Formation of cold gas filaments from colliding supershells Evangelia Ntormousi

Formation of cold gas filaments from colliding supershells Evangelia Ntormousi
Formation of cold gas filaments
from colliding supershells
Evangelia Ntormousi
Formation of cold gas filaments
from colliding supershells
Ph.D Thesis at the faculty of physics
of the Ludwig–Maximilians–Universität München
Presented by Evangelia Ntormousi
from Thessaloniki, Greece
in Munich, on February the 22nd , 2012
1st Evaluator: Prof. Dr. Andreas Burkert
2nd Evaluator: Prof. Dr. Harald Lesch
Contents
Contents
vii
List of Figures
x
Abstract
xi
Zusammenfassung
xiii
1 Preface
1
2 The Interstellar Medium
2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Atomic Interstellar Gas . . . . . . . . . . . . . . . . . . . . .
2.2.1 Molecular Interstellar Gas . . . . . . . . . . . . . . . .
2.2.2 Ionized Interstellar Gas . . . . . . . . . . . . . . . . .
2.3 Dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Cosmic Rays . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 The system as a whole . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Models of the multi-phase ISM . . . . . . . . . . . . .
2.6.2 Cooling and Heating Processes of the Interstellar Gas
2.6.3 The modern view of the ISM: the role of turbulence .
3 Principles of Hydrodynamics
3.1 The equations of hydrodynamics . . . . . . .
3.1.1 The equation of continuity . . . . . .
3.1.2 The force equation . . . . . . . . . . .
3.1.3 The energy equation . . . . . . . . . .
3.2 Hydrodynamical Instabilities . . . . . . . . .
3.2.1 The Non-linear Thin Shell Instability
3.2.2 The Kelvin-Helmholtz Instability . . .
3.2.3 The Thermal Instability . . . . . . . .
3.3 The instabilities combined . . . . . . . . . . .
3.4 Some comments on turbulence . . . . . . . .
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3
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35
viii
CONTENTS
4 Previous Work
4.1 Molecular Cloud Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Shell fragmentation and collapse . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Numerical Method
5.1 Solving the equations of hydrodynamics on a grid . . . . . . .
5.2 The RAMSES code . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Sources and sinks of energy . . . . . . . . . . . . . . . . . . .
5.3.1 Implementation of a new cooling and heating module
5.3.2 Implementation of a winds . . . . . . . . . . . . . . .
5.4 Creating turbulent initial conditions . . . . . . . . . . . . . .
5.5 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Formation of cold filaments from colliding shells
6.1 Shell collision in a uniform diffuse medium . . . . . . . . . . . . . . . . . . . .
6.2 Shell collision in a turbulent diffuse medium . . . . . . . . . . . . . . . . . . .
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7 Morphological features
7.1 General . . . . . . .
7.2 Velocity dispersions .
7.3 Sizes . . . . . . . . .
7.4 Clump evolution . .
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8 Metal enrichment of the clouds
8.1 Setup of the simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Metal enrichment of the gas from the OB associations . . . . . . . . . . . . .
8.3 Metal enrichment of the clumps . . . . . . . . . . . . . . . . . . . . . . . . . .
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84
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9 Summary and Conclusions
91
Bibliography
99
of the cold clumps
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Acknowledgments
101
Curriculum Vitae
103
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Distance calculation ambiguity . . . . . .
The ”Pipe” dark nebula . . . . . . . . . .
The ”Shamrock” nebula . . . . . . . . . .
Dense gas filaments . . . . . . . . . . . . .
The Milky Way in various wavelengths . .
Cooling-heating equilibrium curve . . . .
McKee & Ostriker (1977) view of the ISM
Cooling and heating rates of the local ISM
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6
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18
20
3.1
3.2
3.3
3.4
Illustration of the Vishniac instability. . . . . . . .
Eddies created by the Kelvin-Helmholtz instability
Instabilities in phase space . . . . . . . . . . . . . .
Two-dimensional slab . . . . . . . . . . . . . . . .
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5.1
Energy and mass ejection from an ”average star” . . . . . . . . . . . . . . . .
47
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
Collision snapshots: uniform background . . .
Uniform background: zoom-in . . . . . . . . .
Gas phase histogram: uniform background . .
Gas fractions with time: uniform background
Number of clumps with time . . . . . . . . .
Collision snapshots: turbulent background . .
Turbulent background: zoom-in . . . . . . . .
Filament 1 . . . . . . . . . . . . . . . . . . . .
Filament 2 . . . . . . . . . . . . . . . . . . . .
Filament 3 . . . . . . . . . . . . . . . . . . . .
Filament evolution . . . . . . . . . . . . . . .
Phase histogram: turbulent background . . .
Gas fractions: turbulent background . . . . .
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7.1
7.2
7.3
7.4
Condensing clouds . . . . . . . . .
Rotating clouds . . . . . . . . . . .
Clouds with random motions . . .
Clouds with low internal velocities
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x
LIST OF FIGURES
7.5
7.6
7.7
7.8
7.9
Cloud velocity dispersions . . . . . . . . .
Velocity dispersion with time . . . . . . .
Clump size distributions . . . . . . . . . .
Clump size over Jeans length distributions
Cores in the phase diagram . . . . . . . .
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72
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8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
AMR-unigrid comparison . . . . . . . . . . . . . . . . . .
Density and pressure histograms . . . . . . . . . . . . . .
AMR-unigrid comparison: Mass fractions with time . . .
Shell collision with metal advection . . . . . . . . . . . . .
2d histograms: density-metallicity distribution . . . . . .
Metal distribution of the clumps . . . . . . . . . . . . . .
Metal content of the clumps with distance from the stars
Metal content of the clumps with polar angle . . . . . . .
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Abstract
The aim of this work is to study the dynamics of the interstellar gas, with an eye towards
the formation of the Cold Neutral Medium (CNM), gas with high densities (nH ≃ 100 cm−3 )
and low temperatures (T < 300 K). In particular, we want to explore the mechanisms that
can lead to the filamentary structure typical of these clouds.
A natural mechanism for the formation of large amounts of cold gas in a small region
are converging flows. This type of problem has been studied extensively in the literature in
the form of academic, infinite high-Mach number flows that collide at a perturbed boundary.
The novelty of our approach lies in simulating converging flows by using physically motivated
parameters, extracted from models of young stellar feedback. This feedback creates finite,
structured shocks which are the seeds for turbulence at the shock collision interface.
In our numerical experiments two superbubbles, blown by the violent feedback of OB
associations, collide and fragment to form cold filaments through various fluid instabilities.
The amount of cold gas formed, the morphologies and the kinematics of the cold gas clumps
are roughly in accordance with observed properties of such structures. In particular, our
simulations are able to capture the observed filamentary structure of the cold clouds and
their fractal character.
The metal content of the clumps is tracked through an advected quantity in the code,
inserted in the feedback regions along with the rest of the stellar feedback. Our studies show
that little or no enrichment happens from one stellar generation to the next, if only turbulent
diffusion is considered.
Although further work is needed in order to study the details of the clump structure and
its dependence on the OB association parameters, this work opens a new path to the study
of the ISM dynamics by showing that complex characteristics of the observed structures can
be reproduced by applying physically motivated initial conditions.
xii
LIST OF FIGURES
Zusammenfassung
Das Ziel dieser Arbeit ist es, die Dynamik des interstellaren Gases zu studieren, insbesondere die Entstehung des kalten neutralen Mediums, d.h. Gases mit hohen Dichten (nH ≃ 100
cm−3 ) und niedrigen Temperaturen (T < 300 K). Wir wollen die Mechanismen, die die typische filamentartige Struktur dieser Wolken erzeugen, im Detail verstehen.
Ein natürlicher Mechanismus für die Entstehung großer Mengen kalten Gases in einem
kleinen Gebiet sind konvergente Flüsse. Diese Art von Problem wurde in Form von akademischen Flüssen mit unendlich großen Mach Zahlen, die an einem gestörten Rand kollidieren,
ausgiebig in der Literatur studiert. Unser Ansatz ist insofern neu, als das wir konvergierende
Flüsse simulieren, indem wir physikalisch motivierte Parameter verwenden, die wir aus Modellen für das sogenannte Feedback junger Sterne erhalten, die aus Beobachtungen entstanden
sind. Dieses Feedback erzeugt endliche, strukturierte Schocks, die als Keime fuer Turbulenzen
an den von kollidierenden Schocks erzeugten Grenzflüchen dienen.
In unseren numerischen Experimenten kollidieren zwei Superbubbles, erzeugt durch das
heftige Feedback von OB-Sternansammlungen, miteinander und fragmentieren, wobei sich auf
Grund verschiedener, in Fluessigkeiten typischer, Instabilitäten kalte Filamente ausbilden.
Die Menge an dabei geformtem, kaltem Gas sowie die Morphologie und Kinematik der so
erzeugten kalten Gasklumpen stimmt in guter Näherung mit den für derartige Strukturen
beobachteten Eigenschaften überein. Insbesondere erwähnenswert ist, das unsere Simulationen in der Lage sind, die beobachtete, filamentartige Struktur und den fraktalen Charakter
der kalten Molekülwolken zu reproduzieren.
Innerhalb des Codes verfolgen wir die Metallizität durch eine Größe, die zusammen mit
dem restlichen stellaren Feedback in der OB-Sternansammlung erzeugt wird, aber selbst keine
speziellen physikalischen Eigenschaften hat. Unsere Studien zeigen, das von einer stellaren
Generation zur nächsten, sofern nur die Ausbreitung durch Turbulenzen berücksichtigt wird,
wenig oder gar keine Anreicherung mit Metallen erfolgt.
Obgleich zum Verständnis der Details der Entstehung der klumpigen Strukturen und
deren Abhngigkeit von den Parametern der OB-Sternansammlung weitere, detailiertere Studien nötig sind, öffnet diese Arbeit jedoch neue Wege, die Dynamik des Interstellaren Mediums
zu studieren, da sie beweist, das die komplexen Charakteristika der beobachteten Strukturen
durch das Anwenden physikalisch motivierter Parameter in den Anfangsbediungungen reproduziert werden können.
xiv
LIST OF FIGURES
Chapter
1
Preface
One Sun by Day, by Night ten thousand shine
And light us deep into the Deity.
Edward Young, ”Night Thoughts”
In the late eighteenth century, Sir Frederick William Herschel came across a puzzling fact
in his deep sky observations: there were regions on the sky where there appeared to be no
stars. This was not an easy observation to interpret. Did stars have a tendency to leave
voids in the Universe? As it so ofter happens in science, the explanation was as simple,
elegant and exciting as it was surprising and unexpected. These celestial voids turned out
to be ”dark clouds”, nebulae which obscure the background starlight almost entirely. This
discovery revolutionized the way scientists view the cosmos, as it gave the first indication that
the Galaxy contains non-stellar matter which fills the vast space between the stars.
Support for this interpretation actually came much later, from Johannes Hartmann’s
(1904) discovery of an absorption line in stellar spectra that could only be explained by
interstellar absorption. Three decades later, in 1930, Robert Julius Trumpler showed that,
apart from Herschel’s ”dark clouds”, interstellar space also contained a more diffuse gaseous
material, thus discovering one more of what we now know to be at least four gaseous phases
of Interstellar Matter. In the years that followed, the existence of all the other constituents
of the Interstellar Medium were discovered, dust, magnetic fields, and cosmic rays.
It is nowadays a known fact that the constituents of Interstellar Matter not only affect the
way stellar radiation reaches the Earth, but they are also important ingredients of the Galaxy
as a complex system, especially in due to their active role in the process of star formation.
Interstellar gas is the building material for stars and it carries the energy and mass ejected
by them during their lives. Magnetic fields are thought to play an important role in the
regulation of star formation. Dust, among others, provides an important cooling path for the
gas and is an necessary catalyst for the formation of hydrogen molecules, as well as for other
chemical reactions.
As a complex system the Interstellar Medium makes a fascinating area of study on its
own, but it also holds great part of the solution to problems at much larger scales, as the
laboratory of Galactic evolution. At the time this text is written, the mystery of galaxy
formation and evolution is yet to be solved. Although the ΛCDM model for dark matter has
2
CHAPTER 1. PREFACE
been very successful in reproducing the large-scale structure of the mass in the Universe, we
are still far from understanding the mechanisms by which galaxies form their stars and the
details of metal enrichment from one generation of stars to the next. Moreover, the processes
which shape the distribution of masses in a young stellar cluster in an apparently universal
way are still under debate and are possibly connected to the drivers of turbulence in the
interstellar gas.
I strongly believe the answers to all these questions lie on understanding the physics of
the interstellar medium. However, this thesis does not aspire to answer them. Instead, it
would be considered successful by this author if it managed to break them down in simpler,
more straightforward thought problems. More specifically, this is a thesis about the complex
process of forming cold and dense gas out of the warm diffuse component of the interstellar
medium.
The novel ingredient in this work is that the hot energetic component of the interstellar
medium is controlling this transition, implemented in a self-consistent way, in the form of
winds and supernova explosions from stellar models rather than as academic, perfect shocks.
The non-linear nature of this system is what yields the complex cold structure we observe.
Furthermore, this thesis studies the process of metal mixing between these components, with
an eye towards the possible enrichment of the next generation of stars.
This counting as the first chapter of this work, the second chapter overviews the properties
of interstellar matter as they are known at this time. The third chapter includes the theoretical
background needed to follow the phenomena studied in this thesis, essentially an overview
of hydrodynamics. The fourth chapter presents the current state of the art in the field of
cold cloud formation and hot shell expansion, thus giving a more detailed motivation for this
work. The fifth chapter is dedicated to the description of the techniques used for the numerical
modeling of the systems under study, and the next chapters are dedicated to illustrating our
results. The very last chapter summarizes and concludes the thesis.
Chapter
2
The Interstellar Medium
2.1
General
Thanks to the continuous advances in astronomical observations it is now known that the
space between stars is not empty, but rather filled with gas, dust and energetic particles.
The different ways in which stars interact with this material, which is collectively called the
Interstellar Medium (ISM), leave spectacular tracks for astronomers to follow : HII regions,
reflection nebulae, luminous shocks and supernova remnants. By studying these fascinating
objects it has been discovered, for example, that the matter which fills the space between
stars is a mixture of hydrogen, helium and a small amount of heavier elements, in a gaseous
or in a solid state.
Most of the solid state of the ISM is dust, an ingredient with very important effects on the
radiation we receive from the ISM. The gaseous state itself comprises many phases, ranging
from molecular, to cool atomic, to ionized gas. In this Chapter we give an overview of the
known properties of each of the gas phases, as well some general characteristics of interstellar
dust. We also briefly mention some features of the Galactic magnetic fields and cosmic rays,
since they are important sources of pressure and also tracers of many physical processes in
the ISM.
2.2
Atomic Interstellar Gas
The phase of interstellar gas we refer to as neutral atomic gas is defined by the absence of
Lyman continuum photons, meaning that hydrogen is mostly neutral. However, this term
should not be interpreted in a very strict sense, since other species can still be ionized in this
medium due to the dependence of interstellar extinction on wavelength. In fact, energetic
charged particles traveling in the Galaxy, known as cosmic rays, are known to cause partial
ionization even in the interiors of very dense clouds. What we call neutral atomic gas is
actually not purely atomic, either. Some molecular lines have been detected in this medium,
in the same way that atoms are known to exist in the coronas of molecular clouds.
The information contained in Chapter 2 is mostly a combination of material from the following sources:
Tielens (2005), Lequeux (2005), Ferrière (2001) and Hollenbach & Thronson (1987), unless otherwise stated.
Individual references for the information quoted here can be found in these reviews.
4
CHAPTER 2. THE INTERSTELLAR MEDIUM
There are several ways to study the neutral atomic ISM observationally, in emission and
in absorption. Since interstellar gas is mostly hydrogen, we can study many of its properties
by observing the well-known 21 cm line radio emission from this atom.
This line results from the hyperfine structure of the hydrogen atom, caused by the interaction of the magnetic moments of the electron and the proton within the atom. The transition
is strongly forbidden, with a spontaneous emission probability as low as Aul = 2.87×10−15 s−1 .
Nonetheless, due to the enormous amounts of hydrogen in interstellar space we can indeed
observe it. It is usually detected in absorption in front of continuum radio sources or in
front of 21 cm emission of warmer gas. Moreover, since the lifetime of the upper sublevel is
much longer than the average collision timescale in ISM conditions, local thermodynamical
equilibrium (LTE) can be assumed for the gas when interpreting observations.
The main applications of 21 cm observations are to estimate the mass, the distribution
and the kinematics of atomic hydrogen. From these observations we know, for example, that
atomic hydrogen amounts to at least half of the mass of the interstellar medium of our Galaxy.
This is roughly true for most of the mass in other galaxies as well. Since the 21 cm line is
usually assumed to be optically thin (meaning that we assume all the photons produced in
the emitting source can escape it), which might not always be the case, these masses are
mostly upper limits.
By comparing absorption and emission spectra from the same region of the sky, we can
identify two phases of neutral atomic hydrogen. A dense (nH ≈ 10 − 50 cm−3 ), cold (T ≈
100 − 300 K) phase, which is usually called the cold neutral medium (CNM), and a more
diffuse (nH ≈ 0.1 − 0.3 cm−3 ), warm (T ≈ 10000 K) phase, called the warm neutral medium
(WNM). The WNM is extended in interstellar space and is mostly responsible for what we
observe in emission, while the CNM forms discrete, highly inhomogeneous and structured
filaments, which appear in the spectra as absorption peaks. We should point out that this
extremely inhomogeneous structure of the CNM is believed to be a result of turbulence in
interstellar space, a process which is actually the main subject of this thesis.
The WNM contains as much mass as the CNM in our Galaxy, but it forms a thicker disk,
with a scale height of |z| = 186 pc, compared to |z| = 106 pc of the CNM. From absoption
measurements we know that local cold clouds have a velocity dispersion of 6.9 km/sec and
from emission measurements that the WNM has a velocity dispersion of about 9 km/sec.
The distribution of H I in the Galaxy can also be deduced from 21 cm observations,
although not without some uncertainties. In general, it is very difficult to discern the exact
distribution of interstellar gas in the Galactic disk, the reason for this coming from the
available ways to measure a cosmic object’s distance from us. For example, to measure the
distances to stars, when they are too far away to use parallaxes, we can use the difference
between their apparent (m) and their absolute magnitude (M):
m − M = 5 log
d
−A
10pc
(2.1)
where A is the extinction at the observed wavelength, primarily caused by interstellar dust.
For the gas, however, it is not possible to measure distances this way, unless we know from
the nature of the process that it is intrinsically spatially connected to stars. Examples of such
objects are H II regions, formed by young massive stars as they ionize their surroundings
(–See Figure 2.3 and Section 2.2.2). Another way to measure the distance to an object, which
also comes from equation 2.1, involves determining the distance of stars unobscured by the
cloud. This is applicable to dark, neutral clouds (–See Figure 2.2).
2.2. ATOMIC INTERSTELLAR GAS
5
Of course, a spatial correlation between stars and a region of the ISM only happens for a
few clouds. In absence of such a fortunate coincidence, the only way to determine the distance
to a Galactic gaseous source is through its line-of-sight velocity, induced from its emission
or absorption spectrum. In the case of atomic hydrogen velocity estimates come from 21 cm
hydrogen line observations. By making 21 cm obsrvations for many different lines of sight and
correlating the resulting velocities with a model for the Galaxy’s differential rotation curve,
we can calculate the object’s position in the Galaxy. It should be mentioned here that in
emission this line probes the CNM and the WNM together.
Figure 2.1 illustrates a problem intrinsic to this method. For observations of the inner
Galaxy (regions closer to the center of the Galaxy than the Sun) this method gives two
possible distance estimates for a given object, yielding an uncertainty for the distribution of
atomic hydrogen in these regions. It is still possible, though, to calculate the surface density
of H I. So far it has been found to be constant up to a radius of about 4 kpc from the center of
the Galaxy, and decreasing beyond this distance. The vertical structure of atomic hydrogen
is also estimated to be roughly constant with radius.
The distance uncertainty does not exist for the outer Galaxy, where we can make exact
distance measurements from 21 cm observations. In that part of the Galaxy, we can discern
the spiral arm pattern of the Galactic disk. Until 2008, our Galaxy was thought to have
four spiral arms, named Perseus, Cygnus, Carina and Orion after the constellations where
they were projected. However, high-resolution observations with the Spitzer telescope, gave
significant indications that our Galaxy possesses a bar in the inner parts and only two spiral
arms, like most observed barred galaxies (Churchwell et al., 2009). It is also known that
atomic hydrogen exists at radii of at least 30 kpc from the center of the Galaxy and hat the
outer part of the gaseous Galactic disk is warped.
Another important observable of interstellar neutral gas are the fine-structure lines of
certain atoms, such as C I, C II, N I, N II, O I, O II and O III, to name a few. Fine-structure
interactions are interactions between the orbital momentum of the electrons in an atom and
their total spin and for these species, they are mostly found in the far-infrared. These lines
are the main coolants of the atomic interstellar medium, as we will explain in Section 2.6.2.
Finally, we can infer the chemical composition of interstellar matter from absorption lines
observed in the spectra of stars. We can distinguish the lines coming from interstellar gas
from the intrinsic stellar lines because they have a fixed wavelength, unlike the periodically
Doppler-shifted lines from the star, and because they are much narrower. The abundance of
hydrogen, for example, is calculated by fitting a theoretical profile to the observed Lyα line.
Once the hydrogen column density is known, the elemental abundances of heavier atoms are
expressed in terms of the hydrogen abundance. If the observed line comes from a species
which is expected to be mostly neutral in the atomic ISM, then the procedure is similar to
that for Lyα. If more than one ionization state of the element is observed, then the abundance
is determined by solving the ionization equilibrium for the observed species.
2.2.1
Molecular Interstellar Gas
Although by far the most abundant molecule in the Galaxy is H2 , there are many more
molecules in the ISM (more than 100 known so far) some of them very complex. We can
find molecular emission or absorption due to electronic transitions, like in atoms, but also
due to vibrational and rotational transitions, in the spectra of the ISM, of the envelopes of
asymptotic giant branch stars and of comets.
6
CHAPTER 2. THE INTERSTELLAR MEDIUM
Figure 2.1: Illustration of a line of sight towards the inner galaxy and through the Galactic disk. For a given line
of sight velocity, there are two possible distances. (Image Credit: Ferrière (2001))
Vibrational transitions of molecules come from stretching, bending and deformation modes.
Each vibrational transition can be further decomposed into rotational transitions, so they appear in spectra as ro-vibrational bands. The typical energies for vibrational transitions are of
the order of a fraction of an eV so the bands are observable in the near-infrared. Rotational
transitions, on the other hand, have energies of the order of meV and we can usually observe
them in sub-mm to cm wavelengths.
Molecular electronic transitions are of the order of a few eV and are generally found in the
far-UV. The most important results from such measurements come from H2 emission observations, owing, of course, to the high abundance of this molecule. The highest intensity lines can
yield interstellar H2 abundances. Comparison of molecular to atomic hydrogen abundances
in many lines of sight has shown that, above a gas column density of about N(H I)≥ 1021
cm−2 interstellar gas is almost entirely molecular. This has important consequences for theoretical models of the Galactic ISM where one needs to include some measure of the molecular
2.2. ATOMIC INTERSTELLAR GAS
7
component without explicitly modeling the chemistry of atomic to molecular phase transition.
Ultraviolet observations are possible in relatively diffuse environments, where interstellar
extinction from dust is not significant. Nevertheless, most of the molecular gas in our Galaxy
is located in large and dense structures, called molecular clouds. These clouds contain a
lot of dust due to their high densities, which makes UV observations impossible. This, in
combination with the fact that H2 , being a symmetric molecule, has no permanent moment
of inertia and thus no permitted transitions in the radio regime, has led astronomers to seek
lines from other molecules as tracers of the internal structure of dense molecular clouds.
The next most abundant molecule in these clouds, which in addition has a rotational
transition in the radio regime, is CO. Its J = 1 → 0 rotational transition at 2.6 mm can be
used, in a similar way as the 21 cm line of H I, for mapping the large-scale distribution of
dense molecular gas in the Galaxy, since it is unaffected by dust extinction. Such surveys
have yielded a very high concentration of molecules in a region of 0.4 kpc radius around the
center of the Galaxy, and a ring structure at radii between 3.5 and 7 kpc. It has also been
found that molecular gas follows the Galactic spiral pattern very closely in the outer Galaxy
and less dominantly so in the inner Galaxy, where the ring structure dominates (–see Figure
2.5). Molecular gas is mostly confined at the midplane of the disk, with a scale height of
about 81 pc.
High-resolution CO observations have shown that the mass distribution of molecular
clouds is very hierarchical, with large structures containing ever smaller and denser cores.
Hydrogen densities in these clouds range from 100 to 106 cm−3 , organized in an almost fractal structure.
It is widely believed that molecular clouds are gravitationally bound, although there are
uncertainties in what defines the limits of a cloud. In any case, that the vast majority of
observed molecular clouds are in the process of gravitational collapse. In particular, it is
the dense cores within them that give birth to stars. After star formation has started in a
cloud, the remaining molecular gas is believed to be quickly photodissociated by the intense
radiation from the young stellar objects. The complex subject of star formation from dense
molecular cores will not be treated in this thesis.
We will provide some more observational facts about molecular clouds in Chapter 4, and
we want to point out here that most of them are still derived from CO observations. Another
way to get reliable mass estimates in the densest clouds is from dust extinction itself, a matter
we will explain further in Section 2.3, where we will discuss the properties of interstellar dust.
An important class of molecules in the ISM are Polycyclic Aromatic Hydrocarbons (PAHs).
These are large molecules, the study of which is important, not only for understanding the
chemistry of the ISM, but also for estimating its radiative heating and cooling rates, as we will
see in Section 2.6.2. These molecules are believed to be the origin of observed infrared bands.
This emission is a fluoresence effect, in which a FUV photon absorbed by the molecule leads
to electronic excitations, with the energy re-emitted in the IR through vibrational modes of
the molecule.
2.2.2
Ionized Interstellar Gas
The brightest stars in the Galaxy, O type stars, produce large amounts of high-energy photons,
which are able to ionize hydrogen and helium around them. The energy required to ionize
hydrogen is 13.6 eV, so photons with higher energies (UV wavelengths) will create a region of
ionized hydrogen around the star. The remaining energy of photons which produced ionization
8
CHAPTER 2. THE INTERSTELLAR MEDIUM
Figure 2.2: An example of a dark
cloud is the ”Pipe” nebula, named after its elongated shape which resembles a smoking pipe. This nebula is
currently not star-forming and it has a
filamentary, self-similar structure typical of molecular clouds. Located on
the sky in the constellation of Ophiuchus, it is a small structure, of about
5 pc in projected length. This image shows dust absorption overlayed
on the background starlight. (Image
Credit: ESO/Yuri Beletsky)
is deposited in the free electrons of the plasma as thermal energy, effectively heating the gas.
The ionized regions around O stars are called H II regions, after the fact that hydrogen in
these regions is fully ionized.
From their natural connection to very bright young stars we know that H II regions have
the same distribution as O-type stars in the Galactic disk, following the spiral density pattern
in the disk midplane and with a vertical scale height of about 80 pc.
The simplest description of an H II region, although not entirely accurate in most cases,
is the Strömgen sphere. The Strömgen sphere is defined as a region inside which the number
of atom photoionizations by the stellar radiation per unit time is equal to the number of
recombinations of the ions with free electrons. Since the absorption of UV photons by neutral
hydrogen is very efficient, a Strömgen sphere has very well defined boundaries. Assuming a
uniform medium composed entirely by hydrogen, the radius of a Strömgen sphere is given by
RS = 30pc
N48
nH ne
1
3
(2.2)
where N48 is the number of ionizing photons released by the star per unit time, in units of
1048 s−1 , nH is the neutral hydrogen number density and ne is the electron number density.
H II regions produce continuum radiation as well as line emission. The continuum is thermal Bremsstrahlung in radio wavelengths, produced by free electrons as they are decelerated
by the electric field of the ions. Thermal Brehmstrahlung can give us information on the
temperature of the plasma, which in this case is about 8000 K. Incidentally, the same value
can be obtained by theoretical considerations, assuming that photoelectric heating balances
radiative cooling inside the ionized sphere surrounding the star. This gas is therefore called
”Warm Ionized Medium”, to distinguish it from an even hotter phase we know exists in the
ISM.
In a wide range of wavelengths, free-bound continuum emission is also produced by the
recombinations of free electrons with ions. This process gives rise to characteristic discontinuities , caused by recombinations to a hydrogen energy level. The less probable, but important
for metastable atomic levels mechanism of two-photon radiative de-excitation also becomes
important for H II regions, especially at wavelengths near 400 nm.
Naturally, we can also observe the permitted recombination lines of H, He and other
2.2. ATOMIC INTERSTELLAR GAS
9
elements from H II regions. These lines are very useful for determining dust extinction and
elemental abundances of the ionized gas. Due to the low densities in these environments, we
can also observe forbidden lines if C, O and other species.
However, not all the ionized gas in the Galaxy is confined in H II regions. Indeed, we
can anticipate the existence of diffuse ionized gas in the interstellar medium by raising the
assumption of a uniform medium surrounding the star, which enters the calculation of the
Strömgen radius. Since the position of the ionization front is inversely related to the gas
density, if one side of the surrounding medium is slightly more rarefied, or if the surrounding
gas is stratified in density (like the Galactic ISM along the vertical direction), we expect the
ionized gas to escape in the surrounding space in the direction of the less dense gas, extending
to distances larger than the Strömgen radius given by equation 2.2. This effect is called the
”champagne effect”. Ionized gas from associations of young massive stars can also escape in
surrounding space via the large cavities opened by the combination of their winds. It is not
surprising then that, although stars are created in the very dense environments of molecular
clouds, ionized gas can be found up to relatively high Galactic altitudes.
The distribution of diffuse interstellar gas has been deduced by the scattering of pulsar
signals by the free electrons of the ionized gas, showing that this is actually the case. The
interaction of the pulsar photons with the free electrons reduces the group velocity of the pulse
proportionally to the wavelength. Then the spread in arrival times of different wavelengths
within a pulsar signal will be proportional to the column density of free electrons along the
line of sight to the pulsar. If we know the distance to the pulsar independently, we can deduce
the number density of free electrons in the diffuse medium. With this method it has been
found that the distribution of diffuse ionized gas in the Galaxy comprises a thin, annular
component at about 4 kpc from the center in the radial direction, probably connected to the
molecular annulus at that distance and a more extended, thick disk, with a Gaussian scale
height of at least 20 kpc. The average number density of this medium is of the order of 10−3
cm−3 .
In addition to the warm ionized gas, there is an even hotter ionized phase of the ISM,
usually referred to as the hot interstellar medium (HIM), with a temperature of about 106 K,
which emits radiation in soft X-rays. This gas comes from supernova explosions when stars
reach the end of their lives. It is also produced in large cavities, created by large stellar OB
associations. It is therefore always directly related to the star formation rate of a galaxy. In
these regions, elements such as oxygen get collisionally ionized to give emission and absorption
lines of species such as O VI and O VII. Since interstellar X-ray observations probe emission
from distant hot gas superimposed with absoption from intervening clouds, it is difficult to
discetn the distribution of this hot gas in the Galaxy. We do know, however, that this gas
reaches very high Galactic altitudes and that it occupies some 80-90% of the volume in the
Galactic disk.
X-ray emitting gas, in particular O VI, has also been associated with structures in the
halo of the Galaxy called high-velocity clouds, which were first identified by 21 cm observations. These clouds are moving with velocities of several tens of kilometers per hour and are
believed to be falling into the potential well of our Galaxy. The X-ray gas connected to these
clouds could be coming from photoionization by a diffuse UV background, or it could be a
thermally unstable front between the cold gas contained in these clouds and an even hotter
gaseous halo, which could be detectable in soft X-rays in absorption. For a more detailed
treatment of this matter, which is outside the scope of this thesis, we refer the reader to
Ntormousi & Sommer-Larsen (2010) and references therein.
10
CHAPTER 2. THE INTERSTELLAR MEDIUM
Figure 2.3: The ”Shamrock nebula”,
located at about 10 pc from the Sun,
at the outer edge of our local spiral
arm. It is a typical example of an HII
region, where a central O type star illuminates its parent cloud with strong,
ionizing UV radiation. In this case the
star is CY Camelopardalis. The image is taken in the infrared band, at
wavelengths of 12 and 22 µm, which
is emitted by the dust as it gets heated
by the central star. (Image Credit:
NASA/JPL-Caltech/UCLA)
2.3
Dust
The importance of dust in the physics of the ISM and in the interpretation of astronomical
observations in general cannot be overstated. Since dust is very well mixed with the interstellar gas, it is of extreme importance to understand its effects on the observed radiation and
take them into account both when analyzing observations and when producing theoretical
models of the ISM.
Dust is the main cause of the ISM opacities at wavelengths longer than the Lyman discontinuity. It is also a cause of heavy metal depletion, since it keeps a large portion of the
local ISM metals locked up in its grains. In addition, dust in a very important catalyst for
many chemical reactions in the ISM. It provides not only a surface on which elements can
combine, but also a third body to receive the excess energy from certain chemical reactions.
This is one of the reasons why in many regions the dust content is an indication of the gas
metallicity.
The alignment of elongated dust grains with the magnetic field of the Galaxy causes the
linear polarization of the radiation that passes through the ISM, giving us a means to observe
the Galactic magnetic field.
The most straightforward way to observe dust in the Galaxy is by the reddening effect
it has on background light sources. From equation 2.1 we know that the difference between
the absolute and the observed magnitude of a star depends on the dust extinction along the
line of sight. By measuring the magnitude difference between a reddened star and a nearby
star of the same spectral type, which is unaffected by the intervening dust cloud, we can
calculate the color excess caused by the dust. Since stellar radiation covers a wide range of
wavelengths, from this process we obtain the color excess as a function of wavelength. The
2.4. COSMIC RAYS
11
resulting curve, called extinction curve, for the ISM shows an increase in extinction with
increasing wavelength. This is the reason why dust extinction is also called reddening.
By studying extinction curves we can derive the composition and sizes of interstellar dust
grains. For instance, the presence of a ”bump” in extinction curves in the UV, around 217nm
indicates the presence of graphite particles. Infrared extinction bands, on the other hand,
point to the presence of silicates. Absorption by dust is also a strong indicator of the total
dust volume, if the optical properties of the dust particles are known.
Dust absorbs FUV photons and re-emits them in the infrared. Most of the electrons
excited by the UV photons that hit the dust do not reach the surface of the grain. Their
energy is left in the grain, which is thus heated and emits thermally in the infrared. This
process has important implications for the heating and cooling of the ISM, as we will see in
Section 2.6.2, but it also provides a means of observing the structure of the gas with space
observatories which are unaffected by the Earth’s atmosphere. A spectacular image of the
structure of the ISM as traced by dust emission in the infrared, as seen by the Herschel
satellite, is shown in Figure 2.4.
It follows from simple considerations of the way a solid particle can interact with electromagnetic radiation, that the strongest interaction will happen for particles of similar size to
the incident wavelength. The fact that extinction curves depend on a wide range of wavelengths indicates that interstellar dust particles have a distribution of sizes. Although the
shape of the distribution of dust particle sizes is still debated, the grain radii are known to
vary between about 0.005 and 1 µm.
Solid particles do not only absorb, they also scatter background light. Scattering is dominated by the largest dust grains and is what creates the ”reflection nebulae”, usually bluecolored reflections of starlight by a cloud containing dust.
Although the mechanisms by which dust can be destroyed are numerous, shocks and
ionizing photons being ubiquitous in the ISM and destructive for these particles, the origin
of dust in the ISM is still poorly understood. The standard picture of grain creation is from
planetary nebulae and from cool stellar winds. In both cases, the temperature of the gas
should drop enough for the first nucleation to occur, namely for the bonds between particles
to switch from strong molecular to weaker, inter-molecular bonds. For this to happen, the
metallicity of the gas should be high enough and its cooling rate should be fast. After
nucleation has begun, inter-particle collisions take over the growth of the grains. However,
it is not clear if the production rate from these mechanisms is enough to account for all
interstellar dust, given the rate at which shocks ans supernovae can destroy it Salpeter (1977);
Kochanek (2011); Draine (2011). On the other hand, the apparent element depletion inside
dense molecular clouds could indicate that they are a site where dust could be forming.
2.4
Cosmic Rays
Mostly for historical reasons, we refer to energetic charged particles reaching the Earth from
space as cosmic rays. These particles were initially thought to originate from the Earth itself
or from the atmosphere, until balloon experiments proved their extraterrestrial origin.
These particles are injected into the ISM by supernova explosions and by flares or coronal
mass ejections from late-type stars. Actually, the lowest-energy cosmic rays that reach the
Earth do come from the closest such star to the Earth, the Sun. Cosmic rays are accelerated to
the energies at which we observe them by turbulent magnetic fields and by crossing supernova
12
CHAPTER 2. THE INTERSTELLAR MEDIUM
Figure 2.4: Dense gas filaments in Polaris, as seen by the Herschel satellite at infrared wavelengths 250, 350 and
500 µm,, coming from dust and PAHs.
(Image credit:ESA/Herschel/SPIRE/Ph. André (CEA Saclay) and A. Abergel (IAS Orsay). )
2.4. COSMIC RAYS
13
Figure 2.5: A collection of images of the Milky Way in 10 different wavelength bands. From top to bottom: 1.
408 MHz radio continuum, mostly synchrotron emission as electrons move in the Galactic magnetic field. 2. 21
cm atomic hydrogen emission. 3. 2.4-2.7 GHz radio continuum, tracing hot ionized gas through the synchrotron
emission of free electrons. 4. J = 1 → 0 CO emission in the radio, tracing the Galactic molecular gas. 5.
Composite image of mid- and far-infrared, principal tracer of interstellar dust emission. 6. Mid-infrared, coming
from Polycyclic Aromatic Hydrocarbons. 7. Composite near-infrared image, probing unobscured radiation from
cool star. 8. Optical light, where we can see the regions obscured by dust. 9. X-ray emitting hot gas. Cold clouds
appear as shadows in this image. 10. Gamma rays, produced by cosmic ray collisions with hydrogen atoms in the
Galaxy and by the scattering of lower energy photons to higher energies as a result of collisions with cosmic ray
electrons. (Image and caption credit: NASA Multiwavelength Milky Way project (http://mwmw.gsfc.nasa.gov/)
and references therein)
14
CHAPTER 2. THE INTERSTELLAR MEDIUM
or other types of shocks as they propagate in the Galaxy.
Given their observed constitution and energy distribution, we could say a typical cosmic
ray particle is a proton of energy about 1-10 GeV. Protons actually make up about 85 % of
the cosmic ray particles, the rest being alpha particles and 3% heavier nuclei. Cosmic rays
also contain electrons, positrons and nuclei of elements such as Li, Be and B, which originate
from their interaction with ISM particles. Cosmic ray energies range from 109 eV to values as
hgh as 1020 eV, the latter termed ultra-high energy cosmic rays (UHECR). Their distribution
roughly follows a broken power-law behavior, starting from a slope of -2.75, becoming steeper
(slope=-3) after an energy of about 1015 eV and flatter again, with a slope of -2.5 after about
1018 eV.
The flow of cosmic rays to the Earth, especially at low energies, is regulated by the
magnetic field of the Sun. The solar wind carries a significant magnetic field, which can trap
these particles before they reach our atmosphere. This makes the exact shape of the cosmic
ray energy spectrum quite uncertain. Cosmic rays also follow the magnetic field of the Galaxy,
which makes them interesting tracers for studying its morphology (–see also Section 2.5).
The most important role played by cosmic rays in the ISM is that they provide a source
of pressure. They are also the main cause of ionization in very dense environments, such as
molecular clouds. The small degree of ionization they provide inside molecular clouds is of
extreme importance for the coupling of the gas to the magnetic field.
2.5
Magnetic Fields
Cosmic rays gyrate aroung the field lines of the Galactic magnetic field, emitting synchrotron
radiation (– see Figure 2.5). This radiation is an important probe of the Galactic magnetic
field, if certain assumptions are made for the cosmic ray number densities in the Galaxy. From
such modeling we know that the Galactic field on the plane of the disk can be decomposed
into a structured component, following the spiral density wave pattern, and a turbulent
component, with random direction. In the vertical direction it forms two disks: a thin disk
of scale height of about 150 pc, and a thick disk of scale height about 1500 pc.
Synchrotron emission interpretations are very uncertain since they depend on assumptions
of the cosmic ray number densities and of the magnetic field strength. An independent
measure of the strength of the magnetic field that can be used in such estimates, at least
for its component along the line of sight to the observer, is the Zeeman splitting of certain
atomic levels. Zeeman splitting is the result of the interaction of the magnetic moment of
the electrons in an atom with an external magnetic field. In the Galaxy this method can
be applied to the 21 cm hydrogen line (with some adjustments to account for insufficient
resolution that we will not discuss here). This of course renders it biased towards the densest
regions. From Zeeman splitting measurements we know that the Galactic magnetic field has
a typical strength of a few µG.
When a linearly polarized electromagnetic wave propagates through a magnetized plasma,
its plane of rotation rotates by an angle which is proportional to the square of its wavelength.
This phenomenon, called Faraday rotation, can also be used to probe the magnetic field of
the Galaxy, although only in ionized regions.
The role played by magnetic fields in the physics of the ISM is, on large scales, to confine
interstellar matter by providing a net pressure that can balance the self-gravity of the gas. It
also traps cosmic ray particles in the Galaxy, which is an additional source of pressure.
2.6. THE SYSTEM AS A WHOLE
15
In smaller scales, magnetic fields regulate the expansion of supernovae and superbubbles,
as well as the dynamics of individual clouds, since there is a sufficient degree of ionization in
all ISM phases for magnetic fields to be coupled to the gas. Magnetic fields have also been
argued to support molecular clouds against gravitational collapse.
2.6
The system as a whole
In the previous Sections of this Chapter we introduced the observed properties of the different
constituents of the Interstellar Medium seperately. This is mainly justified by their different
manifestations in observations and by the different physical state of each. Nonetheless, it has
probably become evident to the reader that the different phases of the ISM not only interact
with each other, but they also regulate the behavior of the Galaxy as a system.
Stars, especially massive ones, are the trigger of most of the processes in the ISM. Young
stars, especially of early types, expel large amounts of matter in very energetic winds. They
also provide large amounts of ionizing radiation, heating and ionizing the ISM around them.
The ionization either leads to the formation of well-defined H II regions, or escapes to form
the diffuse ionized medium.
Supernova explosions inject energies of the order of 1051 ergs to the gas surrounding the
star. What is more, in large associations of O and B stars, the combination of the stellar
winds and supernova explosions creates enormous cavities of hot gas, which compress the gas
around them. Such shells, being subject to cooling from various processes that we will explain
in the following Chapters of this thesis, can fragment to give cold gas. However, the violent
passage of such a shock from a clouldlet can instead evaporate it, or trigger it to collapse
gravitationally and give new stars.
2.6.1
Models of the multi-phase ISM
The different phases of the atomic ISM seem to be in rough pressure equilibrium with each
other. This points to a fluid instability known as Thermal Instability (–see Section 3.2.3 for a
detailed description of the instabilty), as a formation mechanism for the CNM and the WNM
phases.
This instability happens when the heating-cooling rate equilibrium curve on the densitytemperature plane of the gas shows wide plateaus, separated by steep steps. An example of
such an equilibrium curve is shown in Figure 2.6. This gives two stable equilibria on either
side of an unstable equilibrium point, defined by a line of constant pressure, for instance, like
the one shown in Figure 2.6. The region above the curve is where cooling is stronger than
heating and the region below the curve is where heating is stronger than cooling. When gas
is perturbed from the unstable equilibrium point to the region above the curve by, say, a
compression, cooling drives it to cool and condense further, until is reaches equilibrium again,
to the equilibrium point on the high density regime. An equivalent process happens to drive
gas that has been perturbed to lower densities and drive it to the plateau on the low density
regime. Depending on the exact shape of the equilibrium curve and the positions of the
plateaus the densities and temperatures of the two phases can be different. This mechanism
for the formation of phases of the ISM was proposed by Field (1965). However, we know that
the ISM possesses at least one more gas phase, the hot component, which is very important
for its dynamics, as well as for its enrichment in metals, but it is not included in this model.
16
CHAPTER 2. THE INTERSTELLAR MEDIUM
The inclusion of the third component in a dynamical model of the ISM was first proposed
by McKee & Ostriker (1977). In their model, the ISM is composed of spherical clouds, each
of them composed of a cold neutral core and surrounded by two envelopes, a warm neutral
medium immediately outside the cold core and a warm ionized corona as the outer surface of
the cloud. The cold clouds are assumed to have a very low filling factor, of about 0.02-0.04.
Their ionized coronas have a filling factor of 0.2, while practically the whole volume of the
ISM, with filling factors of about 0.7-0.8, is composed by a hot dilute medium, created by
supernova explosions. In fact, supernova explosions regulate the behavior of the ISM in this
model, compressing and sometimes evaporating the cold clouds. In their view of the ISM, the
total mass of the gas is conserved and the phases are in pressure equilibrium. A schematic
view of this model is shown in Figure 2.7.
Although this model significantly improved our understanding of the ISM physics, mainly
by introducting the hot component, we now know that the neutral clouds are far from spherical
(Figure 2.2) and that the hot phase of the ISM actually has almost double the pressure of the
other phases and is thus sometimes driven outside the Galactic disk. An improved model for
the ISM, where the cold clouds appear as ”blobby sheets” rather than spheres, was presented
by Heiles & Troland (2003), based on their observations.
2.6.2
Cooling and Heating Processes of the Interstellar Gas
This Subection is in effect a summary of Wolfire et al. (1995). We refer the reader to this
paper for further details on the calculation of the rates.
One very important ingredient for understanding the formation of phases, by the mechanism described briefly in the previous subsection, is the net cooling and heating of the gas
as a function of its density and temperature. The combination of all possible cooling and
heating paths of the gas will give an equilibrium curve like the one shown in Figure 2.6,
which predicts the densities and temperatures of the different phases of the gas. Although we
have mentioned the possible heating and cooling mechanisms on the ISM in passing, here we
explain them in more detail, with a focus only on the atomic phase of the ISM. The heating
and cooling rates of the molecular phase, although of extreme importance for the study of
the physics of molecular clouds, are outside the scope of this thesis.
Possibly the most important heating source of the atomic gas is photoelectric emission
from small dust grains and large molecules, such as PAHs. To remind the reader this wellknown phenomenon, the photoelectric effect happens when a photon carrying an energy larger
than the work function of a material is absorbed by this material. Then it can give this energy
to an internal electron, leading to the ejection of this electron. In the case of dust grains,
the excess energy is mostly consumed in heating the grain, since most electrons liberated by
UV photons do not reach the surface of the solid. However, for the 10% of electrons that do
escape from the grain, the rest of the energy of the photon is available for heating the gas.
The resulting heating rate from this process is
nΓphot = 10−24 nǫG0
(2.3)
in ergs cm−3 sec−1 . ǫ is the fraction of FUV photons absorbed by grains that is converted
to gas heating and G0 is the FUV flux in 1.6 × 10−3 ergs cm−2 . Γphot is called the heating
function for this process and n is the number density of gas atoms.
ǫ is not a constant, but rather a function of temperature and electron density. It depends
on the ionization rate divided by the rate at which electrons recombine on the surface of the
2.6. THE SYSTEM AS A WHOLE
17
Figure 2.6: Thermal instability mechanism. The cooling-heating equilibrium curve as a function of density and
temperature, assuming the cooling and heating rates described in the text, is plotted as a black curve and a straight
line indicating a constant pressure curve is shown in orange. Gas parcels perturbed from the unstable regime A on
the curve towards higher densities will condense further until they reach the stable equilibrium C. In the same way,
a volume of gas heated from A to higher temperatures or lower densities will continue heating and expanding until
it reaches equilibrium point B. We have then, the formation of two gas phases in temperatures and densities given
by points B and C.
18
CHAPTER 2. THE INTERSTELLAR MEDIUM
Figure 2.7: A schematic view of the ISM by McKee & Ostriker (1977). On the left (Fig. 1 of that paper), a
cross-section of a typical cloud in their model. It is composed by a neutral atomic core, enveloped by a warm
neutral and a warm ionized corona. The whole structure is immersed into the hot ionized medium.
On the right (Fig. 2 of that paper),, a small-scale view of the passage of a supernova shock wave, coming from
the upper right, from a region of the ISM filled with clouds such as the one shown on the left. A fraction of these
clouds will be evaporated.
grains. The maximum efficiency happens, of course, for low temperatures, when the grains
are mostly neutral.
The shape of the FUV spectrum has not been found to affect the resulting heating function
a lot, as long as the total flux remains constant. What does affect this heating rate appreciably
is, of course, the distribution of grain sizes. Usually, the distribution is assumed to be a
power-law with a slope of about -3.5, based on models that take into account the observed
dust emission.
The heating rate of the gas by photoelectric emission from dust and PAHs is shown in
Figure 2.8 as the top dashed line. It is evidently the most important heating rate in the local
ISM. However, as we will explain shortly, under certain circumstances the photoelectric effect
can result in a net cooling of the gas.
Additional heating of the ISM comes from cosmic ray ionization of the gas and from diffuse
X-ray radiation. Cosmic rays can ionize the gas to appreciable degrees, with an ionization
value usually adopted for the ISM being nζCR = 1.8 × 10−17 n cm−3 sec−1 . After ionization,
the excess energy is available for heating. Assuming a power-law distribution for the momenta
of cosmic rays, which, as we have seen in Section 2.4, is a well-justified assumption, and taking
into account also secondary ionizations of H and He, the heating rate from this process is
nΓCR = nζCR Eh (E, ne /n)
(2.4)
where Eh (E, ne /n) is the heat provided to the gas by each electron of energy E. This function
depends on the electron fraction in the gas and the energy of the electron and is provided in
the Appendix of Wolfire et al. (1995).
2.6. THE SYSTEM AS A WHOLE
19
A similar process gives the heating rate of the gas from soft X-rays. Soft X-rays in the
Galaxy are produced by energetic stellar feedback, mostly from young massive stars, although
at the highest energies an extragalactic X-ray component dominates, coming mainly from a
hot intergalactic medium. Without going into the specific details of the calculation, we quote
here the heating function calculated for this heating mechanism:
nΓXR
X Z Jv
e−σv Nw σvi Eh (E i , ne /n)dv
= 4πn
hv
(2.5)
i
where the summation is assumed over all elements which experience primary ionization. Nw
is the column density of neutrals, hence the factor e−σv Nw , which acconts for an absorbing
layer of neutral, warm interstellar gas, with the cross-section σv for absorption in frequency
v taking into account the elemental abundances. ne /n is the ionization fraction. Jv , the
assumed X-ray flux spectrum, is calculated by including three X-ray components: A singletemperature component from nearby sources, a single-temperature component from largest
distances which has suffered absorption from interstellar clouds and an extragalactic powerlaw emission component. In Figure 2.8, heating from X-ray ionizations is the second most
important heating mechanism up to hydrogen number densities of about 103 cm−3 , followed
by cosmic ray ionization. At higher densities an important source of heating comes from the
photoionization of C I.
It has already been mentioned in earlier Sections of this Chapter that the most important
coolants of the atomic ISM, especially in the low-temperature regime, are the fine structure
lines of species such as O I and C II (–see Figure 2.8).
Important energy losses, mostly in the low-density regime, come from the recombination
of electrons onto small dust grains and PAH molecules, a process already mentioned when
discussing heating mechanisms. When the thermal energy of the electrons that recombine on
the grains exceeds the energy of the electrons being ejected, then the net effect is an energy loss
from the gas to the grains. It is clear that this effect is important at high temperatures, when
the electron thermal energy, kT, is larger than the typical 1 eV carried by the photoelectrically
ejected electrons. When modeling the gains and losses of the gas caused by the photoelectric
effect, the heating and cooling rates are calculated separately, so the net heating or cooling
is defined as the difference between the two. The cooling by recombination of photoelectric
electrons is shown in Figure 2.8 as a solid black line.
In addition to the fine-structure lines, cooling can also happen due to emission in metastable
or resonance lines. In particular, the collisional excitation of the Lyman α line can contribute
to the energy losses of the gas in the highest temperature regime.
Figure 2.8 summarizes the cooling and heating processes that we have taken into account
in this work. In calculating these rates we have assumed that the lines are optically thin and
that dust extinction is not important. These assumptions are increasingly inaccurate as we
move towards the high density regime, but, since we are only interested in the transition from
warm to cold gas, they provide an adequate description of the average cooling and heating
rates of the gas in this regime. The gas has been assumed to have a constant ionization degree
in the calculation of the rates. The resulting cooling and heating curve for the parameters we
have used is shown in Figure 2.6. The abundances used in this calculation are roughly solar.
20
CHAPTER 2. THE INTERSTELLAR MEDIUM
Figure 2.8: Cooling (solid lines) and heating (dashed lines) rates as a function of density, assuming local ISM
conditions and optically thin lines. Dust absorption is not assumed to be important for the densities and the
radiation frequencies considered.
2.6. THE SYSTEM AS A WHOLE
2.6.3
21
The modern view of the ISM: the role of turbulence
As mentioned in Section 2.6.1, the inclusion of a third, hot phase of the ISM was proposed
by McKee & Ostriker (1977) to account for cloud destruction and to count as a large volumefilling fluid. However, winds and supernova explosions from massive stars introduce a much
more significant ingredient in the complex system of interactions in the ISM: they are a source
of mechanical energy and, as such, the drivers of turbulence.
Turbulence, which will be treated in somewhat more detail in Section 3.4 as a consequence
of the equations of hydrodynamics, is a term to describe the coupling of different scales of
a fluid through the exchange of energy. It is characterized by a stochastic behavior and
the replacement of the local properties of a flow by statistical laws. In principle, one could
roughly describe a turbulent flow by the existence of curved shocks and eddies, appearing and
disappearing unpredictably, almost chaotically.
In this context it is therefore understood that a large-scale energy injection as violent as a
supernova explosion or combined feedback from young OB stars will act as the beginning of an
energy cascade, creating smaller-scale eddies and shocks in the ISM. Although theoretically
such models have been proposed since a long time (ie de Avillez & Breitschwerdt (2004)),
and the turbulent nature of all the phases of the ISM has been evident in observations for
even longer (Schneider & Elmegreen (1979); Lada et al. (1999); Hartmann (2002); Lada et al.
(2007); Molinari et al. (2010), see also Figure 2.4 for a representative image of interstellar
turbulence) it is very difficult to verify that the injection scale of interstellar turbulence has
a supernova origin.
Turbulence in the warm atomic phase of the ISM is thought to be roughly sonic (Verschuur,
2004; Haud & Kalberla, 2007), meaning that the root mean square velocities of the gas are
of the order of its sound speed (∼10 km/sec) while in the cold molecular phase it is thought
to be supersonic, reaching velocities of 1-5 km/sec, while the sound speed in these clouds is
roughly 0.2-0.5 km/sec (–see also Section 4.1 and references therein). This has important
implications not only for global models of the Galactic ISM, but also for the star formation
process, which happens exclusively in molecular clouds.
Interstellar turbulence is a very efficient mechanism for mixing newly-produced metals
from supernovae and stellar winds and dust in the gas. These elements can then participate
in a new cycle of star formation. The velocity dispersion in the diffuse phase of the ISM has
a stabilizing role against the global gravitational instability of the Galaxy. Inside molecular
clouds, the transition scale from supersonic to transonic or subsonic motions is probable
what sets the scale of prestellar cores (Walch et al., 2011a), thus deciding the typical mass
of a protostellar object. It is therefore very useful to study the properties of interstellar
turbulence and look for its drivers at different scales.
The fact that turbulence is maintained at the sonic level for the diffuse phase of the ISM
is consistent with a picture in which the violent, evidently supersonic shocks from the drivers,
possibly supernova explosions, distribute their energy to smaller scales through a cascade,
until it is finally viscously dissipated. The root mean square velocity of such a flow can be
proven to fall to sonic velocities very rapidly. The dissipation rate for sonic turbulence being
much slower, such a turbulent flow can be maintained for many crossing times, in this case,
many Galactic rotation times.
However, the true puzzle in the studies of interstellar turbulence is what maintains the
supersonic velocities inside molecular clouds. Such supersonic motions should die out very
quickly, as we explained before, so there must be a process at molecular cloud scales which
22
CHAPTER 2. THE INTERSTELLAR MEDIUM
keeps driving turbulence in their interior. Star formation, through protostellar outflows has
been proposed as a possible driving mechanism. The formation scenarios of the clouds themselves, however, very often invoke dynamical processes, which could also trigger turbulent
motions in the formed structures. This subject is analyzed addressed in detail in Section 4.1,
as part of the motivation for this work; this thesis is partially aimed at explaining the origin
of turbulence in molecular clouds by proposing a very dynamical model for their formation.
Chapter
3
Principles of Hydrodynamics
3.1
The equations of hydrodynamics
The work in this thesis can be essentially described as solving the equations of hydrodynamics
numerically, for different boundary and initial conditions. It is therefore very important that
we introduce these equations and some of their applications relevant to the phenomena we
will encounter in the following Chapters.
The equations of hydrodynamics provide a macroscopic description of the behavior of
a fluid. This means that the length and time scales of the system considered are much
larger than those of the interactions between individual atoms or molecules. In practice, we
are allowed to use this description of a fluid when the mean free path of the particles is
much smaller than the size of the system under study. For all of the problems treated here,
hydrodynamics is a more than sufficient approximation.
In hydrodynamics, the evolution of certain variables, representative of the state of the gas,
is expressed as a function of space and time. In particular, we follow the velocity of the fluid,
v(r,t) and any two thermodynamic quantities, usually the density ρ(r, t) and the pressure
p(r, t), from which the rest of them, such as the temperature, can be derived.
In the following Sections the equations of hydrodynamics are derived from the conservation
of mass, energy and momentum. We will refer to small volumes in the fluid as fluid elements.
Again, these volumes may be considered infinitesimally small for the sake of the mathematical
derivation, but they are always large enough to contain a very large number of atoms or
molecules and the equations of hydrodynamics are always valid.
3.1.1
The equation of continuity
The first equation we are going to derive expresses the conservation of mass in the system.
Considering a volume element dVRof a larger volume V0 containing the fluid, the total mass
in the system can be expressed as ρ dV , where ρ is the density as a function of position and
the integral is taken over the volume V0 .
Defining dn to be a vector with direction normal to the surface of the fluid element and
magnitude equal to the area of this surface, the mass flux through this surface per unit time
is equal to ρu · dn, where u is the fluid velocity vector. We consider dn to be positive in
The contents of Section 3.1 are from the first chapter of Landau & Lifshitz (1959)).
24
CHAPTER 3. PRINCIPLES OF HYDRODYNAMICS
the outward direction, so that the flux is positive when material is leaving the volume and
negative when material is entering the volume. Then the total flux of mass out of the volume
V0 is
I
ρ u · dn
(3.1)
where the integration is over the entire surface enclosing the volume V0 . The mass flux outside
the volume causes a decrease in the total mass contained in V0 , which is equal to
Z
∂
−
ρ dV
(3.2)
∂t
By equating equations 3.1 and 3.2, we get
Z
I
∂
ρ dV
(3.3)
ρ u · dn = −
∂t
H
R
and replacing ρv · dn by its equivalent ∇(ρv)dV we get the equation of continuity
∂ρ
+ ∇(ρ u) = 0
(3.4)
∂t
where the integral has been dropped, since this equality must hold for any volume in the
fluid.
3.1.2
The force equation
The equation of force balance, in effect the equation of motion of the fluid, is also called the
Euler equation, after L. Euler who derived it first.
Considering, as before, a volume element dV in a fluid of volume V0 , we can express the
total force acting on the fluid as the integral of the pressure over the surface enclosing the
volume:
I
Z
− p dn = − ∇ p dV
(3.5)
By equating this force to the change in momentum of the fluid element, we get
ρ
du
= −∇ p
dt
(3.6)
The derivative du/dt is here the total derivative of the velocity. This means that is carries
the information on both the change of the velocity of the fluid in a given point in space after
a time dt, and the change in velocity experienced by the fluid element because it traveled a
distance dr, to a different point of the fluid. To express this information explicitly, we can
write
∂u
dt
(3.7)
∂t
namely the change in velocity with time when the spatial coordinates, say x, y and z for a
Cartesian description, are held constant. Then the second part of the derivative, which is the
3.1. THE EQUATIONS OF HYDRODYNAMICS
25
part expressing the change in velocity due to the change in coordinates, can be written as
dx
∂u
∂u
∂u
+ dy
+ dz
∂x
∂y
∂z
which is equal to (du · ∇)u. So we have
∂u
du =
dt + (du · ∇) u
∂t
(3.8)
(3.9)
If we divide both sides by dt we can substitute this expression for the total derivative into
equation 3.6 and obtain:
1
∂u
+ (u · ∇) u = − ∇ p
(3.10)
∂t
ρ
Equation 3.10 is the Euler equation. Additional terms are usually included in the right-hand
side of the equation to account for other forces that may be acting on the fluid, such as
gravity.
Here the Euler equation was derived assuming that all the changes in pressure around the
fluid element are consumed in changing its momentum. In other words, we have assumed no
energy losses due to viscosity or other dissipation mechanisms. This form of the equations of
hydrodynamics, when thermal conductivity and viscosity are considered to be negligible, is
called ideal hydrodynamics and the fluids with these properties are called ideal fluids.
Assuming a negligible thermal conductivity is equivalent to saying there is no heat exchange between different parts of the fluid. In other words, the motion of a fluid element is
adiabatic. This condition can be expressed in terms of conservation of the entropy s of the
fluid: ds/dt=0
∂s
+u·∇s=0
(3.11)
∂t
which, in combination with the equation of continuity 3.4, gives an equivalent equation of
continuity for the entropy in the fluid:
∂s
+ ∇(ρ s u) = 0
∂t
(3.12)
If we consider that, at some initial instant the fluid had the same entropy everywhere, equation
3.12 becomes much simpler: s = constant.
3.1.3
The energy equation
The total energy of a fluid element, assumed to be fixed in space, can be expressed as
1 2
ρv + ρ ǫ
2
where ǫ is the internal energy per unit mass.
The change of the kinetic energy of the fluid with time can be written:
1 ∂ρ
∂u
∂ 1 2
ρv = v 2
+ρu
∂t 2
2 ∂t
∂t
(3.13)
(3.14)
26
CHAPTER 3. PRINCIPLES OF HYDRODYNAMICS
In equation 3.14 we can replace the first term of the right-hand side from the continuity
equation, 3.4 and the second term from the Euler equation, 3.10. Then we obtain:
∂ 1 2
1
ρv = − v 2 ∇ (ρ u) − v · ∇ p − ρu · (u · ∇) u
(3.15)
∂t 2
2
In equation 3.15 we can replace the term ρu · (u · ∇) u with 12 u · ∇ ∇2 . If we define w as
the heat function per unit mass, then dw = T ds + V dp. By replacing V = 1/ρ, ∇ p can be
written as ρ ∇ w − ρT ∇ s. Thus we get for the kinetic energy of the volume element:
1 2
1 2
∂ 1 2
ρv = − u ∇ (ρ u) − ρ u · ∇
u + w + ρT u · ∇ s
(3.16)
∂t 2
2
2
The change in the internal energy of the fluid element with time, ∂(ρǫ)/∂t can be derived
from the first law of thermodynamics:
dǫ = T ds − pdV = T ds +
p
dρ
ρ2
(3.17)
and, since ǫ + p/ρ = ǫ + p dV is just the heat function per unit mass w, we have
d(ρǫ) = ǫdρ + ρdǫ = wdρ + ρT ds
(3.18)
where we have also used V = 1/ρ and dV = −dρ/ρ2 . So we can write for the change in the
internal energy of the fluid:
∂ρ
∂s
∂(ρǫ)
=w
+ ρT
= −w ∇(ρ ∇) − ρT u · ∇ s
∂t
∂t
∂t
where equation 3.12 has been used to replace ∂s/∂t.
Combining equations 3.16 and 3.19 we get
∂ 1 2
1 2
1 2
ρu + ρǫ = −
u + w ∇(ρu) − ρu · ∇
u +w
∂t 2
2
2
(3.19)
(3.20)
and finally
∂
∂t
1 2
ρu + ρǫ
2
= −∇ ρu
1 2
u +w
2
(3.21)
Equation 3.21 can be understood as the conservation of energy in each volume element as it
moves in the fluid, so that the energy carried by a fluid element is 12 u2 + w.
3.2
Hydrodynamical Instabilities Related to Shell Fragmentation
An instability is defined in a physical system as the unbounded growth of initially small
perturbations imposed on an equilibrium state. Fluid instabilities are many times identified
by the morphology of the system on which they are developing.
In the following Sections we will illustrate fluid instabilities relevant to the expansion and
collision of shock waves, such as the Non-linear Thin Shell Instability (NTSI), the KelvinHelmholtz Instability (KH) and the Thermal Instability (TI).
3.2. HYDRODYNAMICAL INSTABILITIES
27
We will not provide the full derivation of the perturbation equations or the solutions here,
since such a lengthy analysis is outside the scope of this thesis. We will rather introduce the
methodology and the assumptions that go into deriving the results, which are essential in
order to fully understand the results of our work, contained in the following Chapters. The
rigorous derivations of the instabilities can be found in the references mentioned in the text.
3.2.1
The Non-linear Thin Shell Instability
This instability is very similar to the well-known Rayleigh-Taylor instability, which occurs
when a dense fluid is accelerated by a more rarefied fluid. In this case, however, the instability
occurs without the involvement of gravity, when the surface of a shock, bounded on one side
by thermal pressure and on the other by ram pressure, is perturbed by a ripple.
Let the spherical shock under consideration have an outer radius Rs and a thickness h,
so that, the inner radius denoted by Ri , h = Rs − Ri . The shock is expanding with a
velocity Vs in a uniform medium of density ρE and is driven by a medium of thermal pressure
Pi and negligible density (infinite sound speed). At this point we introduce the thin-shell
approximation, which significantly simplifies the derivation of the equations:
h
< kh ≪ 1
(3.22)
Rs
cs
∂
≪
(3.23)
∂t
h
where k is the wavenumber of the perturbation. Condition 3.23 essentially says that we want
the evolution of the perturbations to be much faster than the sound crossing time of the shock
thinkness, so that the shell reacts quasi-statically to the perturbation.
The equations of hydrodynamics, namely the continuity equation 3.4 and the Euler equation 3.10 have already been introduced in section 3.1. For convenience, though, instead of the
density ρ, the velocities ux , uy and uz and the pressure P, in this analysis here the variables
used are the surface density σ, the radial shock velocity Vr and the tangential shock velocity
VT :
σ =
Vr =
VT =
1
Rs2
Z
1
σRs2
1
σRs2
Rs
ρr2 dr
(3.24)
Ri
Z
Rs
ρr2 ur dr
(3.25)
ρr2 ut dr
(3.26)
[ρu] = 0
P + ρu2 = 0
P +F = 0
(3.27)
Ri
Z Rs
Ri
In the absense of perturbations VT vanishes.
The shock boundary conditions are:
γ
1 2
ρu +
u
2
γ−1
(3.28)
(3.29)
The detailed derivation of the thin-shell instability, as well as the contents of Section 3.2.1, can be found
in Vishniac (1983).
28
CHAPTER 3. PRINCIPLES OF HYDRODYNAMICS
where u is now the fluid velocity relative to the shock, F is the radiative energy flux and the
square brackets denote differences across the shock. For an adiabatic shock F = 0.
Following the definitions of the new variables, 3.24-3.26, that is, integrating the equations
of hydrodynamics 3.4 and 3.10 across the shock and applying the shock jump conditions
3.27-3.29 as boundary conditions, the evolution equations for σ, Vr and Vt are obtained:
∂σ
∂t
∂Vr
∂t
∂VT
∂t
Vs
σ + ρE Vs − σ (∇T · VT )
Rs
1
= − (ρE Vs Vr − Pi )
σ
Z Rs 2 2
ρE V s
Vs
Vs
(cs r ∇T ρ)dr
= −
VT −
VT +
uT (Rs )ρE −
σ
Rs
σ
σRs2
Ri
= −2
(3.30)
(3.31)
(3.32)
where ∇T stands for tangential derivatives, uT (Rs ) is the tangential velocity of the fluid at
the shock surface, as imposed by the boundary conditions, and Vs is the velocity of the shock
surface. In the derivation of equation 3.30 one assumes there is no matter flux through the
inner boundary, that is, all the matter swept up by the shock as it propagates stays on its
surface.
The first term of equation 3.30 just states the obvious fact that, if the mass in the shell
remains the same, the surface density of the shell decreases as its radius increases. The second
term is the increase in surface density caused by material the shock accretes as it propagates
in the surrounding medium. The last term describes the mass transfer on the shell due to
bulk tangential motions.
Equation 3.31 describes the change of the radial velocity of the shell as a result of the
pressure difference at its outer surface, namely the ram pressure of the surrounding medium
ρE Vs Vr and at its inner surface, namely the thermal pressure Pi provided by the hot gas
driving the shock.
In deriving equation 3.32 and, in fact, for the rest of Vishniac’s analysis of this instability,
the assumption is made that the shock velocity is equal to the radial velocity: Vs = Vr , meaning that the motion of the shock is dominated by its radial expansion and not by tangential
motions within it. This stems from the quasi-static assumption 3.23 and simple dimensional
arguments.
Equation 3.32 is of particular interest for the study of this instability. The first term on
the right-hand side is similar to the second term of equation 3.30, expressing the decrease in
tangential motions by the accretion of material from the surrounding medium. The second
term just states the decrease in the tangential velocity as the radius of the shock increases.
The third term tells us that the accreted material can only be accelerated in the direction
of the shock front motion. Using the shock jump conditions, at the same time replacing the
pressure at the outer boundary of the shock with the ram pressure, the last term can be
rewritten and equation 3.32 reads:
∂VT
ρE V s
Vs
cs
Pi
=−
VT −
V T − ∇ T σ − ∇ T Rs
∂t
σ
Rs
σ
σ
(3.33)
where essentially the last two terms of equation 3.32 have been replaced by a term expressing
the effects of gradients in the surface density and a term due to changes in the inner surface
of the shock. It might seem surprising then, that this term contains Rs instead of Ri . The
detailed derivation omitted here, this is justified by the fact that irregularities that come from
3.2. HYDRODYNAMICAL INSTABILITIES
29
variations in the shell thickness are of higher order than irregularities coming from the bulk
motion of the shock.
It can be additionally shown that, if the inner thermal pressure balances the outer ram
pressure almost exactly, the term with ∇σ in equation 3.33 can be omitted. Now we have
two terms, one due to mass accretion and one due to the curvature of the shell, plus one term
(the last one), which accelerates material into lagging regions of the shell.
The appearance of this term in the equations is what causes the instability, in an analysis
that we will not include in this thesis. Figure 3.1, however, illustrates how an instability might
occur due to the presence of this term. The pressures acting on the two surfaces of the shell
are of different nature. The inner surface is supported by the isotropic thermal pressure, but
the outer surface of the shock is supported by ram pressure, which always acts on a direction
normal to the surface. This means that, if there is a ripple on the surface of the shell, the
thermal pressure will push material sideways on the ripples. The ram pressure cannot balance
this motion, since it has no component in that direction. The result will be, to first order, a
tangential force on the shock surface, which will depend on the angle of obliqueness and on
the internal pressure.
The derivation of the perturbation equations from this point is straightforward. Here we
will just mention that, for example, for an isothermal shock and for a perturbation expressed
in the form of spherical harmonics, the shell will be unstable and fragment on scales 6.4 < l <
3M 2 , where l the azimuthal mode of the perturbation and M the Mach number of the shock
expansion velocity relative to the internal sound speed of the shock. The smallest wavelengths
are defined by the ”thin-shell” assumption and the largest wavelengths, comparable to the
shock radius, are stabilized by the bulk motion of the shock. Additional effects that can
stabilize or destabilize the shock are magnetic fields and the consideration of the thickness of
the shock. The growth rate of the unstable modes is roughly cs k(k∆)1/2 where cs the sound
speed, k the wavenumber along the shock, and ∆ the amplitude of the initial perturbation.
We should also note that, although the analysis quoted here is for the linear regime, the
effects we are interested in when we study this problem numerically are related to the nonlinear growth of this instability, when the perturbations affect the expansion of the shock and
the quasi-static approximation is no longer valid. That is why in this text we quote this type
of instability as the ”Non-linear Thin Shell instability”. However, the above analysis is useful
because it shows us why there is an instability in the first place and what its first-order effects
are, which offers useful insight in the problem of shell fragmentation. This instability will be
discussed further in the text, since it is very relevant to our results.
3.2.2
The Kelvin-Helmholtz Instability
In Section 3.2.1 it was shown that when a spherical shock expands in a uniform medium,
ripples on its surface can cause tangential motions of the material in lagging regions. This
type of shear motion can give rise to another well-studied dynamical instability, called the
Kelvin-Helmholtz instability.
The Kelvin-Helmholtz instability occurs when a stratified, heterogeneous fluid with the
different layers in relative motion is perturbed. Its morphological signature are characteristic
eddies, like the ones shown in Figure 3.2. In this Section we are going to present results
relevant to the case of an inviscid fluid, ignoring for sake of brevity the (stabilizing) effects of
A rigorous derivation of the KH instability (Section 3.2.2) can be found in Chandrasekhar (1961).
30
CHAPTER 3. PRINCIPLES OF HYDRODYNAMICS
Figure 3.1: Illustration of the thin shell instability, from
Vishniac (1983) (Figure 1 of that paper). The different
nature of the thermal and the ram pressure cause ripples
on the surface of the shock to grow by tangential motions
of the material towards lagging regions.
surface tension and magnetic fields.
Using Cartesian coordinates, let the equilibrium configuration be that of a stratified fluid
with density ρ0 (z) a function of height z, and a streaming velocity along the x-direction,
ux0 (z) also a function of z. Then the hydrodynamic variables can be expressed as:
ρ = ρ0 (z) + δρ
p = δp
ux = ux0 (z) + δux
(3.34)
uy = δuy
uz = δuz
Inserting the expressions 3.35 for the variables in the equation of continuity 3.4 and the Euler
equation 3.10 and neglecting second order terms yields:
ρ0
∂δux
∂δux
dux0
+ ρ0 ux0
+ ρ0 δuz
∂t
∂x
dz
∂δuy
∂δuy
+ ρ0 ux0
ρ0
∂t
∂x
∂δuz
∂δuz
+ ρ0 ux0
ρ0
∂t
∂x
∂δρ
∂δρ
+ ux0
∂t
∂x
∂δuz
∂δux ∂δuy
+
+
∂x
∂y
∂z
∂δp
∂x
∂δp
= −
∂y
∂δp
= −
− gδρ
∂z
dρ
= −δuz
dz
= −
= 0
(3.35)
(3.36)
(3.37)
(3.38)
(3.39)
where a term due to a gravitational acceleration g = gẑ has been included in equation 3.37
and equation 3.39 expresses the incompressibility of the perturbation.
Looking for perturbations of the form:
exp [i (kx x + ky y + ωt)]
(3.40)
3.2. HYDRODYNAMICAL INSTABILITIES
31
equations 3.35-3.39 become:
dux0
δuz = −ikx δp
dz
iρ0 (ω + kx ux0 ) δuy = −iky δp
dδp
− gδρ
iρ (ω + kx ux0 ) δuz = −
dz
dδuz
i (kx δux + ky δuy ) =
dz
iρ0 (ω + kx ux0 ) δux + ρ0
(3.41)
(3.42)
(3.43)
(3.44)
Now by multiplying equation 3.41 by −ikx and equation 3.42 by −iky and add the resulting
equations, replacing at the same time k 2 = kx2 + ky2 and making use of equation 3.44, one
obtains:
iρ0 (ω + kx ux0 )
dux0
dδuz
+ iρ0 kx
δuz = k 2 δp
dz
dz
(3.45)
and from equations 3.43 and 3.44:
iρ0 (ω + kx ux0 )δuz = −
dρ
dδp
− ig
dz
dz
(3.46)
By eliminating δp from equations 3.45 and 3.46 the following equation is obtained:
d
dδuz
dρ0
δuz
dux0
ρ0 (ω + kx ux0 )
− ρ0 k x
δuz − k 2 ρ0 (ω + kx ux0 )δuz = gk 2
(3.47)
dz
dz
dz
dz ω + kx ux0
The condition for instability is derived by applying appropriate boundary conditions and
considering the specific form of the equilibrium configuration, functions ρ0 (z) and ux0 (z).
The simplest assumption we can make for the equilibrium flow is that of essentially two
fluids, one of density ρ1 and velocity ux1 and one of density ρ2 and velocity ux2 , superimposed
on each other at z=0. (A similar configuration is shown in Figure 3.2).
The boundary conditions at z=-d and z=d (for a system of total size 2d along the z
direction) are just that the vertical components of the fluid velocities vanish on those surfaces.
Additionally, we should ask that the components of the fluid velocities normal to their contact
surface vanish. In other words, the assumption here is that the two fluids do not mix.
Omitting the intervening algebra, we directly present the condition for instability (modes
growing exponentially in time), in the absence of surface tension:
k>
g(α1 − α2 )
α1 α2 (ux1 − ux2 )2
(3.48)
where α1 = ρ1 /(ρ1 + ρ2 ) and α2 = ρ2 /(ρ1 + ρ2 ). Interestingly, relation 3.48 states that, if
one ignores gravity, all wavenumbers are unstable to the Kelvin-Helmholtz instability. The
growth rate of the instability, for a step function in the velocity, is k∆v, where k the unstable
wavenumber and ∆v is the velocity difference between the two layers.
32
CHAPTER 3. PRINCIPLES OF HYDRODYNAMICS
Figure 3.2: Simulated KH instability. A perturbation on the contact layer between two fluids of different density,
shown here in black and white color, moving along the horizontal direction relative to each other, grows to give
characteristic eddies like the ones shown in this figure. This configuration is already in the nonlinear regime of the
instability growth. (Image from wikimedia.org)
3.2.3
The Thermal Instability
The Thermal Instability has already been presented in Section 2.6.1 in a qualitative way.
Figure 2.6 provides a schematic illustration of how this instability works. Here we will give
some dimensional arguments of when the instability takes place and what its overall behavior
is.
Let
L(ρ, T ) = 0
(3.49)
define the heating-cooling equilibrium curve of the gas, where L(ρ, T ) represents the energy
gains minus the energy losses of the gas per unit mass per unit time.
A perturbation of density and temperature is introduced, such that some thermodynamic
quantity A, the pressure for example, remains constant. Then the entropy of the gas will
change by an amount δS and the heating-cooling function L will change by an amount δL:
δL dt = −T d(δS)
(3.50)
where changes are to be calculated with A constant. Then the criterion for instability is
∂L
>0
(3.51)
∂S A
in other words, there is an instability if the gain-loss function L is a monotonous function of
the entropy, so that changes in the entropy due to perturbations always cause a response of
the same sign from the heating-cooling function and the perturbation ccan grow. Assuming
The contents of Section 3.2.3, as well as a full derivation of the TI can be found in Field (1965).
3.3. THE INSTABILITIES COMBINED
33
an ideal gas, for an isochoric perturbation T dS = CV dT and for an isobaric perturbation
T dS = CP dT . Then one obtains two criteria for the instability, an isochoric and an isobaric
one:
∂L
< 0, (isochoric)
∂S ρ
∂L
ρ0 ∂L
∂L
=
−
< 0, (isobaric)
∂S P
∂T ρ T0 ∂ρ T
(3.52)
(3.53)
Criterion 3.52 is in principle incompatible with the force equation, because isochoric temperature variations will cause changes in the density, which then cannot be assumed constant.
Condensations can be shown to be dominated by the isobaric condensation, which is compatible with the force equation. This can also be derived by a more detailed treatment of the
instability.
Burkert & Lin (2000) showed that the growth rate of the thermal instability does not
depend on the size of the perturbation (ie the wavenumber), but is rather determined by the
local cooling time. In particular, the growth of a perturbation is decided by the transition
from isobaric to isochoric cooling. This implicitly means that the smallest perturbations will
grow first. The smallest unstable scale, according to their analysis, is decided by thermal
conduction. In the discussion of our results we will comment more on the role of thermal
conduction and its meaning with respect to our findings.
3.3
The instabilities combined
In the context of shell fragmentation it is very useful to study the effects of all the aforementioned instabilities simultaneously, since each of them will contribute to the dynamics of the
formed phases, if any. We naturally expect, however, that each instability will be dominant
in different regimes, according to the size of the shock and the general properties of the gas.
Heitsch et al. (2008a) explored precisely this subject, separating the phase space of the
gas into different regimes. Figure 3.3 from their paper illustrates the result:
To visualize this result, in Figure 3.4 we plot contours of the gas temperature for an
idealized setup, designed to study the fragmentation of a shock due to these three instabilities.
In this setup, a very thin slab between two infinite, high-Mach flows is perturbed. This is
expected to trigger the NTSI and in turn, the TI and the KHI. The Figure shows the results
of two-dimensional hydrodynamical simulations of this setup after the instabilities have had
significant time to develop. Shown on the left is the result of a Mach 1 flow and on the right
the result of a Mach 3 flow. In accordance with previous findings, we see that different flow
Mach numbers lead to slightly different results: Although both shocks are fragmenting, the
one on the left shows more clumpy structure, signature of the TI, while the shock on the
right is more violently disturbed by the NTSI, as indicated by the large finger-like strucures
growing perpendicularly to the slab.
34
CHAPTER 3. PRINCIPLES OF HYDRODYNAMICS
Figure 3.3: Instability regimes in density and temperature space. The solid black line shows the cooling-heating
equilibrium curve. Cooling instability causes fragmentation in the regions with positive slope and gravity is important
in the regions with negative slope. Red hatching shows where dynamics (KHI, NTSI) dominates over cooling and
blue hatching shows where the TI dominates dynamics. (Figure and caption from Heitsch et al. (2008a))
Figure 3.4: Simulations of a perturbed slab, bounded on each side by an infinite flow. On the left, a Mach 1 flow.
On the right, a Mach 3 flow. The colors indicate different contour levels of the logarithm of temperature. The
physical scale is of 44 pc at each direction.
3.4. SOME COMMENTS ON TURBULENCE
3.4
35
Some comments on turbulence
Considering now a non-ideal fluid, that is, a fluid the kinematic viscosity of which is not negligible, and also assuming incompressibility of the fluid flow, the equations of hydrodynamics
3.4 and 3.10 become:
∇·u = 0
(3.54)
∂u
+ u · ∇u = −∇ p + ν∇2 u
(3.55)
∂t
where ν is the kinematic viscosity of the fluid. Equations 3.54 and 3.55 are usually referred
to as the Navier-Stokes equations.
In principle, the Navier-Stokes equations, supplemented with appropriate boundary and
initial conditions, include all the information we need in order to predict the behavior of a
fluid flow. However, even from everyday life, we are aware of flows which behave in a highly
unpredictable manner, such as the smoke of a cigarette as it rises in the air, or the flow of
water behind a boat. These are examples of flows where the pattern changes rapidly, showing
a dynamical, almost chaotic behavior. These kind of flows are called turbulent flows.
The study of turbulence has puzzled theoretical and experimental physicists for centuries,
for the obvious reason that, in most cases, there appears to be no way to predict the flow
patterns seen in experiments directly from the Navier-Stokes equations, even when we know
the boundary and initial conditions with high accuracy. Fortunately, however, it is possible
to formulate hypotheses based on the experimental data and then, from equations 3.54 and
3.55 derive laws that can help predict properties of the flows.
One very important parameter in the study of turbulent fluid flows is the Reynolds number.
It is defined as
LV
(3.56)
ν
where L and V are a characteristic length and a characteristic velocity of the system under
study, respectively. It can be shown that the Reynolds number is in fact the only cotnrol
parameter for a self-similar, incompressible flow, that is, a flow with defined geometrical
shape. A well-known example of such a flow, the flow behind a solid circular cylinder, has
been studied extensively both in experiments and in simulations. Such studies show that the
transition from a laminar (ordered) flow to a turbulent flow is only controlled by the Reynolds
number. As the Reynolds number increases, the flow gradually loses its symmetries, until it
eventually becomes turbulent.
For high Reynolds numbers, when the fluid is well into the turbulent regime (a state usually
referred to as ”fully developed turbulence”) there are two well-established experimental laws:
The two-thirds law and the law of finite energy dissipation.
The two-thirds law states that,
flow of very high Reynolds number, the
D in a turbulent
E
2
mean square velocity increment, (δu(l)) between two points, seperated by a distance l,
behaves as the two-thirds power of the distance:
D
E
(δu(l))2 ∝ l2/3
(3.57)
R=
The contents of Section 3.4 can be found in Frisch (1996). The phrasing of the experimental laws and
theoretical hypotheses has been quoted almost word for word from that text for exactness.
36
CHAPTER 3. PRINCIPLES OF HYDRODYNAMICS
The law of finite energy dissipation says that, if in an experiment on turbulence all the
control parameters are kept the same and only the viscosity is varied, brought to the lowest
value possible, then the energy dissipation per unit mass of the fluid, dE/dt, behaves in a
way consistent with a finite positive limit ǫ.
These two experimental laws have been included in the form of three hypotheses, in order
to derive the famous Kolmogorov (1941) theory for turbulence. This theory, in accordance to
what we explained above, can only predict some statistical aspects of the flow, under certain
conditions. It is, however, the closest we have to a theoretical description of turbulence.
The first hypothesis is formed to reconcile the apparent inconsistency of the mechanisms
that generate turbulence with the flow symmetries. The scale-invariance typically observed in
turbulent flows must also be reconciled with potential boundary effects. Then the hypothesis
is made that, in the limit of infinite Reynolds number, all the possible symmetries of the
Navier-Stokes equations, usually broken by the mechanisms producing the turbulent flow, are
restored in a statistical sense, at scales small compared to the integral scale and away from
the boundaries.
The term ”integral scale” above defines the length scale on which the turbulence driving
mechanism operates. This, in the case of the flow past a cylinder, for example, would be
the diameter of the cylinder. According to the first hypothesis then, the scales for which the
Kolmogorov theory is valid are always smaller than the integral scale, so that we do not have
to worry about effects due to the distance from the turbulence driver or boundary effects of
the fluid.
The second hypothesis is necessary in order to explicitly express the scale invariance
of turbulence. Since in principle there are infinite possibilities for the scaling exponent of
the flow, giving rise to infinite similarity groups. Under the same assumptions as the first
hypothesis, the turbulent flow is self-similar, in other words it possesses a single scaling
exponent h ∈ R such that
δu(r, λl) = λh δu(r, l)
The third and final hypothesis that goes into the Kolmogorov (1941) theory is that, under
the same assumptions as in the first hypothesis, the turbulent flow has a finite, non-vanishing
mean rate of energy dissipation, ǫ per unit mass.
Using these hypotheses, Kolmogorov derived the first and only exact result obtained so far
to describe a turbulent flow, his famous ”four-fifths law”. Defining the longitudinal structure
function of order p as
Sp (l) ≡
δuk(l)
p (3.58)
we can now express this law: In the limit of infinite Reynolds number, the third-order longitudinal structure function of homogeneous isotropic turbulence, evaluated for length intervals
l small compared to the integral scale, is given in terms of the mean energy dissipation per
unit mass, ǫ (assumed to be finite and non-vanishing) by
δuk (r, l)3
4
= − ǫl
5
(3.59)
Equation 3.59 is not only an exact, non-trivial result of the Navier-Stokes equations, but it
can also be shown to stand, to an order of magnitude approximation, also for compressible
flows.
3.4. SOME COMMENTS ON TURBULENCE
37
Using the self-similarity hypothesis and dimensional arguments, it can be shown that
Kolmogorov’s four-fifths law implies a power-law energy power spectrum in Fourier space:
E(k) ∼ ǫ2/3 k −5/3
(3.60)
Relation 3.60 is an important scaling law for turbulence and we will be using it in our simulations.
Another interesting result of dimensional analysis relates to the numerical resolution necessary for modelling a turbulent flow. Using the velocity u0 and the length l0 at the integral
scale, the expression with dimensions of energy dissipation of unit mass is ǫ = u30 /l0 . The
dissipation scale of turbulence, expected to happen at the viscous scale from molecular in 3 1/4
teractions, is η = νǫ
. Then the ratio of the integral scale to the dissipation scale will
be
3 −1/4
l0
ν
= R3/4 ,
∼
η
l0 u30
a function of the Reynolds number. This means that, for example, in order to simulate a flow
of Reynolds number, say, 10000, we need at least 1000 grid cells per integral scale.
38
CHAPTER 3. PRINCIPLES OF HYDRODYNAMICS
Chapter
4
Previous Work
4.1
Molecular Cloud Formation
In the previous Chapters we introduced some general features of the ISM as they are known
from observations, as well as the basic theoretical background necessary to understand the
gas physics relevant to the dynamics of the ISM. Here we give the motivation for the work in
this thesis, which focuses on the formation mechanisms of the CNM and potentially molecular
clouds as well.
Due to its very high density, molecular gas amounts to only a small fraction of the volume
of the Galactic interstellar medium, but it dominates interstellar mass (Ferrière, 2001). More
specifically, only 1-2% of the gas volume in the Galactic disk is in the molecular phase, but
it makes up about 50% of the total gaseous mass.
In Chapter 2 we mentioned that this phase of the interstellar medium exists in galaxies in highly irregular clouds, with very clumpy, almost fractal structure. The large-scale
morphology of these clouds is very filamentary, according to observations in different wavelengths (Schneider & Elmegreen, 1979; Lada et al., 1999; Hartmann, 2002; Lada et al., 2007;
Molinari et al., 2010), mainly capturing CO transitions and dust absorption or emission and
thus probing slightly different gas temperatures and densities. The internal density distribution of molecular clouds is extremely hierarchical, with large, diffuse structures hosting dense
cores. Molecular clouds, as well as the dense cores in their interior, show a mass distribution that can be very well described by a power-law. For the masses of molecular clouds in
our Galaxy the power-law index is about -1.7 (ie Parmentier (2011)), while for the prestellar
core mass function the slope is shallower, of about -1.3 (ie Alves et al. (2007)). This behavior indicates a very dynamical nature of the interstellar molecular gas. Explaining these
morphological features is the main driver of this work.
Nevertheless, maybe the most important fact about molecular clouds is that they are
the sites of all observed star formation. In fact, there is a very well-known correlation of
the column density of molecular gas with the star formation rate of a galaxy at large scales
This chapter is largely an adaptation of the introduction section of Ntormousi et al. (2011).
40
CHAPTER 4. PREVIOUS WORK
(Wong & Blitz, 2002; Kennicutt et al., 2007). Evidently, they must become gravitationally
unstable at some stage of their evolution. On the other hand, a typical molecular cloud mass
is about 104−5 M⊙ , with a typical temperature of 50 Kelvin, which makes its collapse time
under its own gravitation very short. Any model aiming at explaining star formation should
then account for the fact that, not only will such a cloud give several stars of typical masses
around 1 M⊙ , instead of a single, 104−5 M⊙ object, but it will form them at a gas-to-stars
conversion efficiency of only a few percent (Zuckerman & Evans, 1974).
There are many ideas as to why this should be so. One possible solution is that these clouds
be sustained by strong internal motions, which can provide a support against gravitational
collapse in the form of an additional internal pressure. Luckily, this view is also supported by
the large non-thermal line widths typically observed in molecular clouds, which do indicate
internal supersonic turbulence (Falgarone & Phillips, 1990; Williams et al., 2000). Turbulence
is also an elegant mechanism to fragment the cloud in smaller clumps, which can decouple
from the rest of the cloud and become gravitationally unstable to give lower mass objects
(Larson, 1981; Padoan, 1995; Klessen, 2001).
The interpretation of the aforementioned properties of molecular clouds in combination is
a very challenging task, but it can be somehow alleviated by seeking a common characteristic
to them, one they could inherit from the cloud formation process. If we identify this common quality as non-linearity, then a natural formation mechanism for molecular clouds is a
combination of fluid instabilities (Burkert, 2006).
One general example of an environment that would seed a non-linear fluid evolution, and at
the same time could assemble the large amounts of gas necessary for the formation of molecular
clouds are large-scale converging atomic flows, an idea proposed by Ballesteros-Paredes et al.
(1999) and Hartmann et al. (2001) to explain the fact that stellar populations in local starforming clouds have age spreads that are significantly smaller than the lateral crossing time of
the cloud. Such flows could in principle be anything from expanding shells (Elmegreen & Lada,
1977a; McCray & Kafatos, 1987a), to colliding shells (Nigra et al., 2008), or flows generated
by large-scale gravitational instabilities (Yang et al., 2007; Kim & Ostriker, 2002, 2006).
Pringle et al. (2001) proposed an alternative mechanism for molecular cloud formation by
agglomeration of gas which is already molecular. Although the formation of large clouds from
smaller, already molecular clumps is a possible scenario (Dobbs et al., 2011), there is still no
observational evidence for the existence of a large molecular gas reservoir in the ISM outside
giant molecular clouds.
Heitsch et al. (2006, 2008a) studied the formation of cold and dense clumps between two
infinite flows which collide on a perturbed interface. Their study showed that cold structure
can arise even from initially uniform flows if the conditions favor certain fluid instabilities,
such as the Non-linear Thin shell instability (NTSI, Vishniac (1994)), the Thermal Instability
(Field, 1965) and the Kelvin-Helmholtz instability. We will discuss each of them in detail in
the following chapter.
Vázquez-Semadeni et al. (2006, 2007) used colliding cylindrical flows of tens of parsecs
length, adding random velocity perturbations to the average flow velocity. This allowed them
to study star formation efficiencies and the fates of individual clouds.
These numerical experiments have investigated the mechanisms which lead to the condensation and cooling of atomic to molecular gas and which cause the complex structure of the
resulting clouds, independent of the specific mechanism driving the flows. In a further improvement of these models, in this work we study the possibility for cold structure formation
from colliding flows of limited thickness. Specifically, we model the expansion and collision of
4.2. SHELL FRAGMENTATION AND COLLAPSE
41
thin shells, created by wind- and supernova-blown superbubbles numerically and study the
formed structures.
4.2
Shell fragmentation and collapse
Stellar feedback is a very powerful source of thermal and turbulent energy in the Interstellar
Medium (ISM). Young massive stars, namely stars of O and B spectral types, produce ionizing
photons, expel large amounts of mass in winds and end their lives in supernova explosions, all
processes which shape the matter around them in shells and cavities. Young OB associations,
as groups of between 10 and 100 such stars are commonly called, will create giant shell-like
shocks in their surrounding space from the combined effect of their member stars. Such shells
have been repeatedly observed in our Galaxy in various wavelengths (Heiles, 1979, 1984;
Ehlerová & Palouš, 2005; Churchwell et al., 2006).
These shells are potentially unstable to non-radial perturbations as they expand, as shown
by various analytic studies of thin, shock-bounded or pressure-bounded slabs (Vishniac,
1994; Vishniac & Ryu, 1989). Vishniac’s analytical work on shell fragmentation has been
briefly presented in section 3.2.1. On the other hand, perturbations on large, self-gravitating
shells are able to grow to the point where they can collapse to form stars (Elmegreen,
1994; Whitworth et al., 1994), a scenario usually referred to as the ”collect-and-collapse”
model. This process has been proposed as a trigger of star formation in many environments
(Elmegreen & Lada, 1977b; McCray & Kafatos, 1987b).
An extensive numerical study of the instability of expanding shells, taking into account
nonlinear effects and the different parameters entering the shell expansion was presented in a
series of papers by Dale et al. (2009), Wünsch et al. (2010) and Dale et al. (2011). The focus
of that work being only the gravitational instability of the shells, it is interesting to carry
out an analysis of shell fragmentation due other instabilities, such as the Non-linear Thin
Shell Instability (Vishniac, 1983) for comparison. This is one of the numerical experiments
contained in this thesis.
Giant shells are expected to collide with each other as neighboring stellar associations
evolve (de Avillez & Breitschwerdt, 2004; Nigra et al., 2008), but whether or not these collisions happen before or after the shell has had time to fragment and gravitationally collapse is
not clear. If the shells were to collide before star formation had occurred on their surface, the
collision could trigger a phase change from the atomic gas gathered by the shocks to molecular gas, thus forming molecular clouds by the mechanism described briefly in the previous
Section. The detailed processes which lead to this phase transition will become clearer in the
following Chapter.
42
CHAPTER 4. PREVIOUS WORK
Chapter
5
Numerical Method
5.1
Solving the equations of hydrodynamics on a grid
In order to solve the equations of hydrodynamics numerically, they need to be discretized in
space and in time. There are various approaches to this problem, depending on the choice of
the form of the equations and on the details of the numerical integration scheme. Here we
will briefly discuss the Eulerian class of methods because it contains the algorithm we have
used in our work. Eulerian methods discretize space on a mesh, usually Cartesian, and follow
the evolution of fluid variables on the fixed locations of the grid points.
In principle, this class of methods first calculates the spatial derivatives that appear in
the equations of hydrodynamics, treating them at the right-hand side of the equations, and
then advances these quantities in time by a discrete step dt, treating the temporal derivatives
as the left-hand side f the equations.
It is easy to understand that, when studying the propagation of a fluid on a grid, there is
a point in the calculation at which the flux of material across the grid cell surfaces needs to be
calculated. This is how the advection terms can be approximated in the grid configuration.
The foundation for the class of methods we are going to be discussing here is the so called
”Riemann problem”.
The Riemann problem is defined for an initial state of the fluid where all the variables
are strictly constant for, say, positive x and strictly constant for negative x, but their values
change at x = 0: qi = qL for x < 0 and qi = qR for x > 0, where qi represent each of the
hydrodynamic variables. It is now obvious how this problem relates to solving the equations
of hydrodynamics on a grid; at the smallest scales, each grid cell contains a state of the fluid
which is constant across its length. The neighboring grid cell will, in principle, contain a
different (constant) state of the fluid and the Riemann problem is perfectly defined at their
boundary.
In order to solve the Riemann problem for hydrodynamics, one needs to rewrite the
The short introduction to the Riemann problem contained in Section 5.1 is from ”Finite Volume Methods
for Hyperbolic Problems” by Randall LeVeque.
44
CHAPTER 5. NUMERICAL METHOD
equations 3.4, 3.10 and 3.21 in a matrix form:
 
ρ
ux
∂   1 2
ux +  ρ c −
∂t
e
0

ρ
P
ρΓ
ux
P
ρ
0


ρ
∂  
Γ
ux = 0

∂x
e
ux

(5.1)
where we have written the equations only for one component
of the velocity, ux , e is the total
∂
energy per unit mass of the fluid element, Γ = ρ ∂e ρ is called the ”Grüneisen exponent”
P ∂P
and c2 = ∂P
+
∂ρ e
ρ2 ∂e ρ is the sound speed. This form of the equations makes it a lot
easier to search for the eigenvectors of the system.
From a calculation we will omit here we get three eigenvectors, each representing a quantity
called a ”Riemann invariant”. A numerical solver will have to calculate these quantities at
the cell interfaces. The solution can be pictured as waves propagating in the fluid contained
in a grid cell. One class of waves contains the information for the propagation of the fluid
itself and two are sound waves, one moving in the positive and one in the negative x-direction.
The discrete step for advancing the equations in time will be decided on the condition that
these waves not have time to cross the cell length in one timestep.
This is actually the basis for a numerical method called the Godunov method. In this
method, a grid is defined by the centers of its cells. If we imagine that the x-direction has
been discretized, then the state of the gas qin at time step tn is known at the locations xi , the
centers of N intervals on the one-dimensional space we are considering. The cell interfaces
are then xi+1/2 , each between xi and xi+1 . The Riemann problem can be solved at each one
of these intervals to give the transition of the fluid from one cell to the next. The time step
for the numerical code will be chosen such that solutions from neighboring cells not overlap:
∆t ≤ min(∆ti )
where
∆ti =
xi+1/2 − xi−1/2
max(λi−1/2,k+ ) − min(λi+1/2,k− )
(5.2)
(5.3)
and λi−1/2,k+ denotes the maximum positive eigenvalue of the matrix in equation 5.1 at
interface xi−1/2 and λi+1/2,k− the most negative eigenvalue at interface xi+1/2 . This criterion
is essentially a mathematical expression for what we described in words above: The timestep
should be smaller than the time it takes the fastest waves to cross a cell length, so that the
solutions do not overlap.
In Godunov schemes, in short, the solution advanced in time at position xi from time step
tn to tn+1 is given by
n
n
Fi+1/2
− Fi−1/2
(5.4)
qin+1 = qin + ∆t
xi+1/2 − xi−1/2
n
n
where Fi+1/2
and Fi−1/2
are the fluxes calculated at the cell interfaces by solving the Riemann
problem.
For the numerical simulations in this work we have used the publicly available RAMSES
code (Teyssier, 2002), a Eulerian Cartesian code using a Godunov scheme to solve the equations of hydrodynamics. The code has Adaptive Mesh Refinement (AMR) capabilities, of
which we have made use in some of the simulations in this thesis and which we will shortly
5.2. THE RAMSES CODE
45
discuss in the following Section. We will introduce the general features of the code, as presented by its author, as well as adaptations we have made to selected modules to simulate
physics specific to our purpose.
5.2
The RAMSES code
The RAMSES code has been designed to treat problems relevant to cosmology, so it contains
both an N-body solver, used to calculate the gravitational interaction of collisionless components such as dark matter and stars, and a hydrodynamics solver with AMR capabilities in
order to capture the full dynamical range of cosmological systems.
In this work we have not made use of the N-body solver of the code at all, so we will
not discuss it further. AMR, on the other hand, can be a very useful tool when studying the
physics of the ISM. It is evident from what we have discussed so far that the scales relevant
to the study of interstellar gases range from galactic (kpc), to scales where protostellar disks
are found (10−3 pc). Evidently, the resolution required to capture the evolution of the gas in
all of these scales at once is enormous, making such simulations prohibiting.
Fortunately, in a numerical simulation of such a structured system, the resolution does
not have to be as high in all regions of the computational domain. For instance, the evolution
of the gas within a supernova shell are much more dynamical and structured than inside the
hot gas bubble. AMR gives us the possibility to resolve only the regions of interest in our
simulations, and to let the code decide which these regions are as the simulation advances.
In this way our simulation becomes much more efficient and requires less resources.
In order to achieve AMR, the code is not structured on a basis of single cells, but rather
uses ”octs”, groups of 2dim cells grouped together as the elements of computation, dim denoting the number of dimensions of the simulation. Each oct group belongs to a level of
refinement l, which is advanced in time independently.
When certain criteria are fulfilled at refinement level l, say, then an oct is divided into
its components and the evolution of the fluid variables is not followed on level l + 1. Of
course, the actual implementation of this process in the code is more complicated, due to the
necessary optimization for parallelization and for higher performance in general.
The criteria for the refinement of the grid can be geometrical, they can be based on the
gradients of hydrodynamical quantities, or can relate to the mass contained in an oct. We
will discuss this in more detail in Chapter 8.
The equations of hydrodynamics are solved in the code in their conservative form:
∂ρ
+ ∇ · (ρu) = 0
∂t
(5.5)
∂(ρu)
+ ∇(ρu × u) + ∇P = −ρ∇φ
(5.6)
∂t
∂(ρe)
+ ∇ · [ρu(e + p/ρ)] = −ρu · ∇φ
(5.7)
∂t
with the fluid pressure defined as p = (γ − 1)ρ e − 21 u2 , γ is the adiabatic index and φ the
gravitational potential. Equation 5.7 conserves energy exactly, if we ignore the term due to
Here we limit ourselves to the parts of the code we have used. A detailed description of the code and its
full capabilities can be found in Teyssier (2002).
46
CHAPTER 5. NUMERICAL METHOD
gravity. In Section 5.3 we will discuss additional sources and sinks of energy relevant to our
work.
5.3
5.3.1
Sources and sinks of energy
Implementation of a new cooling and heating module
In Chapter 2 we introduced the cooling and heating processes of interstellar gas and we
explained that they are of extreme importance for the formation of phases in the ISM. It is
therefore essential that we include these processes in our simulations.
The cooling-heating function follows Wolfire et al. (1995) and Dalgarno & McCray (1972)
in the low temperature regime (T < 25000K) and Sutherland & Dopita (1993) in the hightemperature (25000K < T < 108 K) regime. The cooling and heating rates have been discussed in Section 2.6.2 and are shown in Figure 2.8 as a function of density and temperature.
At each timestep of a refinement level and for each cell, the code calculates the gains and
losses of the gas from this tabulated function, according to the state of the gas in the cell.
Then it adds or subtracts the appropriate energy from that cell before advancing the solution
in time. The timestep has to be adjusted accordingly, so that the cell does not acquire a
negative temperature when the cooling rates are very high.
5.3.2
Implementation of a winds
In this work we are studying the impact of hot gas from young massive stars to the surrounding
gas and the triggered formation of structured, cold gas.
The implementation of the energy and mass feedback from such stars for RAMSES has
been done by Katharina Fierlinger as part of her PhD work, so here it will only be shortly
presented. The model for the feedback from massive stars is provided from Voss et al. (2009).
The population synthesis model presented therein provides the energy and mass injection from
an ”average star” of an OB association with time. Such a star represents an entire population
according to a stellar Initial Mass Function, in that its feedback includes the contributions
from stars of all spectral types.
Figure 5.1 shows the energy and mass injection rate from one such star with time, from
the combined effects of stellar winds and supernova explosions.
The domain where feedback is active is defined as a circular region where the cells receive
an extra energy and mass at the beginning of their time step. That energy and mass, taken
from a tabulated form of the data shown in Figure 5.1 are weighted according to the number
of stars the OB association is assumed to contain and to the portion of the cell that is found
inside the region. More details on the wind implementation in RAMSES can be found in
Fierlinger et al. (2011, in prep.)
5.4
Creating turbulent initial conditions
In order to achieve a turbulent initial condition for the diffuse medium, we set up a turbulent
velocity field according to Mac Low et al. (1998), that is, we introduce Gaussian random
perturbations in Fourier space, in a range of wave numbers from k = 1 to k = 4.
The root mean square (rms) Mach number of the turbulent flow is chosen close to unity,
so that the turbulent kinetic energy of the gas is equal to its thermal energy. The energy
5.4. CREATING TURBULENT INITIAL CONDITIONS
47
Figure 5.1: Time dependence of the wind properties for an ”average star”. The solid line shows the energy injection
rate in units of solar luminosity and the dashed line shows the mass injection rate in solar masses per Myr. The
data are from Voss et al. (2009)
48
CHAPTER 5. NUMERICAL METHOD
equipartition assumption between thermal and turbulent kinetic energy is consistent with the
turbulent velocity dispersions calculated for Galactic HI (Verschuur, 2004; Haud & Kalberla,
2007).
After this initial velocity field is calculated, it is introduced as an initial condition to the
code, using a uniform hydrogen density of nH = 1/cm3 and temperature T=8000 K. The
evolution of the gas is followed isothermally and with periodic boundary conditions until the
density-weighted power spectrum of the turbulent velocity field has a power-law slope close to
Kolmogorov (1941) and the density field loses the signature of the initial conditions. During
this time the Mach number is kept constant by driving.
The resulting velocity, density and pressure structure is used as an initial condition for
the simulations of bubble expansion.
In these calculations the driving of turbulence in the diffuse medium has been neglected,
since the turbulence crossing time for our computational domain (tcr ≃ 86 Myrs) is much
longer than the entire simulation. We can therefore safely assume that there are no significant
energy losses due to dissipation on this time scale.
5.5
Simulation setup
We simulate a region of physical size equal to 5002 pc2 , with a resolution of 4096 points at
each dimension. This yields a spatial resolution of about 0.1 pc. An 81922 resolution run
has also been performed but, due to its high computational cost, only for limited integrations
and for testing some resolution effects. Although a 0.1 pc resolution is not sufficient to
resolve the smallest structures in the ISM (see for example, discussion in Hennebelle & Audit
(2007a)), many of the clumps which form in these simulations can still be resolved adequately.
Structures that fall near our resolution limit are not taken into account in our analysis.
As an initial condition we use either a uniform or a turbulent diffuse medium, of hydrogen
density nH = 1/cm3 and temperature T=8000 K. In the case of the turbulent medium these
are, of course, average quantities. We assume the background to be an ideal monoatomic gas
with a ratio of specific heats γ equal to 5/3 and mean molecular weight µ = 1.2mH .
The metallicity assumed in the simulations is solar. Our initial condition is chosen to
lie on a stable point of the heating-cooling equilibrium curve, so that the thermal instability
cannot be triggered unless the medium is externally perturbed. Of course, this is not exactly
true everywhere for the turbulent case, but density fluctuations are not large enough to lead
to a two-phase medium without triggering from the shocks.
In this diffuse medium we insert time-dependent winds that are meant to mimic the
combined effects of winds and supernova explosions in young OB associations. We include
two wind regions, placed on the edges of the computational domain and we assume them to
form simultaneously. We follow their expansion into the diffuse medium until they collide and
a turbulent region arises at their interaction region. Reflecting boundaries are used along the
x-direction and outflow boundaries along the y-direction.
These calculations include no external gravity field or self-gravity of the gas.
Time zero for the simulation is when star formation starts in the wind areas. The system
is evolved in time until boundary effects become potentially important, which is 7 Myrs for
both simulations.
Chapter
6
Formation of cold filaments from colliding
shells
6.1
Shell collision in a uniform diffuse medium
In order to study the expansion of the superbubbles and the development of fluid instabilities
relevant to it, we first simulated the two wind regions in a homogeneous diffuse background
medium. Figure 6.1 shows two snapshots of the simulation in logarithm of temperature and
logarithm of density.
During the expansion of the superbubbles we observe three effects from three different
fluid instabilities. The acceleration of the shock leads to the NTSI, which focuses material
on fluctuation peaks. These condensations are unstable to the Thermal Instability, so they
condense and cool further. The velocity shear caused by the NTSI at the same time also
triggers the Kelvin-Helmholtz instability.
As one can see in Figure 6.1, the NTSI develops faster along the x and y axes, where
we observe more pronounced ”finger-like” structures, characteristic of this instability. This is
clearly a resolution effect. Vishniac & Ryu (1989) showed that for an expanding decelerating
shock there is a critical overdensity with respect to the post-shock gas above which the shell
is unstable. The authors estimated this critical overdensity to be of the order 25 for a windblown shock. Since in our simulations we do not resolve the smallest cooling length adequately,
gas cannot be condensed into as small a volume as it should be according to its cooling rate.
Along the x and y axes though, for a Cartesian grid, the grid cells are closer together than
along the diagonals, leading to an effective smaller distance over which the shock can compress
the gas in one timestep. Thus along the x and y axes the fluid can be compressed slightly
faster, then the critical overdensity can be achieved earlier and the instability arises earlier.
As mentioned in previous sections, we have performed a test run with 81922 resolution for
comparison, but have not been able to suppress this feature. However, the faster growth of
Chapters 6 and 7 are an adaptation of the results section from Ntormousi et al. (2011).
50
CHAPTER 6. FORMATION OF COLD FILAMENTS FROM COLLIDING SHELLS
Figure 6.1: Superbubble collision snapshots in a uniform diffuse medium. Plotted on the top panels is the logarithm
of the hydrogen number density in log(cm−3 ) and on the bottom panels the logarithm of the gas temperature in
log(K). Left: 3 Myrs after star formation, right: 7 Myrs after star formation
6.1. SHELL COLLISION IN A UNIFORM DIFFUSE MEDIUM
51
Figure 6.2: Zoom-in of the last snapshot (7 Myrs after star formation) of the uniform diffuse medium run. Plotted
on the left is the logarithm of the hydrogen number density in log(cm−3 ) and on the right the logarithm of the gas
pressure in log(K cm−3 ). The axes coordinates are in parsecs.
52
CHAPTER 6. FORMATION OF COLD FILAMENTS FROM COLLIDING SHELLS
the instability along these two lines does not affect the average clump formation time and the
clump properties in any measurable way.
We mentioned above that the resulting cold and dense structures are a result of the Thermal Instability. However, our simulations do not include thermal conduction and therefore
cannot, by definition, fulfill the ”Field criterion”. This criterion sets the resolution necessary
for a proper simulation of the Thermal Instability. It essentially states that, in order to accurately represent the smallest structures created by the TI, one needs to resolve the ”Field
length”, λF with at least three grid cells. The Field length is defined as:
λF =
κT
ρ2 Λ
1/2
, where κ is the coefficient of thermal conductivity of the fluid and Λ is the cooling rate of the
gas per unit time. Simulations that do not include thermal conduction may be susceptible to
artificial phenomena near the resolution limit with no apparent convergence with increasing
resolution, since thermal conduction has a stabilizing effect against the Thermal Instability
(Koyama & Inutsuka, 2004).
This stabilization is discussed in Burkert & Lin (2000), who studied the Thermal Instability both analytically and numerically for a generic, power-law cooling function. They find
that density perturbations below a certain wavelength are always damped by thermal conduction. This wavelength depends on the shape of the cooling function, and in our case
would be roughly 4 times the Field length. For typical warm ISM values, that is, a thermal
conductivity of about 104 ergs cm−1 K−1 sec−1 , a temperature of T=8000 K and a hydrogen
number density of n=1 cm−3 , and for a cooling rate of 10−26 ergs sec−1 , the Field length
is about 0.03 pc and four times this length is about our resolution limit. This means that,
even though our resolution is marginal for resolving the smallest unstable fluctuations, higher
resolution without the explicit inclusion of thermal conductivity would produce artificially
small structure. Of course, for lower temperatures and higher densities the Field length is
much smaller, so unstable density fluctuations due to Thermal Instability within our clumps
are not resolved.
Figure 6.2 shows a close-up of the last snapshot of this run, in logarithm of density and
pressure. It is evident from this Figure that all clumps have lower pressures with respect to
the rest of the gas.
Clumps are identified by selecting locations with hydrogen number density greater than
50 cm−3 and temperature smaller than 100 K and using a friends-of friends algorithm to link
such adjacent locations together. This group of cells is then identified as a single clump. The
density and temperature threshold is clearly an approximation, to account for the fact that
these simulations do not include molecule formation and the cooling function we are using
assumes optically thin gas. For the following analysis only structures which contain more
than 16 cells are considered. Smaller structures do not contain sufficient information and are
disregarded as under-resolved.
Figure 6.3 shows the gas phase diagram at the end of the simulation. The plot shows
all the gas in the simulation, color-coded to its corresponding mass fraction in the simulated
box. The solid line is the cooling-heating equilibrium curve for the warm and cold gas and
the red crosses are average clump properties. As an indication for the gas temperatures,
three isotherms have been overplotted. Most of the gas mass seems to be in the cold phase
(– see also Figure 6.4), but there is also about 14% of the gas mass in the warm, thermally
6.1. SHELL COLLISION IN A UNIFORM DIFFUSE MEDIUM
53
Figure 6.3: Pressure versus hydrogen density of the fluid in the box for the uniform medium run. The plot is from
the last snapshot, 7 Myrs after star formation. The gas has been binned and color-coded according to the mass
fraction it represents. The solid line is the cooling-heating equilibrium curve for the warm and cold gas and the
dashed lines show the locations of three isotherms. Red crosses are average clump quantities.
54
CHAPTER 6. FORMATION OF COLD FILAMENTS FROM COLLIDING SHELLS
Figure 6.4: Gas fractions with time for the uniform medium run. The dashed curve shows the mass fraction in
the warm gas phase, the solid curve shows the mass fraction in the hot gas phase and the dotted curve the mass
fraction in the cold phase. (Hot phase: gas with T ≥ 25000K, warm phase: gas with 100K < T < 25000K, cold
phase: gas with T ≤ 100K)
6.1. SHELL COLLISION IN A UNIFORM DIFFUSE MEDIUM
55
Figure 6.5: Number of identified clumps with time. Clumps were only counted if they included more than 16 dense
and cold cells. The solid curve corresponds to the run in a uniform diffuse medium, while the dashed curve to the
run in a turbulent diffuse medium.
unstable regime. This gas is pushed from the stable to the unstable regime by the momentum
inserted by the wind. Some of this thermally unstable gas is located around the cold clumps,
forming a warm corona. We will return to this in the following chapter. The clumps lie on the
equilibrium curve, as does the coldest gas, as expected from clumps formed by the Thermal
Instability.
Figure 6.4 shows the mass fraction of the gas in each phase of the gas with time. We define
cold gas as gas with temperatures T < 100K, warm gas to lie in the temperature regime of
100K < T < 25000K and hot gas to have temperatures T > 25000K. The simulation starts
with almost entirely warm gas in the box. As time goes by more and more cold gas is created.
Since the hot gas is very dilute, it only amounts to approximately 1 percent of the mass
throughout the simulation. In the end of the simulation we have approximately 85 per cent
of the mass in cold clumps and 14 percent in the warm phase. The mass injected by the
OB association amounts to less than 10−5 of the total mass in the domain throughout the
simulation.
Figure 6.5 shows the total number of identified clumps with time for each simulation. At
the end of this simulation more than 400 clumps have formed. This is not due to the new
clumps formed by the TI, but also due to the fragmentation of already existing clumps.
56
6.2
CHAPTER 6. FORMATION OF COLD FILAMENTS FROM COLLIDING SHELLS
Shell collision in a turbulent diffuse medium
In the same way as in the case of a uniform medium, here as well we observe the formation
of cold and dense structures from the combined action of the NTSI, the Kelvin-Helmholtz
and the Thermal Instability. The main difference in this case is the anisotropy caused by
the turbulent density and velocity field to the formation of clumps, as illustrated in Figure
6.6. The shell on the left-hand side fragments very early on in some locations, already at less
than 1 Myr after star formation, but the shell on the right-hand side fragments much later, at
around 3 Myrs after star formation. Rotating the initial conditions by 180 degrees produces
the exactly opposite effect.
This is caused by the difference in the background velocity field and is related to the range
in which the Vishniac instability exists (Vishniac, 1994). The smallest and largest unstable
wavelength both depend on the relative velocity of the shock-bounded slab, in our case the
thin expanding shell, and the background medium. Performing a simple test with a uniform
background velocity field indicated that the instability indeed grows first where the relative
velocity of the shell with respect to the background medium is largest, in agreement with
Vishniac’s analysis.
Figure 6.7 shows a zoom-in of the last snapshot. The picture here very similar to that
in Figure 6.2; the different gas phases are almost in pressure equilibrium and clumps have a
thermal pressure almost an order of magnitude lower then their surroundings.
After the NTSI starts to grow, the condensed material immediately becomes thermally
unstable and we witness the formation of cold and dense clumps, as in the case of the uniform
medium. However, due to the background velocity field, there are now regions where the
NTSI has a faster growth rate with respect to that of a static medium. This leads to a larger
Kelvin-Helmholtz shear. This is the reason why in this simulation longer filaments are formed
compared to the uniform background medium simulation. Some of these structures are shown
in Figures 6.9, 6.10 and 6.11. These figures are zoomed-in fractions of the full domain (Figure
6.8).
By following these filaments from their formation to the end of the simulation, we observe
that they become increasingly more elongated and they constantly fragment into smaller
clumps. Figure 6.11 shows the time evolution of one of these filaments. Before the shells
collide, the structure is a small clump of cold gas. During the collision, this clump gets
caught in a large-scale shear and becomes more and more elongated until, at the end of
the simulation, it has reached a total length of about 100 pc. Note that during the bubble
expansion the clump is not located inside the hot bubble, but rather at the edge of the shock.
The anisotropy in the growth rate of the NTSI leads to the formation of much less clumps
in this simulation with respect to the uniform background simulation, as illustrated in Figure
6.5.
Figure 6.12 shows the phase diagram of the gas in this simulation, in the same way as
Figure 6.3. There are two main differences with respect to the uniform background medium
run; First, the pressure at low densities (n < 0.1 cm−3 ) is higher than in the uniform medium
run. This can be attributed to the additional compression from the turbulence. Second, we
note that the clumps in this simulation are at not only at lower pressures with respect to the
hot medium than the clumps in the uniform background run, but are also at lower absolute
pressures than those clumps. This can be explained by considering the formation mechanism
of the clumps in these simulations. Structures created by the Thermal Instability have lower
pressures than their surroundings. When Thermal Instability has stopped acting on them,
6.2. SHELL COLLISION IN A TURBULENT DIFFUSE MEDIUM
57
Figure 6.6: Super-bubble collision snapshots in a turbulent diffuse medium. Plotted on the top panels is the logarithm of the hydrogen number density in log(cm−3 ) and on the bottom panels the logarithm of the gas temperature
in log(K). Left: 3 Myrs after star formation, right: 7 Myrs after star formation.
58
CHAPTER 6. FORMATION OF COLD FILAMENTS FROM COLLIDING SHELLS
Figure 6.7: Zoom-in of the last snapshot (7 Myrs after star formation) of the turbulent diffuse medium run. Plotted
on the left is the logarithm of the hydrogen number density in log(cm−3 ) and on the right the logarithm of the gas
pressure in log(K cm−3 ). The axes coordinates are in parsecs.
6.2. SHELL COLLISION IN A TURBULENT DIFFUSE MEDIUM
59
Figure 6.8: Simulation snapshot at t=5.3 Myrs after star formation. Plotted here is the logarithm of the hydrogen
number density. The axes are marked in parsecs. The black rectangles show the positions of the filaments shown
in figures 6.9, 6.10 and the 5.3 Myrs snapshot of figure 6.11
60
CHAPTER 6. FORMATION OF COLD FILAMENTS FROM COLLIDING SHELLS
Figure 6.9: A large filament containing several smaller clumps. Plotted here is the logarithm of the hydrogen
number density in log(cm−3 ). The snapshot corresponds to the white box centered at x=250 pc, y=200 pc in the
t=5.3 Myrs snapshot shown in figure 6.8. The axes are marked in parsecs from the axes origin. The black contour
corresponds to nH = 50cm−3
6.2. SHELL COLLISION IN A TURBULENT DIFFUSE MEDIUM
61
Figure 6.10: As in figure 6.9. The snapshot corresponds to the white box centered at x=225 pc, y=314 pc in the
t=5.3 Myrs snapshot shown in figure 6.8.
62
CHAPTER 6. FORMATION OF COLD FILAMENTS FROM COLLIDING SHELLS
Figure 6.11: Time evolution of a single filament. The plots show logarithm of hydrogen number density. The
black contour shows the level nH = 50 cm−3 . From top left to bottom right: 4.3, 4.6, 5.3 and 7 Myrs after star
formation. Note the change in scale between the snapshots.
6.2. SHELL COLLISION IN A TURBULENT DIFFUSE MEDIUM
63
Figure 6.12: Pressure versus hydrogen density of the fluid in the box for the turbulent medium run. The plot is
from the last snapshot, 7 Myrs after star formation. The gas has been binned and color-coded according to the
mass fraction it represents. The solid line is the cooling-heating equilibrium curve for the warm and cold gas and
the dashed lines show the locations of three isotherms. Red crosses are average clump quantities.
64
CHAPTER 6. FORMATION OF COLD FILAMENTS FROM COLLIDING SHELLS
Figure 6.13: Gas fractions with time for the turbulent medium run. The dashed curve shows the mass fraction in
the warm gas phase, the solid curves shows the mass fraction in the hot gas phase and the dotted curve the mass
fraction in the cold phase. (Hot phase: gas with T ≥ 25000K, warm phase: gas with 100K < T < 25000K, cold
phase: gas with T ≤ 100K)
6.2. SHELL COLLISION IN A TURBULENT DIFFUSE MEDIUM
65
they move towards pressure equilibrium with the surrounding medium in approximately a
sound crossing time, cs /L, where cs the sound speed and L the typical size of these clumps.
For a typical density of 10−21 g/cm3 and a typical temperature of 50 K, the sound speed in
these clumps is about 0.8 km/sec. At this velocity sound waves will cross a 2 pc long clump
in about 0.6 Myrs. In the turbulent diffuse background run, clumps are formed later than in
the uniform background run, which means they are less evolved and farther from equilibrium.
According to the above calculations, the time difference between clump formation between
the two simulations is about 3 sound crossing times, which is enough for many the clumps in
the uniform diffuse background simulation to have reached approximate pressure equilibrium
with the hot medium.
Figure 6.13 shows the evolution of the mass fractions of the gas in different temperature
regimes. Unlike the previous run, in this simulation there is a maximum in the hot gas mass
fraction at about 3.5 Myrs after star formation and a later minimum of the warm gas fraction
at around 5 Myrs after star formation. This can be explained by the delayed formation of
cold gas in this simulation. Until the cold gas is formed, the hot gas just compresses the warm
gas. Then the hot gas mass fraction increases, while the warm gas mass fraction decreases.
When cold gas starts to form, it quickly dominates the mass, causing the warm and hot mass
fractions to drop.
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CHAPTER 6. FORMATION OF COLD FILAMENTS FROM COLLIDING SHELLS
Chapter
7
Morphological features of the cold clumps
7.1
General
Figure 6.5 shows that we form hundreds of clumps in each snapshot of the simulations. This
makes it very difficult to study each of them in detail, especially since they are constantly
merging and splitting. However, we can make some general comments on the morphology of
the identified clumps and look into some of their properties.
Clumps are generally at lower pressures with respect to their surrounding gas (– see Figures
7.1, 7.2 7.3 and 7.4). This means that the Thermal Instability is still acting in these regions,
causing them to condense further. Two examples of condensing clumps are shown in Figure
7.1. Others (about 11% in the last snapshot of the uniform background run and 16% in the
turbulent background run) are rotating. Two examples of rotating structures are shown in
Figure 7.2. Rotation is either combined with compression or it introduces a centrifugal force
which makes the clump expand. Of the rotating cores in the last snapshots of the uniform
and the turbulent background runs, about 25% and about 35% respectively are at the same
time condensing and the rest are expanding. In very few cases, less than 1%, the centrifugal
force exactly balances the force due to the pressure difference between the interior and the
exterior of the clump.
A small fraction of the identified clumps are in pressure equilibrium with their surroundings, at least at their most central parts. These clumps do not tend to host significant internal
motions (– Figure 7.4).
All clumps are surrounded by a warm corona which is more dilute than their cold central
regions and at an intermediate pressure between their pressure and the one of the surrounding
gas. This corona usually surrounds more than one clump, indicating that they are parts of
larger structures, such as the ones illustrated in Figures 6.9, 6.10 and 6.11.
In the following we focus on the velocity dispersions, sizes and possible evolution of the
clumps.
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CHAPTER 7. MORPHOLOGICAL FEATURES OF THE COLD CLUMPS
Figure 7.1: Two examples of condensing clouds. Top panels show the logarithm of hydrogen number density in
log(cm−3 ) and bottom panels show the logarithm of thermal pressure in log(K cm−3 ). The black arrows show the
velocity field with the mean velocity of the central clump subtracted. Overplotted in black are the contour levels
for nH equal to 50 cm−3 , 100 cm−3 and 1000 cm−3 .
7.1. GENERAL
69
Figure 7.2: Two examples of rotating clumps. Top panels show the logarithm of hydrogen number density in
log(cm−3 ) and bottom panels show the logarithm of thermal pressure in log(K cm−3 ). The black arrows show the
velocity field with the mean velocity of the central clump subtracted. Overplotted in black are the contour levels
for nH equal to 50 cm−3 , 100 cm−3 and 1000 cm−3 .
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CHAPTER 7. MORPHOLOGICAL FEATURES OF THE COLD CLUMPS
Figure 7.3: Two examples of clumps hosting random motions. Top panels show the logarithm of hydrogen number
density in log(cm−3 ) and bottom panels show the logarithm of thermal pressure in log(K cm−3 ). The black arrows
show the velocity field with the mean velocity of the central clump subtracted. Overplotted in black are the contour
levels for nH equal to 50 cm−3 , 100 cm−3 and 1000 cm−3 .
7.1. GENERAL
71
Figure 7.4: Two examples of clumps with small internal velocities. Top panels show the logarithm of hydrogen
number density in log(cm−3 ) and bottom panels show the logarithm of thermal pressure in log(K cm−3 ). The black
arrows show the velocity field with the mean velocity of the central clump subtracted. Overplotted in black are the
contour levels for nH equal to 50 cm−3 , 100 cm−3 and 1000 cm−3 .
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CHAPTER 7. MORPHOLOGICAL FEATURES OF THE COLD CLUMPS
Figure 7.5: Clump internal velocity dispersion versus their sound speed at different times. From top to bottom, 3,
5 and 7 Myrs after star formation. The left panel corresponds to the uniform background medium run, the right
panel to the turbulent background medium run.
7.1. GENERAL
73
Figure 7.6: Time evolution of the velocity dispersions over sound speed for three isolated cores. The green line
corresponds to a condensing clump, the blue line to a rotating clump and the red line to a clump with random
motions. The data for each clump begin from the snapshot where we can still identify the clump as being the same.
74
7.2
CHAPTER 7. MORPHOLOGICAL FEATURES OF THE COLD CLUMPS
Velocity dispersions
Figure 7.5 shows the internal velocity dispersions of the clumps at different times for both
simulations. The velocity dispersion in this context is defined as the square root of the variance
among all locations which compose the clump. This means that for many clumps a velocity
dispersion may indicate rotation, compression, expansion or random motions. As random
motions here we define any combination of rotation and compression or expansion. Since the
clumps have no significant internal density fluctuations, mass-weighted velocity dispersions
are not very different from the ones presented here.
Although many of the formed clumps host supersonic motions in their interior at all times,
there is an indication that these motions decrease at later times, as shown in the velocity
dispersion plot of the last snapshots. This effect seems slightly more pronounced for the
turbulent background run, where the largest internal velocity dispersions have disappeared
in the last snapshot.
In the case of compression or expansion, this means that the clumps gradually move to
pressure equilibrium with their surrounding gas. In the case of random motions, it indicates
that these motions were inherited by the turbulent environment that created the clumps but,
in absence of any mechanism to sustain them, they die out. In the case of rotation, though,
the situation is a bit more complicated. Rotation could be an effect of the large-scale KelvinHelmholtz shear, it can originate from Thermal Instability accretion or can be a result of
structures splitting or merging. Angular momentum conservation sustains these motions for
longer, so we would only expect them to decrease on a viscous timescale.
In order to study if the decrease in internal velocity dispersion is observable for individual
clumps, we tracked some of them back in time and plotted their velocity dispersion evolution.
Since dense structures are created all the time and clumps merge or split at each snapshot, it
is very difficult to construct an algorithm able to automatically identify a clump in different
snapshots. Instead we identified the clumps by eye according to their positions and translational velocities. We focus here on three examples: a rotating clump, a contracting clump
and a clump with random internal motions. Their velocity dispersions over their corresponding sound speed as a function of time are shown in Figure 7.6. Since clumps evolve almost
isothermally, their sound speed does not vary significantly during the interval shown in the
plot. The green line corresponds to a contracting clump, the blue line to a rotating clump
and the red line to a clump with random motions. All three clumps show some decrease in
velocity dispersion with time. Although we are not able to make a complete study at this
stage, we observe that the rotating clump practically maintains the same velocity dispersion
throughout its existence, showing only a slight decrease, while the condensing clump and the
clump with random motions show a decrease in velocity dispersion. For the condensing clump
this decrease is especially pronounced.
7.3
Sizes
Figure 7.7 shows the size distribution of the clumps at different times for both simulations.
The sizes are calculated as the square root of the area occupied by the clump. Although we
do not form clumps larger than approximately 3 pc, the maximum clump length can reach
about 10 pc. This means that structures may have one very large dimension, but they occupy
a very small area.
7.3. SIZES
75
Figure 7.7: Clump size distributions in parsecs at different times. The plot on the left-hand side corresponds to
the run in a uniform diffuse medium and the plot on the right to the run in a turbulent diffuse medium. The solid
black, dashed red and dash-dotted blue histograms correspond to the 3 Myrs, 5 Myrs and 7 Myrs, respectively.
Figure 7.8: Clump size over clump Jeans length distributions at different times. The plot on the left-hand side
corresponds to the run in a uniform diffuse medium and the plot on the right to the run in a turbulent diffuse
medium. The solid black, dashed red and dash-dotted blue histograms correspond to the 3 Myrs, 5 Myrs and 7
Myrs, respectively.
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CHAPTER 7. MORPHOLOGICAL FEATURES OF THE COLD CLUMPS
Figure 7.8 shows the distributions of the ratios of clump sizes over their corresponding
Jeans length, at different times. The size distributions of the clumps have a very similar
shape and range between the two simulations. As clumps are fairly uniform in density and
temperature, their Jeans length does not vary significantly within a single clump. Clumps in
general seem to be smaller than their corresponding Jeans length but, as time advances, some
of them become large enough to be potentially unstable to gravitational collapse. Of course,
as we have not included gravity in these simulations we cannot know if this would actually
be the case.
As mentioned earlier, since the formed structures are very filamentary, they are likely to
contain more Jeans lengths along a single dimension. Note also that the algorithm we use to
find clumps favors the identification of the smallest possible structures as separate entities.
As mentioned above however, the clumps we identify are usually parts of larger structures
which are dynamically interacting or surrounded by a common warm and more diffuse corona.
7.4
Clump evolution
Although most of the clumps are in low-pressure regions in the simulation, there are some
clumps with approximately the same pressure as their surrounding gas. All of the clumps are
surrounded by an intermediate pressure corona, which is also thermally unstable.
This, in combination with the fact that clumps show a tendency of decreasing their internal
motions with time and that clouds in pressure equilibrium tend to host smaller internal
motions leads us to believe that there might be an evolution from clumps out of equilibrium,
with strong internal motions to more quiescent clumps.
Figure 7.9 shows the tracks of the same three clumps we traced back in time, on the
pressure-density diagram. The dotted line is the cooling-heating equilibrium curve. The
cold structures form in an area where cooling dominates, possibly from thermally unstable
gas, at the left of the figure. As time goes by they move on the equilibrium curve and
gradually increase their density and pressure, always staying on the curve. The timescale
of this evolution is about 2-3 Myrs, which corresponds to about 4 - 5 clump sound crossing
times.
7.4. CLUMP EVOLUTION
77
Figure 7.9: Isolated core tracks on a pressure-density diagram. The green triangles correspond to a condensing
clump, the blue crosses to a rotating clump and the red diamonds to a clump with random motions. The solid black
line is part the cooling-heating equilibrium curve which was shown in figures 6.3 and 6.12.
78
CHAPTER 7. MORPHOLOGICAL FEATURES OF THE COLD CLUMPS
Chapter
8
Metal enrichment of the clouds
8.1
Setup of the simulation
A two-dimensional, high-resolution simulation very similar to those presented in the previous
Chapters, is performed using the same setup, namely two young OB associations placed at a
certain distance from each other in a warm diffuse background. These cold and dense shells
eventually collide in the middle of the computational domain. In this case though, each OB
association comprises 20 ”average” stars, placed in a circular region of 5 pc radius and the
physical size of the domain is half that of the simulations in Chapters 6 and 7.
An important increase in efficiency in comparison to the previous simulations is achieved
with the use of Adaptive Mesh Refinement (AMR). Given the nature of the problem under
study, the most adequate refinement policy is to trigger the division of a oct when the difference in the gradients of pressure and density exceeds a certain threshold. We have used
a threshold equal to 1% in this work. Experimenting with the value of the threshold gave
no significant differences in the grid structure, although thresholds higher than 10% failed to
capture the shock structure properly.
The supershells in this setup are expanding in a uniform diffuse background, which means
that any perturbations that are expected to seed the fluid instabilities we want to study must
arise at the grid level. In order then to make this simulation comparable to previous results,
we should also seed the perturbations at the smallest grid level and the physical scale for
seeding the perturbation should be the same between different runs. For this reason, we
initiate the simulation with a nested grid configuration, where the highest resolution region
is located at the center of the simulation box. Once the first seeds of the perturbation start
to grow, we switch to the adaptive refinement policy described above.
Figure 8.1 shows the behavior of the three different grid configurations. The top row shows
the logarithm of hydrogen number density and the bottom row shows the corresponding grid
structure. The grid structure for the uniform grid run is, of course, trivial, but it is shown here
This chapter is an adaptation of the paper ”On the inefficiency of metal enrichment of cold gas in colliding
flow simulations”, by E.Ntormousi and A. Burkert, to be submitted to the Astrophysical Journal.
80
CHAPTER 8. METAL ENRICHMENT OF THE CLOUDS
Figure 8.1: Snapshots of three runs using different refinement techniques, taken at the same timestep, about 3
Myrs after star formation. From left to right, uniform grid, geometry-based refined and gradient-based refined grid.
The plots on the top row show the logarithm of hydrogen number density and the plots on the bottom row show
the corresponding grid structure. The axes coordinates are in parsecs.
8.1. SETUP OF THE SIMULATION
81
Figure 8.2: Density (top) and pressure (bottom) histograms of the same snapshots shown in Figure 8.1, about 3
Myrs after star formation. The solid black histograms correspond to the run with a uniform grid, the dashed red
histograms to the gradient-based AMR run and the dashed-dotted blue histograms to the geometry-based refinement
run.
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CHAPTER 8. METAL ENRICHMENT OF THE CLOUDS
Figure 8.3: Gas mass fractions with time for three models. The black lines correspond to the run with a uniform
grid, the red lines to the gradient-based refinement run and the blue lines to the geometry-based refinement run.
Dashed lines show the mass fraction of the gas in the warm phase, solid lines show the gas mass fraction in the hot
phase and dashed-dotted lines show the gas mass fraction in the cold phase. (Hot phase: gas with T ≥ 25000 K,
warm phase: gas with 100 K< T <25000 K, cold phase: gas with T < 100 K).
8.1. SETUP OF THE SIMULATION
83
anyway for completeness. The density and pressure gradient-based refinement works very well
in following the shocks and the locations where cold structure is formed (rightmost panel of
Figure 8.1). However, the small-scale noise introduced by the interpolation, in combination
with the nonlinear nature of the gas dynamics in this environment leads to slight differences
in the morphology of the gas between the uniform and the adaptively refined grid at the same
timestep.
There is a minimum resolution below which cold gas does not form at the edges of the
shells at all, evident in the run with a nested grid configuration (middle panel of Figure 8.1).
In the regions where the grid size is too large the shocks are smoothed out, suppressing the
Vishniac Instability and preventing the gas at the edges of the shells from becoming thermally
unstable.
The volume fractions of the gas in each phase are practically identical between the uniform
grid and the gradient-refined grid simulations (Figure 8.2), however the gas in the uniform
grid run is allowed to condense to slightly higher densities, as shown by the density histogram
on the top panel of Figure 8.2. The nested grid simulation has significantly less gas in the
high-density regime in comparison, due to the lower resolution in the top and bottom parts
of the grid.
The corresponding pressure histograms for the three runs are shown on the bottom panel
of Figure 8.2. Again, the uniform grid run and the gradient-defined AMR runs seem to agree
almost completely, but the geometry-based refinement run has a slightly different configuration, with a lot of gas in the thermally unstable regime, unable to condense and cool further.
This is indicative of the result one might get from cooling shock simulations if the resolution
is insufficient to capture the initial density enhancements.
The mass fractions of the gas in different temperature regimes are plotted in Figure 8.3.
The dotted-dashed lines in this Figure correspond to the cold gas (T < 100 K), which forms
small dense clumps, the dashed lines correspond to the warm gas (100 K< T <25000 K),
mainly consisting of the background gas and the coronas around the cold clumps, and the
solid lines correspond to the hot gas (T ≥ 25000 K), mainly located in the wind and the
shocked areas. Black color denotes the uniform grid run, blue the geometry-based refinement
run and red the gradient-based refinement run. The mass fractions in different phases seem to
agree between the uniform grid and the gradient-based grid runs during the whole simulation
time, with the cold gas dominating the mass towards the end of the simulation (about 80%
of the total mass). However, the nested grid run fails to reach these high mass fractions due
to the very low resolution in part of the domain.
In conclusion, the comparison between simulations of this setup with AMR and with
uniform grid simulations have shown no difference in the amount of cold gas, the position of
the shocks and the sizes or velocity dispersions of the formed clumps. Small differences in the
shock morphology are, of course, always present, due to the very nonlinear nature of these
phenomena.
For this particular simulation we use a box of 250 pc physical size, at an effective resolution
of 20482 . The choice of a smaller box with respect to Paper I is not only more physical, in
terms of the average distance between OB associations in the Galaxy but it also yields a
smaller computational volume for the same physical resolution, significantly reducing the
computational cost of the simulation.
We stop the calculation when the turbulence in the collision area starts to expand towards
the inflow boundaries. In this particular case this happens 4.36 Mys since star formation in
the OB associations.
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CHAPTER 8. METAL ENRICHMENT OF THE CLOUDS
As mentioned before, the aim of this work is to study the advection of metals from the
OB associations to the cold gas formed at the shock wake. This is done by means of a passive
advected quantity, representing the metal injection from the stars. A constant amount of
metals, equal to 10−3 metal particles/cm3 is added at the wind region at each coarse timestep.
As this value is totally arbitrary, it can be weighted to simulate different environments.
The amount of metals introduced by the OB associations is assumed here to have a
negligible effect on the amount of cold gas formed. In principle, though, extra metals could
affect our results due to their contribution to the cooling, which would also change the regime
where the gas becomes thermally unstable. In our calculations the most important coolant of
the gas is line emission from carbon and oxygen. A decrease in the abundance of these elements
would cause the area where we can have phase formation due to the Thermal Instability to
shrink, and an enrichment would enlarge the Thermal Instability regime (Wolfire et al., 1995).
Support for the approximation we are making comes from Walch et al. (2011b), who
studied the effect of metallicity on the formation of cold gas from Thermal Instability in
simulations of turbulence. They found that, for driven turbulence (which is the case in our
models), the total amount of cold gas in the simulation is not significantly affected by changes
in the metallicity. This means that, as long as the metallicity of the gas we are simulating
is high enough to capture the Thermal Instability regime of the ISM, we are not making
significant errors in the total amount of cold gas in the domain by ignoring the enrichment
from the OB stars in the cooling function.
8.2
Metal enrichment of the gas from the OB associations
Figure 8.4 shows snapshots of the simulation before and after the shell collision. The top
panels of this Figure are contours of the logarithm of the gas temperature and the bottom
panels show the logarithm of the ratio of metal to hydrogen atoms in the cell.
The general picture of the simulation is the same as in Paper I. The spherical shocks created by the stellar feedback are unstable to the Vishniac instability (Vishniac, 1983, 1994) as
small-scale wind imperfections create ripples on their surface. The result of the gas condensation at the peaks of these ripples is to trigger the Thermal Instability (Field, 1965), creating
cold and dense clumps at the shock wake. The shear on the shell surface, also caused by the
Vishniac Instability, gives rise to characteristic Kelvin-Helmholtz eddies, thus contributing to
the dynamics of the newly-formed cold clumps (left panel of Figure 8.4).
When the shells collide, the combination of the large-scale shear by the collision and
the small-scale structure already present in the shells gives rise to a turbulent region at
the collision interface which contains both warm and cold gas (right panel of Figure 8.4).
Turbulence is a very efficient mixing mechanism, so we naturally expect an enhancement in
metallicity of the warm gas after the shell collision.
In Figure 8.4, we can indeed see that the warm gas has enhanced metal content. The
same can be shown more clearly by plotting the mass fraction of the gas in the computational
domain in density-metallicity bins. In Figure 8.5, showing such plots for two snapshots of the
simulation, we can see that the dense gas dominates the mass of the gas in the computational
domain. At the same time we see that it never reaches relative enrichment of more than 10−4 .
For comparison, we note that, were the metals ejected by the stars to be instantaneously and
homogeneously mixed in the diffuse gas phase, the relative enrichment would be 10−2 and
if all the metals from the stars ended up in the cold phase, the relative enrichment of that
8.2. METAL ENRICHMENT OF THE GAS FROM THE OB ASSOCIATIONS
85
Figure 8.4: Logarithm of temperature (top) and logarithm of relative metal content (bottom) for two snapshots.
On the left, 1.22 Myrs and on the right, 4.46 Myrs after star formation took place in the OB associations.
86
CHAPTER 8. METAL ENRICHMENT OF THE CLOUDS
phase would be of about 5 · 10−2 .
Throughout the simulation practically all the metals injected by the OB associations stay
in the hot wind, despite the fact that most of the mass is in the cold gas component. At
late times a small fraction of the metals (1-5%) mixes into the slightly denser, warm gas
(nH ≃ 10−1 , T ≃ 105 ) due to the shell collision that causes turbulent mixing.
8.3
Metal enrichment of the clumps
A clump is identified as a collection of adjacent cells with densities above 50 cm−3 and
temperatures lower than 100 K. By this definition alone, Figure 8.5 already indicates that
the clumps do not contain significant amounts of material from the OB associations.
To look at the clump metallicities in more detail, we plot their metallicity distributions,
shown in Figure 8.6. The top panels in this Figure show the distributions of the mean
absolute metal content of the clumps, in the arbitrary units chosen in this simulation for the
metallicity (10−3 /cm3 injected in the wind domain at each timestep) and the bottom panels
show distributions of the mean metallicity over the mean hydrogen number density of the
clumps. The plots on the left-hand side correspond to the snapshot at 1.22 Myrs and the
plots on the right-hand side to the final snapshot, at 4.36 Myrs.
Even though the numbers can be rescaled to mean different absolute metal content in the
clumps, the important fact here is that the cold phase will always receive at least two orders
of magnitude less metals than the diffuse warm phase.
Even though the metal injection from the OB associations does not stop during the simulation time, the metal content of the clumps does not seem to increase significantly. The
spread of the distribution of the relative metal content of the clumps seems to increase with
time. As the system evolves, new clumps are formed at relatively lower metallicities. The
little metals they accumulate over time leads to the formation of a peak in the distribution. However, the maximum value of the distribution does not increase, meaning there is no
significant enrichment.
Figures 8.7 and 8.8 show the mean number density of metals in a clump as a function
of distance from its closest OB association and as a function of the polar angle with respect
to the horizontal line in the middle of the domain, respectively. The amount of metals in a
clump does not seem to depend on its position with respect to the OB associations, pointing
to a very uniform distribution of metallicities around the young associations.
8.3. METAL ENRICHMENT OF THE CLUMPS
87
Figure 8.5: Mass fractions in density-metallicity bins at two snapshots, 1.22 Myrs (top) and 4.36 Myrs (bottom)
after star formation.
88
CHAPTER 8. METAL ENRICHMENT OF THE CLOUDS
Figure 8.6: Top: Distributions of the absolute metal content of the clumps. Bottom: Distributions of the metal
content of the clumps over their hydrogen number density. The data are from snapshots 1.22 Myrs (left) and 4.36
Myrs (right) after star formation.
Figure 8.7: Dependence of the clump metal content on their distance from the closest association. The plots are
shown at 1.22 Myrs (left) and 4.36 Myrs (right) since the beginning of the simulation.
8.3. METAL ENRICHMENT OF THE CLUMPS
89
Figure 8.8: Dependence of the clump metal content on their polar angle calculated with respect to the horizontal
line at the center of the computational domain. The plots are shown at 1.22 Myrs (left) and 4.36 Myrs (right) since
the beginning of the simulation.
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CHAPTER 8. METAL ENRICHMENT OF THE CLOUDS
Chapter
9
Summary and Conclusions
In this text we have presented our work on the formation of cold structure from large-scale
colliding flows. In particular, we have shown the results of detailed numerical simulations of
colliding superbubbles. These large cavities of hot gas are created by the violent feedback
from young massive stars, implemented as a time-dependent energy and mass source in a
public hydrodynamics code. The extremely dynamical evolution of the gas was followed to
the point of the fragmentation and collision of the dense shells.
Our results provide a picture of the ISM similar to that in Audit & Hennebelle (2005)
and Hennebelle & Audit (2007b), where the ISM phases are tightly interwoven, with sharp
thermal interfaces between them. Of course, a more detailed comparison is not possible, since
that work was done with a much higher resolution, representing a smaller region of the ISM
and not including the hot phase. In our simulations, elongated cold structures sit in warm,
thermally unstable coronas, submerged in a hot dilute medium that the stellar winds and
supernovae from young OB stars create. Clumps are dynamical entities, constantly merging
and splitting and in general not in pressure equilibrium with their surrounding gas. In that
sense, the picture we see in our simulations is very different from the classical ISM model
proposed by McKee & Ostriker (1977), where the clouds are treated as quasi-static spheres.
The structures we find are rather long thin filaments, very similar to the large ”blobby sheets”
proposed by Heiles & Troland (2003).
We identify individual clumps in the simulations by setting density and temperature
thresholds and connecting adjacent locations in the simulation domain which exceed these
thresholds. This method favors the identification of the smallest cold and dense structures.
The clumps tend to be connected in groups within a common warm corona, forming long
filaments and are mostly located in areas of lower pressure with respect to their surroundings.
The latter causes many of them to condense to higher densities and smaller volumes, almost
isothermally. Apart from condensing clumps we find rotating clumps, as well as clumps
hosting irregular motions. All these internal motions are reflected in the velocity dispersion as
supersonic internal motions. Approximately 11 to 16% of the identified clumps are rotating
and are interesting candidates for forming protostellar disks. However, rotation should be
studied in three dimensions for more reliable results.
Internal clump motions tend to decrease with time, as the clumps come closer to thermal
pressure equilibrium with their warm surroundings. However, if gravity and star formation
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CHAPTER 9. SUMMARY AND CONCLUSIONS
was included in our simulations, we would expect these motions to be sustained for longer
times due to gravitational collapse and feedback from the newly formed stars in their interior.
Cold clumps in our simulations form by Thermal Instability condensation of the ambient
diffuse medium when it is perturbed by the expanding shells. This, in combination with the
fact that the mass injected by the OB associations is a very small fraction of the total mass
in our domain throughout the simulation, shows that the material forming the new clumps
could not have been enriched directly by the supernova explosions. The material ejected from
the OB associations will probably enrich the interstellar matter on much larger timescales,
after the hot bubbles have mixed with the diffuse medium.
In order to study of metal advection from the hot to the cold ISM, we have presented
a simulation of the same setup, using an advected quantity. This quantity was inserted in
the region of the domain representing the wind, to represent the metals ejected by the winds
and supernova explosions in these OB associations. In this way, we were able to distinguish
between material originating from the stars and material originating from the diffuse ISM in
the composition of the cold clumps.
We have found that the metal content of the clumps is very low throughout the simulation.
The maximum metal to hydrogen number ratio reached in the simulation is about 10−4 , two
orders of magnitude lower than the value of the same ratio for the warm diffuse medium. The
fact that the diffuse gas receives a significant fraction of the injected metals implies that, if
molecular clouds were to form in this environment, significant enrichment would be delayed
by at least one stellar generation. This effect is even more relevant if we consider that the
free-fall time for each of these dense clumps is about 1 Myr, which means that many of them
would be collapsing before the end of this simulation, had gravity been considered, leaving
even less time for enrichment.
The metal content of the clumps seems to be independent of their position with respect to
the OB association. This, in combination with the small spread in cloud metallicities, means
that the next stellar generation, formed by the clumps created in such an environment, would
be very uniform in its metal content, provided the diffuse component is also well mixed.
Of course, there are many effects that have not been included in this work. For instance,
we have assumed that the metal injection from the OB associations is roughly constant with
time and that it is uniformly distributed in the wind region. Both these assumptions are
questionable. We would, in principle, expect the metals to be contained in small clumps,
as part of clumpy winds or fast supernova ejecta, possibly making mixing more efficient. In
addition, the wind material should of course differ in quantities and composition from the
supernova material, although this would still mostly end up in the diffuse rather than the
dense cold phase. These are all complications that should be taken into account in future
work.
Thermally unstable gas amounts to about 8 to 10% of the total gas mass in the last
snapshots of our simulations, far from the almost 50% that is commonly detected in observations (Heiles & Troland, 2003) or found in simulations of turbulent, thermally bistable flows
(Gazol et al., 2001, 2005). This is because our simulations are dominated by what we identify
as CNM, that is, gas with temperatures lower than 100K. Due to our very high resolution,
this gas is also very dense, reaching hydrogen number densities of the order of 105 cm−3 .
This gas would be mostly molecular, so it would not be identified as cold HI in observations.
In simulations with gravity we would not expect to encounter this issue, since gravity would
eventually dominate the dynamics of the formed clumps and turn them into stars once they
became dense enough, so thermal instability would no longer be responsible for their evolution
93
at these densities.
The simulations presented here are only a first step to modeling triggered molecular cloud
formation using physically motivated colliding flow parameters. We have not attempted a
parameter study in this paper, but it will be the object of future work to study the effect of
varying the distance between the superbubbles, the number of OB stars creating the superbubbles and the metallicity of the gas.
In this work we have also not taken into account galactic shear or density stratification.
Assuming a galactic rotation rate of 26 km sec −1 kpc −1 , the relative shear in our computational box would be 13 km sec−1 . This velocity would have a crossing time of approximately
37 Myrs, which is much longer than what our simulations last. Density stratification with
a scale height of H=150pc would cause the superbubbles to expand faster along the vertical
direction, but since we are interested mostly in what happens at the superbubble collision
interface, we can neglect this effect as a first approximation.
Our calculations also ignore magnetic fields and gravity. s We would expect magnetic
fields to play a significant role in the dynamics of the problem, both during the expansion of
the superbubbles and also during the more complex collision phase, if they had a preferred
orientation, but in two dimensions we would not be able to model all relevant phenomena
properly anyway. Gravity is essential for studying the evolution of the clumps and for estimating their star forming efficiency. Combining with the modeling of a third dimension, it
would give us useful mass estimates of the formed structures. Moving to three dimensions
and including gravity is work in process and will be presented in a future paper.
We find the main limitations of our work to be the lack of resolution and bidimensionality.
As we have pointed out, our simulations do not reach the resolution required to capture all
the relevant physical processes on the smallest scales. We have accounted for small-scale
effects by only studying clumps which contain more than 16 grid cells. Higher numerical
resolution will certainly provide more insight on the number of formed clumps and their
internal structure. Although numerical effects do introduce a thermal conduction effect,
explicitly modeling thermal conduction would help set the minimum scale for formed clumps
and achieve convergence with increasing resolution. On the other hand, if gravity is simulated
it will probably already become important at length scales larger than the Field length.
The restriction of the presented models to two dimensions is emphasizing the formation of
filaments. In three dimensions, these structures could have a sheet-like morphology. Gravity
could then be responsible for turning these sheets into filaments (Burkert & Hartmann, 2004;
Hartmann & Burkert, 2007; Heitsch et al., 2008b).
94
CHAPTER 9. SUMMARY AND CONCLUSIONS
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Acknowledgments
This thesis represents the efforts of the last four years and signifies the beginning of my
research career. It would have been impossible for me though to reach this point without the
support of several people in my environment. I take this opportunity to thank them.
I am grateful to my PhD advisor, Andreas Burkert, for all our interesting discussions,
for sharing his excitement for science with me and for helping my growth as an independent
researcher with his sincere support.
I received a lot of practical help for my thesis work from Fabian Heitsch. Not only did
he help me gain insight into the physics of the ISM by sharing his experience with me, but
he also trusted me with some of his codes. In particular, the calculations of the cooling and
heating rates, as explained in Chapter 2 have been calculated with his code.
Katharina Fierlinger shared her PhD work on implementing stellar feedback in RAMSES
with me. Without her contribution a lot of this work would not have been possible. In
addition, her careful observations significantly improved this work. Of course, I am also
grateful to Rasmus Voss for providing the stellar feedback data for this work and for being
always available for questions regarding his calculations.
I also thank all my friends and colleagues here at the LMU Observatory, Thorsten Naab,
Peter Johansson and Roland Jesseit, not only for techical advice, but also for all the fun we
have had these years and for their kind words of advice at hard times.
To my Physics Diploma supervisor at the Aristotle University of Thessaloniki, Loukas
Vlahos, an eternal ”thank you” for believing in me from the start and for encouraging me to
pursue a career as a scientist.
To my family, all my love and gratitude for always being there for me and for all the
sacrifices they’ve made so that I can fulifill my dreams.
Finally, to my partner, Jose Luis Pinar, a big thank you from my heart for your faith in
me, for your support and understanding. I would never have reached this far without you.
Curriculum Vitae
Personal
Name:
Evangelia Ntormousi
Born:
31st of March, 1984
Citizenship:
Greek
Address:
Scheinerstr. 1, Munich, Germany D-81679
Education
2007–date
Ph.D in Astrophysics
2001–2007
Bachelor and Masters in Physics
International Max Planck Research School
(IMPRS), USM-LMU
Aristotle University Thessaloniki, Greece
Publications
• E. Ntormousi, A. Burkert, On the inefficiency of metal enrichment in colliding flow
simulations, in prep.
• E. Ntormousi, A. Burkert, K. Fierlinger and F. Heitsch, Formation of cold filamentary
structure from wind-blown superbubbles, ApJ, 731, 13
• E. Ntormousi, J. Sommer-Larsen, Hot gas haloes around disc galaxies, 2010, MNRAS,
409, 1049
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