Dissertation submitted to the Naturwissenschaftlich-mathematische Gesamtfakultat¨ of the Ruprechts-Karl-Universitat¨ of Heidelberg,Germany for the degree of Doctor of Natural Sciences Put foward by Gabriel-Dominique Marleau, M. Sc. born in Verdun,Quebec´ (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 ober- sten 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 In- formationen Direktbeobachtungen über die Anfangsentropie liefern können. Erstens versuchen wir, genaue Kühlungskurven durch eine selbstkonsistente Einbindung der komplexen BT-Settl-2010- Atmosphärenmodelle zu berechenen. Wir stellen jedoch fest, daß dies aufgrund unkonvergierter Staubphasenübergänge bei Teff ≈ 1500 K unmöglich ist. Zweitens benutzen wir einfache Küh- lungskurven 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 Teff ≈ 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 – Abstracti Contents iii 1. Introduction 1 1.1. Shock physics . .2 1.1.1. Shock structure . .3 1.1.2. Conservation equations across the shock . .4 1.1.3. Estimate of the shock temperature . .7 1.2. Radiative flux . .8 1.3. The radiation quantity R and the reduced flux . .9 2. Shocks with 1-T radiation transport 11 2.1. Methods . 11 2.1.1. Model . 11 2.1.2. Initial set-up . 12 2.1.3. Boundary conditions . 12 2.2. Microphysics . 13 2.2.1. Equation of state . 13 2.2.2. Opacities . 14 2.3. Quantities of interest . 15 2.4. Relevant parameter space . 16 2.4.1. Formation parameters . 16 2.4.2. Microphysics parameters . 17 2.5. Results: Efficiencies and downstream quantities . 17 2.5.1. Dependence of the structure and efficiency on optical depth . 20 2.5.2. Results for selected cases with tabulated opacities . 24 2.5.3. Results for a large parameter space and constant opacity . 27 2.5.4. Results for a large parameter space and tabulated opacities . 31 2.6. Estimate of the coupling . 31 2.7. Discussion . 36 2.8. Summary . 38 3. Shocks with 2-T radiation transport 41 3.1. Introduction . 41 3.2. Selected cases with tabulated opacities . 42 3.2.1. Gas–radiation equilibrium . 42 3.3. Accreting a whole planet . 46 3.3.1. First results . 48 3.4. Conclusion . 50 4. Cooling tracks for gas giants using BT-Settl atmospheres 51 4.1. Introduction . 51 4.2. Background . 52 4.2.1. Overview of the cooling code . 52 ~ iii ~ 4.2.2. Description of the coupling . 53 4.2.3. Overview of BT-Settl atmospheres . 53 4.3. Results . 55 4.3.1. Input from the BT-Settl atmospheres . 55 4.3.2. Cooling tracks with BT-Settl . 55 4.4. Summary . 58 5. Constraining the initial entropy of gas giants 59 5.1. Introduction . 60 5.2. Cooling models with arbitrary initial entropy . 61 5.2.1. Calculation of time evolution . 61 5.2.2. Calculation of gas giant models . 62 5.2.3. Luminosity as a function of mass and entropy . 63 5.2.4. Luminosity as a function of helium fraction . 68 5.2.5. Cooling . 69 5.2.6. Comparison with classical hot starts and other work . 69 5.3. General constraints from luminosity measurements . 72 5.3.1. Shape of the M-Si constraints . 72 5.3.2. Solutions on the hot- vs. cold-start branch . 73 5.3.3. Definition of ‘hot-start mass’ . 73 5.4. General constraints from gravity and effective temperature . 75 5.5. Comparison with observed objects . 77 5.5.1. Directly-detected objects . 77 5.5.2. 2M1207 . 80 5.5.3. HR 8799 . 82 5.5.4. beta Pic . 83 5.6. Summary . 85 5.7. Appendix A: Radii . 88 5.8. Appendix B: Entropy offset in SCVH . 88 5.9. Appendix C: Age, luminosity, and mass constraints . 90 5.9.1. 2M1207 . 90 5.9.2. HR 8799 . 92 5.9.3. beta Pic . 93 5.10. Update to beta Pic b . 94 5.10.1. Summary . 95 5.10.2. Discussion . 95 5.11. Analysis of kappa Andromeda b . 95 5.11.1. Summary . 96 5.11.2. The mass of kappa And b: Warm-start models . 97 6. Conclusion 101 7. Acknowledgements 103 A. Summary and test of the radiation hydrodynamics code 105 A.1. Numerical settings . 105 A.2. Basic equations . 105 A.2.1. Internal energy equation . 106 A.2.2. Mechanical energy equation . 106 A.2.3. Radiation energy equation . 106 A.3. Operator splitting for the energy equation . 107 A.4. Radiation-hydrodynamics shock test . 108 A.4.1. Subcritical case . 109 ~ iv ~ A.4.2. Supercritical case . 110 A.4.3. Summary . 110 B. Additional material 113 B.1. Flux limiters . 113 B.2. Effect of the opacity . 113 B.3. Reduced fluxes for Bell & Lin (1994) opacities . 113 B.4. Parameter exploration . 113 B.5. Estimated Mach numbers for population synthesis planets . 114 B.6. Density inversions . 114 B.7. Extracting an atmospheric structure from a log file . 116 B.8. Coupled cooling tracks: Effective temperature evolution . 122 References 123 ~ 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.