 13 C18 O column densities of 4.1 × 1011 , 1.6 × 1012, 5.9 × 109
and 2.3 ×1010 cm−2 , consistent with observational constraints. For the centre of the cloud,
we adopt half of these values. We set b(CO) = 0.48 km s−1 and T ex (CO) = 4.2 K (Sheffer
et al. 1992), and we populate the H2 rotational levels explicitly according to the observed
distribution. The cloud is illuminated by three times the Draine field (χ = 3).
Table 5.4 lists the relative contribution of the most important bands at the edge and
centre for each isotopologue, as well as the overall photodissociation rate at each point.
The column densities are small, but isotope-selective shielding already occurs: the 12 CO
rate at the centre is lower than that of the other isotopologues by factors of three to six.
135
Band
λ0 (Å)b
33
1076.1
28
985.6
24
970.4
23
968.9
22
968.3
20
956.2
19
950.0
16
941.2
15
940.0
13
933.1
Totalc
12
d
Edge (%)
Centre (%)d
Shieldinge
32.2
2.8
0.0084
4.7
0.4
0.0072
8.7
5.7
0.063
3.0
9.3
0.30
Edge (%)
Centre (%)
Shielding
32.0
22.0
0.21
4.7
0.1
0.0067
8.7
5.3
0.19
3.0
3.5
0.36
Edge (%)
Centre (%)
Shielding
31.5
35.3
0.58
5.1
0.1
0.0067
6.2
4.7
0.39
1.5
2.6
0.88
Edge (%)
Centre (%)
Shielding
28.1
18.5
0.23
6.5
0.0
0.0007
8.3
5.3
0.23
1.8
4.4
0.87
Edge (%)
Centre (%)
Shielding
28.0
32.5
0.63
6.3
0.0
0.0007
8.3
7.4
0.48
1.8
2.9
0.88
Edge (%)
Centre (%)
Shielding
30.1
39.0
0.71
7.0
0.0
0.0007
5.9
5.8
0.54
1.5
2.4
0.87
CO
3.6
3.4
0.8
12.1
0.022
0.34
17
C O
3.6
3.4
0.8
8.7
0.065
0.79
C18 O
4.9
3.7
7.8
3.3
0.82
0.46
13
CO
5.2
3.4
11.2
4.1
0.76
0.42
13 17
C O
5.2
3.4
7.5
5.0
0.79
0.78
13 18
C O
4.7
3.6
5.7
1.7
0.67
0.26
4.3
13.8
0.31
5.1
6.1
0.12
3.5
11.0
0.30
3.1
7.0
0.22
7.8(-10)
7.5(-11)
0.10
4.3
10.3
0.74
5.1
14.2
0.85
3.5
9.1
0.81
3.1
2.4
0.23
7.8(-10)
2.4(-10)
0.31
4.7
7.7
0.83
5.2
8.1
0.80
3.8
5.9
0.81
3.4
3.0
0.45
7.2(-10)
3.7(-10)
0.52
4.3
10.7
0.87
4.9
4.4
0.32
3.9
9.1
0.83
2.4
0.8
0.13
7.8(-10)
2.8(-10)
0.35
4.3
7.2
0.91
4.9
8.5
0.94
3.9
6.1
0.85
2.4
1.0
0.24
7.8(-10)
4.2(-10)
0.54
4.4
7.5
0.92
5.0
8.5
0.94
4.0
6.3
0.85
2.4
1.7
0.38
7.5(-10)
4.1(-10)
0.55
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
136
Table 5.4 – Relative and absolute shielding effects for the ten most important bands in the ζ Oph cloud.a
5.3 Depth-dependent photodissociation
The E0 band is the most important contributor at the edge. This was also found by
vDB88, and the higher oscillator strength now adopted makes it even stronger. Going to
the centre, it saturates rapidly for 12 CO: its absolute contribution to the total dissociation
rate decreases by two orders of magnitude, and it goes from the strongest band to the 12th
strongest band. The five strongest bands at the centre are the same as in vDB88: Nos. 13,
15, 19, 20 and 23.
Figure 5.1 illustrates the isotope-selective shielding. The left panel is centred on the
R(1) line of the E0 band (No. 33 from Table 5.1). This line is fully saturated in 12 CO and
the relative intensity of the radiation field (I/I0 , with I0 the intensity at the edge of the
cloud) goes to zero. 13 CO and C18 O also visibly reduce the intensity, to I/I0 = 0.50 and
0.72, but the other three isotopologues are not abundant enough to do so. Consequently,
these three are not self-shielded in the ζ Oph cloud, but they are shielded by 12 CO, 13 CO
and C18 O. The weaker shielding of 13 C17 O and 13 C18 O in the E0 band compared to C17 O
is due to their lines having less overlap with the 12 CO lines.
The right panel of Fig. 5.1 contains the R(0) line of the W1 band (No. 20), with the
R(1) line present as a shoulder on the red wing. Also visible is the saturated B13 R(2)
line of H2 at 956.58 Å. The W1 band is weaker than the E0 band, so 12 CO is the only
isotopologue to cause any appreciable reduction in the radiation field and to be (partially)
self-shielded. The shielding of the other five isotopologues is dominated by overlap with
the 12 CO and H2 lines. This figure also shows the need for accurate line positions: if the
13 18
C O line were shifted by 0.1 Å in either direction, it would no longer overlap with the
H2 line and be less strongly shielded. Note that the position of the W1 band has only been
measured for 12 CO, 13 CO and C18 O, so we have to compute the position for the other
isotopologues from theoretical isotopic relations. This causes the C17 O line to appear
longwards of the C18 O line.
5.3.4 Continuum shielding by dust
Dust can provide a very strong attenuation of the radiation field. This effect is largely
independent of wavelength for the 912–1118 Å radiation available to dissociate CO, so it
affects all isotopologues to the same extent. It can be expressed as an exponential function
of the visual extinction, as in Eq. (5.2). For typical interstellar dust grains (radius of 0.1
µm and optical properties from Roberge et al. 1991), the extinction coefficient γ is 3.53
for CO (van Dishoeck et al. 2006). Larger grains have less opacity in the UV and do
not shield CO as strongly. For ice-coated grains with a mean radius of 1 µm, appropriate
for circumstellar disks (Jonkheid et al. 2006), the extinction coefficient is only 0.6. The
effects of dust shielding are discussed more fully in Sect. 5.6.
Table 5.4 – footnotes.
a
See the text for the adopted column densities, Doppler widths, excitation temperatures and radiation field.
b 12
CO band head position.
c
Total photodissociation rate in s−1 at the edge and the centre, and the shielding factor at the centre.
d
Relative contribution per band to the overall photodissociation rate at the edge and the centre of the cloud.
e
Shielding factor per band: the absolute contribution at the centre divided by that at the edge.
137
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
Figure 5.1 – Relative intensity of the radiation field (I/I0 ) and intrinsic line profiles for the six CO
isotopologues (φ, in arbitrary linear units) at the centre of the ζ Oph cloud in two wavelength ranges.
Photodissociation of CO may still take place even in highly extincted regions. Cosmic
rays or energetic electrons generated by cosmic rays can excite H2 , allowing it to emit
in a multitude of bands, including the Lyman and Werner systems (Prasad & Tarafdar
1983). The resulting UV photons can dissociate CO at a rate of about 10−15 s−1 (Gredel
et al. 1987), independent of depth. That is enough to increase the atomic C abundance
by some three orders of magnitude compared to a situation where the photodissociation
rate is absolutely zero. The cosmic-ray–induced photodissociation rate is sensitive to the
spectroscopic constants of CO, especially where it concerns the overlap between CO and
H2 lines, so it would be interesting to redo the calculations of Gredel et al. with the new
data from Table 5.1. However, that is beyond the scope of this chapter.
5.3.5 Uncertainties
The uncertainties in the molecular data are echoed in the model results. When coupled
to a chemical network, as in Sect. 5.6, the main observables produced by the model are
the column densities of the CO isotopologues for a given astrophysical environment. The
accuracy of the photodissociation rates is only relevant in a specific range of depths; in
the average interstellar UV field, this range runs from an AV of ∼0.2 to ∼2 mag. Photo138
5.4 Excitation temperature and Doppler width
processes are so dominant at lower extinctions and so slow at higher extinctions that the
exact rate does not matter. In the intermediate regime, both the absolute photodissociation rates and the differences between the rates for individual isotopologues are important.
The oscillator strengths are the key variable in both cases and these are generally known
rather accurately. Taking account of the experimental uncertainties in the band oscillator
strengths and of the theoretical uncertainties in computing the properties for individual
lines, and identifying which bands are important contributors (Table 5.4), we estimate the
absolute photodissociation rates to be accurate to about 20%. This error margin carries
over into the absolute CO abundances and column densities for the AV ≈ 0.2–2 mag range
when the rates are put into a chemical model. The accuracy on the rates and abundances of
the isotopologues relative to each other is estimated to be about 10% when summed over
all states, even when we allow for the kind of isotope effects suggested by Chakraborty
et al. (2008).
5.4 Excitation temperature and Doppler width
The calculations of vDB88 were only done for low excitation temperatures of CO and H2 .
Here, we extend this work to higher temperatures, as required for PDRs and disks, and
we re-examine the effect of the Doppler widths of CO, H2 and H on the photodissociation
rates. We first treat four cases separately, increasing either T ex (CO), b(CO), T ex (H2 ) or
b(H2 ). At the end of this section we combine these effects in a grid of excitation temperatures and Doppler widths. As a template model we take the centre of the ζ Oph cloud,
with column densities and other parameters as described in Sect. 5.3.3.
5.4.1 Increasing Tex (CO)
As the excitation temperature of CO increases, additional rotational levels are populated
and photodissociation is spread across more lines. Figures 5.2 and 5.3 visualise this for
band 13 of 12 CO. At 4 K, only four lines are active: the R(0), R(1), P(1) and P(2) lines
at 933.02, 932.98, 933.09 and 933.12 Å. The R(0) and P(1) lines are both fully selfshielded at the line centre. Going to 16 K, the R(0) line loses about 70% of its intrinsic
intensity and ceases to be self-shielding. In addition, the R(2), P(3) and higher-J lines
start to absorb. The combination of less saturated low-J lines and more active higher-J
lines yields a 39% higher photodissociation rate at 16 K compared to 4 K.
A higher CO excitation temperature has the same favourable effect for 13 CO, which is
partially self-shielded at the centre of the ζ Oph cloud. Its photodissociation rates increase
by 16% when going from 4 to 16 K. C18 O is also partially self-shielded, but less so than
13
CO, so the favourable effect is smaller. At the same time, it suffers from increased
overlap by 12 CO. The net result is a small increase in the photodissociation rate of 0.2%.
The two heaviest isotopologues, 13 C17 O and 13 C18 O, are not abundant enough to be
self-shielded. Their J ′′ <2 lines generally have little overlap with the corresponding 12 CO
lines, especially in the E0 band near 1076 Å. This band, whose lines are amongst the
narrowest in our data set, is the strongest contributor to the photodissociation rate at the
139
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
Figure 5.2 – Illustration of the effect of increasing T ex (CO) (left) or b(CO) (right) in our ζ Oph
cloud model (Sect. 5.3.3). Top: relative intensity of the radiation field, including absorption by
12
CO only. Middle: intrinsic line profile for band 13 (933.1 Å) of 12 CO. Bottom: photodissociation
rate per unit wavelength, multiplied by a constant as indicated.
centre of the cloud for 13 C17 O and 13 C18 O (Table 5.4). In fact, its narrow lines are part of
the reason it is the strongest contributor. The J ′′ =3 and 4 lines that become active at 16 K
do have some overlap with 12 CO. Without the favourable effect of less self-shielding, this
causes the photodissociation rate for 13 C17 O and 13 C18 O to decrease for higher excitation
140
5.4 Excitation temperature and Doppler width
Figure 5.3 – Illustration of the effect of increasing T ex (H2 ) (left) or b(H2 ) (right) in our ζ Oph cloud
model (Sect. 5.3.3). Top: relative intensity of the radiation field, including absorption by H2 only.
Middle: intrinsic line profile for band 30 (1002.6 Å) of 12 CO. Bottom: photodissociation rate per
unit wavelength, multiplied by a constant as indicated.
temperatures. The change is only small, though: 0.4% for 13 C17 O and 2% for 13 C18 O.
Finally, C17 O experiences an increase of 18% in its photodissociation rate. Its lines
lie closer to those of 12 CO than do the 13 C17 O and 13 C18 O lines, so it is generally more
strongly shielded. At 4 K, most of the shielding is due to the saturated R(0) lines of 12 CO.
141
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
These become partially unsaturated at higher T ex (CO), so the corresponding R(0) lines of
C17 O become a stronger contributor to the photodissociation rate, even though the shift
towards higher-J lines make them intrinsically weaker. Overall, increasing T ex (CO) from
4 to 16 K thus results in a higher C17 O photodissociation rate.
5.4.2 Increasing b(CO)
The width of the absorption lines is due to Doppler broadening and natural (or lifetime)
broadening. The integrated intensity in each line remains the same when b(CO) increases,
so a larger width is accompanied by a lower peak intensity. The resulting reduction in
self-shielding then causes a higher 12 CO photodissociation rate, as shown in Fig. 5.2 for
band 13. However, the effect is rather small because the Doppler width is smaller than
the natural width for most lines at typical b values. Natural broadening is the dominant
broadening mechanism up to b(CO) ≈ 6 × 10−12 Atot , with both parameters in their normal
units. The R(0) line of band 13 has an inverse lifetime of 1.8 × 1011 s−1 (Tables 5.1 and
5.3), so Doppler broadening becomes important at about 1 km s−1 . From 0.3 to 3 km s−1 ,
as in Fig. 5.2, the line width only increases by a factor of 1.9. Integrated over all lines, the
12
CO photodissociation rate becomes 26% higher.
The rates of the other five isotopologues decrease along this b(CO) interval due to
increased shielding by the E0 lines of 12 CO. With an inverse lifetime of only 1 × 109
s−1 , Doppler broadening is this band’s dominant broadening mechanism in the regime of
interest. A tenfold increase in the Doppler parameter from 0.3 to 3 km s−1 results in a
nearly tenfold increase in the line widths. At 0.3 km s−1 , the E0 lines of 12 CO are still
sufficiently narrow that they do not strongly shield the lines of the other isotopologues.
This is no longer the case at 3 km s−1 . 13 CO still benefits somewhat from reduced selfshielding in other bands, but it is not enough to overcome the reduced strength of the E0
band, and its photodissociation rates decrease by 2%. The decrease is 13% for C17 O and
26–28% for the remaining three isotopologues. The relatively small decrease for C17 O is
due to its E0 band being already partially shielded by 12 CO at 0.3 km s−1 , so the stronger
shielding at 3 km s−1 has less of an effect.
5.4.3 Increasing Tex (H2 ) or b(H2 )
Increasing the excitation temperature of H2 , while keeping the CO parameters constant,
results in a decreased photodissociation rate for all six isotopologues. The cause, as illustrated in Fig. 5.3, is the activation of more H2 lines. At T ex (H2 ) = 10 K, the R(1) line of
the B8 band at 1002.45 Å is very narrow and does not shield the F0 band (No. 30) of the
CO isotopologues. (The continuum-like shielding visible in Fig. 5.3 is due to the strongly
saturated B8 R(0) line at 1001.82 Å.) It becomes much more intense at 30 K and widens
due to being saturated, thereby shielding part of the F0 band. The same thing happens to
other CO bands, resulting in an overall rate decrease of 0.6–2.5%. There is no particular
trend visible amongst the isotopologues; the magnitude of the rate change depends purely
on the chance that a given CO band overlaps with an H2 line.
142
5.4 Excitation temperature and Doppler width
Similar decreases of one or two percent in the CO photodissociation rates are seen
when the H2 Doppler width is changed from 1 to 10 km s−1 . As the H2 lines become
broader, the amount of overlap with CO increases across the entire wavelength range. As
an example, Fig. 5.3 shows again the 1002.0–1002.8 Å region, where the B8 R(1) line of
H2 further reduces the contribution of the F0 band to the 12 CO photodissociation rate.
5.4.4 Grid of Tex and b
We now combine the four individual cases into a grid of excitation temperatures and
Doppler widths to see how they influence each other. T ex (CO) is raised from 4 to 512 K
in steps of factors of two. The v′′ =1 vibrational level of 12 CO lies at 2143 cm−1 above
the v′′ =0 level, so it starts to be thermally populated at ∼500 K. No data are available
on dissociative transitions out of this level, so we choose not to go to higher excitation
temperatures. We increase the range of rotational levels up to J ′′ =39, at 2984 cm−1 above
the J ′′ =0 level for 12 CO. At T ex (CO) = 512 K, the normalised population distribution
peaks at J ′′ =9 and decreases to 9.7 × 10−5 at J ′′ =39. The H2 excitation temperature is
set to [T ex (CO)]1.5 to take account of the fact that its critical densities for thermalisation
are lower than those of CO. Where necessary, absorption by rotational levels above our
normal limit of J ′′ =7 and by non-zero vibrational levels is taken into account (Dabrowski
1984, Abgrall et al. 1993a,b). All H2 rovibrational levels are strictly thermally populated;
no UV pumping is included. The
widths of 0.1, 0.3, 1.0 and
√ grid is run for CO Doppler
√
3.2 km s−1 ; we set b(H2 ) = 14b(CO) and b(H) = 28b(CO), corresponding to the
differences appropriate for thermal broadening.
The top set of panels in Fig. 5.4 shows the photodissociation rate of the six isotopologues at the centre of the ζ Oph cloud as a function of excitation temperature for the
different Doppler widths. The rates are normalised to the rate at 4 K. The 12 CO rate increases from 4 to 16 K, as described in Sect. 5.4.1. At higher temperatures the increased
overlap with H2 lines takes over and the rate goes down. As long as the CO excitation
temperature is less than ∼100 K, the 12 CO rate remains constant up to b(CO) = 0.3 km s−1
and increases as b(CO) goes from 0.3 to 3.2 km s−1 (bottom part of Fig. 5.4). At higher
temperatures there is so much shielding by H2 that reduced self-shielding in the CO lines
has no discernible effect on the rate. Instead, the rate goes down with b due to stronger
shielding by the broadened H2 lines.
The 13 CO rate also increases initially with T ex (CO) and then goes down as H2 shielding takes over. The rate increases from b(CO) = 0.1 to 0.3 km s−1 , but decreases for
higher values as described in Sect. 5.4.2. For the remaining four isotopologues, the plotted curves likewise result from a combination of weaker shielding by 12 CO and stronger
shielding by H2 . At CO excitation temperatures between 4 and 8 K, the rates typically
change by a few per cent either way. Going to higher temperatures, all rates decrease
monotonically. Likewise, the rates generally decrease towards higher b(CO) values.
A change in behaviour is seen when increasing T ex (CO) from 256 to 512 K. It is at this
point that the v′′ >0 levels of H2 become populated. Less energy is now needed to excite
H2 to the B and C states, so absorption shifts towards longer wavelengths. This causes
even stronger shielding in the heavy CO isotopologues, for whom the E0 band at 1076 Å
143
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
Figure 5.4 – Top: photodissociation rate of the CO isotopologues as a function of excitation temperature, normalised to the rate at 4 K, at four different Doppler widths, for parameters corresponding to the centre of the ζ Oph diffuse cloud. Bottom: photodissociation rates as a function of
Doppler width, normalised to the rate at 0.1 km s−1 , at four different
excitation temperatures.
The
√
√
other model parameters are as in Sect. 5.3.3, except b(H2 ) = 14b(CO), b(H) = 28b(CO) and
T ex (H2 ) = [T ex (CO)]1.5 .
144
5.4 Excitation temperature and Doppler width
Figure 5.5 – As Fig. 5.4, but with all rates normalised to the 12 CO rate.
145
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
is still an important contributor to the photodissociation rate, at least as long as b(CO)
does not exceed 0.3 km s−1 . The E0 band is strongly self-shielded in 12 CO (Table 5.4), so
the shift of the H2 absorption to longer wavelengths does not reduce its contribution by
much. In fact, the weaker H2 absorption at shorter wavelengths allows for an increased
contribution of bands like Nos. 13 and 16 at 933 and 941 Å, causing a net increase in
the 12 CO photodissociation rate from 256 to 512 K. The situation changes somewhat
when the CO Doppler width increases to 1.0 km s−1 or more. The E0 band of the heavy
isotopologues is now much less of a contributor, because it is shielded by the broader
12
CO lines. The shifting H2 absorption does not cause any additional shielding, so the
rates remain almost the same. Furthermore, the H2 lines are also broader and continue to
shield the 12 CO bands at shorter wavelengths, preventing its photodissociation rate from
increasing like it does in the low-b cases.
The main astrophysical consequence becomes clear when we look at the dissociation
rates of the five heavier isotopologues with respect to that of 12 CO. Figure 5.5 shows this
ki /k(12 CO) ratio as a function of T ex (CO) and b(CO). The ratios generally decrease with
both parameters: a higher excitation temperature and a larger Doppler width both cause
less self-shielding in 12 CO, so the rate differences between the isotopologues become
smaller. Shielding by H2 increases at the same time, further reducing the differences
between the isotopologues. This means that photodissociation of CO is more strongly
isotope-selective in cold sources than in hot sources.
5.5 Shielding function approximations
It is unpractical for many astrophysical applications to do the full integration of all 5622
lines in our model every time a photodissociation rate is required. Therefore, we present
approximations to the shielding functions Θ introduced in Eq. (5.2). The approximations
are derived for several sets of model parameters and are valid across a wide range of
astrophysical environments (Sect. 5.5.2).
5.5.1 Shielding functions on a grid of N(CO) and N(H2 )
The transition from atomic to molecular hydrogen occurs much closer to the edge of the
cloud than the C+ -C-CO transition, so the column density of atomic H is roughly constant
at the depths where shielding of CO is important. In addition, H shields CO by only a few
per cent. Therefore, it is a good approximation to compute the shielding functions on a
grid of CO and H2 column densities, while taking a constant column of H. It is sufficient
to express the shielding of all CO isotopologues as a function of N(12 CO), because selfshielding of the heavier CO isotopologues is a small effect compared to shielding by
12
CO.
Table 5.5 presents the shielding functions in the same manner as vDB88 did, but for
somewhat different model parameters: b(CO) = 0.3 instead of 1.0 km s−1 , T ex (CO) = 5
instead of 10 K, and T ex (H2 ) = 51.5 instead of 101.5 K. The present parameters correspond
more closely to what is observed in diffuse and translucent clouds. The column density
146
5.5 Shielding function approximations
ratios for the six isotopologues are kept constant at the elemental isotope ratios from
Wilson (1999): [12 C]/[13 C] = 69, [16 O]/[18 O] = 557 and [18 O]/[17 O] = 3.6. A small
column of 5.2 × 1015 cm−2 of H2 at J ′′ =4–7 is included throughout (except at log N(H2 ) =
0) to account for UV pumping (Sect. 5.3.1).3 Shielding functions for larger values of b
and T ex and for other isotope ratios are given in the online appendix to the paper that this
chapter is based on (Visser et al. 2009). For ease of use, we have also set up a webpage4
where the shielding functions can be downloaded in plain text format. This webpage
offers shielding functions for a wider variety of parameters than is possible to include in
this chapter or the paper. In addition, it uses a grid of N(12 CO) and N(H2 ) values that is
five times finer than the grid in Table 5.5, allowing for more accurate interpolation.
For column densities of up to 1017 cm−2 of CO and 1021 cm−2 of H2 , our shielding
functions are generally within a few per cent of the vDB88 values when corrected for
the difference in b and T ex . Larger differences occur for larger columns: we predict the
shielding to be five times weaker at N(CO) = 1019 cm−2 and more than a hundred times
stronger at N(H2 ) = 1023 cm−2 . The 912-1118 Å wavelength range was divided into 23
bins by vDB88, and most lines were included only in one bin to speed up the computation.
We integrate all lines over all wavelengths. As the lines get strongly saturated at high
column depths, absorption in the line wings becomes important. Thus, H2 lines can cause
substantial shielding over a range of more than 10 Å, while CO lines may still absorb
several Å away from the line centre. The binned integration method of vDB88 did not
take these effects into account, so they underpredicted shielding at large H2 columns and
overpredicted shielding at large CO columns (Fig. 5.6). It should be noted, however, that
photodissociation at these depths is typically already so slow a process that it is no longer
the dominant destruction pathway for CO. In addition, a large CO column is usually
accompanied by a large H2 column, so the two effects partially cancel each other.
5.5.2 Comparison between the full model and the approximations
Despite being computed for a limited number of model parameters, the shielding functions from Table 5.5 and the online appendix to Visser et al. (2009) provide a good approximation to the rates from the full model for a wide range of astrophysical environments.
Section 5.6.1 presents a grid of translucent cloud models, where the photodissociation
model is coupled to a chemical network and CO is traced as a function of depth. This
presents a large range of column densities, with the ratios between the isotopologues deviating from the fixed values adopted for Table 5.5. The grid covers gas densities from
100 to 1000 cm−3 and gas temperatures from 15 to 100 K, while keeping the excitation
temperatures and Doppler widths constant at the values used for Table 5.5. Altogether, the
grid contains 2880 points per isotopologue where the photodissociation rate is computed.
Here, we compare the photodissociaton rates from the approximate method to the full
integration for each of these points. The sensitivity of k(13 CO) to the N(12 CO)/N(13 CO)
ratio can be corrected for in a simple manner: we use the shielding functions from Table
3
4
Although not mentioned explicitly by vDB88, their tabulated shielding functions also include this extra column
of J ′′ =4–7 H2 .
http://www.strw.leidenuniv.nl/∼ewine/photo
147
log N(H2 )
(cm−2 )
0
0
19
20
21
22
23
1.000
8.176(-1)
7.223(-1)
3.260(-1)
1.108(-2)
3.938(-7)
0
19
20
21
22
23
1.000
8.459(-1)
7.337(-1)
3.335(-1)
1.193(-2)
3.959(-7)
0
19
20
21
22
23
1.000
8.571(-1)
7.554(-1)
3.559(-1)
1.214(-2)
4.251(-7)
log 12 CO (cm−2 )
13
14
15
16
12
CO: unattenuated rate k0,i = 2.592 × 10−10
8.080(-1) 5.250(-1) 2.434(-1) 5.467(-2)
6.347(-1) 3.891(-1) 1.787(-1) 4.297(-2)
5.624(-1) 3.434(-1) 1.540(-1) 3.515(-2)
2.810(-1) 1.953(-1) 8.726(-2) 1.907(-2)
1.081(-2) 9.033(-3) 4.441(-3) 1.102(-3)
3.938(-7) 3.936(-7) 3.923(-7) 3.901(-7)
C17 O: unattenuated rate k0,i = 2.607 × 10−10
9.823(-1) 8.911(-1) 6.149(-1) 3.924(-1)
8.298(-1) 7.490(-1) 5.009(-1) 3.196(-1)
7.195(-1) 6.481(-1) 4.306(-1) 2.741(-1)
3.290(-1) 3.039(-1) 2.293(-1) 1.685(-1)
1.191(-2) 1.172(-2) 1.095(-2) 9.395(-3)
3.959(-7) 3.959(-7) 3.959(-7) 3.958(-7)
C18 O: unattenuated rate k0,i = 2.392 × 10−10
9.974(-1) 9.777(-1) 8.519(-1) 5.060(-1)
8.547(-1) 8.368(-1) 7.219(-1) 4.095(-1)
7.532(-1) 7.371(-1) 6.336(-1) 3.572(-1)
3.549(-1) 3.477(-1) 3.035(-1) 1.948(-1)
1.212(-2) 1.199(-2) 1.105(-2) 8.233(-3)
4.251(-7) 4.251(-7) 4.251(-7) 4.249(-7)
17
s−1
1.362(-2)
1.152(-2)
9.231(-3)
4.768(-3)
2.644(-4)
3.893(-7)
s−1
2.169(-1)
1.850(-1)
1.556(-1)
9.464(-2)
5.644(-3)
3.954(-7)
s−1
1.959(-1)
1.581(-1)
1.372(-1)
7.701(-2)
3.324(-3)
4.233(-7)
18
19
3.378(-3)
2.922(-3)
2.388(-3)
1.150(-3)
7.329(-5)
3.890(-7)
5.240(-4)
4.662(-4)
3.899(-4)
1.941(-4)
1.437(-5)
3.875(-7)
4.167(-2)
3.509(-2)
2.645(-2)
1.460(-2)
1.183(-3)
3.924(-7)
2.150(-3)
1.984(-3)
1.411(-3)
6.823(-4)
2.835(-5)
3.873(-7)
2.764(-2)
2.224(-2)
1.889(-2)
1.071(-2)
6.148(-4)
4.180(-7)
1.742(-3)
1.618(-3)
1.383(-3)
6.863(-4)
3.225(-5)
4.142(-7)
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
148
Table 5.5 – Two-dimensional shielding functions Θ[N(12 CO), N(H2 )].a
Table 5.5 – continued.
0
0
19
20
21
22
23
1.000
8.447(-1)
7.415(-1)
3.546(-1)
1.180(-2)
2.385(-7)
0
19
20
21
22
23
1.000
8.540(-1)
7.405(-1)
3.502(-1)
1.279(-2)
2.370(-7)
0
19
20
21
22
23
1.000
8.744(-1)
7.572(-1)
3.546(-1)
1.561(-2)
2.490(-7)
log 12 CO (cm−2 )
14
15
16
17
13
−10 −1
CO: unattenuated rate k0,i = 2.595 × 10 s
9.824(-1) 9.019(-1) 6.462(-1) 3.547(-1) 9.907(-2)
8.276(-1) 7.502(-1) 5.113(-1) 2.745(-1) 7.652(-2)
7.266(-1) 6.581(-1) 4.451(-1) 2.360(-1) 6.574(-2)
3.502(-1) 3.270(-1) 2.452(-1) 1.398(-1) 3.750(-2)
1.177(-2) 1.153(-2) 1.023(-2) 6.728(-3) 1.955(-3)
2.385(-7) 2.385(-7) 2.384(-7) 2.379(-7) 2.348(-7)
13 17
C O: unattenuated rate k0,i = 2.598 × 10−10 s−1
9.979(-1) 9.820(-1) 8.832(-1) 5.942(-1) 3.177(-1)
8.520(-1) 8.374(-1) 7.469(-1) 4.901(-1) 2.677(-1)
7.387(-1) 7.254(-1) 6.439(-1) 4.198(-1) 2.333(-1)
3.494(-1) 3.434(-1) 3.076(-1) 2.214(-1) 1.386(-1)
1.278(-2) 1.267(-2) 1.198(-2) 1.045(-2) 7.743(-3)
2.370(-7) 2.370(-7) 2.370(-7) 2.369(-7) 2.368(-7)
13 18
C O: unattenuated rate k0,i = 2.503 × 10−10 s−1
9.988(-1) 9.900(-1) 9.329(-1) 7.253(-1) 3.856(-1)
8.734(-1) 8.656(-1) 8.164(-1) 6.403(-1) 3.441(-1)
7.562(-1) 7.492(-1) 7.047(-1) 5.518(-1) 3.006(-1)
3.542(-1) 3.506(-1) 3.283(-1) 2.638(-1) 1.666(-1)
1.560(-2) 1.550(-2) 1.475(-2) 1.235(-2) 7.850(-3)
2.490(-7) 2.490(-7) 2.489(-7) 2.487(-7) 2.482(-7)
13
18
19
1.131(-2)
8.635(-3)
7.187(-3)
3.973(-3)
2.665(-4)
2.310(-7)
7.591(-4)
6.747(-4)
5.429(-4)
2.703(-4)
1.471(-5)
2.292(-7)
1.523(-1)
1.302(-1)
1.142(-1)
6.941(-2)
4.088(-3)
2.359(-7)
3.885(-2)
3.135(-2)
2.607(-2)
1.195(-2)
4.581(-4)
2.312(-7)
1.524(-1)
1.347(-1)
1.185(-1)
6.887(-2)
3.416(-3)
2.471(-7)
2.664(-2)
2.491(-2)
2.224(-2)
1.149(-2)
5.290(-4)
2.421(-7)
These shielding functions were computed for the Draine (1978) radiation field (χ = 1) and the following set of parameters: b(CO) = 0.3 km s−1 , b(H2 ) = 3.0 km
s−1 and b(H) = 5.0 km s−1 ; T ex (CO) = 5 K and T ex (H2 ) = 51.5 K; N(H) = 5×1020 cm−2 ; N(12 CO)/N(13 CO) = 69, N(12 CO)/N(C18 O) = N(13 CO)/N(13 C18 O) =
557 and N(C18 O)/N(C17 O) = N(13 C18 O)/N(13 C17 O) = 3.6. Self-shielding is mostly negligible for the heavier isotopologues, so all shielding functions are
expressed as a function of the 12 CO column density. Continuum attenuation by dust is not included in this table (see Eq. (5.2)).
149
5.5 Shielding function approximations
a
log N(H2 )
(cm−2 )
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
Figure 5.6 – Illustration of the effect of limited integration ranges. Solid black curve: relative
intensity of the radiation field at N(CO) = 1018 and N(H2 ) = 1022 cm−2 for the parameters of Table
5.5, computed with a full integration of all lines over all wavelengths. Grey curve: the same, but
computed with the binned integration method of vDB88. Dotted curve: attenuation due to the C3
R(0) line of H2 , centred on 946.42 Å. This line was not included by vDB88 when computing the
shielding in the 938.9–943.1 Å bin, thus overestimating the photodissociation rate in the W2 band
at 941.1 Å.
8 of Visser et al. (2009) when the ratio is closer to 35 than to 69, and those from Table 5.5
(this chapter) otherwise.
The rate from our approximate method is within 10% of the “real” rate in 98.3% of all
points (Fig. 5.7). In no cases is the difference between the approximate rates and the full
model more than 40%. Perhaps even more important than the absolute photodissociation
rates are the ratios, Q, between the rates of 12 CO and the other five isotopologues x Cy O:
Qi =
k( x Cy O)
.
k(12 CO)
(5.3)
The shielding functions from Table 5.5 and Visser et al. (2009) together reproduce the
ratios from the full method to the same accuracy as the absolute rates. At 98.4% of all
points in the grid of translucent clouds, the ratios are off by less than 10% (Fig. 5.7).
Similar scores can be obtained for models of PDRs or other environments if one uses
shielding functions computed for the right combination of parameters. Otherwise, the
accuracy goes down. For example, the shielding functions from Table 5.5 can easily give
rates off by a factor of two when applied to a high-density, high-temperature PDR.
150
5.6 Chemistry of CO: astrophysical implications
Figure 5.7 – Histogram of the absolute relative difference between the approximate method (see
text for details) and the full computation for the absolute photodissociation rates (k, black line) and
the isotopologue rate ratios (Q, grey line) in a grid of translucent cloud models. The data for all
six isotopologues are taken together, but the Q values for 12 CO (which are unity regardless of the
method) are omitted.
5.6 Chemistry of CO: astrophysical implications
Photodissociation is an important destruction mechanism for CO in many environments.
In this section, we couple the photodissociation model to a small chemical network in
order to explore abundances and column densities. Specifically, we model the CO chemistry in translucent clouds, PDRs and circumstellar disks, and we compare our results
to observations of such objects. The photodissociation rates are computed with our full
model throughout this section.
5.6.1 Translucent clouds
5.6.1.1 Model setup
Translucent clouds, with visual extinctions between 1 and 5 mag, form an excellent test
case for our CO photodissociation model. Two recent studies of several dozen lines of
sight through diffuse (AV < 1 mag) and translucent clouds provide a set of CO and H2
151
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
column densities for comparison (Sonnentrucker et al. 2007, Sheffer et al. 2008). These
surveys show a clear correlation between N(H2 ) and N(12 CO), with distinctly different
slopes for H2 columns of less and more than 2.5 × 1020 cm−2 . Sheffer et al. attributed
this break to a change in the formation mechanism of CO. Their models show that the
two-step conversion from C+ to CH+ and CO+ ,
C+ + H2 → CH+ + H ,
(5.4)
CH+ + O → CO + + H ,
(5.5)
followed by reaction with H (forming CO directly) or H2 (forming HCO+ , which then recombines with an electron to give CO) is the dominant pathway at low column densities.
However, the highly endothermic Reaction (5.4) is not fast enough at gas kinetic temperatures typical for these environments to explain the observed abundances of CH+ and CO.
Suprathermal chemistry has been suggested as a solution to this problem. Sheffer et al.
followed the approach of Federman et al. (1996), who argued that Alfvén waves entering
the cloud from the outside result in non-thermal motions between ions and neutrals. Other
mechanisms have been suggested by Joulain et al. (1998) and Pety & Falgarone (2000).
The effect of the Alfvén waves can be incorporated into a chemical model by replacing
the kinetic temperature in the rate equation for Reaction (5.4) and all other ion-neutral
reactions by an effective temperature:
T eff = T gas +
µv2A
.
3kB
(5.6)
Here, kB is the Boltzmann constant, µ is the reduced mass of the reactants and vA is the
Alfvén speed. The Alfvén waves reach a depth of a few 1020 cm−2 of H2 , corresponding
to an AV of a few tenths of a magnitude, beyond which suprathermal chemistry ceases
to be important. CO can therefore no longer be formed efficiently through Reactions
(5.4) and (5.5), and the reaction between C+ and OH (producing CO either directly or
via a CO+ intermediate) takes over as the key route to CO. The identification of these
two different chemistry regimes supports the conclusion of Zsargó & Federman (2003)
that suprathermal chemistry is required to explain observed CO abundances in diffuse
environments. Suprathermal chemistry also drives up the HCO+ abundance, confirming
the conclusion of Liszt & Lucas (1994) and Liszt (2007) that HCO+ is the dominant
precursor to CO in diffuse clouds.
We present here a grid of translucent cloud models to see how well the new photodissociation results match the observations. We set the Alfvén speed to 3.3 km s−1 for
N(H2 ) < 4 × 1020 cm−2 and to zero for larger column densities (Sheffer et al. 2008). The
grid comprises densities (nH = n(H) + 2n(H2 )) of 100, 300, 500, 700 and 1000 cm−3 , gas
temperatures of 15, 30, 50 and 100 K, and relative UV intensities (χ) of 1, 3 and 10. The
dust temperature is assumed to stay low for all models (∼ 15 K), so the H2 formation rate
does not change. The ionisation rate of H2 due to cosmic rays is set to a constant value
of 1.3 × 10−17 s−1 . Attenuation by 0.1 µm dust grains (Sect. 5.3.4) is taken into account.
The Doppler widths and level populations are as described in Sect. 5.3.1, with CO and H2
152
5.6 Chemistry of CO: astrophysical implications
Table 5.6 – Elemental abundances.
Element
He
12
C
13
C
Abundance relative to nH
1.00 × 10−1
1.40 × 10−4
2.03 × 10−6
Element
16
O
17
O
18
O
Abundance relative to nH
3.10 × 10−4
1.55 × 10−7
5.57 × 10−7
excitation temperatures of 5 and 51.5 K. Taking other b or T ex values plausible for these
environments does not alter our results significantly. All models are run to an AV of 5
mag; results are also presented for a range of smaller extinctions.
The models require a chemical network to compute the abundances at each depth
step. Since we are only interested in CO, the number of relevant species and reactions
is limited. We adopt the network from a recent PDR benchmark study (Röllig et al.
2007), which includes only 31 species consisting of H, He, C and O. We duplicate all
C- and O-containing species and reactions for 13 C, 17 O and 18 O. Freeze-out and thermal
evaporation are added for all neutral species, but no grain-surface reactions are included
other than H2 formation according to Black & van Dishoeck (1987). We add ion-molecule
exchange reactions such as
12
CO +
13
C+ ⇄
H12 CO+ +
13
CO ⇄
12
C+ +
12
13
CO + 35 K ,
CO + H13 CO+ + 9 K ,
(5.7)
(5.8)
which can enhance the abundances of the heavy isotopologues of CO and HCO+ (Watson
et al. 1976, Smith & Adams 1980, Langer et al. 1984). The temperature dependence of
the rate of these two reactions was fitted by Liszt (2007); the alternative equations from
Woods & Willacy (2009) give the same results. The effective temperature from Eq. (5.6) is
used instead of the kinetic temperature for all ion-neutral reactions, including Reactions
(5.7) and (5.8). Altogether, the network contains 118 species and 1723 reactions. We
adopt the elemental abundances of Cardelli et al. (1996) and isotope ratios appropriate
for the local ISM ([12 C]/[13 C] = 69, [16 O]/[18 O] = 557 and [18 O]/[17 O] = 3.6; Wilson
1999); the complete list of elemental abundances is given in Table 5.6. Chemical steady
state is reached at all depths after ∼1 Myr, regardless of whether the gas starts in atomic
or molecular form.
5.6.1.2
12
CO
The left panel of Fig. 5.8 shows N(12 CO) versus N(H2 ) for all depth steps in our grid.
These and other column densities are also listed for a set of selected AV values in the
online appendix to the paper that this chapter is based on (Visser et al. 2009). The scatter
in the data is due to the different physical parameters. For a given N(H2 ), N(12 CO) is
about two times larger at T gas = 100 K than at 15 K. The formation rate of H2 increases
with temperature; through the chain of reactions starting with Reaction (5.4), that results
in a larger CO abundance and column density. Up to N(H2 ) = 1021 cm−2 , increasing the
153
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
Figure 5.8 – Column densities of 12 CO versus those of H2 . Left: data for all points with AV ≤ 5 mag
from our grid of translucent cloud models. The vertical grey band indicates the range over which
photodissociation ceases to be the dominant destruction mechanism of 12 CO; the dotted line is the
median. Likewise, the horizontal band indicates the range over which 12 CO becomes self-shielding.
Right: 100-point means of our model results (black squares), fitted with two straight lines. Also
shown are the fits to the observations by Sheffer et al. (2008, grey lines) and the results from the
translucent cloud models of vDB88 for χ = 0.5, 1 and 10 (dashed lines, left to right).
gas density by a factor of ten increases N(12 CO)/N(H2 ) also by about a factor of ten. This
is due to the photodissociation rate being mostly independent of density, while the rates
of the two-body reactions forming CO are not. Photodissociation ceases to be the main
destruction mechanism for CO deeper into the cloud, so increasing nH has a smaller effect
there. Increasing the UV intensity from χ = 1 to 10 has the simple effect of decreasing
N(12 CO)/N(H2 ) roughly tenfold for N(H2 ) < 1021 cm−2 . For larger depths, changing χ
only has a small effect. These dependencies on the physical parameters are consistent
with the observations of Sheffer et al. (2008).
The full set of points already gives some indication that the data are not well represented by a global single-slope correlation. The right panel of Fig. 5.8 shows the 100-point
means of the full set, revealing the same two distinct regimes as found by Sheffer et al.
(2008). Taking the uncertainties in both the observations and the models into account, the
two sets of power-law fits (indicated by the black and grey lines) are identical. The break
154
5.6 Chemistry of CO: astrophysical implications
between the two slopes occurs at N(H2 ) = 2.5 × 1020 cm−2 and is due to the switch from
suprathermal to normal chemistry. It is clearly unrelated to the switch from UV photons to
He+ or H+3 as the main destroyer of 12 CO, which does not occur until N(H2 ) = 1–2 × 1021
cm−2 . When we extend the grid to higher AV (not plotted), we quickly reach the point
where the 12 CO abundance equals the elemental 12 C abundance and the plot of N(12 CO)
versus N(H2 ) continues with a slope of unity. The model results from vDB88 (dashed
lines, corresponding to χ = 0.5, 1 and 10) do not extend to low enough N(H2 ) to show the
suprathermal regime, but they do show at the upper end the transition to the regime where
all gas-phase carbon is in CO.
5.6.1.3
13
CO
With our model able to reproduce the observational relationship between H2 and 12 CO, the
next step is to look at the heavier isotopologues. Sonnentrucker et al. (2007), Burgh et al.
(2007) and Sheffer et al. (2007) together presented a sample of 29 diffuse and translucent
sources with derived column densities for both 12 CO and 13 CO. These data are plotted in
the left panel of Fig. 5.9 (grey crosses) as N(13 CO) versus N(H2 ). The 100-point means
from our model grid (black squares) agree with the observations to within a factor of three.
The model results are again best fitted with two power laws, but the 13 CO observations do
not extend to low enough N(H2 ) to confirm this.
The observations show considerable scatter in the N(12 CO)/N(13 CO) ratio, with some
sources deviating by a factor of two either way from the local elemental 12 C-13 C ratio of
69±6 (Wilson 1999). It has long been assumed that the enhanced ratios are due to isotopeselective photodissociation and that the reduced ratios are due to isotope-exchange reactions. However, no answer has been found so far to the question of why one process
dominates in some sources, and the other process in other sources (Langer et al. 1980,
McCutcheon et al. 1980, Sheffer et al. 1992, Lambert et al. 1994, Federman et al. 2003,
Sonnentrucker et al. 2007, Sheffer et al. 2007). Chemical models consistently show that
isotope exchange is more efficient than selective photodissociation for observed cloud
temperatures of up to ∼60 K. The models can easily reproduce N(12 CO)/N(13 CO) ratios
of less than 69, but they leave the ratios of more than 69 unexplained.
Our grid of models suggests that the answer lies in suprathermal chemistry. The
right panel of Fig. 5.9 shows a plot of N(12 CO)/N(13 CO) versus N(H2 ) for the observations (grey) and for all depth steps at AV ≤ 5 from our models (black plus signs). Also
shown are the 100-point means from the model, offset by a factor of three for clarity. The
N(12 CO)/N(13 CO) ratio initially increases from 69 to a mean of 151 at N(H2 ) = 4 × 1020
cm−2 , then decreases to a mean of 51 at 1 × 1021 cm−2 , and gradually increases again for
larger depths. The turnover from the initial rise to the rapid drop is due to the transition
from suprathermal to normal chemistry.
At an Alfvén speed of vA = 3.3 km s−1 , the effective temperature for Reaction (5.7) is
about 4000 K, so the forward and backward reactions are equally fast. Photodissociation
is therefore the only active fractionation process and it enhances the amount of 12 CO
relative to 13 CO. As we move beyond the depth to which the Alfvén waves reach (N(H2 ) =
4 × 1020 cm−2 in our model), suprathermal chemistry is brought to a halt. The forward
155
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
Figure 5.9 – Column densities of 13 CO (left) and column density ratios of 12 CO to 13 CO (right)
plotted against the column density of H2 . Grey crosses: observations of Sonnentrucker et al. (2007),
Burgh et al. (2007) and Sheffer et al. (2007). Black squares and straight lines: 100-point means
from our grid of translucent cloud models, offset by a factor of three in the right panel. Dotted line:
median depth at which 12 CO becomes self-shielding. Black plus signs: model data for all points
with AV ≤ 5 mag. Dashed lines: data from the translucent cloud models of vDB88 for χ = 0.5, 1
and 10 (left to right).
channel of Reaction (5.7) becomes faster than the backward channel and isotope exchange
now reverses the fractionation. When we reach a depth of N(H2 ) = 1–2 × 1021 cm−2 , the
13 +
C abundance drops too low to sustain further fractionation and the N(12 CO)/N(13 CO)
ratio gradually returns to the elemental ratio of 69.
The observations show a downward trend in the N(12 CO)/N(13 CO) ratio in the N(H2 )
range from 2×1020 to 2×1021 cm−2 . This supports a rapid switch from selective photodissociation to isotope-exchange reactions being the dominant fractionation mechanism. A
firm test of our model predictions requires observational data at both smaller and larger
N(H2 ) than are currently available.
5.6.1.4 Other isotopologues
Moving to the next two isotopologues, C17 O and C18 O, the number of observed column
densities goes down to three. Lambert et al. (1994) determined a ratio between N(12 CO)
156
5.6 Chemistry of CO: astrophysical implications
and N(C18 O) of 1550 ± 440 for the ζ Oph diffuse cloud, some three times larger than the
elemental 16 O-18 O ratio of 557 ± 30 (Wilson 1999). An even higher ratio of 3000 ± 600
was recorded by Sheffer et al. (2002) for the X Per translucent cloud, along with an
N(12 CO)/N(C17 O) ratio of 8700 ± 3600. The elemental 16 O-18 O ratio is about four times
smaller: 2000 ± 200.
The H2 column density towards X Per (HD 24534) is 8.5 × 1020 cm−2 (Sonnentrucker
et al. 2007, Sheffer et al. 2007). Our grid of models gives N(12 CO)/N(C18 O) = 680–
3880 (median 1080) and N(12 CO)/N(C17 O) = 2520–10 600 (median 3660) for that value,
consistent with the observations. Out of the three parameters that we varied, χ affects the
ratios most strongly. Both ratios decrease by a factor of two when χ is increased from 1
to 10. The onset of self-shielding of 12 CO, and with it the onset of isotopic fractionation,
occurs deeper into the cloud for a stronger UV field. Therefore, N(12 CO)/N(C18 O) and
N(12 CO)/N(C17 O) are smaller at any given depth.
In order to further explore the behaviour of C17 O and C18 O, we take a more detailed
look at our results for nH = 300 cm−3 , T gas = 50 K and χ = 1 (model C2α from Table 12
of Visser et al. 2009). This also allows for a comparison with 13 C17 O and 13 C18 O. Both
of these have been detected in interstellar clouds (Langer et al. 1980, Bensch et al. 2001),
but column densities have not yet been derived.
The photodissociation rates of CO and H2 for this combination of parameters are
plotted in Fig. 5.10 as a function of depth (z) into the cloud. The isotope-selective nature
of the photodissociation is clearly visible. At a depth of 4.0 pc (AV = 2.3 mag, N(H2 ) =
1.7 × 1021 cm−2 ), 12 CO photodissociates about 7 times slower than 13 CO, 15–18 times
slower than C17 O and C18 O, and 23–26 times slower than 13 C17 O and 13 C18 O. At depths
between 0.4 and 3.4 pc (AV = 0.3–2.0 mag), the photodissociation of C18 O proceeds
faster than that of C17 O. The lines of the latter lie closer to those of 12 CO and are therefore
more strongly shielded, as was the case for the diffuse ζ Oph cloud (Table 5.4). As the
CO column grows larger in our model cloud, C18 O becomes self-shielding and its rate
drops below that of C17 O around 3.4 pc.
The extent to which each isotopologue is fractionated can easily be seen from the
cumulative column density ratios when they are normalised to 12 CO and the isotope ratios:
Nz ( x Cy O) [12 C] [16 O]
,
Nz (12 CO) [ x C] [y O]
(5.9)
with [Z] the elemental abundance of isotope Z and
Z z
x y
Nz ( C O)(z) =
n( x Cy O)dz′
(5.10)
Ri (z) =
0
the cumulative column density. A plot of R(13 CO) as function of depth (Fig. 5.11) shows
the same trends as the model data in Fig. 5.9: the amount of 13 CO is reduced (R < 1) up to
a certain depth and is enhanced (R > 1) farther in by isotope-exchange reactions. We only
see R < 1 for C17 O and C18 O because 17 O+ and 18 O+ never become abundant enough to
convert C16 O in a reaction similar to Reaction (5.7). Photodissociation ceases to be the
dominant destruction mechanism for C17 O and C18 O around z = 4 pc (Fig. 5.10), so it
can no longer cause substantial fractionation and R gradually returns to unity.
157
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
Figure 5.10 – Photodissociation rates of H2 and the six CO isotopologues as a function of depth into
translucent cloud model C2α (Table 12 of Visser et al. 2009) with nH = 300 cm−3 , T gas = 50 K and
χ = 1. Attenuation by dust is included. The diamonds and corresponding markers on the bottom
axis indicate the depth at which He+ takes over from UV radiation as the dominant destroyer for
each isotopologue.
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5.6 Chemistry of CO: astrophysical implications
Figure 5.11 – Cumulative column densities of the CO isotopologues normalised to 12 CO and the
elemental isotopic abundances (Eq. (5.9)) as a function of depth into translucent cloud model C2α
(Table 12 of Visser et al. 2009).
The remaining two isotopologues, 13 C17 O and 13 C18 O, experience the fractionation
of C and 17 O/18 O simultaneously. Isotope exchange enhances their abundances relative
to C17 O and C18 O, in the same manner that it enhances the abundance of 13 CO relative
to 12 CO. However, isotope-selective photodissociation is a stronger effect for the two
heaviest isotopologues, so R remains less than unity. This result holds across our entire
grid of translucent cloud models.
13
5.6.2 Photon-dominated regions
Photon-dominated regions (PDRs) form another type of objects where photodissociation
is a key process in controlling the CO abundance. Their densities are higher (nH up to
∼106 cm−3 ) than those of diffuse and translucent clouds and they are exposed to stronger
UV fields (χ up to ∼106 ; Hollenbach & Tielens 1997). Observational efforts have mostly
gone into mapping the structure of PDRs, so CO column densities are tabulated only for
a small number of sources. For most of these the column density of neutral carbon has
also been determined. There are no direct measurements of H2 column densities; instead,
N(H2 ) is usually obtained from N(12 CO) using a typical abundance ratio between the two
species (e.g., Frerking et al. 1982).
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Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
Figure 5.12 – Column density of 12 CO versus that of 12 C on two different scales. Black: results
from our grid of PDR models with nH = 103 cm−3 (plus signs) and 105 cm−3 (diamonds). The line
traces the results for different depth steps at nH = 105 cm−3 , T gas = 50 K and χ = 104 . Grey squares:
observations of Beuther et al. (2000), Kamegai et al. (2003), Kramer et al. (2008) and Sun et al.
(2008).
We present here the results from our combined photodissociation and chemistry model
for a grid of physical parameters appropriate for PDRs: T gas = 50, 80 and 100 K, nH =
103 , 104 , 105 and 106 cm−3 , and χ = 103 , 104 and 105 . The excitation temperatures of
CO and H2 are set equal to T gas. Where necessary, we expand the number of rotational
levels of CO beyond the default limit of J ′′ =7, and likewise for H2 (see Sect. 5.4.4). UV
pumping increases the population in the v′′ =0, J ′′ >3 and v′′ >0 levels of H2 . However,
this does not affect the CO photodissociation rates at these temperatures, so the level
population of H2 is simply taken as fully thermal. The models are run to an AV of 30 mag
and are otherwise unchanged from the previous section.
Figure 5.12 shows the relationship between the column densities of 12 CO and 12 C for
a large series of depths steps from our grid. The data for nH = 104 and 106 cm−3 are
omitted for reasons of clarity. Overplotted are column densities determined towards a
number of positions in different PDRs (Beuther et al. 2000, Kamegai et al. 2003, Kramer
et al. 2008, Sun et al. 2008). Figure 5a of Mookerjea et al. (2006) contains additional data
for several dozen positions in the PDRs in Cepheus B. In none of these cases was the
12
CO column density measured directly; instead, it was derived from the 13 CO or C18 O
column density using standard abundance ratios. The spread in both the model predictions
and the observational data is large. The column of 12 C for each individual model is nearly
constant for N(12 CO) > 3 × 1017 cm−2 , as shown by the solid line connecting the various
depth steps for nH = 105 cm−3 , T gas = 50 K and χ = 104 . The observations seem
160
5.6 Chemistry of CO: astrophysical implications
Figure 5.13 – Column densities of C18 O (left) and 13 CO (right) versus that of 12 C. Symbols are as
in Fig. 5.12. See text for references to the observations.
to correspond more to the lower-density end of the model grid (the plus signs), but the
sample is too small and is too scattered to be conclusive.
The onset of shielding is seen to occur around N(12 CO) = 1015 cm−2 in the models. From that point onwards, the low-density models gain more atomic carbon than do
the high-density models. For a given column of 12 CO, the photodissociation rates (or, in
other words, the 12 C formation rates) are roughly independent of density, but the destruction of 12 C occurs faster in higher densities. The same argument explained the density
dependence of the N(12 CO)/N(H2 ) ratio in the translucent clouds.
The two isotopologues for which column densities have been determined directly are
C18 O and 13 CO (Loren 1989, Gerin et al. 1998, Plume et al. 1999, Beuther et al. 2000,
Schneider et al. 2003, Mookerjea et al. 2006, Kramer et al. 2008, Sun et al. 2008). Again,
both the observations and the models show a large spread in column densities (Fig. 5.13).
It seems the observations do not trace the high-extinction (AV > 10 mag), high-density
(nH > 105 cm−3 ) material shown in the upper left corner of each panel, but a more detailed study is required to draw any firm conclusions on this point. N(12 CO), N(C18 O)
and N(13 CO) are mutually well correlated in both the observations and the models, but
significant deviations can easily occur at specific depths due to the different photodissociation rates. When using standard abundance ratios to derive the column density of 12 CO
from that of another isotopologue, as was done for all cases in Fig. 5.12, the result will
generally be accurate to at best a factor of two.
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Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
5.6.3 Circumstellar disks
5.6.3.1 Model setup
Low-mass stars like our own Sun are formed through the gravitational collapse of a cold
molecular cloud. As the young star grows at the centre, it gathers part of the remaining
cloud material into a circumstellar disk. It is inside this disk that planets are formed, so
its chemical composition is of great interest. The physical structure of the disk may be
simplified as consisting of a cold, dense region near the midplane, covered by a warmer
region of lower density. This surface layer intercepts the star’s radiation and shows many
similarities to common PDRs.
We adopt the standard disk model of D’Alessio et al. (1999), whose chemistry has
been studied extensively (Aikawa et al. 2002, van Zadelhoff et al. 2003, Jonkheid et al.
2004). The star, a T Tauri type, has a mass of 0.5 M⊙ , a radius of 2 R⊙ and an effective
temperature of 4000 K. It is surrounded by a disk of mass 0.07 M⊙ and outer radius
400 AU. The disk is accreting onto the star at a constant rate of 10−8 M⊙ yr−1 and its
viscosity is characterised by α = 0.01. We focus on a vertical slice through the disk at a
radius of 105 AU, where the surface is located at a height of z = 120 AU. This slice is
irradiated from the top by a radiation field with an effective UV intensity of 516 times that
of the interstellar radiation field. Following van Zadelhoff et al., we adopt two spectral
shapes: the Draine field and a 4000 K blackbody spectrum. The latter is very weak in the
wavelength range where CO is photodissociated. We adopt again the chemical network
from Sect. 5.6.1.1, and we set the gas temperature equal to the dust temperature from the
D’Alessio et al. model. Given the high densities (2 × 104 cm−3 at the surface and more
than 109 cm−3 at the midplane), we also use T gas for T ex (CO) and T ex (H2 ). As in the PDR
models, UV pumping of H2 does not affect the CO photodissociation rates, so the H2 level
populations are taken as fully thermal.
In reality, disks are not irradiated by the Draine field or a pure blackbody. For example,
T Tauri stars typically show UV emission in excess of these simple model spectra. Many
of them also emit X-rays from accretion hotspots. As shown by van Zadelhoff et al.
(2003), the photochemistry of the disk can be sensitive to the details of the radiation field.
Moreover, the excess emission can be concentrated in a few specific lines such as H 
Lyman-α. CO cannot be dissociated by Ly-α, but it can be by the H  Ly-γ line at 972.54
Å through overlap with the W0 band at 972.70 Å (No. 25 from Table 5.1). A detailed
treatment of these effects is beyond the scope of the current work.
5.6.3.2 Isotopologue ratios
The depth dependence of the dissociation rates in the disk, illuminated by the Draine field,
is qualitatively the same as in the clouds and PDRs from Sects. 5.6.1 and 5.6.2. Several
quantitative differences arise because of the different UV flux and the higher densities.
The profiles of the normalised column density ratios, R (Eq. (5.9)), do not show any enhancement in 13 CO (Fig. 5.14). Moving down from the surface, suprathermal chemistry
initially prevents the formation of 13 CO from 12 CO and 13 C+ . The Alfvén waves penetrate
to N(H2 ) = 4 × 1020 cm−2 or a height of about 50 AU. The absence of suprathermal chem162
5.6 Chemistry of CO: astrophysical implications
Figure 5.14 – Cumulative column densities of the CO isotopologues normalised to 12 CO and the
elemental isotopic abundances (Eq. (5.9)) as a function of height at a radius of 105 AU in our disk
model with the Draine field and dust grains of 0.1 µm (left) or 1 µm (right). CO freezes out below
24 AU. The diamond symbols indicate where H+3 takes over from UV photons as the main destroyer
of each isotopologue.
istry below that point allows R(13 CO) to increase again. However, the 13 C+ abundance
decreases at the same time, and R(13 CO) never gets above unity. This is consistent with
the N(12 CO)/N(13 CO) ratios of more than 69 determined for the disks around HL Tau,
AB Aur and VV CrA (Brittain et al. 2005, Schreyer et al. 2008, Smith et al. 2009).
In terms of local abundances, our results show a trend opposite to the model by
Woods & Willacy (2009). Moving up vertically from the midplane at a given radius,
their n(12 CO)/n(13 CO) abundance ratio initially remains nearly constant, then drops to a
lower value, and finally shows a shallow rise for the remainder of the column. We also
find a constant ratio at first, but then we get a rise (equivalent to R(13 CO) < 1) and a
gradual decrease back to the elemental ratio of 69. The difference is due to the absence
of suprathermal chemistry in the Woods & Willacy model. Suprathermal chemistry effectively shuts down the isotope-exchange reactions, leaving photodissociation (which
increases n(12 CO)/n(13 CO)) as the only fractionation mechanism. Additional observations of 12 CO and 13 CO in circumstellar disks are required to ascertain whether isotope
exchange or selective photodissociation is the key process in controlling the ratio of the
abundances of these two species.
For C17 O and C18 O, R is ∼1 from the disk surface down to 70 AU. It then gradually
drops to ∼0.1 due to their dissociation being faster than that of 12 CO. Shielding of C17 O
and C18 O sets in around 32 AU, so photodissociation no longer causes isotope fractionation and R increases again. However, it does not reach a value of unity. Below 24 AU,
163
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
the temperature is low enough that most of the CO freezes out onto the dust grains. The
tiny amount remaining in the gas phase does not add to the column density, so R is effectively frozen at 0.75 and 0.73 (Fig. 5.14). The corresponding column density ratios are
N(12 CO)/N(C17 O) = 2700 and N(12 CO)/N(C18 O) = 760, consistent with the degree of
fractionation determined for the disk around the T Tauri star VV CrA (Smith et al. 2009).
5.6.3.3 Model variations
If we illuminate the disk by a 4000 K blackbody (spectrum C from van Zadelhoff et al.
2003), it does not receive enough photons in the 912–1118 Å range to cause substantial
photodissociation of CO. Instead, the non–isotope-selective reactions with He+ and H+3
are the main destruction mechanisms even at the top of our vertical slice at R = 105 AU.
The normalised column density ratios therefore remain unity at all heights. Adding the
interstellar UV field at χ = 1 to the 4000 K blackbody spectrum allows CO to be photodissociated again in the surface layers. The isotope fractionation returns, but only partially, because the UV field is still weaker than in the original case. Freeze-out now locks
R(C17 O) and R(C18 O) at 0.86 and 0.85 instead of ∼0.75. Clearly, some UV component is
required in the irradiating spectrum to get isotope fractionation through the photodissociation of CO, but it does not have to be a very strong component: the interstellar UV field
already has a significant effect in this model.
Another parameter to be varied is the grain size. The disk model so far contained
small dust grains, with an average size of 0.1 µm. Grain growth is an inevitable step
towards planet formation and is indeed known to occur in disks (van Boekel et al. 2003,
Przygodda et al. 2003, Lommen et al. 2007). Larger grains cause less extinction in the
UV (Shen et al. 2004), so CO can be photodissociated deeper into the disk. Grain growth
also affects the H2 formation rate: the rate is inversely proportional to approximately the
square of the grain radius (Jonkheid et al. 2006). The right panel of Fig. 5.14 shows the
normalised vertical column density ratios at a radius of 105 AU for an average grain size
of 1 µm. The onset of fractionation occurs at a lower altitude than in the 0.1 µm case:
55 AU instead of 75 AU. The slower formation rate of H2 has a direct influence on the
formation of CO through the pathway starting with Reaction (5.4). There is less H2 and
CO in the disk’s surface layers when we increase the grain size, so we have to go down
deeper before 12 CO becomes self-shielding and isotope fractionation sets in. At the same
time, the weaker continuum extinction allows the UV radiation to penetrate deeper, so the
isotope-selective photodissociation goes on to lower altitudes. The spike in R(13 CO) to
1.4 at 30 AU is due to a narrow zone where the temperature is relatively low (35 K), the
Alfvén waves do not reach, and the 13 C+ abundance is high enough to convert 12 CO into
13
CO. The model with 0.1 µm grains also has a narrow zone with a high 13 C+ abundance
and no suprathermal chemistry, but the kinetic temperature in this zone is 50 K. There is
still some conversion of 12 CO into 13 CO at that temperature, but not enough to show as a
spike in the normalised column density ratio.
The bigger dust grains also affect the fractionation of C17 O and C18 O. The point at
which photodissociation ceases to be the main destruction mechanism now nearly coincides with the point where CO freezes out (Fig. 5.14). Both species have R ≈ 0.15 when
164
5.6 Chemistry of CO: astrophysical implications
this happens, so the ratios are locked at that value. In the small-dust model, the ratios
were locked at ∼0.75. Observations of face-on disks in principle probe CO all the way
to the midplane, so our model predicts that grain growth from 0.1 to 1 µm can increase
the observed fractionation of C17 O and C18 O by a factor of five. The fractionation in the
local abundances, which is relevant for incorporating material into comets, increases by
the same amount from 0.1 to 1 µm grains.
All the results so far in this section are for a T Tauri disk. Herbig Ae/Be stars are more
massive and more luminous and tend to have warmer disks, where CO may not be frozen
out at all. That would prevent R from being locked as it is in Fig. 5.14; instead, it would
rapidly return to unity for all isotopologues. Observations probing the midplane of such
a disk would not find any fractionation in 17 O and 18 O relative to 16 O.
5.6.3.4 Implications for meteorites
Regardless of the grain size and the absence or presence of freeze-out, the model shows
17
O and 18 O to be nearly equally fractionated: R has very similar values for C17 O and
18
C O. The equal photodissociation rates implied by this result are partly due to our choice
of molecular parameters. Spectroscopic data on C17 O are still very scarce, so we assume
many of its oscillator strengths and predissociation probabilities to have the same value
as for C18 O (Sects. 5.2.3 and 5.2.4). Still, two important differences remain. First, the
elemental abundance of 18 O is higher than that of 17 O, resulting in some self-shielding for
C18 O (Fig. 5.1). Second, the shifts of the C17 O lines with respect to 12 CO are generally
somewhat smaller than those of the C18 O lines, so C17 O is more strongly shielded by
12
CO. Both effects are minor, however, and the photodissociation rates end up being
nearly the same.
According to the experiments of Chakraborty et al. (2008), the photodissociation rate
due to the E0 and E1 bands may be up to ∼40% higher for C17 O than for C18 O. If we
mimic this by artificially increasing the E0 and E1 oscillator strengths of C17 O by 40%,
the overall change in R(C17 O) is less than 10% at any depth. When the two oscillator
strengths are doubled from their standard value, R(C17 O) changes by at most 20%. In
both cases, the change in R(C17 O) is largest at the top of the disk and gradually decreases
for lower altitudes; it is less than 1% below 50 AU.
Observational evidence for equal fractionation of 17 O and 18 O in disks was recently
obtained by Smith et al. (2009) for the T Tauri star VV CrA. Their analysis points to
isotope-selective photodissociation of CO as the most likely explanation, although they
cautioned that more observations are needed to confirm that conclusion. Meanwhile, their
observations are consistent with our model results.
A long-standing puzzle in our solar system involves the equal fractionation of 17 O and
18
O in the most refractory phases of meteorites (Clayton et al. 1973). Current evidence
points at isotope-selective photodissociation of CO in the solar nebula playing a key role
(Clayton 2002, Lyons & Young 2005, Lee et al. 2008). It is generally accepted that the
resulting atomic oxygen eventually makes it into the refractory material to produce the observed isotope ratios, but it remains unknown how this actually happens. Any additional
mass-independent fractionation during that process must be limited, because photodisso165
Chapter 5 – The photodissociation and chemistry of interstellar CO isotopologues
ciation of CO already reproduces the observed isotope abundance ratios (Young 2007).
One important requirement here is to have C17 O and C18 O photodissociate at the same
rate, an assumption that had gone untested up to now. Our model confirms they do so in
many different environments – including diffuse and translucent clouds, PDRs, and circumstellar disks – even when allowing for the isotope effects suggested by Chakraborty
et al. (2008).
Equal (or nearly equal) 17 O and 18 O fractionation also occurs for other oxygencontaining molecules in the model, such as water. The equal fractionation is a clear
sign of CO photodissociation playing a key role in the fractionation process. Other fractionation processes, such as isotope-exchange reactions, diffusion, freeze-out and evaporation, depend on the vibrational energies of the isotopologues and are therefore massdependent. Specifically, such processes would lead to 17 O being 0.52 times as fractionated as 18 O (Matsuhisa et al. 1978). If mass-independent fractionation processes other
than isotope-selective photodissociation play a role in circumstellar disks, they have not
been discovered so far (Clayton 2002, Yurimoto & Kuramoto 2004, Lyons & Young
2005). Recently, Kimura et al. (2007) did report on a set of chemical experiments yielding
mass-independently fractionated silicates, but they were unable to identify the underlying reaction mechanism. Hence, it remains unknown if this would be relevant for disks.
Self-shielding in O2 (Thiemens & Heidenreich 1983) is unimportant because O2 never
becomes abundant enough to become a significant source of atomic O.
5.7 Conclusions
This chapter presents an updated model, based on the method of van Dishoeck & Black
(1988), for the photodissociation of carbon monoxide (CO) and its heavier isotopologues.
It contains recent spectroscopic data from the literature and produces a photodissociation
rate of 2.6 × 10−10 s−1 in the interstellar medium, 30% higher than currently adopted
values. Our model is the first to include C17 O and 13 C17 O and we apply it to a broader
range of model parameters than has been done before. The main results are as follows:
• Self-shielding is very important for 12 CO and somewhat so for C18 O and 13 CO. The
rare isotopologues 13 C17 O and 13 C18 O have the highest photodissociation rate at most
depths into a cloud or other object. The rates of C17 O and C18 O are very similar at all
depths, but details in the line overlap and self-shielding can cause mutual differences
of up to 30% (Sects. 5.3.3 and 5.6.1.4).
• When coupled to a chemical network, the model reproduces column densities observed towards diffuse clouds and PDRs. It shows that the large spread in observed
N(12 CO)/N(13 CO) ratios may be due to a combination of isotope-selective photodissociation and suprathermal chemistry (Sect. 5.6.1.3).
• Photodissociation of CO is more strongly isotope-selective in cold gas than in warm
gas (Sect. 5.4).
• The results from the full calculation are well approximated by a grid of pre-computed
shielding functions, intended for easy use in various other models. Shielding functions are provided for a range of astrophysical parameters (Sect. 5.5).
166
5.7 Conclusions
• Grain growth in circumstellar disks increases the vertical range in which CO can be
photodissociated. If photodissociation is still important at the point where CO freezes
out onto the grains, the observable gas-phase column density ratios N(12 CO)/N(C17 O)
and N(12 CO)/N(C18 O) may become an order of magnitude larger than the initial isotopic abundance ratios (Sect. 5.6.3.3).
• Without a far-UV component in the irradiating spectrum, the photodissociation of CO
cannot cause isotope fractionation. The interstellar radiation field can already cause
substantial fractionation in disks (Sect. 5.6.3.3).
• The isotope-selective nature of the CO photodissociation results in mass-independent
fractionation of 17 O and 18 O. Column density ratios computed for a circumstellar
disk agree well with recent observations. Our model supports the hypothesis that the
photodissociation of CO is responsible for the anomalous 17 O and 18 O abundances in
meteorites (Sect. 5.6.3.4).
167
6
PAH chemistry and IR emission from
circumstellar disks
R. Visser, V. C. Geers, C. P. Dullemond, J.-C. Augereau,
K. M. Pontoppidan and E. F. van Dishoeck
Astronomy & Astrophysics, 2007, 466, 229
169
Chapter 6 – PAH chemistry and IR emission from circumstellar disks
Abstract
Aims. The chemistry of polycyclic aromatic hydrocarbons (PAHs) in disks around Herbig Ae/Be
and T Tauri stars is investigated, along with the infrared emission from these species. PAHs can exist
in different charge states and they can bear different numbers of hydrogen atoms. The equilibrium
(steady-state) distribution over all possible charge and hydrogenation states depends on the size and
shape of the PAHs and on the physical properties of the star and surrounding disk.
Methods. A chemistry model is created to calculate the equilibrium charge and hydrogenation
distribution. Destruction of PAHs by ultraviolet (UV) photons, possibly in multi-photon absorption
events, is taken into account. The chemistry model is coupled to a radiative transfer code to provide
the physical parameters and to combine the PAH emission with the spectral energy distribution from
the star+disk system.
Results. Normally hydrogenated PAHs in Herbig Ae/Be disks account for most of the observed
PAH emission, with neutral and positively ionised species contributing in roughly equal amounts.
Close to the midplane, the PAHs are more strongly hydrogenated and negatively ionised, but these
species do not contribute to the overall emission because of the low UV/optical flux deep inside
the disk. PAHs of 50 carbon atoms are destroyed out to 100 AU in the disk’s surface layer, and
the resulting spatial extent of the emission does not agree well with observations. Rather, PAHs
of about 100 carbon atoms or more are predicted to cause most of the observed emission. The
emission is extended on a scale similar to that of the size of the disk, with the short-wavelength
features less extended than the long-wavelength features. The continuum emission is less extended
than the PAH emission at the same wavelength. Furthermore, the emission from T Tauri disks is
much weaker and concentrated more towards the central star than that from Herbig Ae/Be disks.
Positively ionised PAHs are predicted to be largely absent in T Tauri disks because of the weaker
radiation field.
170
6.1 Introduction
6.1 Introduction
Polycyclic aromatic hydrocarbons (PAHs; Léger & Puget 1984, Allamandola et al. 1989)
are ubiquitous in space and are seen in emission from a wide variety of sources, including
the diffuse interstellar medium, photon-dominated regions, circumstellar envelopes, and
(proto)planetary nebulae (Peeters et al. 2004 and references therein). The PAHs in these
sources are electronically excited by ultraviolet (UV) photons. Following internal conversion to a high vibrational level of the electronic ground state, they cool by emission in the
C–H and C–C stretching and bending modes at 3.3, 6.2, 7.7, 8.6, 11.3, 12.8 and 16.4 µm.
Using the Infrared Space Observatory (ISO; Kessler et al. 1996), the Spitzer Space
Telescope (Werner et al. 2004a, Houck et al. 2004) and various ground-based telescopes,
PAH features have also been observed in disks around Herbig Ae/Be and T Tauri stars
(Van Kerckhoven et al. 2000, Hony et al. 2001, Peeters et al. 2002, Przygodda et al. 2003,
van Boekel et al. 2004, Acke & van den Ancker 2004, Geers et al. 2006). Spatially resolved observations confirm that the emission comes from regions whose size is consistent
with that of a circumstellar disk (van Boekel et al. 2004, Habart et al. 2006, Geers et al.
2007). Because of the optical or UV radiation required to excite the PAHs, their emission
is thought to come mostly from the surface layers of the disks (Habart et al. 2004b). Acke
& van den Ancker (2004) showed that PAH emission is generally stronger from flared
disks (Meeus et al. 2001, Dominik et al. 2003) than from flat or self-shadowed disks.
Although the presence of such large molecules in disks and other astronomical environments is intrinsically interesting, it is also important to study PAHs for other reasons.
They are a good diagnostic of the stellar radiation field and can be used to trace small dust
particles in the surface layers of disks, both near the centre and farther out (Habart et al.
2004b). In addition, they are strongly involved in the physical and chemical processes
in disks. For instance, photoionisation of PAHs produces energetic electrons, which are
a major heating source of the gas (Bakes & Tielens 1994, Kamp & Dullemond 2004,
Jonkheid et al. 2004). The absorption of UV radiation by PAHs in the surface layers influences radiation-driven processes closer to the midplane. Charge transfer of C+ with
neutral and negatively charged PAHs affects the carbon chemistry. Finally, Habart et al.
(2004a) proposed PAHs as an important site of H2 formation in photon-dominated regions, a process which is also important in disk chemistry (Jonkheid et al. 2006, 2007).
Although this process is probably more efficient on grains and very large PAHs than it is
on PAHs of up to 100 carbon atoms, the latter may play an important role if the grains
have grown to large sizes.
Many PAHs and related species that are originally present in the parent molecular
cloud, are able to survive the star formation process and eventually end up on planetary
bodies (Allamandola & Hudgins 2003). They add to the richness of the organochemical
“broth” on planets in habitable zones (Kasting et al. 1993), from which life may originate.
Further enrichment is believed to come from the impact of comets. PAHs have now been
detected in cometary material during the Deep Impact mission (Lisse et al. 2006) and
returned to Earth by the Stardust mission (Sandford et al. 2006). The icy grains that
constitute comets also contain a variety of other molecules (Ehrenfreund & Fraser 2003).
Radiation-induced chemical reactions between frozen-out PAHs and these molecules lead
171
Chapter 6 – PAH chemistry and IR emission from circumstellar disks
to a large variety of complex species, including some that are found in life on Earth
(Bernstein et al. 1999, Ehrenfreund & Sephton 2006). These possibilities are another
reason why it is important to study the presence and chemistry of PAHs in disks.
The chemistry of PAHs in an astronomical context has been studied with increasingly
complex and accurate models since the 1980s (Omont 1986, Lepp et al. 1988, Bakes &
Tielens 1994, Salama et al. 1996, Dartois & d’Hendecourt 1997, Vuong & Foing 2000,
Le Page et al. 2001, 2003, Weingartner & Draine 2001, Bakes et al. 2001a,b); however,
none of these were specifically targeted at PAHs in circumstellar disks. Disk chemistry
models that do include PAHs only treat them in a very simple manner (e.g., Jonkheid
et al. 2004, Habart et al. 2004b). In this chapter, an extensive PAH chemistry model
is coupled to a radiative transfer model for circumstellar disks (Dullemond & Dominik
2004a, Geers et al. 2006, Dullemond et al. 2007a). The chemistry part includes ionisation
(photoelectric emission), electron recombination and attachment, photodissociation with
loss of hydrogen and/or carbon, and hydrogen addition. Infrared (IR) emission from the
PAHs is calculated taking multi-photon excitation into account, and added to the spectral
energy distribution (SED) of the star+disk system. The model can in principle also be
used to examine PAH chemistry and emission in other astronomical environments.
We present the chemistry model in Sect. 6.2, followed by a brief review of the radiative
transfer model in Sect. 6.3. The results are discussed in Sect. 6.4 and our conclusions are
summarised in Sect. 6.5.
6.2 PAH model
The chemistry part of our model is a combination of the models developed by Le Page
et al. (2001, hereafter LPSB01) and Weingartner & Draine (2001, hereafter WD01). Photodissociation is treated according to Léger et al. (1989). Where possible, theoretical rates
are compared to recent experimental data. Our model employs the new PAH cross sections of Draine & Li (2007, hereafter DL07), which are an update of Li & Draine (2001)
based on experimental data (Mattioda et al. 2005a,b) and IR observations (e.g., Smith
et al. 2004, Werner et al. 2004b). In this section, we present the main characteristics of
our model.
6.2.1 Characterisation of PAHs
The PAHs in our model are characterised by their number of carbon atoms, NC . A PAH
bears the normal number of hydrogen atoms, NH◦ , when one hydrogen atom is attached
to each peripheral carbon atom bonded to exactly two other carbon atoms (e.g., 12 for
coronene, C24 H12 ). The ratio between NH◦ and NC is taken as (e.g., DL07):
172


0.5 p



f ◦ = NH◦ /NC = 
25/NC
0.5


 0.25
NC ≤ 25 ,
25 < NC ≤ 100 ,
NC > 100 .
(6.1)
6.2 PAH model
This formula produces values appropriate for compact (pericondensed) PAHs. Elongated
(catacondensed) PAHs have a higher hydrogen coverage; however, they are believed to
be less stable and to convert into a more compact geometry (Wang et al. 1997, Dartois &
d’Hendecourt 1997), so only compact PAHs are assumed to be present.
Every carbon atom that bears one hydrogen atom in the normal case is assumed to be
able to bear two in extreme conditions, so each PAH can exist in 2NH◦ + 1 possible hydrogenation states (0 ≤ NH ≤ 2NH◦ ). Furthermore, each PAH can exist in two or more charge
states Z. The maximum and minimum attainable charge depend on the radiation field and
the PAH’s ionisation and autoionisation potentials (Bakes & Tielens 1994, WD01). The
number of accessible charge states increases with PAH size.
For a number of properties, the radius of the PAH is more important than NC . The
PAHs of interest are assumed to be spherically symmetric and, following e.g. Draine &
Li (2001) and WD01, are assigned an effective radius a in Å:
NC
a=
0.468
!1/3
.
(6.2)
This equation does not give the actual geometric radius; rather, it gives the radius of a
pure graphite sphere containing the same number of carbon atoms. Equations (6.1) and
(6.2) do not apply to very small PAHs (NC <
∼ 20), but like irregularly shaped PAHs, they
are assumed not to be abundant enough to contribute to the emission (Sect. 6.2.6).
6.2.2 Photoprocesses
The absorption of a UV or visible photon of sufficient energy by a PAH causes either the
emission of an electron (ionisation or detachment) or a transition to an excited electronic
and vibrational state. In the second case, internal conversion and fluorescence rapidly
bring the molecule to a high vibrational level of the ground electronic state. From here,
several processes can take place (Léger et al. 1988, 1989): (1) dissociation with loss of
atomic or molecular hydrogen; (2) dissociation with loss of a carbon-bearing fragment;
or (3) cooling by infrared emission. The cooling rate constant for a PAH with internal
energy Eint is given by Li & Lunine (2003):
Z ∞
4πBλσabs
kcool =
dλ ,
(6.3)
hc/λ
912 Å
with Bλ(T [Eint ]) the Planck function at the PAH’s vibrational temperature, T [Eint ] (Draine
& Li 2001). The cross sections, σabs , are treated in Sect. 6.2.3.
The yield of a single dissociation process i, Yi , depends on the rate constants, k, of all
possible processes:
ki
.
(6.4)
Yi = P
j kj
To determine the rate Γi of dissociation process i, the yield is multiplied by the absorption rate, taking into account the possibility of electron emission, and integrated over the
173
Chapter 6 – PAH chemistry and IR emission from circumstellar disks
relevant energy range:
Γi =
Z
0
∞
(1 − Yem )Yi σabs Nph dE ,
(6.5)
where Yem is the photoelectric emission yield (Eq. (6.8)) and Nph is the number of photons
in units of cm−2 s−1 erg−1 . The radiation field is treated explicitly at every point in the disk
in a 2D axisymmetric geometry, i.e., the wavelength dependence is taken into account as
well as the magnitude.
Photoelectric emission is the ejection of an electron from a PAH due to the absorption
of a UV photon. The electron can come either from the valence band (photoionisation;
possible for all charge states) or from an energy level above the valence band (photodetachment; only for negatively charged PAHs). The photoelectric emission rate is given by
WD01 as
Z ∞
Z ∞
Ydet σdet Nph dE ,
(6.6)
Yion σabs Nph dE +
Γem =
hνdet
hνion
where h is Planck’s constant. Photoelectric emission can only occur when the photon
energy exceeds the threshold of hνion or hνdet . The photodetachment yield, Ydet , is taken
to be unity: every absorption of a photon with hν > hνdet leads to the ejection of an
electron. The photoionisation yield, Yion , has a value between 0.1 and 1 for hν >
∼ 8 eV and
drops rapidly for lower energies. For photons with hν > hνion , the photodetachment cross
section, σdet , is about two orders of magnitude smaller for NC = 50 than the ionisation
cross section, which is assumed equal to σabs .
In order to derive Yem as used in Eq. (6.5), Eq. (6.6) is rewritten:
!
Z ∞
σdet
Yion +
Γem =
(6.7)
Ydet σabs Nph dE .
σabs
0
The quantity between brackets is the electron emission yield, Yem :
Yem = Yion +
σdet
Ydet ,
σabs
(6.8)
taking Yion and Ydet to be zero for photon energies less than hνion and hνdet , respectively.
Furthermore, Yem is not allowed to exceed unity: each absorbed photon can eject only one
electron.
If no photoelectric emission takes place, the PAH can undergo dissociation with loss
of carbon or hydrogen. Several theoretical schemes exist to calculate the dissociation
rates. LPSB01 employed the Rice-Ramsperger-Kassel-Marcus quasi-equilibrium theory
(RRKM-QET) and obtained hydrogen loss rates close to those determined experimentally
for benzene (C6 H6 ), naphthalene (C10 H8 ) and anthracene (C14 H10 ). Léger et al. (1989)
investigated the loss of carbon as well as hydrogen, using an inverse Laplace transform of
the Arrhenius law (Forst 1972) to determine the rates. In this method, the rate constant is
zero when the PAH’s internal energy, Eint , originating from one or more UV photons, is
less than the critical energy, E0 , for a particular loss channel. When Eint exceeds E0 ,
kdiss,X = AX
174
ρ(Eint − E0,X )
,
ρ(Eint )
(6.9)
6.2 PAH model
Table 6.1 – Arrhenius parameters for the loss of carbon and hydrogen fragments from PAHs.a
Fragment
C
C2
C3
H
H
H
H2
H2
a
NH
all
all
all
≤ NH◦
> NH◦
> NH◦
> NH◦
> NH◦
Z
all
all
all
all
≤0
>0
≤0
>0
E0 (eV)
7.37
8.49
7.97
4.65
1.1
2.8
1.5
3.1
A (s−1 )
6.2 × 1015
3.5 × 1017
1.5 × 1018
1.5 × 1015
4 × 1013
1 × 1014
4 × 1013
1 × 1014
E0 and A for C, C2 and C3 are from Léger et al. (1989). E0 for H and H2 is based on the RRKM-QET
parameters from Le Page et al. (2001) and modified slightly to obtain a better match with their rates. A for
H is modified from Léger et al. (1989) and used also for H2 .
where AX is the pre-exponential Arrhenius factor for channel X and ρ(E) is the density
of vibrational states at energy E. Léger et al. (1989) considered the loss of H, C, C2 and
C3 ; LPSB01 also took H2 loss into account for PAHs with NH > NH◦ . Our model uses
the method of Léger et al. (1989) for all loss channels, with values for A and E0 given in
Table 6.1.
The intensity of the UV radiation field inside the disk is characterised in our model by
G′0 = uUV /uHab , where
uUV =
Z
13.6 eV
uν dν =
6 eV
Z
13.6 eV
(h/c)hνNph dν
(6.10)
6 eV
and uHab = 5.33 × 10−14 erg cm−3 is the UV energy density in the mean interstellar
radiation field (Habing 1968). However, the radiation field inside the disk does not have
the same spectral shape as the interstellar radiation field, so G′0 is used instead of G0 to
denote its integrated intensity. The exact UV field at every point and every wavelength is
calculated by a Monte Carlo code, which follows the photons from the star into the disk
in an 2D axisymmetric geometry (see also Sect. 6.3.1). As a result, regions of high optical
depth may not receive enough photons between 6 and 13.6 eV to calculate G′0 . In those
cases, the lower limit of the integration range is extended in 1-eV steps and the energy
density of the Habing field is recalculated accordingly. If G′0 still cannot be calculated
between 1 and 13.6 eV, it is set to a value of 10−6 .
In the strong radiation fields present in circumstellar disks (up to G′0 = 1010 in the
inner disk around a Herbig Ae/Be star), multiple photons are absorbed by a PAH before it can cool through emission of IR radiation. These multi-photon events result in
higher dissociation rates than given by Eqs. (6.5) and (6.9). For example, a PAH of 50
carbon atoms is photodestroyed in a radiation field of G′0 ≈ 105 if multi-photon events
are allowed, while G′0 ≈ 1014 is required in a pure single-photon treatment (Sect. 6.2.6).
However, the multi-photon treatment is much more computationally demanding, hence
175
Chapter 6 – PAH chemistry and IR emission from circumstellar disks
our model is limited to single-photon treatments for all processes but photodestruction;
the latter is discussed in detail in Sect. 6.2.6. The errors introduced by not including a full
multi-photon treatment are discussed in Sect. 6.4.5.
As long as the internal energy of the PAH exceeds E0 for any of the loss channels from
Table 6.1, there is competition between dissociation and radiative stabilisation. Typically,
the emission of a single IR photon is not sufficient to bring Eint below E0 . If the emitted
IR photons are assumed to have an average energy qIR of 0.18 eV (6.9 µm; LPSB01), a
total of n = (Eint − E0 )/qIR photons are required to stabilise the PAH. The ith photon in
this cooling process is emitted at an approximate rate (Herbst & Dunbar 1991)
krad,i = 73
[Eint − (i − 1)qIR ]1.5
,
s0.5
(6.11)
with Eint in eV and s = 3(NC + NH ) − 6 the number of vibrational degrees of freedom. The
competition between dissociation and IR emission occurs for every intermediate state.
Hence, the total radiative stabilisation rate is (Herbst & Le Page 1999)
krad = krad,1
n
Y
i=1
krad,i
,
krad,i + kdiss,H + kdiss,H2 + kdiss,C
(6.12)
where kdiss,C denotes the sum of all three carbon loss channels. Note that Eqs. (6.11) and
(6.12) are used only in the chemical part of the code, where the details of the cooling
rate function are not important (LPSB01). The radiative transfer part employs the more
accurate Eq. (6.3).
Alternative stabilisation pathways such as inverse internal conversion, inverse fluorescence and Poincaré fluorescence (Leach 1987, Léger et al. 1988) are probably of minor
importance. Their effect can be approximated by increasing qIR , thus creating an “effective” krad that is somewhat larger than the “old” krad . However, this has no discernible
effect on the rates from Eq. (6.5), so these alternative processes are ignored altogether.
6.2.3 Absorption cross sections
Li & Draine (2001) performed a thorough examination of the absorption cross sections,
σabs , of PAHs across a large range of wavelengths. Following new experimental data
(Mattioda et al. 2005a,b) and IR observations (e.g., Smith et al. 2004, Werner et al. 2004b),
an updated model was published in DL07, providing a set of equations that can readily be
used in our model. The cross sections consist of a continuum contribution that decreases
towards longer wavelengths, superposed onto which are a number of Drude profiles to
account for the σ-σ∗ transition at 72.2 nm, the π-π∗ transition at 217.5 nm, the C–H
stretching mode at 3.3 µm, the C–C stretching modes at 6.2 and 7.7 µm, the C–H in-plane
bending mode at 8.6 µm, and the C–H out-of-plane bending mode at 11.2–11.3 µm. Some
of these primary features are split into two or three subfeatures, and several minor features
are included in the 5–20 µm range to give a better agreement with recent observations
(DL07). The additional absorption for ions in the near IR measured by Mattioda et al. is
included as a continuum term and three Drude profiles.
176
6.2 PAH model
The absorption properties of a PAH depend on its charge. Neutral PAHs have 6.2, 7.7
and 8.6 µm features that are a factor of a few weaker than do cations. The 3.3 µm band
strength increases from cations to neutrals (Langhoff 1996, Hudgins et al. 2000, 2001,
Bauschlicher 2002). The other features have similar intensities for each charge state.
While the cation-neutral band ratios at 6.2, 7.7 and 8.6 µm are mostly independent of
size, those at 3.3 µm decrease for larger PAHs. DL07 provide integrated band strengths,
σint , for neutral and ionised PAHs, but they do not account for the size dependence of this
Z=0
feature in cations. Rather than taking σZ=1
int,3.3 = 0.227σint,3.3 as DL07 do, our model uses
Z=0
σZ=1
int,3.3 = σint,3.3 1 +
41
NC − 14
!−1
.
(6.13)
This way, the cation-neutral band ratio approaches unity for large PAHs. The theoretical
ratios for NC = 24, 54 and 96 from Bauschlicher are well reproduced by Eq. (6.13).
LPSB01 based their opacities on experiments by Joblin (1992) on PAHs in soot extracts. They fitted the cross section of an extract with an average PAH mass of 365 amu
(corresponding to NC ≈ 30) with a collection of Gaussian curves, obtaining practically
the same values for λ <
∼ 0.25 µm (E >
∼ 5 eV) as DL07. Between 0.25 and 0.5 µm, the
integrated cross section of LPSB01 is about 2.5 times stronger. Since LPSB01 were only
interested in the UV opacities, their cross section throughout the rest of the visible and all
of the infrared is zero. For our purposes, the different opacities in the 0.25–0.5 µm range
only affect the hydrogen dissociation rates, and, as shown in Sect. 6.4.5, the effects are
negligible.
The details of the IR cross sections of negatively charged PAHs are not well understood, with very different values to be found in the literature (Langhoff 1996, Bauschlicher
& Bakes 2000). However, this is not a problem for our model. Because of the small electron affinities, photodetachment is a very efficient process and absorption of a photon by
a PAH anion leads to ejection of an electron rather than substantial emission in the infrared. Rapid electron attachment retrieves the original anion before the transient neutral
PAH can absorb a second photon. Hence, if anions are present in steady state, they are
excluded altogether when the infrared emission is calculated.
Large PAHs can carry a double or triple positive charge in those regions of the disk
where the radiation field is strong and the electron density is low. The differences between
the cross sections of singly charged and neutral PAHs generally increase when going to
multiply charged PAHs (Bauschlicher & Bakes 2000, Bakes et al. 2001a). However, no
cross sections for multiply charged species exist that can be directly used in our model,
so the cross sections for singly charged species are used instead. Since multiply charged
species only constitute a very small part of the PAH population (Sect. 6.4.1), this approximation is not expected to cause major errors.
6.2.4 Electron recombination and attachment
Free electrons inside the disk can recombine with PAH cations or attach to PAH neutrals
and anions; both processes are referred to here as electron attachment. The electron at177
Chapter 6 – PAH chemistry and IR emission from circumstellar disks
tachment rate, Γea = kea ne , with ne the electron number density, depends on the frequency
of collisions between electrons and PAHs and on the probability that a colliding electron
sticks, as expressed by WD01:
r
8kT
e
kea = πa se J
,
πme
(6.14)
se = 0.5(1 − e−a/le )
(6.15)
2
where k is Boltzmann’s constant and me the electron mass. The sticking coefficient, se , has
a maximum value of 0.5 to allow for the possibility of elastic scattering. Non-scattering
electrons have to be retained by the PAH before their momentum carries them back to
infinity. The probability of retention is approximately 1 − e−a/le , where le ≈ 10 Å can
be considered the mean free path of the electron inside the PAH. WD01 included an
additional factor 1/(1 + e20−NC ) for PAH neutrals and anions to take into account the
possibility that the PAH is dissociated by the electron’s excess energy, but this is only
important for PAHs smaller than 24 carbon atoms (see also LPSB01) and they are assumed
not to be present in the disk (Sect. 6.2.6). Thus, our model uses
for all PAHs with a charge Z > Zmin .
e a dimensionless factor that takes into account the charge and size
Expressions for J,
of the PAH and the temperature of the gas, can be found in Draine & Sutin (1987). For
neutral PAHs, Je ∝ T −1/2 , so the attachment rate does not depend significantly on the
temperature (Bakes & Tielens 1994). Je changes as T −1 for PAH cations in the PAH size
and gas temperature regimes of interest, leading to Γea ∝ T −1/2 . Finally, for PAH anions,
both Jeand the attachment rate vary as e−1/T .
Recombination rates at 300 K for PAH cations of up to sixteen carbon atoms have been
determined experimentally (Abouelaziz et al. 1993, Rebrion-Rowe et al. 2003, Hassouna
et al. 2003, Novotný et al. 2005, Biennier et al. 2006). The experimental and theoretical rates agree to within a factor of two for these small species, except for naphthalene
(C10 H8 ), where the theoretical value is larger by a factor of seven. Experimental data on
larger PAHs are needed to ascertain the accuracy of Eq. (6.14) for the sizes used in our
model.
Experimental data on electron attachment to PAH neutrals are scarce. Rates for anthracene (C14 H10 ) have been reported from 9 × 10−10 to 4.5 × 10−9 cm3 s−1 (Tobita et al.
1992, Moustefaoui et al. 1998), with no apparent dependence on temperature. Equation
(6.14) predicts a rate of 2 × 10−10 cm3 s−1 , about an order of magnitude lower, for temperatures between 10 and 1000 K. For pyrene (C16 H10 ), however, the theoretical value,
2 × 10−9 cm3 s−1 , is 5–10 times larger than the available experimental values of 2 and
4.2 × 10−10 cm3 s−1 (Tobita et al. 1992). Hence, we can only estimate that Eq. (6.14) is
accurate to within about an order of magnitude. No experimental data are available on
electron attachment to negatively charged PAHs or on electron recombination with PAHs
carrying a multiple positive charge, so the uncertainties in the theoretical rates for these
reactions are at least as large.
178
6.2 PAH model
6.2.5 Hydrogen addition
The addition of atomic hydrogen to neutral and ionised PAHs is an exothermic process
requiring little or no activation energy (Bauschlicher 1998, Hirama et al. 2004). For addition to cations, we use the temperature-independent rates from LPSB01, which are based
primarily on experimental work by Snow et al. (1998). Addition to neutrals is about two
orders of magnitude slower for benzene (Mebel et al. 1997, Triebert et al. 1998) and has
not been measured for larger PAHs. No rates are known for the addition of hydrogen to
Z<1
Z=1
Z>1
Z=1
anions or to cations with Z > 1. We take kadd,H
= 10−2 kadd,H
and kadd,H
= kadd,H
. The rate
depends on NH as described by LPSB01.
It is assumed that molecular hydrogen can only attach to PAHs with NH < NH◦ . Again,
the rate for addition to cations from LPSB01 is used and divided by 100 for the neutral
and anion rates.
For naphthalene and larger PAHs, addition of atomic hydrogen (PAH + H → PAH+1 )
is assumed to be much faster than the bimolecular abstraction channel (PAH + H →
PAH−1 + H2 ; Herbst & Le Page 1999), so the latter is not included in our model.
6.2.6 PAH growth and destruction
The PAHs observed in disks are generally not believed to have been formed in situ after the collapse and main infall phases. Formation and growth of PAHs requires a high
temperature (∼1000 K), density and acetylene abundance (Frenklach & Feigelson 1989,
Cherchneff et al. 1992) and, especially for the smallest PAHs, a weak UV radiation field
to prevent rapid photodissociation. If these conditions exist at all in a disk, it is only in a
thin slice (less than 0.1 AU) right behind the inner rim, and it is unlikely that this affects
the PAH population at larger radii. Another possible method of late-stage PAH formation
is in accretion shocks (Desch & Connolly 2002), but too little is known about these events
to include them in the model. Hence, no in situ formation or growth of PAHs is assumed
to take place.
PAH destruction is governed by the loss of carbon fragments upon absorption of one
or more UV photons. C–C bonds are a few eV stronger than C–H bonds (e.g., Table 6.1),
so carbon is lost only from completely dehydrogenated PAHs. In order to get an accurate
destruction rate, multi-photon events have to be taken into account, as first recognised
by Guhathakurta & Draine (1989) and Siebenmorgen et al. (1992). Our model uses a
procedure based on Habart et al. (2004b). A PAH ensemble residing in a given radiation
field assumes a statistical distribution over a range of internal energies. This distribution
is represented by P(Eint ), which is normalised so that P(Eint )dEint is the probability to find
the PAH in the energy interval from Eint to Eint + dEint . At every Eint , there is competition
between cooling and dissociation with loss of a carbon fragment. The probability for
dissociation depends on the ratio between kC (the sum of the rate constants for all carbon
loss channels in Eq. (6.9)) and kIR (the instantaneous IR emission rate, comparable to Eq.
(6.11)) at Eint :
kC
ηdes (Eint ) =
.
(6.16)
kC + kIR
179
Chapter 6 – PAH chemistry and IR emission from circumstellar disks
kIR is a very flat function in the relevant energy regime, while kC is very steep, so ηdes is
approximately a step function.
The probability, pdes , that a PAH in a certain radiation field is destroyed is then found
by integrating P(Eint )ηdes (Eint ) over all energies:
Z ∞
P(Eint )ηdes (Eint )dEint ,
(6.17)
pdes =
0
with P(Eint ) calculated according to Guhathakurta & Draine (1989). This is equal to the
formula in Habart et al. (2004b) if ηdes is replaced by a step function and the internal
energy is converted to a temperature using the PAH’s heat capacity (e.g., Draine & Li
2001). Destruction is assumed to take place if pdes exceeds a value of 10−8 .
We define a critical radiation intensity, G∗0 , which is the intensity required to cause
photodestruction of a given PAH within a typical disk lifetime of 3 Myr, i.e., the intensity
required to get τdiss,C = 1/Γdiss,C = 3 Myr. The G∗0 for both single-photon and multiphoton destruction is plotted in Fig. 6.1. Knowledge of the radiation field at every point
of the disk (Fig. 6.3) is required to determine where PAHs of a certain size are destroyed.
An approximate approach is to trace the radiation field along the τvis = 1 surface, where
the intensity decreases almost as a power law (see Sect. 6.4.2). In our Herbig Ae/Be
model (Sect. 6.3.2), PAHs of 50 carbon atoms (G∗0 = 1.2 × 105 ) are destroyed out to
100 AU on the τvis = 1 surface. The destruction radius is larger for smaller PAHs and
vice versa; for example, PAHs with NC = 100 are only destroyed in the inner 5 AU.
PAHs with less than 24 carbon atoms are not taken into account at all, because they are
already destroyed when G′0 ≈ 1. There are regions inside the disk where the UV intensity
is below that limit, but the PAHs in such regions do not contribute significantly to the
emission spectrum. The radiation field in a T Tauri disk is much weaker than for a Herbig
Ae/Be disk, so the destruction radius is smaller. In our model T Tauri disk, 50-C PAHs
can survive everywhere but in the disk’s inner 0.01 AU, while 100-C PAHs can survive
even there (Sect. 6.4.6).
6.2.7 Other chemical processes
No reactions between PAHs and species other than H and H2 are included in our model.
Although the second-order rate coefficients for the addition of, e.g., atomic nitrogen and
oxygen are comparable to that for atomic hydrogen (Snow et al. 1998, LPSB01), the
abundances of these heavier elements are not high enough to affect the chemical equilibrium. Formation of dimers and clusters (Rapacioli et al. 2005 and references therein) and
trapping of PAHs onto grains and ices (Gudipati & Allamandola 2003) are also left out.
The midplane of the disk, where densities are high enough and temperatures low enough
for these processes to play a role, does not contribute significantly to the IR emission
spectrum.
180
6.2 PAH model
Figure 6.1 – Photodestruction of PAHs. (a) The UV intensity for which the destruction timescale
is shorter than the disk lifetime, with (solid line) or without (dashed line) multi-photon events
included. The right axis shows the corresponding “destruction radius” along the τvis = 1 surface in
our model Herbig Ae/Be disk. The disk’s inner and outer radius are indicated by the dotted lines.
(b,c) The τvis = 1 surface in a vertical cut through our two model disks. The tick marks denote the
radius inwards of which PAHs of a given size are destroyed within the disk lifetime; for the Herbig
Ae/Be model, this corresponds to the right vertical axis of panel (a).
181
Chapter 6 – PAH chemistry and IR emission from circumstellar disks
6.3 Disk model
6.3.1 Computational code
The Monte Carlo radiative transfer code RADMC (Dullemond & Dominik 2004a) is used
in combination with the more general code RADICAL (Dullemond & Turolla 2000) to
produce the IR spectra from PAHs in circumstellar disks. Using an axisymmetric density
structure, but following photons in all three dimensions, RADMC determines the dust
temperature and radiation field at every point of the disk. RADICAL then calculates a
spectrum from all or part of the disk, or an image at any given wavelength.
The calculations in RADMC and RADICAL are based on the optical properties of a
collection of carbon and silicate dust grains. Recently, PAHs were added as another type
of grain to model the emission from the Herbig Ae star VV Ser and the surrounding nebulosity (Pontoppidan et al. 2007), from a sample of Herbig Ae/Be and T Tauri stars (Geers
et al. 2006), and to study the effects of dust sedimentation (Dullemond et al. 2007a). The
PAHs are excited in a quantised fashion by UV and, to a lesser degree, visible photons
(Li & Draine 2002), and cool in a classical way according to the “continuous cooling”
approximation (Guhathakurta & Draine 1989), which was found by Draine & Li (2001)
to be accurate even for small PAHs. A detailed description of the PAH emission module
is given in Pontoppidan et al. (2007), whereas tests against other codes are described in
Geers et al. (2006).
In the models used by Geers et al. (2006), Pontoppidan et al. (2007) and Dullemond
et al. (2007a), no PAH chemistry was included. PAHs of a given size existed in the
same charge and hydrogenation state everywhere, or in a fixed ratio between a limited
number of states (e.g., 50% neutral, 50% ionised). In the current model, the chemistry is
included in the following way. First, a single charge and hydrogenation state for a given
NC is included at a given abundance in the radiative transfer procedure, to calculate the
disk structure and radiation field at every point. Using these physical parameters, the
equilibrium distribution of the PAHs over all possible charge and hydrogenation states is
then determined. After an optional second iteration of the radiative transfer to take into
account the heating of thermal grains by emission from the PAHs, the spectrum or image
is calculated.
Some additional chemistry is added to the model in order to determine the electron
and atomic and molecular hydrogen densities, which are needed to calculate the chemical
equilibrium of the PAHs. The electron abundance, xe = ne /nH , is set equal to the C+
abundance, based on a simple equilibrium between the photoionisation of neutral C and
the recombination of C+ (Le Teuff et al. 2000, Bergin et al. 2003, 2007). All hydrogen
is in atomic form at the edges of the disk and is converted to molecular form inside the
disk as the amount of dissociating photons decreases due to self-shielding and shielding
by dust (Draine & Bertoldi 1996, van Zadelhoff et al. 2003). The H2 formation rate is
taken from Black & van Dishoeck (1987). For the outer parts of the disk, which receive
little radiation from the star, an interstellar radiation field with G0 = 1 is included.
The RADMC and RADICAL codes only treat isotropic scattering of photons. We verified with a different radiative transfer code (van Zadelhoff et al. 2003) that no significant
182
6.4 Results
changes occur in the results when using a more realistic anisotropic scattering function.
Different PAHs (i.e., different sizes) can be included at the same time, and for each
one, the equilibrium distribution over all charge and hydrogenation states is calculated.
The abundance of each PAH with a given NC is equal throughout the disk, except in those
regions where G′0 > G∗0 (NC ); there, the abundance is set to zero. As discussed in Sect.
6.2.6, mixing processes are ignored.
6.3.2 Template disk with PAHs
For most of the calculations, a template Herbig Ae/Be star+disk model is used with the
following parameters. The star has radius 2.79 R⊙ , mass 2.91 M⊙ and effective temperature 10 000 K, and its spectrum is described by a Kurucz model. No UV excess due to
accretion or other processes is present. The mass of the disk is 0.01 M⊙ , with inner and
outer radii of 0.48 and 300 AU. The inner radius corresponds to a dust evaporation temperature of 1700 K. The disk is in vertical hydrostatic equilibrium, with a flaring shape
and a slightly puffed-up inner rim (Dullemond & Dominik 2004a). The dust temperature
is calculated explicitly, whereas the gas temperature is put to a constant value of 300 K
everywhere, appropriate for the upper layers from which most of the PAH emission originates. The results are not sensitive to the exact value of the gas temperature (see also
Sect. 6.4.5).
The same parameters are used for a template T Tauri star+disk model, except that the
star has mass 0.58 M⊙ and effective temperature 4000 K. The dust evaporation temperature is kept at 1700 K, so the disk’s inner radius moves inwards to 0.077 AU.
C50 H18 is present as a prototypical PAH, at a high abundance of 1.6 × 10−6 PAH
molecules per hydrogen nucleus to maximise the effects of changes in the model parameters. This abundance corresponds to 10% of the total dust mass and to 36% of the total
amount of carbon in dust being locked up in this PAH, assuming an abundance of carbon
in dust of 2.22 × 10−4 with respect to hydrogen (Habart et al. 2004b). The model was also
run for PAHs of 24 and 96 carbon atoms.
6.4 Results
This chapter focuses on the chemistry of the PAHs in a circumstellar disk and on its
effects on the mid-IR emission, as well as on the differences between disks around Herbig
Ae/Be and T Tauri stars. Geers et al. (2006) analysed the effects of changing various disk
parameters, such as the PAH abundance and the disk geometry; settling of PAHs and dust
was analysed by Dullemond et al. (2007a).
6.4.1 PAH chemistry
Due to the large variations in density and UV intensity throughout the disk, the PAHs are
present in a large number of charge and hydrogenation states (Tables 6.2 and 6.3). When a
disk containing only 50-C PAHs (NH◦ = 18) is in steady state, our model predicts that 56%
183
Chapter 6 – PAH chemistry and IR emission from circumstellar disks
Table 6.2 – Abundances of the dominant charge and hydrogenation states for three PAHs in the
model Herbig Ae/Be star+disk system.a,b
a
b
NH
Z
12
12
23
24
24
−1
0
0
−1
0
17
17
17
18
18
18
18
19
19
19
36
36
0
+1
+2
−1
0
+1
+2
−1
0
+1
−1
0
48
48
48
48
48
−2
−1
0
+1
+2
N.R.
100Γea 0.01Γea
NC = 24, NH◦ = 12, −1 ≤ Z
6.6(+1) 9.2(+1) 3.5(+0)
2.8(+1) 6.0(−1) 8.9(+1)
1.0(−1) 6.4(−5) 6.3(−3)
3.6(+0) 7.4(+0) 3.7(−1)
7.3(−1) 5.1(−2) 7.1(+0)
NC = 50, NH◦ = 18, −1 ≤ Z
2.2(−1) 8.1(−2) 1.6(−1)
3.5(−2) 2.0(−3) 2.6(−2)
4.3(−4) 8.7(−6) 9.7(−4)
2.8(+1) 4.4(+1) 1.1(+0)
2.1(+1) 6.2(−1) 4.8(+1)
3.3(−2) 1.3(−3) 3.7(−1)
7.5(−4) 8.9(−6) 8.3(−3)
9.6(−1) 1.1(+0) 7.6(−2)
1.8(−1) 3.3(−3) 1.1(+0)
5.6(−3) 3.0(−6) 4.7(−1)
4.5(+1) 5.3(+1) 4.2(+1)
4.0(−1) 3.0(−2) 5.6(+0)
NC = 96, NH◦ = 24, −2 ≤ Z
2.1(+1) 2.0(+1) 2.3(+1)
5.7(+1) 7.9(+1) 2.4(+1)
2.2(+1) 7.1(−1) 5.0(+1)
1.3(−1) 1.0(−2) 1.1(+0)
2.0(−2) 3.8(−4) 2.1(−1)
100Γdiss,H
≤ +2
6.7(+1)
2.7(+1)
1.0(−1)
3.7(+0)
7.5(−1)
≤ +3
2.1(−1)
2.2(−2)
1.5(−4)
3.0(+1)
2.1(+1)
4.9(−2)
1.1(−3)
3.2(−1)
5.3(−2)
3.1(−3)
4.5(+1)
3.8(−1)
≤ +4
2.1(+1)
5.7(+1)
2.2(+1)
1.3(−1)
2.0(−2)
0.01Γdiss,H
6.5(+1)
2.8(+1)
1.0(−1)
3.6(+0)
7.3(−1)
2.2(−1)
3.6(−2)
6.1(−4)
2.6(+1)
2.0(+1)
3.1(−2)
4.3(−4)
5.6(−1)
2.0(−1)
7.5(−3)
4.8(+1)
8.3(−1)
2.1(+1)
5.7(+1)
2.2(+1)
1.3(−1)
2.0(−2)
In per cent of the entire PAH population in the disk.
N.R.: normal rates, i.e., the rates as discussed in Sect. 6.2. 100ΓX and 0.01ΓX : the normal rate for process
X (ea: electron attachment; diss,H: photodissociation with loss of H or H2 ) in/decreased by a factor of 100.
The effect of increasing a certain rate is the same as decreasing the rate of the reverse process (photoelectric
emission or addition of H or H2 ).
of the observed PAH emission between 2.5 and 13.5 µm originates from neutral C50 H18 ,
with another 9% from C50 H+18 . PAHs missing one hydrogen atom also contribute to the
emission: 22% comes from C50 H17 and 12% from C50 H+17 .
A strong contribution to the observed emission does not imply a high abundance
throughout the disk, because the contribution of each state to the spectrum also depends
on its spatial distribution. Figure 6.2 shows a cut through the disk and indicates the abundance of the six most important states with respect to the total C50 HZy population. Near
the surface, where the radiation field is strong, the PAHs are ionised and some of them
184
6.4 Results
Table 6.3 – Emissivity contributions from 2.5 to 13.5 µm for the dominant charge and hydrogenation states for three PAHs in the model Herbig Ae/Be star+disk system.a
a
NH
Z
12
12
23
24
24
−1
0
0
−1
0
17
17
17
18
18
18
18
19
19
19
36
36
0
+1
+2
−1
0
+1
+2
−1
0
+1
−1
0
48
48
48
48
48
−2
−1
0
+1
+2
N.R.
100Γea 0.01Γea
NC = 24, NH◦ = 12, −1 ≤ Z
−
−
−
1.0(+2) 1.0(+2) 1.0(+2)
1.2(−3)
−
2.7(−5)
−
−
−
3.6(−2) 3.6(−1) 5.1(−2)
NC = 50, NH◦ = 18, −1 ≤ Z
2.2(+1) 5.1(+1) 2.9(+0)
1.2(+1) 2.0(+0) 3.0(+0)
1.3(−1) 4.9(−3) 2.4(−1)
−
−
−
5.6(+1) 4.6(+1) 2.9(+1)
9.0(+0) 1.2(+0) 1.0(+1)
2.3(−1) 5.4(−3) 2.4(+0)
−
−
−
5.3(−6) 6.9(−6) 9.1(−6)
1.7(+0)
−
4.5(+1)
−
−
−
5.0(−5) 3.5(−5) 6.2(−5)
NC = 96, NH◦ = 24, −2 ≤ Z
−
−
−
−
−
−
4.4(+1) 8.7(+1) 1.3(+1)
4.2(+1) 1.2(+1) 2.9(+1)
1.4(+1) 5.0(−1) 5.6(+1)
100Γdiss,H
≤ +2
−
1.0(+2)
1.9(−3)
−
5.5(−2)
≤ +3
1.8(+1)
6.8(+0)
2.4(−2)
−
5.9(+1)
1.5(+1)
3.5(−1)
−
3.9(−7)
8.1(−1)
−
5.0(−5)
≤ +4
−
−
4.4(+1)
4.2(+1)
1.4(+1)
0.01Γdiss,H
−
1.0(+2)
1.2(−3)
−
3.6(−2)
2.1(+1)
1.2(+1)
1.9(−1)
−
5.5(+1)
8.3(+0)
1.3(−1)
−
8.2(−1)
2.4(+0)
−
5.2(−5)
−
−
4.4(+1)
4.2(+1)
1.4(+1)
See footnotes to Table 6.2.
have lost a hydrogen atom. Going to lower altitudes, the PAHs first become neutral and
then negatively ionised. Still lower, the increasing density and optical depth lead to further hydrogenation, resulting in a high abundance of the completely hydrogenated anion,
C50 H−36 , around the midplane.
Normally hydrogenated and positively ionised states occur in more strongly irradiated
regions than do completely hydrogenated and negatively ionised states, so the former emit
more strongly. For example, the state responsible for more than half of the emission,
C50 H18 , constitutes only 21% of all PAHs in the entire disk (Tables 6.2 and 6.3). About
half of the PAHs (45%) are expected to be in the form of C50 H−36 . This state, which is
assumed not to emit at all (Sect. 6.2.3), dominates the high-density regions close to the
midplane. The normally hydrogenated anion, C50 H−18 , accounts for 28% of all PAHs,
185
Chapter 6 – PAH chemistry and IR emission from circumstellar disks
Figure 6.2 – Steady-state distribution of the most important charge and hydrogenation states of
C50 Hy (NH◦ = 18) in a disk around a Herbig Ae/Be star (104 K). Each panel shows a cut through the
disk, with the equator on the horizontal axis and the pole on the vertical axis. The grey scale denotes
the fraction at which a state is present compared to all possible states. The dashed line denotes the
τvis = 1 surface. The dotted√“emission line” connects the points where the PAH emission is strongest
for a given distance r = R2 + z2 from the star. The thick black contour lines denote the region
responsible for most of the PAH emission; from the outside inward, the contours contain 95, 90, 76,
52 and 28% of the total emitted power between 2.5 and 13.5 µm. They are omitted from the last
frame for clarity.
and the remaining PAHs are mostly present as anions with 18 < NH < 36. States like
C50 H−17 and C50 H+36 are entirely absent: the same environmental parameters that favour
dehydrogenation (strong radiation, low density), also favour ionisation.
Species with NH < 17 are also predicted to be absent. They have to be formed from
C50 HZ17 , but wherever these states exist, the ratio between the UV intensity and the hydrogen density favours hydrogen addition. In those regions where the ratio is favourable
to photodissociation with H loss, the radiation field is also strong enough to destroy the
carbon skeleton. The abundance of C50 HZ17 is boosted by H2 loss from C50 HZ19 , which is a
much faster process than H loss from C50 HZ18 . Moving from the surface to the midplane,
the increasing nH /G′0 ratio allows all hydrogenation states from NH◦ + 1 to 2NH◦ to exist.
However, since all of them except NH = 19 are negatively charged, they are assumed not
to emit.
6.4.2 PAH emission
The integrated UV intensity between 6 and 13.6 eV, characterised by G′0 (Eq. (6.10)),
varies by many orders of magnitude from the disk’s surface to the midplane. The radiation
186
6.4 Results
Figure 6.3 – The radiation field (characterised by G′0 ; Eq. (6.10)) for a flaring disk around a Herbig
Ae/Be star (104 K), containing PAHs of 50 carbon atoms. The dashed line is the “emission line”
from Fig. 6.2. The white contour lines are the same as the black contour lines in Fig. 6.2.
field for the model Herbig Ae/Be system is shown in Fig. 6.3, with the region responsible
for most of the 2.5–13.5 µm PAH emission indicated by contour lines. Part of this region
is truncated sharply due to photodestruction of the PAHs, marking the G′0 = G∗0 line. An
analysis of the disk structure shows that some 95% of the PAH emission originates from
a region with 101 < G′0 < 105 , 103 < nH (cm−3 ) < 109 , and 10−1 < ne (cm−3 ) < 104 (see
also Fig. 6.4).
When the radiation field is traced along the τvis = 1 surface, and the disk’s inner 0.01
AU is disregarded, the integrated intensity decreases almost as a power law. Specifically,
for any point on the τvis = 1 surface a distance rτ (in AU) away from the star,
G′0 ≈ 3.0 × 108 rτ−1.74 .
(6.18)
The power-law exponent differs from the value of −2 expected based on geometrical
considerations because of the curvature in the τvis = 1 surface. This relationship allows
one to predict to what radius rdes PAHs of a certain size are destroyed, as was done in
Sect. 6.2.6 and Fig. 6.1. It should be noted, however, that this only applies to the τvis = 1
surface. PAHs can survive and contribute to the emission from r < rdes when they are at
lower altitudes (e.g., Figs. 6.3 and 6.7).
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Chapter 6 – PAH chemistry and IR emission from circumstellar disks
Figure 6.4 – Normalised cumulative PAH emission for the Herbig Ae/Be model as function of total
hydrogen density (nH , left panel), radiation field (G′0 , middle panel) and electron density (ne , right
panel).
The abundances of the charge and hydrogenation states in the 95% emission region are
different from those in the entire disk. Although they agree more closely to the emissivity
contributions, a perfect correspondence is not achieved. For instance, only 1.0% of the
PAHs in this region are in the form of C50 H+18 , but they account for 8.0% of the emission
(Tables 6.2 and 6.3). This is again due to the positive ions occurring in a more intense
radiation field than the neutrals and negative ions.
In a vertical cut through the disk, a point exists for every distance rem from the star
where the PAH emission is strongest. These points form the “emission lines” in Figs. 6.2
and 6.3. The conditions along this line determine the charge and hydrogenation of the
PAHs responsible for most of the observed emission. Its altitude depends mostly on two
competing factors: the intensity of the UV field and the PAH number density. A strong
UV field leads to stronger emission, but a strong UV field can only exist in regions of low
density, where the total emission is weaker.
The variation of nH , ne and G′0 along the emission line is plotted in Fig. 6.5. In the
disk’s inner 100 AU (disregarding the actual inner rim), photodestruction of PAHs causes
the emission line to lie below the τvis = 1 surface, where nH and ne are relatively high (1011
and 104 cm−3 , respectively) and G′0 is relatively low (102 –103 ). The resulting ne /G′0 and
nH /G′0 ratios favour the neutral normally hydrogenated species, C50 H18 . Going to larger
radii, the emission line gradually moves up and crosses the τvis = 1 surface at around
100 AU. As a consequence, nH and ne decrease while G′0 increases. Because the electron
abundance with respect to hydrogen also decreases, the PAHs first lose a hydrogen atom
to become C50 H17 . Farther out, ionisation takes place to produce C50 H+17 . At still larger
188
6.4 Results
Figure 6.5 – Total hydrogen density (nH ), electron density (ne ), radiation field (G′0 ) and ratio between electron density and radiation field (ne /G′0 ) along the emission line defined in Fig. 6.2 and
Sect. 6.4.2. The bars at the bottom indicate the main emitter at each distance along the emission
line. The parameters are those of the template Herbig Ae/Be model.
radii, the radiation field starts to lose intensity and C50 H+18 becomes the dominant species.
The deviation from the r−1.74 power law for the radiation field (Eq. (6.18)) is due to the
emission line not following the τvis = 1 surface.
It should be noted that ionised and dehydrogenated species do exist (and emit) in the
189
Chapter 6 – PAH chemistry and IR emission from circumstellar disks
disk’s inner part. However, for any given distance to the star, they only become the most
abundant species at around 100 AU.
6.4.3 Other PAHs
If C96 H24 is put into the disk instead of C50 H18 , the abundance of the dehydrogenated
and normally hydrogenated states essentially goes to zero. This is due to the hydrogen
addition rates being larger and the hydrogen dissociation rates being smaller for larger
PAHs. Most of the 2.5–13.5 µm emission in the NC = 96 case comes from C96 H48 (44%),
−
C96 H+48 (42%) and C96 H2+
48 (14%), while the anion, C96 H48 , is the most abundant overall
(Tables 6.2 and 6.3).
If only C24 H12 is put in, the PAH emission becomes very weak. The critical radiation
intensity for this PAH is less than 1 (Fig. 6.1), so it can only survive in strongly shielded
areas. There, 66% of the PAHs are present as C24 H−12 , 28% as C24 H12 and 4% as C24 H−24 .
Only the neutral species contributes to the emission; however, since this PAH only emits
from regions where G′0 <
∼ 1, no PAH features are visible in the calculated spectrum,
despite the high abundance used in our model. This is exemplified in Fig. 6.6, where the
calculated spectra for the model star+disk system are compared for the three PAH sizes.
A disk around a Herbig Ae/Be star containing PAHs of 50 or 96 carbon atoms shows
strong PAH features, but the 24-C spectrum contains only thermal dust emission.
The goal of Fig. 6.6 is to show the differences arising from the photochemical modelling of the three PAH sizes in model Herbig Ae/Be and T Tauri disks, rather than to
provide realistic spectra from such objects. In order to fit our model results to observations, one would need to include a range of PAH sizes in one model (Li & Lunine 2003)
and execute a larger parameter study (Habart et al. 2004b, Geers et al. 2006) than was
done in this work.
6.4.4 Spatial extent of the PAH emission
The top panel in Figure 6.7 presents the cumulative intensity of the five main PAH features
and the continua at 3.1 and 19.6 µm as a function of radius for NC = 50. In accordance
with Figs. 6.2 and 6.3, more than 95% of the power radiated in the features originates
from outside the inner 10 AU, and some 80% from outside 100 AU. This is largely due to
these PAHs being destroyed closer to the star. The 3.3 and 11.3 µm features are somewhat
less extended than the other three features. The continua are much more confined than the
features, especially at 3.1 µm, where 75% comes from the disk’s inner rim.
The PAH emission from a disk with NC = 96 (Fig. 6.7, middle panel) is less extended
than that from the same disk with NC = 50, because larger PAHs can survive at smaller
radii. About 80% of the integrated intensity of the 3.3 µm feature originates from within
100 AU, and so does about 60% of the other, less energetic features. The continuum at 20
µm has the same spatial behaviour as the 3.3 µm PAH feature, while the continuum at 3.1
µm is very much confined towards the centre. These results are in good agreement with
the spatial extent modelled by Habart et al. (2004b). The spatial extent of the continuum
190
6.4 Results
Figure 6.6 – Model spectra (flux at 1 pc) for disks around a Herbig Ae/Be (black) and a T Tauri star
(grey) containing either C24 H12 , C50 H18 (shifted by a factor of two) or C96 H24 (shifted by a factor
of four), with the charge and hydrogenation balance calculated by the chemistry model.
emission from the C96 H24 disk is identical to that from the C50 H18 disk, so the thermal
dust emission appears to be unaffected by the details of the PAHs and their chemistry.
Our model also agrees well with a number of spatially resolved observations (van
Boekel et al. 2004, Habart et al. 2006, Geers et al. 2007), where the PAH features were
consistently found to be more extended than the adjacent continuum. Furthermore, PAH
emission is typically observed on scales of tens of AU, which cannot be well explained
by our model if only PAHs of 50 carbon atoms are present. Hence, the observed emission
is probably due to PAHs of at least about 100 carbon atoms.
6.4.5 Sensitivity analysis
The rates for most of the chemical reactions discussed in Sect. 6.2 are not yet well known.
In order to gauge the importance of having an accurate rate for a given reaction, the
equilibrium distributions were calculated for rates increased or decreased by a factor of
100 from their normal model values. These results are also presented in Tables 6.2 and 6.3.
Modifying the hydrogen dissociation and addition rates leads to no significant changes, so
treating photodissociation with loss of hydrogen in a purely single-photon fashion likely
191
Chapter 6 – PAH chemistry and IR emission from circumstellar disks
Figure 6.7 – Normalised cumulative integrated intensity of the five main PAH features (black) and
the continua at 3.1 and 19.6 µm (grey) for C50 H18 around a Herbig Ae/Be star (top), C96 H24 around
a Herbig Ae/Be star (middle) and C50 H18 around a T Tauri star (bottom). In each case, the charge
and hydrogenation balance are calculated by the chemistry model.
does not introduce large errors. The fact that most of the PAH emission comes from
regions with a relatively weak UV field (G′0 < 105 ), where multi-photon events play only
a minor role, further justifies the single-photon dehydrogenation treatment.
The ionisation and electron attachment rates are more important to know accurately.
192
6.4 Results
For instance, the contribution from the cations to the spectrum decreases by a factor of
a few when taking an ionisation rate that is 0.01 times the normal model rate. The shift
away from positively charged species also affects hydrogenation (addition of hydrogen
is faster to cations than to neutrals), resulting, e.g., in a smaller emissivity contribution
from C50 H18 and C50 H+18 with respect to C50 H17 and C50 H+17 . Further laboratory work on
ionisation and electron attachment rates, especially for larger PAHs, will help to better
constrain this part of the model.
As shown by e.g. Kamp & Dullemond (2004) and Jonkheid et al. (2004), the gas temperature is not constant throughout the disk. The recombination rates between electrons
and cations decrease for higher temperatures, whereas the attachment rates of electrons to
neutrals are almost independent of temperature (Sect. 6.2.4). Taking a higher temperature
would result in a slightly larger cation abundance. However, even for an extreme temperature of 5000 K, the recombination rates decrease by only a factor of four, and the effects
are smaller than what is depicted in Tables 6.2 and 6.3. Hence, we believe it is justified to
take a constant temperature of 300 K throughout the disk.
6.4.6 T Tauri stars
The results discussed so far are for a star with an effective temperature of 1 × 104 K,
appropriate for a Herbig Ae/Be type. Colder stars, like T Tauri types, are less efficient
in inducing IR emission in PAHs, as shown with both observations and models by Geers
et al. (2006), unless they have excess UV over the stellar atmosphere. Hence, when the
model parameters are otherwise unchanged, the continuum flux from the star+disk system
and the absolute intensity of the PAH features become weaker (Fig. 6.6).
It was shown in Sect. 6.4.2 that the integrated intensity of the UV field along the
τvis = 1 surface of a disk around a Herbig Ae/Be star decreases approximately as a power
law. This is also true for the T Tauri case, if the disk’s inner 0.01 AU are again disregarded.
The exponent is slightly larger:
G′0 ≈ 12rτ−1.93 ,
(6.19)
with rτ in AU. The UV field peaks at G′0 = 6 × 105 at the inner rim and drops by two
orders of magnitude within 0.01 AU, so a PAH of 50 carbon atoms (G∗0 = 1.2 × 105 ) can
survive almost everywhere (see also Fig. 6.1).
The smaller destruction radii also lead to the PAH emission being more concentrated
towards the inner disk (Fig. 6.7, bottom panel). The 3.3 µm feature is particularly confined, with 70% originating from the inner 10 AU and 25% from the inner rim. As was
the case for the Herbig Ae/Be disks (Sect. 6.4.4), the inner rim contributes strongly to the
continuum emission at 3.1 µm. The 19.6 µm continuum also behaves very similarly to that
from the Herbig Ae/Be disks, except that it is stretched inwards because of the smaller
Rin . Thus, the spatial extent of the thermal dust emission seems to be largely unaffected
by the temperature of the central star.
Cations are practically absent in the model T Tauri disk with any kind of PAH, due
to the weaker radiation field. For NC = 24 and 50, almost all of the PAH emission originates from the normally hydrogenated neutral species, C24 H12 or C50 H18 . Less than 0.1%
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Chapter 6 – PAH chemistry and IR emission from circumstellar disks
originates from other neutral states, primarily those missing one hydrogen atom (C24 H11 ,
C50 H17 ) or having twice the normal number of hydrogen atoms (C24 H24 , C50 H36 ). If a
disk around a T Tauri star contains only 96-C PAHs, all of the emission is due to C96 H48 .
The absence of cations could help explain the weak 7.7 and 8.6 µm features in observed
spectra (Geers et al. 2006), because these two features are weaker in neutral PAHs.
Anions are abundant for all three PAH sizes, accounting for about half of the entire
PAH population. However, they are again assumed not to contribute to the emission.
6.4.7 Comparison with observations
Acke & van den Ancker (2004) performed a comprehensive analysis of the PAH features
in a large sample of Herbig Ae/Be stars observed with ISO, measuring line fluxes and
comparing them to each other. Figure 6.8 recreates their Fig. 9, plotting the ratio of the
integrated fluxes in the 8.6 and 6.2 µm bands against the ratio of the integrated fluxes in
the 3.3 and 6.2 µm bands. In order to gauge the plausibility of the numerous charge and
hydrogenation states a PAH can in principle attain, the same ratios are also plotted for a
sample of models containing one PAH in one specific state only.
The 3.3-6.2 ratio from the model is very sensitive to the charge of the PAH and increases by an order of magnitude when going from ionised to neutral species. The 3.3
and 8.6 µm features are due to C–H vibrational modes, while the 6.2 µm feature is due to
a C–C mode (Sect. 6.2.3), so both ratios in Fig. 6.8 increase with NH . The observations
fall mostly in between the model points for neutral and ionised PAHs, so both charge
states appear to contribute to the observed emission. This strengthens the model results
presented in this work, although the observed emissivity contribution from neutral species
seems to be somewhat less than the predicted ∼50%. Most of the observations agree with
the model prediction that the emission originates from multiple hydrogenation states, with
a lower limit of NH = NH◦ − 1.
6.5 Conclusions
We studied the chemistry of polycyclic aromatic hydrocarbons (PAHs) in disks around
Herbig Ae/Be and T Tauri stars, as well as the infrared (IR) emission from these species.
We created an extensive PAH chemistry model, based primarily on the models of Le
Page et al. (2001) and Weingartner & Draine (2001), with absorption cross sections from
Draine & Li (2007). This model includes reactions affecting the charge (ionisation, electron recombination, electron attachment) and hydrogen coverage (photodissociation with
hydrogen loss, hydrogen addition) of PAHs in an astronomical environment. Destruction
of PAHs by UV radiation is also taken into account, including destruction by multi-photon
absorption events. By coupling the chemistry model to an existing radiative transfer
model, we obtained equilibrium charge and hydrogenation distributions throughout the
disks. The main results are as follows:
• Very small PAHs (24 carbon atoms) are destroyed within a typical disk lifetime of
3 Myr even in regions of low UV intensity (G′0 ≈ 1). No features are seen in the
194
6.5 Conclusions
Figure 6.8 – The ratio of the fluxes of the 8.6 and 6.2 µm bands against the ratio of the fluxes of
the 3.3 and 6.2 µm bands. Filled circles and arrows are detections and upper limits from Acke &
van den Ancker (2004). Open diamonds and lines are predictions from the template Herbig Ae/Be
model. The cross in the lower right shows the typical error bars for both the observations and the
model results.
calculated spectrum for either a Herbig Ae/Be or a T Tauri disk, despite a high PAH
abundance.
• PAHs of intermediate size (50 carbon atoms) do produce clearly visible features, even
though they are still photodestroyed out to about 100 AU in the surface layers of
a disk around a Herbig Ae/Be star. The model predicts that most of the emission
arises from the surface layers and from large radii (more than 100 AU). Neutral and
positively ionised species, bearing the normal number of hydrogen atoms or one less,
contribute in roughly equal amounts. Negatively charged species are also present, but
are assumed not to contribute to the emission.
• Going to still larger PAHs (96 carbon atoms), photodestruction becomes a slower
process and the PAHs can survive down to 5 AU from a Herbig Ae/Be star. The slower
photodissociation rates also mean that these PAHs are fully hydrogenated everywhere
in the disk. Neutral and ionised species still contribute in comparable amounts to the
emission, with some 15% originating from doubly ionised PAHs.
• The PAH emission is predicted to be extended on a scale similar to the size of the
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Chapter 6 – PAH chemistry and IR emission from circumstellar disks
disk, with the features at longer wavelengths contributing more in the outer parts
and the features at shorter wavelengths contributing more in the inner parts. The
continuum emission is less extended than the emission from the PAH features at the
same wavelength.
• Disks around T Tauri stars show weaker PAH features than do disks around Herbig
Ae/Be stars because of the weaker radiation from T Tauri stars, assuming they have
no excess UV over the stellar atmosphere. The PAH emission from T Tauri disks is
considerably more confined towards the centre than that from disks around Herbig
Ae/Be stars, because PAHs can survive much closer to the star. For instance, a 50C PAH survives everywhere but in the disk’s innermost 0.01 AU. Furthermore, the
radiation field is no longer strong enough to ionise the PAHs, and all the PAH emission
originates from neutral species for all three PAH sizes. This could help explain the
weak 7.7 and 8.6 µm features in observed spectra. About half of all PAHs in a T Tauri
disk are predicted to be negatively ionised.
• Comparing the model results to spatially resolved observations (van Boekel et al.
2004, Habart et al. 2006, Geers et al. 2007) for Herbig Ae/Be stars, it appears that
PAHs of at least about 100 carbon atoms are responsible for most of the emission.
The emission from smaller species is predicted to be too extended. Other observations
(Acke & van den Ancker 2004) support the conclusion that the emission is due to a
mixture of neutral and singly positively ionised species.
196
Nederlandse samenvatting:
Chemische evolutie
van kernen tot schijven
Astrochemie: scheikunde in de ruimte
Chemie is overal. Auto’s worden aangedreven door de chemische reactie tussen benzine
en zuurstof. Planten en bomen zetten koolstofdioxide en water met behulp van zonlicht
om in zuurstof en glucose. In ons eigen lichaam vindt het omgekeerde proces plaats om
energie te leveren voor onze spieren. Zo zijn er nog talloze chemische reacties die de Aarde en het leven daarop maken wat ze zijn. De leus “chemie is overal” beperkt zich echter
niet tot de Aarde. De rode kleur van Mars is bijvoorbeeld afkomstig van ijzeroxides, een
groep chemische verbindingen bestaande uit ijzer- en zuurstofatomen. De verschillende
kleuren in de atmosfeer van Jupiter worden veroorzaakt door allerlei stikstof-, zwavel- en
fosforhoudende verbindingen. Zelfs in de schijnbaar lege ruimte tussen de planeten en de
sterren blijkt zich een scala aan chemische verbindingen te bevinden.
De wetenschap die zich bezighoudt met deze verbindingen – met deze scheikunde in
de ruimte – is de astrochemie. Zij is ontstaan in de jaren twintig van de vorige eeuw,
toen langzaam maar zeker duidelijk werd dat de ruimte tussen de sterren niet geheel leeg
is. Er bevindt zich een zeer ijl gas, ook wel het interstellaire medium genoemd, dat op
de meeste plaatsen een dichtheid heeft van minder dan één atoom per kubieke centimeter
(cc). Ter vergelijking: een kubieke centimeter van de lucht om ons heen bevat ruim
10.000.000.000.000.000.000 (10 triljoen) atomen. Een klein deel van het interstellaire
medium – ongeveer 1% van het totale volume – bestaat uit zogeheten moleculaire wolken.
Dit zijn gebieden met een dichtheid van ongeveer 100 atomen per cc. De toevoeging
“moleculair” slaat op het feit dat bij deze dichtheid de atomen elkaar al vaak genoeg
tegenkomen om chemische bindingen te vormen en zo moleculen te maken.
Het eerste molecuul in de ruimte werd pas in 1937 gevonden: CH, een combinatie
van één atoom koolstof (C) en één atoom waterstof (H). Inmiddels zijn er zo’n 150 interstellaire moleculen ontdekt, en ieder jaar komen er nog nieuwe bij. Op de lijst staan
heel gewone verbindingen zoals H2 O (water), CH4 (methaan, het hoofdbestanddeel van
aardgas) en CH3 CH2 OH (ethanol, de bekendste vorm van alcohol), maar ook exotische
moleculen zoals C6 H− en HC11 N. Een van de uitdagingen voor astrochemici is om te
begrijpen hoe al deze stoffen in de ruimte worden gevormd, en waarom de ene stof in veel
197
Nederlandse samenvatting
hogere concentratie voorkomt dan de andere. In dit proefschrift zijn deze vragen vooral
gericht op moleculen die we waarnemen in de omgeving rond pas gevormde sterren.
De vorming van sterren en planeten
De moleculaire wolken uit de vorige paragraaf zijn tot enkele tientallen lichtjaren groot en
hebben dus, ondanks hun enorm lage dichtheid, een aanzienlijke massa: 1000 à 10.000
keer de massa van de Zon.1,2 Ze bevatten kleinere gebieden met hogere dichtheid. De
dichtste hiervan worden kernen genoemd; zij zijn rond de 10.000 astronomische eenheden (AE) groot,3 een paar zonsmassa’s zwaar, en hebben een dichtheid van ongeveer een
miljoen atomen of moleculen per cc. De wolk en de kernen bestaan voor 99 massa-% uit
gas met een betrekkelijk eenvoudige chemische samenstelling (zie de paragraaf “Chemische evolutie”). Voor de rest bestaan ze uit stof: minuscule zandkorreltjes van ongeveer
0,1 micrometer (of 0,0001 millimeter), omgeven met een dun laagje ijs.
Sterren zoals de Zon ontstaan uit kernen. In figuur 1 is schematisch aangegeven hoe
dit proces verloopt. Paneel a toont een moleculaire wolk met daarin een aantal kernen
van hogere dichtheid, aangegeven met een donkerdere tint grijs. Paneel b zoomt in op één
zo’n kern, met een typische afmeting van 10.000 AE. Voor het gemak doen we hier alsof
de kern mooi bolvorming is, zodat we een cirkelvormige dwarsdoorsnede zien. In werkelijkheid zijn kernen echter zelden precies rond. Ongeacht de vorm verliest de kern op een
gegeven moment haar stabiliteit en begint ze onder invloed van haar eigen zwaartekracht
ineen te storten. Dit moment wordt beschouwd als het begin van het stervormingsproces.
Door het instorten wordt de dichtheid in het midden van de kern steeds hoger, zoals
wederom aangegeven met de verschillende grijstinten. Na ongeveer 10.000 jaar4 is de
dichtheid zo hoog dat we kunnen spreken van een jonge ster (paneel c). De ronde vorm uit
paneel b heeft nu plaatsgemaakt voor een plattere dichtheidsverdeling. Dit komt doordat
de kern, net als de omringende moleculaire wolk, langzaam ronddraait. Een gevolg van
deze draaiing is dat het materiaal dat van de buitenkant van de kern naar binnen valt,
niet precies in het midden uitkomt. In plaats daarvan vormt het een platte schijf rond de
jonge ster, meestal aangeduid als een circumstellaire schijf. Veel van het materiaal dat in
de schijf terechtkomt, beweegt verder naar binnen en valt uiteindelijk alsnog in de ster.
Ondertussen wordt een deel van het materiaal weer uitgestoten in een sterke straalstroom,
in een richting loodrecht op het vlak van de schijf. Deze straalstroom creëert een holle
ruimte in het omringende materiaal, die we in paneel c zien als de witte zandlopervorm.
Na zo’n 100.000 jaar (105 jr, paneel d) is de kern volledig ingestort. Wat overblijft is
een jonge ster omringd door een circumstellaire schijf van ongeveer 100 AE groot. De
1
2
3
4
Een lichtjaar is de afstand die het licht in een jaar aflegt. Licht reist met een snelheid van 300.000 km per
seconde, dus een lichtjaar is 10.000.000.000.000 (10 biljoen) km. Na de Zon staat de dichtstbijzijnde ster,
Proxima Centauri, op 4,2 keer deze afstand.
De massa van de Zon is 2 × 1030 kg, oftewel een 2 met 30 nullen. De Zon is ruim 300.000 keer zo zwaar als
de Aarde.
Een astronomische eenheid is de gemiddelde afstand tussen de Zon en de Aarde, oftewel 150 miljoen km. Een
afstand van 10.000 AE is gelijk aan 0,16 lichtjaar.
In wetenschappelijke notatie is dit 104 : een 1 met vier nullen. Een miljoen is dus bijvoorbeeld 106 .
198
Chemische evolutie
Figuur 1 – Schematische weergave van de vorming van een ster met planeten. De tijd benodigd
voor iedere fase is rechtsboven in de panelen aangegeven. Linksonder staan de ruimtelijke schalen;
lj staat voor lichtjaar (10 biljoen km) en AE voor astronomische eenheid (150 miljoen km).
dichtheid van de schijf loopt op tot meer dan een biljoen moleculen per cc, een miljoen
keer hoger dan de kern waaruit ze is ontstaan. De minuscule stofdeeltjes die in de kern
aanwezig waren, beginnen zich nu te concentreren in het midden van de schijf. Ze botsen
daar tegen elkaar aan en blijven soms plakken om steeds grotere stofdeeltjes te vormen.
Zo krijgen we op een gegeven moment kiezels van een paar centimeter, die weer verder
groeien tot rotsblokken van een paar meter, en nog verder tot planetaire embryo’s van een
paar kilometer. Ze zijn dan zo groot en zwaar dat hun zwaartekracht sterk genoeg is om
het gas in de schijf als het ware op te zuigen. Zo belanden we na ongeveer een miljoen
jaar in paneel e. De schijf is nu grotendeels verdwenen; wat er nog van over is bevat grote
gaten waar het gas is opgenomen in de planeten-in-wording. Het proces van planeetgroei
duurt nog enkele miljoenen jaren en resulteert tot slot in een volwassen zonnestelsel zoals
getoond in paneel f.
Chemische evolutie
De rode draad in dit proefschrift is de chemische samenstelling van het gas en het stof
tijdens de verschillende fases in de vorming van een jonge ster. We bestuderen deze
199
Nederlandse samenvatting
samenstelling met behulp van astrochemische modellen. Zoals eerder opgemerkt is de
chemische samenstelling van het gas in wolkenkernen relatief eenvoudig. Dat komt omdat ze een lage temperatuur hebben (10 Kelvin5 ) en nog niet blootstaan aan het sterke
stralingsveld van een ster. De lage temperatuur leidt ertoe dat veel moleculen vastvriezen
op de stofdeeltjes. Zodra de kern begint in te storten loopt de temperatuur op. De vluchtigste verbindingen in het ijs verdampen weer, en de algehele chemische samenstelling
van het gas en het ijs verandert. Als even later de jonge ster begint te schijnen, zorgt het
stralingsveld voor nog meer veranderingen. Dankzij al deze processen, die zijn samengevat in figuur 2, ziet de circumstellaire schijf er chemisch dus heel anders uit dan de kern
waarmee we begonnen. Een belangrijke reden dat we zeer geïnteresseerd zijn in de chemie van de schijf, is dat het materiaal uit de schijf in en op de planeten terechtkomt. Het
is nog steeds onduidelijk waar de chemische bouwstenen voor het eerste leven op Aarde
vandaan komen. Als we begrijpen in wat voor omgeving de Aarde is gevormd, en uit wat
voor materiaal, kunnen we die vraag in de toekomst wellicht beantwoorden.
De meest directe manier om iets te weten te komen over de chemische samenstelling
van moleculaire wolken en circumstellaire schijven, is door ze te bestuderen met een
telescoop. Ieder molecuul, zowel op Aarde als in de ruimte, absorbeert licht en andere
vormen van straling op een unieke manier. Door met telescopen te zoeken naar deze
“streepjescodes” kunnen we vaststellen of een bepaalde stof wel of niet in een wolk of
een schijf voorkomt. Deze methode heeft echter een aantal beperkingen. In de eerste
plaats hebben we te maken met de enorme afstanden waarop stervormingsgebieden zich
bevinden, waardoor we slechts weinig details kunnen onderscheiden. Een ander probleem
is dat jonge sterren en schijven in het begin nog zijn ingebed in de moleculaire wolk
(panelen b en c in figuur 1), zodat we ze niet rechtstreeks kunnen waarnemen. Tot slot
vertellen waarnemingen ons alleen iets over de chemische samenstelling van een wolk of
schijf op dit moment, en niet hoe de samenstelling vroeger was of in de toekomst zal zijn.
Een geschikte methode om de chemische evolutie in de tijd te volgen, en meteen ook
om meer details te kunnen zien, is het gebruik van astrochemische modellen. Zo’n model
bevat een aantal moleculen, die bepaalde reacties met elkaar aangaan. De snelheid van
elke reactie hangt af van de fysische omstandigheden, zoals dichtheid, temperatuur en
stralingsveld. Aan elk molecuul in het model wordt een beginconcentratie gegeven, die
meestal berust op waargenomen concentraties. Vervolgens wordt een stelsel wiskundige
vergelijkingen opgelost om te kijken hoe de concentraties van alle moleculen veranderen
in de tijd. Dit kan gedaan worden voor constante of veranderende fysische condities. In
het laatste geval kunnen we bijvoorbeeld kijken hoe de concentraties veranderen als een
wolkenkern instort om een jonge ster en bijbehorende schijf te vormen.
Dit proefschrift
Zoals gezegd staat in dit proefschrift de verandering van de chemische samenstelling tijdens het stervormingsproces centraal. In het verleden is hier door anderen al veel aan5
Een temperatuur in Kelvin kan worden omgerekend in graden Celsius via T (◦ C) = T (K) − 273. Een temperatuur van 10 K is dus -263 ◦ C.
200
Dit proefschrift
Figuur 2 – Schematische weergave van de chemische evolutie tijdens de vorming van een ster.
Bovenin zijn de dichtheid (n, in atomen per cc) en temperatuur (T , in Kelvin) aangegeven; beide
nemen toe van buiten naar binnen. De grijze bol stelt de instortende kern voor, met binnenin de
jonge ster en de omringende schijf. De pijl (langs de punten 0, 1 en 2) is een mogelijke baan
die het materiaal kan volgen vanuit de buitendelen van de kern naar de schijf. Langs deze baan
is aangegeven hoe de samenstelling van de ijslaag rond de stofdeeltjes (niet op schaal) globaal
verandert. Voor de kern begint in te storten is de temperatuur laag en vriezen stoffen als water (H2 O)
en koolstofmonoxide (CO) vast op het stof. Het CO wordt deels omgezet in methanol (CH3 OH).
Tijdens het instorten wordt het ijs opgewarmd. Het CO verdampt, maar het minder vluchtige water
en methanol blijven bevroren. In het ijs worden radicalen (instabiele verbindingen) gevormd die
snel verder reageren tot complexere organische moleculen zoals azijnzuur (CH3 COOH) en ether
(CH3 OCH3 ). De circumstellaire schijf is relatief koud, dus als het materiaal daarin terechtkomt kan
CO weer bevriezen. Dichtbij de ster is de temperatuur zo hoog dat al het ijs verdampt.
201
Nederlandse samenvatting
dacht aan besteed. Dit is echter altijd met eendimensionale modellen gedaan. Daarmee
kan prima de chemische evolutie in de instortende kern worden gevolgd, maar voor een
goede beschrijving van de circumstellaire schijf zijn tweedimensionale modellen nodig.
Dit proefschrift bevat het eerste astrochemische model dat de gehele chemische evolutie
vanaf de kern tot aan de schijf volgt in twee dimensies.
Na een algemene inleiding in hoofdstuk 1 geven we in hoofdstuk 2 een gedetailleerde
beschrijving van het model. We gebruiken een zogeheten semi-analytische methode om
de dichtheden in de kern en de schijf te berekenen, alsmede de snelheden waarmee het
materiaal naar de ster toe valt. De dichtheden en snelheden komen goed overeen met wat
anderen hebben berekend met een meer gedetailleerde numerieke methode. Het voordeel
van onze semi-analytische methode is dat ze sneller is en dat we makkelijker het model
kunnen herhalen voor bijvoorbeeld een kern met een andere massa of andere draaisnelheid. De temperatuur heeft een grote invloed op de chemische evolutie. Zij kan niet goed
semi-analytisch worden berekend, dus gebruiken we hier een nauwkeurige numerieke
methode.
Als een eerste voorbeeld voor de chemische evolutie volgen we de twee veel voorkomende moleculen water en koolstofmonoxide (CO). Door de lage temperatuur in de
kern vriezen ze allebei vast op de stofdeeltjes voordat de instorting begint. Tijdens het
instorten verdampt CO al snel omdat het een lage bindingsenergie heeft. In een later stadium kan het weer bevriezen in de diepste delen van de schijf. Zij ontvangen geen directe
straling van de ster en zijn dus erg koud. Water heeft een hogere bindingsenergie dan CO
en verdampt pas als het binnen ongeveer 10 AE van de jonge ster komt.
In hoofdstuk 4 kijken we in meer detail naar de draaisnelheid waarmee materiaal vanuit de kern op de schijf terechtkomt. Het is al sinds de jaren tachtig bekend dat het neerkomende materiaal minder snel rond de ster draait dan het materiaal in de schijf. Hoofdstuk
2 bevat een beperkte oplossing voor dit probleem; hier leiden we een betere af, die vervolgens ook in hoofdstuk 3 gebruikt wordt. De nieuwe oplossing levert nieuwe inzichten
in de kwestie van kristallijn stof in schijven. Het stof in moleculaire wolken is geheel of
vrijwel geheel amorf, maar in schijven is het tot zo’n 30% kristallijn. De overgang van
amorf naar kristallijn vereist een temperatuur die veel hoger is dan de temperatuur waarbij
het kristallijne materiaal wordt waargenomen. Ons model laat zien dat stof dicht bij de
ster op de schijf terecht kan komen, daar heet genoeg wordt om van amorf over te gaan in
kristallijn, en vervolgens door de schijf heen weer van de ster af beweegt en zo in koudere
gebieden belandt.
De chemie uit hoofdstuk 2 wordt in hoofdstuk 3 sterk uitgebreid tot een netwerk van
ruim 400 moleculen en ruim 5200 reacties. We kijken nu niet alleen meer naar CO en
water, maar naar een twintigtal belangrijke zuurstof-, koolstof- en stikstofhoudende verbindingen. Het blijkt dat de meeste veranderingen in hun concentraties het gevolg zijn van
een klein aantal sleutelprocessen, zoals het verdampen van CO of de fotodissociatie van
water. Deze sleutelprocessen zijn weer terug te voeren op veranderingen in bijvoorbeeld
de temperatuur of de intensiteit van het stralingsveld.
In hoofdstuk 3 en 4 besteden we speciale aandacht aan kometen. Dit zijn rotsblokken
ter grootte van een paar tot enkele tientallen kilometers, die tijdens de vorming van het
zonnestelsel niet verder zijn gegroeid tot planeten. Op grond van de modelresultaten uit
202
Dit proefschrift
hoofdstuk 4 en waarnemingen gedaan door anderen, concluderen we dat kometen worden
gevormd uit kleinere brokken met verschillende oorsprong. Een deel van deze brokken is
nooit sterk verwarmd, zodat water altijd in ijsvorm is gebleven en de gehele chemische
samenstelling vrijwel hetzelfde is als die van moleculaire wolken. Een ander deel is juist
wel sterk verwarmd doordat het dichtbij de ster op de schijf terechtkwam. Daarna is het
naar het koudere gebied getransporteerd waar kometen worden gevormd. In dit materiaal
is al het water verdampt en later weer bevroren, en het heeft een chemische samenstelling
die duidelijk verschilt van die van een moleculaire wolk.
Hoofdstuk 5 beperkt zich tot CO en zijn isotoopvarianten.6 Onder invloed van ultraviolette straling (UV-straling) kan een molecuul CO uiteenvallen in de losse atomen C
en O. We herzien een model uit 1988 met nieuwe laboratoriumdata om de snelheid van
dit uiteenvallen te berekenen en om te kijken hoe de snelheid afhangt van de positie in
bijvoorbeeld een moleculaire wolk of een circumstellaire schijf. De meest voorkomende
isotoopvariant 12 C16 O blijkt in de meeste gevallen het langzaamst uiteen te vallen. Daardoor komen er relatief meer 13 C-, 17 O- en 18 O-atomen vrij. Dit gebeurt in een dusdanige
verhouding dat we de hoeveelheden 17 O en 18 O in meteorieten kunnen verklaren.
Het laatste hoofdstuk draait om de chemie van polycyclische aromatische koolwaterstoffen (PAK’s) in circumstellaire schijven. PAK’s zijn een groep grote, stabiele moleculen bestaande uit een kippengaasachtig skelet van koolstofatomen omringd door een
enkele rand waterstofatomen. We berekenen onder andere waar in de schijf de PAK’s
elektrisch geladen zijn en waar ze worden vernietigd door de UV-straling van de jonge
ster. We concluderen dat waargenomen straling van PAK’s in schijven in ruwweg gelijke mate afkomstig is van neutrale en geladen PAK’s. De PAK’s kunnen bovendien niet
kleiner zijn dan honderd koolstofatomen.
6
Isotopen zijn verschillende atomen van hetzelfde element, maar met een afwijkende massa. Twee van nature
voorkomende stabiele koolstofisotopen zijn 12 C en 13 C, met atoommassa’s van respectievelijk 12 en 13. De
drie natuurlijke zuurstofisotopen zijn 16 O, 17 O en 18 O.
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Publicaties
Refereed papers
1. The chemical history of molecules in circumstellar disks. II. Gas-phase species
R. Visser, S. D. Doty and E. F. van Dishoeck
Astronomy & Astrophysics, to be submitted (Chapter 3)
2. Sub-Keplerian accretion onto circumstellar disks
R. Visser and C. P. Dullemond
Astronomy & Astrophysics, to be submitted (Chapter 4)
3. PROSAC: a Submillimeter Array survey of low-mass protostars. II. The mass
evolution of envelopes, disks and stars from the Class 0 through I stages
J. K. Jørgensen, E. F. van Dishoeck, R. Visser, T. L. Bourke, D. J. Wilner,
D. J. P. Lommen, M. R. Hogerheijde and P. C. Myers
Astronomy & Astrophysics, in press
4. The photodissociation and chemistry of CO isotopologues: applications to
interstellar clouds and circumstellar disks
R. Visser, E. F. van Dishoeck and J. H. Black
Astronomy & Astrophysics, 2009, 503, 323–343 (Chapter 5)
5. The steric nature of the bite angle
W.-J. van Zeist, R. Visser and F. M. Bickelhaupt
Chemistry: A European Journal, 2009, 15, 6112–6115
6. Photodesorption of ices. II. H2 O and D2 O
K. I. Öberg, H. V. J. Linnartz, R. Visser and E. F. van Dishoeck
The Astrophysical Journal, 2009, 693, 1209–1218
7. The chemical history of molecules in circumstellar disks. I. Ices
R. Visser, E. F. van Dishoeck, S. D. Doty and C. P. Dullemond
Astronomy & Astrophysics, 2009, 495, 881–897 (Chapter 2)
8. Spatially extended PAHs in circumstellar disks around T Tauri and Herbig Ae stars
V. C. Geers, E. F. van Dishoeck, R. Visser, K. M. Pontoppidan, J.-C. Augereau,
E. Habart and A. M. Lagrange
Astronomy & Astrophysics, 2007, 476, 279–289
9. Dust sedimentation in protoplanetary disks with PAHs
C. P. Dullemond, T. Henning, R. Visser, V. C. Geers, E. F. van Dishoeck and
K. M. Pontoppidan
Astronomy & Astrophysics, 2007, 473, 457–466
219
Publicaties
10. PAH chemistry and IR emission from circumstellar disks
R. Visser, V. C. Geers, C. P. Dullemond, J.-C. Augereau, K. M. Pontoppidan and
E. F. van Dishoeck
Astronomy & Astrophysics, 2007, 466, 229–241 (Chapter 6)
11. Polycyclic benzenoids: Why kinked is more stable than straight
J. Poater, R. Visser, M. Solà and F. M. Bickelhaupt
Journal of Organic Chemistry, 2007, 1134–1142
12. c2d Spitzer-IRS spectra of disks around T Tauri stars. II. PAH emission features
V. C. Geers, J.-C. Augereau, K. M. Pontoppidan, C. P. Dullemond, R. Visser,
J. E. Kessler-Silacci, N. J. Evans, II, E. F. van Dishoeck, G. A. Blake,
A. C. A. Boogert, J. M. Brown, F. Lahuis, B. M. Merín
Astronomy & Astrophysics, 2006, 459, 545–556
13. Oxidative addition to main group versus transition metals: insights from the
Activation Strain model
G. T. de Jong, R. Visser and F. M. Bickelhaupt
Journal of Organometallic Chemistry, 2006, 691, 4341–4349
Conference proceedings
1. Chemical changes during transport from cloud to disk
R. Visser, E. F. van Dishoeck and S. D. Doty
IAU Symposium 251: Organic Matter in Space, 2008, 111–115
2. PAH chemistry and IR emission from circumstellar disks
R. Visser, V. C. Geers, C. P. Dullemond, J.-C. Augereau, K. M. Pontoppidan and
E. F. van Dishoeck
Molecules in Space and Laboratory, 2007, 102–105
220
Curriculum vitae
Ik ben geboren op 15 januari 1983, tien dagen voor de lancering van de Infrared Astronomical Satellite (IRAS), de eerste ruimtetelescoop ooit die volledig in het infrarood
werkte. Na negen maanden verhuisde ik met mijn ouders van Amersfoort naar Uithoorn,
waar later ook mijn broer en zus zijn geboren. In Uithoorn behaalde ik in 2000 mijn gymnasiumdiploma aan het Alkwin Kollege. Dat najaar begon ik met een studie scheikunde
aan de Vrije Universiteit Amsterdam. In augustus 2003 behaalde ik mijn bachelor, twee
jaar later gevolgd door mijn master; beide waren cum laude. Mijn hoofdvakonderzoek
voerde ik uit in de sectie Theoretische Chemie onder begeleiding van dr. Matthias Bickelhaupt (sindsdien benoemd tot hoogleraar) en drs. Theodoor de Jong. De resultaten hebben
geleid tot twee wetenschappelijke publicaties. Een vijfweeks bezoek aan prof. dr. Miquel
Solà op de Universitat de Girona in Spanje vormde de basis voor een derde publicatie.
Tijdens mijn bachelor was ik op een studentensymposium aanwezig bij de lezing
“Astrochemie: Van exotische ionen tot ‘geladen’ vragen”. Tot dan toe had ik sterrenkunde
altijd wel leuk gevonden, maar afgedaan als te natuurkundig om me er professioneel in te
verdiepen. Nu bleek sterrenkunde opeens een stuk scheikundiger te zijn! Deze hernieuwde kennismaking mondde in 2004 uit in een bijvakonderzoek op de Sterrewacht Leiden.
Na mijn afstuderen in september 2005 keerde ik daar terug voor een promotieplaats.
De Sterrewacht Leiden is een internationeel georiënteerd instituut, en daar heb ik
dankbaar gebruik van gemaakt. Ik ben op werkbezoek geweest bij Denison University in
Granville, Ohio, het MPIA in Heidelberg, het MPE in Garching, en de ETH in Zürich.
Een bijzondere ervaring was de waarneemsessie met de Very Large Telescope in Chili in
2006. Ik heb mijn werk gepresenteerd op congressen in St. Jacut de la Mer (Frankrijk),
Vidago (Portugal), Belfast, Parijs, Hong Kong, Amsterdam, Dalfsen (Nederland), Londen
en Rolduc (Nederland). Ook tijdens de werkbezoeken aan Granville en Zürich heb ik
lezingen gehouden over mijn onderzoek. Tevens heb ik publiekslezingen gegeven op de
Oude Sterrewacht in Leiden en op volkssterrenwachten in Amersfoort en Overveen. Als
werkcollegeassistent was ik tweemaal betrokken bij het college stralingsprocessen.
Als ik niet met astrochemie bezig ben, ben ik vaak te vinden op het honkbalveld van
de Keytown Hitters in Leiden of op een van de andere velden in Nederland. Ik honkbal
sinds mijn zevende, afgezien van een onderbreking van een paar jaar tijdens mijn bachelor. In 2006 heb ik mijn scheidsrechterslicentie behaald en sindsdien ben ik steeds vaker
actief als scheidsrechter en minder als speler. Ik heb inmiddels wedstrijden geleid tot en
met de eerste klasse. In oktober hoop in een tweede licentie te behalen waarmee ik kan
doorstromen naar de hoofdklasse, het hoogste niveau dat we in Nederland hebben. Mijn
wens is om het uiteindelijk te schoppen tot internationale toernooien als EK’s en WK’s.
Na mijn promotie blijf ik als postdoc nog een jaar in Leiden. Ik zal dan betrokken
zijn bij het project Water in Star-Forming Regions with Herschel (WISH) op de in mei
gelanceerde Herschel-ruimtetelescoop.
221
Nawoord
Hoewel er maar één naam op de kaft van dit proefschrift staat, is het geenszins het werk
van slechts één persoon. Deze laatste pagina is een dankwoord aan iedereen die iets
heeft bijgedragen. Om te beginnen is er natuurlijk de Sterrewacht Leiden als geheel. De
Sterrewacht is een groot instituut, maar dankzij de vele gezamenlijke activiteiten blijft
het prettig kleinschalig. De computergroep, het secretariaat en het andere ondersteunend
personeel staan altijd klaar om te helpen en zorgen ervoor dat alles soepel blijft draaien.
De astrochemiegroep is een mix van waarnemers, theoretici en experimentalisten, en
biedt daarmee een geweldige omgeving voor wetenschappelijk onderzoek. Vincent and
Jean-Charles, when I first arrived in Leiden, I couldn’t tell a T Tauri star from a supernova.
You turned me from a chemist into a proper astrochemist. Bastiaan, jouw hulp met de
chemische modellen – ook nog lang na je vertrek – heb ik zeer gewaardeerd. Lars, the
hot water and all the merde helped keep me sane during the final months. Edith, the same
goes for all the laughs we shared at coffee, lunch and other occasions. Karin, I enjoyed
our collaboration on the water photodesorption paper. Christian, Dave, meneer Dominic,
Elena, Guido, Herma, Isa, Jeanette, Nadine, Olja and all other Sterrewachters past and
present, it was a pleasure to share these corridors with you. To the staff at the MPE, the
MPIA, Denison and the ETH: the hospitality during my visits is much appreciated.
Steve, your name deserves to be printed here in capitals. If it weren’t for you, the
model never would have worked. I look back fondly on our many long discussions in
Granville and elsewhere. Kees, ook jij verdient een eervolle vermelding. Mijn tweede
bezoek aan Heidelberg vormde de basis voor het stervormingsmodel zoals het uiteindelijk
is geworden. Daarnaast ben ik je zeer erkentelijk voor de vele hulp met RADMC.
In de sectie Theoretische Chemie van de Vrije Universiteit Amsterdam leerde ik voor
het eerst hoe leuk wetenschappelijk onderzoek is. Matthias en Theodoor, jullie begeleiding was onmisbaar bij die eerste stappen. Aan mijn studietijd op de VU heb ik een aantal
goede vrienden overgehouden. Petra, Dianne, Galvin, Philip, Michiel, Danièle, Maaike
en Nanda, alle etentjes, bioscoopbezoeken en potjes Kolonisten hebben de afgelopen jaren voor de broodnodige afleiding gezorgd. Carlos, Els, Gideon, Jeroen, Manon, Petra,
Mike, Bart, Bart, Lennart, Robin en alle andere Keytown Hitters, met jullie beleefde ik
vele uren plezier op het honkbalveld. Fred, Henk, Jeroen, Rinus, Renée en alle collegascheidsrechters, jullie zorgden voor nog vele uren meer honkbalplezier.
Rolf, Tom, ik ben blij dat ik jullie al zo lang tot mijn vrienden mag rekenen. Onze
vakanties waren ieder jaar iets om naar uit te kijken en zorgden ervoor dat ik daarna weer
met nieuwe energie verder kon. Martin, Suzanne, ik bof dat ik zo’n geweldige broer en
zus heb. Zonder al die potjes Kolonisten, Carcassonne en Fase 10, zonder al dat bowlen,
poolen, PSone’en en PS3’en, zonder al die wederzijdse etentjes – kortom: zonder jullie –
zou het leven maar saai zijn. Papa, mama, bij jullie kon ik altijd terecht om de goede en
minder goede momenten te delen. Jullie zijn geweldig.
223
“If you have an apple and I have an apple and we exchange these apples then you and I
will still each have one apple. But if you have an idea and I have an idea and we
exchange these ideas, then each of us will have two ideas.”
– George Bernard Shaw
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