Doktorarbeit Uni

Doktorarbeit Uni
Dissertation
submitted to the
Naturwissenschaftlich-mathematische Gesamtfakultät
of the
Ruprechts-Karl-Universität of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
Put foward by
Gabriel-Dominique Marleau, M. Sc.
born in Verdun, Québec (Canada)
Oral examination: 1st February 2016
The post-formation entropy of gas giants:
Radiative properties of the accretion shock and
constraints from observations
Referees:
Priv.-Doz. Dr. Hubert Klahr
Priv.-Doz. Dr. Simon Glover
Kurzfassung
Wir untersuchen den Strahlungstransport und insbesondere die Strahlungsverlusteffizienz η bei der
planetarischen Akkretionsstoßfront mittels strahlungshydrodynamischer Simulationen der obersten Schichten eines Planeten und wählen dafür Momentaufnahmen während seiner Entstehung
aus. Wir benutzen den PLUTO-Code in sphärischer Symmetrie und flußbegrenzte Diffusion mit den
Ein- und Zweitemperaturenverfahren (1- und 2-T ). Tabellierte Gas- und Staubopazitäten werden
herangezogen und eine konstante Zustandsgleichung wird benutzt um Strahlungstransporteffekte
zu isolieren.
Die Simulationen zeigen, daß in einem bedeutsamen Teil des Entstehungsparameterraumes die
Stoßfront isotherm ist und die ganze kinetische Energie in Strahlung umgewandelt wird und daß der
in den Planeten hineingebrachte Anteil mit der internen Leuchtkraft verglichen vernachlässigbar ist.
Der Befund scheint zu sogenannten Kalten Anfängen zu führen.
Wir untersuchen auch die Frage aus der Perspektive der Beobachtungen und bestimmen, welche Informationen Direktbeobachtungen über die Anfangsentropie liefern können. Erstens versuchen wir,
genaue Kühlungskurven durch eine selbstkonsistente Einbindung der komplexen BT-Settl-2010Atmosphärenmodelle zu berechenen. Wir stellen jedoch fest, daß dies aufgrund unkonvergierter
Staubphasenübergänge bei T eff ≈ 1500 K unmöglich ist. Zweitens benutzen wir einfache Kühlungskurven um Niedrigstwerte der Anfangsentropie von Gasriesen herzuleiten. Wir finden, daß
die untersuchten Beispiele wahrscheinlich nicht als klassiche Kalte Anfängen entstanden sind, aber
daß κ Andromeda b möglicherweise einen ‘Deuteriumblitz’ erlebt.
Abstract
We study the radiative properties and in particular the radiation loss efficiency of the planetary
accretion shock by performing radiation-hydrodynamics simulations of accretion onto a planet at
snapshots during its formation. We use the state-of-the-art code PLUTO in spherical symmetry and
both one- and two-temperature flux limited diffusion. We take tabulated gas and dust opacities and
use a constant equation of state to isolate radiation transport effects.
We find that, for a significant subset of the formation parameter space, the schock is isothermal
and essentially the entire kinetic energy is converted to radiation. The fraction brought into the
planet is negligible compared to the internal luminosity, which appears to favour the so-called
‘cold-start’ assumption.
We also study what constraints direct detections can provide on the post-formation entropy.
First, we attempt to produce accurate cooling tracks by using self-consistently the sophisticated
BT-Settl-2010 atmospheric models. However, we find that this is not possible because of due
to dust phase transitions at T eff ≈ 1500 K. Secondly, we set lower bounds on the post-formation
entropy of some well-studied gas giants. We find that they most likely did not form as classical
coldest starts, but that κ Andromeda b may be undergoing a ‘deuterium flash’.
Contents
Kurzfassung – Abstract
i
Contents
iii
1. Introduction
1
1.1. Shock physics . . . . . . . . . . . . . . . . . .
1.1.1. Shock structure . . . . . . . . . . . . .
1.1.2. Conservation equations across the shock
1.1.3. Estimate of the shock temperature . . .
1.2. Radiative flux . . . . . . . . . . . . . . . . . .
1.3. The radiation quantity R and the reduced flux .
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2.1. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2. Initial set-up . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.3. Boundary conditions . . . . . . . . . . . . . . . . . . . . .
2.2. Microphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1. Equation of state . . . . . . . . . . . . . . . . . . . . . . .
2.2.2. Opacities . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Quantities of interest . . . . . . . . . . . . . . . . . . . . . . . . .
2.4. Relevant parameter space . . . . . . . . . . . . . . . . . . . . . . .
2.4.1. Formation parameters . . . . . . . . . . . . . . . . . . . .
2.4.2. Microphysics parameters . . . . . . . . . . . . . . . . . . .
2.5. Results: Efficiencies and downstream quantities . . . . . . . . . . .
2.5.1. Dependence of the structure and efficiency on optical depth
2.5.2. Results for selected cases with tabulated opacities . . . . . .
2.5.3. Results for a large parameter space and constant opacity . .
2.5.4. Results for a large parameter space and tabulated opacities .
2.6. Estimate of the coupling . . . . . . . . . . . . . . . . . . . . . . .
2.7. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2. Shocks with 1-T radiation transport
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3. Shocks with 2-T radiation transport
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41
3.1. Introduction . . . . . . . . . . . . . .
3.2. Selected cases with tabulated opacities
3.2.1. Gas–radiation equilibrium . .
3.3. Accreting a whole planet . . . . . . .
3.3.1. First results . . . . . . . . . .
3.4. Conclusion . . . . . . . . . . . . . .
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4. Cooling tracks for gas giants using BT-Settl atmospheres
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1. Overview of the cooling code . . . . . . . . . . . . . . . . . . . . . . . .
~ iii ~
41
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52
52
4.2.2. Description of the coupling . . . . .
4.2.3. Overview of BT-Settl atmospheres .
4.3. Results . . . . . . . . . . . . . . . . . . . .
4.3.1. Input from the BT-Settl atmospheres
4.3.2. Cooling tracks with BT-Settl . . . .
4.4. Summary . . . . . . . . . . . . . . . . . .
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5. Constraining the initial entropy of gas giants
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5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Cooling models with arbitrary initial entropy . . . . . . . .
5.2.1. Calculation of time evolution . . . . . . . . . . . . .
5.2.2. Calculation of gas giant models . . . . . . . . . . .
5.2.3. Luminosity as a function of mass and entropy . . . .
5.2.4. Luminosity as a function of helium fraction . . . . .
5.2.5. Cooling . . . . . . . . . . . . . . . . . . . . . . . .
5.2.6. Comparison with classical hot starts and other work .
5.3. General constraints from luminosity measurements . . . . .
5.3.1. Shape of the M-Si constraints . . . . . . . . . . . .
5.3.2. Solutions on the hot- vs. cold-start branch . . . . . .
5.3.3. Definition of ‘hot-start mass’ . . . . . . . . . . . . .
5.4. General constraints from gravity and effective temperature .
5.5. Comparison with observed objects . . . . . . . . . . . . . .
5.5.1. Directly-detected objects . . . . . . . . . . . . . . .
5.5.2. 2M1207 . . . . . . . . . . . . . . . . . . . . . . . .
5.5.3. HR 8799 . . . . . . . . . . . . . . . . . . . . . . .
5.5.4. beta Pic . . . . . . . . . . . . . . . . . . . . . . . .
5.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.7. Appendix A: Radii . . . . . . . . . . . . . . . . . . . . . .
5.8. Appendix B: Entropy offset in SCVH . . . . . . . . . . . .
5.9. Appendix C: Age, luminosity, and mass constraints . . . . .
5.9.1. 2M1207 . . . . . . . . . . . . . . . . . . . . . . . .
5.9.2. HR 8799 . . . . . . . . . . . . . . . . . . . . . . .
5.9.3. beta Pic . . . . . . . . . . . . . . . . . . . . . . . .
5.10. Update to beta Pic b . . . . . . . . . . . . . . . . . . . . . .
5.10.1. Summary . . . . . . . . . . . . . . . . . . . . . . .
5.10.2. Discussion . . . . . . . . . . . . . . . . . . . . . .
5.11. Analysis of kappa Andromeda b . . . . . . . . . . . . . . .
5.11.1. Summary . . . . . . . . . . . . . . . . . . . . . . .
5.11.2. The mass of kappa And b: Warm-start models . . . .
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6. Conclusion
101
7. Acknowledgements
103
A. Summary and test of the radiation hydrodynamics code
105
A.1. Numerical settings . . . . . . . . . . . .
A.2. Basic equations . . . . . . . . . . . . . .
A.2.1. Internal energy equation . . . . .
A.2.2. Mechanical energy equation . . .
A.2.3. Radiation energy equation . . . .
A.3. Operator splitting for the energy equation
A.4. Radiation-hydrodynamics shock test . . .
A.4.1. Subcritical case . . . . . . . . . .
~ iv ~
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A.4.2. Supercritical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
A.4.3. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
B. Additional material
B.1.
B.2.
B.3.
B.4.
B.5.
B.6.
B.7.
B.8.
113
Flux limiters . . . . . . . . . . . . . . . . . . . . . . . . .
Effect of the opacity . . . . . . . . . . . . . . . . . . . . .
Reduced fluxes for Bell & Lin (1994) opacities . . . . . .
Parameter exploration . . . . . . . . . . . . . . . . . . . .
Estimated Mach numbers for population synthesis planets .
Density inversions . . . . . . . . . . . . . . . . . . . . . .
Extracting an atmospheric structure from a log file . . . . .
Coupled cooling tracks: Effective temperature evolution . .
References
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~v~
We are so small between the stars,
so large against the sky
— Leonard Cohen
1
Introduction
Of all fields of astrophysical research, the science of extrasolar planets (exoplanets) is a relatively
new and particularly interesting one. While philosophical musings about the nature and origin
of the planets can be traced back to ancient Greece, it has only recently become possible to start
addressing these issues observationally. Detailed research on our Solar System planets, which are
rocky (Mercury, Venus, Earth, Mars), gaseous (Jupiter, Saturn), or icy (Uranus, Neptune), only
became possible in the second half of the 20th century with probes passing by some planets or
even entering into their atmosphere. This has brought a wealth of information and allowed detailed
studies, but it was only with the discovery of the first exoplanet orbiting a main-sequence star
twenty years ago (Mayor & Queloz, 1995) that the current, modern era of exoplanet science began.
This has revealed an even richer diversity of planet types and system architectures and also allows
statistically meaningful statements about these characteristics and, still in its early phase, about the
physical processes which shape them (e.g., Ford, 2014). Moreover, exoplanet discoveries usually
elicit a particularly positive response across wide segments of society, also from some who are
not typically attracted to science. This is due perhaps to the fact that many exoplanetary physical
parameters can be, relative to other subfields of astrophysics, more easily brought in relation with
our experience as inhabitants of a planet, and to the possibility of discovering that life does exist
somewhere else in the cosmos.
As supported by strong observational evidence, planets form in young stars’ dust and gas discs,
which are leftover material from their formation. Both bottom-up (hiarchical) and top-down formation processes are considered: core accretion (CA), in which a solid protoplanetary core of several
Earth masses (M⊕ = 5.97 × 1028 g) first forms and triggers runaway gas accretion (Pollack et al.,
1996, described in more detail below), and gravitational instability (GI), in which the local selfgravity of a region of the gas disc begins to dominate over its pressure support and collapse ensues
(e.g., Boss, 1997). Most planets, in particular planets close to their hosts, seemed to be formed by
CA (Janson et al., 2012; Grether & Lineweaver, 2006). A number of observational diagnostics can
help distinguish between the two modes, both population-wide and, tentatively, at the planet level
(Dodson-Robinson et al., 2009), but most are uncertain and a robust interpretation of observations
does not yet seem possible.
There are several open questions in planet formation, one of which concerns the accretion of gas
in CA’s runaway gas accretion phase. In this model, which reproduces quite well observations of
several independent observables (e.g., mass–period distribution, frequency as a function of hoststar metallicity, etc.; Mordasini et al., 2009, 2015), the accretion rate onto a planet starts increasing
exponentially once the solid core has accreted a gaseous envelope equal roughly to its mass (the
‘crossover’ mass) (Pollack et al., 1996; Hubickyj et al., 2005). This early envelope is still connected
~1~
CHAPTER 1. INTRODUCTION
to the surrounding disc, the ‘nebula’ (Stevenson, 1982), and the planet is able to accrete more mass
as it contracts on its Kelvin–Helmholtz timescale
tKH
GMp 2
=
,
Lp Rp
(1.1)
where Mp , Lp , and Rp are the planet’s mass, radius, and luminosity. The outer radius of the planet
is thus on order of the Hill or Bondi radii, defined below, and equal to several tens to hundreds of
Jupiter radii RJ (RJ = 7.15 × 109 cm). Soon after the start of the runaway accretion, however, the
planet is contracting faster than the local disc is able to provide material, and it shrinks down to
a new equilibrium with a much smaller radius. This state is that of a gaseous body in hydrostatic
equilibrium which is accreting gas in free-fall from the accretion radius the Hill or Bondi radius.
Thus there is a shock defining the planet’s surface, with boundary conditions provided by this
infalling gas. The important unanswered question concerning this stage is therefore the following
(Marley et al., 2007): What fraction of the kinetic energy of the gas is not radiated away but added
to the planet instead? To the best of our knowledge, this question has not yet been studied in the
context of planet formation.
Indeed, the two extreme situations, where all the kinetic energy is kept or, conversely, fully
radiated away, lead to very different outcomes in terms of planetary properties. One convenient
parameterisation of this is in terms of the entropy S of the gas giant (Marley et al., 2007; Spiegel &
Burrows, 2012). Classically, and to a very good approximation, gas giants are thought to have an
isentropic (i.e., adiabatic) interior (e.g., Baraffe et al., 2014), and both their luminosity and effective
temperature T eff as well as their cooling Lp (t) depend solely on their mass and entropy. Thus the
traditional and useful way of framing the question is: What is the post-formation entropy of a gas
planet of a given mass? This post-formation entropy is also termed ‘initial’, as it is the value the
planet has at the beginning of its cooling phase.
Marley et al. (2007) drew attention to the fact that the consequences of the assumption about
the post-shock state of the gas are dramatic at early ages, with differences of several orders of
magnitude in luminosity. Since direct observations, which began in the last decade (the first examples being Chauvin et al., 2004; Lafrenière et al., 2008; Marois et al., 2008; Lagrange et al.,
2009), provide only a brightness measurement, determining a directly-observed object’s mass requires a conversion through planetary cooling curves, which need to assume some initial value for
the entropy. However, what this should be has not yet been predicted by theory and, before the
author’s Master’s thesis, (Marleau, 2012), not constrained by observations. We therefore approach
the question from both sides in the present dissertation.
This thesis is structured as follows: The remainder of this chapter presents basic shock theory
which will be needed in the first few chapters. Then, Chapter 2 presents shock calculations using
one-temperature radiation transport, followed by selected comparison cases for two-temperature radiation transport in Chapter 3. In Chapter 4, we present efforts to produce self-consistent planetary
cooling curves which use a state-of-the-art and popular set of detailed atmosphere models since they
are needed to interpret direct observations. Finally, Chapter 5 presents published work on deriving
information about the post-formation entropy of planets from observational data, and a summary
and outlook are presented in Chapter 6. Background on and tests of the radiation-hydrodynamics
code used in Chapters 2 and 3 are presented in Appendix A, and additional material is given in
Appendix B.
1.1. Shock physics
A shock is characterised by an abrupt change in velocity of the fluid, from supersonic to subsonic.
Radiative shocks are shocks in which at least the radiation flux Frad and, at higher temperatures, also
the radiation energy density Erad and pressure Prad play a role. They are common in astrophysics,
whether in stellar explosions (novæ, supernovæ) or in envelopes of pulsating stars (see, e.g., Drake,
2006 and references therein). Usually, the radiation energy density may be neglected because
~2~
1.1. SHOCK PHYSICS
Erad Eint , whereas the flux ratio, the Boltzmann number
Bo =
Eint v
Eint v
∼
× ,
Frad
Erad c
(1.2)
may be close to unity since we are considering non-relativistic flows (v c). For the approximate
second equation, we used the free-streaming case Frad = cErad (i.e., that the photons effectively
move at the speed of light; in general, the flux of quantity X can always be written as the density of
X times the propagation velocity). Eint is the internal energy density, with dimensions of energy per
volume, with Eint = %cv T only in the case that the heat capacity at constant volume cv is constant.
One must distinguish between ‘one-’ and ‘two-temperature’ (1- and 2-T ) radiation transport, also
known as the equilibrium and non-equilibrium approaches. In both cases, one is solving moments
of the radiative transfer equation, but makes different assumptions about the radiation field (as discussed in more detail in Appendix A). In 2-T radiation transport, Erad is taken to be independent of
the gas temperature T gas (hereafter usually denoted with T ), whereas 1-T assumes that the radiation and gas couple sufficiently well that Erad = aT 4 , where a = 7.5657 × 10−15 erg cm−3 K4 is the
radiation constant. One could also go one step further and consider 3-T radiation transport, where
the electron and ion temperatures are not assumed to be equal, but for the densities relevant to the
astrophysical context (and here in particular) both equilibrate in the downstream region of a shock
over a small distance (Zel’dovich & Raizer, 1967; Mihalas & Mihalas, 1984; Drake, 2006) and
they will be assumed equal. The ‘radiation temperature’ is defined as the equilibrium temperature
corresponding to the radiation energy density, i.e., T rad = (Erad /a)0.25 .
1.1.1. Shock structure
The schematic structure of a radiative shock is shown in Figure 1.1. In the frame in which the shock
is at rest, gas in an initial state (T 0 , %0 ) flows in from the right in the upstream (or pre-shock) region, passes through a precursor region (described below) before encountering the hydrodynamical
shock, and flows away in the downstream (or post-shock) region in a final state (T 1 , %1 ) > (T 0 , %0 ),
i.e., heated up and compressed. The hydrodynamical shock occurs over the viscous length scale,
which is equal to a few gas mean free paths ` = 1/nσ (where n is the volume number density and σ
is the cross-section of the gas constituents) and is therefore embedded in the much larger ‘radiation
exchange zone’ (Mihalas & Mihalas, 1984) of several optical depths λphot = 1/κ%, where κ is the
opacity (with dimensions of cross-sectional area per mass).
The radiation precursor arises because the hot post-shock (downstream) region, which in the
planetary or stellar context is optically thick as it corresponds to the planetary or stellar interior,
emits radiation which penetrates into the upstream material and pre-heats it. The radiation and gas
temperatures are decoupled in the ‘transmissive’ parts of the precursor, where radiation flows at
close to the streaming limit (Frad ≈ cErad ), as well as in the optically thin cooling layer behind
the shock (see Figure 1.1). The increase of temperature in the precursor leads to a small pressure
gradient which slightly slows down the gas; in turn, its density must increase slightly to have a
spatially constant accretion mass flux ṁ = %v (neglecting non-planar geometric effects here). In
the downstream region and in the ‘diffusive’ parts of the precursor, where Frad cErad , matter and
radiation are in equilibrium.
The classical shock literature distinguishes between ‘subcritical’ and ‘(super)critical’ shocks depending on the strength of the shock (e.g., Zel’dovich & Raizer, 1967; Mihalas & Mihalas, 1984;
Drake, 2006, see also Heaslet & Baldwin, 1963). As the post-shock temperature T 1 increases, so
does the generated flux F ∝ σT 1 4 , and for sufficiently strong shocks (i.e., for a velocity equal to
or greater than the ‘critical velocity’), this flux is able to raise the temperature of the gas in the
precursor to approximately the value it would have attained downstream. This is the supercritical
regime, which is a is a limiting case (Drake, 2007). While in both sub- and supercritical shocks the
immediate post-shock temperature T + is larger than the equilibrium value T 1 due to preheating, any
increase above the critical velocity only serves to increase T + but cannot drive T − to be higher than
~3~
CHAPTER 1. INTRODUCTION
Fig. 1.1.: Schematic structure of radiative shocks, showing the temperature (top halves) of the gas (solid
line) and the radiation (dotted line), as well as the density (bottom halves). Left panel: Subcritical shock
(T − < T 1 ). Right panel: Supercritical shock (T ≈ T 1 ). The hydrodynamical shock is highlighted (vertical
dashed line). The relative sizes of the regions are only illustrative. Figure reproduced from Vaytet et al.
(2013).
T 1 . (As discussed in Drake (2007) and Vaytet et al. (2013), there is however a number of interesting
refinements which can be made to this description but they are left out for the sake of brevity.)
A 1-T analysis of the shock yields essentially the same profiles, except in the regions where
both temperatures would not be equal. In particular, the immediate post-shock temperature T + ,
known as the Zel’dovich spike, cannot be modeled. Also, in the subcritical case, the pre-shock gas
temperature T − will in general be different from that of the 2-T case.
We now derive useful conservation relations for steady-state shocks, which will lead to the definition of an efficiency.
1.1.2. Conservation equations across the shock
Consider the usual equations of radiation hydrodynamics, expressing conservation of mass, momentum, and total energy along the one-dimensional flow (Mihalas & Mihalas, 1984, see also
Appendix A):
∂%
+ ∇ (%v) = 0
∂t
∂%v
+ ∇ %v2 + P = −%g
∂t
∂Etot
+ ∇ ([Eint + Ekin + P] v + Frad ) = −%vg,
∂t
(1.3a)
(1.3b)
(1.3c)
where as usual %, P, v, g, Eint = %eint , Ekin = 21 %v2 , and Frad are respectively the density, pressure,
velocity, magnitude of the gravitational acceleration, internal and kinetic energy volume density
(eint and ekin being the massic energy densities), and radiative flux, and where in spherical coordinates the divergence operator ∇(·) = ∂(·)/∂r for scalars and ∇(·) = 1/r2 ∂(r2 ·)/∂r for vectors, with r
the radial coordinate. The total energy density is Etot = Eint + Eint + Erad but we assumed that Erad
is negligible compared to Eint in the other terms.
~4~
1.1. SHOCK PHYSICS
It is easy to show that the shock front1 is moving out in the laboratory frame with velocity
vfront =
%1 v1 − %− v−
.
%1 − %−
(1.4)
Jumping into the shock frame and assuming steady-state (i.e., on the downstream side, this implies
looking only close to the shock, before the rmin boundary is felt), the mass, momentum, energy
conservation Equations (1.3) yield (Marshak, 1958; Mihalas & Mihalas, 1984; Commerçon et al.,
2011)
%− v− = %1 v1
(1.5a)
%− v− + P− = %1 v1 + P1
!
!
1 2
1 2
ṁ h− + v− + F− = ṁ h1 + v1 + F1 ,
2
2
2
2
(1.5b)
(1.5c)
where h is the specific enthalpy (per unit mass) and ṁ = %v, and where the ‘−’ subscript denotes
immediately upstream of the shock and the ‘1’ subscript denotes downstream of the radiative relaxation (cooling) layer (see Figure 1.1)2 . As mentioned above, the radiation flux plays in general
an important role. Note that these equations are valid in general, independently of the details of
the radiation transfer. The enthalpy is given by h = γ/(γ − 1)P/% = γeint and we will work with a
constant equation of state (EOS), which means in particular that the mean molecular weight µ and
the heat capacity ratio γ are assumed constant.
Following Commerçon et al. (2011), we divide the momentum conservation equation (Equation 1.5b) by the upstream pressure P− to obtain
!
1
2
Π = 1 + γM− 1 − ,
(1.6)
r
where the pressure and compression ratios are defined as Π = P1 /P− and r = %1 /%− , respectively.
Similarly, making the energy conservation equation (Equation 1.5c) dimensionless yields
!
2
∆F
1
Π
−
=
1
−
−
1
,
(1.7)
2
1
2 r
3
r
%
v
(γ
−
1)M
−
−
−
2
where ∆F = F− −F1 is the jump in flux at the shock, i.e., the net amount of radiation lost at the shock
and an important quantity in this work. These derivations have made use of mass conservation
(Equation 1.5a) and the definition of the Mach number M,
M2 =
v2 %
v2
=
,
γP
cs 2
(1.8)
where cs is the sound speed. Note that at low Mach numbers, it becomes important to distinguish
between the laboratory and shock frames since vfront approaches v− . Finally, the dimensionless
momentum and energy conservation expressions (Equations 1.6 and 1.7) can be combined to yield
ηkin ≡
∆F
1
3
2 %− v−
=1+
2
(γ − 1)M−
2
r−1 γ+1 1
+
.
r
γ − 1 r2
(1.9)
This defines the kinetic-energy-normalised shock loss efficiency ηkin , the central quantity of the first
few chapters of this thesis. It is the efficiency with which the incoming kinetic energy is converted
to radiation, which is then lost from the shock into the upstream region, nevertheless interacting
with the incoming gas to generate a precursor.
1
2
This expression is redundant, since a shock is already a front, but helps distinguish the velocity of the shock in the
laboratory frame from that of the material upstream of the shock in its frame.
Note that in equation (3) of Commerçon et al. (2011) %1 and %2 should both be replaced by 12 but that this is only a
typesetting error, their following equations being correct.
~5~
CHAPTER 1. INTRODUCTION
100
Energy transmission fraction
Shock loss efficiency
100 %
75 %
50 %
ηkin , γ = 1.44
γ = 1 .1
ηphys , γ = 1.44
γ = 1 .1
25 %
0%
1
10−1
10−2
10−3
10−4
10−5
10−6
ηkin , γ = 1.44
γ = 1 .1
10−7
10
1
Shock Mach number
10
Shock Mach number
Fig. 1.2.: Left panel: Shock loss efficiency against the shock-frame Mach number according to the classical,
kinetic-energy normalised definition (Equation 1.11, as in Commerçon et al. 2011; upper two curves) and
according to the physical, total-energy definition (Equation 1.13; bottom two curves), for a constant γ = 1.44
(solid curves) and γ = 1.1 (dashed). Right panel: Fraction (1−ηkin ) of the kinetic energy not lost as radiation
and thus transmitted to the downstream region; (1 − ηkin )Lacc, max is the shock heating.
Anticipating that our simulation results will always show (very nearly) isothermal shocks, we
can look at the case where the pre- and post-shock temperatures are the same (T 1 = T − ). We
recall that this is classically referred to as the supercritical case. In that case, the compression ratio
depends quadratically on the Mach number:
r = γM− 2 ,
(1.10)
since Π = r, so that the efficiency becomes
ηkin
isoth = 1 −
1
γ 2 M4
,
(1.11)
one of the key equations of this work. Note that there is effectively no dependence on γ when
√
the Mach number is written out since cs ∝ γ. The basic theory was presented by Marshak
(1958) and Mihalas & Mihalas (1984), and Drake (2006) gave a derivation but without writing
down the explicit form of Equation (1.11). While he showed efficiency curves as a function of the
shock velocity for examples in high-energy research, credit for providing and plotting the general
ηkin
isoth (M) function goes, to the best of our knowledge, to Commerçon et al. (2011). It is shown in
Figure 1.2. As pointed out by Commerçon et al. (2011), ηkin
isoth ≈ 100 per cent for M & 2. This
is valid only for isothermal shocks but since our shocks are always isothermal, we will drop the
subcript for convenience.
Of course, the gas flowing into the shock brings not only kinetic but also internal energy with it,
and this total energy flux is
!
1
%vetot = %v v2 + h
(1.12a)
2
!
1 3
2
1
= %v 1 +
.
(1.12b)
2
γ − 1 M2
Naively, one might expect in supersonic flows (M = v/cs 1) the internal energy (measured by
cs 2 ) to be negligible compared to the kinetic energy (measured by v2 ), but the 2/(γ − 1) factor can
change somewhat the qualitative situation, especially for low γ values; e.g., for γ = 1.1 and even a
~6~
1.1. SHOCK PHYSICS
high Mach number M = 10, the factor 2/M2 (γ − 1) = 0.2, i.e., a 20 per cent contribution. From
this, one can then define a ‘physical’ shock efficiency in the isothermal case as
!−1
1
2
phys
kin
ηisoth = ηisoth × 1 +
,
(1.13)
γ − 1 M2
which expresses what fraction of the total (kinetic and internal) energy is converted to radiation.
This is also shown in Figure 1.2. However, this efficiency definition is only of limited use since an
isothermal shock is not able to tap into the internal energy when the heat capacity is constant. The
‘physical efficiency’ will therefore only be used for pragmatic display purposes, as the dependence
on γ of the efficiency is quite clear on a linear scale even for M ∼ 30 (see Figure 1.2).
In the planetary (or, similarly, stellar) formation context, the maximal ‘accretion luminosity’ can
be written as
GMp Ṁ
1
1
Lacc, max = 4πRp 2 %v3 = Ṁv2 ≈
,
(1.14)
2
2
Rp
where Ṁ = 4πr2 %v is the mass accretion rate (neglecting the sign of v) and the last expression is
valid for free fall from a large radius (see Equation 2.7). Therefore, the energy actually radiated
away at the shock is usually defined as
Lacc = ηkin
GMp Ṁ
,
Rp
(1.15)
with ηkin set to 1 (e.g., Hartmann et al., 1997; Baraffe et al., 2012; Zhu, 2015). (For completeness,
note that our loss efficiency ηkin corresponds to the quantity (1 − η) of Spiegel & Burrows (2012),
αh of Mordasini et al. (2012), (1−α) of Hartmann et al. (1997), and X of Commerçon et al. (2011).)
The case ηkin = 1 is termed a ‘cold start’ and ηkin = 0 a ‘hot start’, as the resulting planets are then
respectively colder or hotter (Marley et al., 2007). In this work, we set out to determine ηkin by
radiation hydrodynamical simulations.
While Lacc escapes from the shock and is therefore relevant for the disc structure and observations
of this accretion luminosity (Zhou et al., 2014; Eisner, 2015; Sallum et al., 2015), the quantity of
interest for the planetary structure is the amount of energy which goes into the planet. By energy
conservation, this ‘shock heating’ is Q+shock = (1 − ηkin )Lacc, max (see Figure 1.2b). The internal
energy being the same across the shock for a constant-heat capacity gas, this energy remains as
kinetic energy, in the (subsonic) settling velocity. The increase of ηkin with M is due to the fact
that higher Mach numbers lead to higher compression ratios, which, by mass conservation, imply
smaller settling velocities. As the gas is brought to rest in the settling region and compresses, it
gives up its energy as volume work, which is thus spread over a small region of the atmosphere. The
detailed structure of this heating will not be important when one is interested only in the energetics
of the shock for coupling purposes.
1.1.3. Estimate of the shock temperature
We return to shock physics to derive an estimate of the shock temperature which will prove very
useful for interpreting the numerical results. When all the incoming kinetic energy is converted to
radiation, i.e., ηkin = 1, the jump in radiative flux is given by
1
∆Frad = %vshock 3 .
2
(1.16)
If the upstream region is in the free-streaming limit (discussed also in the next section), as should
be the case for sufficiently optically thin gas, we have Frad = cErad ; but in general
Frad 6 cErad = acT shock 4 ,
(1.17)
where T shock is the temperature at the shock (which is the same up- and downstream) and we
assumed equilibrium radiation transport (T gas = T rad ). If the luminosity Ldownstr downstream of the
~7~
CHAPTER 1. INTRODUCTION
shock is positive (i.e., radiation is flowing in the positive r direction), Frad > ∆Frad . Combining
these results, one obtains
%vshock 3
σT shock 4 >
,
(1.18)
8
since ac = 4σ, with σ the Stefan–Boltzmann radiation constant. This thus provides an approximate
lower bound on the shock temperature. Commerçon et al. (2011) give a similar formula but based
only on heuristic arguments and do not include the factor of 4 (their equation 22 or 53), which
makes their lower bound higher. It is approximate because ηkin can be smaller than 1 and because
Ldownstr might not be negligible compared to the upstream luminosity.
For reference, the strong-shock, pure-hydrodynamical limit of the shock temperature is given by
(Drake, 2006)
γ−1
kB T = 2
(1.19)
µmH vshock 2 ,
(γ + 1)2
where vshock is the pre-shock velocity. For γ = 5/3, appropriate for a monoatomic gas, the prefactor
has the commonly quoted value of 2(γ − 1)/(γ + 1)2 = 3/16 (e.g., Stahler et al., 1980).
1.2. Radiative flux
The energy equation (Equation 1.3c) involves the zeroth moment Erad and the first moment Frad of
the radiation intensity (see Equation A.7) and thus requires a further relation to close the system.
One common choice for this is to assume flux-limited diffusion, in which the radiative flux is
written as a diffusion (Fick-like) equation for the radiation density with diffusion coefficient DF :
Frad = −DF ∇Erad , .
(1.20)
To prevent superluminal propagation speeds when the Erad gradient becomes steep enough and to
recover the free-streaming limit, a ‘flux limiter’ λ is introduced:
λc
,
κR %
λ = f (R),
k∇ ln Erad k
R≡
,
κR %
DF ≡
(1.21a)
(1.21b)
(1.21c)
where the radiation quantity R is discussed in more detail in Section 1.3. There is some freedom in
the choice of the flux limiter but it is required to behave asymptotically as

1


3,
λ(R) → 

1,
R
R 1 (diffusion limit)
R 1 (free-streaming limit)
(1.22)
to recover the diffusion and free-streaming limits. In this work, we take the flux limiter from
Levermore & Pomraning (1981) given by
λ=
2+λ
,
6 + 3R + R2
(1.23)
but note that a simpler expression, with the same required limits, is given by (e.g., Ensman, 1994)
λ=
1
.
3+R
(1.24)
They differ only up to some tens of per cent in the transition region between the optically thin and
thick limits (see Section B.1).
~8~
1.3. THE RADIATION QUANTITY R AND THE REDUCED FLUX
1.3. The radiation quantity R and the reduced flux
The local radiation quantity R(%, T, Erad ) defined in Equation (1.21c) compares the photon mean
free path λphot = 1/κ% to the ‘radiation energy density scale height’ HErad ; in spherical coordinates
it is given by
1 ∂Erad ∂ ln Erad
=
κ%Erad ∂r
∂τ
1/κ%
=
Erad /(dErad /dr)
λphot
=
.
HErad
R≡
(1.25a)
(1.25b)
(1.25c)
Large R values mean that the radiation energy density (which can be identified with temperature in
the 1-T approximation) changes over a shorter distance than photons get absorbed and re-emitted.
In spherical geometry, a radially constant luminosity in the free-streaming regime implies L ∝
2
r F ∝ r2 Erad = constant, so that
1 ∂ ln Erad 2
=
Rfree stream =
(1.26)
κ%
∂r
κ%r
1.
Therefore, κ%r, which is usually used as a crude estimate of the optical depth of a region as discussed around Equation (2.15), is in fact a diagnostic quantity which indicates whether the radiation
is locally propagating by free streaming (κ%r 1) or not. That the local quantity R depends on an
absolute coordinate r may appear at first surprising but only reflects the geometry.
The ‘reduced flux’ is defined as
fred ≡
Frad
= Rλ(R)
cErad
(1.27)
and is a (local) measure of the extent to which radiation is streaming freely ( fred → 1) or diffusing
( fred → 0), with the effective speed of propagation of the photons ceff = fred c. Obviously, fred 6 1.
Note that fred is different from the optical depth between two points, although there exists a relation
between both as discussed around Equation (2.16a).
Finally, note that in the case of diffusion, whether the luminosity is spatially constant or not, R
simply becomes three times the reduced flux:
Rdiffusion = 3
and the flux can be written as
Frad
= 3 fred 1,
cErad
4
Frad = RσT 4 .
3
This concludes our presentation of the necessary background material.
~9~
(1.28)
(1.29)
– I’m shocked, shocked to find that gambling is
going on in here!
– Your winnings, sir.
– (sotto voce) Oh, thank you very much.
— Casablanca (1942)
2
The gas giant accretion shock with
one-temperature radiation transport
2.1. Methods
For our one-temperature radiation hydrodynamics simulations, we use the static-grid version of
the versatile and modular (magneto-)hydrodynamics code PLUTO (version 3; Mignone et al., 2007,
2012) combined (by R. Kuiper) with the grey, one-temperature flux-limited diffusion (FLD) radiation transport module Makemake described in Kuiper et al. (2010), without ray tracing. The
only significant difference from Kuiper et al. (2010) is an update of the energy scheme, which is
described and tested in Appendix A.
2.1.1. Model
Each simulation is meant as a snapshot of the accretion process when the planet has grown to
Rp = rshock . To follow gas accretion onto a growing planet which is detached from the nebula
(see Chapter 1), we let our simulation box extend from the top-most layers of the planet to a large
fraction of its accretion radius Racc , defined through (Bodenheimer et al., 2000)
1
1
1
=
+
,
Racc RHill RBondi
where
(2.1)
!
Mp 1/3
GMp
RHill = a
, RBondi =
(2.2)
3M∗
c∞ 2
are the Hill and Bondi radii, respectively, with a the semi-major axis of the planet to the star of mass
M∗ and c∞ the sound speed in the disc at the planet’s location. Racc is thus the approximate region
where gas should be bound to the planet, both in terms of gravitational force compared to the star
(RHill ) and thermal energy compared to the planet’s potential (RBondi ). While global and local disc
simulations have shown that the accretion onto the protoplanet is highly three-dimensional (Ayliffe
& Bate, 2009; Tanigawa et al., 2012; D’Angelo & Bodenheimer, 2013; Ormel et al., 2015; Fung
et al., 2015) and possibly affected by magnetic fields in the gap and protoplanetary disc (e.g., Uribe
et al., 2013; Keith & Wardle, 2015), we take a first step here by using a spherically-symmetric
set-up and neglecting magnetic fields.
Note that, in the detached runaway phase, the continued accretion of solids (dust and planetesimals) by the planet is important for setting the final mass of the core (Mordasini, 2013). However,
this accretion rate of solid is several orders of magnitude smaller than that of gas and is therefore
neglected here.
~ 11 ~
CHAPTER 2. SHOCKS WITH 1-T RADIATION TRANSPORT
2.1.2. Initial set-up
Concretely, we use a semi-open box fixed at some height in the atmosphere of the planet, with a
closed left, inner edge (towards the centre of the planet) at r = rmin , and start with an atmosphere
of some arbitrary small height (e.g., 0.5 RJ ), onto which gas falls from the outer edge of the grid
at rmax . For the initial set-up, we calculate an atmosphere in hydrostatic equilibrium with a constant luminosity Lp = 10−3 L using the usual equations of stellar structure (but Equation 2.3d as
appropriate for an atmosphere)
dmr
dr
dT
dr
dP
dr
dL
dr
= 4πr2 %,
T dP
,
P dr
Gmr
= −% 2 ,
r
!
dmr
dS
=
ε−T
,
dr
dr
= ∇act
= 0,
(2.3a)
(2.3b)
(2.3c)
(2.3d)
where mr is the mass interior to r (dominated by the Mp ), S the entropy, and ε the energy generation
rate, and where the actual, adiabatic, and radiative gradients are given respectively by
∇act = min(∇ad , ∇rad )
γ−1
∇ad =
,
γ
3LPκ
∇rad =
.
64πσGmr T 4
(2.4a)
(2.4b)
(2.4c)
Equation (2.4a) is the Schwarzschild criterion and could have been replaced with ∇act = ∇rad since
the actual simulations do not include convection (see also the discussion in Section 2.7). We use
an adaptive step size ∆r for the integration to resolve accurately the pressure and temperature gradients. This atmosphere is then smoothly joined onto a calculated accretion flow for % and v. (See
Equations 2.7ff below.) The goal of these efforts is (i) to provide a numerically sufficiently smooth
initial profile while (ii) beginning with a certain atmospheric mass to speed up the computation.
‘Production data’ is measured only after an adjustment phase, once the actual accretion rate at the
shock is equal to the one set through the outer boundary conditions, described below.
We take a uniform grid segment from rmin to rmin + ∆r to resolve sufficiently well the pressure
gradient in the innermost part, using by default ∆r = 0.5 RJ and N = 500 cells there. The other grid
patch is a stretched segment out to rmax , with usually also N = 500, i.e., a much smaller resolution.
This has proven to be stable and accurate.
2.1.3. Boundary conditions
For the hydrodynamics, reflective boundary (zero-gradient) conditions are used in the density, pressure, and velocity, i.e.,
dP
=0
dr
d%
=0
dr
dv
= 0.
dr
(2.5a)
(2.5b)
(2.5c)
It proved numerically not feasible to enforce hydrostatic equilibrium at rmin , i.e., to set dP/dr
to −%g, where g = Gmr /r2 is the local gravitational acceleration. Since g acts in all grid cells,
~ 12 ~
2.2. MICROPHYSICS
Equation (2.5a) implies an acceleration at rmin . However, this is unproblematic as Equation (2.5c)
ensures that no mass flows over the boundary. In the radiation transport also, we prevent flow of
energy over rmin by using (since the flux limiter λ = 1/3 always)
dErad
= 0.
(2.6)
dr
The outer edge of the grid rmax is set well outside of the atmosphere and away from the shock.
For the hydrodynamics, we choose an accretion rate and approximate the velocity as the free-fall
velocity:
s
!
1
1
vff (rmax ) = 2GMp
−
,
(2.7)
rmax Racc
with Racc defined in equation (2.1). Mass conservation then determines the ‘free-fall density’:
%ff (rmax ) =
Ṁ
4πr2 |v(rmax )|
The pressure gradient here too is required to vanish:
dP = 0,
dr r
.
(2.8)
(2.9)
max
although for some simulations we considered a Dirichlet boundary condition with P = P(%ff (rmax ),
T neb ) for a nebula temperature T neb , taken as T neb = 150 K (e.g., Mizuno, 1980) unless stated
otherwise.
Finally, the radiation outer boundary condition is usually set to the flux-divergence-free condition
∂r2 Erad
= 0,
(2.10)
∂r
which corresponds to a constant luminosity if fred is sufficiently close to 1. However, even when
the flux at the outer edge is rather in the diffusion regime, we obtain similar results for a simple
Dirichlet boundary condition on the radiation temperature.
2.2. Microphysics
In this chapter, we use a constant equation of state, but we still present in this section the effective
heat capacity ratio γeff (usually abbreviated to γ) and the mean molecular weigth µ of a hydrodenhelium mixture as a function of density and temperature. This will serve to justify our choice for
values of γ and µ. We take a helium mass fraction Y = 0.25.
2.2.1. Equation of state
Figure 2.1 shows the effective heat capacity ratio γ, computed with the ideal-gas equation of state
(e.g., Black & Bodenheimer, 1975; D’Angelo & Bodenheimer, 2013) as implemented Vaidya et al.
(2015), as function of density and temperature. At hydrogen dissociation and ionisation, γ is seen to
sink, and one can also see the effect of helium ionisation at higher temperatures. Below roughly 40–
100 K, depending on the assumption for the ratio of the ortho- to parahydrogen (e.g. Vaytet et al.,
2014), rotational degrees of freedom of molecular hydrogen are frozen (not shown), with γ ≈ 1.4
for higher temperatures until 1000–2000 K (increasing with density). There, H2 dissociation lets γ
drop to 1.1 (for % = 10−8 g cm−3 , but γ = 1.07 for % = 10−13 ). Moving further up in temperature,
it reincreases to γ = 1.2 only1 before hydrogen ionisation again sinks it. At T = 3 × 104 K, the
adiabatic index has reincreased to γ ≈ 1.3. Thus, values of γ = 1.1 and γ = 1.44 bracket the
relevant range for planet formation given the densities and temperatures found later.
1
While these figures agree with Saumon et al. (1995) and Vaytet et al. (2013), they contrast with figure 1 of Tomida
et al. (2013) and figure 2 of Vaytet et al. (2014), who apparently find that γ = 5/3 is reached again between hydrogen
dissociation and hydrogen ionisation and helium ionisation. The source of this is not clear, and their slightly higher
helium mass fraction (Y = 0.28 or Y = 0.30 with unaccounted-for metals) cannot explain the discrepancy.
~ 13 ~
Temperature (K)
CHAPTER 2. SHOCKS WITH 1-T RADIATION TRANSPORT
H+
104
H
10
3
X H = 0 .5
X H = 0.1, 0.9
γeff (Y = 0.25)
H2
10−12
10−10
10−8
10−6
10−4
10−2
Density (g cm−3 )
Fig. 2.1.: Contours of constant effective heat capacity ratio γeff for an ideal hydrogen–helium gas with
Y = 0.25 (solid lines) and ionisation or dissociation fraction for hydrogen XH (dashed lines). Heat capacity
ratio contours are for γeff = 1.10–1.40 (blue to red) in steps of 0.05, starting at % = 10−13 g cm−3 near
3000 K or 104 K and increasing to higher densities. The shift in the γeff curves at low densities and high
temperatures are due to helium ionisation. The dominant hydrogen species (H2 , H i, H+ ) is indicated in each
region separated by the dashed lines.
2.2.2. Opacities
We consider both constant and tabulated opacities. The contribution of the dust to the opacity
dominates below approximately 1400–1600 K, at which temperature its refractory components
(olivine, silicates, pyroxene, etc.) evaporate (Pollack et al., 1994; Semenov et al., 2003). The
standard opacity tables we use are the smoothed Bell & Lin (1994, hereafter BL94) tables. We can
also make use of the Malygin et al. (2014) gas opacities combined with the dust opacities from
Semenov et al. (2003) and compare these in Figure 2.2.
BL94 model the destruction of dust (metal grains) with a very steep slope of κ(T ) ∝ T −24 between
two temperatures obtainable analytically from their table,
T dest, beg = 813 K
T dest, end
%
10−11 g cm−3
!1/24.5
%
= 1485 K
−11
10 g cm−3
,
(2.11a)
,
(2.11b)
!1/81
where the dust begins melting at T = T dest, beg and is completely removed at T dest, end . Using
directly the tabulated dust opacity at a given temperature effectively assumes that the dust melts
instantaneously but, in reality, this melting process might be time-dependent, so that the opacity
in this region is not certain. This destruction region is shown as a grey band in the %–T diagrams
in the following sections where appropriate. Note however that above it (i.e, at higher T ) and up
to roughly 4000 K, the BL94 opacity is unrealistically low compared to more recent opacities (cf.
with the Malygin et al., 2014 data in Figure 2.2; Helling et al., 2000; Ferguson et al., 2005), due to
missing H2 O lines or alkali line wings (M. Malygin, priv. comm.).
~ 14 ~
2.3. QUANTITIES OF INTEREST
Malygin et al. (2014)
Semenov et al. (2003), ‘nrm.h.s’
Bell & Lin (1994)
Rosseland mean opacity (cm2 g−1)
103
102
101
100
10−1
10−2
̺cgs = 10−13,−11,−9
10−3
10−4
10−5
10−6
10−7
10−8
102
103
104
Temperature (K)
Fig. 2.2.: Gas, dust, and total Rosseland mean opacities from Malygin et al. (2014), BL94, and Semenov
et al. (2003). Three densities are shown: % = 10−13,−11,−9 g cm−3 . For the Semenov et al. (2003) opacities, we
use their ‘nrm.h.s’ model, with dust grains made of ‘normal silicates’ ([Fe/(Fe+Mg)] = 0.4) as homegeneous
spheres. The Malygin et al. (2014) opacities are kept constant above the table limit of T = 2 × 104 K.
2.3. Quantities of interest
The goal of these simulations is to determine the appropriate boundary conditions for an accreting
protoplanet by measuring the temperature and pressure downstream of the shock, as well as the
amount of kinetic energy not converted to radiation at the shock. Through the equation of state,
knowledge of (T, P) can be recast as of (T, %) and vice versa, and also provides the entropy S (T, P).
To make a prediction about the post-formation entropy of a planet, however, requires coupling to
formation calculations, which can follow the thermal response of the entire planet over the complete
formation phase (Mordasini et al., 2012). Since we model only the upper layers of the atmosphere
and thus do not include the bulk of the mass of the planet, we focus on the luminosity, and defer
the coupling to subsequent work.
We will find that the shock Mach number is always M & 2, implying efficiencies close to 100 per
cent. Since the relevant quantity for the planet is 1 − ηkin , this would require a very high precision
(better than parts in a hundred) in the measurement of ηkin . This is however not possible due to the
presence of (small) temporal oscillations in the temperature profile and inaccuracies in determining
the shock location (spread over a few cells).
Therefore, we adopt a different approach. Since we find isothermal shocks, with T − = T 1 , we
will determine the precise value of the efficiency by measuring the shock temperature, calculating
the upstream Mach number, and using Equation (1.11). Explicitly,
1−η
kin
1
kB T +
= 2 4 =
γ M
vshock 2 µmH
!2
,
(2.12)
which provides the shock heating as Q+shock = (1 − ηkin )Lacc, max . The fractional uncertainty in
1 − ηkin is only twice that on T . Even though this is an exact equation directly following from
~ 15 ~
CHAPTER 2. SHOCKS WITH 1-T RADIATION TRANSPORT
Hot, in situ
Cold, in situ
Hot, migration
Cold, migration
10
10−2
Lluminosity (L ⊙ )
Accretion rate ( M⊕ an−1 )
Radius (R J )
100
10−1
10−2
10−3
10−4
10−5
0.1
1
10
10−4
10−5
10−6
1
10−3
10−6
0.1
Mass ( MJ )
1
Mass ( MJ )
10
0.1
1
10
Mass ( MJ )
Fig. 2.3.: Radius (left panel), accretion rate (middle), and internal luminosity (right) against mass (growing
with time) of a 10-MJ planet (final mass) forming in situ at 5.2 au with a constant accretion rate during
the detached phase assuming hot or cold accretion (violet and green curves, respectively), and for a 5 MJ
planet with a more realistic accretion rate history due to migration in the disc, assuming hot or cold accretion
(red and blue curves, respectively). (Cold and hot starts refer to the post-formation condition and are used
synonymously.) Dotted lines in the three panels show, respectively, the accretion radius from which the gas
falls (same for cold and hot starts), the solids accretion rate, and the maximal accretion luminosity due to
kinetic energy Lacc, max = GMp Ṁ(1/Rp − 1/Racc ). The post-formation luminosity of the migrating planets is
highlighted (filled circle).
conservation of mass, momentum, and energy, it will be explicitly verified, within errorbars, by
comparing the jump in L at the shock, ∆L, to the incoming Lacc, max .
The effect of using 1-T radiation transport will be assessed in Chapter 3 and found to be negligible in most cases.
2.4. Relevant parameter space
Before addressing the shock problem itself, we show the values of the ‘formation parameters’ (defined below) which occur in formation calculations during the detached phase, when the accretion
shock can play a significant role. While coupling of our shock results to these formation calculations will be needed, considering hot and cold starts (i.e., hot and cold accretion; the terms are used
interchangeably) should provide a bracketing of the expected range of results. Secondly, based on
these, we estimate the shock temperature and density to assess what values for the ‘microphysical
parameters’ µ, γ, and κ are likely to be relevant.
2.4.1. Formation parameters
The key shock parameters are Mp , Rp , Ṁ, and Lp , i.e., mass, radius, accretion rate, and internal
luminosity. Their range is shown in Figure 2.3 for the in situ formation at 5.2 au of a planet with
a final mass of 10 MJ (Mordasini et al., 2012) with a fixed accretion rate in the detached phase, as
in Pollack et al. (1996). (Because of this fixed accretion rate, tracks for planets with smaller final
masses simply cut off earlier.) For comparison, planets with a final mass of 5 MJ accreting with
ηkin = 0 or ηkin = 1 and with an accretion rate set by their migration in the disc are also displayed.
When the planet detaches from the nebula, its radius very quickly decreases from ≈ 100 RJ for
the in situ case or ≈ 20 RJ for the more realistic migrating one to several RJ over a small mass
range, from ≈ 0.2 to 1 MJ . The radii are smaller (around 1–2 RJ ) in the case with a lower accretion
rate and show a smaller difference between hot and cold accretion than for the fixed- Ṁ cases. The
luminosities also, especially the post-formation values, are very different. The fixed- Ṁ provide a
more conservative upper bound.
~ 16 ~
2.5. RESULTS: EFFICIENCIES AND DOWNSTREAM QUANTITIES
We now turn to full population synthesis calculations Mordasini et al. (2012), which display a
similar but larger diversity, as shown in Figures 2.4 and 2.5 for cold and hot accretion, respectively.
This will help us define the relevant region of parameter space. For Mp > 1.3 MJ , the radii range
from 1 to ≈ 4 RJ (6 RJ ) for cold (hot) starts, and, in general, the lower the accretion rate, the lower
the radii are at a given mass. In the cold-accretion case, for most points at Ṁ > 10−3 M⊕ an−1 ,
Lacc, max dominates by a factor of at least 3, with a significant number where Lacc, max > 10Lp , up
to a few orders of magnitude. In the case of hot accretion, however, Lacc, max is always comparable
(with ±0.5 dex) to Lp , and for many planets 3–10 times smaller for Ṁ 6 10−5 M⊕ an−1 .
Although there are clear correlations between Mp , Rp , Ṁ, and Lp , Figures 2.4 and 2.5 show that
they there is a certain scatter, i.e., a significant portion of the four-dimensional formation parameter
space, which spans several orders of magnitude in each dimension, needs to be covered. As a first
step, we will focus on those cases where Lp is negligible compared to Lacc, max . Since the luminosity
right of the shock is Lp + ηkin Lacc, max and we find ηkin ≈ 1, this means that we have to choose for
the simulations only the desired (Mp , Rp , Ṁ) and do not have to worry about obtaining a particular
Ldownstr , which would be the effective Lp . To evaluate the effects of boundary conditions and other
simulation details, however, we will make use of the full grids of simulations, whether the resulting
Ldownstr match or not the Lp found in the current population syntheses.
2.4.2. Microphysics parameters
Figure 2.6 shows the estimated shock temperature from Equation (1.18), using the free-fall velocity
(Equation 2.7), and the pre-shock density, given by Equation 2.8, for the migrating forming planet
of Figure 2.3 and for planets from the cold-start population synthesis, shown in Figure 2.4. We
focus on the formation phase when the shock could be important. Comparing to the contours of
constant γ and the rough %–T region were dust is destroyed and the opacity drops from ∼ 1 to
∼ 10−2 cm2 g−1 , one can expect for Ṁ . 10−5 M⊕ an−1 the hydrogen to remain molecular and
dust to be only partially destroyed, but for higher accretion rates (i.e., for most of the parameter
space of interest here) great importance of both dissociation and dust destruction. For hot-start
population synthesis, the highest temperature reached is, for increasing Ṁ, ≈ 700–2000 K instead
of 1000–5000 K, due to the radii being larger and thus the shock velocity lower.
From the temperature and densities covered, both hydrogen and helium ionisation should be
entirely negligible at the shock. While these estimates will have to be compared with a full, selfconsistent calculation, they do indicate that reasonable values include
γ ≈ 1.1–1.44,
µ = 1.23–2.353,
κ ∼ 10−4 –1 cm2 g.
In other words, the gas giant formation shock problem occupies an interesting region of the microphysical parameter space, including phase transitions and clearly non-constant opacities. We wish
to address all these effects simultaneously in future work but begin here with calculations performed
using a constant equation of state, nevertheless with either constant or tabulated opacities.
2.5. Results: Efficiencies and downstream quantities
The key output quantities which will serve to couple to formation calculations are the luminosity
going into the planet (and thus not radiated away at the shock) as well as the temperature and
pressure left of the shock. We begin (Section 2.5.1) by looking at shock structures and efficiencies
as a function of the optical depth to rmax by varying the value of the constant opacity. We also
confirm that Equation (1.11), which is a local relation, holds independently of the optical depth
to rmax . This will allow us to avoid the large numerical noise in 1 − ηkin , given that ηkin ≈ 1
almost always, by focussing on determining the temperature upstream of the shock to obtain the
~ 17 ~
CHAPTER 2. SHOCKS WITH 1-T RADIATION TRANSPORT
Internal luminosity (L ⊙ ) Internal luminosity (L ⊙ )
−2
10−2
−1
0
1
Ṁ ≈ 2 × 10−2 M⊕ an−1
Ṁ ≈ 10−2 M⊕ an−1
Ṁ ≈ 10−4 M⊕ an−1
Ṁ ≈ 10−5 M⊕ an−1
2
Ṁ ≈ 10−3 M⊕ an−1
10−3
10−4
10−5
10−6
10−2
0 .1
1
10
Planet mass ( MJ)
10−3
Cold-start
population synthesis
10−4
log Ṁ 12 vff2 /L p
10−5
Sample: 5 MJ , ηkin = 0 %
Sample: 5 MJ , ηkin = 100 %
10−6
0 .1
1
10
0 .1
Planet mass ( MJ)
−2
Radius (R J )
100
1
10
Planet mass ( MJ)
−1
0
1
Ṁ ≈ 2 × 10−2 M⊕ an−1
Ṁ ≈ 10−2 M⊕ an−1
Ṁ ≈ 10−4 M⊕ an−1
Ṁ ≈ 10−5 M⊕ an−1
2
Ṁ ≈ 10−3 M⊕ an−1
10
1
Radius (R J )
100
0 .1
1
10
Planet mass ( MJ)
Cold-start
population synthesis
10
log Ṁ 21 vff 2 /L int
Sample: 5 MJ , ηkin = 0 %
Sample: 5 MJ , ηkin = 100 %
1
0 .1
1
10
Planet mass ( MJ)
0 .1
1
10
Planet mass ( MJ)
Fig. 2.4.: Parameter space covered by a standard cold-accretion population synthesis: Ratio of the maximal
kinetic energy influx to the internal luminosity (colour) as a function of luminosity (top group of panels) or
radius (bottom group) and mass for different gas accretion rates (subpanels). The points selected are within
0.05 dex of the accretion rate indicated. The grey curves (identical in all panels) reproduce the hot- and
cold-start 5-MJ formation tracks shown in Figure 2.3.
~ 18 ~
2.5. RESULTS: EFFICIENCIES AND DOWNSTREAM QUANTITIES
−1
Internal luminosity (L ⊙ ) Internal luminosity (L ⊙ )
−2
10−2
0
1
Ṁ ≈ 2 × 10−2 M⊕ an−1
Ṁ ≈ 10−2 M⊕ an−1
Ṁ ≈ 10−4 M⊕ an−1
Ṁ ≈ 10−5 M⊕ an−1
2
Ṁ ≈ 10−3 M⊕ an−1
10−3
10−4
10−5
10−6
10−2
0 .1
1
10
Planet mass ( MJ)
10−3
Hot-start
population synthesis
10−4
log Ṁ 21 vff 2 /L int
10−5
Sample: 5 MJ , ηkin = 0 %
Sample: 5 MJ , ηkin = 100 %
10−6
0 .1
1
10
0 .1
Planet mass ( MJ)
−2
−1
100
Radius (R J )
1
10
Planet mass ( MJ)
0
1
Ṁ ≈ 2 × 10−2 M⊕ an−1
Ṁ ≈ 10−2 M⊕ an−1
Ṁ ≈ 10−4 M⊕ an−1
Ṁ ≈ 10−5 M⊕ an−1
2
Ṁ ≈ 10−3 M⊕ an−1
10
1
Radius (R J )
100
0 .1
1
10
Planet mass ( MJ)
Hot-start
population synthesis
10
log Ṁ 21 vff 2 /L int
Sample: 5 MJ , ηkin = 0 %
Sample: 5 MJ , ηkin = 100 %
1
0 .1
1
10
Planet mass ( MJ)
0 .1
1
10
Planet mass ( MJ)
Fig. 2.5.: As in Figure 2.4 but for hot starts, i.e., always assuming that ηkin = 0.
~ 19 ~
CHAPTER 2. SHOCKS WITH 1-T RADIATION TRANSPORT
104
C
103
Temperature (K)
Temperature (K)
104
D
B
102
10−13
A
10
−12
Dust destruction region
X H = 0 .5
γeff (Y = 0.25)
Estimated pre-shock
Examples
10−11
10−10
10−9
10−8
−3
103
102
Dust destruction region
X H = 0 .5
γeff (Y = 0.25)
Estimated pre-shock ( Ṁ = 10−5 –10−2)
10−13
10−12
10−11
10−10
10−9
10−8
−3
Density (g cm )
Density (g cm )
Fig. 2.6.: Left panel: A posteriori estimate of the shock temperature and density upstream for for the migrating case of Figure 2.3, made by assuming ηkin = 1, fred = 1 upstream, and L(rshock − ) L(rshock + ), which
4
leads to 4πRp 2 caT shock
≈ GMp Ṁ/Rp , with % given by the free-fall density. The symbol size is proportional
to the Mach number using the estimated temperature, with the size in the legend corresponding to M = 10.
Points are shown in steps of 0.1 dex in Mp /MJ , starting at Mp = 0.2 at the bottom, with 0.5, 1, 2, and 4 MJ
highlighted in red. No point is shown at 5 MJ . Shown are contours of γ = 1.1–1.65 in steps of 0.05 (from
≈ 100 to 1000 K is γ ≈ 1.4), as in Figure 2.1 (coloured solid lines), contours of XH = 0.1, 0.5, and 0.9
(black dashed lines), and the region of dust destruction in BL94 (grey band), with κ of order 1 cm2 g−1 at
lower T (cf. Figure 2.2). The fuchsia squares are examples from Section 2.5.2 and will be discussed there.
Right panel: Same but for the population synthesis planets of Figure 2.4, however showing all times and not
displaying the Mach number. The groups of points are, from left to right, for log Ṁ/(M⊕ an−1 ) = −5, −4,
−3, and −2.
Mach number. Results will then be presented for a few representative cases for a forming planet
(Section 2.5.2) and then for large grids in parameter space (Section 2.5.3).
2.5.1. Dependence of the structure and efficiency on optical depth
We first explore how the shock profiles depend on the optical depth between the shock and the
outer edge. For this purpose, unrealistically high opacity values are also considered to help span
a large range of optical depths (∆τ = 10−2 –100), but plausible values are chosen for the other
parameters. As an example, in Figure 2.7 are shown results for a low-mass (Mp = 0.316 MJ ), large
(rmin = 15.85 RJ ) planet with Q+atm = 0 and with an accretion rate of Ṁ = 10−2 M⊕ an−1 . The grid
extends out to 0.9 Racc with as outer boundary conditions dP/dr = 0 for the hydrodynamics and
dr2 Erad /dr = 0 for the radiation transport.
Overall, the radial structures are as expected and show a number of typical features, discussed
next.
Density and velocity The density reveals a nearly hydrostatic atmosphere abruptly cut off at the
shock, followed upstream by accelerating gas falling almost at free fall from Racc due to the
planet’s gravity and a lack of pressure support2 . The density jumps are in part much larger
than for a hydrodynamical shock. As it falls deeper in the potential well of the planet, the
gas slows down to sub-free-fall speeds due to the pressure gradient caused by the increasing
temperature (and density). Even in the case with the hottest accretion envelope, this causes
only a decrease of some ten per cent in the pre-shock |v| at the time shown. For the case
with the highest opacity, the precursor (discussed below) is contained within the simulation,
2
The radial re-increase of density at r > 43 Racc is due the decrease in velocity there (see Equation 2.7), with the velocity
at Racc zero and thus formally the density infinite; all grids end at rmax 6 0.9 Racc .
~ 20 ~
2.5. RESULTS: EFFICIENCIES AND DOWNSTREAM QUANTITIES
10−6
10−8
10−9
10−10
10−11
T ∝ r −1/2
102
−12
20
100
10
Opacity (g cm−2 )
1
0
−1
−2
−3
−4
−5
−6
−7
−8
50
Radius (R J )
50
10
1
50
−1.5
10−2
−2
20
50
Shock loss efficiency
Temperature (K)
100 %
20
60 %
40 %
20 %
100
Radius (R J )
1 .2
1
0 .8
0 .6
0 .4
0 .2
0
L = L 0 e−∆τ
−0.2
−5 −4 −3 −2 −1 0
10 10 10 10 10 10 101 102 103
120 %
100
1
0 .8
0 .6
0 .4
0 .2
0
1
10
100
20
ηphys
100 %
80 %
60 %
40 %
20 %
0%
1
10
Shock Mach number
50
100
Radius (R J )
Pressure (bar)
104
103
102
101
100
10−1
10−2
10−3
50
Radius (R J )
Time (106 s)
Shock loss efficiency
Optical depth from r max
100
80 %
Density (g cm−3 )
50
−2.5
ηphys
0%
0 .1
10−12 10−11 10−10 10−9 10−8 10−7
20
0
Radius (R J )
102
100
−1
10−1
100
103
50
Radius (R J )
−0.5
100
10−3
20
20
0 .5
102
Radius (R J )
Luminosity (L max )
100
Radius (R J )
3
Material
¡ energy
¢
flux 10−4 L ⊙
20
Velocity (km s−1 )
103
Reduced flux |F |/ cE
10
1 .4
1 .2
1
0 .8
0 .6
0 .4
0 .2
0
−0.2
¡
¢
Luminosity 10−4 L ⊙
Temperature (K)
Density (g cm−3 )
10−7
10−2
10−3
10−4
10−5
10−6
10−7
10−8
10−9
20
50
100
Radius (R J )
Mp = 0.32 MJ , Lnom
= 0, Ṁ = 10−2 M⊕ an−1
p
γ = 1.1; r max : d r 2 E /d r = 0, dP /d r = 0
κ = 0.01 cm2 g−1 , t = 25 × 106 s
κ = 0.1 cm2 g−1 , t = 56 × 106 s
κ = 1 cm2 g−1 , t = 15 × 106 s
κ = 10 cm2 g−1 , t = 10 × 106 s
κ = 30 cm2 g−1 , t = 6.4 × 106 s
κ = 102 cm2 g−1 , t = 10 × 106 s
Optical depth upstream of shock
Fig. 2.7.: Profiles for runs with different optical depths between the shock and rmax for a constant EOS.
See the legend for the parameter values and the axis labels for a description of the quantities shown. Only
a few comments are needed: The temperature panel also shows the strong-shock, non-radiating post-shock
value (Equation 1.19; open dots), the free-streaming lower bound (Equation 1.18; filled dots), as well as
T ∝ r−1/2 (grey dashed line segment); the velocity and energy fluxes are evaluated in the laboratory frame,
with the shock expansion velocity (usually positive) shown in the velocity panel (filled dots); the material
energy flux refers to Ṁ(ekin + eint + P/%) (i.e., a ‘material luminosity’); the velocity panel displays also the
free-fall velocity from Racc (the same for all simulations; grey dashed line); the solid dots in the temperature–
density phase diagram mark the up- and downstream conditions of the hydrodynamical shock; the shock
loss efficiency shown is the physical one (Equation 1.13) directly across the shock; the reduced-flux panel
shows the relativity ratio v/c as dotted lines; the shock loss efficiency as a function of the Mach number of
the gas immediately upstream of the shock is compared to grey lines showing the analytical optically-thin,
isothermal result (Equation 1.13) for γ = 1.44 (solid curve) and 1.1 (dashed grey); the luminosity, against
the optical depth measured upstreamward from the shock, is normalised to the maximum value in the profile,
which is compared to steady-state exponential absorption (dashed grey curve); and the pressure panel also
displays the ram pressure (dashed curves). Note that 1 bar = 106 erg cm−3 , and that the first two simulations
have the same Mach number and efficiency.
~ 21 ~
CHAPTER 2. SHOCKS WITH 1-T RADIATION TRANSPORT
and this feature can be seen as a slight additional slow-down. Due to our simulating only the
upper layers of the atmosphere, the shocked matter can settle down only over a limited radial
extent, so that the shock moves out over time as mass accumulates below3 . This is subsonic
(e.g., ∼ 0.3 km s−1 ) and thus usually, for higher Mach numbers, negligible compared to the
free-fall velocity. At small Mach numbers, unless otherwise stated, we take into account the
shock velocity and evaluate all relevant quantities in the frame in which the shock is at rest.
Note that during runaway gas accretion, the planet contracts at first at a rate dRp /dt ∼ 1 m s−1 ,
decreasing quickly to ∼ 1 cm s−1 . Thus the velocities are usually decoupled by several
orders of magnitude. All the same, to test whether this changes the shock structure, after an
adjustment phase mass was removed from the innermost cell of a simulation at the same rate
as the accretion rate, thus keeping the total mass constant. The shock remained stationary
and the structure did not change.
Temperature profile In the examples in Figure 2.7, the temperature always decreases radially.
Only for the lowest-opacity case (κ = 10−2 cm−2 g−1 ) is the reduced flux fred ≡ Frad /cErad =
1 (with fred ≈ 0.9 for κ = 0.1), i.e., the accretion envelope is in the free-streaming regime.
At higher opacities, the effective speed of light is less than c, nevertheless remaining several
orders of magnitude larger than the gas flow speed. Interestingly, since fred is rather constant
radially for all simulations, T ∝ r−1/2 at least approximately, even when fred 1. Finally,
note that if the temperature increased solely due to adiabatic compression, i.e., at constant
entropy (for which there would need to be no radiation transport), we would have T ∝ %γ−1 ∝
r−1.5(γ−1) , i.e., T ∝ r−0.15 or T ∝ r−0.66 for γ = 1.1 or 1.44, respectively. Thus, when
T ∝ r−1/2 , entropy decreases inward if γ > 4/3 ≈ 1.33.
Shock temperature The temperatures immediately up- and downstream of the hydrodynamics
shocks are equal to within roughly one per cent or much less, depending on the opacity. An
approximate lower bound to this temperature was obtained in Equation (1.18),
σT 4 >
%v3
.
8
The simulations whose pre-shock region is almost in the free-streaming regime indeed have
a temperature given by Equation (1.18), whereas in the other cases a higher temperature is
needed to carry a similar luminosity. One way of thinking about this is that the effective speed
of light is lower than c, so that Erad must increase in order to have the same Frad = ceff Erad .
(See the filled circles in Figure 2.7.)
Pressure The radial profile of the pressure reveals clearly the precursor as an increase in the pre-
shock region. Also, the post-shock pressure is given very accurately by the ram pressure of
the incoming gas,
Ppost = Pram = %v2 .
(2.13)
This differs slightly from the strong-shock, non-radiating case where Ppost = 2/(γ + 1)%v2
(see equation 4.18 of Drake, 2006), as we verified with a simulation using γ = 5/3 to increase
the potential difference.
Optical depth and regime According to the classical definition of the optical depthR as the numr
ber of photon mean free paths between two points r1 and r2 , ∆τ(r1 , r2 ) = r 2 κ% dr, the
1
simulations of Figure 2.7 have ∆τ(rshock , rmax ) ≈ 10−2 –102 at the times shown, while the
region below the shock and down to rmin has ∆τ(rmin , rshock ) ≈ 4–104 . The effective speed of
light Frad /Erad , which is the diffusion velocity, ranges from ≈ c for the least optically deep
3
For some (but not all) of the simulations with M & 15, the shock position actually moves inward while the layers
between rmin and the shock cool over time, the density increasing close to rmin but decreasing closer to the shock.
However, this velocity too is small compared to the shock velocity and does not affect our results.
~ 22 ~
2.5. RESULTS: EFFICIENCIES AND DOWNSTREAM QUANTITIES
cases to ≈ 10−3 c for the thickest, however always remaining much larger—at least by a factor 100—than the gas speed. This is termed ‘static diffusion’ by Mihalas & Mihalas (1984).
(For κ = 104 cm2 g−1 (not shown), the diffusion velocity becomes equal to the gas velocity,
yet there is no qualitative change in the profiles.)
The reduced flux directly upstream of the shock is compared to the optical depth to rmax in
Figure 2.8. The tight correlation can be understood thus: First, for an opacity which depends
on radius as κ ∝ r−α , the optical depth through an arbitrary density profile % ∝ r−β is given
by
Z ∞
∆τ∞ ≡ ∆τ(rshock , ∞) =
κ% dr
(2.14a)
rshock
=
1
κ%(rshock )rshock ,
α+β−1
(2.14b)
where for a free-fall profile β = 3/2 exactly when Racc rshock , and assuming that α+β−1 >
0; for a sufficiently constant opacity (α β − 1 = 0.5), this implies
∆τ∞ =
1
κ%ff rshock ≈ 2κ%ff rshock .
0.5 + α
(2.15)
We note in passing that this justifies (within a factor of a few) the estimate4 of Stahler et al.
(1980) of the optical depth upstream of the second-core accretion shock as κ%rshock . A similar
estimate is used by Mordasini et al. (2012) for their boundary conditions.
Secondly, as discussed above (Equation 1.26), the radiation quantity R is given as a function
of radius by Rfree stream = 2/%κr when fred = λ(R)R is not too much smaller than 1 (the
criterion will be discussed below), i.e., in the free-streaming regime, and with a radially
constant luminosity. Therefore, R(rshock ) = 4/∆τ for a free-fall, constant-L profile in the
free-streaming regime with a constant opacity, which yields
!
4
4
fred = λ
×
(2.16a)
∆τ
∆τ
!−1
3
≈ ∆τ + 1 ,
(2.16b)
4
where the last expression uses the simple flux limiter λ(R) = 1/(3 + R) taken in Ensman
(1994). Equation (2.16a) is compared to the data of Figure 2.7 in Figure 2.8 and matches
very well, especially taking the non-constancy of the luminosity profiles into account.
Luminosity The luminosity increases approximately linearly from rmin to the shock where it
jumps by a finite amount, then decreasing with radius. The value of L downstream of the
shock reflects in part the cooling of the layers below it, which will probably have a different
thermal history than if the layers were allowed to sink further down into the planet instead of
stopping at most at rmin . Since we are interested only in the jump at the shock and the radial
behaviour of L has no bearing upon the efficiency, we do not investigate it further.
Efficiency We find that for this small subset of parameter space, the kinetic efficiency measured
directly across the shock is ηkin ≈ 100 per cent, i.e., all the incoming kinetic energy is
converted into radiation. The nominal values range from 100.6 to 101.9 per cent but this is
due to small oscillations in the temperature profile, which lead to changes in the luminosity,
as well as to simulation- and resolution-dependent ringing downstream of the shock and the
details of a robust but simple analysis algorithm.
4
This had puzzled Vaytet et al. (2013), who took κ%rshock to be at best an estimate of the downstream optical depth.
The estimate is certainly rough for a non-constant opacity but at least it does estimate the optical depth in the correct
(upstream) direction.
~ 23 ~
CHAPTER 2. SHOCKS WITH 1-T RADIATION TRANSPORT
Reduced flux at r +shock
100
10−1
10
−2
10−3
10−2
10−2 cm2 g−1
10−1 cm2 g−1
100 cm2 g−1
101 cm2 g−1
LP81, const. κ
E94, const. κ
10−1
100
101
102
∆τ( r shock , r max )
Fig. 2.8.: Reduced flux fred directly upstream of the shock compared to the optical depth between the shock
and rmax for simulations from Figure 2.7. The highest-opacity cases were left out. Several times are included
for simulation, which is visible as ‘noise’ only in the κ = 10 cm2 g−1 case. In all cases, the gas velocity is
near v/c = 10−5 , which is much smaller than Frad /Erad , the radiation diffusion velocity. Equation (2.16a) is
shown for the Levermore & Pomraning (1981) (solid grey line) and Ensman (1994) (λ = 1/(3 + R); dashed
grey) flux limiters.
Given that the Mach numbers are all M & 3, the efficiency values thus match convincingly
Equation (1.11), as expected. See Figure 2.7. (The small deviation at the highest opacity
is due to a slight inaccurary in determining the shock front velocity, which affects both the
denominator of the efficiency and the Mach number.)
For our actual profiles with a finite Racc , we can compute ∆τ to rmax , either analytically or
numerically. The estimate ∆τ∞ ≈ 2/κ%ff rshock is 1.4–1.8 times larger than the integrated ∆τ from
rshock to rmax . This is, of course, in part a function of time as the shock moves out. Moreover, since
κ%r re-increases for r & 0.5, the integral to rmax does depend somewhat sensitively on the value of
rmax . Thus the estimate is approximate only.
In summary, we have shown that the profiles we obtain are sensible and have confirmed, up
to small, essentially numerical fluctuations, the validity of Equation (1.11), independently of the
optical depth regime upstream of the shock.
2.5.2. Results for selected cases with tabulated opacities
We now turn to parameter combinations likely to be directly relevant to formation calculations.
From Figures 2.3 and 2.4, we select four combinations of formation parameters given in Table 2.1
and provide estimates of the shock quantities in Table 2.2. The pre-shock density %pre is the freefall density at the chosen radius (Equation 2.8), and the shock temperature T shock is estimated from
Equation (1.18). By placing these values in the %–T diagram, one can easily estimate the mean
molecular weight, effective heat capacity ratio, and opacity. This is done in Figure 2.6a and the
results are displayed in Table 2.2.
The pre-shock densities are around 10−11 g cm−3 , and for Cases A and B, T shock ≈ 150–500 K.
At those densities, dissociation starts becoming important near 1300 K, so that even though T shock
is a lower limit, we should indeed find that the hydrogen remains molecular, assuming fred (rshock + )
~ 24 ~
2.5. RESULTS: EFFICIENCIES AND DOWNSTREAM QUANTITIES
Table 2.1.: Selected points in parameter space for forming planets as shown in Figures 2.4–2.5. The
values are chosen approximately. The column ‘Origin’ refers to (1c) in situ formation with cold accretion
(Figure 2.3); (1h) idem, hot accretion; (2) cold accretion with migration (Figure 2.3); (3c) cold accretion
population synthesis (Figure 2.4); and (3h) hot accretion population synthesis (Figure 2.5). In the notation
a(b), a is the mantissa and b is the base-10 exponent. Entropy values are provided for reference; S cent refers
to the central entropy of the planet and S surf to the value at the photosphere.
Name,
Origin
A: (1c/h)
B: (3c)
C: (2)
D: (3h)
Mp
(MJ )
Rp
(RJ )
Ṁ
(M⊕ an−1 )
S cent
(kB /baryon)
S surf
(kB /baryon)
Lp
(L )
Lacc, max
(L )
0.3
0.3
1.3
2.5
70/81
5–6
1.8
4
10−2
10−3
10−3
10−2
10.7
11.0
10.2
12.0
11.27/11.31
11–12.5
10.4
12.1
1(−4)
1–3(−4)
9(−5)
4(−3)
4/3(−5)
5–4(−5)
6(−4)
5(−3)
Table 2.2.: For the cases of Table 2.1, estimated (left of the vertical bar) and measured (right thereof )
shock quantities. The pre-shock density %pre is the free-fall density at the chosen radius. T + = T (rshock + ) and
M+ are the actual pre-shock temperature and Mach number in the shock frame. Note that the simulations
actually used all have µ = 2.353 and γ = 1.44 but that this does not make a significant difference (and does
not change the estimate of % and T shock ). The measured Mach number and Equation (1.18) serve to compute
ηkin . See text for details.
Name,
Origin
A: (1c/h)
B: (3c)
C: (2)
D: (3h)
%pre
(g cm−3 )
T shock
(K)
µ
3(−12)
8(−12)
2(−11)
4(−11)
150
480
1400
1600
2.353
2.353
2.353
1.5
M+
(cm2 g−1 )
T+
(K)
(1 − ηkin )Lacc, max
(L )
5
2
10−6
10−2
210
550
1630
1670
3.5
8.5
17
16
2(−7)
5(−9)
3(−9)
4(−8)
γ
κ
1.44
1.44
1.44
1.3
is not orders of magnitude smaller than 1. For the standard Y = 0.3 H–He mixture, this means that
µ = 2.353 and γ = 1.44. For Case C, the estimate implies also molecular hydrogen, although less
securely so, whereas Case D is clearly in the phase transition from molecular to atomic hydrogen.
We quote for Case D µ = 1.5 but this is highly uncertain due to the steep slope of µ(T ) there.
Likewise, γ ≈ 1.4 for Case C but γ = 1.3 for Case D, here again with high uncertainty; it could be
as low as γ ≈ 1.1 if at a slightly (∼ 0.1 dex) higher temperature.
For the opacities, Cases A and B are clearly in the dust-dominated region, with κ ≈ 5 and
2 cm2 g−1 , respectively (see Figure 2.2). The estimate of κ for Case C is very approximate since it is
in the dust destruction region, where κ decreases by several orders of magnitude over a small temperature range of a few 100 K Semenov et al. (2003). For Case D we quote an approximate value
for the region dominated by gas opacities. Note that these estimates of κ serve only as reference
and will not be used any further.
Figure 2.9 shows structures for the examples of Table 2.1 with the BL94 opacities. They are
qualitatively similar to all but the optically thickest of Figure 2.7; the optical depth from the shock
to rmax is ∆τ ≈ 0.4–2 but in no case does the luminosity upstream of the shock decrease markedly
towards rmax . The reduced flux is fred & 0.2 v/c for all simulations.
Consider first the opacity profiles. In the accretion envelope (i.e., upstream of the shock) of
Case B, temperatures remain below approximately 500 K, so that the dust grains contribute to
the opacity, with κ ∼ 3 cm2 g−1 out to rmax ≈ 100 RJ . Cases C and D are different: there, the
dust is destroyed ahead of the shock, letting the opacity drop some six orders of magnitude (see
shaded region in the %–T panel; the opacity remains low up to ∼ 3000 K). Thus there is a ‘dust
destruction front’ (DDF) as is the case for protostellar collapse (Stahler et al., 1980; see, e.g., also
~ 25 ~
10−4
10−5
10−6
10−7
10−8
10−9
10−10
10−11
10−12
10−13
10−14
50 100
10
−20
−30
−40
−50
10 20
10
−2
10−4
Shock loss efficiency
−1
10 20
10−6
10−7
50 100
1
Radius (R J )
10 20 50 100
Radius (R J )
1
ηkin
120 %
100 %
80 %
60 %
40 %
20 %
0 .8
0 .6
0 .4
0 .2
0
1
10
100
1
10 20 50 100
Radius (R J )
102
120 %
Pressure (bar)
ηkin
100 %
80 %
60 %
40 %
20 %
100
10−2
10−4
10−6
10−8
10−10
0%
1
50 100
Radius (R J )
Upstream reduced flux
10−5
0%
Shock loss efficiency
Optical depth from r max
100
1
10−4
Time (106 s)
101
50 100
10−8
10 20
Density (g cm−3 )
10
10 20
Radius (R J )
−2
10−3
−6
140 %
102
10−1310−1210−1110−1010−9 10−8 10−7
10
0
1
103
1
Radius (R J )
6
50 100
Radius (R J )
50 100
102
10
1
10 20
104
10
10−5
−Material energy
flux (L ⊙ )
Opacity (g cm−2 )
−10
10−4
10−6
1
0
10−3
Reduced flux |F |/ cE
10 20
Radius (R J )
10
Velocity (km s−1 )
T ∝ r −1/2
102
1
Temperature (K)
103
Luminosity (L ⊙ )
10−2
Temperature (K)
Density (g cm−3 )
CHAPTER 2. SHOCKS WITH 1-T RADIATION TRANSPORT
10
1
Shock Mach number
10 20
50 100
Radius (R J )
1 .2
Bell & Lin (1994), γ = 1.44, µ = 2.353
1
A:
B:
C:
D:
0 .8
0 .6
0 .4
0 .2
0
0 .1
Mp = 0.3, R p = 50 R J , Ṁ = 10−2
Mp = 0.3, R p = 5 R J , Ṁ = 10−3
Mp = 1.3, R p = 1.7 R J , Ṁ = 10−3
Mp = 2.5, R p = 4.2 R J , Ṁ = 10−2
λLP81 (4/∆τ) × 4/∆τ
1
10
Shock optical depth
Fig. 2.9.: As in Figure 2.7 but for the four cases of Table 2.1, using the BL94 opacities. The grey band in the
%–T plane shows the zone of dust destruction region in BL94. The outer boundary conditions are dP/dr = 0
and dr2 Erad /dr = 0.
~ 26 ~
2.5. RESULTS: EFFICIENCIES AND DOWNSTREAM QUANTITIES
figure 8 of Vaytet et al. (2013)). Since BL94 are missing opacity sources in that temperature region,
as discussed in Section 2.2.2, more realistic opacities (e.g., Ferguson et al., 2005; Malygin et al.,
2014) should lead to a less pronounced ‘opacity gap’ (Stahler et al., 1980) but the qualitative feature
should remain.
Because of the steeply changing opacity as a function of temperature in BL94, the flux limiter is
somewhat noisy, leading to jumps between adjacent grid cells in the reduced flux ( fred = λ × R(λ))
and the temperature profile. In effect, this assumes that the dust evaporates instantaneously compared to the flow timescale. One way of mitigating the jumps would be to introduce a computational time lag between the computation of the opacities and the temperature or to use explicitly
time-dependent dust destruction. Using other opacities with a smaller gradient might also prove to
be sufficient.
Despite the noise in the C case, as we verified with other simulations (shown in Section B.2),
the respective shock temperatures are similar whether using the constant or BL94 opacities. Since
all Ldownstr are negligible compared to the luminosity right of the shock, the small differences in
temperature are due only to the change in fred . The largest difference is in Case D, where T shock is
higher in the constant-opacity case by about 18 per cent, which is consistent with fred being lower
by 50 per cent (0.5−1/4 ≈ 1.19). The respective Mach numbers change only slightly and the shock
heating Q+shock (discussed below) remains around 10−5 Lacc, max (see Figure 1.2b).
The last panel of Figure 2.9 shows the reduced flux at the shock compared to the optical depth
to rmax . For a constant opacity, we were able to write down a unique analytical relation between
the two (Equation 2.16a) and found that the simulations matched very well. Ignoring Case A, we
see that this relation is in fact a lower bound to fred ; if upstream of the shock the opacity is much
lower, photons have a longer mean free path, i.e., their effective diffusion velocity is closer to c,
and fred = ceff /c is closer to one. This is a local effect; the optical depth, a path-integral quantity,
can still remain similar, as is the case in these examples.
For Case A, it appears that the shock is too close to rmax for Equation (2.15) to apply, as we
formally integrated out to infinity, taking also Racc Rp to avoid a diverging density at Racc , where
vff = 0 (this is a limitation of this simple one-dimensional picture).
Note that the presence of a DDF does not necessarily imply that the accretion envelope is optically thin; Cases C and D both have a DDF but ∆τ = 0.5 for Case C and ∆τ ≈ 1.5 for Case D.
To assess the importance of the residual kinetic energy on the structure of the planet one can
compare (1 − ηkin )Lacc, max to the respective internal luminosity. Looking at the respective columns
of Tables 2.1 and 2.2, it is clear that the fraction of the energy that is not lost radiation is entirely
negligible, being at most 0.2 per cent of the internal luminosity (Case A). As discussed in Section 2.3, the luminosity is not very sensitive to possible errors in the temperature determination
(choice of grid cell, etc.), so that, given the orders-of-magnitude difference, our conclusion is a
robust one for these four cases.
We defer a discussion of the post-shock pressure and temperature to the next section, in which
we look at a much larger parameter space.
2.5.3. Results for a large parameter space and constant opacity
In this section, we consider a large grid of parameters as appropriate for the phase of detached
runaway accretion where the accretion shock might have a significant effect on the planet’s growh.
We cover
0.1 MJ < Mp < 10 MJ
1 RJ . Rp . 20 MJ
10−5 M⊕ an−1 < Ṁ < 10−2 M⊕ an−1 ,
using γ = 1.1 and a constant opacity κ = 1 cm2 g−1 . The grid extends out to 0.7 Racc (to have
|v| monotonically increasing inward) and has the outer boundary conditions dP/dr = 0 for the
hydrodynamics and dr2 Erad /dr = 0 for the radiation transport.
~ 27 ~
CHAPTER 2. SHOCKS WITH 1-T RADIATION TRANSPORT
Figures 2.10–2.12 display key results of this study: the shock temperature T shock , the resulting
upstream Mach number M, the effective heating from the shock Q+shock = (1 − ηkin )Lacc , and the
post-shock pressure, as well as the upstream reduced flux and optical depth, and the luminosity
jump across the shock compared to the Mach number. Each point represents one simulation, with
starting radii rmin = 1, 1.6, 2.5, 3.9, 6.3, 10, and 16 RJ . Only a subset of the masses is shown.
For each simulation, we choose one snapshot when the actual accretion rate 4πrshock 2 %(v − vfront ) is
within one per cent of the desired value, after an early adjustment phase. (Usually the actual rate
deviates by much less than this.) The radius, which usually decreases over time during the detached
accretion stage, is the shock position at that time and we use rshock and Rp interchangeably.
As expected, the shock temperature is equal to or greater than the estimate (Equation 1.18), and
it agrees overall with it, although there are some deviations. Almost all temperatures for Ṁ =
10−2 M⊕ an−2 are slightly higher (≈ 0.1–0.2 dex) than predicted, especially at low masses, while
the match is very good for Ṁ = 10−3 M⊕ an−2 . At Ṁ = 10−4 M⊕ an−2 there is a slight deviation for
temperatures around 1000 K, with however a strong departure from the trend by the points with the
smallest radius for the two highest masses. Perhaps surprisingly, despite their high Mach numbers
(M ≈ 30–50), their Lacc, max is much smaller than the luminosity left of the shock Ldownstr , so that
the actual temperature greatly exceeds the lower bound (Equation 1.18). The situation is similar
for the Ṁ = 10−5 M⊕ an−2 cases5 . We also verified that the temperatures left and right are indeed
equal, to less than 4 per cent (not shown).
The post-shock pressure range roughly from P = 10−8 to 10−2 bar and matches quite well the ram
pressure Pram = %vshock 2 . The very small simulation-dependent but always positive offset is simply
due to resolution effects (the choice of the ‘post-shock cell’), while the more noticeable deviations
at very low accretion rates ( Ṁ = 10−5 M⊕ an−1 ) are associated with simulations in which the shock
position oscillates.
The top panel of Figure 2.11 displays the reduced flux fred = Frad /cErad against the optical depth
upstream of the shock. The higher the accretion rate, the lower the reduced flux, i.e., the less the
upstream position is in the free-streaming regime. Nevertheless, the lowest value is fred ≈ 0.1,
which is greater than the transition near fred ≈ 0.01 seen in Figure 2.8. We can also see that the
optical depth ahead of the shock is at most ∆τ(rshock , rmax ) ∼ 10, and closer to ∆τ ∼ 0.01–0.1
for the lower accretion rates. (Note that material between rmax and infinity, ignoring the accretion
radius, would essentially not contribute at all to the ∆τ integral.) At face value, this means that for
some simulations, the shock is located below the photosphere (∆τ ≈ 2/3). However, the opacity
(κ = 1 cm2 g−1 ) is typical of the dust-dominated regime; comparing to the temperature panel of
Figure 2.10 shows that those simulations with the highest optical depths also have the highest shock
temperatures, so that with realistic opacities the dust might be destroyed, lowering the optical depth
by a few to several orders of magnitude. This is discussed in the next section.
Figure 2.11 also presents the post-shock luminosity Ldownstr . The post-shock luminosity is, in
some sense, the effective internal luminosity of the planet, but it is has proven difficult to control
since it reflects the thermal history of the accreted gas below the shock; for a given shock radius, this
depends on the choice of rmin . The values we obtain differ in part by several orders of magnitude
from the actual Lp of population-synthesis planets (see Figures 2.4 and 2.5). For instance, at Ṁ =
10−3 M⊕ an−1 , Lp ≈ 10−4 –10−3 L for 0.6 . Mp . 13 MJ in the cold-accretion case, whereas
from our simulations Ldownstr ≈ 10−5 –10−4 L for those masses and the corresponding radius range
(1.3 . Rp 3 RJ ). However, for cold accretion, Figure 2.4 reveals that for a large portion of the
parameter space Lp is much smaller than the maximum accretion luminosity. Since we find that
ηkin ≈ 100 per cent as discussed below, this means that the interior luminosity (here, Ldownstr ) is
mostly irrelevant in determining the shock structure. Work is planned to extend the parameter space
to cover those cases with Lp ∼ Lacc, max .
We compare in Figure 2.12 ηkin as measured from ∆L and the gas properties directly upstream
5
Note that some points are missing because they were still in the adjustment phase and the time of analysis; the
numerical timestep being independent of the accretion rate (since vff does not depend on Ṁ), the wall-clock time to
adjustment is larger at lower Ṁ.
~ 28 ~
2.5. RESULTS: EFFICIENCIES AND DOWNSTREAM QUANTITIES
Shock temperature (K)
10
4
γ = 1.1, κ = 1 g cm−2 , dP /d r ( r max ) = 0, d r 2 E rad /d r ( r max ) = 0
Ṁ = 10−2 M⊕ an−1
103
10.0 MJ
5 .0 M J
1 .3 M J
0 .3 M J
min
Rough Tshock
102
104
Shock temperature (K)
Ṁ = 10−3 M⊕ an−1
Ṁ = 10−4 M⊕ an−1
Ṁ = 10−5 M⊕ an−1
103
102
10
1
Planet radius (R J )
Post-shock pressure (bar)
10−2
Post-shock pressure (bar)
10
−1
10−2
−2
γ = 1.1, κ = 1 g cm
10
1
Planet radius (R J )
, dP /d r ( r max ) = 0, d r 2 E rad /d r ( r max ) = 0
Ṁ = 10−2 M⊕ an−1
Ṁ = 10−3 M⊕ an−1
10−3
10−4
10−5
10.0 MJ
5 .0 M J
1 .3 M J
0 .3 M J
Pram = ̺v2
10−6
10−7
10−8
10−1
Ṁ = 10−4 M⊕ an−1
Ṁ = 10−5 M⊕ an−1
10−3
10−4
10−5
10−6
10−7
10−8
10
1
Planet radius (R J )
10
1
Planet radius (R J )
Fig. 2.10.: Shock results for a constant equation of state and opacity (γ = 1.1, µ = 2.353, κ = 1 cm2 g−1 ). Top
panel: Temperature at the shock (filled circles), compared to the estimated lower bound of Equation (1.18;
open circles). Bottom panel: Downstream pressure (filled circles), compared to the ram pressure (open
circles). Each panel group shows the quantities as a function of accretion rate (subpanels), planet mass, and
shock position (i.e., planet radius). The boundary conditions at rmax are indicated in the titles, with % and v
as always given by their free-fall values from Racc .
~ 29 ~
CHAPTER 2. SHOCKS WITH 1-T RADIATION TRANSPORT
γ = 1.1, κ = 1 g cm−2 , dP /d r ( r max ) = 0, d r 2 E rad /d r ( r max ) = 0
Pre-shock reduced flux
Ṁ = 10−3 M⊕ an−1
Ṁ = 10−2 M⊕ an−1
1
0 .8
0 .6
10.0 MJ
5 .0 M J
1 .3 M J
0 .3 M J
λLP81 (4/∆τ) × 4/∆τ
0 .4
0 .2
0
Ṁ = 10−4 M⊕ an−1
Pre-shock reduced flux
1
Ṁ = 10−5 M⊕ an−1
0 .8
0 .6
0 .4
0 .2
0
10−3 10−2 10−1 100
101 10−3 10−2 10−1 100
Optical depth r shock – r max
101
Optical depth r shock – r max
γ = 1.1, κ = 1 g cm−2 , dP /d r ( r max ) = 0, d r 2 E rad /d r ( r max ) = 0
Post-shock luminosity (L ⊙ )
10−2
10
−3
Ṁ = 10−2 M⊕ an−1
Ṁ = 10−3 M⊕ an−1
10−4
10−5
10−6
10.0
5 .0
1 .3
0 .3
10−7
Post-shock luminosity (L ⊙ )
10−8
10−2
10
−3
MJ
MJ
MJ
MJ
Ṁ = 10−4 M⊕ an−1
Ṁ = 10−5 M⊕ an−1
10−4
10−5
10−6
10−7
10−8
10
1
Planet radius (R J )
10
1
Planet radius (R J )
Fig. 2.11.: (Continued from Figure 2.10.) Top panel: Reduced flux upstream of the shock against the
optical depth to rmax , compared to Equation (2.16a) with the flux limiter used in this work (dashed line). The
∆τ = 2/3 location marking the photosphere is highlighted (vertical dashed line). Bottom panel: Post-shock
(i.e., downstream) luminosity.
~ 30 ~
2.6. ESTIMATE OF THE COUPLING
of the shock to Equation (1.11). Since most upstream regions are in the free-streaming regime,
small temperature fluctuations are amplified into larger flux variations. For this reason, we average
the efficiency over several measurements and display the standard deviation as an errorbar. Some
simulations were quite unstable and have large root-mean square values (around ten percentage
points), which is possibly due to resolution effects. Since this affects only a small number of cases,
it was not studied further. Most simulations, however, are quite stable and show fluctuations of
order three percentage points, with several much smaller than one percentage point, with a trend
to more stability at lower Ṁ. All simulations agree within their errorbars with the theoretical
prediction.
The Mach numbers range from M ≈ 5 to 50, with√the lowest values at the highest accretion rates
as expected from the temperature scaling and cs ∝ T . This means that, overall, 99.9 per cent or
more of the kinetic energy is radiated away, with only a small part kept as internal heat by the gas.
This heating by the shock, Q+shock = (1 − ηkin )Lacc , is shown in the bottom panel. It is an insignificant Q+shock ≈ 10−13 L , going up to only 10−6 L at the highest accretion rate Ṁ = 10−2 M⊕ an−1 ,
with approximately Q+shock ∝ Ṁ 3/2 as expected. (Note that Q+shock was again derived from the
measured Mach number and the expression for energy conservation in Equation (1.11) because the
deviation of ηkin from 100 per cent is extremely small.) Thus the gas only brings its internal energy
through the shock and into the planet.
2.5.4. Results for a large parameter space and tabulated opacities
Figure 2.13 displays the shock temperature for the same points from the grid as above. The shock
temperature increases with increasing mass, increasing accretion rate, and decreasing radius, and
is again close to its expected minimum value (Equation 1.18). There are more deviations from
the theory, especially at low accretion rates. These seem to be due for some simulations to the
extremely steep temperature derivative of the opacity in the dust destruction region in BL94, which
leads to orders-of-magnitude jumps in temperature over a few grid cells, or also to the relatively
large contrast in opacity from up- to downstream, even when the transition is resolved numerically.
Usually, at low accretion rates, the pre-shock luminosity is a large fraction of the downstream
luminosity since the shock contribution is small.
The upstream reduced flux (shown in Figure B.3 for completeness) is always greater than v/c,
so that we are in the same physical regime as in the previous sections, static diffusion (Mihalas &
Mihalas, 1984).
The post-shock pressure is equal to the ram pressure, augmented by a small amount which reflects the numerical thickness of the shock and the choice of the post-shock cell as discussed above,
leading to an offset of a few grid cells, which represents 0.001–0.1 RJ depending on rmin , Mp , and
the shock location. Those simulations which show a large deviation from the trend are those with
non-converged structures.
The lowest Mach numbers reached are for the largest planets with the smallest mass accreting at
the largest rate. Out to a radius of ≈ 20 RJ , this is M ≈ 4, so that again approximately the whole
Lacc, max is radiated away at the shock. The effective shock heating, as in the constant-opacity case,
is therefore negligible. It thus appears to be a robust conclusion that over the range of formation
parameters we considered and which covers a large part of the relevant parameter space, the shock
is efficient in converting kinetic energy to radiation.
2.6. Estimate of the coupling
Self-consistent and exact coupling of our results with formation calculations will require the use
of ηkin , Pdownstr , and T shock as provided by our simulations. Before doing so, however, one can
simply compare these values to the ones from formation tracks to assess how much of an influence
this could have. Since we find that, over the range of parameters considered (valid, roughly, for
~ 31 ~
CHAPTER 2. SHOCKS WITH 1-T RADIATION TRANSPORT
Shock loss efficiency (%)
105
Shock loss efficiency (%)
γ = 1.1, κ = 1 g cm−2 , dP /d r ( r max ) = 0, d r 2 E rad /d r ( r max ) = 0
105
Ṁ = 10−2 M⊕ an−1
Ṁ = 10−3 M⊕ an−1
100
95
10.0
5 .0
1 .3
0 .3
90
MJ
MJ
MJ
MJ
Ṁ = 10−4 M⊕ an−1
Ṁ = 10−5 M⊕ an−1
100
95
90
1
10
1
Mach number
10
Mach number
γ = 1.1, κ = 1 g cm−2 , dP /d r ( r max ) = 0, d r 2 E rad /d r ( r max ) = 0
Effective shock heating (L ⊙ ) Effective shock heating (L ⊙ )
10−6
10
−7
Ṁ = 10−2 M⊕ an−1
Ṁ = 10−3 M⊕ an−1
10−8
10−9
10−10
10.0
5 .0
1 .3
0 .3
10−11
10−12
10−6
10
−7
MJ
MJ
MJ
MJ
Ṁ = 10−4 M⊕ an−1
Ṁ = 10−5 M⊕ an−1
10−8
10−9
10−10
10−11
10−12
10
1
Planet radius (R J )
10
1
Planet radius (R J )
Fig. 2.12.: (Continued from Figure 2.11.) Top panel: Kinetic loss efficiency ηkin against Mach number.
Each point is a time average, with a standard deviation given by the errorbars. Notice the small range of the
vertical axis. Bottom panel: Effective heating from the shock Q+shock = (1 − ηkin )Lacc , i.e., what is not lost as
radiation, using the measured M and Equation (1.11).
~ 32 ~
2.6. ESTIMATE OF THE COUPLING
Shock temperature (K)
10
4
Ṁ = 10−2 M⊕ an−1
Ṁ = 10−3 M⊕ an−1
103
102
104
Shock temperature (K)
γ = 1.44, Bell & Lin (1994), dP /d r ( r max ) = 0, d r 2 E rad /d r ( r max ) = 0
10.0 MJ
5 .0 M J
1 .3 M J
0 .3 M J
min
Rough Tshock
Ṁ = 10−4 M⊕ an−1
Ṁ = 10−5 M⊕ an−1
103
102
10
1
Planet radius (R J )
Post-shock pressure (bar)
10−2
Post-shock pressure (bar)
10−1
10−2
10
1
Planet radius (R J )
γ = 1.44, Bell & Lin (1994), dP /d r ( r max ) = 0, d r 2 E rad /d r ( r max ) = 0
Ṁ = 10−2 M⊕ an−1
Ṁ = 10−3 M⊕ an−1
10−3
10−4
10−5
10.0 MJ
5 .0 M J
1 .3 M J
0 .3 M J
Pram = ̺v2
10−6
10−7
10−8
10−1
Ṁ = 10−4 M⊕ an−1
Ṁ = 10−5 M⊕ an−1
10−3
10−4
10−5
10−6
10−7
10−8
10
1
Planet radius (R J )
10
Planet radius (R J )
Fig. 2.13.: As in Figure 2.10 but for the BL94 opacities and with γ = 1.44.
~ 33 ~
1
10−3
50
10−4
45
40
10−5
10
35
L int
L acc, max
(1 − ηkin )L acc, max
M
Tshock /100 K
−6
10−7
10−8
30
25
20
15
10−9
10
−10
5
10−11
0
10
0.1
1
Mach number, Shock temperature
Luminosity (L ⊙ )
CHAPTER 2. SHOCKS WITH 1-T RADIATION TRANSPORT
10
Mass ( MJ)
Fig. 2.14.: Approximate comparison of the luminosity brought into the planet through the shock with the
interior one for a planet with a final mass of 5 MJ migrating. The luminosity does not depend on γ but does
on µ. The raw tracks (mass, radius, accretion rate, intrinsic luminosity as a function of time) were provided
by Ch. Mordasini. For the temperature and Mach number, γ = 1.1, and µ = 2.353.
Rp . 20 RJ ), the shock efficiency ηkin ≈ 100 per cent, we use the cold-accretion (cold-start)
formation tracks as the basis for our estimates.
First we look at a single planet taken from a population synthesis. Figure 2.14 compares the
shock heating Q+shock = (1 − ηkin )Lacc with the interior luminosity of the cold-accretion, 5-MJ planet
whose evolution is shown in Figure 2.3. The tracks are displayed starting from the detached phase,
when the local disc cannot provide gas anymore at the rate at which the planet is contracting and
the radius shrinks significantly below the Hill sphere radius, and ending once the complete mass
is assembled. The planet migrates in the disc and experiences a time-varying accretion rate but,
from 0.2 MJ onward, Ṁ is slowly decreasing from Ṁ = 10−3 to 10−4 M⊕ an−1 . Estimating the
shock temperature with Equation (1.18), we find that it is a few 100 K at small masses (early
times), increases with time to 1700 K when the planet has reached ≈ 3 MJ , and decreases slightly
afterward.
The estimated Mach number reaches almost 50 and does not drop below M ≈ 7, and taking
the effective temperature of the planet into account only lowers the mininum Mach number to
5. Therefore, over the whole accretion history, the interior luminosity is at least four orders of
magnitude larger than the part of the kinetic energy which was not lost as radiation. Thus, this is
very securely cold accretion and the assumption ηkin = 1 turns out to be very well justified.
Figure 2.15 shows the same luminosity comparison as in Figure 2.14 for the whole population
synthesis. (Explicit temperatures and Mach numbers are displayed in Figure B.7.) We find similar
results as for the single-planet example, namely that for no planet is 1−ηkin important, so that we do
not expect the shock heating Q+shock to play a direct role, at least for planets where Lp is negligible
compared to Lacc, max (see also the discussion of this point in the next section). As Figure B.7
shows, This conclusion is not sensitive to the value of µ used to estimate the shock temperature.
~ 34 ~
2.6. ESTIMATE OF THE COUPLING
10−1
10
−2
10
−3
10−1
Cold accretion
10
10−5
L int
L acc, max
(1 − ηkin )L acc, max
10−6
10−7
−3
10−4
Luminosity (L ⊙ )
10−4
Luminosity (L ⊙ )
Hot accretion
10−2
10−5
L int
L acc, max
(1 − ηkin )L acc, max
10−6
10−7
10
−8
10
−9
10−8
10−9
10
−10
10−10
10
−11
10−11
0.1
1
10
0.1
1
Mass ( MJ)
100
100
Cold accretion
Hot accretion
−1
10
10−2
Pressure (bar)
Pressure (bar)
10
10−3
10−4
10−5
10
−6
1
−1
10−2
10−3
10−4
10−5
P (R P)
Pram
0.1
10
P (R P)
Pram
−6
10
0.1
Mass ( MJ)
1
10
Mass ( MJ)
1.2
1.8
Cold accretion
1.6
1.1
1.4
1
Temperature ratio
Temperature ratio
10
Mass ( MJ)
1.2
1
0.8
0.6
0.4
Hot accretion
0.9
0.8
0.7
0.6
0.5
0.4
min
Tshock
/T (R P)
0.2
0.1
1
min
Tshock
/T (R P)
0.3
10
Mass ( MJ)
0.1
1
10
Mass ( MJ)
Fig. 2.15.: Approximate comparison of the shock effects on cold-accretion (left column) and hot-accretion
(right column) population synthesis planets (data as in Mordasini et al., 2012; provided by Ch. Mordasini).
Three accretion rates ( Ṁ ≈ 10−4 to ≈ 10−2 M⊕ an−1 ; clusters of points from bottom to top) are selected as in
Figures 2.4 and 2.5. Here also, for the temperature and Mach number estimates, γ = 1.1, and µ = 2.353. Top
row: Luminosity brought into the planet through the shock compared to the interior one. For cold accretion,
there is a larger range of interior luminosities at a given Ṁ, whereas Lacc, max dominates for the hot accretion.
Middle row: Ram pressure and actual pressure in cold-start (left) and hot-start (right) population synthesis
for Ṁ ≈ 10−4 to 10−2 M⊕ an−1 (groups of points from bottom to top). Bottom row: Ratio of the estimated
pre-shock temperature (Equation 1.18) to the actual temperature in the same formation calculations. For the
mininum shock-temperature estimate, γ = 1.1.
~ 35 ~
CHAPTER 2. SHOCKS WITH 1-T RADIATION TRANSPORT
We next turn to the post-shock pressure and temperature. The pressure is estimated with the
ram pressure Pram = %v2 , where v is the pre-shock velocity, and the shock temperature as usual by
Equation (1.18), which actually provides only a lower bound. We find that Pram is always much
smaller than the photospheric pressure, as mentioned in Mordasini et al. (2012). If the temperature
structure and thus opacity at the photosphere were to remain the same, this would mean that the
shock happens well above the photosphere.
The bottom row of Figure 2.15 compares the obtained shock temperatures to the effective temperatures of the population synthesis planets. For most cold-start planets, the shock temperature is
larger than T eff , but only up to a factor of two, with also a significant group for which T shock < T eff ,
down to a factor near 0.2. In the hot-start case, as expected, the shock temperature is very similar
to the effective temperatures, with T shock ≈ 0.3–0.8T eff . This will have to be studied further but our
results seem to indicate that the shock is heating the atmosphere not through an inward radiative
flux (Q+shock ) but by making the pre-shock gas hot: the shock being isothermal, this internal energy
is brought into the planet, effectively heating it.
2.7. Discussion
As shown in Section 2.4.1, the parameter space of planet formation is large and practical considerations (runtime of the simulations) have forced us to restrict ourselves to a subset. We offer a few
comments on these restrictions:
Neglect of the interior luminosity We have explicitly focussed on cases where the luminosity
coming from the deep interior of the planet Lp is negligible compared to the luminosity coming from the shock, Lacc = ηkin Lacc, max ≈ Lacc, max = GMp Ṁ/Rp . As shown in Figure 2.4,
this approximation is appropriate for the highest accretion rates and towards higher masses,
and less so for the early phases of detached runaway accretion.
In cases where Ldownstr (an effective Lp ) was not negligible, the shock temperature was higher
than given by the simple estimate (Equation 1.18). Thus taking Lp into account should raise
the temperature, thereby lowering the Mach number and leading to a lower ηkin value. We
can try to estimate how important this effect can be: The smallest Mach number we found for
realistic opacities was around M ≈ 5 (see Figure 2.12 and also Figure B.7) with temperatures
around 500 K. To have a significant shock heating with, e.g., 1 − ηkin = 99 per cent, the Mach
number would need to be around 2.7–3.0 (for γ = 1.44–1.1), i.e., a factor of two smaller.
Keeping vff constant, this would require a temperature at the shock approximately four times
higher, or around 2000 K, which seems plausible given the hot- and cold-start T eff in the
currently available (non-coupled) population syntheses.
A priori, the reduction in the Mach number could be weaker because of two effects: (i) The
higher temperature could lead to a more pronounced upstream pressure gradient and therefore
to a slow-down of the gas, thereby diminishing M. (ii) Secondly, the shock expansion speed
vfront is related to the compression ratio, which itself is a function of the Mach number,
and thus could become more important at lower Mach numbers. However, both effects are
probably small.
Note that should the efficiencies at early times turn out to be much smaller than 100 per cent,
one could imagine a self-amplifying mechanism very similar to that discussed by Mordasini
(2013): a low efficiency would lead to hotter material being added to the planet, which
would make it larger, thereby reducing the free-fall speed of the material accreted next and
thus lowering further the efficiency. Whether this amplication, if it takes place, stops with
transition into the ‘Lacc, max -dominated regime’ will also have to be investigated.
Choice of rmin It was realised that, at least in principle, the choice of rmin for a given Rp = rshock
can have a certain influence on the temperature at the shock. This occurs through the luminosity structure, determined in part by the amount of mass that has been accreted below
~ 36 ~
2.7. DISCUSSION
rshock , which is greater for lower rmin . We look at the size of this effect in Section B.4 for
one example and find that it is modest; however, it will be necessary to explore this more
systematically to assess the influence of rmin on the shock temperature and thus efficiency.
We now discuss some of the simplifications made both in our approach but also in the underlying
model.
Choice for the radiation transport We have made three main simplifications concerning the ra-
diation transport (RT):
(i) We have used flux-limited diffusion (FLD), for which a flux limiter λ is neeed to solve
approximately the RT equation. The flux limiter is essentially a mathematical trick designed
to recover the free-streaming and diffusion limits, with no guarantee that the transition behaviour is correctly modeled. How important this is in our work would need to be assessed
for instance by using a different functional form for λ.
Because of its diffusion ansatz, FLD leads in 3D to unphysical behaviour, in which the radiation field diffuses around an opaque obstacle instead of casting a hard shadow (e.g., Klassen
et al., 2014). However, in 1D, the work of Ensman (1994) and Commerçon et al. (2011)
show that it yields results comparable to more sophisticated RT schemes.
(ii) Secondly, we have used 1-T RT, which assumes that the radiation field and the gas are
always in equilibrium. We study this assumption by considering the 2-T case in the next
chapter.
(iii) Finally, we computed frequency-averaged, i.e., grey RT. While Vaytet et al. (2013) show
that the detailed shock structure, including the extent of the precursor, depends on the number
of frequency groups (i.e., bins), these differences are on lengthscales which are not relevant
in the astrophysical context. Indeed, studying the collapse of a cloud core with M1 radiation transfer, a scheme more accurate than FLD, Vaytet et al. (2013) found no significant
difference between grey and frequency-dependent RT.
Neglect of convection The assumption of spherical symmetry prevents the development of con-
vective instabilities, which in a 3D situation could transport a significant portion of the energy flux in the downstream region. As a result, the actual temperature gradient we obtain
can be larger than the adiabatic one, which leads in some cases to density inversions (see
Section B.6). Clearly, this is not realistic and, while it does not influence directly the energy
conversion at the shock, this density structure could lead to a different shock temperature, in
particular when using temperature- and density-dependent opacities.
However, short of performing computationally expensive 3D simulations, it should be possible to simulate the effects of convection by adding a flux term ∇Fconv to the right-hand side of
energy equation (Equation 1.3c), where Fconv ∝ (∇act − ∇ad )3/2 when ∇act > ∇ad and Fconv = 0
otherwise (e.g., Hansen & Kawaler, 1994). For numerical stability, it would probably be best
to write this flux implicitly by increasing the diffusion coefficient
Frad = − (DF + Dconv ) ∇Erad n+1 ,
Fconv
Dconv n ≡
.
∇Erad n
(2.17a)
(2.17b)
That this is approximate should not matter as convection is very efficienct at cancelling superadiabatic temperature gradients (e.g., Mihalas & Mihalas, 1984).
Equation of state In star formation, the endothermic transition from molecular to atomic hydro-
gen in a collapsing core is a key process which leads to the formation of the ‘second core’
(Larson, 1969; Commerçon et al., 2011; Tomida et al., 2013). Here, we assumed a constant
equation of state (EOS), and preliminary studies such as displayed in Figure B.6 seem to
show that the dependence of the efficiency on γ and µ is weak. Nevertheless, considering
~ 37 ~
CHAPTER 2. SHOCKS WITH 1-T RADIATION TRANSPORT
a non-constant EOS (i.e., µ = µ(%, T ), γ = γ(%, T )) could lead to interesting effects at the
shock. We have begun investigating this and will present the results in a dedicated study.
Geometry Finally, the most important simplification is that of a spherically symmetric accretion
geometry as in the models of Stevenson (1982); Pollack et al. (1996); Bodenheimer et al.
(2000). As for low-mass star formation (Hartmann et al., 1997; Zhou et al., 2014), accretion
onto a forming gas giant can be expected to be highly non-spherically symmetric but rather
along magnetic field lines (‘magnetospheric accretion’), due to interactions with the local
disc which are hydrodynamical and magnetic in nature (Lovelace et al., 2011; Ormel et al.,
2015; Uribe et al., 2013; Keith & Wardle, 2015). Thus the accretion shock is probably
relevant only for a small fraction of the planetary surface, where the accretion column hits
the stellar surface.
In the low-mass star case, this fraction f ranges from 0.1 to 10 per cent (Calvet & Gullbring,
1998), and in our 1D model, we would need to increase the accretion rate by a factor 1/ f =
1000–10 to obtain the corresponding pre-shock densities. This would significantly raise the
optical depth ∆τ upstream of the shock, which could possibly lead to a non-isothermal shock
and, in general, to a different (i.e., lower) shock efficiency.
Moreover, since only a small fraction of the planetary surface is affected by the shock, the
rest is free to radiate into space with usual photospheric boundary conditions. Hartmann
et al. (1997) found that this consideration made a significant difference on the observational
appearance of protostars at the end of formation compared to spherically-symmetric accretion. On the planetary side, Eisner (2015) and Zhu (2015) discuss possible signatures of this
accretion (and of the circumplanetary discuss) during formation. Given recent first (possible) observations of accretion onto planets (Zhou et al., 2014; Sallum et al., 2015), it would
be particularly interesting to look at the influence of the different efficiency on the accretion
luminosity, especially in the H α band.
2.8. Summary
We have studied spherically symmetric gas accretion onto a gas giant planet during the detached
runaway phase, when the gas falls in free-fall from the accretion radius (of order of the Hill or Bondi
radius). We set out primarily to determine the radiative efficiency of the shock defining the planet’s
surface, i.e., what fraction of the incoming kinetic energy flux, the maximal accretion luminosity
Lacc, max ≈ GMp Ṁ/Rp , is lost at the shock, being however interested in the complementary fraction
as this gets added to the planet. Further quantities of interest are the post-shock temperature and
pressure of the gas as these set the boundary conditions for the calculation of the interior structure.
To the best of our knowledge the energetics of the shock have not yet been studied in detail as we
have, yet are thought to be key in determining the post-formation thermal state of gas giants, with
several orders of magnitude of difference in luminosity betwee the two extreme cases.
We have focussed on planets whose accretion luminosity dominates over their internal luminosities and used a constant equation of state. Both constant and tabulated opacities (Bell & Lin, 1994)
were considered. Summarising, we find the following:
• Our radiation hydrodynamics simulations confirm, over a large range of Mach numbers,
the theoretical expression for the efficiency given by Commerçon et al. (2011, our Equation 1.11).
• Unrealistically high opacity values (κ ∼ 102 cm2 g−1 ) were needed to cause the luminosity
generated at the shock to be completely absorbed in the precursor, ahead of the shock region.
For realistic constant or tabulated opacities, all luminosity profiles are qualitatively similar,
decreasing by some amount with increasing radius or staying rather constant.
~ 38 ~
2.8. SUMMARY
• The effective speed of light of the escaping photons, even for the smallest fred values, is
always much larger than the gas flow speed, so that the upstream region is in the ‘static
diffusion’ regime (Mihalas & Mihalas, 1984).
• We found a relation between the reduced flux directly upstream of the shock and the optical
depth out to rmax (Equation 2.16a). For constant opacities, this is an exact correspondence,
as can be shown analytically, whereas in the case of tabulated opacities it provides a lower
limit on fred given ∆τ or vice versa.
• The shock was observed to always be isothermal, which corresponds in the classical terminology to a supercritical shock (Stahler et al., 1980; Mihalas & Mihalas, 1984).
• This shock temperature at the shock was found to be well approximated by Equation (1.18)
since fred is usually not much smaller than 1. This estimate is less accurate for planets whose
internal luminosity Lp is a significant fraction of their accretion luminosity.
• Most importantly, we find that for most of the formation parameter space, the Mach number
at the shock is sufficiently high that very nearly all of the kinetic energy is radiated away at
the shock, i.e., ηkin ≈ 100 per cent. This shock luminosity, which can be quite large, should
be observable, and may even have already been detected for a few objects (Quanz et al.,
2015; Sallum et al., 2015).
• The associated shock heating of the planet Q+shock = (1 − ηkin Lacc, max ) was compared to
the range of internal luminosities seen in formation calculations. The shock contribution
was found to be several orders of magnitude smaller than Lp , at least for planets up to a
moderately large size. Thus the planetary accretion shock appears to lead to cold accretion.
• We also compared the shock temperatures and pressures to the effective temperature and
photospheric pressure of modeled planet formation and found that T shock was between a
factor 0.2 and 2 of T eff , while the post-shock pressure Ppost , quite accurately given by the
ram pressure Pram = %vff 2 , was orders of magnitude smaller than the photospheric pressure.
The next step will be to use these results self-consistently in planet formation calculations by
using tabulated values of ηkin , T shock , and Ppost at every step. Interestingly, Mordasini (2013) show
the results of formation calculations in which, as a test, the dominant photospheric pressure was
dropped, leaving only Pram . The resulting planets were a factor of a few brighter than planets
calculated with the usual boundary conditions. We expect that this corresponds approximately to
our situation, although the difference in temperature at the planet surface—i.e., at the shock—may
have a non-negligible effect.
In parallel, we will address the effects of a non-constant equation of state, which may be important if the shock is isothermal, since a jump in density at constant temperature will lead in general to
a change in internal energy. We will also extend our analysis to cases where the internal luminosity
is not negligible compared to Lacc, max . This should increase the temperature at the shock and thus,
at a constant Mp /Rp ∝ vff , lead to lower Mach numbers and thus possibly higher shock heating.
Whether this has an influence on the early evolution of gas giants will also be addressed in future
work.
~ 39 ~
‘Epsilon’ in Greek means small;
that’s why we call our children ‘my little epsilons’.
— A McGill physics professor
3
Shock calculations with two-temperature
radiation transport
3.1. Introduction
The description of radiative shocks as presented in Section 1.1 showed that, in the precursor as
well as immediately behind the shock, the radiation temperature T rad = (Erad /a)0.25 can be different
from the gas temperature T gas . To capture this, it is necessary to have non-equilibrium radiation
transport by evolving the gas and the radiation temperatures separately. For flux-limited diffusion
(FLD), the energy equation (Equation 1.3c) becomes (Mihalas & Mihalas, 1984; Turner & Stone,
2001)
∂(Ekin + Eint )
+ ∇ ([Eint + Ekin + P] v) = −cκP 4πB + cκE Erad − %vg,
∂t
∂Erad
+ ∇Frad = +cκP 4πB − cκE Erad ,
∂t
(3.1a)
(3.1b)
where the radiation pressure was omitted since it is negligible here and where κP and κE are, respectively, the Planck and energy mean opacities. They will be assumed to be equal. (See also the
derivation in Appendix A.) B is the Planck function and thus assumes the radiation to be thermal.
Equations (3.1a) and (3.1b) state that the material (gas or dust) is losing energy at the rate κP %4πB
but absorbing photons at the rate cκE %Erad , and conversely for the radiation.
In this chapter, we perform shock calculations in the planetary formation context discussed in
Chapter 2 but using 2-T radiation transport as this is more accurate. We wish to assess the consequences this has on the pre-shock temperature, which sets the efficiency, and use for this the
module, Belt, which was written and coupled to PLUTO v.4.1 by R. Kuiper and is as presented
in Klassen et al. (2014). Since Belt has a number differences from Makemake beyond the energy
equation (e.g., in the choice of the stencil for the flux), these can be considered to be two semiindependent codes. Note also that is uses a slightly updated version of PLUTO (4.1 instead of 3.0),
although the changes to the hydrodynamics module are small.
Additionally, we look at a slightly different approach to the accretion shock calculation. Instead
of focussing only on the top layers of the atmosphere, we simulate the planetary structure down to
the rocky core and accrete as much of the envelope as possible. We can then compare the resulting
downstream luminosities to the ones from formation calculations.
~ 41 ~
CHAPTER 3. SHOCKS WITH 2-T RADIATION TRANSPORT
3.2. Selected cases with tabulated opacities
We consider the same examples as in Section 2.5.2, as listed in Table 2.1 and shown in Figure 2.9.
The resulting structures are shown in Figures 3.1–3.3 for both the 1-T code of Chapter 2, Makemake,
and for Belt with 1-T and 2-T radiation transport. The simulations are set up exactly in the same
way, with three exceptions: (i) we start only with a free-fall density profile reaching down to
rmin instead of adding a hydrostatic atmosphere; (ii) it was necessary to halve the CFL number to
CFL = 0.4 for the 1-T C and D cases to avoid code crashes in the early adjustment phase; and
(iii) because Bell & Lin (1994) does not provide Planck mean opacities, it was necessary to turn to
an alternative data set. The dust contribution to the opacity is provided by Weingartner & Draine
(2001) and is modeled to vanish sharply over 200 K around an evaporation temperature given by
Pollack et al. (1994). Since the opacities of Malygin et al. (2014) were not available for Belt yet,
we took the gas opacity of Helling et al. (2000). As pointed out by Malygin et al. (2014), note
however that the Helling et al. (2000) Planck opacities are underestimated by up to four orders of
magnitude between 3300 K and 10000 K. We take again γ = 1.44 and µ = 2.353 for all cases.
In all cases, the qualitative features of the results are similar:
• The gas and radiation temperatures are everywhere equal except for a sharp peak in T gas at the
shock location, whereas T rad remains smooth. This peak can be very dramatic and represent
up to a factor-ten increase in temperature. However, it does not affect the rest of the profile.
We note in passing that, as in Vaytet et al. (2013), this is clearly not the real Zel’dovich spike
as its physical size should be orders of magnitude smaller, on order of the gas mean free path.
• The velocity and pressure profiles are almost identical between all three models.
• The opacities (and thus resulting optical depths) are within a factor of a few of each other
for the different table. The only exception is for Case C, where the solution goes through the
dust destruction region, and in particular, the minimum of the Bell & Lin (1994) opacities is
reached. As discussed above, however, these extremely low values are not realistic.
• The Mach numbers and thus efficiencies are very similar between the 1-T and 2-T runs. Note
however that the Belt 1-T efficiency was left out for Case B because the luminosity profile
was actually not converged at the time the desired radius was reached.
Thus both Makemake and the newer code Belt yield very similar results, despite further differences
(in opacity and initial set-up). Most importantly, the 1-T and 2-T shock upstream temperatures (and
thus efficiencies) are found to be essentially the same.
3.2.1. Gas–radiation equilibrium
To address the accuracy of the one-temperature approximation is extreme cases, we look at Mach
numbers M ≈ 3–4 (correcting to the shock frame, which can be important at lower M) with a high
constant opacity in Figure 3.4 and with a lower opacity in Figure 3.5. All other parameters (see
legend) are the same and standard. We will compare the flow timescale from rmax = 0.9 Racc ≈
300 RJ down to the shock, which is roughly
τflow = ∆r/v ∼ 200 RJ /2 km s−1 ≈ 107 s,
(3.2)
to the radiation–gas equilibration time. If the latter is much shorter than the flow time, 1-T and 2-T
computations should give the same results, while in the other case the gas and radiation will not be
able to come into equilibrium before the gas reaches the downstream region.
To derive the time needed by the gas to equilibrate with the radiation, one assumes that the gas
is initially at T T rad0 , where T rad0 is the initial and constant radiation temperature, and that the
density does not change temporally, which, in our case, is equivalent to assuming that τst. st. τflow
~ 42 ~
3.2. SELECTED CASES WITH TABULATED OPACITIES
10−10
10−12
40
60
70
80
3
2
1
0
40
50
70
80
Opacity (g cm−2 )
−2
−3
−4
40
10
8
6
4
2
0
−2
−4
−6
100
40
50
60
70
40
80
50
Radius (R J )
Temperature (K)
10−1
10−2
Radius (R J )
80
60
70
80
40
50
60
70
80
Radius (R J )
1
ηkin
80 %
60 %
40 %
20 %
0 .8
0 .6
0 .4
0 .2
0
1
10
100
40
50
120 %
60
70
80
Radius (R J )
10−4
ηkin
Pressure (bar)
100
70
80
0%
Shock loss efficiency
101
60
70
Time (106 s)
102
50
60
100 %
Density (g cm−3 )
40
50
Radius (R J )
Radius (R J )
Shock loss efficiency
3
102
10−1310−1210−1110−1010−9 10−8 10−7
Optical depth from r max
60
101
−1
10
4
Radius (R J )
0
−5
5
Radius (R J )
1
Velocity (km s−1 )
50
6
Kinetic
¡ energy
¢
flux 10−5 L ⊙
10
−11
400
350
300
250
200
150
100
50
0
¡
¢
Luminosity 10−5 L ⊙
−9
Reduced flux |F |/ cE
10
Temperature (K)
Density (g cm−3 )
10−8
100 %
80 %
60 %
40 %
20 %
10−5
10−6
10−7
10−8
10−9
0%
1
10
Shock Mach number
40
50
60
70
80
Radius (R J )
A, Makemake, BL94, t = 1.5 × 107 s
A, Belt, 1-T , H00+WD01, t = 1.1 × 107 s
A, Belt, 2-T , H00+WD01, t = 1.2 × 107 s
Fig. 3.1.: Comparison between 1-T and 2-T for Case A of Table 2.1, as in Figure 2.9. The curve labeled
‘Makemake’ is from that figure (see legend). The shock temperature estimate (Equation 1.18) was left out for
the Belt results. In the panel of efficiency against Mach number, the different points for a single simulation
correspond to different times.
(which is to be checked a posteriori). Thus, one writes the energy equation as (see the matter–
radiation coupling test of Ramsey & Dullemond, 2015 or equation 7.108 of Drake, 2006)
d%cv T
! E int (T eff ) − 0
= cκ%(Erad − aT 4 ) ≈
,
dt
τst. st.
(3.3)
uses that Erad aT 4 , and obtains what Drake (2006) calls the steady-state timescale, the time
~ 43 ~
CHAPTER 3. SHOCKS WITH 2-T RADIATION TRANSPORT
6000
10−8
10−9
10−10
10−11
5000
5
4000
4
3000
3
2000
2
1000
10−12
1
0
4
Radius (R J )
10
Opacity (g cm−2 )
2
0
−2
−4
−6
−8
−10
−12
−14
0
8
0
−1
−2
10−1
10
−3
−4
−2
−5
4
8
102
10−1310−1210−1110−1010−9 10−8 10−7
100 %
−1
10−2
4
Reduced flux |F |/ cE
80 %
60 %
40 %
20 %
0 .8
0 .6
0 .4
0 .2
0
1
10
100
4
Time (106 s)
120 %
ηkin
100 %
80 %
60 %
40 %
20 %
0%
1
8
Radius (R J )
8
Radius (R J )
Pressure (bar)
100
8
1
0%
Shock loss efficiency
Optical depth from r max
101
4
Radius (R J )
ηkin
Density (g cm−3 )
102
−6
8
Radius (R J )
Shock loss efficiency
Temperature (K)
3
8
Radius (R J )
1
100
Radius (R J )
10
4
Radius (R J )
1
10−3
4
10
8
Kinetic
¡ energy
¢
flux 10−5 L ⊙
4
Velocity (km s−1 )
6
¡
¢
Luminosity 10−5 L ⊙
10−7
Temperature (K)
Density (g cm−3 )
10−6
10−1
10−2
10−3
10−4
10−5
10−6
10−7
10−8
10
Shock Mach number
4
8
Radius (R J )
B, Makemake, BL94, t = 3.0 × 107 s
B, Belt, 1-T , H00+WD01, t = 2.2 × 107 s
B, Belt, 2-T , H00+WD01, t = 2.5 × 107 s
Fig. 3.2.: As in Figure 3.1 but for Case B of Table 2.1.
required for the gas to reach T = T rad0 = T eff , as
τst. st. =
1 Eint (T eff )
cv
=
.
κ%c Erad (T eff ) κcaT eff 3
(3.4)
Equation 3.4 differs by a factor Eint /Erad ≈ 103 from 1/κ%c, the expression given by Kuiper et al.
(2010) for the energy exchange time, based on the radiation energy equation.
In Figure 3.4, one thus finds τst. st. = cv /(κσT eff 3 ) ∼ 0.3 × 106 (1 cm2 g−1 /κ) s for γ = 1.44,
µ = 2.353, and T eff = 160 K. Since, for κ = 1 cm2 g−1 , we have that τst. st. ≈ 0.03τflow , the gas
should be in equilibrium with the radiation, whereas for lower values of κ (at the same post-shock
temperature T eff ) this is not so certain.
~ 44 ~
103
2
6
−10
−20
−30
−40
−50
2
4
10
101
100
10−1
10−2
10−3
10−4
10−5
10−6
−3
−4
−5
Shock loss efficiency
Temperature (K)
2
4
1
80 %
60 %
40 %
20 %
0 .8
0 .6
0 .4
0 .2
0
1
10
100
2
−1
10−2
6
ηkin
100 %
80 %
60 %
40 %
20 %
0%
1
10
Shock Mach number
4
6
Radius (R J )
Pressure (bar)
100
120 %
6
Radius (R J )
ηkin
0%
0 .1
Shock loss efficiency
Optical depth from r max
−6
6
Time (106 s)
101
Radius (R J )
4
100 %
Density (g cm−3 )
102
6
−2
2
102
10−1310−1210−1110−1010−9 10−8 10−7
4
1
Radius (R J )
3
2
4
Radius (R J )
0
6
104
10
2
−1
Radius (R J )
10
6
Kinetic
¡ energy
¢
flux 10−3 L ⊙
Opacity (g cm−2 )
0
4
Radius (R J )
2
Reduced flux |F |/ cE
4
Radius (R J )
10
Velocity (km s−1 )
104
102
2
16
14
12
10
8
6
4
2
0
−2
¡
¢
Luminosity 10−3 L ⊙
10−4
10−5
10−6
10−7
10−8
10−9
10−10
10−11
10−12
Temperature (K)
Density (g cm−3 )
3.2. SELECTED CASES WITH TABULATED OPACITIES
102
101
100
10−1
10−2
10−3
10−4
10−5
10−6
10−7
10−8
2
4
6
Radius (R J )
C, Makemake, BL94, t = 2.3 × 107 s
C, Belt, 1-T , H00+WD01, t = 1.3 × 107 s
C, Belt, 2-T , H00+WD01, t = 2.8 × 107 s
D, Makemake, BL94, t = 2.5 × 107 s
D, Belt, 1-T , H00+WD01, t = 1.5 × 107 s
D, Belt, 2-T , H00+WD01, t = 3.2 × 107 s
Fig. 3.3.: As in Figure 3.1 but for Cases C and D of Table 2.1. Note that Case C has not quite reached the
desired radius yet.
Indeed, in Figure 3.5 we have κ = 10−2 cm2 g−1 and again T eff = 150 K (which actually reflects
the radiation Dirichlet boundary condition) with the same flow timescale of ∼ 107 s but a timescale
for equilibrium of 107.5 s. Thus, gas and radiation should not be in equilibrium, and they are not,
which means that the 1-T approach in this case would not be valid. The 1-T (gas) temperature can
be seen to match the radiation temperature upstream of the shock and in the downstream cooling
region.
~ 45 ~
100
10
−10−3
−10−2
−10−1
0
0
50
100
Temperature (K)
102
10−1310−1210−1110−1010−9 10−8 10−7 10−6
¡
¢
Luminosity 10−5 L ⊙
4
3
2
1
0
0
Shock loss efficiency
Number fraction of H I
1
0 .6
0 .4
0 .2
0
100
50
50
7
150
4
3
2
1
0
50
100
150
0
50
100 %
80 %
60 %
40 %
20 %
0%
100
1000
1
0 .6
0 .4
0 .2
0
0
50
100
Radius (R J )
100
150
Radius (R J )
Shock loss efficiency
Radius (R J )
50
150
0 .8
10000
106
104
102
100
10−2
10−4
10−6
0
100
Radius (R J )
Time (106 s)
150
100
Radius (R J )
5
0
Optical depth from r max
100
0
150
6
Radius (R J )
50
100
Radius (R J )
1 .8
1 .7
1 .6
1 .5
1 .4
1 .3
1 .2
1 .1
1
150
7
6 .5
6
5 .5
5
4 .5
4
3 .5
150
Reconstructed flux
Radius (R J )
0 .8
100
Radius (R J )
7
100
Density (g cm )
50
50
5
−3
Entropy ( k B baryon−1 )
0
6
150
103
0
150
10−1
Radius (R J )
0
100
Radius (R J )
¡
¢
Enthalpy flux 10−5 L ⊙
−101
50
1
Reduced flux |F |/ cE
−10
0
150
Opacity (g cm−2 )
Velocity (km s−1 )
50
Radius (R J )
−4
7
6
5
4
3
2
1
0
Trad
Material
¡ energy
¢
flux 10−5 L ⊙
0
−10
300
280
260
240
220
200
180
160
140
Temperature (K)
10−6
10−7
10−8
10−9
10−10
10−11
10−12
10−13
Effective heat capacity ratio
Density (g cm−3 )
CHAPTER 3. SHOCKS WITH 2-T RADIATION TRANSPORT
150
100 %
ηphys
80 %
60 %
40 %
20 %
0%
1
10
Shock Mach number
Mp = 0.32 MJ , Lnom
= 0, Ṁ = 10−2 M⊕ an−1
p
γ = 1.44, N = 1000
Mkmk, 1-T , t = 1.6 × 109 s, N = 2000
Mkmk, 1-T , t = 1.6 × 109 s, N = 1000
Belt, 2-T , t = 2.7 × 109 s, N = 1000
Fig. 3.4.: Comparison between 1-T and 2-T for a high opacity. The shock was not identified in the 2-T
case. The Mach numbers and efficiencies are clearly the same (same ∆L), even though the luminosity left of
the shock is different for 2-T . Note that the long-dashed curve in the %–T phase diagram shows the radiation
temperature and thus belongs to the Belt curve.
3.3. Accreting a whole planet
One of the issues we have seen in the previous chapter is that we cannot easily control the luminosity left of the shock. In this section, we try to see what luminosity structure arises if the whole
atmosphere is accreted. Specifically, we start with a small-mass, solid core and accrete as much
atmosphere as possible, taking values typical for protoplanets at the onset of runaway accretion:
rmin = 2 R⊕ and Mp = 10 M⊕ , which is thus the core mass. For simplicity, we let the gas fall from
~ 46 ~
Temperature (K)
350
−10−2
−10−1
0
50
3
2
1
0
0
Shock loss efficiency
Number fraction of H I
1
0 .6
0 .4
0 .2
0
50
0
100
Radius (R J )
1 .8
1 .7
1 .6
1 .5
1 .4
1 .3
1 .2
1 .1
1
50
7
5
4
3
2
1
0
50
100
0
50
100 %
80 %
60 %
40 %
20 %
0%
1000
10000
1
0 .8
0 .6
0 .4
0 .2
0
0
100000
50
Shock loss efficiency
Optical depth from r max
102
100
10−2
10−4
10−6
50
Radius (R J )
100
Radius (R J )
104
0
100
Radius (R J )
Time (106 s)
100
100
Radius (R J )
6
0
100
7
6 .5
6
5 .5
5
4 .5
4
3 .5
3
2 .5
Radius (R J )
Reconstructed flux
Radius (R J )
0 .8
100
4
Radius (R J )
Entropy ( k B baryon−1 )
7
10−2
Density (g cm )
50
50
Radius (R J )
5
−3
50
0
6
100
102
10−1310−1210−1110−1010−9 10−8 10−7 10−6
0
100
¡
¢
Enthalpy flux 10−5 L ⊙
0
103
0
50
Radius (R J )
−1
10−3
Radius (R J )
Temperature (K)
150
Reduced flux |F |/ cE
−101
200
10
−10−3
−10
250
0
Radius (R J )
−4
300
100
Opacity (g cm−2 )
Velocity (km s−1 )
−10
50
Trad
Material
¡ energy
¢
flux 10−5 L ⊙
0
7
6
5
4
3
2
1
0
−1
¡
¢
Luminosity 10−5 L ⊙
10−4
10−5
10−6
10−7
10−8
10−9
10−10
10−11
10−12
Effective heat capacity ratio
Density (g cm−3 )
3.3. ACCRETING A WHOLE PLANET
100
100 %
ηphys
80 %
60 %
40 %
20 %
0%
1
10
Shock Mach number
Mp = 0.32 MJ , Lnom
= 0, Ṁ = 10−2 M⊕ an−1
p
γ = 1.44, N = 1000
Mkmk, 1-T , t = 1.6 × 109 s, N = 1000
Belt, 2-T , t = 2.7 × 109 s, N = 1000
Fig. 3.5.: Same as Figure 3.4 but for κ = 10−2 cm2 s−1 . Notice in the phase diagram the very clear cooling
zone downstream of the shock The 1-T shock moves out at a higher velocity than the 2-T case (∼ 90 RJ in
1.6 × 109 s vs. in 2.7 × 109 s). in the 2-T case.
120 RJ , which is much larger than actual Hill radius (the planet is still attached in the early runaway
phase) but avoids moving the outer edge as a function of time as the Hill radius grows. In the spirit
of these simplifications, we take a constant accretion rate Ṁ = 10−3 or even 10−2 M⊕ an−1 to speed
up the process in wall-clock time.
Since the mass of the atmosphere will eventually dominate, we take into account spherically
~ 47 ~
CHAPTER 3. SHOCKS WITH 2-T RADIATION TRANSPORT
symmetric self-gravity1 by including a potential Φ(r) given by
Z rmax
G%
Φ(r) = −
dV 0 ,
0 , r)
max(r
rmin
(3.5)
where the volume element dV = 4πr2 dr, to yield an acceleration g(r) = −∇Φ(r). The effect of the
fixed core mass is added as a body force as in the main work.
Two important elements are missing: convection and a stiffer (less compressible) equation of
state. Convection would provide an upper limit to the temperature gradient to transport a given
amount of energy, thus leading to lower central temperatures if the current gradient were higher.
Given that the pressure at a given mass coordinate is set by the mass of the layers above, lowering
the temperature would cause the density to be higher and thus the planet more compact.
We compare 1-T and 2-T radiation transport and, to speed up the computation, try cases without
self-gravity and cases where Φ is calculated only every 100th step, which is much shorter than the
flow time across a cell. We take N = 500 and 1000 cells to asses the effect of resolution. It was
noticed that the choice of γ does not strongly influence the shock radius although, perhaps surprisingly, lower γ values lead to slightly more compact structures. We consider γ = 1.05 and 1.44.
The dust and gas opacities are from Weingartner & Draine (2001) and Helling et al. (2000), respectively, since Malygin et al. (2014) is not currently available for Belt and BL94 do not provide
Planck opacities, needed for 2-T radiation transport. As pointed out by Malygin et al. (2014), note
however that the Helling et al. (2000) Planck opacities are underestimated by up to four orders of
magnitude between 3300 K and 10000 K.
The boundary conditions are as in Chapter 2: at rmin , reflective in the hydrodynamics and the
radiation transport (P0 = %0 = v0 = Erad 0 = 0, where X 0 = dX/dr), and at rmax , % = %ff , v = vff ,
P0 = 0, and dr2 Erad /dr = 0. We initialise the domain with v = 0 everywhere, a constant gas and
radiation temperature T rad = T gas = T neb , and a free-fall density profile from infinity. The total
mass above the core is negligible. For simplicity, the core mass is kept constant, with the 30-M⊕
cases providing an estimate of the effects of a core mass increasing in time. The cases are listed in
Table 3.1 and the simulations are currently running.
3.3.1. First results
Figure 3.6 shows preliminary results for some cases from Table 3.1. Only the innermost parts of
the grid are shown. We can observe the following:
• The temperature spike at the shock in the 2-T cases is resolved and is relatively small (∆T ∼
100 K at 1500 K) for Ṁ = 10−2 M⊕ an−1 but very pronounced (∆T ∼ 4000 K at 1500 K) for
Ṁ = 10−3 M⊕ an−1 .
• The velocities follows a free-fall profile, as expected, and show some ringing downstream of
the shock.
• All simulations pass through the dust destruction region.
• The reduced fluxes at the shock front are fred ∼ 0.2, which means that radiation is flowing
almost in free streaming. Nevertheless, the optical depths to rmax are gigantic, with ∆τ ∼
1020 .
• The luminosity profiles increase somewhat radially and are non-monotonic with radius downstream of the shock. It is not yet clear whether this is a numerical or a physical effect since
doubling the resolution changed the flux by 20 per cent upstream (leaving however the gas
temperature, velocity, and density unchanged) but did not change the qualitative appearance
of the solution.
1
Included in R. Kuiper’s Belt.
~ 48 ~
3.3. ACCRETING A WHOLE PLANET
Name
Core mass
γ
Opacity
Ṁ
Self-gravity
N
RT
10ME.1T
10ME.2T
10ME.2T.1k
10ME.2T.gam
10ME.2T.SG10
10ME.1T.noSG
10ME.2T.Mdot
10
10
10
10
10
10
10
M⊕
M⊕
M⊕
M⊕
M⊕
M⊕
M⊕
1.44
1.44
1.44
1.05
1.44
1.44
1.44
Std
Std
Std
Std
Std
Std
Std
10−2
10−2
10−2
10−2
10−2
10−2
10−3
1
1
1
1
10
0
1
500
500
1000
500
500
1000∗
1000
1-T
2-T
2-T
2-T
2-T
1-T
2-T
30ME.1T
30ME.2T
30ME.2T.2k
30ME.1T.noSG
30ME.2T.Mdot
30
30
30
30
30
M⊕
M⊕
M⊕
M⊕
M⊕
1.44
1.44
1.44
1.44
1.44
Std
Std
Std
Std
Std
10−2
10−2
10−2
10−2
10−3
1
1
1
0
1
500
500
2000
1000∗
1000
1-T
2-T
2-T
1-T
2-T
1
3
4
0
10
109
108
107
106
105
104
103
102
101
100
10−1
Velocity (km −1 )
Pressure (erg cm−3 )
2
Radius (R J )
0
1
2
3
4
2
0
−2
−4
−6
−8
−10
−12
−14
−16
−18
1
102
10−1110−1010−9 10−8 10−7 10−6 10−5 10−4
Density (g cm−3 )
3
4
0
1
2
3
4
3
4
Radius (R J )
1
0 .8
0 .6
0 .4
0 .2
0
0
1
2
3
4
0
Radius (R J )
Optical depth from r max
Temperature (K)
103
2
Radius (R J )
Radius (R J )
104
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0
−0.1
−0.2
Reduced flux |F |/ cE
0
10
8000
7000
6000
5000
4000
3000
2000
1000
0
¡
¢
Luminosity 10−4 L ⊙
10−1
10−2
10−3
10−4
10−5
10−6
10−7
10−8
10−9
10−10
10−11
Temperature (K)
Density (g cm−3 )
Table 3.1.: Small parameter study. The opacity is Weingartner & Draine (2001) with smoothed dust destruction and Helling et al. (2000). The number in the ‘Self-gravity’ column refers to the how often the potential
is computed (see discussion below), in numbers of steps (1: every step). The column ‘RT’ refers to radiation
transport. Number of cells with a star indicate that the simulation is run in parallel. All core radii are 1.9 R⊕ .
Recall that R⊕ = 0.089RJ = 6.38×8 cm and M⊕ = 0.0031MJ = 5.97 × 1027 g.
1
2
Radius (R J )
1025
t = 4.2 × 106 s
1020
10
10ME.1T
10ME.2T
10ME.2T.1k
10ME.2T.gam
10ME.2T.Mdot
15
1010
105
100
10−5
0
1
2
3
4
Radius (R J )
Fig. 3.6.: Same as Figure 3.4 but for five selected cases from Table 3.1. To ease the comparison, they are all
measured at 4.2 × 106 s. Notice in the phase diagram the very clear cooling zone downstream of the shock
(shock region indicated by points) in the 2-T case.
In 48h of wallclock time, simulation 10ME.2T.SG10 has reached 5 × 108 s and thus accreted a
mass of 0.2 M⊕ . Therefore, it seems possible to reach on a reasonable timescale a mass where the
~ 49 ~
CHAPTER 3. SHOCKS WITH 2-T RADIATION TRANSPORT
self-gravity could have an effect (compared to the mass of the core if 20 M⊕ ), albeit perhaps only
at the ten-per cent level.
3.4. Conclusion
To assess how good our approximation that the gas and radiation are perfectly coupled (1-T radiation transport), we have compared for selected representative cases the shock structures and
efficiencies obtained with 1-T and 2-T transport. At the same time, we recomputed the 1-T cases
from Chapter 2 using a new, semi-independent version of the radiation-hydrodynamics code.
Our main and most important finding is that the 2-T results agree extremely well with the 1-T
results and thus that the equilibrium approximation seems justified in the context of the planetary
accretion shock. While we found a sharp peak in the gas temperature at the shock for the 2-T
case, while the radiation temperature remains smooth, this feature (akin to but not to be identified
with the Zel’dovich peak) is of no consequence for the gas temperature down- or upstream of the
shock. In particular, we find in that sense isothermal shocks also in 2-T , again justifying our use
of Equation (1.11). The small differences seen in the profiles can be confidently mainly attributed
to the use of different opacity tables. Thus the shock heating Q+shock , the post-shock temperature,
and the post-shock pressure are expected to be independent of the approximation of 1-T radiation
transport.
As an exercise, we looked at the valididy of the 1-T approach for a low and for a high constant
opacity. The results were as expected theoretically from the comparison of the gas–radiation equilibration timescale and the flow timescale, with the gas and radiation temperatures being equal at
high opacity and not otherwise. However, it is important to emphasise that this situation of nonequilibrium was obtained for a contrived example with a low constant opacity which we did not see
with realistic opacities.
We observed that in the low-opacity case, there is no clear jump in the radiative flux at the shock
when using non-equilibrium (2-T ) radiation transport. This contrasts with the 1-T case, and the
precise cause would need to be investigated. Incidentally, the absence of a clear jump in the flux
means that the shock loss efficiency could not be measured as ηkin = ∆L/Lacc, max ; rather, use
would have to be made of Equation (1.11), which is valid independently of the radiation transport.
However, this remains presently a question purely of academic interest.
Two concluding remarks concerning the main part of this chapter are in order. We caution that
we have not yet attempted to cover the full extent of the relevant formation parameter space, so
that definitive conclusions about the appropriateness of 1-T radiation transport will have to await a
more detailed study. Secondly, using a variable equation of state in the 2-T approach might lead to
interesting behaviour; dissociation or, more likely given the temperatures attained, ionisation could
change markedly the energy budget at the shock since the peak temperature is currently very large.
This additional microphysics could possibly lead to efficiencies very different from the constantEOS value at high Mach numbers, ηkin ≈ 100 per cent.
Finally, we also presented first results from accreting a large amount of the atmosphere, starting
with a solid core. We found that the resolution may not be high enough to determine the absolute
value of the luminosity but that the other quantities (gas temperature, density, velocity) did not
depend on the resolution. Once the simulations have had more time to run, it will be interesting to
see what luminosity structures develop.
~ 50 ~
This is the most difficult part of the calculation;
the rest is pure routine and could be entrusted
to even an unskilled computer.
— Kassel, J. Chem. Phys. 1, 576–585 (1933)
4
Cooling tracks for gas giants using BT-Settl
atmospheres
4.1. Introduction
The atmosphere of a gas giant (planet or brown dwarf) is the bottleneck through which the energy
flux coming from the deep interior must pass to allow the object to cool. Moreover, the spectral
appearance of sub-stellar mass objects is clearly sculpted by the molecular lines and bands, for
T eff . 4000 K (Baraffe, 2014), and condensates for T eff . 2000 K, in the photospheric layers of the
atmosphere (i.e., where the optical depth to infinity τ at a given wavelength λ is τλ ∼ 1). Therefore,
realistic atmospheres are necessary both to compute accurately the cooling of gas giants and to
predict their spectra.
Work in this field has a long history, starting with studies of the Solar System gas giants (e.g.,
Hubbard, 1977; Lunine et al., 1989). Observations of extrasolar and brown dwarfs, starting twenty
years ago (e.g., Rebolo et al., 1995; Oppenheimer et al., 1995), revealed chemically complex atmospheres and motivated the development of models with increasing numbers of species (and in particular molecular lines) and an increasing level of detail of the modeled physical processes (amongst
others, dust formation, (patchy) cloud formation, vertical mixing leading to non-equilibrium chemistry, and even lightning; e.g., Tsuji et al., 1996; Allard et al., 2001; Ackerman & Marley, 2001;
Barman et al., 2011b; Helling et al., 2011; Morley et al., 2014). This in turn has spurned progress
in fundamental-physics calculations of atomic and molecular line lists and continua (see Mollière
et al., 2015 for an up-to-date compilation). The interested reader is referred to a number of detailed
reviews (Allard et al., 1997, 2012; Helling & Casewell, 2014; Baraffe, 2014; Marley & Robinson,
2015) as well as a thorough comparison of the different cloud models in Helling et al. (2008).
There exists a number of sets of cooling tracks, which provide the bolometric luminosity Lbol ,
T eff , and magnitudes as a function of time for a gas giant of a given mass and assuming a certain
state at the beginning of the cooling. The classical works by Burrows et al. (1997), Chabrier et al.
(2000b), Baraffe et al. (2003), and Saumon & Marley (2008), which all use detailed atmospheres,
are appropriate for the planetary and brown-dwarf regime and publicly available. More recently,
Spiegel & Burrows (2012) computed cooling curves which begin with an explicitly-specified entropy available in a range covering the extreme outcomes of planetary formation; it was the first
work to do so.
Nevertheless, several reasons motivate us to perform our own coupling between detailed atmospheres and a cooling code:
• First, current population synthesis calculations (e.g., Mordasini et al., 2012) make use of
simple boundary conditions (detailed below) to compute the evolution of gas giants of masses
~ 51 ~
CHAPTER 4. COOLING TRACKS FOR GAS GIANTS USING BT-SETTL ATMOSPHERES
up to ∼ 50 MJ . Luminosity predictions (Mordasini et al., in prep.) of population synthesis
would therefore gain from coupling to realistic atmospheres, especially in the light of directdetection surveys (e.g., Beuzit et al., 2006; Vigan et al., 2012; Brandt et al., 2014).
• The widely-used models of Burrows et al. (1997) and Baraffe et al. (2003) assume ‘hot
starts’, i.e., a high initial entropy (see Chapters 2 and 5), and there have been significant
developments in atmospheric modeling since these works. The recently-published cooling
curves of Baraffe et al. (2015) represent the latest instalment from the Lyon group (Baraffe
et al., 2003) but focus explicitly on the stellar and high-mass brown-dwarf regime, since
models at lower effective temperature are not ready yet.
• While we use BT-Settl because it is a state-of-the-art set of atmospheres, which fare well
when used to interpret observations (Bonnefoy et al., 2014; Baraffe, 2014), performing our
own coupling also allows us to use any other atmosphere.
We note that a common approach in the literature consists of taking published tracks such as
Baraffe et al. (2003) and interpolating the (log g, T eff ) pairs as a function of time in the X =
X(log g, T eff ) mapping provided by another (in particular, more recent) atmospheric model, where
log g is the surface gravity (g = GMp /Rp 2 , always using cgs units and base ten) and X here stands
for an arbitrary observable (magnitude in some band, luminosity, effective temperature). However, this is not self-consistent, in the sense that the same atmosphere model is not used to regulate
the cooling and predict the brightness. Moreover, while this a posteriori approach does provide a
useful estimate, it is possible and conceptually simple to perform an exact coupling.
The remainder of this chapter is structured as follows: Section 4.2 presents the existing code
which calculates the cooling, the coupling approach. and an overview of the BT-Settl atmospheres. The extracted BT-Settl-2010 grid is shown in Section 4.3.1, with the resulting cooling
tracks in Section 4.3.2. A brief summary (Section 4.4) closes this chapter.
4.2. Background
4.2.1. Overview of the cooling code
The planetary structure and evolution code, Rapax 21, is the Ṁ = 0 limit of the code described
in Alibert et al. (2005), Mordasini et al. (2012), and Mollière & Mordasini (2012) and is sketched
here. At each timestep t, the radial structure [T (r), P(r), m(r), L(r)] is determined by inward
integration of the usual equations of hydrostatic equilibrium (Equations 2.3). The idea is to find by
bisection the outer radius, from which the integration begins, which yields the desired total mass,
i.e., such that m = 0 at r = 0. Integrating the dT/dr equation requires the luminosity Lp , assumed
spatially constant in the planet (which was verified to be an excellent approximation as discussed
in section 3.4 of Mordasini et al., 2012). It is provided in one of two ways: (i) for the first time
step, Lp is simply the desired initial value, chosen by the user; (ii) for all subsequent steps, Lp is
determined at each trial radius Rp 0 in the root-finding iterations by the corresponding difference in
the total energy, the contraction luminosity: Lp (Rp 0 , t) = −[Etot (Rp 0 , t + ∆t) − Etot (t)]/∆t, where
∆t is the size of the timestep between t and the new time, andR the
Etot is the sum of
totalGmenergy
the gravitational and internal energies: Etot = Egrav + Eint =
eint − r r dm. Since Etot is not
available for the new step since the structure has not yet been determined, it needs to be estimated.
The presence of deuterium burning does not change this basic picture and the details are given in
Mollière & Mordasini (2012).
The mass outer boundary condition for any trial Rp 0 is trivially M(Rp 0 ) = Mp , and the atmosphere model provides the remaining conditions T (Rp 0 ) and P(Rp 0 ). Rapax 21 currently uses the
simple classical relations provided by the Eddington approximation1 , which yields a photosphere
1
This assumes, to solve the radiative transfer equation in a plane-parallel atmosphere, that the radiation pressure
everywhere is related to the radiation energy density by Prad = 13 Erad , which is appropriate both in the diffusion limit
and the two-stream approximation; see e.g., Mihalas & Mihalas (1984).
~ 52 ~
4.2. BACKGROUND
at T = T eff with the photospheric pressure Pphot = 2g/3κ. These simple boundary conditions yield
reasonably good results when compared to detailed atmospheres (see Section 5.2.6) and will serve
as a comparison to the coupling with the BT-Settl atmospheres.
4.2.2. Description of the coupling
The idea behind the coupling of interior calculations to model atmospheres is simple and consists
of matching the adiabatic convective layers of the atmosphere to the convective zone of the planet,
as described, e.g., in Chabrier & Baraffe (1997). This connection can be made at a fixed pressure of
10 bar (Saumon et al., 1996), although there is the possibility that the atmosphere not be adiabatic
there (Burrows et al., 1997). A safer choice is to take a layer at a high optical depth, τ = 30 or
100 where structures are always convective (Chabrier & Baraffe, 1997). In both cases, the distance
to the photospheric radius Rp (at τ = 2/3) is less than 1 percent and thus negligible (Chabrier &
Baraffe, 1997), and so is the amount of mass contained in the upper layers, as can be easily shown2 .
When information about the interface is shown by atmospheric modelers, it is usually in the
form of T 10 , the temperature at 10 bar on the adiabat, whether the atmosphere is convective there
or not (e.g., Saumon et al., 1996; Burrows et al., 1997; Saumon & Marley, 2008; Fortney et al.,
2011). The entropy can also directly be shown (Saumon et al., 1994). Ultimately, however, for a
given (trial) g = GMp /Rp 02 and T eff ∝ (Lp /Rp 02 )1/4 , one needs a pressure and temperature point
(P, T ), which is what we will extract from the BT-Settl structures. Ideally, the atmosphere and
the interior model would need to be computed with exactly the same equation of state, in particular,
with the same helium mass fraction Y and metals distribution. However, in practice, the latter is
usually neglected in the interior equation of state, and small changes in Y are of not consequence;
for instance, Baraffe et al. (2015) report that a mismatch in Y of 0.01 yields less than 1 per cent
differences in T eff and 0.01 mag for all colours.
4.2.3. Overview of BT-Settl atmospheres
As summarised by Bonnefoy et al. (2013) and Baraffe (2014), the BT-Settl models (Allard et al.,
2003, 2012, 2013) use a cloud description which solves for the formation and gravitational settling
of dust model and solve frequency-dependent radiative transfer self-consistently in spherical geometry. The cloud model, based on Rossow (1978, see also Mordasini, 2014), compares the timescales
for mixing, sedimentation, condensation, coalescence, and coagulation of the dust particles to find
the cloud structure, using for the mixing the results of dedicated 2D radiation hydrodynamics simulations (Freytag et al., 2010). The radiative transfer is computed with PHOENIX (Allard et al.,
2001, 2003), and iterations are performed on the structure, cloud model, and radiative transport
modules until convergence. Different releases of BT-Settl use different elemental abundances:
unless noted otherwise, we use the BT-Settl-2010 grid, which took elemental abundances from
Asplund et al. (2009), but also compare to the CIFIST grid, which uses the Caffau et al. (2011)
abundances.
Atmospheric structures were not directly available and had to be extracted from the log files
as available at http://phoenix.ens-lyon.fr/fallard/. (The awk script written for this is
reproduced in Section B.7.) A few examples are shown in Figure 4.1a and reveal a complex atmospheric structure where zones dominated by radiative transport alternate with convective layers.
Figure 4.1b compares structures from different versions of BT-Settl and shows that, while the upper atmospheric layers differ noticeably (by a few hundred kelvins for P . 1 bar), these differences
mostly at levels above the photosphere do not affect significantly the deep adiabat.
2
Combining Equations (2.3) yields dm/dP = 4πr4 /Gmr , and using the one-zone estimate of the central pressure
Pc ∼ Mp 2 /Rp 4 , this implies that the mass ∆m above a (small) pressure ∆P is ∆m/Mp ∼ ∆P/Pc . Given typical central
pressures (see Figure 5.1), this is entirely negligible.
~ 53 ~
CHAPTER 4. COOLING TRACKS FOR GAS GIANTS USING BT-SETTL ATMOSPHERES
Temperature gradient
0.2
0.3
0
0.1
∇act
∇ad
Teff = 1200 K
∇rad
-2
-3
100
100
101
102
103
Rosseland optical depth
-4
Radiative
Convective
τR ∼ 2/3
τstd ∼ 100
102
0
500
1000
log κR (cm−2 g−1 )
Teff = 1200 K
Teff = 600 K
10−2
10−1
0
-1
10−1
101
Pressure (bar)
10−3
10−4
BT-Settl 2010, log g = 3.0 (cm s−2 )
-5
-6
1500
2000
2500
3000
3500
Temperature (K)
log g = 4.0 (cm s−2 ), Teff = 2200 K
-1
0
CIFIST 6b
10−2
10−1
100
τR
101
102
-1.5
CIFIST 6b
CIFIST 5
BT-Settl 2010
τR ∼ 2/3
τstd ∼ 100
500
log κR (cm−2 g−1 )
0.2
-0.5
0.1
∇act , ∇ad , ∇rad
102 101 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7
Pressure (bar)
[M/H] = 0.0 (C2011/A2009), Y = 0.27
-2
1000 1500 2000 2500 3000 3500 4000
Temperature (K)
Fig. 4.1.: Top panel: Pressure–temperature profiles of BT-Settl-2010 (i.e., with Asplund et al. (2009)
abundances) for log g = 3 and T eff = 600 K and 1200 K. Radiative parts of the structures are shown by
full lines and convective parts, in the sense that Fconv /Ftot > 0.5, by dotted segments. Apparent gaps in
the profiles are due to a change in the dominant flux transport mode over a single cell. The Rosseland mean
opacity in each radiative layer is indicated by the colour of the curve. Indicated are the photosphere τR ∼ 2/3
(where τR is the Rosseland optical depth) and the deep layer at τstd ∼ 100 used for the coupling (the index
stands for ‘standard’ and refers to 2.15 µm; F. Allard, priv. comm.) (open and filled circle, respectively).
Compare to figures 2ff of Burrows et al. (1997). Note that the pressure increases downward as is usual
when showing atmospheric structures as it corresponds to the physical situation. Inset: Actual, adiabatic,
and radiative gradients as a function of the Rosseland optical depth for the T eff = 2000 K case. Bottom
panel: Similar but comparing the structure of a log g = 4.0, T eff = 2200 K atmosphere for two versions of
BT-Settl-CIFIST and for BT-Settl-2010 (see legend).
~ 54 ~
4.3. RESULTS
4.3. Results
4.3.1. Input from the BT-Settl atmospheres
Figure 4.2a shows the pressure and temperature in the convective zone of the BT-Settl-2010 atmospheres as a function of log g and T eff . The lowest available T eff is 600 K and we show only up
to 3000 K since this is the domain relevant for Mp . 50 MJ (see e.g., figure 9 of Burrows et al.,
1997). (As it cools, the T eff of a planet drops, while the surface gravity stays roughly constant,
increasing (typically by less than 1 dex) slightly as it contracts.) The curves are smooth and mostly
monotonic for decreasing T eff , and the rough P ∝ g scaling (especially around 3000 K) reflects the
atmospheric opacities (e.g., Chabrier & Baraffe, 1997). A closer inspection of the figure reveals,
however, that the pressure and temperature reincrease with decreasing T eff around 1500 K. This
behaviour is more clearly seen in the entropy, shown in the bottom panel. This will have a noticeable effect on the cooling curves and contrasts with the result from simple Eddington boundary
conditions.
Figure 4.2b shows the corresponding entropies. As a planet cools, its entropy decreases over
time (see e.g., Section 5.2.1), which is why the entropy is shown as the independent variable. As
expected, the curves are smooth for T eff & 2000 K and also for T . 1000 K, but in the intermediate
region, especially around T eff = 1400–1500 K, the T eff in atmospheric models does not decrease
monotonically with entropy but rather is multiple-valued.
This behaviour seems to be due to a cycle of dust condensation (D. Homeier and F. Allard, priv.
comm.). If in the course of the iterations the temperature in the atmosphere drops sufficiently, solids
start condensing out of the gas phase and thus increase the opacity; this rise in opacity causes the
temperature in the atmosphere to reincrease to carry the same flux σT eff 4 , which in turn causes
the dust to evaporate. Due to the different mixing and formation timescales, there is a certain
amount of hysteresis, and the resulting non-monotonicity may be compounded by the fact that each
atmospheric model (i.e., each point in Figure 4.2) is calculated independently.
This feature, which indicates non-perfect convergence of the models, was not noticed before
because the atmospheres had not been analysed in this way. Moreover, the common a posteriori
interpolation of colours using the (log g, T eff ) provided by a simpler cooling track (e.g., Baraffe
et al., 2003) will not exhibit unusual features if the underlying track is monotonic in T eff .
Before proceeding to showing coupled cooling curves, we note that the latest BT-Settl-CIFIST
atmospheres, as used for Baraffe et al. (2015) and from D. Homeier (priv. comm., 2015 June 26),
have a monotonic T eff (S ) but there are steep regions where T eff decreases very quickly for a small
S decrement. This behaviour should be, qualitatively, physical, but whether the slope is as steep as
predicted remains to be seen. In any case, since this new grid does not cover temperatures lower
than ≈ 1500 K, we refrain from using it in its current version.
4.3.2. Cooling tracks with BT-Settl
Figure 4.3 shows hot-start cooling tracks for Mp = 3, 5, 10, and 15 MJ . We compare our selfconsistent coupling with the results from an Eddington boundary condition as well as to Baraffe
et al. (2003), Burrows et al. (1997), and Spiegel & Burrows (2012). Of relevance to direct detections
(e.g., Bonnefoy et al., 2014), both the (bolometric) luminosity and the absolute K s -band magnitude
are shown. (The effective temperature as a fuction of time is shown in Figure B.8.) The zero-point
for the time is the begin of the cooling, i.e., the end time of the formation, which, in principle,
should not be identified directly with the age of an observed system (Fortney et al., 2005).
There are noticeable differences between the BT-Settl-2010 (hereafter simply BT-Settl) and
the Eddington (grey) luminosities but these are not systematic and are relatively small. However,
the effects of the non-converged atmospheres around T eff ≈ 1300–1700 K are clearly visible as upand-down jumps, as the high- or the low-T eff solutions at given neighbouring entropies are chosen
by the cooling algorithm. These jumps affect most clearly the 10-MJ planet at 3–10 Myr, as well
as the 15-MJ track at 70–100 Myr after its deuterium-burning ‘shoulder’. The effect is also slightly
~ 55 ~
CHAPTER 4. COOLING TRACKS FOR GAS GIANTS USING BT-SETTL ATMOSPHERES
10−1
3000
BT-Settl-2010 (AGSS09)
=
log g
2 .5
2500
2000
Teff (K)
Pressure (bar)
100
101
1500
10
g
log
2
=5
.0
1000
103
500
2500
3000
3500
4000
4500
Temperature (K)
2000
1500
1000
.0
=2
.5
BT-Settl-2010 (AGSS09)
=5
g
log
g
log
Effective temperature (K)
2500
500
15
14
13
12
11
10
9
8
7
−1
Entropy ( k B baryon )
Fig. 4.2.: Top panel: Pressure and temperature at τ ∼ 139, as used for the coupling, for the BT-Settl2010 grid as a function of log g and T eff . Each curve is for models with the same surface gravity, from
log g = 2.5 to 5.0 (cm2 s−1 ) (bottom to top), and each point is a different effective temperature (colour), with
T eff = 600 to 3000 K from left to right but the last points not shown for the smaller log g. Multiples of
500 K are highlighted (open circles). Notice the non-monotonicity around 1500 K for low surface gravities.
Bottom panel: Corresponding T eff as function of the entropy S at the point shown in the left panel, with S
decreasing to the right to represent a time evolution. (For reference, the proton spin term (see Section 5.8) is
not included.) The curves are, from left to right, for log g = 2.5 to 5.0 in steps of 0.5 dex.
~ 56 ~
4.3. RESULTS
rapax 21 + BT-Settl 2010
rapax 21 + Eddington
10−3
Luminosity (L ⊙ )
COND03
B97
10−4
10−5
10−6
1
10
100
Time (Myr)
rapax 21 + BT-Settl 2010
rapax 21 + Eddington
7
SB12, no clouds, 1 × ⊙
SB12, clouds, 3 × ⊙
COND03
M K s (mag)
8
9
10
11
12
13
14
1
10
100
Time (Myr)
Fig. 4.3.: Hot-start cooling curves in luminosity (top panel) and K s band absolute magnitude (bottom panel)
for Mp = 3, 5, 10, and 15 MJ . Results for the self-consistent use of the BT-Settl-2010 atmospheres
(solid blue curves; shown only down to the grid edge of T eff = 600 K) are compared to tracks using the
Eddington boundary condition (solid red), Baraffe et al. (2003) (‘COND03’; dotted grey), Burrows et al.
(1997) (dashed grey), and Spiegel & Burrows (2012) (dashed fuchsia and grey for, respectively, their cloudfree solar-metallicity and cloudy 3 × solar-metallicity models). The non-labeled, pale grey curves are from
http://perso.ens-lyon.fr/isabelle.baraffe/BHAC15dir/ and are available as cooling tracks only
for the highest masses and down to T eff ≈ 1600 K (which is more restrictive than what is shown in Baraffe
et al., 2015). As an example, β Pic b is shown (dot with errorbars; Bonnefoy et al., 2014b).
~ 57 ~
CHAPTER 4. COOLING TRACKS FOR GAS GIANTS USING BT-SETTL ATMOSPHERES
visible for 3 MJ , though only at a very early time (≈ 2 Myr).
Overall, and despite the unphysical features, our BT-Settl tracks agree relatively well with
those of Burrows et al. (1997) and Baraffe et al. (2003). Nevertheless, there is both in luminosity
and magnitude the non-constant but systematic trend that we predict fainter objects at a given time.
In particular, at early times for the highest masses (where all other tracks agree), is lower by roughly
40 per cent. However, at later times, there is a larger spread between the different models, and our
systematic offset is comparable to this spread. For instance, at 10 Myr for 5 MJ , our BT-Settl
curve is a factor of two (0.3 dex) smaller in luminosity compared to Baraffe et al. (2003).
In Figure 4.3, we focus on the tracks of absolute K s magnitude for 10 and 15 MJ as interpolated
as a function of (log g, T eff ) in the BT-Settl-2010 grid (Ch. Mordasini and M. Bonnefoy, priv.
comm.). Interestingly, the difference between Eddington and the BT-Settl tracks is negligible
at early times even though the effective temperatures and surface gravities differ. We verified that
the situation is similar in the J and H bands. Moreover, both 10-MJ tracks join onto the Spiegel
& Burrows (2012) cloud-free, solar-metallicity curve at later times, namely t > 10 and ≈ 40 Myr
for the BT-Settl and Eddington models, respectively. Given the numerous differences between
the atmospheric models (especially the absence of clouds), it is difficult to determine whether this
agreement is fortuitous or has a deeper cause.
In any case, we note that the curves for various models differ by up to a magnitude at a given age,
which is much larger than typical errorbars. Thus it is useful to work with bolometric magnitudes,
especially when they can be constrained empirically as done recently for β Pic b by Morzinski et al.
(2015).
4.4. Summary
The choice of outer boundary conditions for the calculation of gas giant cooling curves, which is
set by the atmosphere, has measurable consequences on both the cooling rate and the observable
spectra and magnitudes. For this reason, and with in mind the on-going direct-detection surveys, the
attempted to couple our in-house cooling track code (Rapax 21) with the detailed BT-Settl-2010
atmospheres. This produced cooling tracks in reasonable agreement with previously published
work using both very simple or detailed atmospheres, but revealed unphysical jumps in the cooling
tracks that occur at ages relevant to direct observations. This is due to non-converged atmosphere
models because of dust condensation for T eff ≈ 1500 K, leading to a multi-valued relation between
T eff and the internal entropy S . Given these issues with the input data, we focussed on hot starts,
especially since cold starts only differ by a time offset in the cooling tracks.
The BT-Settl models are continuously being updated but the other grids (e.g., BT-SettlCIFIST_2011) have several ‘holes’ in (log g, T eff ) which made it impossible to perform the coupling. Therefore, the production of useful and accurate, self-consistent cooling tracks will have to
await the completion of the BT-Settl grid with fully converged models.
~ 58 ~
Wait here for further instructions.
— Sticker on a wall in Berlin
5
Constraining the initial entropy of gas giants
This chapter includes work published in Marleau & Cumming, MNRAS 437, 1378–1399 (2014),
“Constraining the initial entropy of directly detected exoplanets” (Marleau & Cumming, 2014).
Some figure have therefore already been presented in author’s Master’s thesis (Marleau, 2012),
but several results and figures go beyond that. More specifically:
• A small part of the text (roughly 20 per cent by page number) containing the key ideas was
written originally by A. Cumming but this was thoroughly rewritten and very significantly
extended (in terms of content) by the author.
• All figures except for Figures 5.4b, 5.6, 5.7, and 5.12 were present in the author’s Master’s
thesis. They have been slightly updated, however, with a handful of data points added to
Figure 5.8 and the MCMC simulations (see Sections 5.5.2ff) added to Figures 5.9ff.
• A small part of the literature review on the 2M 1207, HR 8799 and β Pic systems was included
in the author’s Master’s thesis, but it was considerably extended (by roughly more than a
factor of two by page count) and updated.
Many sections, representing a sizeable effort, are entirely original work by the author and not
present in his Master’s thesis: Sections 5.2.3, 5.2.4, 5.2.5, 5.3, 5.4, as well as 5.7 and 5.8.
The second, minor part of this chapter consists of a reevaluation of the companion to β Pic
(Section 5.10) as work published subsequently, in Bonnefoy et al. (2014b), and of a similar analysis
performed on κ Andromeda b (Section 5.11), published in Bonnefoy et al. (2014a). In both cases,
only the author’s contribution is included, with the strictly necessary information from the rest of
the publications summarised.
Abstract
The post-formation, initial entropy S i of a gas giant planet is a key witness to its mass-assembly
history and a crucial quantity for its early evolution. However, formation models are not yet able
to predict reliably S i , making unjustified the use solely of traditional, ‘hot-start’ cooling tracks to
interpret direct imaging results and calling for an observational determination of initial entropies
to guide formation scenarios. Using a grid of models in mass and entropy, we show how to place
joint constraints on the mass and initial entropy of an object from its observed luminosity and
age. This generalises the usual estimate of only a lower bound on the real mass through hotstart tracks. Moreover, we demonstrate that with mass information, e.g. from dynamical stability
analyses or radial velocity, tighter bounds can be set on the initial entropy. We apply this procedure
~ 59 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
to 2M1207 b and find that its initial entropy is at least 9.2 kB /baryon, assuming that it does not burn
deuterium. For the planets of the HR 8799 system, we infer that they must have formed with S i >
9.2 kB /baryon, independent of uncertainties about the age of the star. Finally, a similar analysis for
β Pic b reveals that it must have formed with S i > 10.5 kB /baryon, using the radial-velocity mass
upper limit. These initial entropy values are respectively ca. 0.7, 0.5, and 1.5 kB /baryon higher
than the ones obtained from core accretion models by Marley et al., thereby quantitatively ruling
out the coldest starts for these objects and constraining warm starts, especially for β Pic b.
5.1. Introduction
While only a handful of directly-detected exoplanets is currently known, the near future should
bring a statistically significant sample of directly-imaged exoplanets, thanks to a number of surveys
underway or coming online soon. Examples include VLT/SPHERE, Gemini/GPI, Subaru/HiCIAO,
Hale/P1640 (Vigan et al., 2010; Chauvin et al., 2010; McBride et al., 2011; McElwain et al., 2012;
Yamamoto et al., 2013; Hinkley et al., 2011). An important input for predicting the yields of such
surveys and for interpreting their results is the cooling history of gas giant planets. In the traditional
approach (Stevenson, 1982; Burrows et al., 1997; Baraffe et al., 2003), objects begin their cooling,
fully formed, with an arbitrarily high specific entropy1 and hence radius and luminosity. In the
past, the precise choice of initial conditions for the cooling has been of no practical consequence
because only the 4.5-Gyr-old Solar System’s gas giants were known, while high initial entropies
are forgotten on the short Kelvin–Helmholtz time-scale GM 2 /RL (Stevenson, 1982; Marley et al.,
2007). However, direct-detection surveys are aiming specifically at young (6 500 Myr) systems,
and the traditional models, as their authors explicitly recognised, are not reliable at early ages.
Using the standard core-accretion formation model (Pollack et al., 1996; Bodenheimer et al.,
2000; Hubickyj et al., 2005), Marley et al. (2007, hereafter M07) found that newly-formed gas
giants produced by core accretion should be substantially colder than what the usual cooling tracks
that begin with arbitrarily hot initial conditions assume. These outcomes are known as ‘cold start’
and ‘hot start’, respectively. M07 however noted that there are a number of assumptions and approximations that go into the core accretion models that make the predicted entropy uncertain. In
particular, the accretion shock at the surface of the planet is suspected to play a key role as most of
the mass is processed through it (M07; Baraffe, Chabrier & Barman, 2010) but there does not yet
exist a satisfactory treatment of it. Furthermore, there may also be an accretion shock in the other
main formation scenario, gravitational instability, such that it too could yield planets cooler than
usually expected (see section 8 of Mordasini et al., 2012). Conversely, Mordasini (2013) recently
found that the initial entropy varies strongly with the mass of the core, leading nearly to warm starts
for reasonable values of core masses in the framework of core accretion.
The most reasonable viewpoint for now is therefore to consider that the initial entropy is highly
uncertain and may lie almost anywhere between the cold values of M07 and the hot starts. In fact,
M07 calculated ‘warm start’ models that were intermediate between the cold and hot starts, and recently, Spiegel & Burrows (2012) calculated cooling tracks and spectra of giant planets beginning
with a range of initial entropies. This uncertainty in the initial entropy means that observations
are in a privileged position to inform models of planet formation, for which the entropy (or luminosity) at the end of the accretion phase is an easily accessible output. Its determination both
on an individual basis as well as for a statistically meaningful sample of planets would be very
valuable, with the latter allowing quantitative comparisons to planet population synthesis (Ida &
Lin, 2004; Mordasini et al., 2012). Moreover, upcoming and ongoing surveys should bring, in a
near future, both qualitative and quantitative changes to the collection of observed planets to reveal
a number of light objects at moderate separations from their parent star (i.e. Jupiter-like planets),
which contrasts with the few currently known direct detections. With this in mind, we stress the importance of interpreting direct observations in a model-independent fashion and of thinking about
1
The ‘initial’ entropy refers to the entropy at the beginning of the pure cooling phase, once the planet’s mass is assembled.
~ 60 ~
5.2. COOLING MODELS WITH ARBITRARY INITIAL ENTROPY
the information that these yield about the initial (i.e. post-formation) conditions of gas giants.
In this paper, we investigate the constraints on the mass M and initial, post-formation entropy S i
that come from a luminosity and age determination for a directly-detected object, focusing on exoplanets. Since the current luminosity increases monotonically both with mass and initial entropy,
these constraints take the form of a ‘trade-off curve’ between M and S i . We show that, in the planetary regime, the allowed values of M and S i can be divided into two regions. The first is a hot-start
region where the initial entropy can be arbitrarily high but where the mass is essentially unique.
This corresponds to the usual mass determination by fitting hot-start cooling curves. The second
corresponds to solutions with a lower entropy in a narrow range and for which the planet mass has
to be larger than the hot-start mass. (A priori, this may even reach into the mass regime where deuterium burning is important, which is discussed in a forthcoming paper.) The degeneracy between
mass and initial entropy means that in general the mass and entropy cannot be constrained independently from a measurement of luminosity alone. When additional mass information is available,
for example for multiple-planet systems or systems with radial velocity, it is possible to constrain
the formation entropy more tightly.
We start in Section 5.2 by describing our gas-giant cooling models, discussing the luminosity scalings with mass and entropy, and comparing to previous calculations in the literature. In
Section 5.3, we show in general how to derive constraints on the mass and initial entropy of a
directly-detected exoplanet by comparing its measured luminosity and age to cooling curves with
a range of initial entropies. We briefly consider in Section 5.4 similar constraints based on a (spectral) determination of the effective temperature and surface gravity. After a brief discussion of the
luminosity–age diagram of directly-detected objects, we apply the procedure based on bolometric luminosities in Section 5.5 to three particularly interesting systems – 2M1207, HR 8799, and
β Pictoris – and derive lower bounds on the initial entropies of the companions. For HR 8799 and
β Pic, we use the available information on the masses to constrain the initial entropies more tightly.
Finally, we offer a summary and concluding remarks in Section 5.6.
5.2. Cooling models with arbitrary initial entropy
Given its importance in the initial stages of a planet’s cooling, we focus in this paper on the internal
entropy of a gas giant as a fundamental parameter which, along with its mass, determines its structure and controls its evolution. Very few authors have explicitly considered entropy as the second
fundamental quantity even though this approach is more transparent than the usual, more intuitive
use of time as the second independent variable. In this section, before comparing our models to
standard ones, we discuss interior temperature–pressure profiles and semi-analytically explain the
different scalings of luminosity on mass and entropy which appear in the models.
5.2.1. Calculation of time evolution
We calculate the evolution of cooling gas giants by a ‘following the adiabats’ approach (Hubbard,
1977; Fortney & Hubbard, 2004; Arras & Bildsten, 2006). We generate a large grid of planet
models with ranges of interior specific entropy S and mass M and determine for each model the
luminosity L = L(M, S ) at the top of the convection zone. We then use it to calculate the rate of
change of the convective zone’s entropy dS /dt, which is defined by writing the entropy equation
∂L/∂m = −T ∂S /∂t as
Z
dS
L=−
T dm + LD ,
(5.1)
dt
with LD the luminosity due to deuterium burning in the convection zone. With the current S and
dS /dt in hand, calculating a cooling curve becomes a simple matter of stepping through the grid of
models at a fixed mass. This is computationally expeditious and gives results nearly identical to the
usual procedure based on the energy equation (Henyey et al., 1964; Kippenhahn & Weigert, 1990),
as discussed in Section 5.2.6.
~ 61 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
The assumptions in the ‘following the adiabats’ approach are that ∂S /∂t is constant throughout
the convective zone and that no luminosity is generated in the atmosphere. For these to hold, we
require both the thermal time-scale ttherm (r) = PcP T/gF (Arras & Bildsten, 2006) – where cP
is the specific heat capacity, g the gravity, and F the local flux – in the radiative zone overlying
the convective core and the convective turnover2 time-scale to be much shorter than the timescale on which the entropy is changing, τS = −S /(dS /dt) = MT S /L, with T the mass-weighted
mean temperature in the convection zone. A similar approach was used to study the evolution of
ohmically-heated irradiated gas giants by Huang & Cumming (2012).
We do not include deuterium burning directly in the planet models since it always occurs inside
the convection zone. Instead, we calculate the deuterium burning luminosity per unit deuterium
mass fraction for each model in the grid and use it in equation (5.1) to follow the cooling of the
planet and the time evolution of the deuterium mass fraction averaged over the convective region.
However, we focus in this work on masses below the (parameter-dependent) deuterium-burning
limit near 12–14 MJ (Spiegel, Burrows & Milsom, 2011; Mollière & Mordasini, 2012; Bodenheimer et al., 2013), such that deuterium burning does not play a role in the objects’ evolution.
We defer a detailed exploration of deuterium burning in our models to an ulterior publication but
already mention the interesting result that cooling curves starting at low entropy can exhibit an initial increase in their luminosity due to deuterium burning. We subsequently noticed that the colder
starts in fig. 8 of Mollière & Mordasini (2012) show a similar behaviour, and that Bodenheimer
et al. (2013) also find in their formation simulations the possibility of deuterium ‘flashes’.
5.2.2. Calculation of gas giant models
To construct a model with a given mass M and internal entropy S , we integrate inwards from
the photosphere and outwards from the centre, adjusting the central pressure, the cooling time
τS , and the luminosity L and the radius R at the top of the inner zone until the two integrations
match at a pressure of 30 kbar. We use the Eddington approximation at the photosphere, setting
T = T eff at Pphot = 2g/3κ, and take the solar-metallicity (based on Lodders, 2003) radiative opacity
from Freedman, Marley & Lodders (2008). In the deep interior, the contribution from the electron
conductive opacity of Cassisi et al. (2007) is also included. The equation of state for the hydrogen–
helium mixture is that of Saumon, Chabrier & van Horn (1995, hereafter SCvH) with a helium
mass fraction Y = 0.25. Since we focus on gas giants, we do not include a rocky core, which was
found in a test grid to increase the luminosities by at most a few per cent at the lower masses, as in
Saumon et al. (1996).
The grid has a lower entropy limit of S ' 7–7.5 kB /baryon3 . The upper S limit in the grid is set
by the requirement that R . 7 RJ , a value found to make numerical convergence straightforward.
The upper limits are S max ' 12 near 1 MJ and S max ' 14 for the higher masses.
Fig. 5.1 shows interior profiles in the T –P plane for a range of entropies and masses. Schematically, since hydrostatic balance dictates Pc ∼ M 2 /R4 , increasing the mass at a fixed entropy extends
the centre to a higher pressure along the adiabat. This is exacerbated at high entropies, where the
planet substantially shrinks with increasing mass, while low-entropy objects have a roughly constant radius. (Radii as a function of M and S are presented in Appendix 5.7.) At fixed mass,
increasing the entropy mostly shifts the centre to higher T or to lower P. The first case obtains for
low-entropy objects, which are essentially at zero temperature in the sense that kB T c EF , where
T c is the central temperature and EF is the Fermi energy level at the centre, taken to approximate
the electron chemical potential. Increasing the entropy partially lifts the degeneracy since the degeneracy parameter y ≈ kB T c /EF ∝ T c /%c 2/3 is a monotonic function of S , and Pc remains constant
2
We do make the standard assumption that the interior is fully convective, even though stabilising compositional gradients have been suggested to possibly shut off large-scale convective motions (Stevenson, 1979; Leconte & Chabrier,
2012, 2013).
3
Entropy values in this work are given in the usual units, written explicitly or not, of multiples of Boltzmann’s constant
kB per baryon (i.e. per mass of hydrogen atom). For comparison, 4.5-Gyr-old Jupiter has a current entropy of
7 kB /baryon (Marley et al., 2007, but see Appendix 5.8).
~ 62 ~
5.2. COOLING MODELS WITH ARBITRARY INITIAL ENTROPY
because of the constant radius. At entropies higher than a turn-over value of ' 10.4 kB /baryon, the
central temperature does not increase (and even decreases) with entropy. As pointed out in Paxton
et al. (2013), this entropy value is given by kB T c ∼ EF – we find that y ' 0.15 for S = 10.4 – and is
thus independent of mass. As for Pc , it decreases because the radius increases. These behaviours
also hold at higher masses not shown in the figure.
As for the 1-MJ planet with S = 9 shown in Fig. 5.1, some models with entropy S ' 8–9.5 show
a second, ‘detached’ convective zone at lower pressures, which follows from a re-increase of the
Rosseland mean opacity (see section 3.1 of Burrows et al., 1997). This second convection zone,
which is at most at an entropy 0.2 kB /baryon higher than the convective core, will not affect the
evolution of the object since the radiative thermal time-scale is much shorter than the cooling time
τS throughout the atmosphere. This holds in particular at the inner radiative-convective boundary
(RCB); for example, the 1-MJ , S = 10 model has τS = 14 Myr and ttherm = 0.2 Myr at its RCB.
Planets with a second convective zone can equivalently be thought of as having a radiative shell
interrupting their convective zone, as originally predicted for Jupiter’s adiabat by Guillot et al.
(1994) and based on too low opacities (see e.g. the brief review in Freedman et al., 2008).
Finally, we note that the higher-entropy objects (S & 11.5, with some dependence on mass) are
convective from the centre all the way to the photosphere.
5.2.3. Luminosity as a function of mass and entropy
To provide a model-independent way of thinking about an object’s brightness and thus to facilitate
comparison with other models, we show in Fig. 5.2 the luminosity of the planet as a function of its
internal entropy. We focus on objects without significant deuterium burning. Two main regimes are
apparent: at lower entropies, the scaling with mass is roughly L ∼ M, while at high entropies, the
luminosity becomes almost insensitive to mass. Looking more closely, the high-luminosity regime
is described by L ∝ M 0.3 (for M & 1 MJ ), and the brief steepening of the luminosity slope with
respect to entropy between L ' 10−6 and 10−4 L (at S ∼ 8.2–9, depending on mass) marks a
transition from L ∝ M 1 to L ∝ M 0.7 at fixed intermediate and low entropy respectively.
To try to understand these luminosity scalings, we firstly note that the radiative luminosity at
the radiative-convective boundary, which is equal to the total luminosity, can always be written as
(Arras & Bildsten, 2006)
!
4 ∇
3
16πGac MT RCB
adRCB
2 4acT dT
L = 4πr
=
,
(5.2)
3κρ dr RCB
3
κRCB PRCB
approximating the convection zone to contain the whole mass (cf. the 99-per-cent mass labels in
Fig. 5.1) and radius. Thus, one way of obtaining L(M, S ) is to express the four quantities PRCB ,
T RCB , ∇adRCB and κRCB in terms of M and S . This will now be done for the three regimes in turn,
starting with low entropies.
Low-entropy regime: L ∝ M 0.7 flow (S )
To begin, consider the entropy dependence of luminosity at a fixed mass when S . 9–8 (at 1–10 MJ
respectively). Fig. 5.1 reveals that the opacity at the radiative-convective boundary κRCB remains
constant at a given mass, which provides a first relation. Secondly, over the temperature and pressure range of interest in this regime, the hydrogen and helium remain respectively molecular and
neutral. This implies that ∇adRCB is constant and that the entropy has a simple functional form in
the ideal-gas approximation, given by the Sackur–Tetrode expression (e.g. Callen, 1985). For an
H2 –He i mixture with Y = 41 , this is
S = 9.6 +
45
7
ln(T/1600 K) −
ln(P/3 bar)
32
16
(5.3)
(hence ∇ad = 0.31), where S is the entropy per baryon in multiples of kB , ln 10 = 2.3, and the
reference T and P values were chosen for Section 5.2.3. Finally, T RCB is approximately fitted by
~ 63 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
10 MJ
3 MJ
1 MJ
Temperature (K)
105
0 % Mtot
10
4
S = 12
99 % Mtot
S = 7.5
103
102
10−2
100
102
104
106
Pressure (bar)
108
1010
3000
Temperature (K)
2000
1000
500
10 MJ
3 MJ
1 MJ
200
0.01
0.10
1.00
Pressure (bar)
10.00
100.00
Fig. 5.1.: Top panel: Temperature–pressure profiles of core-less planets with entropy S = 7.5–12 kB /baryon
(bottom to top), in steps of 0.5 kB /baryon, and M = 1, 3, and 10 MJ (full, dashed, and dotted lines, respectively). Squares indicate the centre and triangles mark where the enclosed mass is 99 per cent of the total
mass for each model. Curves end at the photosphere in the Eddington approximation, and the thick part(s)
of each curve show the radiative zone(s), while the thin part(s) of the curve are convective; the radiativeconvective boundary relevant for the thermal evolution is highlighted by a ring. Thin grey lines at low P
and T indicate contours of constant Rosseland mean opacity (Freedman et al., 2008), with log κ = −3.5 to
−1.0 (cm2 g−1 ) in steps of 0.5 dex, increasing with P. See text for a mention of the dominant sources of
opacity in some parts of the diagram. Bottom panel: Same as top panel but focussing on the atmospheres.
The three diagonal dashed lines mark log R ≡ log %/T 6 3 = 2, 4, and 6 (g cm−3 K−3 ), from top to bottom.
~ 64 ~
5.2. COOLING MODELS WITH ARBITRARY INITIAL ENTROPY
Luminosity (LSun)
10−3
10−4
10 MJ
5 MJ
3 MJ
1 MJ
10−5
L = M × f(S)
10−6
10−7
10−8
7
8
9
10
Entropy (kB/baryon)
11
12
Fig. 5.2.: Luminosity as a function of entropy for our planet models. The masses shown are (bottom to top)
M = 1, 3, 5, and 10 MJ .The two dashed lines approximately indicate the mass-dependent entropy boundaries
between the low-, intermediate- and high-entropy regimes of the luminosity L = L(M, S ). At intermediate
entropies, the atmosphere reaching the radiative-zero solution leads to the simple result that L/M depends
only on S (see text for details). Here as throughout, LSun = 3.86 × 1033 erg s−1 .
T RCB ∝ 100.19S . Combining the four relations (constant ∇adRCB and the three non-trivial ones) with
equation (5.2) yields L ∝ 101.04S . This is quite close to a power-law fit of Fig. 5.2, which gives
L ∝ 101.3S .
Next, consider the position of the RCB at fixed low entropy for different masses. Fixed entropy
immediately implies T ∝ P∇ad , and the constant ∇ad = 0.31 gives the second relation. The profiles of
Fig. 5.1 indicate that in this low-entropy regime (for T ' 200–400 K, P ' 0.01–10 bar), the opacity
depends approximately only on the pressure, with κ ∝ P0.9 . Finally, we find that PRCB ∝ M 0.5 , thus
providing the fourth relation. Combining these and using equation (5.2) gives L ∝ M 0.8 , in good
agreement with a direct fit, which gives L ∝ M 0.7 .
Thus, a constant adiabatic gradient, an RCB opacity dependent only on the mass, and a few
power laws suffice to show that in the low-entropy regime
Llow = 10
−7.7
L
M
MJ
!0.7
101.3(S −7.5) ,
(5.4)
which defines flow (S ) up to a constant. For Jupiter’s adiabat with T = 165–170 K at 1 bar (Saumon
& Guillot, 2004) and thus an entropy of 6.71–6.75 kB /baryon, this predicts ' 1.9 × 10−9 L , in
good agreement with the current value LJ = 8.7 × 10−10 L .
Intermediate-entropy regime: L ∝ M frz (S )
We now look at intermediate entropies, which are in the range 9–10.2 at 1 MJ to 8.2–9.6 kB /baryon
at 10 MJ . (More concisely, this corresponds to planets with L ∼ 10−6 –10−4 L .) As explained by
Arras & Bildsten (2006), equation (5.2) is of the form L = M f (S ), where f (S ) is a function of the
entropy in the convection zone, only if the quantity T 4 ∇ad /κP at the RCB is a function of a unique
variable, S . The intermediate-S behaviour can then be understood by noticing in Fig. 5.1 that
~ 65 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
objects at those entropies have a T RCB independent of internal entropy and an extended atmosphere
(interrupted or not by a second convection zone, i.e. including the deeper radiative window when
present). For these planets, the photosphere is sufficiently far from the RCB (PRCB & 10 Pphot ,
T RCB 4 & 5 T eff 4 ) that the atmosphere merges on to the radiative-zero profile, which solves (Cox &
Giuli, 1968)
dT
3 L κ
=
,
(5.5)
dP 16acG M T 3
where the boundary condition is by definition of no consequence. The solution is a T (P) relation,
which yields the radiative gradient ∇rad = d ln T/d ln P along it. The important point is that both the
atmosphere profile and its gradient depend only on the quantity L/M. Now, choosing an internal
entropy fixes the adiabat, i.e. sets a second T (P) relation. We require that at the intersection of the
atmosphere and the adiabat ∇rad be equal to ∇ad , which is the slope of the chosen adiabat. This thus
pins down ∇rad in the atmosphere and hence L/M (and the atmosphere profile itself). Therefore,
there is a unique L/M associated with an S , which means that L must be of the form4 L/M = f (S ).
Since we determine convective instability through the Schwarzschild criterion ∇ad < ∇rad , where
∇ad is relatively constant and the radiative gradient is given by
∇rad =
3LP
κ
16πacGMT 4
(5.6)
near the surface, a slow inward increase of κ will ensure a deep RCB. Consequently, the L ∝ M f (S )
scaling will hold for those M and S such that, starting at the photosphere with Pphot ' f (M) and
T eff ' f (Pphot , S ), the opacity increases only slowly along the adiabats. (For a power-law opacity
κ = κ0 Pn T −n−s , this means n/(n + s) ' ∇ad .) This is indeed the case for intermediate-S models
in Fig. 5.1, where the T (P) profiles are nearly aligned with contour lines of constant opacity. For
this argument to hold, the radius must be rather independent of the mass (cf. Zapolsky & Salpeter,
1969) and of the entropy. Also, one needs to approximate the entropy at the photosphere to be that
of the interior, which is reasonable: even in the extended atmospheres, the entropy increases by at
most ' 0.5 kB /baryon over our grid of models.
The functional form of f (S ) can be obtained by fixing T RCB and using equation (5.3) for the
entropy of an ideal gas. The reference temperature of 1600 K was chosen based on the radiativeconvective boundaries of Fig. 5.1. For convenience, S (3 bar, 1600 K) was computed using the interpolated SCvH tables. These include the contribution from the ideal entropy of mixing5 , a remarkably constant S mix = 0.18 kB /baryon in a large region away from H2 dissociation. Combining this
S (P, T ) with equation (5.2), fixing T = 1600 K, and taking κ(P, 1600 K) ' 0.0104(P/3 bar)0.2 cm2 g−1
from the interpolated table yields
log10
Lrz /L
= −5.05 + 1.51(S − 9.6),
M/MJ
(5.7)
i.e. L = M frz (S ) with frz (S ) = 3.4×1028 erg s−1 MJ −1 101.51(S −9.6) . (Thus, if κ ∝ Pn and dS /d ln P =
−b, log10 frz (S ) ∝ (n + 1)/b ln 10.) The subscript ‘rz’ highlights that the solution applies when the
radiative-zero solution is reached. This fits excellently (being mainly only 0.1 dex too high in log L)
the L/M–S relationship found in our grid of models at intermediate entropies.
High-entropy regime: L ∝ M 0.3 fhigh (S )
For S & 10–9 (at 3–10 MJ , respectively), the luminosity becomes almost independent of mass at a
given entropy. This indicates that the radiative solution does not hold anymore, and indeed Fig. 5.1
shows that planets with high entropy have atmospheres extending only over a small pressure range,
with the more massive objects fully convective from the centre to the photosphere. The shortness
4
This regime was obtained by Arras & Bildsten (2006) when looking at irradiated planets. This can be roughly thought
to fix T RCB to the irradiation temperature, so that T 4 ∇ad /κP at the RCB is automatically a function of only one
thermodynamic variable, for instance S , for all entropies.
5
We use the corrected version of the equations in SCvH; see the appendix of Saumon & Marley (2008).
~ 66 ~
5.2. COOLING MODELS WITH ARBITRARY INITIAL ENTROPY
of the atmosphere is due to the opacity’s rapid increase inward, as constant-κ contours are almost
perpendicular to T (P) profiles in that region. As in Section 5.2.3, we look at the behaviour of T , P,
κ, and ∇ad at the RCB as a function of M and S .
Fig. 5.1 shows that at a fixed mass, T RCB is almost independent of the (high) entropy, with the
actual scaling closer to T RCB 4 ∝ 100.2S . Also, κRCB is again mostly independent of S , as for the
low entropies. At high entropies, ∇adRCB drops continuously with increasing S (decreasing PRCB ),
with a very rough ∇ad ∝ 10−0.13S . For the fourth relation, we can fit PRCB ∝ 10−0.6S . Combining
all this with equation (5.2) then gives L ∝ 100.7S , which is quite close to a fit L ∝ 100.6S .
At fixed high entropy, ∇ad is somewhat constant at 0.25–0.13 for S = 10.5–12. Also, above
' 1500 K and at P < 1 bar, the opacity is approximately independent of pressure, scaling only with
temperature as κ ∼ T 4 . Finally, as in the low-entropy regime, PRCB ∝ M 0.5 ; combining the four
relations, we should have L ∝ M a with a ' 0.5–0.4. This is not far from the direct fit L ∝ M 0.3 .
Therefore, the luminosity in the high-entropy regime is
Lhigh = 10
−3.88
L
M
MJ
!0.3
100.6(S −10.5) ,
(5.8)
which defines fhigh (S ) up to a multiplicative constant. Fig. 5.2 indicates that the luminosity at the
lower masses (M . 2 MJ ) depends more steeply on S (' 100.85 S at 1 MJ ), but this was ignored
when obtaining equation (5.8).
Before summarising, let us briefly digress about the PRCB ∝ M 0.5 scaling seen both at fixed
low entropy and at fixed high entropy. In both cases, the model grid shows that, as a reasonable
approximation, PRCB ∝ Pphot . At low entropies, R is constant, such that PRCB ∝ Pphot = 2g/3κ ∝
M/κ. Then, the opacity’s scaling of κ ∼ P (see Section 5.2.3; the exponent is actually closer
to ' 0.95) immediately implies roughly PRCB ∝ M 0.5 . At high entropies, planetary radii are
significantly larger and vary, such that the radius dependence of the photospheric pressure should
not be neglected; thus Pphot ∝ M/R2 κ. Fitting the R(M, fixed high S ) relation in our models, we
find R ∼ M −1.1 or R ∼ M −1.2 , depending on the entropy. Then, ∇ad ' 0.2 and κ ∼ T 4 yield κ ∼ P0.8
at fixed S , and thus, isolating, PRCB ∼ M 0.6 or PRCB ∼ M 0.7 . This is a slightly stronger dependence
on mass than what is found in the grid, but the argument shows how the rough scaling can be
derived.
Summary of luminosity scalings
In summary, we found from fitting the L(M, S ) relation across our planet models that L ∝ M a 10λS ,
with a = (0.7, 1, 0.3) and λ = (1.3, 1.5, 0.6) at low (S . 9–8, for 1–10 MJ respectively), intermediate (S . 10.2–9.6), or high entropy, respectively, from considering the behaviour of the different
factors in equation (5.2). Fitting directly the relations in the grid gives very similarly
Llow = 1.5 × 10−7 L M̃ 0.72 101.3 (S −8.2) ,
Lrz = 7.2 × 10
−6
L M̃
0.98
1.58 (S −9.6)
10
(5.9a)
,
Lhigh = 8.7 × 10−5 L M̃ 0.29 100.58 (S −10.2) ,
(5.9b)
(5.9c)
with M̃ ≡ M/MJ , to ' 0.01 dex in log L except for M . 2 MJ at high entropies.
Our approximate understanding of the different regimes is the following. For the conditions
found in the atmospheres of intermediate-entropy planets, contours of constant opacity are almost
parallel to adiabats, which is equivalent to saying that κ is almost constant along lines of constant
R (see Fig. 5.1), where log R ≡ log %/T 6 3 and T 6 = T/106 K. Since ∇rad ∝ κ by equation (5.6) and,
for a constant ∇ad , only ∇rad determines when the atmosphere becomes convective, a slow inward
increase of the opacity causes the radiative zone to extend over a large pressure range. This in turn
means that the atmosphere can reach the radiative-zero solution, which we have shown necessarily
implies L = M f (S ). For the conditions found in the atmospheres of planets with high and low
entropy, however, opacity increases relatively quickly along an adiabat. Since the T (P) slope in the
~ 67 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
atmosphere is not too different from that of the convective zone’s adiabat, this means that opacity
increases rapidly in the atmosphere, which therefore cannot join on to the radiative-zero solution
before becoming convective. It is interesting to note that the transition from low to intermediate
entropies is accompanied by a ‘second-order’ (i.e. relatively small) change in a and λ, while the
physical explanation changes to ‘zeroth order’.
The different L(M, S ) scalings then reflect in part the approximate temperature- and pressureindependence of the opacity at low and high S , respectively, and the fact that opacity increases
relatively little along adiabats at intermediate S . While developing even a rough analytical understanding of the various opacity scalings would be interesting but outside the scope of this work,
we briefly indicate the major contributors in the high- and intermediate-S regimes. (The following temperatures should all be understood as somewhat approximate; cf. Fig. 5.1). Moving from
2800 K (or 3000 K at higher P), above which continuum sources dominate, down to 1300 K, the
decrease in opacity is due to the settling of the rovibrational levels of H2 O. Similarly, the settling of
the rovibrational levels of CH4 dominates from 800 K down to 480 K, with H2 O also contributing.
Around 2000 K, where opacity is nearly independent of pressure across a wide pressure range, H2 O
dominates the opacity, with its abundance remaining rather constant. In the intermediate-entropy
range, from 1300 or 1600 K to 800 K, H2 dominates the composition but it is the appearance of CH4
which is crucial for the opacity. The same qualitative behaviours can be found in the data of Ferguson et al. (2005) (J. Ferguson 2013, priv. comm., who also provided the information just presented).
Ferguson et al. (2005) do not include the powerful alkali Na, K, Cs, Rd, and Li as Freedman et al.
(2008) do, which can raise the opacity by some 0.2 dex near the RCB of intermediate-entropy
planets.
5.2.4. Luminosity as a function of helium fraction
The standard grid used for analyses in this work uses a helium mass fraction Y = 0.25 but results
can be easily scaled to a different Y. Following a suggestion by D. Saumon (2013, priv. comm.),
one can write
!
!
dL
∂L
d ln S ≈
×
,
(5.10)
dY M,S
∂ ln S M,Y
dY (P0 ,T0 )
where (P0 , T 0 ) is some appropriate location, and compute γS ≡ dS /dY = (S H − S He ) + dS mix /dY,
where S = (1−Y)S H +YS He +S mix , S H , S He , S mix are respectively the total, hydrogen, helium, and
mixing entropies per baryon (SCvH). Given equation (5.2), one might heuristically expect (P0 , T 0 )
to be near the radiative–convective boundary (RCB), at least when determining dS /dY at constant
L. Indeed, we find this to be the case, with γ ' −0.63 at the RCB of the Y = 0.25 or 0.30 models
for intermediate to high entropies. Thus, for Y 0 sufficiently close to 0.25,
X(M, S 0 , Y 0 ) = X(M, S , Y)
(5.11)
S 0 = S × 1 − 0.63 Y 0 − 0.25 ,
(5.12)
if
where X is a planet property such as L or τS . For Y = 0.27, this is at worse accurate only to 0.05 dex
in luminosity towards high entropies for masses below ' 12 MJ when S . 8. This rescaling is also
adequate (in the same domain) for X = R, to 15 per cent towards high entropies, but a more accurate
fit can be obtained with γ ' −0.3 ± 0.1. Interestingly, nowhere within a planet structure in the grid
does |γ| drop below 0.55; the explanation in this case (in analogy to γ being evaluated at the RCB
for L) is not clear.
Since at the RCBs in the grid the hydrogen is molecular, one might try to obtain γ analytically
with the Sackur–Tetrode formula. Neglecting the subdominant contribution from the rotational
degrees of freedom of H2 yields S H /S He ≈ µHe /µH2 = 2, where µX is the molecular mass of X.
This leads to γ = −0.57, which is not far from the accurate result −0.63 yet shows that the dS mix /dY
term cannot be neglected.
~ 68 ~
5.2. COOLING MODELS WITH ARBITRARY INITIAL ENTROPY
Equation (5.12) makes it simple to convert e.g. constraints on the initial entropy based on cooling
curves with a particular Y to another set with a different Y, regardless of the approach used for the
cooling. The typically small change in the entropy (' 3 per cent) is nevertheless significant because
of the strong dependence of L on S .
5.2.5. Cooling
One can also derive the functional form of the cooling tracks with a few simple arguments. With
L ∝ M a 10λS , equation (5.1) implies dS /dt = −M a−1 f (S )/T . Since hydrostatic balance yields
Pc ∼ GM 2 /R4 and the convective core is adiabatic, we expect T ' T c ∝ (M 2 /R4 )∇ad . (Across
the whole grid, T /T c = 0.55–0.63.) Therefore, ignoring the radius dependence (since 4∇ad is not
large and R is rather constant at lower entropies), −dS / f (S ) ∝ M −2∇ad −1+a dt. This means that
the entropy of a cooling planet should be approximately a function of t/M 2∇ad +1−a , and that more
massive planets cool more slowly since a is always smaller than 2∇ad + 1 ' 5/3. In fact, using that
f (S ) = 10λS , one can compute the integral to obtain an analytic expression for the cooling tracks:
1
1
1
=
+
,
L(t) Li Lhs (t)
Lhs (t) =
M 2∇ad +1
,
βt
(5.13)
where Li and Lhs are the initial and hot-start luminosities, respectively, and β = Cλ ln 10 with C a
dimensional constant grouping prefactors. We have assumed that a, λ, and ∇ad do not change as the
planet cools. Thus, L ∝ M a at fixed entropy, with a = 0.3–1, but at a fixed time the luminosity has
a steeper dependence on M, L ∝ M 2∇ad +1 ' L ∝ M 2 for intermediate entropies. A more detailed
analytic understanding of the cooling curves for irradiated planets along these lines was developed
by Arras & Bildsten (2006), and a careful, approximate but surprisingly accurate analysis for brown
dwarfs may be found in Burrows & Liebert (1993).
Equation (5.13) approximately describes cooling tracks with arbitrary initial entropy, with our
cooling curves well fitted (to roughly 0.1 dex) below 10−5 L by C = 3 × 10−2 cgs when λ = 1.5
and ∇ad = 1/3. At higher luminosities, in particular for hot starts, a better fit (with the restriction of
L & 10−6 L if M . 2.5 MJ ) which captures the average shape and spacing of the cooling curves is
provided directly by the classical result of Burrows & Liebert (1993), who find
Lhs = 7.85 × 10−6 L
(M/3 MJ )2.641
t/10 Myr 1.297
(5.14)
(for κ = 0.01 cm2 g−1 ), i.e. a somewhat different mass and time dependence.
5.2.6. Comparison with classical hot starts and other work
We now compare our cooling curves to classical hot starts. Fig. 5.3 shows cooling curves for
large initial entropy compared to the hot-start models of Marley et al. (2007), the COND03 models
(Baraffe et al., 2003) and those of Burrows et al. (1997), all of which use non-grey atmospheres
with detailed opacities. The agreement is excellent, with our luminosities within the first 3 Gyr
approximately within −30 and 20 per cent and −30 and 60 per cent above those of Baraffe et al.
and Burrows et al.. (Of interest for the example of Section 5.4, our radii along the cooling sequence
are at most approximately two to five per cent greater at a given time. This difference is comparable
to the effect of neglecting heavy elements in the equation of state (EOS) or not including a solid
core (Saumon et al., 1996) and not significant for our purposes. See also Appendix 5.7.) Our
deuterium-burning phase at 20 MJ ends slightly earlier than in Burrows et al. (1997) but this might
be due to our simplified treatment of the screening factor.
Fig. 5.4 shows cooling curves for lower initial entropies than in Fig. 5.3. The cooling curves
show the behaviour found by M07 and Spiegel & Burrows (2012) in which the luminosity initially varies very slowly, with the cooling time at the initial entropy much larger than the age of
the planet. Eventually, the cooling curve joins the hot start cooling curve once the cooling time
becomes comparable to the age.
~ 69 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
10−2
20 MJ
Luminosity (LSun)
10−3
10−4
10−5
1 MJ
10−6
This work
Marley et al. (2007)
Baraffe et al. (2003)
Burrows et al. (1997)
−7
10
10−8
106
107
108
Time (yr)
109
Fig. 5.3.: Cooling curves for (bottom to top) M = 1, 2, 3, 5, 8, 10, 15, and 20 MJ (solid curves) compared
with the Marley et al. (2007) hot starts (for M = 1, 2, and 10 MJ only; dash-dotted curves), the COND03
tracks (dots) and the Burrows et al. (1997) models (dashed curves). The COND03 data is shown as dots
because of insufficient sampling of the cooling tracks at higher masses.
Comparing to the cold starts in fig. 4 of M07, our models are a factor of ' 3.5–3.9 lower in
luminosity for the same initial entropies, as given in their fig. 2. Increasing our initial entropies by
0.38 kB /baryon brings our cooling curves into agreement with theirs when the planet has not yet
started cooling. (For this comparison, we do not correct the time offset for the higher masses (see
figs. 2 and 4 of M07), which are already on hot-start cooling curves at the earliest times shown.) As
mentioned in Appendix 5.8, this implies a real difference between our L(S ) of only 0.14 kB /baryon.
Spiegel & Burrows (2012) computed the evolution of gas giants starting with a wide range of
initial entropies. To compute the bolometric luminosity of their models, we take the published
spectra and integrate the flux in the wavelength range given, 0.8–15 µm. The bottom panel of
Fig. 5.4 shows the comparison to all four model types, with or without clouds and at solar or three
times solar metallicity. Increasing our entropy by 0.45 kB /baryon – e.g. comparing the Spiegel
& Burrows (2012) model with S = 9.0 to ours with S = 9.45 – yields very good agreement,
with our luminosities overlying their curves or within the spread due to the different atmospheres.
As discussed in Appendix 5.8, this is mainly due to a constant entropy offset of 0.52 kB /baryon
between the tables used by the Burrows et al. group and the published SCvH tables used in the
present work, leaving a net offset of merely 0.07 kB /baryon.
The apparent disagreement with Spiegel & Burrows (2012) at late times comes from the increasing fraction of the flux in the Rayleigh–Jeans tail of the spectrum beyond 15 µm. For comparison,
the implied required bolometric correction is equal to 10 to 50 per cent of the flux in 0.8–15 µm
for a blackbody with T eff ' 700–300 K. From this and the hot-start T eff tracks shown in Spiegel
& Burrows (2012), we estimate that integrating the spectrum should give a reasonable estimate
(to ca. 30 per cent) of the bolometric luminosity only up to ' 50, 200, and 1000 Myr for objects
with M = 1, 3, and 10 MJ respectively. This is indeed seen in Fig. 5.4. Bolometric luminosities
kindly provided by the authors (D. Spiegel 2013, priv. comm.), which are rather insensitive to the
atmosphere type, are also shown for a more direct comparison and confirm the reasonable match
of our cooling curves with those of Spiegel & Burrows (2012).
~ 70 ~
5.2. COOLING MODELS WITH ARBITRARY INITIAL ENTROPY
Luminosity (LSun)
10−5
This work; ∆Sthis − M07 = 0.38
MESA; ∆Sthis − MESA = −0.09
Marley et al. (2007)
10−6
1 MJ
10−7
107
2 MJ
4 MJ
108
Time (yr)
10 MJ
109
10−2
Luminosity (LSun)
10−3
= 11
SSB12
i
This work; ∆Sthis − SB12 = 0.45
Spiegel & Burrows (2012), int.
D. Spiegel (2013, priv. comm.)
10−4
10−5
10−6
10 MJ
SB12
i
S
= 9.0
10−7
3 MJ
1 MJ
10−8
106
107
108
Time (yr)
109
Fig. 5.4.: Top panel: Cooling curves for M = 1, 2, 4, 10 MJ and initial entropies equal to the Marley et al.
(2007, M07) ‘tuning-fork diagram’ (their fig. 2) values S i = 9.23, 9.00, 8.60, 8.23. The initial entropies for
our cooling tracks were increased by 0.38 kB /baryon to compare with Marley et al. (2007), with those of
mesa further increased by 0.09 kB /baryon. Bottom panel: Cooling curves for M = 1, 3, 10 MJ (bottom to
top) with initial entropies S i = 9.0 and S i = 11 increased by 0.45 kB /baryon mainly for ‘thermodynamic
reasons’ (see text; solid lines), compared to those of Spiegel & Burrows (2012, SB12) with S i = 9.0 and
S i = 11 (dashed and dotted lines). For each mass and initial entropy, the four dotted lines (some overlap)
correspond to the integrated spectra for the four atmosphere types in no particular order, serving only to
indicate a range. The red-side truncation of the Spiegel & Burrows (2012) spectra causes an underestimation
of the bolometric luminosity at late times (see text). The dashed line at each mass and initial entropy shows
the bolometric luminosity directly from the models.
~ 71 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
We have also computed cooling curves with the mesa stellar evolution code (Paxton et al., 2011,
2013, revision 4723), and they are in excellent agreement with our results. We compared our
L(S ) relation to ones obtained from mesa, (also with Y = 0.25) at different masses and found
that they are very nearly the same, with an entropy offset ∆S . 0.1. Moreover, Fig. 5.4 shows
that the agreement of the time evolution is quite good, with in particular the late-time ‘bumps’
due to opacity when the cooling curves enter the intermediate-entropy regime (cf. Figs. 5.1, 5.2,
and 5.6). We also produced grids with other opacities, using the mesa tables with the default
Y = 0.28 and Z = 0.02 (for the opacity calculation only), and the Freedman et al. (2008) tables
with [M/H] = ±0.3 dex, and found that this changed the luminosity by at most ' 10 per cent at a
given mass and internal entropy. Similarly, small differences were found to result from a changed
helium mass fraction in the bulk of the planet at a given mass and entropy per nucleus.
The upshot of these comparisons to classical, non-grey-atmosphere hot and cold starts is that
cooling tracks computed with the simple and numerically swift cooling approach described above
can reproduce models which explicitly calculate the time dependence of the luminosity. When
comparing models from different groups, one should keep in mind that there can be a systematic
offset in the entropy values of ∆S syst = 0.52 (at Y = 0.25) due to different versions of the SCvH
EOS, which however has no physical consequence for the cooling. Moreover, the remaining difference |∆S | . 0.15 is small compared to the entropy range between hot and cold starts.
5.3. General constraints from luminosity measurements
Masses of directly-detected exoplanets are usually inferred by fitting hot-start cooling curves (e.g.
Burrows et al., 1997; Baraffe et al., 2003) to the measured luminosity of the planet, using the
stellar age as the cooling time. Since the hot-start luminosity at a given age is a function only of
the planet mass, the measured luminosity determines the planet mass. Equation (5.14) provides a
quick estimate of this ‘hot-start mass’:
Mhs = 3 MJ
L
7.85 × 10−6 L
!0.379
t
10 Myr
!0.491
.
(5.15)
Moreover, a planet’s luminosity, at a given time and for a given mass, can never exceed that of the
hot starts, since a larger initial entropy would have merely cooled on to the hot-start cooling track
at an earlier age.
However, we have seen above that the luminosity at a given mass can be lowered by considering
a sufficiently smaller initial entropy, which might be the outcome of more realistic formation scenarios (Marley et al., 2007; Spiegel & Burrows, 2012). With the fact that luminosity increases with
planet mass at a given entropy, this simple statement has important consequences for the interpretation of direct-detection measurements, namely that there is not a unique mass which has a given
luminosity at a given age. Cold-start solutions correspond to planets not having forgotten their initial conditions, specifically their initial entropy S i , and every different initial entropy is associated
to a different mass. In other words, a point in (t, L) space – a single brightness measurement – is
mapped to a curve in (M, S i ) space. Since Marley et al. (2007) (but see also Baraffe et al., 2002;
Fortney et al., 2005), it is generally recognised that direct detections should not be interpreted to
yield a unique mass solution, but, with the exception of Bonnefoy et al. (2013), who used infrared
photometry, this is the first time that this degeneracy is calculated explicitly.
5.3.1. Shape of the M–Si constraints
The top panel of Fig. 5.5 shows the allowed masses and initial entropies for different values of
luminosity Lbol /L = 10−7 , 10−6 and 10−5 , at ages of 10, 30, and 100 Myr. Below the deuteriumburning mass, constant-luminosity curves in the M–S i plane have two regimes. At high initial
entropies, the derived mass is the hot-start mass independent of S i since all S i greater than a certain value have cooling times shorter than the age of the system. There, uncertainty in the stellar
~ 72 ~
5.3. GENERAL CONSTRAINTS FROM LUMINOSITY MEASUREMENTS
age translates directly into uncertainty in the planet mass: since L ∝ 1/t and L ∼ M 2 , the mass
uncertainty is ∆M/M ' 21 ∆t/t. At lower entropies, the luminosity measurement occurs during the
early, almost constant-luminosity evolution phase. Given a luminosity match in this region, one
can obtain another by assuming a lower (higher) initial entropy and compensating by increasing
(decreasing) the mass. As seen in Section 5.2, L is a very sensitive function of S at low and intermediate entropies, so that a small decrease in initial entropy must be compensated by a large
increase in mass to yield the same luminosity at a given time; this yields the approximately flat portion of the curves. As long as the cooling time for a range of masses and entropies remains shorter
than the age, the entropy constraints do not significantly depend on the age. The uncertainty in
the initial entropy is ∆S i ' 1/λ ∆ log10 Lbol , where 1/λ ' 0.7 or 1.7 at low or high entropy (see
Section 5.2.3).
The circles in the top panel of Fig. 5.5 show the initial entropies for cold- and hot-start models
from M07, the ‘tuning-fork diagram of exoplanets’. The entropies were increased by 0.38 kB /baryon
as in the top panel of Fig. 5.4 to match our models. Since the cooling time increases with mass
(see Section 5.2.5), heavier planets of the hot-start (upper) branch, i.e. with arbitrarily high initial
luminosity, have cooled less and are therefore at higher S . For their part, the cold-start entropies,
which are still the post-formation ones, lie close to a curve of constant luminosity L ' 2 × 10−6 L .
This reflects the fact that the post-formation luminosities in M07, as seen in their fig. 3, have similar
values for all masses. The cause of this ‘coincidence’ is presently not clear; it might be a physical
process or an artefact of the procedure used to form planets of different masses. Putting uncertainties in the precise values aside, the two prongs of the tuning fork in Fig. 5.5 give an approximate
bracket within which just-formed planets might be found.
Mass information for a directly-detected planet can put useful constraints on its initial entropy
and also potentially on its age and luminosity simultaneously. For instance, dynamical-stability
analyses and radial-velocity observations (see Section 5.5 and Appendix 5.9) typically provide
upper bounds on the masses. Since S i decreases monotonically with M at a fixed luminosity, this
translates into a lower limit on the initial entropy. This has the potential of excluding the coldeststart formation scenarios. Conversely, a lower limit on the planet’s mass implies an upper bound
on S i . If it is greater than the hot-start mass, this lower limit on the mass of a planet would be
very powerful, due to the verticalness of the hot-star branch. Combined with the flatness of the
‘cold branch’ of the M(S i ) curve, this could easily restrict the initial entropy to a dramatically
small ∆S i ' 0.5 ∼ ∆S syst . Also, the top panel of Fig. 5.5 shows that not all age and luminosity
combinations are consistent with a given mass upper limit. Given the often important uncertainties
in the age and the bolometric luminosity, this may represent a valuable input.
5.3.2. Solutions on the hot- vs. cold-start branch
The bottom panel of Fig. 5.5 shows lightcurves illustrating the two regimes of the M(S i ) curves
discussed above. The hot-start mass is 1.85 MJ , whereas the selected cold-start case (S i = 8.33)
has M = 12 MJ – i.e. a six times larger mass – and both reach log Lbol /L = −6 at t = 30 Myr,
with the cold-start values essentially independent of age. The hatched region around the 1.85-MJ
curve comes from hot-start solutions between 1.35 and 2.35 MJ and is within a factor of two of the
target luminosity, showing the moderate sensitivity of the cooling curves to the mass. However, in
the cold-start phase, a variation by a factor of two can also be obtained by varying at a fixed mass
the initial entropy from 8.23 to 8.43 kB /baryon. This great sensitivity implies that combining a
luminosity measurement with information on the mass would yield, if some of the hot-start masses
can be excluded, tight constraints on the initial entropy.
5.3.3. Definition of ‘hot-start mass’
By showing the entropy of hot starts as a function of time, Fig. 5.6 provides another way of looking
at Fig. 5.5. Given a mass obtained from hot-start cooling tracks, the entropy value at that time read
off from the curves indicates what ‘hot’ is, i.e. provides a lower bound on the initial entropy if the
~ 73 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
Initial entropy (kB/baryon)
12
11
10
10 Myr
100 Myr
9
8
7
0
10−4
Luminosity (LSun)
30 Myr
2
4
6
8
Mass (MJ)
10
12
1.85 ± 0.50 MJ
Si = 11.9
10−5
10−6
12 MJ
Si = 8.33 ± 0.10
10−7
106
107
Time (yr)
108
109
Fig. 5.5.: Top panel: Allowed values of planet mass M and initial entropy S i corresponding to Lbol = 10−7 ,
10−6 , and 10−5 L (bottom to top sets of curves) and ages t = 10, 30, and 100 Myr (left to right or dotted,
full and dashed lines). The circles show the results from Marley et al. (2007), increased by 0.38 kB /baryon
(cf. Fig. 5.4), for cold starts (filled circles) and hot starts (open circles), where the entropy is 1 Myr after
the onset of cooling. Red and blue squares indicate the illustrative M and S i used for the lightcurves in the
bottom panel, with errorbars corresponding to the hatched regions there. Bottom panel: Two examples of
cooling curves that have L = 10−6 L at 30 Myr. The dot-dashed (red) line has M = 1.85 MJ and S i = 11.9
(hot start), while the full (blue) one has M = 12 MJ and S i = 8.33 (cold start). Hashed regions indicate
a spread of 0.5 MJ (0.1 kB /baryon) for the hot- (cold-)start curve. The respective relative uncertainties are
∆M/M ' 21 ∆t/t and ∆S i ' 0.7∆ log10 Lbol for the cold- and hot-start cases, respectively (see text).
~ 74 ~
5.4. GENERAL CONSTRAINTS FROM GRAVITY AND EFFECTIVE TEMPERATURE
13
Entropy (kB/baryon)
12
10 MJ
11
3 MJ
1 MJ
10
9
8
7
105
106
107
108
Time (yr)
109
1010
Fig. 5.6.: Entropy of hot starts as a function of time for planet masses of 1, 3, and 10 MJ (bottom to top). At
a given age, the curve indicates the value of initial entropy above which the ‘hot-start mass’ applies. For a
planet mass larger than the hot-start value, the initial entropy must be lower than the hot-start entropy at the
current age.
hot-start mass is the true mass. If however the planet is more massive, this entropy value is instead
an upper bound on the post-formation entropy. As a rule of thumb, the entropy slope is ' −2 or
−1 kB /baryon per time decade at early or late times, approximately, with the break coming from
the change in the entropy regime (cf. Fig. 5.2).
5.4. General constraints from gravity and effective-temperature
measurements
Before applying the analysis described in the previous section to observed systems, it is worth discussing a second way by which constraints on the mass and initial entropy can be obtained. The
idea is to firstly derive an object’s effective temperature and surface gravity by fitting its photometry and spectra with atmosphere models. Integrating the best-fitting model spectrum gives the
luminosity, and this can be combined with T eff and log g to yield the radius and the mass. This procedure was described and carefully applied by Mohanty et al. (2004) and Mohanty, Jayawardhana
& Basri (2004). However, one can go further: considering models coupling detailed atmospheres
with interiors at an arbitrary entropy, the mass and radius translate into a mass and current entropy.
Then, using cooling tracks beginning with a range of initial conditions, the initial state of the object
can be deduced given the age. Thus, in contrast to the case when only luminosity is used, M and
S i can both be determined without any degeneracy between the two.
In practice, however, there seems to be too much uncertainty in atmosphere models for this
method to be currently reliable, as the work of Mohanty et al. shows. Their sample comprised a
dozen young (∼ 5 Myr) objects with T eff ' 2600–2900 K6 with high-resolution optical spectra.
6
The spectral types are ' M–M7.5, but Mohanty, Jayawardhana & Basri (2004) stress that the correspondence between
the spectral type and effective temperature of young objects has not yet been empirically established and thus that
calibration work (in continuation of theirs) remains to be done.
~ 75 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
The combined presence of a gravity-sensitive Na i doublet and effective-temperature-sensitive TiO
band near 0.8 µm allowed a relatively precise determination of log g and T eff for most objects, with
uncertainties of 0.25 dex and 150 K (Mohanty et al., 2004). However, there were significant offsets
in the gravity (' 0.5 dex) of the two coldest objects with respect to model predictions of Baraffe
et al. (1998) and Chabrier et al. (2000a). Mohanty et al. (2004) came to the conclusion that the
models’ treatment of deuterium burning, convection7 or accretion – i.e. the assumed initial conditions – are most likely responsible for this disagreement at lower T eff . Moreover, the more recent
work of Barman et al. (2011b) (see Section 5.9.1 below) indicates that ‘unexpected’ cloud thickness and non-equilibrium chemistry may compromise a straightforward intepretation of spectra in
terms of gravity and temperature for young, low-mass objects. (See also Moses (2014) for a review
of photochemistry and transport-induced quenching in cool exoplanet atmospheres.) Nevertheless,
with the hope that future observations will allow a reliable calibration of atmosphere models, we
illustrate with an example how M and S i can be determined for an object from its measured T eff
and log g.
Fig. 5.7 shows the constraints on the current mass and entropy of a planet with log g = 4.00 ±
0.25 (cm s−2 ) and T eff = 900±50 K, where the uncertainties in the gravity and effective temperature
are the possible accuracy reported by Mohanty et al. (2004) and thus correspond to an optimistic
scenario. The constraints are obtained by simultaneously solving for M, S , R, and L given T eff
and log g, with the L(M, S ) and R(M, S ) relations given by our model grid. The required mass
and current entropy are M = 6.5 MJ and S = 9.2, with the 1-σ ellipse within 4.0–10.1 MJ and
9.0–9.3 kB /baryon. The large uncertainty in the mass (the width of the ellipse) is dominated by
the uncertainty in the gravity, since radius is roughly constant at these entropies. We note that nonGaußian errorbars are trivial to propagate through when determining the mass and entropy in this
way since it is only a matter of mapping each (log g, T eff ) pair to an (M, S ) point.
Since this M–S determination concerns only the current state, it is independent of the cooling
sequence, in particular of the ‘hot vs. cold start’ issue. Nevertheless, with this approach, it is
immediately apparent what constraints the age imposes on the initial conditions. Indeed, not all
(M, S ) are consistent with an age since no planet of a given mass can be at a higher entropy than the
hot-start model at that time (see Fig. 5.6). The entropy of hot starts is shown in Fig. 5.7 after 10, 30,
and 50 Myr. The 30-Myr age excludes objects with M . 6 MJ , which is the hot-start mass of this
example. Considering only hot-start evolution sequences would have been equivalent to requiring
the solution to be on one of the hot-start (grey) curves. This is however a restriction which currently
could not be justified, given our ignorance about the outcome of the formation processes.
The solid line in Fig. 5.7 indicate the derived constraints on the initial entropy, assuming an age of
30 Myr. These constraints are similar to ones based on luminosity (see Fig. 5.5) but are somewhat
tighter since an upper mass limit is provided by the measurement of log g. Even within a set of
models, i.e. putting aside possible systematic issues with the atmospheres of young objects, it is
however often the case that the surface gravity is rather ill determined (as for the objects discussed
below in Section 5.5). In this case, provided a sufficiently large portion of the spectrum is covered,
we expect the approach based on the bolometric luminosity presented in this work to be more
robust than the derivation of constraints from effectively only the surface temperature. Indeed,
the former avoids compounding uncertainties in T eff with those in the radius in an evolutionary
sequence, which can be further affected by the presence of a core (of unknown mass). With both
sustained modelling efforts and the detailed characterisation of an increasing number of detections,
one may hope that reliable atmosphere models for young objects will become available in a near
future, allowing accurate determinations of the mass, radius, and initial entropy of directly-detected
exoplanets.
7
Qualitatively, their finding that theoretical tracks predict too quick cooling at low masses might be evidence for the
argument of Leconte & Chabrier (2012) that convection in the interior of these objects could be less efficient than
usually assumed.
~ 76 ~
5.5. COMPARISON WITH OBSERVED OBJECTS
Entropy (kB/baryon)
12
11
10
Hot starts
9
8
7
0
Current entropy (1 σ)
Initial entropy (1 σ) if 30 Myr
2
4
6
Mass (MJ)
8
10
12
Fig. 5.7.: Constraints on the mass and entropy of an example planet from its gravity and effective temperature, with log g = 4.00 ± 0.25 (cm s−2 ) and T eff = 900 ± 50 K (and thus optimistic errorbars; see text).
The mass and current entropy corresponding to log g = 4.00 and T eff = 900 K are shown by a filled circle.
The narrow ellipse (dotted line) shows the 1-σ confidence region in the mass and current entropy, whereas
the deformed ellipse (solid line) shows the constraints on the initial entropy assuming an age of 30 Myr.
The open circle corresponds to the ‘measured’ central value. The three solid grey lines indicate the entropy
of hot starts at 20, 30, and 50 Myr (top to bottom). Higher masses can be seen to cool more slowly (see
Section 5.2.5).
5.5. Comparison with observed objects
5.5.1. Directly-detected objects
Neuhäuser & Schmidt (2012) recently compiled and homogenously analysed photometric and spectral data for directly-imaged objects and candidates, selecting only those for which a mass below
' 25 MJ is possible8 . They report luminosities and effective temperatures, which they either take
from the discovery articles or calculate, usually from bolometric corrections when a spectral type or
colour index is available or brightness difference with the primary when not. Neuhäuser & Schmidt
then use a number of hot-start cooling models to derive (hot-start) mass values along with errorbars,
while recognising that hot starts suffer from uncertainties at early ages.
In this section, we determine joint constraints on the masses and initial entropies of directlydetected objects, focusing on the ones for which (tentative) additional mass information is available. A proper statistical analysis of the set of M(S i ) curves would be challenging at this point
due to the inhomogeneity of the observational campaign designs. However, upcoming surveys
should produce sets of observations with well-understood and homogenous biases, convenient for
a statistical treatment.
Before turning to specific objects, we display in Fig. 5.8 the data collected and computed by
Neuhäuser & Schmidt (2012), as well as more recent detections, along with hot- and cold-start
cooling tracks for different masses. This is an update of the analogous figures of Marley et al.
8
This value was chosen by Schneider et al. (2011) as an approximation to the ‘brown-dwarf desert’, which is a gap in
the mass spectrum between ' 25–90 MJ (Marcy & Butler, 2000; Grether & Lineweaver, 2006; Luhman et al., 2007;
Dieterich et al., 2012).
~ 77 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
(2007) and Janson et al. (2011), which had only a handful of data points. Given doubts about
its nature (e.g. Janson et al., 2012; Currie et al., 2012; Kalas et al., 2013; Kenworthy et al., 2013;
Galicher et al., 2013; Currie et al., 2013), we do not include the reported upper limit for Fomalhaut b
in this plot. Since their uncertainties are large, the central age values are taken as the geometric
mean of the upper and lower bounds reported if no value is given. For RXJ1609 B/b, we use instead
11 ± 2 Myr (Pecaut, Mamajek & Bubar, 2012). The errorbars for the HR 8799 planets go up to
160 Myr and do not include the controversial Moya et al. (2010) asteroseismology9 measurement
of 1.1–1.6 Gyr since it is not used in our analysis (see discussion in Section 5.9.2). Finally, since
no luminosity was given for WD 0806-661 B/b, we crudely estimate from the hot-start mass of
6–9 MJ from Luhman et al. (2012) and the Spiegel & Burrows (2012) models a luminosity of
log L/L = −7.0 ± 0.3 at 1.5–2.5 Gyr.
Fig. 5.8 also includes four recent objects discovered since the analysis of Neuhäuser & Schmidt
(2012): 2M0122 b (Bowler et al., 2013), GJ 504 b (Kuzuhara et al., 2013), 2M0103 ABb (Delorme
et al., 2013), and κ And b (Carson et al., 2013; Bonnefoy et al., 2014a). For 2M0103 ABb, we
estimate a bolometric luminosity of log L/L = −4.87 ± 0.12 as done in Appendix 5.9.3 for β Pic b.
The same approach with κ And b yields log L/L = −3.83 ± 0.15, which is entirely consistent with
the published value of −3.76 ± 0.06 dex (Bonnefoy et al., 2014a). We show the conservative age
range of 30+120
−10 Myr for κ And b.
A recent detection which is not included in Fig. 5.8 is a candidate companion to HD 95086
with a hot-start mass near 4 MJ (Rameau et al., 2013; Meshkat et al., 2013), since only an L0 -band
measurement is available. Nevertheless, we report a prediction for its luminosity of log Lbol /L '
−4.8 ± 0.4 from the estimated T eff = 1000 ± 200 K and log g = 3.9 ± 0.5 (cm s−2 ), with the Lbol
errorbars entirely dominated by those on T eff and ignoring that the atmospheric parameters were in
fact estimated from hot-start models.
Fig. 5.8 shows that there are already many data points which – at least based solely on their
luminosity – could be explained by cold, warm, or hot starts, highlighting the importance of being
open-minded about the initial entropy when interpreting these observations. Indeed, as Mordasini
et al. (2012) carefully argue, it is presently not warranted to assume a unique mapping between
core accretion (CA) and cold starts on the one hand, and gravitational instability and hot starts
on the other hand. (Even in the case of a weak correlation, planets found beyond ' 50 AU, the
farthest location where CA should be possible (Rafikov, 2011), could still in principle have formed
by core accretion and then migrated outward (e.g. Ida et al., 2013).) As an extreme example of
a cold start, we also display a cooling curve for a deuterium-burning object with a low S i which
undergoes a ‘flash’ at late times, somewhat arbitrarily chosen to pass near the data point of Ross
458 C (Burgasser et al., 2010); this contrasts with a monotonic hot-start cooling track at ' 12 MJ
which would also match the data point. Such solutions will be explored in a forthcoming work
but we already note that, very recently, Bodenheimer et al. (2013) independently found lightcurves
with flashes to be a possible outcome of a realistic formation process.
There are two features of the data distribution in Fig. 5.8 which immediately stand out. The first
is that the faintest young objects are brighter than the faintest oldest objects, i.e. that the minimum
detected luminosity decreases with the age of the companion. Moreover, this minimum, with the
exception of data points (7), (31), and (30) (2M1207 b and WD 0806-661 B/b, which are particular
for different reasons, and GJ 504 b) approximately follows the cooling track of a hot-start planet of
' 10 MJ . The interpretation of this fact is not obvious given that the data points forming the lower
envelope come from multiple surveys and that different observational biases apply at different ages
(e.g. due to the relatively low number of young objects in the solar neighbourhood).
The second feature is the absence of detections between the hot-start 10- and 15-MJ cooling
curves, roughly between 20 and 100 Myr. More accurately, there is, in a given age bin in that
range, a lower density of data points with luminosities around 10−4 L than at higher or lower
luminosities. A proper assesment of the statistical significance of this ‘gap’ in the data points
9
The linguistically inclined reader will delight in the communication of Gough (1996) about the term’s prefix ‘ast(e)ro’.
~ 78 ~
5.5. COMPARISON WITH OBSERVED OBJECTS
1
10−2
6
21
12
10
−3
Hot starts
Si = 9.5
16 MJ, Si = 8.2
25 2 27
5 26
Luminosity (LSun)
23
20
10
−4
19
24
18
4
8
30
9
28 3
10
16
7
10−5
14 15
31
11
13
25 MJ
10
22
−6
17
29
10−7
1 MJ
106
107
108
Host−star age (yr)
32
109
1010
Fig. 5.8.: Directly-observed objects with a hot-start mass below ∼ 25 MJ compared with cooling curves
for M = 1, 2, 3, 5, 10, 15, 20, and 25 MJ (bottom to top), for S i = 9.5 (dashed) and hot starts (dash-dotted);
the 25-MJ curve is from Burrows et al. (1997) and has no cold-start equivalent. A lightcurve with a latetime deuterium ‘flash’ (at 400 Myr for M = 16 MJ , S i = 8.2) is also shown to draw attention that some
of these objects could, in principle, be undergoing rapid deuterium burning (Marleau & Cumming, in prep.,
but see Bodenheimer et al., 2013, who independently found such flashes). The data points are based on
Neuhäuser & Schmidt, 2012 (see text for details) and are, with in boldface those investigated more closely
below: (1) GG Tau Bb, (2) TWA 5 B, (3) GJ 417 BC, (4) GSC 8047 B/b, (5) DH Tau B/b, (6) GQ Lup b,
(7) 2M1207 b, (8) AB Pic B/b, (9) LP 261-75 B/b, (10) HD 203030 B/b, (11) HN Peg B/b, (12) CT Cha b,
(13, 14, 15, 16) HR 8799 bcde, (17) Wolf 940 B/b, (18) G 196-3 B/b, (19) β Pic b, (20) RXJ1609 B/b,
(21) PZ Tel B/b, (22) Ross 458 C, (23) GSC 06214 B/b, (24) CD-35 2722 B/b, (25) HIP 78530 B/b, (26) SR
12 C, (27) HR 7329 B/b, (28) 2M0122 b, (29) GJ 504 b, (30) κ And b, (31) 2M0103 ABb, (32) WD 0806661 B/b. Note that some luminosity errorbars were shifted for clarity and that those of Ross 458 C (0.03 dex)
are smaller than its symbol.
~ 79 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
would require taking both the smallness of the number of detections around 40 Myr and the biases
and non-detections of the various surveys into account. However, it would not be surprising if the
underdensity in the luminosity function dN/dL of the data points, at a fixed age, proved to be real,
since there is also a suggestive underdensity in the cooling curves. Indeed, the onset of deuterium
burning near 13.6 MJ slows down the cooling, which breaks the hot-start scaling L ∝ t−1 (see
Section 5.2.5) and leads to a greater distance between the hot-start cooling curves for 10 and 15 MJ
than for 5 and 10 MJ . (This is clearly visible in fig. 1 of Burrows et al. (2001), which also shows
that there is a similar gap for low-mass stars at 10−4 L and 1–10 Gyr, due to the hydrogen main
sequence.) In particular, cooling tracks for objects of 15, 20, and 25 MJ nearly overlap at ∼ 100 Myr
around 10−4 L , where data points and their errorbars collect too. This tentative indication of an
agreement between the detections and the cooling tracks suggests that the latter might be consistent
with the data10 . It will therefore be interesting to see how significant the ‘gap’ is and how it evolves
as data points are added to this diagram.
In Fig. 5.8, the best-fitting age is calculated as the geometric mean of the reported upper and
lower bounds since, in most cases, no best-fitting value is provided, and the bounds are typically
estimates from different methods, which cannot be easily combined. In fact, the ages of young
(. 500 Myr) stars are in general a challenge to determine, as Soderblom (2010) reviews, and
represent the main uncertainty in direct observations. Moreover, as Fortney et al. (2005) point out,
assuming co-evality of the companion and its primary may be problematic for the youngest objects.
Indeed, a formation time-scale of ' 1–10 Myr in the core-accretion scenario would mean that some
data points of Fig. 5.8 below ∼ 10 Myr may need to be shifted to significantly lower ages, by an
unknown amount. This consideration is thus particularly relevant for GG Tau Bb, DH Tau B/b,
GQ Lup b, and CT Cha b (data points 1, 5, 6, and 12), which are all possibly younger than 1 Myr,
and would require a closer investigation.
We now provide detailed constraints for three planetary systems, chosen for the low hot-start
mass of the companion (2M1207) or because additional mass information is available (HR 8799
and β Pic).
5.5.2. 2M1207
The companion to the brown dwarf 2MASSWJ 1207334–393254 (2M1207 A, also known as
TWA 27 A; Gizis, 2002) is the first directly-imaged object with a hot-start planetary mass (Chauvin
et al., 2004, 2005). Since the age and luminosity of 2M107 b are the inputs for our analysis, they are
discussed in some detail in Section 5.9.1, along with tentative information on the mass. We adopt an
age of 8+4
−3 Myr (Chauvin et al., 2004; Song et al., 2006) and a luminosity of log L/L = −4.68±0.05
(Barman et al., 2011b), and assume that deuterium-burning masses above ' 13 MJ (Spiegel et al.,
2011) are excluded.
Fig. 5.9 shows the joint constraints on the mass and initial entropy of 2M1207 b based on its
luminosity and age. We recover the hot-start mass of 3–5 MJ (Barman et al., 2011b), with equation
(5.15) predicting ' 3.9 MJ , but also find solutions at higher masses. If deuterium-burning masses
can be excluded, the formation of 2M1207 b must have led to an initial entropy of S i > 9.2, with
an approximate formal uncertainty on this lower bound of 0.04 kB /baryon (see Section 5.3.1) due
solely to the luminosity’s statistical error, independent of the age’s. This initial entropy implies that
the M07 cold starts are too cold by 0.7 kB /baryon, roughly independently of the mass, to explain
the formation of this planet. This is consistent with the time-scale-based conclusion of Lodato,
Delgado-Donate & Clarke (2005) that core accretion cannot be responsible for the formation of this
system if one also accepts the received wisdom that core accretion necessarily leads to the coldest
10
Cold-start curves too show this gap, to the extent that the luminosity rise due to deuterium burning is very sensitive
to the initial entropy (see Fig. 5.8), which would need to be set accordingly finely to have the lightcurves pass
through the data gap. The implicit assumptions here are that the observed distribution of masses is uniform in the
approximate range 5–25 MJ (as are the mass values chosen for Fig. 5.8), and that the same applies to the initial
entropy. The former cannot currently be validated but the latter seems reasonable, as the entropy interval over which
cooling curves change from going above to below the gap is very narrow.
~ 80 ~
5.5. COMPARISON WITH OBSERVED OBJECTS
12
Initial entropy (kB/baryon)
5 Myr
8 Myr
11
12 Myr
10
9
8
2M1207 b
7
0
2
4
6
8
Mass (MJ)
10
12
Fig. 5.9.: Allowed values of planet mass M and initial entropy S i for 2M1207 b for assumed ages of
8+4
−3 Myr. The luminosity is that of Barman et al. (2011b), log Lbol /L = −4.68 ± 0.05, with the hatched
regions corresponding to curves within ±1 σ of the luminosity values and the thick lines to the central value.
The curves along the vertical axis show posterior distributions from MCMCs with different mass priors and
assumed luminosity distributions. The solid curves only have an upper mass limit of 13 MJ , while the dashed
curves include solutions in the deuterium-burning region, up to 20 MJ . For the blue curve (further from the
left axis) and the corresponding
dashed grey posterior (practically identical), a lognormal error distribution
√
in time centred at 5 × 12 ' 7.7 Myr and of width 0.19 dex was used, while the red curve and corresponding
dashed line are for a flat distribution in time between 5 and 12 Myr (and zero otherwise). The circles show
the results from Marley et al. (2007), increased by 0.38 kB /baryon (cf. Fig. 5.4), for cold starts (filled circles)
and hot starts (open circles), where the entropy is 1 Myr after the onset of cooling.
starts. However, our robust quantitative finding is more general, in that it provides constraints on
the result of the formation process which are model-independent.
To show how these M(S i ) constraints can easily be made even more quantitative and thus suitable
for statistical analyses, we ran a Metropolis–Hastings Markov-chain Monte Carlo (MCMC; e.g.
Gregory, 2005) in mass and entropy with constant priors on these quantities. Uncertainties in
the luminosity L and age t were included in the calculation of the χ2 by randomly choosing an
‘observed’ Lobs and a stopping time for the cooling curve tstop at every step in the chain. The
quantity log Lobs was drawn from a Gaußian defined by the reported best value and its errorbars,
and tstop from a distribution which is either constant in t between the adopted upper and lower limits
√
t1 = 5 Myr and t2 = 12 Myr and zero otherwise, or lognormal in t, centred at t0 = t1 t2 = 7.7 Myr
and with σlog t = log t2 /t1 = 0.19 dex. The results are shown along the vertical axis of Fig. 5.9 for
four different assumptions. To obtain the two solid lines, we applied an upper mass cut at 13 MJ
and took a lognormal (less peaked curve) or a top-hat (more peaked) distribution in time. The
dashed curves come from the same MCMC chains but with a mass cut-off of 20 MJ , i.e. including
deuterium-burning objects. Because of the delayed cooling due to deuterium burning, the required
initial entropy drops down faster with mass than in the cold-start branch, such that lower S i values
are possible. At 20 MJ , the required S i is 8.2, but it is still only 9 kB /baryon at 15 MJ . However,
the phase space for high masses is very small since the initial entropy needs to be extremely finely
tuned; hence the smallness of the effect on the posteriors. As Fig. 5.9 shows, different assumptions
on the luminosity, age, and mass priors all lead to similar results for the initial entropy, namely that
~ 81 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
S i > 9.2 and that there are more solutions near this lower limit.
5.5.3. HR 8799
We now turn to the only directly-imaged system with multiple objects for which planetary masses
are possible, HR 8799. The age of the system and the luminosities of the companions are discussed
in Section 5.9.2, along with information on the mass. We consider ages of 20 to 160 Myr, close to
the values of Marois et al. (2008), and use the standard luminosities of log Lbol /L = −5.1 ± 0.1
(HR 8799 b), −4.7 ± 0.1 (cd) and −4.7 ± 0.2 (e) from Marois et al. (2008, 2010). It also seems
reasonable to assume that deuterium-burning masses can be excluded for all objects, thanks to the
(preliminary) results from simulations of the system’s dynamical stability.
Fig. 5.10 shows the joint constraints on the masses and initial entropies of HR 8799 b, d, and
e. Uncertainties in the age are taken into account by considering the two extremes of 20 Myr and
160 Myr separately, while the 1-σ errors in the luminosities are reflected by the width of the hatched
regions. We find hot-start masses for 20 Myr of 4.4+0.4
(b), 6.3 ± 0.6 (cd), and 6 ± 1 MJ (e), where
−0.5
the errorbars here come only from those on the luminosity, fully consistent with the prediction by
equation (5.15) of ' 4.2 or 6 MJ . These values are in good agreement with Neuhäuser & Schmidt
(2012) and are similar to the usually-cited 30-Myr values of (5, 7, 7, 7) MJ (Marois et al., 2010). The
hot-start masses for 160 Myr are above 12 (HR 8799 b) and 13 MJ (cde), and the M(S i ) constraints
for the latter three are not shown since they are within the deuterium-burning regime.
Excluding deuterium-burning masses for all objects and using only the luminosity measurements, Fig. 5.10 shows that all planets of the HR 8799 system must have formed with an initial
entropy greater than 9 kB /baryon, with S i & 8.9 for b, S i & 9.2 for c and d, and S i & 9.1 for e
(using as throughout this work the published SCvH entropy table; see Appendix 5.8). Using tentative upper mass limits of 7 and 10 MJ , respectively, the lower bounds on the initial entropies can be
raised to 9.2 (b) and 9.3 kB /baryon (cde) if one takes the conservative scenario of the 1-σ lower luminosities value at 20 Myr. These lower bounds on the entropy are however mostly independent of
the age because they are set by cold-start solutions, where the age is much smaller than the cooling
time at that entropy. Formal uncertainties on the lower bounds due to those in the luminosities are
(see Section 5.3.1) approximately 0.07 (bcd) or 0.14 kB /baryon (e) and thus negligible.
Here too we ran an MCMC to derive quantitative constraints on the initial entropy of each planet.
The quantity log Lobs was drawn from a Gaußian defined by the reported best value and its errorbars,
and tstop from a distribution which is either constant in t between the adopted upper and lower limits
√
t1 = 20 Myr and t2 = 160 Myr and zero otherwise, or lognormal in t, centred at t0 = t1 t2 = 57 Myr
and with σlog t = log t2 /t1 = 0.9 dex. Posteriors on the initial entropy for each of the HR 8799
planets are shown along the vertical axis in Fig. 5.10, using a flat prior in S i and a mass prior
constant up to an Mmax and zero afterward. The cases ‘without mass constraints’ (Mmax = 13 MJ )
are nearly constant in S i , especially for planet b, but show a peak near cold-branch values of 9
and 9.5 kB /baryon for HR 8799 b and cde. Adding mass information from dynamical-stability
simulations by taking Mmax = (7, 10, 10, 10) MJ flattens the S i posterior and shifts the minimum
bounds at half-maximum from (8.9, 9.2, 9.2, 9.2) to (9.3, 9.5, 9.5, 9.4) kB /baryon, respectively. We
note that these results are insensitive to both the form of the uncertainty in t and the use of a non-flat
prior in mass (as shown below for β Pic b in Section 5.5.4).
Comparing to the ‘tuning fork’ entropy values reproduced in Fig. 5.10, we find that the coldeststart models of M07 cannot explain the luminosity measurements for the HR 8799 planets. Spiegel
& Burrows (2012) and Marley et al. (2012) also came to this conclusion, with the latter noting that
‘warm starts’ match the luminosity constraints. It is now possible to say specifically that the Marley
et al. (2007) cold starts would need to be made ∆S ' 0.5 kB /baryon hotter to explain the formation
of the HR 8799 planets. Given that the precise outcomes of the core accretion and gravitational
instability scenarios are uncertain and that this system represents a challenge for both (as Marois
et al., 2010 and Currie et al., 2011 review), quantitative comparisons such as our procedure allows
should be welcome to help evaluate the plausibility of the one or the other.
We note in passing that one needs to take care also when interpreting the measurements of Hink-
~ 82 ~
5.5. COMPARISON WITH OBSERVED OBJECTS
Initial entropy (kB/baryon)
12
11
unstable?
10
9
b
8
cd
HR 8799 bcde
7
0
2
4
e
6
8
Mass (MJ)
10
12
Fig. 5.10.: Allowed values of planet mass M and initial entropy S i for HR 8799 b (log Lbol /L = −5.1 ± 0.1;
blue bands), HR 8799 cd (−4.7 ± 0.1; purple), and HR 8799e (−4.7 ± 0.2; orange) for assumed ages of
20 (lower-mass group) and 160 Myr (higher-mass band; planets cde are outside the plotting range). Luminosities are from Marois et al. (2008, 2010), and nearly in agreement with Marley et al. (2012). The lines
correspond to the best-fitting luminosity values, while the hatched regions use the 1-σ errorbars. Along the
left axis are shown the posterior distributions in S i (colours as for the M–S i hashed bands), with no particular
relative normalisation. These were obtained from an MCMC, taking luminosity and age uncertainties into
account based on lognormal distributions (see text), and using flat priors in S i and M. The mass upper cutoff is Mmax = 13 MJ (full lines) or Mmax = (7, 10, 10, 10) MJ (for planets bcde, respectively; dashed lines),
since analyses of the system’s dynamical stability seem to indicate that higher masses are unstable. This is
indicated by the vertical line segments with arrows in the direction of the excluded masses. The circles show
the results from M07, increased by 0.38 kB /baryon (cf. Fig. 5.4), for cold starts (filled circles) and hot starts
(open circles), where the entropy is 1 Myr after the onset of cooling.
ley et al. (2011) and Close & Males (2010). These authors measured upper limits on the brightness
of companions within 10 AU and between 200 and 600 AU from the star, respectively. However,
both groups then used the hot-start models of Baraffe et al. (2003) to translate the brightness limits
into masses (11 MJ at 3–10 AU and 3 MJ within 600 AU, respectively). Therefore, since colderstart companions would need to be more massive to have the same luminosity, what they provide
are really “lower upper limits” on the mass of possible companions. How much higher the masses
could realistically be in this case is difficult to estimate without a bolometric luminosity, but there is
an important general point: without the restriction of considering only hot-start evolutionary tracks,
luminosity upper limits do not provide unambiguous mass constraints. Incidentally, this more general view of the results of Hinkley et al. (2011) means that the unseen companion evoked by Su
et al. (2009) as the possible cause of the inner hole (at . 6 AU) does not have to be of small mass.
However, this inner object would nevertheless have to be consistent with the results of dynamical
stability simulations, with those of Goździewski & Migaszewski (2014) indicating a mass less than
' 1–8 MJ .
5.5.4. β Pic
A companion to the well-studied star β Pic was first observed in 2009 (Lagrange et al., 2009; Bonnefoy et al., 2011) and, very recently, it became the first directly-detected object with a planetary
~ 83 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
12
Initial entropy (kB/baryon)
8 Myr
12 Myr
11
20 Myr
10
9
excluded by
radial velocity
8
β Pic b
7
0
5
10
Mass (MJ)
15
20
Fig. 5.11.: Allowed values of planet mass M and initial entropy S i for β Pic b using our estimated
+8
log Lbol /L = −3.90+0.05
−0.12 and an age of 12−4 Myr. The hatched regions correspond to luminosity values
within the (asymmetrical) 1-σ interval, with the central values marked by thick lines. The curves along the
vertical show the marginalised posterior distribution on S i from an MCMC simulation using the luminosity
and age values and their uncertainties. These
were taken to be lognormal but asymmetric for the luminosity,
√
and lognormal in time, centred at t0 = 8 × 20 = 12.7 Myr with a standard deviation of 0.35 dex. The full
blue line (closer to the vertical axis) results from the (M, S i ) distribution being multiplied by a dN/dM ∝ M α
prior, with α = −1.3 (Cumming et al., 2008; Nielsen & Close, 2010), and zero for M > 12 MJ , while the
full grey posterior has only a mass cut at 12 MJ , from the Lagrange et al. (2012) radial-velocity constraint.
The dashed and dotted curves both have an upper mass cut of 20 MJ , with and without the dN/dM prior
applied, respectively. The circles show the results from Marley et al. (2007), increased by 0.38 kB /baryon
(cf. Fig. 5.4), for cold starts (filled circles) and hot starts (open circles), where the entropy is 1 Myr after the
onset of cooling.
mass for which radial-velocity data are also available. The age of the system is taken as 12+8
−4 Myr
(Zuckerman et al., 2001), and we discuss in detail in Section 5.9.3 how we derive a bolometric
luminosity11 of log Lbol /L = −3.90+0.05
−0.12 .
Fig. 5.11 shows the M(S i ) constraints available for β Pic b from our luminosity estimate and
the radial-velocity (RV) constraint. We recover a hot-start mass ' 9.5 ± 2.5 MJ (cf. ' 9.4 MJ
from equation 5.15), in agreement with Quanz et al. (2010) and Neuhäuser & Schmidt (2012), but
additionally find that higher masses are consistent with the luminosity measurement. Excluding
solutions where deuterium burning plays an important role in the evolution of the object (recognisable by the extreme thinness of the constant-luminosity M(S i ) curve) implies that S i > 9.8. Using
the RV mass upper limit, these constraints on the initial entropy can be made tighter: with an age
of 12 Myr, it must be that S i > 10.5. Since this corresponds to a warm start, both uncertainties on
the age and on the luminosity contribute to that on the minimum S i , of order 0.5 kB /baryon.
11
As this manuscript was being prepared, we became aware of the first robust estimate of the bolometric luminosity, by
Bonnefoy et al. (2013). They find log Lbol /L = −3.87 ± 0.08, which excellently agrees with our value and thus does
not change our conclusions. In particular, they find similar constraints on the initial entropy of β Pic b, although this
depends on which band they use (cf. their fig. 11 with our Fig. 5.11). We also note the more recent estimate by Currie
et al. (2013) of log Lbol /L = −3.80 ± 0.02 (very near our approximate 1-σ upper limit), from which they estimate a
(hot-start) mass in the 3–11-MJ range. Importantly, they obtain the lower masses by considering an age of 7 Myr for
β Pic b, i.e. by relaxing the assumption that it is co-eval with its star (see Section 5.5.1).
~ 84 ~
5.6. SUMMARY
This lower limit on S i implies that coldest-start objects of any mass are too cold by a significant
1.5–2.0 kB /baryon. Various authors (e.g. Quanz et al., 2010; Bonnefoy et al., 2011) recognised
that the classical cold starts (M07; Fortney et al., 2008) cannot explain the observations, but it is
now possible to quantify this. These results are mostly insensitive to the uncertainty on the age
range; using instead 12–22 Myr as summarised by Fernández et al. (2008) would not change the
conclusions.
As for the objects in the 2M1207 and HR 8799 systems, we ran an MCMC to obtain a posterior
distribution on the initial entropy. This is shown along the vertical axis of Fig. 5.11 for four different
assumptions. In all cases, we assumed lognormal uncertainties on the age and the luminosity, with
asymmetric upper and lower errorbars for the latter. For the full curve closer to the vertical axis
(in blue), we applied an upper mass cut at 12 MJ and took into account that the underlying (real)
mass distribution is possibly biased towards lower masses, as radial velocity measurements indicate
(Cumming et al., 2008; Nielsen & Close, 2010). Out of simplicity, this was done by taking the
(M, S i ) distribution obtained with a flat prior in mass and using importance sampling to put in a
posteriori a dN/dM ∝ M α prior, with α = −1.3, thus weighing lower masses more12 . Of course,
the value of α might depend on the formation mechanism relevant to the object but this serves to
illustrate the effects of a non-constant prior on mass. The other three posterior distributions on S i
of Fig. 5.11 (in grey) correspond to the remaining combinations of ‘with mass cut or not’ and ‘with
power-law mass prior or not’. These curves are all similar, with the radial-velocity measurement
increasing the minimum bound at half maximum from 10.2 to 10.5 kB /baryon, quite insensitively
to the use of the dN/dM prior.
At the distance from its primary where β Pic b is currently located (' 9 AU), core accretion
is expected to be efficient and thus a likely mechanism for its formation (Lagrange et al., 2011;
Bonnefoy et al., 2013). Thus, the question posed to formation models is whether core accretion
can be made hotter (by 1.5–2 kB /baryon) than what traditional cold starts predict. Very recently,
Bodenheimer et al. (2013) and Mordasini (2013) showed that in the framework of formation models
(which seek to predict S i ), different rocky core masses are associated with a significantly different
initial entropies at a fixed total mass; for instance, Bodenheimer et al. (2013) found S i = 7.5 for
a 12 MJ object with a core of 5 M⊕ but S i = 9.1 when a different choice of parameters lead to a
core mass of 31 M⊕ . Since these coldest starts assume that all the accretion energy is radiated away
at the shock, the constraints on the initial entropy stress the need to investigate the physics of the
shock (and its dependence on physical quantities such as the accretion rate), when the initial energy
content of the planet is claimed to be set.
5.6. Summary
The entropy of a gas giant planet immediately following its formation is a key parameter that can
be used to help distinguish planet formation models (Marley et al., 2007). In this paper, we have
explored the constraints on the initial entropy that can be obtained for directly-detected exoplanets
with a measured bolometric luminosity and age. When the initial entropy is assumed to be very
large, a ‘hot start’ evolution, the measured luminosity and age translate into a constraint on the
planet mass. In contrast, when a range of initial entropies are considered (‘cold starts’ or ‘warm
starts’), the hot-start mass is in fact only a lower limit on the planet mass: larger-mass planets with
lower initial entropies can also reproduce a given observed luminosity and age. Fig. 5.5 shows the
12
Wahhaj et al. (2013) derive in a recent analysis of the NICI campaign results for debris-disc stars a similar slope: if
the linear semi-major axis power-law index βa = −0.61 as Cumming et al. (2008) found for radial-velocity planets
within 3 AU, the 66 non-detections combined with the Vigan et al. (2012) survey imply α < −1.7 to 2 σ, with the
most likely values at 0.3α + βa −1. When however β Pic b and HR 8799 bcd are included in the analysis, α > 2.2
for βa = −0.61, with more solutions at βa = −2.1, α > 2.2. (Too few detected objects prevent Biller et al. (2013)
from inferring constraints on α and β in a similar analysis of young moving-group stars.) Note finally that hot-start
models were used to convert magnitudes to masses and that most targets are less than 100 Myr old, with a significant
fraction near 10 Myr; ignoring cold starts at these ages can skew the inferred (limits on the) mass distribution. Also
considering colder starts should yield more negative constraints on α given the same luminosity constraints.
~ 85 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
allowed values of mass and entropy for different ages and luminosities, and can be used to quickly
obtain estimates of mass and initial entropy for any given system.
To derive these constraints, we constructed a grid of gas giant models as a function of mass and
internal entropy which can then be stepped through to calculate the time evolution of a given planet.
In a hot-start evolution, the structure and luminosity of cooling gas giant planets are usually thought
of as being a function of mass and time only; this leads to a ‘hot-start mass’ as given by equation
(5.15). Once the assumption of hot initial conditions is removed, however, a more convenient
variable is the entropy of the planet. One way to think of this is that gas giants obey a Vogt–Russell
theorem in which the internal structure, luminosity, and radius of a planet depend only on its mass
and entropy (as well as its composition, as for stars). Fig. 5.2 shows the luminosity as a function
of entropy for different masses, and a general fitting formula for L(M, S ) is given by equation
(5.9). (Similarly, cooling tracks with arbitrary initial entropy are well described analytically by
equations (5.13–5.14).) A noteworthy result is that in the intermediate-entropy regime (S ' 8.5–
10 kB /baryon), where the outer radiative zone is thick and follows a radiative-zero solution, the
luminosity obeys L = M f (S ) as found for irradiated gas giants by Arras & Bildsten (2006), with
f (S ) a steeply increasing function of the entropy. We also note that constraints obtained for models
with a particular helium mass fraction Y can be easily translated to another Y with equation (5.12)
as it provides an approximate value for dS /dY at constant L. This is general and independent of the
approach used to compute the cooling.
We find that our models are in good agreement (within tens of per cent) with the hot-start models
of Burrows et al. (1997) and Baraffe et al. (2003), as well as the cold-start models of Marley
et al. (2007), and cooling models calculated with the mesa stellar evolution code (Paxton et al.,
2011, 2013). We caution that the Spiegel & Burrows (2012) and Mollière & Mordasini (2012)
models, for example, use a version of the Saumon et al. (1995) equation of state whose entropy
is offset by a constant 0.52 kB /baryon from the published tables (which the present work uses);
this difference is not significant physically but needs to be taken into account when comparing
results of various groups. Details and (Y, P, T, %, S ) points for a quick comparison are provided
in Appendix 5.8. The remaining intrinsic difference in entropy between our models and those
just cited is then approximately |∆S | ' 0.15 at worse. This is more important than differences in
opacities or composition; for example, we estimate from Saumon et al. (1996) that the uncertainty
in the helium (Y) and metal (Z) mass fractions introduces variations of at most 10 per cent in the
luminosity at a given age.
We stress again that when the initial entropy is allowed to take a range of values, the hot-start
mass (equation (5.15)) is only a lower limit on the companion mass. The larger range of allowed
masses means that the hot-start mass could actually lead to the mischaracterization of an object,
with a hot-start mass in the planetary regime actually corresponding to an object with a mass above
the deuterium-burning limit for low enough entropies. One way to break the degeneracy between
mass and entropy is an accurate determination of the radius from spectral fitting (or actually a determination of log g and T eff from the spectrum), which would yield the mass and (current) entropy of
the object without any degeneracy (see e.g. Fig. 5.7). As discussed in Section 5.4, however, current
atmosphere models have significant uncertainties that make this approach difficult. Another possibility is to obtain independent constraints on the mass of a companion, for example from dynamical
considerations.
In Section 5.5, we applied our models to three directly-imaged objects which have hot-start
masses in the planetary-mass regime and for some of which there are additional constraints on
the planet mass. We find that the initial entropy of 2M1207 b is at least 9.2 kB /baryon, assuming
that it does not burn deuterium. For the planets of the HR 8799 system, we infer that they must
have formed with S i > 9.2 kB /baryon, independent of the age uncertainties for the star. Finally,
a similar analysis for β Pic b reveals that it must have formed with S i > 10.5 kB /baryon, using
the radial-velocity mass upper limit of 12 MJ . These initial entropy values are respectively ca. 0.7,
0.5, and 1.5 kB /baryon higher than the ones obtained from core accretion models by Marley et al.
(2007). This quantitatively rules out the coldest starts for these objects and constrains warm starts,
~ 86 ~
5.6. SUMMARY
especially for β Pic b.
An important point is that the uncertainties in age and luminosity impact the derived hot-start
mass and the lower bound on initial entropy in different ways. The major uncertainty in direct
detections is the age of the star and, relevant for very young systems, all the more that of the planet.
This age uncertainty translates into errorbars for the hot-start mass which are ∆M/M ' 12 ∆t/t.
However, the uncertainty on the initial entropy on the mass-independent branch of the M(S i ) curve
is due only to that in the bolometric luminosity, with ∆S i ' 1/λ∆ log Lbol where λ ' 0.7 when
S i . 9.6 (or, less accurately, when L ∼ 10−6 –10−4 L ). This uncertainty ∆S i is typically very small,
which means that, up to systematic errors, the initial entropy can be determined quite accurately.
It has been pointed out before that current direct-imaging detections are all inconsistent with the
cold-start predictions from core-accretion models. It is important to stress however that the cold
starts are in some sense an extreme case, as they assume complete radiation efficiency at the shock
during runaway accretion, which is argued to set the low initial entropy of planet. By varying
the nebula temperature or the accretion rate, Marley et al. (2007) were able to change the entropy
by barely ∆S ' 0.1; however, Mordasini (2013) and Bodenheimer et al. (2013) report that it is
possible within the core-accretion scenario to obtain considerably higher entropies, increased by
as much as 1–2 kB /baryon, when considering different masses for the solid core (through a selfamplifying process explained in Mordasini, 2013). With β Pic b a likely candidate for formation by
core accretion, this indicates that it is essential to gain a deeper understanding about what sets the
initial entropy, for instance by looking in more detail at the properties of the shock during runaway
accretion.
The derived bounds on S i for the HR 8799 objects and β Pic b made use of information on the
mass, which comes from dynamical stability analyses and radial velocity, respectively. Radialvelocity data of directly-detected planets are currently available only for β Pic b, but this should
change in a near future as close-in planets start being detected directly. In the absence of dynamical
information, an upper limit to the mass (and thus a lower limit on the initial entropy) should be
obtainable from log g, even if its errorbars are large; thus, in practice, the M(S i ) constraint curve
does not extend to arbitrarily high masses as a pure luminosity measurement would imply.
Finally, we ran Markov-chain Monte Carlo simulations to derive more detailed quantitative constraints on the mass and entropy of directly-detected objects. When taking the uncertainties in the
age and luminosity into account, we considered normal, lognormal, and flat distributions and found
the chosen form to make little difference. The advantage of this approach is that it allows one to
derive posterior distributions on the initial entropy, which are suitable for statistical comparisons.
Given the small semi-major axis of β Pic b (9 AU), we also tried a prior dN/dM ∝ M −1.3 (which
describes the population of radial-velocity planets) in addition to a flat prior on M. The latter case
yielded posterior distributions on S i with a more pronounced peak.
The benefits that the expected large increase in the number of directly-detected exoplanets in the
near future should bring are at least twofold. Firstly, each new detection will yield a new constraint
on the initial entropy and therefore formation mechanism. Particularly with the ability to detect
lower mass gas giants at small semi-major axes (. 20 AU), which should be possible with instruments such as GPI or SPHERE, there will be an opportunity to constrain the state of the gas giant
immediately following core accretion. Secondly, with a larger sample of objects comes a chance for
a statistical comparison with formation models. For example, as noted in Section 5.5.1, the onset
of deuterium burning leads to a relative underdensity of planets with luminosities ∼ 10−4 L and
ages of tens of Myr (assuming that the masses of substellar objects are smoothly distributed near
the deuterium-burning limit). Indeed, it is interesting that there appears to be such an underdensity
in the current data sample (see Fig. 8), although the small number of detections so far means that
this could be due to a statistical fluctuation. The best constraints on initial conditions for planet
cooling will come from improved spectral models that can give reliable determinations of log g and
T eff .
~ 87 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
Acknowledgments
We thank P. Bodenheimer, D. Saumon, J. Ferguson, K. Goździewski, A. Burrows, J. Fortney, A.
Showman, M. Marley, X. Huang, C. Mordasini, D. Spiegel, T. Guillot, T. Schmidt, M. Bonnefoy,
B. Biller, and J. Carson for useful and often detailed discussions, helpful and rapid answers to
inquiries, and generosity with data. This work was supported by the National Sciences and Engineering Research Council of Canada (NSERC) and the Canadian Institute for Advanced Research
(CIFAR), and by a scholarship from the Fonds de recherche du Québec – Nature et technologies
(FRQNT). GDM warmly thanks the MPIA for support during the last stages of this work.
5.7. Appendix A: Radii
For completeness, we present and compare the radii R in our models as a function of mass M
and entropy S . This comparison thus separates out possible differences in the treatment of deuterium burning since the planetary structure at a given (M, S ) is independent of the nuclear energy
generation, which only affects the time evolution.
Fig. 5.12a shows R(S ) for different masses, using the standard grid with a helium mass fraction
Y = 0.25 and without a solid core. At low entropies, objects of a given mass have a roughly
constant radius asymptotically tending to the zero-temperature value (Zapolsky & Salpeter, 1969;
Hubbard, 1977; Arras & Bildsten, 2006), while the radius increases at high entropies. This increase
is less pronounced for higher masses, with more massive objects being smaller at any given entropy.
Finally, there is at a given mass a maximum entropy for which a finite radius is possible, as the
upturn of the curves suggests. In fact, each curve turns over on to a hot branch (not calculated)
where the ion thermal pressure dominates, leading to distended objects; see for instance the analytic
one-zone model of Deloye & Bildsten (2003).
The inset of Fig. 5.12a shows the relative difference ∆R/Rref = (R−Rref )/Rref between our radius
at a given mass and entropy and that from more detailed calculations, either Burrows et al. (1997)
or mesa (revision 4723; Paxton et al., 2011, 2013), over the range where data are available. There
are larger deviation at higher entropies but the overall agreement for masses between 1 and 20 MJ
is excellent, with our models systematically a few per cent smaller.
In Fig. 5.12b are displayed radii as a function of mass for entropies from 14 down to 7 kB /baryon.
Objects with S < 10 have R < 2 RJ for all masses, with radii rapidly increasing at higher entropies.
Low-entropy planets approach the zero-temperature limit (Zapolsky & Salpeter, 1969), with a maximum13 radius of 1.08 RJ at 2.1 MJ for S = 7 (cf. the T = 0 result of Rmax = 0.98 RJ at 2.77 MJ ).
At higher entropies, there are no solutions below a certain mass, preventing the existence of a finite
maximum to R(M) (cf. Deloye & Bildsten, 2003).
Fig. 5.12b also shows R(M) curves at fixed S in a grid with a 20-M⊕ core at a constant density
of 8 g cm−3 and in a grid with a helium mass fraction Y = 0.30 (and no core). Differences are
small when including a solid core but larger when varying the helium fraction, with differences
of the order of tens of per cent at smaller entropies. Note that a realistic equation of state for the
core, such as a half-half rock–ice mixture from ANEOS (Marley et al., 2007) as formation by core
accretion may produce, would yield average core densities closer to ' 10–100 g cm−3 . The effect
on the radius should however still be within tens of per cent.
5.8. Appendix B: A systematic entropy offset in different versions
of the SCvH EOS
Investigating the entropy offset between the models of Spiegel & Burrows (2012) and our work
(see Section 5.2.6), we noticed that there is a nearly constant entropy offset of 0.51–0.52 kB /baryon
13
The presence of a turnover in the mass–radius relationship is due to the balance between the attractive Coulomb forces
in the ion-electron plasma and the repulsive Fermi forces between the degenerate electrons.
~ 88 ~
5.8. APPENDIX B: ENTROPY OFFSET IN SCVH
6
Radius (RJ)
∆R/Rref (%)
5
0
4
−2
1 MJ
20 MJ
−4
3
1 MJ
20 MJ
−6
2 7 8 9 10 11 12 13
This work
Burrows et al. (1997)
MESA; ∆S = −0.09
1
7
8
9
10
11
12
Entropy (kB/baryon)
13
14
S = 14
Radius (RJ)
5
2
1
No core, Y = 0.25
20−ME core at 8 g cm−3
Y = 0.30
1
T=0
10
S=7
20
Mass (MJ)
Fig. 5.12.: Top panel: Radii as a function of entropy for M = 1, 3, 10, and 20 MJ from top to bottom in our
models (solid lines) compared to those of Burrows et al. (1997) (dashed) and mesa (revision 4723; dotted).
The entropy values of Burrows et al. (1997) were shifted by (1−Y) ln 2 = 0.52 kB /baryon for ‘thermodynamic
reasons’ as discussed in Appendix 5.8, while those of mesa were shifted by the −0.09 kB /baryon offset
needed to match L(S ) (see Fig. 5.4). The inset shows the percentage difference with Burrows et al. (1997)
(dashed lines) and mesa (dotted lines) against entropy. We use for Jupiter’s radius RJ = 7.15 × 109 cm.
Bottom panel: Radii as a function of mass for S = 7–14 in steps of 1 kB /baryon from top to bottom in the
standard grid (without a core and with a helium mass fraction Y = 0.25; solid lines). Note the logarithmic
vertical scale. The lowest curve is the T = 0 result of Zapolsky & Salpeter (1969) for Y = 0.25, with the
peak of Rmax = 0.96 RJ at 2.6 RJ indicated by a filled circle (note that their table 1 indicates instead 0.98 RJ
at 2.77 MJ ). A ring symbol indicates the peak for S = 7 in the standard grid. For comparison, radii in grids
with a 20-M⊕ core and Y = 0.25 (dashed) or without a core but with Y = 0.30 (dotted) are also shown.
~ 89 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
over a large portion of the %–T plane between the entropy of Burrows et al. (1997)14 and that of our
SCvH data15 . As pointed out by D. Saumon (2013, priv. comm.), this is very probably due to the
absence, in some versions of the EOS, of the statistical weight due to the spin states of the proton
in the hydrogen partition function. The term omitted in the entropy (to facilitate comparisons with
other equations of state, which usually do not include it; D. Saumon 2013, priv. comm.) is of kB ln 2
per proton, which implies an offset of (1 − Y) ln 2 = 0.52 kB /baryon for Y = 0.25. This is almost
what we find from the direct comparison, leaving only a small residual difference ∆S ' 0.07 from
the luminosity comparison (∆S ' 0.14 when comparing with Marley et al., 2007). Thus Burrows
et al. (1997), Spiegel & Burrows (2012), and Mollière & Mordasini (2012) all use a version which
does not include the contribution, while the published tables include it.
We emphasize however that an additive constant in the entropy is not important physically, nor
for the evolution of the planet, since only differences in the entropy are meaningful. The relevance
here is only when comparing (initial) entropies as determined using different models, with the
entropy being a convenient label for the adiabat of the interior. In this spirit, we provide a few values
to allow a quick determination of whether a given EOS includes the term or not: for Y = 0.25 at P =
5.5×1012 erg cm−3 and T = 5×104 K (hence % = 1.001 erg cm−3 ), S = 10.557 kB /baryon according
to the interpolated published SCvH tables. For Y = 0.243 (0.30), % = 0.9954 (1.044) erg cm−3 and
S = 10.617 (10.128) at the same (P, T ). To avoid confusion, we encourage publications using the
Saumon et al. (1995) equation of state to quote entropy values as given in the published tables, and
to make clear that this is the case.
5.9. Appendix C: Age, luminosity, and mass constraints
5.9.1. 2M1207
Age and luminosity
The first directly-imaged object with a hot-start planetary mass (Chauvin et al., 2004, 2005) is
located 0.80 from its M8 brown-dwarf primary, a well-studied (see Gizis, 2002 for the report of
the discovery and Skemer et al., 2011 for a summary) member of the young (8+4
−3 Myr) TW Hydræ
association (Chauvin et al., 2004; Song et al., 2006) at a distance of 53±1 pc (as averaged by Skemer
et al., 2011 from Gizis, 2002; Gizis et al., 2007; Biller & Close, 2007; Mamajek, 2005; Mamajek
& Meyer, 2007; Ducourant et al., 2008), implying a projected orbital separation of 41 ± 1 AU.
Determining the luminosity of 2M1207 b is not straightforward. Photometry is available in the
J Mohanty et al. (2007), H, K s , and L0 bands from NaCo observations at the VLT (Chauvin et al.,
2004; Mohanty et al., 2007), at 0.8–1.0, 1.0–1.2, 1.35–1.55, and 1.4–1.8 µm (similar to H) from
the Hubble Space Telescope (Song et al., 2006), and in the Herschel SPIRE bands of 250 and
350 µm Riaz et al. (2012b,a). Skemer et al. (2011) also obtained an upper limit at 8.3–9.1 µm.
Spectroscopy is available at 1.1–1.35 µm, 1.4–1.8 µm (Chauvin et al., 2004; Mohanty et al., 2007;
Patience et al., 2010), and 1.95–2.5 µm (Patience et al., 2010) and 2.0–2.4 µm (Mohanty et al.,
2007), i.e. in JHK s and slightly redwards. Summing up the fluxes in J, H, K s , and L0 listed in
Mohanty et al. (2007) and using the known distance gives a luminosity of 2.1 × 1028 erg s−1 , or
27–30 per cent of a bolometric luminosity log Lbol /L = −4.68 to −4.74 (see below). A ‘direct
luminosity determination’ is thus not possible.
As summarised by Barman et al. (2011b), there are inconsistencies between the luminosity of
2M1207 b and its spectral type, determined by Chauvin et al. (2004) to be L5–L9.5. With the
corresponding bolometric correction BCK of Golimowski et al. (2004), the K s -band magnitude
implies a luminosity of log L/L = −4.7 ± 0.1 (Barman et al., 2011b). The hot-start, equilibrium
models of Baraffe et al. (2003) then yield from the age and luminosity an effective temperature of
1010 ± 80 K (Barman et al., 2011b) in disagreement with T eff ' 1600 K implied by the spectral
type (Mohanty et al., 2007; Patience et al., 2010, 2012). However, this second value is questionable.
14
15
From the data available at http://www.astro.princeton.edu/~burrows/dat-html/data/.
As available at http://aas.org/archives/cdrom/volume5/doc/files5.htm.
~ 90 ~
5.9. APPENDIX C: AGE, LUMINOSITY, AND MASS CONSTRAINTS
Indeed, the high effective temperature and low luminosity would require an improbably small radius
of ' 0.6 RJ .
There are two distinct approaches to the solution. Mohanty et al. (2007) argue that the actual
luminosity has been underestimated due to grey extinction by an almost edge-on disc (the ' 25-MJ
brown-dwarf primary is accreting; e.g. Stelzer, Scholz & Jayawardhana, 2007). The correspondingly higher luminosity is reported by Ducourant et al. (2008) as log Lbol /L = −3.8 ± 0.1. In
contrast, Skemer et al. (2011) argue against the disc explanation and suggest that thicker clouds
than what might be naïvely expected16 are needed. Barman et al. (2011a,b) go further and explicitly claim that the problem is with the derived T eff . They show that an atmosphere model
with T eff = 1000 K can fit very well the photometry and spectroscopy if clouds of typical thickness and also, crucially, non-equilibrium chemisty are included. The latter leads to a heavily reduced methane abundance (by ca. two orders of magnitude at photospheric depths) compared to
the chemical-equilibrium clouds and thus to redder colours than expected. From their best-fitting
models, Barman et al. (2011b) estimate a luminosity of log L/L = −4.68 ± 0.05, in agreement
with the luminosity derived from BCK . As Luhman (2012) notes, differing bolometric corrections
for old (field) and young brown dwarfs are thus not the sole explanation.
We adopt the luminosity of Barman et al. (2011b), in agreement with Neuhäuser & Schmidt
(2012) who report log L/L = −4.74 ± 0.06 (with however T eff = 1590 ± 280 K). Using the age
of 5–12 Myr and the Baraffe et al. (2003) cooling tracks, this luminosity yields a hot-start mass of
2–5 MJ as Barman et al. (2011b) state. Skemer et al. (2011) give a slightly higher hot-start mass
between 5 and 7 MJ based on T eff = 1000 K and the Burrows et al. (1997) models. We note that
these errorbars match the estimate ∆M/M ' 21 ∆t/t ' 0.5 from Section 5.3.1.
Mass information
Mass information of dynamical origin for this two-body system is not available. Indeed, since the
separation implies a period of at least (depending on the eccentricity) 1700 yr, detectable orbital
evolution or change of the velocity amplitude are not expected in the near future (Mamajek, 2005),
precluding both astrometry and radial-velocity measurements.
However, the surface gravity is somewhat constrained, which can be used to set an approximate
mass upper limit. Barman et al. (2011b) state that their best-fitting model has g = 104 cm s−2 but
do not provide any sense of how large the uncertainty on this value might be. However, typical
errorbars (as in their similar analysis for HR 8799 b; Barman et al., 2011a) are at least of 0.5 dex.
Mohanty et al. (2007) found that the fit to both the photometric and spectroscopic data is rather
insensitive to log g within 3.5–4.5 (cm s−2 ), in agreement with the indications of low gravity from
the triangular H-band spectral shape and relatively weak Na i absorption (e.g. Allers et al., 2007;
Mohanty et al., 2007). Also, Patience et al. (2012) fit J, H, and K spectra with five grids of
atmosphere models, including BT-Settl (Allard, Homeier & Freytag, 2011), Drift-phoenix (Helling
et al., 2008), and those of Marley et al. (Ackerman & Marley, 2001). The Marley et al. models
yielded a gravity on the edge of their grid (g = 105.0 cm s−2 ), but the others gave log g ' 3.5, 3.5,
4.3, and 5.0 (cm s−2 ), respectively. (With the best-fitting T eff = 1500–1650 K, the implied radii
are of 0.4–0.7 RJ , well below any theoretical cooling track.) Even though systematic issues with
atmosphere models of young, low-mass brown dwarfs are expected, we will take these results to
suggest tentatively that 2M 1207 b has a low gravity. As discussed below, the initial entropy on the
cold-start branch is S i = 9.2, which is thus an upper limit to the current entropy. With S < 9.2
and a reasonable upper limit log g < 4.0 (cm s−2 ), the upper bound on the mass is ' 7 MJ , and for
log g = 4.35 (cm s−2 ) it is 12.7 MJ . Therefore, we shall assume that the estimates of the surface
gravity imply a mass below the deuterium-burning limit, near 13 MJ (Spiegel et al., 2011; Mollière
& Mordasini, 2012; Bodenheimer et al., 2013; Marleau & Cumming, in prep.).
16
See comment in section 1.2.2 of Marley et al. (2012).
~ 91 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
5.9.2. HR 8799
Age and luminosities
Several properties of HR 8799 let its age be estimated: variability from non-radial oscillations, low
abundance of iron-peak elements, and far-IR excess due to circumstellar dust (Marois et al., 2008).
Along with its Galactic space motion and position in a Hertzsprung–Russell diagram, these lead
Marois et al. (2008) to estimate an age range of 30–160 Myr with a preferred value of 60 Myr,
consistent with the 20–150 Myr range of Moór et al. (2006) based on membership in the Local
Association. Recently, Baines et al. (2012) used interferometric measurements of HR 8799’s radius
to derive a stellar mass and age. They found best-fitting ages of 33 or 90 Myr, depending on whether
the star is approaching or moving away from the main sequence. However, the statistical errorbars,
which do not take uncertainties in the stellar models into account, are considerable in the second
case (the 1-σ ranges are 20–40 and 40–471 Myr, respectively). Nevertheless, if Baines et al.’s
measurement of the stellar radius and the deduced metallicity are correct, the age range of 1.1–
1.6 Gyr from the asteroseismological analysis of Moya et al. (2010) would be compromised, as
Baines et al. (2012) point out. Indeed, they estimate a near-solar metallicity, which contrasts with
the [M/H] ' −0.3 or −0.1 result of Moya et al. (2010), while metallicity is an important input of
asteroseismological analyses. Moreover, there is the statistical argument put forth by Marois et al.
(2008) that massive discs (such as HR 8799’s of 0.1 M⊕ ; Su et al., 2009) are unlikely to be found
around older stars. In our analysis, we shall therefore ignore the 1.1 Gyr result and instead use 20
and 160 Myr as lower and upper limits, which brackets the ranges reviewed in Moya et al. (2010)
and Baines et al. (2012).
Marois et al. (2008, 2010) estimated luminosities of log Lbol /L = −5.1±0.1 (HR 8799 b), −4.7±
0.1 (cd) and −4.7 ± 0.2 (e) from the known distance of 39.4 ± 0.1 pc and six infrared magnitudes,
covering ' 40 per cent of the bolometric luminosity, and also from bolometric corrections for
brown dwarfs. For their part, Marley et al. (2012) recently derived luminosities of log Lbol /L =
−4.95 ± 0.06, −4.90 ± 0.10, and −4.80 ± 0.09 for planets b, c17 , and d, respectively, by selfconsistently obtaining the radius from evolutionary models. This contrasts with the usual procedure
of optimising (R/d)2 along with T eff and log g to fit the photometry, which yields unphysically
small radii of ' 0.8 RJ (Barman et al., 2011a; Marley et al., 2012). Similarly, in a recent study18
using “atmospheric retrieval” (non-parametric determination of the P–T and composition structure)
including a simple cloud model, Lee, Heng & Irwin (2013) also find for HR 8799 b a small radius
+30
+0.1
−2
of 0.66+0.07
−0.04 RJ , which implies with their T eff = 900−90 K and log g = 5.0−0.2 (cm s ) a bolometric
luminosity log L/L = −5.57. Since Marley et al. considered only hot starts, i.e. fixed S i = high,
the cooling tracks are Lbol (M, t) and R(M, t) relations, which let R and t be uniquely determined
from log g and T eff . This gives an age of 360 Myr for HR 8799 b and age ranges of 40–100 Myr and
30–100 Myr for c and d, consistent with other literature estimates. Given the difficulties in obtaining
a reasonable fit, Marley et al. (2012) warn that the first result should not be taken seriously, and
note that the tension in the age would be reduced by considering colder initial conditions. In our
analysis, the luminosity values of Marois et al. (2008, 2010) will be used since they are standard
and almost or marginally consistent with those of Marley et al. (2012).
Finally, an age of 30 or 60 Myr for the system leads Marois et al. (2010) to derive from the
luminosity and the cooling tracks of Baraffe et al. (2003) masses of (5, 7, 7, 7) or (7, 10, 10, 10) MJ .
The uncertainty in the age implies (see Section 5.3) ∆M ' 1–1.3 MJ for the hot-start values.
Dynamical stability
Since it is to date the only directly-imaged multiple-planet system, HR 8799 has received a considerable amount of attention with regard to its dynamical stability (e.g. Marois et al., 2008; Reidemeister et al., 2009; Fabrycky & Murray-Clay, 2010; Moro-Martín et al., 2010; Marois et al., 2010;
17
In fact, the errorbars on the luminosity of HR 8799 c are clearly non-Gaußian, but this will not be taken into account
out of simplicity.
18
Note also their careful and detailed review of atmospheric modelling efforts for HR 8799 b and their problems.
~ 92 ~
5.9. APPENDIX C: AGE, LUMINOSITY, AND MASS CONSTRAINTS
Bergfors et al., 2011; Currie et al., 2011; Sudol & Haghighipour, 2012; Esposito et al., 2013; Currie
et al., 2012; see reviews in Sudol & Haghighipour, 2012 and Goździewski & Migaszewski, 2014).
However, only the most recent studies were able to consider all four planets. Crucial questions include whether there are two- or three-planet mean-motion resonances (MMRs; as Goździewski &
Migaszewski, 2009 and Fabrycky & Murray-Clay, 2010 suggest), what the inclination and eccentricities of the orbits are (for instance, Lafrenière et al., 2009 estimated 13–23° for the inclination
of HR 8799 b with respect to the plane of the sky) and whether they are co-planar (against which
Currie et al., 2012 recently provided evidence, while Kennedy et al. (2013) argues in favour),
and, naturally, what the masses (including that of the star) are and how long the system should
be required to survive. Solutions are very sensitive to these parameters and even to the numerical
integrator used, as Esposito et al. (2013) note. The parameter space’s high dimensionality makes
a proper exploration – i.e. without artificially-imposed restrictions as all authors had to assume –
, computationally prohibitive, and trying to include information about the disc would only make
matters worse.
Stability is estimated by using astrometric constraints and numerically evolving the system over
time, requiring that it be stable (without collisions nor ejections) for a period equal to its age.
However, Goździewski & Migaszewski (2009) and Fabrycky & Murray-Clay (2010) point out that
if it is young with respect to its main-sequence lifetime, HR 8799 could indeed be a transient system
undergoing dynamical relaxation. Therefore, it may not be possible to draw firm conclusions even
from the results of a complete analysis.
Nevertheless, if the direction in which these studies point is correct, the planets should have
as low masses as allowed, with however somewhat higher masses permitted if some orbits are
resonant. For this reason, we shall consider as approximate upper limits from stability analyses19
masses of 7, 10, 10, and 10 MJ (bcde). In particular, as Goździewski & Migaszewski (2014) also
find, it seems very likely that none is a deuterium-burning object.
5.9.3. β Pic
Age and luminosity
The namesake A5 dwarf of the nearby (9–73 pc; Malo et al., 2013) β Pictoris moving group has an
age of 12+8
−4 Myr (Zuckerman et al., 2001) and asymmetric outer and warped inner discs, which have
been observed for more than two decades (Lagrange et al., 2009, 2012, see review in Lagrange et al.,
2011). A companion was first detected in L0 (Lagrange et al., 2009) and subsequently confirmed
at 4 µm (Quanz et al., 2010) and in K s (Bonnefoy et al., 2011). Very recently, Bonnefoy et al.
(2013) added to these observations photometry in J, H, and M 0 . The distance of 19.44 ± 0.05 pc20
to β Pic b implies an orbital separation of 8–9 AU (Chauvin et al., 2012; Bonnefoy et al., 2013),
which is the smallest of all low-mass directly-detected objects. An object at this position had been
predicted from the disc morphology by Freistetter, Krivov & Löhne (2007).
Until recently (Bonnefoy et al., 2013; Currie et al., 2013), the only bolometric luminosity estimate for β Pic b was due to Neuhäuser & Schmidt (2012), who report log Lbol /L = −3.90+0.07
−0.40 .
They firstly derived, from the T eff = 1700 ± 300 K of Bonnefoy et al. (2011), a spectral type
SpT ' L2–T4 using the T eff –SpT relation of Golimowski et al. (2004, hereafter G04). They then
estimated from their SpT–BCK curve a bolometric correction BCK = 3.3+0.15
−1.00 (T. Schmidt 2012,
priv. comm.). Thus, the large, asymmetric lower errorbar on the luminosity comes from the large,
asymmetric lower errorbar on the bolometric correction, which itself is due to the flat T eff –SpT
relation between L7 and T4 in G04. A more direct approach to the bolometric luminosity consists
of converting the colour to a spectral type and obtaining from this a bolometric correction. Also
19
Note that in a very recent study, Goździewski & Migaszewski (2014) find, using a novel approach which assumes
multiple MMRs but yields masses independently, that broad mass rangeswhich include the hot-start values (though
not perfectly for HR 8799 c and d) are possible: 4–8, 8–12, 8–12, and 7–10.5 MJ (bcde). Using these values instead
would barely change our derived minimum bounds on S i , lowering some by ' 0.1 kB /baryon.
20
This is the value obtained from a re-reduction of Hipparcos data by van Leeuwen (2007). However, a number of
recent studies still use the value of 19.3 ± 0.2 pc (Crifo et al., 1997).
~ 93 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
using the fits21 of G04, this gives the same luminosity as found by Neuhäuser & Schmidt (2012)
but with a smaller lower errorbar of 0.12 dex. One should however note that BCK (SpT) is not a
monotonic function (see fig. 6a of G04), so that the errorbars are strongly non-Gaußian. With a
maximum BCK of 3.3 mag near L3.5, log Lbol /L cannot formally be above −3.9 dex. However,
the BCK –SpT relation of G04 was derived for field dwarfs (see also Stephens et al., 2009), and the
spectral classification of young objects is not yet well understood nor, in fact, well defined (see e.g.
Liu et al., 2011; Faherty et al., 2012). Keeping in mind these uncertainties in interpreting the photometry, we shall use for the analysis log Lbol /L = −3.90+0.05
−0.12 , where the upper errorbar reflects
the residuals of the G04 fit. This is at some variance with the value of log L/L = −3.80 ± 0.02 of
Currie et al. (2013) but compares favourably with log L/L = −3.87 ± 0.08 from Bonnefoy et al.
(2013).
Mass information
The object β Pic b is particularly interesting because it is the first directly-imaged companion for
which radial-velocity data are also available (Lagrange et al., 2012). Using new and archival data
spanning eight years and thanks to the high inclination of the system (88 ± 2°; Chauvin et al.,
2012), Lagrange et al. (2012) were able to place tentative lower mass limits of 1–2 MJ , which is
fully consistent with all reasonable age and luminosity combinations, even allowing for very large
lower errorbars on the latter. However, Lagrange et al.’s upper limit of 10–25 MJ , with 12 MJ for
the most probable orbit of 9 AU (Lagrange et al., 2009; Chauvin et al., 2012) is an important result
which excludes high-mass solutions and puts the object quite likely in the planetary (non-deuterium
burning) range.
We conclude with a brief digression. To the rarity of objects observable simultaneously in radial
velocity (RV) and in direct imaging contribute both intrinsic detection biases – direct imaging
favours planets further out from their star, resulting in a small RV signal – as well as selection
biases – target stars are usually chosen based on the presence of a disc, which implies that the
systems are preferentially seen face-on. A further hindrance is that young stars – young systems
being of greater interest because of a smaller brightness contrast – are usually active and thus less
amenable to radial-velocity measurements. See also Lagrange et al. (2013) for a discussion of
radial-velocity searches around young nearby stars and example prospects of coupling to direct
imaging.
This ends Marleau & Cumming (2014, hereafter MC14). The next two sections present the
author’s contribution to two publications, one on an updated measurement of β Pic b and the other
on κ And b, preceded by their respective abstracts.
5.10. Update to β Pic b
Section 5.10.2 reproduces the contribution of the author to Bonnefoy, Marleau et al., A&A 567, L9
(2014), “Physical and orbital properties of β Pictoris b” (Bonnefoy et al., 2014b). As a summary
of the article, the abstract is first provided. Note that Section 5.10.2 represents a segment of the
section “Physical properties and initial conditions” and the entire “Discussion” of Bonnefoy et al.
(2014b).
21
It was brought to our attention that there is an extension of G04 by Liu et al. (2010), who use updated spectral types
and removed binary systems from the sample. However, differences in BCK only begin appearing later than ' L1,
whereas the SpT we consider for β Pic b is L1–T0 (Bonnefoy et al., 2011; cf. the constraints of L0–L4 by Bonnefoy
et al., 2013 or L2–L5 by Currie et al., 2013). According to the fit of Liu et al. (2010), BCK (L) = 3.05 mag, while G04
gives 2.99 mag; the difference (0.06 mag) is less than the root-mean-square fit residuals (0.08 mag and 0.13 mag,
respectively).
~ 94 ~
5.11. ANALYSIS OF KAPPA ANDROMEDA B
5.10.1. Summary
The intermediate-mass star β Pictoris is known to be surrounded by a structured edge-on debris
disk within which a gas giant planet was discovered orbiting at 8–10 au. The physical properties
of β Pic b were previously inferred from broad- and narrow-band 0.9–4.8 µm photometry. We
used commissioning data of the Gemini Planet Imager (GPI) to obtain new astrometry and a lowresolution (R ∼ 35–39) J-band (1.12–1.35 µm) spectrum of the planet. We find that the planet
has passed quadrature. We constrain its semi-major axis to 6 10 au (90% prob.) with a peak at
8.9+0.4
au. The joint fit of the planet astrometry and the most recent radial velocity measurements
−0.6
of the star yields a planet dynamical mass lower than 20 MJ (> 96% probability). The extracted
spectrum of β Pic b is similar to those of young L1+1
−1.5 dwarfs. We used the spectral type estimate to
revise the planet luminosity to log(L/L ) = −3.90±0.07. The 0.9–4.8 µm photometry and spectrum
are reproduced for T eff = 1650 ± 150 K and a log g 6 4.7 dex by 12 grids of PHOENIX-based and
LESIA atmospheric models. For the most recent system age estimate (21 ± 4 Myr), the bolometric
luminosity and the constraints on the dynamical mass of β Pic b are only reproduced by warmand hot-start tracks with initial entropies S i > 10.5 kB baryon−1 . These initial conditions may
result from an inefficient accretion shock and/or a planetesimal density at formation higher than
in the classical core-accretion model. Considering a younger age for the system or a conservative
formation time for β Pic b does not change these conclusions.
5.10.2. Discussion
To derive quantitative constraints on the initial entropy S i of β Pic b, we used the method of MC14
and performed an MCMC in mass and S i using their evolutionary models up to masses of 17 MJ .
The models have gray atmospheres, include deuterium burning, and span in S i the extreme outcomes of any formation process. Figure 5.13 shows the allowed M and S i combinations that match
the luminosity and age taking Gaussian errorbars into account.
If the system if truly 21 ± 4 Myr old, shows that β Pic b cannot have formed according to the
classic (Marley et al., 2007) parameters of core accretion, which include a supercritical accretion
shock (coldest starts) and an initial planetesimal density leading to a 15-M⊕ core. Indeed, the 95-%level lower bound22 on the post-formation entropy of S i, min = 10.4 kB baryon−1 is ≈ 2 kB baryon−1
warmer than the supercritical 15-M⊕ prediction.
An on average inefficient shock and/or a higher planetesimal density (Mordasini, 2013) must
be invoked to lead to warmer starts. For a completely efficient accretion shock, the predicted core
would need to be & 65 M⊕ , which remains possible given the system parameters. These conclusions
are nearly unchanged even assuming an extreme duration for the planet’s formation phase of 9 Myr
(Figure 5.13).
Finally, for masses within the 68.3-% contour, the MC14 cooling curves predict β Pic b to not be
affected by deuterium flashes (Bodenheimer et al., 2000), where the luminosity and T eff of massive
objects increase, possibly at very late times (MC14; Marleau & Cumming, in prep.). However, due
to differences in boundary conditions and nuclear rate details, and given the high precision of the
luminosity measurement, using other cooling tracks can somewhat affect the mass constraints and
the importance of deuterium burning in the cooling history of β Pic b.
5.11. Analysis of κ Andromeda b
The contribution of the author to Bonnefoy, Currie, Marleau et al., A&A 562, A111 (2014), “Characterization of the gaseous companion κ Andromedae b” (Bonnefoy et al., 2014a) is presented in
Section 5.11.2. First, the abstract is reproduced but slightly expanded, and two relevant figures not
prepared by the author are included.
22
An upper limit is given by the fact that the radius starts diverging when S increases above ' 16.5. Varying the upper
bound of the cumulative integral barely varies the quoted figures.
~ 95 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
14
hot
ts
star
Initial entropy [kB/baryon]
13
127 M
12
E
90 %
65 M
E
11
95 %
10
9
15 ME
Flat mass prior
RV−based prior
Long formation
dN/dSi
dN/dM
8
Priors
6
8
10
12
Mass [MJup]
14
16
Fig. 5.13.: Joint probability contours (68.3, 95, and 99%) on the mass and post-formation entropy of β
Pic b from log(L/L ) = −3.90 ± 0.07 and an age of 21 ± 4 Myr using a flat prior on S i and a prior flat in
mass (dashed black curves) or given by the radial-velocity and astrometry constraints for a linear prior on
the radial velocity amplitude K (solid dark-blue curves). Using the logarithmic-K prior of Figure 1 gives
nearly identical results. The long-dashed orange line shows the combinations matching log(L/L ) = −3.90
at 12 Myr, i.e., allowing for an extreme 9-Myr formation delay. The open circles indicate the cold-start postformation entropies for different final core masses (labeled) or for hot starts (Mordasini, 2013, App. B of
MC14). The bottom panel displays the mass priors (dotted lines) and the marginalized posterior distributions
(black and blue lines), whereas the side panel shows the marginalized S i posterior and the non-flat mass
prior’s 90- and 95-% lower limits (up to S i ∼ 14).
5.11.1. Summary
Context. We previously reported the direct detection of a low-mass companion at a projected separation of 55 ± 2 au around the B9-type star κ Andromedae, the first around such a massive star
(2.5 ± 0.1 M ). The properties of the system (mass ratio, separation) make it a benchmark for
understanding the formation and evolution of gas giant planets and brown dwarfs on wide orbits.
Aims. We present new angular differential imaging (ADI) images of the system at 2.146 (K s ),
3.776 (L0 ), 4.052 (NB4.05 ), and 4.78 µm (M) obtained with Keck/NIRC2 and LBTI/LMIRCam, as
well as more accurate near-infrared photometry of the star with the MIMIR instrument. We aim to
determine the near-infrared spectral energy distribution of the companion and use it to characterize
the object.
Methods. We used analysis methods adapted to ADI to extract the companion flux. We compared
the photometry of the object to reference young and old objects and to a set of seven PHOENIXbased atmospheric models of cool objects accounting for the formation of dust (AMES-DUSTY,
AMES-COND, BT-DUSTY, BT-COND, BT-Settl 2010, BT-Settl 2012, DRIFT-PHOENIX). We
used evolutionary models to derive mass estimates considering a wide range of plausible initial
conditions. Finally, we used dedicated formation models to discuss the possible origin of the companion. For this, we identify regions in the clump-mass–semi-major-axis plane for which (i) the
local gravity of a gas clump dominates over the gas pressure (Toomre criterion) and (ii) the cooling
~ 96 ~
5.11. ANALYSIS OF KAPPA ANDROMEDA B
-1
Stars
-2
Brown dwarfs
78 M
log L/LSun
-3
Jup
30
-4
M
Ju
p
13
.6
M
Ju
-5
-6
p
κ And b
2M1207 b
-7
10
M
β Pic b
AB Pic b
5M
HR8799 e
Jup
Jup
HR8799 c & d
3M
HR8799 b
Jup
1RXS1609 b
6.0
6.5
7.0
7.5
log (age) [yr]
8.0
8.5
Fig. 5.14.: Left panel: LBTI/LMIRCam high-contrast image of the κ And system at M band. The companion is recovered. Right panel: Evolution of the luminosity of gaseous objects predicted by the COND
models (solid black line) and by the MC14 models with typical ‘hot-start’ (dashed light pink curve; 3, 5,
10, 13.6 MJ ) and ‘cold-start’ initial conditions (dashed dark blue curve; 3, 5, 10, 13.6 MJ ). We overlaid
measured luminosity of young low-mass companions. A more complete version of this figure can be found
in MC14.
timescale of the clump is smaller than the shearing time, which would allow formation by gravitational instability.
Results. We derive a more accurate J = 15.86 ± 0.21, H = 14.95 ± 0.13, K s = 14.32 ± 0.09 mag
for κ And b. We detect the companion in all our high-contrast observations. We confirm previous
contrasts obtained at K s and L band. We derive NB4.05 = 13.0 ± 0.2, and M = 13.3 ± 0.3 mag and
estimate log(L/L ) = −3.76 ± 0.06. Atmospheric models yield T eff = 1900+100
−200 K but do not set any
constraint on the surface gravity. Models with dust in the photosphere of the object better reproduce
the SED. ‘Hot-start’ evolutionary models predict masses of 14+25
−2 MJ based on the luminosity and
temperature estimates, and when considering a conservative age range for the system (30+120
−10 Myr),
‘warm-start’ evolutionary tracks constrain the mass to M > 10 MJ .
Conclusions. The mass of κ Andromedae b mostly falls in the brown-dwarf regime, owing to remaining uncertainties in age and in mass-luminosity models. According to the formation models,
disk instability in a primordial disk may account for the position and a wide range of plausible
masses of κ And b.
Figure 5.14 reproduces figures 2 and 10 of Bonnefoy et al. (2014a) for interest and convenience,
respectively.
5.11.2. The mass of κ And b: Warm-start models
This is the author’s contribution to section 3.4.2 of Bonnefoy et al. (2014a). Note that it was
shortened for publication.
To assess the dependence of the mass–initial-entropy constraints on the choice of the cooling
curves, we also used the bolometric luminosity and effective temperature of κ And b to derive
joint constraints on its mass and initial entropy using the grid of MC14. As in Spiegel & Burrows
~ 97 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
13
13
L/LSun
Teff
L/LSun
Teff
12
Initial entropy [kB/baryon]
Initial entropy [kB/baryon]
12
11
10
9
8
0
11
10
9
5
10
Mass [MJup]
8
0
15
5
10
Mass [MJup]
15
Fig. 5.15.: Joint posteriors on the mass and intial entropy of κ And b obtained through MCMC runs with the
models of MC14, which include deuterium burning, from the bolometric luminosity log L/L = −3.76±0.04
(solid lines) or effective temperature T eff = 1800 ± 100 K (dashed lines), assuming Gaußian errorbars on
either quantity. The colour is proportional to the joint posterior probability at each M and S i location in
+120
the T eff -based run. The age was taken as t = 30+20
−10 Myr (top panel) and t = 30−10 Myr, with Gaußian
errorbars in the first case and lognormal ones in the second. Flat priors on the mass and initial entropy were
used. Open and closed circles are as in Marley et al. (2007), showing the approximate range of entropies
spanned by hot and coldest starts, respectively (the ‘tuning fork’ diagram of exoplanets), but shifted upwards
by +0.38 kB /baryon to match the luminosity in the models of MC14, see therein. Similarly, the offset with
the models of Spiegel & Burrows (2012) is ca. ∆S = +0.45.
(2012), these models were computed by assuming a constant-entropy interior with a near-solar
helium mass fraction Y = 0.25 and include deuterium burning. However, they have a much simpler
outer boundary condition, using a grey, solar-metallicity atmosphere terminating at the photosphere
in the Eddington approximation, instead of calculating the radiative transfer in the upper layers
(Burrows et al., 1997). More importantly, the time evolution is handled very differently. Cooling
tracks are rapidly calculated by moving in a grid of models labeled by their mass and entropy,
in the ‘following the adiabats’ approach (see MC14 and references therein). This contrasts with
the Henyey-based approach of Spiegel & Burrows (2012). Nevertheless, despite the simplicity of
the structure and time-evolution calculation, the models of MC14 match within some ten percent
independent cold-, warm- and hot-starts models (Burrows et al., 1997; Baraffe et al., 2003; Marley
et al., 2007; Spiegel & Burrows, 2012; Paxton et al., 2013).
When comparing the joint constraints on the mass and initial entropy based on the models of
Spiegel & Burrows (2012) (shown in Fig. 5.14) with those of MC14, we also took the uncertainty
in the age and the luminosity explicitly into account. For this, we ran Markov Chain Monte Carlo
simulations (MCMCs) in mass and initial entropy as in MC14 and with their models, drawing
at every step a luminosity or effective temperature, as well as an age, according to the values
+120
log L/L = −3.76 ± 0.04 or T eff = 1800 ± 100 K, and t = 30+20
−10 Myr or t = 30−10 Myr. We
assumed Gaußian distributions on L and T eff and, respectively, normal or lognormal errorbars for
the age, and took flat priors in S i and M. (See MC14 for the effect of a non-flat, radial-velocitybased prior in the case of β Pic b.)
Figure 5.15 displays the 68-, 95- and 99-percent joint confidence regions from the MCMC runs
using L or T eff , for both age groups. The results are entirely consistent with those of Figure 5.14.
Moreover, both L and T eff lead to similar contours, with differences due to the effective temperature
T eff = 1800 ± 100 K corresponding to a slightly different luminosity; the mean log L/L of the
points in the T eff chain is −3.66 instead of −3.72 for the L-based chain, and with a slight skewness.
Even for the low-age group, the phase space for masses below 13 or 14 MJ is very small, i.e.,
~ 98 ~
5.11. ANALYSIS OF KAPPA ANDROMEDA B
almost all solutions are in the mass regime where deuterium burning is important for the evolution
of the object. A dramatic illustration of this lies in the solutions found at lower entropies. Whereas
the models of Spiegel & Burrows (2012) allowed, for an age of 30+20
−10 Myr, initial entropies down
to only 11 kB /baryon (see Figure 5.14), the models of MC14 find that κ And b could have formed
with an entropy as low as ≈ 9.0 kB /baryon, correcting for the entropy offset of ≈ 0.5 kB /baryon
between the two models (see MC14).
The low-S i solutions are possible only if the models include the possibility of a rise in the object’s luminosity due to deuterium burning, as mentioned in MC14 and discussed in Marleau &
Cumming (in prep.). (As seen in Figure 5.14, these were not found by the models of Spiegel &
Burrows (2012) as none of their lightcurves are of this type.) Very recently, Bodenheimer et al.
(2013) independently found these ‘flashes’23 occurring in the cooling of objects formed by core
accretion for some combinations of parameters, as also seen but in a milder form by Mollière &
Mordasini (2012). Figure 5.16 shows cooling tracks for different (M, S i ) combinations which all
reach log L/L = −3.76 at 30 Myr, with the lowest initial-entropy solutions undergoing a flash.
While the exact low-entropy solutions can depend on the details of deuterium burning, for instance
on the (unknown) initial deuterium content or metallicty of the object (Spiegel et al., 2011; Mollière & Mordasini, 2012), the main conclusion that the combustion of deuterium plays a significant
role in the cooling history of κ And b is a solid one, irrespective of the precise age of the system.
We warn that while the lowest-S i are comparable to the extrapolation of the coldest starts (Marley
et al., 2007) to higher masses, what this implies for the formation mechanism is not clear given the
major uncertainties about their outcome.
23
This term reflects the fact that the entropy increase happens on a relatively short timescale and can lead to a dramatic
increase in the luminosity and radius.
~ 99 ~
CHAPTER 5. CONSTRAINING THE INITIAL ENTROPY OF GAS GIANTS
10−2
13
.5
M
Ju
p
10
Luminosity (LSun)
,S
ini
−3
t
=1
4.0
14.0 MJup, Sinit = 10.0
10−4
S = 9.1
15.1 MJup, init
10−5
16.2 MJup, Sinit
10−6
106
= 8.6
107
108
109
Time (yr)
Fig. 5.16.: Examples of possible cooling curves for κ And b using log L/L = −3.76 at 30 Myr, with
the corresponding mass and initial entropy of each curve indicated in the figure; from top to bottom, the
mass increases (up to 16.2 MJ ) and the initial entropy decreases (down to 8.6 kB /baryon) to compensate.
For the lower entropies, we find that κ And b could be presently undergoing a deuterium ‘flash’ (Salpeter,
1992; Bodenheimer et al., 2013), where the burning of deuterium drives an increase in its luminosity and
radius. Similar curves and a more general discussion of M–S i constraints from luminosity measurements
when considering deuterium-burning masses can be found in Marleau & Cumming (in prep.).
~ 100 ~
so much depends
upon
a red wheel
barrow
glazed with rain
water
6
besides the white
chickens.
— William Carlos Williams
Conclusion
Up to now, the post-formation thermal state—i.e., entropy and thus luminosity—of gas giant planets
was neither predicted by theory nor constrained by observations, yet has significant observational
consequences. We addressed this issue from both sides. The modeling effort was directed at the
accretion shock at the surface of a forming planet to determine the radiation efficiency, i.e., the
fraction of the incoming kinetic energy which is lost as radiation, the rest being added to the planet.
We used equilibrium (1-T ) flux-limited diffusion in spherical symmetry in Chapter 2 and focussed
on cases where the contribution of the internal luminosity is negligible. We found that the shock
is always isothermal with a Mach number sufficiently high that almost the entire kinetic energy
is radiated away. The fraction brought into the planet was seen to be negligible compared to the
planet’s internal luminosity and therefore unable to heat the planet up. The ram pressure is much
smaller than the photospheric pressure but the shock temperature can be smaller or larger by a
factor of several to a few. The implications of this will have to be determined with self-consistent
coupling to formation calculations.
For selected cases from the large formation parameter space, we verified in Chapter 3 that the use
of non-equilibrium (2-T ) radiation transport does not change the Mach numbers as the radiation
and gas temperatures were found to be equal. However, it remains to be investigated what the effect
of other assumptions is, in particular that of a spherically symmetric geometry and of a constant
equation of state. Work concerning the latter is already underway.
In a related aside (Chapter 4), we attempted to produce self-consistent cooling tracks for gas
giants using the sophisticated BT-Settl atmospheres. However, we found that the only available
grid of models which is complete down to sufficienctly low temperatures, BT-Settl-2010, has
multiple values of T eff corresponding to a given entropy for T eff ≈ 1500 K. This is due to convergence issues of the models due to hysteresis in the formation and destruction of dust, with the
consequence that the grid cannot be used for computing cooling tracks.
On the observational side (Chapter 5), we showed how to convert a measurement of the luminosity and age of a gas giant (planet or brown dwarf) to joint constraints on its mass and initital
entropy. The shape of the ‘trade-off’ curve is set by the behaviour of the cooling curves, which
show a constant luminosity (or entropy) at early times before turning off onto the powerlaw cooling
track common to hot starts. Useful constraints were derived on a number of objects which showed
that most objects form with an entropy clearly above the cold starts predictions of Marley et al.
(2007).
Extrapolating our simulation results, we seem to predict only cold starts, while observations
seem more favourable to at least warm starts. One possibility of reconciling both statements is
if the accretion shock is not the sole or principal physical process setting the initial entropy of a
planet. An alternative is provided by the effect of the core mass, as shown in Mordasini (2013).
~ 101 ~
CHAPTER 6. CONCLUSION
This is a promising venue as it in turn depends somewhat sensitively on a number of ill-constrained
quantities such as the initial planetesimal surface density in the disc. However, it will still be
necessary to first clearly isolate the effects of the accretion shock though detailed coupling and,
ultimately, full three-dimensional radiation hydrodynamical simulations including the effects of
(non-ideal) magnetic fields.
~ 102 ~
Vernunft und Glaube sind wie die beiden Flügel,
mit denen sich der menschliche Geist
zur Betrachtung der Wahrheit erhebt.
— Fides et ratio (1998)
7
Acknowledgements
Mein erster Dank gilt meinem Betreuer Hubert Klahr und unserem Institutsdirektor Thomas Henning, die es mir ermöglichten, bei ihnen spannende Forschung zu betreiben. Außerdem darf ich
Christoph Mordasini und Rolf Kuiper für ihre großzügig gespendeten Hilfe und Zeit und Ermutigungen danken.
Einem Leben ohne Freunde und gute Kollegen fehlte etwas, was mich gerne folgenden Personen
in alphabetischer Reihenfolge danken läßt: S. Ciceri (grazie!), S. Cielo, K.-M. Dittkrist, A. Gomes
(obrigado!), C. Hansen, Th. Hansen (takk!), S. Jin, N. Malygin (spasybi!), J. Menu (dank u!),
P. Mollière (danke!), M. Schulik, M. Wöllert (merci!), S.-W. Wu und die vielen anderen, die mir
erst nach dem Drucken dieser Arbeit einfallen werden.
Der Exoplanetenkaffee hat mir viel gelehrt, wofür ich unter anderem E. Buenzli, M. Bonnefoy,
J. Schlieder, N. Zimmerman, I. Crossfield, V. Jörgens, B. Biller, N. Deacon, A. Bayo, T. Kopytova
und F. Rodler danken möchte.
Es war ein großes Privileg, am Max-Planck-Institut zu sein und ein herzliches Dankeschön geht
an das ganze Personal für ihren mit Hilfsbereitschaft und Professionalität – und doch Freundlichkeit
– ausgeführten Dienst.
Schließlich möge auch Dir, liebe Mona, sowie meinen Eltern und naher Familie und so vielen
‚persönlichen‘ Bekannten und Freunden für ihre vielfältige und geduldige Unterstützung gedankt
werden – merci, danke, thanks!
~ 103 ~
A
Summary and test of the radiation
hydrodynamics code
In this chapter, we briefly list our numerical settings and proceed in Section A.2 to summarise the
equations of radiation hydrodynamics solved numerically in Chapters 2 and 3, as implemented in
Mignone et al. (2007) and Kuiper et al. (2010). Details of the operator splitting for the energy
equation are given in Section A.3 as this differs slightly from Kuiper et al. (2010). We verified
our update of the code relative to Kuiper et al. (2010) with the standard Ensman (1994) sub- and
supercritical shock tests and present these results in Section A.4.
A.1. Numerical settings
The following settings concern mainly PLUTO (see Mignone et al., 2007 and the detailed documentation available under http://plutocode.ph.unito.it/). We use the hll solver with the
default slope limiter and linear interpolation. Time integration is performed with a second-order
Runge–Kutta scheme and we take a Courant–Friedrich–Lax number CFL = 0.8 unless indicated
otherwise. Thermal conduction and artificial viscosity are switched off.
For the implicit FLD solver (see below), we use the default PETSc pre-conditioner and take a
relative convergence tolerance on the temperature of ε = 10−15 , i.e., the iterations to solve the
implicit equation terminate when the relative change in temperature in each grid cell is 10−15 or
less. Typically, after an adjustment phase, only one iteration is needed.
A.2. Basic equations
The hydrodynamics module of PLUTO (Mignone et al., 2007, 2012) solve the Navier–Stokes equations with a finite-difference scheme. In the absence of (explicit) viscosity, these can be written as
(see Equations 1.3)
∂%
+ ∇ · (%v) = 0
∂t
∂%v
+ ∇ · (%vv + P1) = %g
∂t
∂Etot
+ ∇ · ([Eint + Ekin + P] v + Frad ) = %v · g,
∂t
(A.1a)
(A.1b)
(A.1c)
where symbols have their usual meaning (and in particular Etot = Eint +Ekin +Erad ) and the equations
were written in vector form for generality’s sake, with g = −gêr . We now briefly derive the
~ 105 ~
APPENDIX A. SUMMARY AND TEST OF THE RADIATION HYDRODYNAMICS CODE
combined energy equation by deriving the equations for the conservation of internal, mechanical,
and radiative energy.
A.2.1. Internal energy equation
The gas energy equation comes directly from the first law of thermodynamics and reads in Langrangian form (Castor, 2004, and see below):
!
Deint
D 1
%
+ %P
= c(−4πκP B + κE Erad ) + %,
(A.2)
Dt
Dt %
where = 0 is an energy generation rate (e.g., from thermonuclear reactions), and the terms
c(−4πκP B + κE Erad ), which represent respectively absorption of and losses to the radiation field,
are defined below. The ‘advective derivative’ is defined as D(·)/Dt ≡ ∂(·)/∂t + v · ∇(·) and is the
derivative at fixed mass element instead of position. Using the continuity equation, one can show
that for an arbitrary function f ,
Df
∂% f
%
=
+ ∇ · (% f v) ;
(A.3)
Dt
∂t
with this, the gas energy equation is in Eulerian form
∂Eint
+ P∇ · v + ∇ · (Eint v) = c(−4πκP B + κE Erad ) + %,
∂t
(A.4)
where, as defined around Equations (1.3), Eint = %eint .
A.2.2. Mechanical energy equation
The mechanical energy equation can be derived from the inner product of v with the momentum
equation (Equation A.1b), which yields
!
D 1 2
%
v = %v · g − v · ∇P
(A.5)
Dt 2
and thus, using Equation (A.3),
∂Ekin
+ ∇ · (Ekin v) + v · ∇P = %v · g.
∂t
(A.6)
A.2.3. Radiation energy equation
Finally, consider the radiative transfer equation (e.g., Mihalas & Mihalas, 1984)
!
1∂
+ n · ∇ I = η − χI,
c ∂t
(A.7)
where I = I(x, t, n, ν) is the specific intensity, which depends on the position x, time t, propagation direction n, and frequency ν, and where η = ην and χ = χν are the (in general frequencydependent) emission and absorption coefficients. In thermodynamic equilibrium, neglecting scattering, emission and absorption of energy must be equal, which leads to the Kirchhoff–Planck
relation ην = κν Bν , where Bν is the Planck function. Taking the angular average of Equation (A.7)
then yields the zeroth moment equation,
∂Erad
+ ∇ · Frad = c (4πκP B − κE Erad ) ,
∂t
(A.8)
where Erad = 4π
c J and Frad
R = 4πH are the (astrophysical)
R radiation energy density and flux, for
1
1
a mean intensity J = 4π I dΩ and mean flux H = 4π In dΩ (Mihalas & Mihalas, 1984), and
~ 106 ~
A.3. OPERATOR SPLITTING FOR THE ENERGY EQUATION
where κP and κE are, respectively, the Planck and energy mean opacities. The subscript on Frad is
usually dropped for simplicity.
Thus, summing Equations (A.4), (A.6) and (A.8), we obtain the equation for the total energy,
Equation (A.1c). The needed closure relation, for instance, an expression for Frad as a function of
Erad , was discussed in Section 1.2.
We note in passing that, to have two-temperature radiation transport (where Erad is not necessarily given by the Planck function B), it is necessary to keep seperate the equation for the combined gas internal and kinetic energy from the radiation energy equation to prevent the ±Λ =
∓c (4πκP B − κE Erad ) terms from cancelling. It is usually assumed that κE = κP for convenience.
As it is the main focus of our work (Chapter 2), we next discuss only the 1-T case and refer
the interested reader to Klassen et al. (2014) for details of the energy equations when evolving
seperately Erad and Eint + Ekin .
A.3. Operator splitting for the energy equation
The energy equation is solved in an operator-split fashion, where the combined internal and kinetic
energy are updated in the hydrodynamics module, before the radiation transport modifies the internal energy to account for heating and cooling by the radiation field. Explicitly, the hydrodynamics
module updates the kinetic and internal components of the energy at time tn as
(Eint + Ekin )? − (Eint + Ekin )n
= −∇ · ([Eint + Ekin + P] v)n + (%v · g)n ,
∆t
(A.9)
where the superscripts indicate the time step but ? denote the state after the hydrodynamics but
before the radiation transport. This yields the internal and kinetic energies Eint ? and Ekin ? .
To derive the required energy equation for the radiation transport, we first write
∂ (Eint + Erad )
= Qhydro − ∇ · F,
∂t
(A.10)
where Qhydro is a source term from the hydrodynamics (see below). Since both Eint and Erad are
power laws of temperature, their derivatives are proportional to ∂T/∂t, so that (holding % constant
for radiation transport)
h
i ∂T
∂ (Erad + Eint ) 3
(A.11a)
= 4aT + %cv
∂t
∂t
%
"
#
%cv T ∂Erad
= 1+
(A.11b)
4Erad ∂t
1 ∂Erad
≡
,
(A.11c)
fc ∂t
where by definition the constant-volume heat capacity is, for a general equation of state (EOS),
!
∂(Eint /%)
cv (%, T ) ≡
.
(A.12)
∂T
%
Note that Equation (A.11) is valid for an EOS which is not necessarily constant, i.e, when cv (%, T )%T
cannot be automatically identified with Eint , and that it implicitly defined the factor fc . Equation (A.10) then becomes with a first-order time discretisation
1 Erad n+1 − Erad n
= Qhydro − ∇ · F.
fc
∆t
(A.13)
To solve for Erad n+1 , one must choose how to evaluate fc , Qhydro , and F. The following choices
were made (R. Kuiper & H. Klahr, priv. comm.):
~ 107 ~
APPENDIX A. SUMMARY AND TEST OF THE RADIATION HYDRODYNAMICS CODE
• For fc , the updated density %? after the hydrodynamics step but the ‘old’ temperature T n are
used, yielding:
1
cv (%? , T n )%?
=1+
.
(A.14)
fc
4a(T n )3
• The procedure for the flux is similar:
F = −DF (%? , T n )∇Erad n+1 ,
(A.15)
where writing Erad n+1 leads to an implicit algorithm, and the diffusion coefficient is calculated
as
λ(R(%? , T n ))c
DF =
,
(A.16)
κ(%? , T n )%?
with λ the flux limiter and R the radiation quantity as discussed in Sections 1.2 and 1.3.
• Finally, the hydrodynamics energy source term Qnhydro is taken to be
!
Eint (%? , T ? ) Eint (%n , T n )
n
Qhydro = %
−
,
%?
%n
1 ?
%=
% + %n .
2
(A.17a)
(A.17b)
For a constant-γ and -µ EOS, Qnhydro ∝ (P? /%? − Pn /%n ) ∝ (T ? − T n ).
Since Ekin is not updated during the radiation transport, Ekin n+1 = Ekin ? , and summing Equations (A.9) and (A.13) gives, using Equation (A.17),
(Etot )n+1 − (Etot )n
= −∇ · ([Eint + Ekin + P] v)n + (%v · g)n − ∇ · −DF ∇Erad n+1 + Q+atm , (A.18)
∆t
which is a correct discretisation of Equation (A.1c). As mentioned above, since Erad n+1 appears
in Equation (A.13) both on the left-hand side and in a gradient, one must solve Equation (A.13)
implicitly. Details are given in Kuiper et al. (2010).
A.4. Radiation-hydrodynamics shock test
As in Kuiper et al. (2010), we perform the classical one-temperature (1-T ) radiative shock test
described in Ensman (1994, hereafter E94) to test our set-up. Her results are for 2-T radiation
transport but she does show that there are only little differences between 1-T and 2-T for these
tests. Briefly, we consider a uniformly-spaced spherically-symmetric grid from r = 8–8.7×106 km.
At the inner grid edge, as in the main work, we use reflective boundary conditions in the hydrodynamical quantities and impose zero radiative flux, whereas the outer edge has ‘outflow’ (i.e.,
zero-gradient) hydrodynamical boundary conditions and Erad = 0. The main differences with the
main work are that (i) the accretion boundary conditions are not used, and (ii) the external potential
(due to the planet’s mass) is set to zero. We neglect self-gravity as in the main study.
We take as in E94 a constant opacity κ = 0.4 cm2 g−1 , a constant mean molecular weight µ = 0.5,
and a constant number of degrees of freedom f = 3 (needed for cV , where cV = 1 + 2/ f in the
ideal-gas approximation, used here). We use the simple flux limiter of Equation (1.24) used by
E94.
As in E94, the problem is set up with a constant initial density %0 = 7.78 × 10−10 g cm−3 and a
temperature decreasing linearly from 85 K at the inner edge to 10 K at the outer edge. The initial
velocity of the gas is −6 or −20 km s−1 (i.e., directed towards the left) for the sub- or supercritical
case, respectively. Section 1.1.1 reviewed the two shock types, in which the post-shock flux is,
respectively, not able or able to heat the pre-shock material to its post-shock value.
The results of the radiative shock tests (sub- and supercritical) are presented in the next subsections along with a comparison to E94.
~ 108 ~
A.4. RADIATION-HYDRODYNAMICS SHOCK TEST
F /107
2
This work (1-T )
E94 (2-T )
Subcritical
1
0
60
50
40
30
20
10
0
Subcritical
2
1
8
T /102
600
6
4
400
2
200
0
4
0
E94, Tgas
E94, Trad
This work, 6λ
3
400
̺/10−9
Pgas (erg cm−3 )
3
0
800
T (K)
Pgas /102
F (Merg s−1 cm−2 )
4
300
200
100
2
1
0
-2
0
8
8.1
-1
0
1
2
Optical depth from shock
Radius (106 km)
Fig. A.1.: Left column: Radial variation of the (top panel) flux, (middle) temperature, and (bottom) gas
pressure in the subcritical case (vshock = 6 km s−1 ). The four profiles in each panel are, from left to right,
taken at 5.4, 17, 28, and 38.3 ks after the initial time, as in figure 8 of E94, while the grey dashed line
shows the initial profile. The circles in the third profile of the top panel indicate the grid resolution in the
spike. There are 1000 cells in the whole grid. Right column: From top
R to bottom, (a) flux, (b) gas pressure,
(c) temperature, and (d) density as a function of the optical depth κρ r. from the shock front for a small
region in the subcritical case (vshock = 6 km s−1 ) at t = 3.83 × 104 s as in figure 9 of E94. Her data (for
2-T radiation transport) is also shown (green dashed line). The units are in all cases cgs and are omitted for
clarity. The bottom panel also shows as a dashed line six times the flux limiter (see Equation 1.23).
A.4.1. Subcritical case
Figure A.1a shows the radial dependence of the radiative flux, temperature and gas pressure at four
different times in the subcritical case with vshock = 6 km s−1 , which corresponds to laboratoryframe Mach numbers M ≈ 4–5 at early to late times, respectively. (Note that, in accordance with
Equation (1.4), the shock in our reference frame advances with a speed of vfront = 2.1 km s−1 .) The
results are qualitatively similar to the ones of E94, with (i) some pre-heating but still (ii) T − T 1 ,
reflecting the quick absorption (over only a few grid cells; see also Figure A.1b) of the flux, which
leads to a sharp peak, and (iii) the negative flux just after the shock in an early profile, although this
is at 17 ks for E94 but in the 5.4 ks-profile for us. Our 1–T diffusion approximation does not let
us reproduce the temperature overshoot behind the shock (the Zel’dovich spike; see Chapter 1.1.1)
but apart from that the overall shape of the curves is in very good agreement, especially given the
differences between 1- and 2-T radiation transport.
There quantitative differences between E94 and our results in the subcritical case are small.
The post-temperature (T 2 ≈ 700 K, with some decay over time due to radiative losses) is similar
to within ten per cent, while our peak flux decreases faster over the four snapshots shown, from
F = 3.4 to 2.2 × 107 erg s−1 cm−2 compared to her F = 2.5 to 2.4 × 107 erg s−1 cm−2 . This is
probably due to the much sharper drop in temperature directly upstream of the shock in the 1–T
case, as expected from E94.
~ 109 ~
APPENDIX A. SUMMARY AND TEST OF THE RADIATION HYDRODYNAMICS CODE
Figure A.1b shows profiles at t = 3.83 × 104 s as a function of the optical depth from the
shock. The main difference with the full-radiation transport case or the 2-T approximation is in
the thickness, in optical depths, of the flux peak at the shock: in the 1–T case, F decreases within
only a few grid cells and
in much less than one optical depth, in contrast to E94 where the flux
√
−
3∆τ≈1
decreases with F ∝ e
as a simple semianalytical model predicts (Mihalas & Mihalas, 1984).
Temperatures and pressures are however very similar, with also a density jump %1 /%− ≈ 3.9. A pure
hydrodynamic jump would yield (for a shock-frame Mach number M0 = 6.6) a ratio of
%1
(γ + 1)M0 2
=
≈ 3.7.
%− (γ − 1)M0 2 + 2
(A.19)
Thus the interaction of the radiation with the gas increases the density jump slightly, as in discussed
in Mihalas & Mihalas (1984, see also equation 10 of Commerçon et al., 2011). Notice finally how
her artificial viscosity spreads the shock over a few grid cells.
A.4.2. Supercritical case
Figure A.2a shows the radial dependence at four different times of the radiative flux, temperature
and gas pressure in the supercritical case (vshock = 20 km s−1 , which corresponds to a laboratoryframe M ≈ 10–25 at early to late times, respectively). The profiles are very similar to those of E94,
with (i) a negative flux directly downstream of the shock (i.e., flowing downstream), (ii) T 1 ≈ T − ,
and (iii) a precursor ahead of the shock stretching out to several optical depths. The Zel’dovich
peak is of course absent in our 1-T calculations and the temperature drops sharply at the leading
edge of the precursor, as discussed in E94.
The profiles are also quantitatively similar, with only slightly smaller peak temperatures: while
E94’s peak radiation temperature increases from 3200 to 4900 K over the four snapshots, ours goes
from 2800 to 4000 K, which nevertheless represents a similar relative increase. Our peak fluxes are
20 per cent smaller but also roughly constant in time.
At t = 7.5 × 103 s, our precursor has a thickness ∆r = 0.9 × 1010 cm, which is smaller than the
∆r = 2.0 × 1010 cm, found by E94 but similar to the ∆r ≈ 1.3 × 1010 cm of Kuiper et al. (2010).
Nevertheless, as shown in Figure A.2b, the agreement with E94 around the shock is very good. In
particular, the pressure structures match almost perfectly, both in the pre- and post-shock regions.
A.4.3. Summary
Overall, our results for the Ensman (1994) radiation-hydrodynamics shocks match her results.
Given that the pressure structures are nearly identical, differences in the temperatures translate
into differences in the density structure (since P ∝ %T for an ideal-gas equation of state). These
moderate temperature differences are possibly due to our use of 1-T radiation transport, whereas
Ensman (1994) performed 2-T calculations. Note finally that, as used in Chapter 3, the 2-T radiation transport module was tested with the same set-ups by Klassen et al. (2014) and is therefore not
discussed here.
~ 110 ~
Pgas /103
Supercritical
T /103
F (Gerg s−1 cm−2 )
5000
4000
3000
2000
1000
0
5
4
3
2
1
0
Supercritical
This work (1-T )
E94 (2-T )
5
4
3
2
1
0
5
4
3
2
1
0
E94, Tgas
E94, Trad
This work, 6λ
6
4000
̺/10−9
Pgas (erg cm−3 )
5
4
3
2
1
0
T (K)
F /109
A.4. RADIATION-HYDRODYNAMICS SHOCK TEST
3000
2000
1000
4
2
0
-2
0
8
8.2
-1
0
1
2
Optical depth from shock
Radius (106 km)
Fig. A.2.: Left column: Radial profiles for the supercritical shock (vshock = 20 km s−1 ). The panels and lines
are as in Figure A.1a but for t = 0.9, 4, 7.5, and 13 ks; cf. figure 10 of E94. Note the different reference
frame when comparing to E94. Right column: As in Figure A.1b (note however the different normalisations)
but for the supercritical case (vshock = 20 km s−1 ) at t = 7.5 × 103 s. Compare to figure 13 of E94.
~ 111 ~
B
Additional material
This chapter provides additional material, mostly figures, which did not fit into the main body of
this thesis but which might be useful as reference.
B.1. Flux limiters
The flux limiters of Levermore & Pomraning (1981) and Ensman (1994) are compared in Figure B.1. They differ at most by 20 per cent, at R ≈ 2.5, and by less than ten per cent for R . 0.5 or
R & 15. Another common choice, the flux limiter of Minerbo (1978), is also shown.
B.2. Effect of the opacity
We compare in Figure B.2 the structures for a constant opacity and using Bell & Lin (1994), as in
Section 2.5.2. Only the three last cases of Table 2.1 are considered because of the available data.
The results are very similar to Figure 2.9.
B.3. Reduced fluxes for Bell & Lin (1994) opacities
For completeness, Figure B.3 shows the reduced fluxes directly upstream of the shocks for the
simulations of Section 2.5.4.
B.4. Parameter exploration
In Figures B.4 and B.5, we vary, respectively, the outer pressure boundary condition as well as the
number of grid cells, or the choice of rmin . The former is seen to not make a significant difference
even when using tabulated opacities.
Due to the inner radiative boundary condition (dErad /dr = 0) and the slight deviation from
hydrostatic equilibrium caused by the settling of the gas, planets with a smaller rmin are slightly
hotter at the shock. The trend in the shock temperature is dT shock /drmin = −1 K RJ −1 for κ =
10−2 cm2 g−1 around rmin = 10 RJ , with T shock (rmin = 20 RJ , κ = 10−2 ) = 170 K. The slope is
similar for κ = 10−1 for rmin = 20, with T ≈ 170 K. Thus, taking rmin → 0 here should lead to a
difference on the order of 20 K, and probably less for smaller planets. This is a crude estimate but
it seems to indicate that the choice of rmin is not crucial; uncertainties in the opacities, for instance,
are likely to have much more pronounced consequences.
~ 113 ~
APPENDIX B. ADDITIONAL MATERIAL
0
10−1
λM78
λLP81
λ′ = 1/(3 + R )
20
30
′
|1 − λ /λM78|
|1 − λ′/λLP81 |
|1 − λLP81/λM78 |
10−2
0.01
0.1
1
Difference (%)
Flux limiter λ
10
40
10
50
100
R = |∇E rad |/(E rad κR ̺)
Fig. B.1.: Rational approximation of the Levermore & Pomraning (1981) flux limiter and flux limiter of
Ensman (1994) (solid lines) as a function of the logarithmic optical derivative of the radiation energy density
R. Their ratio is displayed as a dashed curve (with the right axis scale).
Finally, in Figure B.6 we assess the effect of the choice of κ, µ and γ by looking at the shock
temperature for a mass and radius grid with Ṁ = 10−3 M⊕ an−1 and considering extreme cases: a
constant opacity of the BL94 table; µ = 1.23 (appropriate when hydrogen is atomic) or µ = 2.353;
and γ = 1.1 or 1.44. (The number of grid cells is varied simultaneously from N = 1000 to 3000 but
several other simulations have shown this to be without consequence.) The shock temperature can
be seen to not depend at all on N and barely on µ and γ, but noticeably on κ, as found in the main
work.
B.5. Estimated Mach numbers for population synthesis planets
Figure B.7 shows the estimated shock temperature and the Mach number obtained with Equation (1.18) and the data of Figures 2.4 and 2.5, as indirectly used in Figure 2.15. The shock temperature increases with planet mass. For the Mach number, we consider two values of µ, where
µ = 1.23 is more appropriate at higher temperatures and µ = 2.353 at lower temperatures.
B.6. Density inversions
Since we do not model convection, density inversions can occur. Indeed, considering a profile, for
which it holds that
%(P)T (P) kB
P=
,
(B.1)
µ(P) mH
it is easy to show that the density–pressure stratification is (for a general idea gas with variable
mean molecular weight)
!
!
d ln %
d ln µ
= 1 − ∇act +
,
(B.2)
d ln P act
d ln P act
where the actual T (P) gradient is
∇act ≡
d ln T
.
d ln P
~ 114 ~
(B.3)
10−2
1500
10−3
1000
10−4
500
10−5
15
10−6
0
Opacity (g cm−2 )
0
−10
−20
−30
−40
−50
0
5
10
5
Shock loss efficiency
103
102
10−1310−1210−1110−1010−9 10−8 10−7
10−6
10−7
5
10−1
−2
0
5
10
15
Radius (R J )
Upstream reduced flux
10
15
0
5
Radius (R J )
15
1
ηkin
120 %
10
Radius (R J )
100 %
80 %
60 %
40 %
20 %
0 .8
0 .6
0 .4
0 .2
0
1
10
100
0
5
10
15
Radius (R J )
104
120 %
ηkin
Pressure (bar)
0
15
10−5
0%
0 .1
Shock loss efficiency
Optical depth from r max
101
10
10−1
Time (106 s)
102
10
Radius (R J )
10−4
Density (g cm−3 )
10
5
10−3
140 %
104
0
10−2
0
Radius (R J )
15
Radius (R J )
103
102
101
100
10−1
10−2
10−3
10−4
10−5
10−6
15
10
−Material
energy
¡
¢
flux 10−1 L ⊙
10
Reduced flux |F |/ cE
5
Radius (R J )
10
Velocity (km s−1 )
2000
0
0
Temperature (K)
10−1
¡
¢
Luminosity 10−1 L ⊙
10−3
10−4
10−5
10−6
10−7
10−8
10−9
10−10
10−11
10−12
10−13
Temperature (K)
Density (g cm−3 )
B.6. DENSITY INVERSIONS
100 %
80 %
60 %
40 %
20 %
102
100
10−2
10−4
10−6
10−8
0%
1
10
Shock Mach number
0
2
4
6
8 10 12 14 16
Radius (R J )
1 .2
γ = 1 .1
1
B, Bell & Lin (1994), t = 2.7 × 107 s
B, κ = 1 cm2 g−1 , t = 2.7 × 107 s
C, Bell & Lin (1994), t = 4.2 × 107 s
C, κ = 1 cm2 g−1 , t = 0.4 × 107 s
D, Bell & Lin (1994), t = 1.1 × 107 s
D, κ = 1 cm2 g−1 , t = 1.1 × 107 s
0 .8
0 .6
0 .4
0 .2
0
0 .1
λLP81 (4/∆τ) × 4/∆τ
1
10
Shock optical depth
Fig. B.2.: As in Figure 2.9 but for the three last cases of Table 2.1, using γ = 1.1, and compared to the same
cases but using the BL94 opacities. The last panel is as in Figure 2.11, where, as in the efficiency against the
Mach number, full (open) circles correspond to solid (dotted) data curves. Only Cases A and B are compared
to a free-fall velocity profile (dashed grey curve) and only for Case C with BL94 is an errorbar shown for the
upstream reduced flux fred (last panel), which comes from time averaging. The outer boundary conditions
are dP/dr = 0 and dr2 Erad /dr = 0.
~ 115 ~
APPENDIX B. ADDITIONAL MATERIAL
Pre-shock reduced flux
Pre-shock reduced flux
γ = 1.44, Bell & Lin (1994), dP /d r ( r max ) = 0, d r 2 E rad /d r ( r max ) = 0
1
0 .8
0 .6
0 .4
0 .2
0
1
10.0
5 .0
1 .3
M⊕ an−1 0.3
Ṁ = 10−2 M⊕ an−1
Ṁ = 10−3
Ṁ = 10−4 M⊕ an−1
Ṁ = 10−5 M⊕ an−1
MJ
MJ
MJ
MJ
0 .8
0 .6
0 .4
0 .2
0
10
10
1
Planet radius (R J )
1
Planet radius (R J )
Fig. B.3.: As in Figure 2.10 but for the BL94 opacities and with γ = 1.44.
The µ derivative being negative but very small (e.g., 6 0.001), it is the size of ∇act which is decisive.
If one considers convection by the usual Schwarzschild criterion
∇act = min(∇ad , ∇rad ),
(B.4)
the fact that ∇ad (which is equal to 1 − 1/γ for a constant-γ gas) is < 0.5 for hydrogen and helium
ensures that d%/dP > 0. If however there is no convection and the temperature gradient is given by
∇act = ∇rad ≡
3LP
κ,
64πσGmr T 4
(B.5)
large enough opacity values will lead to a density inversion.
B.7. Extracting an atmospheric structure from a Phoenix log file
The script shown in Listing B.1 was used to extract structures for the coupling work to BT-Settl
(Chapter 4). It was written by analysing a typical log output file (http://phoenix.ens-lyon.
fr/Grids/BT-Settl/AGSS2009/STRUCTURES/lte006-3.0-0.0.BT-Settl.6) from the Phoenix
atmosphere calculator to identify the different parts of the (typically 80000-line long) file and derive criteria to extract the desired blocks. Although not strictly necessary for the coupling, which
requires only a single (P, T ) point in the convective part, the complete atmospheric structure was
extracted to check for anomalities and allow comparisons to other work (see Chapter 4). (It is not
available under another form, as confirmed with the author F. Allard.) Note that there are slight differences in the output of different versions of the atmosphere code and that a single log file usually
contained all 6 or 12 iterations, which somewhat complexified the extraction process.
~ 116 ~
600
Temperature (K)
400
300
200
100
100
20
Radius (R J )
50
100
20
102
100 %
40 %
20 %
10
100
100
20
50
0 .6
0 .4
0 .2
0
20
Luminosity (L max )
100
50
100
Radius (R J )
100
120 %
100 %
80 %
60 %
40 %
20 %
0%
ηphys
1
Radius (R J )
10−1
10−2
10−3
10−4
10−5
10−6
10−7
10−8
100
1
1000
104
103
102
101
100
10−1
10−2
10−3
Radius (R J )
50
50
0 .8
0%
Shock loss efficiency
15
100
Radius (R J )
60 %
Optical depth from r max
20
Radius (R J )
20
Time (106 s)
25
20
100
ηphys
Density (g cm−3 )
50
50
80 %
10−12 10−11 10−10 10−9 10−8 10−7 10−6
20
50
Radius (R J )
Radius (R J )
Shock loss efficiency
Temperature (K)
20
0 .1
0
−0.1
−0.2
−0.3
−0.4
−0.5
−0.6
10−3
20
103
Entropy ( k B baryon−1 )
100
10−2
Radius (R J )
Pressure (bar)
50
Radius (R J )
10−1
Material
¡ energy
¢
flux 10−4 L ⊙
1
0
−1
−2
−3
−4
−5
−6
−7
−8
50
Opacity (g cm−2 )
Velocity (km s−1 )
20
0 .9
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0
500
Reduced flux |F |/ cE
Density (g cm−3 )
10−6
10−7
10−8
10−9
10−10
10−11
10−12
¡
¢
Luminosity 10−4 L ⊙
B.7. EXTRACTING AN ATMOSPHERIC STRUCTURE FROM A LOG FILE
10
Shock Mach number
1 .2
Mp = 0.32 MJ , Ṁ = 10−2 M⊕ an−1 , κ = 10−2 cm2 g−1
γ = 1.1; r max : d r 2 E /d r = 0, dP /d r = 0
1
0 .8
10−5
L = L 0 e−∆τ
10−4
10−3
10−2
r min = 15.85 R J ,
r min = 16.85 R J ,
r min = 17.85 R J ,
r min = 18.85 R J ,
r min = 19.85 R J ,
r min = 19.85 R J ,
t = 9.0 × 108 s
t = 7.8 × 108 s
t = 6.7 × 108 s
t = 5.7 × 108 s
t = 4.7 × 108 s
t = 3.2 × 108 s
Optical depth upstream of shock
Fig. B.4.: Comparison of the outer boundary conditions. See parameter settings in the caption.
Listing B.1: Code to extract a structure from a Phoenix log file.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
#! /usr/bin/awk -f
# extrahiert die relevanten Blöcke aus PHOENIX -.6- Dateien (z.B. für BT -Settl)
# - entweder nur die letzte Iteration
#
oder alle Iterationen (mit einer leeren Zeile dazwischen )
#
wird/ werden gespeichert
# - die .7- Dateien enthalten nur eine Iteration
#
# Beispielruf : ./ extra.awk
BT -Settl/ AGSS2009 /lte006 -3.0 -0.0.BT -Settl .6
#
Die Ausgabedateien lte006 -3.0 -0.0.BT -Settl .6 _Info und lte006 -3.0 -0.0.BT -Settl .6 _Strukt
#
werden in BT -Settl/ AGSS2009 / ( Vorläufige Dateien aber im gegenwärtigen Verzeichnis ) erzeugt
#
# Beispielruf : ./ extra.awk -v alle =1 BT -Settl/ AGSS2009 /lte006 -3.0 -0.0.BT -Settl .6
#
Die Ausgabedateien lte006 -3.0 -0.0.BT -Settl .6 _Info und lte006 -3.0 -0.0.BT -Settl .6 _Strukt_alleIter
#
werden in BT -Settl/ AGSS2009 / ( Vorläufige Dateien aber im gegenwärtigen Verzeichnis ) erzeugt
#
#
#
# sample call: ./ extra.awk somedir /lte006 -3.0 -0.0.BT -Settl .6
#
The output files lte006 -3.0 -0.0.BT -Settl .6 _Info (not really useful ) and lte006 -3.0 -0.0.BT -Settl .6 _Strukt
#
will be created in somedir / ( temporary files however in current directory )
#
# sample call: ./ extra.awk -v alle =1 somedir /lte006 -3.0 -0.0.BT -Settl .6
#
The output files lte006 -3.0 -0.0.BT -Settl .6 _Info (not really useful ) and lte006 -3.0 -0.0.BT -Settl .6 _Strukt_alleIter
#
will be created in somedir / ( temporary files however in current directory )
#
#
# G.-D. Marleau -- MPIA
#
(c) 9.01.2013
# letzte Änderung : 02.06.2014
BEGIN {
~ 117 ~
10−4
10−5
10−6
10−7
10−8
6
7
8
4
5
6
7
8
4
5
9
4
2
10−6
5
10−5
10−4
100 %
15
8
9
20 %
0%
0 .1
1
Pressure (bar)
Luminosity (L max )
7
4
8
9
7
8
9
5
4
5
6
7
6
7
8
9
8
9
1
0 .8
0 .6
0 .4
0 .2
0
10
4
5
6
7
Radius (R J )
104
103
102
101
100
10−1
10−2
10−3
8
9
120 %
100 %
80 %
60 %
40 %
20 %
0%
−20 %
ηphys
1
10
Shock Mach number
1 .2
1
0 .8
0 .6
0 .4
0 .2
0
L = L 0 e−∆τ
−0.2
−3
−2
10 10 10−1 100 101 102 103 104
Radius (R J )
6
Radius (R J )
Radius (R J )
102
101
100
10−1
10−2
10−3
10−4
6
5
Radius (R J )
ηphys
Radius (R J )
5
8
Shock loss efficiency
20
4
7
40 %
Optical depth from r max
Entropy ( k B baryon−1 )
25
7
4
Time (106 s)
30
6
6
60 %
10−3
35
5
9
0
−2
−4
−6
−8
−10
−12
−14
−16
80 %
Density (g cm−3 )
4
8
Reduced flux |F |/ cE
Shock loss efficiency
Temperature (K)
103
10−7
7
Radius (R J )
104
10−8
6
Radius (R J )
104
103
102
101
100
10−1
10−2
10−3
10−4
Radius (R J )
10
9
102
9
Radius (R J )
5
0
−5
−10
−15
−20
−25
−30
103
Material
¡ energy
¢
flux 10−1 L ⊙
5
Opacity (g cm−2 )
Velocity (km s−1 )
4
7
6
5
4
3
2
1
0
−1
104
Temperature (K)
Density (g cm−3 )
10−3
¡
¢
Luminosity 10−1 L ⊙
APPENDIX B. ADDITIONAL MATERIAL
Mp = 0.63 MJ , Ṁ = 10 M⊕ an−1 , Bell & Lin (1994)
γ = 1.1; r max : d r 2 E /d r = 0
dP /d r ( r max ) = 0, N = 5000, t = 2.0 × 106 s
dP /d r ( r max ) = 0, N = 10000, t = 2.0 × 106 s
P ( r max ) = P (̺ff , Tneb ), N = 5000, t = 6.5 × 106 s
P ( r max ) = P (̺ff , Tneb ), N = 10000, t = 6.5 × 106 s
Optical depth upstream of shock
Fig. B.5.: Comparison of the effect of the choice of rmin . The time of each simulation snapshot is chosen to
yied the same shock position.
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
gewollteIt = 0
# gewollte Iteration : die letzte
# Informationsblock liefert T_eff , R_eff , L (je einen Wert)
InfoText = "This is for effective radius and is physical "
Info = 0
# 1. Block liefert i, tau , T, P, rho , mu
einsText = "i
tau
temp
Nrho
conv"
eins = 0
pg
# um den Block zu erkennen
pe
rho
# 2. Block liefert S, Stgrad ( del_rad ), gradient , adgrad ( del_ad )
zweiText = "i
tau
z
temp
pg
gradient
adgrad
Fconv"
zwei = 0
# 3. Block liefert kappa_Rosseland
#
(1/ cm)
dreiText = "layer
tstd
b-mean
ross -corr"
drei = 0
# 4. Block liefert tau_Rosseland
vierText = "tau averages " #layer
vier = 0
wartevier = 1
j-mean
tstd
f-mean
b-mean
mu
rho
mu
ross -mean
j-mean
vel
(cm **2/g)
b-mean
f-mean
mdot
S
j-mean
Hp
f-mean
stgrad
ross -mean
ross -mean"
# 5. Block liefert den bolometrischen radiativen Fluß (suf) (tk=tau= taustd : tau bei lambda = 1.25 um)
# ** suf wird mit pi/1e8 multipliziert um den Fluß in erg/s/cm^2 zu bekommen **
# * (F/sigma)^0.25 nicht ganz = Teff weil Fehler != 0 *
fuenfText = "no
tk
sub
subk
suj
sujk
suf
sufk
Leerzeichen bei CIFIST 6... der Rest einfach weg] deltaE \\(%\\)
fehler \\(%\\)
aenderung "
~ 118 ~
lg \\( g_eff \\)
sulb" # [ weniger
B.7. EXTRACTING AN ATMOSPHERIC STRUCTURE FROM A LOG FILE
Ṁ = 10−3 M⊕ an−1 , dP /d r ( r max ) = 0, d r 2 E rad /d r ( r max ) = 0
Shock temperature (K)
104
BL94, γ = 1.44, µ = 2.353, N = 1000
103
10.0 MJ
5 .0 M J
1 .3 M J
0 .3 M J
min
Rough Tshock
102
104
Shock temperature (K)
N = 2000
γ = 1.1, µ = 2.353, κ = 1 cm2 g−1 , N = 3000
γ = 1.1, µ = 1.23
103
102
Shock temperature (K)
104
10
1
Planet radius (R J )
103
BL94, γ = 1.44, µ = 2.353, N = 1000
N = 2000
γ = 1.1, µ = 1.23
102
γ = 1.1, µ = 2.353, κ = 1 cm2 g−1 , N = 3000
10
1
Planet radius (R J )
Fig. B.6.: Shock temperatures for different combinations of κ, γ, µ, and number of grid cells N, as indicated
in the panels, with only the differences to the top left panel indicated. The last panel combines all results for
a direct comparison.
61
fuenf = 0
62 }
63
64 {
65
if (NR == 1) {
66
InfoDatei = ( FILENAME "_Info")
67
einsDatei = ( FILENAME ". TPrhomu ")
68
zweiDatei = ( FILENAME ".Sdel")
69
dreiDatei = ( FILENAME ".kappa")
70
vierDatei = ( FILENAME ".tau")
71
fuenfDatei = ( FILENAME ".suf")
72
ausDateib = ( FILENAME " _Strukt ")
# Basis
73
74
# bei den .spec .6- Datein gibt es nur eine Iteration
75
if ( FILENAME ~ ".spec" ) {
76
gewollteIt = 1
77
}
78
79
if( system ( "[ -f " InfoDatei " ] " ) == 0 ) { system ( "rm " InfoDatei ) }
80
print "# i
tau
T
P
rho
mu
81
print "# i\t
tau\t
S\t
Hp\t
delrad \t
del\t
82
print "# i
tau
kappa_Rosseland " > dreiDatei
83
print "# i
tau(_std)
tau_Rosseland " > vierDatei
84
print "# i
tau(_std)
bolflux " > fuenfDatei
85
}
86
87
if ( $0 ~ "max. no. of iterations :" ) {
88
Itanz = $(NF);
# Iterationenanzahl
89
print " Iterationenanzahl : " Itanz # "; gewollte Iteration : " Itnum;
90
if ( alle ) {
~ 119 ~
log(g_eff)" > einsDatei
delad" > zweiDatei
APPENDIX B. ADDITIONAL MATERIAL
104
104
Hot accretion
Temperature (K)
Temperature (K)
Cold accretion
103
10
T (R P )
Tshock approximation
2
0.1
1
103
10
T (R P )
Tshock approximation
2
10
0.1
1
Mass ( MJ)
102
102
Hot accretion
Mach number
Mach number
Cold accretion
101
10
10
Mass ( MJ)
µ = 1.23
µ = 2.353
0
0.1
1
101
10
µ = 1.23
µ = 2.353
0
10
0.1
1
Mass ( MJ)
10
Mass ( MJ)
Fig. B.7.: As in Figure 2.15 but showing explicitly the temperatures (top row) and Mach numbers (bottom
row). The effective temperature is shown as T (Rp ) and the estimate of the shock temperature as T shock . In
the Mach number plots, the groups of points correspond to Ṁ = 10−2 to 10−4 M⊕ an−1 (bottom to top).
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
ausDatei = ( ausDateib " _alleIter ");
gewollteIt = 1
# wir wollen alle
} else {
ausDatei = ausDateib
}
}
if ( $0 ~ "no. of depth points " ) {
Schichtenanzahl = $(NF)
}
# für den Fall , wo wir nur eine Iteration wollen :
if ( ( $0 ~ "next iteration is no." ) && $0 ~ ( Itanz ) ) {
, denn Info , eins und zwei kommen davor
gewollteIt = 1
}
#
; print " Schichtenanzahl : " Schichtenanzahl ;
# man kann nicht " results of iteration no:
}
8 :" benutzen
if ( gewollteIt ) {
if ( Info ) {
print $0 >> InfoDatei
if ($0 ~ "R_eff is") {
}
Info = 0
}
# letzter Eintrag des Blockes
if (eins || zwei || drei || vier || fuenf) {
gsub (/D/,"E", $0) # Fortran - Format konvertieren
}
if (eins) {
#
i
tau
T
P
rho
mu
log(g_eff)
# Achtung ! log(g_eff) kann mit dem nächsten Feld
(Nrho) zusammengeschweißt sein ... deswegen substr
print $1 "\t" $2 "\t" $3 "\t" $4 "\t" $6 "\t" $7 "\t" substr ($10 ,1 ,10) >> einsDatei
# log(g_eff) ist 10 breit
}
if (zwei) {
#
i
tau
S
Hp
delrad
sein
# Fconv muß mit 4*pi multipliziert werden
del
delad
Fconv
# wenn del_rad negativ ist:
if ( substr ($9 ,12 ,1) == "-"){
Hp = substr ($9 ,1 ,11)
strad = "-" substr ($9 ,13 ,11)
print " strad = " strad " < 0 bei i = " $1 " ( tau = " $2 ") in " FILENAME
~ 120 ~
# Achtung ! letztes Feld Fconv kann negativ
B.7. EXTRACTING AN ATMOSPHERIC STRUCTURE FROM A LOG FILE
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
# gradient sollte auch < sein (weil adgrad > 0 und somit > strad bleibt )
if ( substr ($9 ,24 ,1) == "-"){
gradient = "-" substr ($9 ,25 ,11)
print " ( gradient = " gradient " bei tau = " $2 " in " FILENAME ")"
#
# prüfen , ob Fconv negativ ist
if ( substr ($10 ,12 ,1) == "-"){
adgrad = substr ($10 ,1 ,11)
Fconv = substr ($10 ,12 ,12)
# mit Vorzeichen
print " Fconv = " Fconv " bei tau = " $2 " in " FILENAME
} else {
adgrad = $10
Fconv = $11
}
#
}else {
print " gradient = " gradient " in " FILENAME " aber != " strad " < 0? Problem !"
exit
}
# delrad nicht negativ (und somit gradient auch nicht)
} else {
Hp = $9
strad = $10
gradient = $11
# prüfen , ob Fconv negativ ist
if ( substr ($12 ,12 ,1) == "-"){
adgrad = substr ($12 ,1 ,11)
Fconv = substr ($12 ,12 ,12)
# hat Vorzeichen
if($1 ~"77") print " ** Fconv = " Fconv " bei tau = " $2 " (i=" $1 ") in " FILENAME
} else {
adgrad = $12
Fconv = $13
}
#
}
print $1 "\t" $2 "\t" $8 "\t" Hp "\t" strad "\t" gradient "\t" adgrad "\t" Fconv >> zweiDatei
}
if (drei) {
#
i
tau
kappa_Rosseland kappa_P kappa_J
kappa_H
## std: nur kappa_Ross
#print $1 "\t" $2 "\t" $10 >> dreiDatei
## Pauls Spezialversion : mit anderen kappas
print $1 "\t" $2 "\t" $10 "\t" $7 "\t" $8 "\t" $9 >> dreiDatei
}
if (vier && wartevier == 0) {
#
i
tau
tau_Rosseland
print $1 "\t" $2 "\t" $6 >> vierDatei
} else if (vier) {
print "1
0
0.0000 E+00" >> vierDatei
wartevier = 0
}
# keine 1- Schicht im Block ...
# die erste Zeile nach "tau averages " beinhaltet die Spaltennamen
if (fuenf) {
#
i
tk
suf
if ($0 !~ " convection ") { print $1 "\t" $2 "\t" $7 >> fuenfDatei }
}
# Konvektionsanfang und -beginn wird angegeben
if ($1 ~ Schichtenanzahl ) {
if (eins) { print "" >> einsDatei ; }
if (zwei) { print "" >> zweiDatei ; }
if (drei) { print "" >> dreiDatei ; }
if (vier) { print "" >> vierDatei ; }
if (fuenf) { print "" >> fuenfDatei ; }
if (vier) wartevier = 1
# für die nächste Iteration , falls wir mehrere wollen
eins = 0; zwei = 0; drei = 0; vier = 0; fuenf = 0;
}
if
if
if
if
if
if
($0
($0
($0
($0
($0
($0
~
~
~
~
~
~
InfoText ) {
einsText ) {
zweiText ) {
dreiText ) {
vierText ) {
fuenfText ) {
Info =
eins =
zwei =
drei =
vier =
fuenf
1 } #; print "Info " NR }
1 } #; print "eins " NR }
1 } #; print "2 " NR }
1 } #; print "3 " NR }
1 } #; print "4 " NR }
= 1 } #; print "5 " NR }
}
}
END {
N = split(FILENAME ,Vorl ,"/")
# Vorläufige Dateien im gegenwärtigen Verzeichnis speichern
Name = Vorl[N]
system ( "join --header " einsDatei " " zweiDatei " > vorl1." Name )
system ( "join --header vorl1."Name " " dreiDatei " > vorl2." Name )
system ( "join --header vorl2."Name " " vierDatei " > vorl3." Name )
system ( "join --header vorl3."Name " " fuenfDatei " > vorl4." Name )
# die Reihenfolge von vorl4 ist (mit ,join ' wird das Vergleichsfeld nur einmal gedruckt !):
#
i
tau
T
P
rho
mu
log(g_eff)
tau
S
Hp
Fconv
tau
kappa_Rosseland
tau
tau_Rosseland
tau
suf
#
1
2
3
4
5
6
7
8
9
10
14
14+1
15+1
1+16
17
1+18
1+19
## Pauls Spezialversion :
#
i
tau
T
P
rho
mu
log(g_eff)
tau
S
Hp
Fconv
tau
kappa_Rosseland
kappa_P kappa_J
kappa_H
tau
tau_Rosseland
#
1
2
3
4
5
6
7
8
9
10
14
14+1
15+1
17
18
19
20
21
system ( "awk 'BEGIN{print \"" \
~ 121 ~
delrad
11
del
12
delrad
del
tau
suf
11
12
22
23
delad
13
delad
13
APPENDIX B. ADDITIONAL MATERIAL
3000
rapax 21 + BT-Settl 2010
rapax 21 + Eddington
Teff (K)
2500
COND03
B97
2000
1500
1000
500
1
10
100
Time (Myr)
Fig. B.8.: As in Figure 4.3, hot-start cooling tracks for Mp = 3, 5, 10, 15 MJ (bottom to top) for different
models including our self-consistent coupling. The lower edge of the grid is marked (pale grey dashed line).
229
"# 1
2
5
6
7
8
12
13
14
15
16
17
19\"; print \"" \
"# i
tau
P
log(g_eff)
Hp
rho
S
mu
kappaRoss
delrad
del
delad
tauRoss
bolradflux
convflux
kappa_P
kappa_J
kappa_H \"; b = \"
\";}; " \
"(NR >= 2) { if(NF){ print $1 b $2 b $4 b $3 b $7 b $10 b $5 b $9 b $6 b $16 b $11 b $12 b $13 b $21 b $23 /1e8
*3.1415926 b $14 *4*3.1415926 b $17 b $18 b $19 } else { print $0 } }' vorl4." Name " > " ausDatei ) # NF (leere
Zeile): um nicht " "/4 e8 = 0 zu bekommen
9
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
3
10
4
11
18
T
Befehl = ("wc -l < " ausDatei )
Befehl | getline Zeilenanz
close( Befehl )
if ( (! alle && abs(Zeilenanz - Schichtenanzahl ) > 4) || (alle && abs( Zeilenanz - Itanz *( Schichtenanzahl +1)) > 4) ) {
# einige
zusätzliche Zeilen ist nicht unbedingt ein Problem
print
print " *** Seltsam ! `" ausDatei "' hat " Zeilenanz " Zeilen für eine " Schichtenanzahl " schichtige Atmosphäre ... *** = "
} else {
Teff = substr (Name , 4, 3) *100
# ohne die Verzeichnisnamen davor; Name = lte *** -*** -...
if ( substr (Name , 8, 3) !~ "-") # die neueren haben eine genauere Teff
logg = substr (Name , 8, 3)
else
logg = substr (Name , 10, 3)
if ( Zeilenanz < 100 ) {
print " ** [ " Teff " K, " logg " dex ] ** Etwas ist schief gelaufen ! < 100 Zeilen "
} else {
print " [ " Teff " K, " logg " dex ] Struktur extrahiert und in " ausDatei " gespeichert ! " Itanz " Iterationen und "
Schichtenanzahl " Schichten "
}
}
system ( "rm vorl1." Name " vorl2." Name " vorl3." Name " vorl4." Name " " einsDatei " " zweiDatei " " dreiDatei " " vierDatei "
" fuenfDatei )
253 }
254
255 function abs(value) { return (value <0?- value:value); }
B.8. Coupled cooling tracks: Effective temperature evolution
Figure B.8 shows the effective temperature as a function of time for the coupled cooling tracks with
BT-Settl-2010 and compares to the simple Eddington boundary conditions and other commonlyused tracks (Baraffe et al., 2003; Burrows et al., 1997). The effects of the non-converged atmospheres around T eff ≈ 1300–1700 K are clearly visible.
~ 122 ~
References
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Biller B. A. et al., 2013, ApJ, 777, 160
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Declaration
I hereby declare that, except where reference to the work of others is made, this thesis and its
contents represent my own and original work. It was conducted at the Max-Planck-Institut für
Astronomie (Heidelberg) under the supervision of Priv. Doz. H. Klahr and therefore benefited from
his assistance and suggestions, as well as from those of Prof. Dr. Ch. Mordasini and Dr. R. Kuiper.
As carefully detailed there, a fraction of the work presented in Chapter 5 was submitted as part of
the requirements for the obtention of the degree of M. Sc. at McGill University (Montréal, Canada),
but no other part of the present thesis has been submitted at any other institution for any degree or
qualification.
Heidelberg, 7 December 2015
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