these de doctorat – phd thesis` judit szulagyijudits/thesisszulagyi.pdf · universite nice sophia...

Universite Nice Sophia Antipolis – Observatoire de la Cote d’Azur

Ecole Doctorale des Sciences Fondamentales et Appliquees

These de Doctorat – PhD Thesisspecialite Sciences de l’Univers

Judit Szulagyi

Accretion du Gaz sur Planetes Geantes

Gas Accretion onto Giant Planets

Soutenue le 19 Novembre 2015 a l’Observatoire de la Cote d’Azur devant le jury compose de :

Defended November 19th 2015 at the Observatoire de la Cote d’Azur in front of the committee:

Gennaro D’Angelo Carl Sagan Center, SETI Institute Rapporteur/Reviewer

Richard Nelson Queen Mary, University of London Rapporteur/Reviewer

Yann Alibert Universitat Bern Examinateur/Examiner

Stephane Guilloteau Laboratoire d’Astrophysique de Bordeaux, Examinateur/Examiner

Universite Bordeaux

Michael R. Meyer Institute for Astronomy, ETH Zurich Examinateur/Examiner

Bruno Lopez Observatoire de la Cote d’Azur President/President

Alessandro Morbidelli Observatoire de la Cote d’Azur Directeur de these/PhD advisor

Aurelien Crida Observatoire de la Cote d’Azur Co-Directeur de these/PhD co-advisor

Cover picture: The image shows a circumplanetary disk formed around a Jupiter-mass planet which

is the focus of my thesis. The hydrodynamic simulation was carried out with JUPITER-code isothermal

version. The plot shows the volume density map via a vertical slice cut through the planet. Thanks to the

nested meshing technique, the resolution is higher approching the planet. Therefore, this plot does not

include the entire simulation box, just a small zoom-in to the planet vicinity.

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Contents

Introduction 9

0.1 Introduction in English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

0.1.1 Astronomy is the most ancient natural science . . . . . . . . . . . . . . . . . . . 10

0.1.2 Motivation of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

0.2 Introduction en francais . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

0.2.1 Astronomy is the most ancient natural science . . . . . . . . . . . . . . . . . . . 15

0.2.2 Motivation de cette these . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1 Overview 21

1.1 Young Stellar Objects and their Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.1.1 When do planets form? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.1.2 Disk Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.1.2.1 Disk Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.1.3 Structure of the Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.1.3.1 Vertical Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.1.3.2 Radial Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.2 Core Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.2.1 Pebble Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.3 Disk Instability Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.4 The missing population of super-giants . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.5 The circumplanetary disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.5.1 What we learned about the subdisk from hydrodynamic simulations so far? . . . 39

1.5.1.1 Circumplanetary Disk Qualitative Description . . . . . . . . . . . . . 40

1.5.1.2 Accretion Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.5.2 What we learned about the subdisk from observations? . . . . . . . . . . . . . . 43

2 Development of modules into the JUPITER hydro-code 49

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.2 The JUPITER code in its isothermal version . . . . . . . . . . . . . . . . . . . . . . . . 52

2.2.1 Basic description of the JUPITER code . . . . . . . . . . . . . . . . . . . . . . 52

2.3 New Modules for the JUPITER code: energy equation, heating, cooling . . . . . . . . . 55

2.3.1 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.3.2 The Radiative Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.3.3 Heating Effects – Stellar Heating . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.3.4 Heating Effects – Viscous heating . . . . . . . . . . . . . . . . . . . . . . . . . 63

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Contents Contents

2.3.5 Boundary Conditions for Cooling . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.3.6 Equilibrium & Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.4 Parameters of the code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.4.1 The parameter file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.4.2 Predefined variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.4.3 The arguments of the code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.5 Testing the code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.5.1 Testing the Riemann Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.5.2 Testing the Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.5.3 Testing the Radiative Module . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.5.4 The Ultimate Test: the Dimensional Homogeneity Test . . . . . . . . . . . . . . 81

2.6 Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2.6.1 The stability issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2.6.2 The cold circumplanetary disk problem . . . . . . . . . . . . . . . . . . . . . . 83

2.7 Fun-facts about the code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3 Accretion of Jupiter-mass Planets in the Limit of Vanishing Viscosity 89

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.2 Setup of the Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.2.1 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.2.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.3 Structure of the Circumplanetary Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.3.1 The Vertical Structure of the Circumplanetary Disk . . . . . . . . . . . . . . . . 94

3.3.2 Radial Structure of the Circumplanetary Disk . . . . . . . . . . . . . . . . . . . 101

3.3.3 Flow in the Midplane of the Circumplanetary Disk . . . . . . . . . . . . . . . . 102

3.4 Planetary Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.5 Discussion and Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.7 Appendix material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4 Meridional circulation of gas into gaps opened by giant planets in three-dimensional low-

viscosity disks 113

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.2 Gaps in 3D disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.3 Implications on planet’s accretion from the flow of gas into the gap . . . . . . . . . . . 120

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5 Circumplanetary Disk or Circumplanetary Envelope? 123

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.2 Physical Model and Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 125


5.3.1 Units, Frame and Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3.3 Disk Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3.4 Planetary Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

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5.3.5 Simulation sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.4.1 Circumplanetary disk or circumplanetary envelope? . . . . . . . . . . . . . . . . 129

5.4.2 Planet Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.4.3 Velocity and Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6 Mass of the Circumplanetary Disk and Accretion 139

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.2 Mass evolution inside the Hill-sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.2.1 Envelope Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.2.1.1 Comparison with FARGOCA . . . . . . . . . . . . . . . . . . . . . . 143

6.2.2 Disk Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.3 Mass of the Circumplanetary Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Own Publications on this Thesis Work 151

Own Publications Outside this Thesis Work 153

Conclusion 161

6.4 Conclusion in English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.5 Conclusion en franccais . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

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“When I dipt into the future far as human eye could see;

Saw the Vision of the world and all the wonder that would be.”

Locksley Hall by Alfred Lord Tennyson

Introduction

0.1 Introduction in English

The year of 2015, when these lines are written is UNESCO’s International Year of Light. This cannot be

overlooked, when such amazing successes have been made this year in astronomy, like the ROSETTA

mission’s Philae lander landing on a comet, the striking, detailed pictures on Pluto and Charon by the

New Horizons mission, and the beautiful images of HL Tau’s dust disk with the Atacama Large Mil-

limeter Array, where we see maybe a forming planetary system. The International Year of Light closing

ceremony will be in Chichen Itza in Mexico. The choice of the place is not incidental. This ancient Maya

city hosts one of the world most ancient, still existing observatory, which is about 1000 years old (Fig.

6).

Figure 1. The 1000 years old observatory in Chichen Itza, Mexico, as I saw it in 2010.

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0.1. Introduction in English Introduction

0.1.1 Astronomy is the most ancient natural science

Astronomy is not only in the focus today, it was always important throughout the history of humans.

We have evidences that humans acquired basic astronomical knowledge even in the pre-historic times.

Observing the lunar phases, the season changes, the daily and yearly passage of Sun was inevitable for

the time measurements and for the agriculture. There are several ruins and archeological artifacts that

witness the astronomical knowledge acquired in ancient times. One of the most famous examples are the

Stonehenge (built between 3000 BC - 2000 BC), the Nebra Sky Disc (circa 1,600 BC; Fig. 7), or the

pyramids in Giza (circa 3000 BC).

Figure 2. The Nebra Disk, artifact dated around 1,600 BC showing the sky with Sun, Moon, the

Pleiades and a few other stars from other constellations. The disk was found on the Mittelberg

mountain in Germany, along with other objects from the bronze-age era.

Early human civilizations discovered that there are fast moving, luminous objects apart from the

steady bright spots on the night sky. The Babylonians and Sumerians called them “stray sheep”, but

our current word “planet” comes from the ancient Greek word “planetes”, meaning nomad, wanderer,

traveler. But observing the planet’s passage on the sky is not the only thing Sumerians have mastered. As

early as around 4000 BC they already had a very modern cosmic picture, they put the Sun in the center of

the solar system with planets orbiting around it (Figure 8). Nevertheless they overestimated the number

of planets, they believed there are 10 planets in our planetary system. Their fascination about astronomy

is well represented in the nearly 25,000 astronomical texts in the famous Nineveh library. In fact, twelve

major constellations today origins from the Sumerian era, when they called them the “Shiny herd”. Table

2 summarizes their ancient and modern names. But thanks to Sumerians we measure the time in years, a

year in 12 months, the day in 24 hours, an hour in 60 minutes, a minute in 60 seconds. They really laid

the foundations of modern astronomy, and astrometry. The Sumerians even observed and described an

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Sumerian Translation Modern Name

GU.AN.NA Heavenly Bull Taurus

MASH.TAB.BA Twins Gemini

DUB Pincers, Tongs Cancer

UR.GULA Lion Leo

AB.SIN Her father was Sin Virgo

ZI.BA.AN.NA Heavenly Fate Libra

GIR.TAB Which claws and cuts Scorpio

PA.BIL (Archer) Defender Sagittarius

SUHUR.MASH Goat-Fish Capricorn

GU Lord of the waters Aquarius

SIM.MAH Fishes Pisces

KU.MAL Field dweller Aries

Table 1. Constellations from the Sumerian Culture

asteroid approach and collision with our planet.

Figure 3. A Sumerian seal showing the cosmic picture of the Sumerians: the Sun in the center on

the top-left part of the artifact with planets orbiting around it.

We can thank the Babylonians for the seven-day week, and the names of the days in a week, which

each connect with a celestial body. Latin, French, English names of the days show that they named Mon-

day after the Moon, Tuesday after Mars (“Mardi” in French), Wednesday after Mercury (“Mercredi” in

French), Thursday after Jupiter (french “Jeudi”), Friday after Venus (“Vendredi” in French), Saturday

after Saturn, and obviously, Sunday after the Sun. The Egyptian civilization was also very advanced

in astronomical skills. They knew five planets and recorded their passage on the sky with great preci-

sion. Thanks to them we use decimal counting system and Egyptians even used fractions. The people

of Pharaohs know about the precession of the equinoxes and there are evidences which suggest they

understood that Earth is spherical.

In the Americas, Maya have constructed calendars, and aligned their buildings, streets often with

celestial objects. They computed that the year is 365.25 days long, even though they used 365 days long

calendars without leap years. One of their major breakthroughs in astronomy was that they were able to

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predict eclipses very precisely, even hundreds years ahead.

Ancient Greeks believed the spherical Earth and measured first its radius. They had very advanced

mathematical skills for their era, even if we think about the “first computer”, the Antikythera Mechanism

from 83 BC, which was used to work out the motions of the five naked-eye planets, the Sun and the

Moon. Even though most of them believed that Earth is in the center of our world, Aristarchus already

believed in heliocentric system. Their observations about the stars made into stellar catalogs, one of the

most famous created by Hipparchus.

Sources:

• http://ephemeris.com/history/

• http://www.ancient-wisdom.co.uk/astronomy.htm

• https://en.wikipedia.org/wiki/Ancient˙Greek˙astronomy

0.1.2 Motivation of this thesis

Even though astronomy is probably the most ancient natural science, there are still a lot of unsolved

questions. How planets, and especially the gas-giants form is a focus of my studies. We are capable to

observe exoplanetary systems since more than 20 years ago, so we know that our planetary system is

not unique. In 1992 the first exoplanet was discovered (Wolszczan & Frail 1992) from irregularities of

a pulsar’s otherwise atomic clock sharp pulsations. Then, in 1995 the first giant planet was discovered

(Mayor & Queloz 1995) around a solar-type star thanks to the periodic oscillations of its radial velocity.

The subsequent years brought a boost of exoplanet discoveries similarly from the space, like from the

ground of Earth, thanks to missions like Corot, and Kepler. In mid-2015, the number of exoplanets is

close to 2000, and possibly pass over this number by the end of the year. The variety of the observed

planets and planetary systems are striking. There are odd systems where the giant planet is closer to its

star than Mercury from our Sun, while in others the planets are so tightly packed that it is impossible to

imagine that they are dynamically stable. There are giant planets discovered, which are lighter than our

Uranus, while others are 20 times more massive than Jupiter or even beyond that.

When a new star is born, the remaining gas around the young star is collapsing into a gas-dust disk.

This is the place where planets form (Fig. 9). Giant planets form differently from rocky planets. Gas

giants are thought to form in the outer planetary systems where the temperature is constantly below the

water’s freezing point. Still today we do not know how exactly giant planets form, there are multiple

competing theories in the literature, however it could well be that there is not only one unique way to

form these planets.

The most popular giant planet formation scenario, in the frame of which this thesis lies, is the core

accretion scenario (Bodenheimer & Pollack 1986) (see more details in Chapter 1). It predicts that first a

rocky core forms, which – after its mass is high enough – begins to attract gas around it. The gas piles

up around the rocky core, and eventually forms a gaseous envelope. However, at the final stages of

the formation, when the envelope mass overshoot the core mass, the planet grows in a runaway fashion

(Pollack et al. 1996). This is such a high rate, that our Jupiter would double its mass in this evolutionary

stages during 10,000 years. This is an extremely short time in astronomical sense. We know that gas-dust

disks around young stars have a lifetime of about 3 million years, which is a 300 times longer timescale

than the above mentioned 10,000 years. One would expect that as long as there is gas supply in this disk

for the giant planet to acquire more and more mass – so during 3 million years – the planet will grow. In

other words, if this core-accretion model with this runaway growing phase is true, an average gas disk

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Figure 4. Artistic conception about a dust-gas disk surrounding a young star and a forming giant

planet in the inner cavity. Giant planets are able to form a disk around them, which we call the

circumplanetary disk and is the focus of my studies. Studying the circumplanetary disk is very

important to understand better the late stages of giant planet formation, but also to understand how

the satellites of these gas-giants form. Image credit: P. Marenfeld & NOAO/AURA/NSF

would build up giant planets which are several ten or hundred times more massive than our Jupiter. In

fact, everywhere we would see, we should observe giant planets with these huge masses ... but this is

absolutely not what we see in the universe. Observations show the distribution of planetary masses as

shown on Fig. 10, where it can be seen that only few observed gas giants exceed above 3 Jupiter-masses.

Clearly, there is a mismatch between the observations and the theoretical model. Something seems

to limit or cut-off the last runaway growth phase, and this is the focus of my thesis. It is believed that

as a young star form its own disk, similarly, giant planets at the late stages of their formation are also

capable to form a so-called circumplanetary disk around themselves (Fig. 9). This disk around the giant

planet is maybe the key to limit the runaway growth. Understanding its role in the planet formation

process, determining how much gas it can channel down to the planet in a given period of time, and what

are the characteristics of this disk can reveal how giant planets form at their late evolutionary stages,

and how fast they form. Furthermore, regular satellites of giant planets are also believed to form in

these circumplanetary disks, so by studying them, we could even understand better the satellite forming

environment.

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Figure 5. The distribution of planetary masses of the observed exoplanet population. Clearly, the

distribution beyond Jupiter-size sharply declines and only fewer planets exceed above 3 Jupiter-

masses. Figure and data from exoplanet.eu .

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0.2. Introduction en francais Introduction

0.2 Introduction en francais

L’annee 2015, ou ces lignes sont ecrites, est l’annee internationnale de la lumiere de l’UNESCO. L’importance

considerable de la lumiere en astronomie a ete illustree par les succes extraordinaires de la mission

Rosetta (dont l’atterrissage de Philae sur une comete), les photos detaillees de Pluton et Charon par la

sonde New Horizons, et les superbes images du disque de poussieres de HL Tau avec ALMA (Atacama

Large Milimeter Array) dans lequel nous assistons peut-etre a la naissance d’un systeme planetaire. La

ceemonie de cloture de l’annee de la lumiere se tiendra a Chichen Itza au Mexique. Ce lieu n’est pas

choisi par hasard : cette ancienne cite Maya abrite l’un des plus anciens observatoires du monde, vieux

d’environ 1000 ans (Fig. 6).

Figure 6. L’observatoire de Chichen Itza, au Mexique, vieux de 1000 ans, tel que je l’ai vu en

2010.

0.2.1 Astronomy is the most ancient natural science

L’astronomie n’est pas seulement sous le feu des projecteurs aujourd’hui, elle a toujours ete tres impor-

tante pour l’humanite. Il y a des preuves que les hommes ont acquis des savoirs astronomiques de base

des la prehistoire. Observer les phases de la Lune, les passages journaliers et annuels du Soleil, etait

indispensable pour la mesure du temps et pour l’agriculture. Plusieurs ruines et restes archeologiques

temoignent du savoir astronomique acquis dans ces temps anciens. Parmis les plus celbres exemples, on

citera Stonehenge (3000 a 2000 avant J.C.), le disque de Nebra (∼ −1600 ; Fig. 7), ou les pyramides de

Gizeh (∼ −3000).

Les premieres civilisations humaines ont decouvert qu’il y a des objets lumineux qui se deplacent

rapidement, en plus des points brillants fixes sur le ciel nocturne. Les babyloniens et les sumeriens

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Figure 7. Le disque de Nebra, date d’environ -1600, montre le ciel avec le Soleil, la Lune, les

Pleiades et quelques autres etoiles et constellations. Ce disque a ete trouve dans les montagnes de

Mittelberg en Allemagne, avec d’autres objets de l’age du bronze.

les appelaient des “brebis egarees”, mais notre mot actuel, “planetes” vient du grec ancien “planetes”

qui signifie nomade, errant, voyageur. Mais observer le passage des planetes sur le ciel n’est pas la

seule chose que les sumeriens ont maitrise. Des 4000 av. JC, ils avaient une conception de l’univers

tres moderne, avec le Soleil au centre du systeme solaire, et les planetes en orbite autour (Figure 8).

Toutefois, ils ont surestime le nombre des planetes, pensant qu’il y en avait 10 dans notre systeme. Leur

fascination pour l’astronomie est bien representee par les quelque 25 000 textes astronomiques de la

fameuse bibliotheque de Ninive. En fait, douze constellations majeures d’aujourd’hui ont leur origine

dans l’ere sumerienne, quand ils les appelaient “la horde scintillante”. La table 2 resume leurs noms

anciens et modernes. Mais grace aux sumeriens nous comptons le temps en annees de douze mois, en

jours de 24 heures de soixante minutes de soixante seconde chacune. Ils ont vraiment pose les bases

de l’astronomie et de l’astrometrie moderne. Les sumeriens ont meme observe et decrit l’approche et la

collision d’un asteroıde avec notre planete.

Nous devons aux babyloniens la semaine de sept jours, et les noms des jours de la semaine, qui sont

chacun relies a un corps celeste. Les noms latins, francais et anglais montrent que Lundi est nomme

d’apres la Lune (Monday – Moon), Mardi d’apres Mars, Mercredi renvoie a Mercure, Jeudi a Jupiter,

Vendredi a Venus, Samedi a Saturne (Saturday = Saturn day en anglais), et dimanche (Sunday en anglais)

correspond au Soleil. La civilisation egyptienne avait aussi des talents astronomiques tres developpes.

Ils connaissaient 5 planetes, dont ils consignaient les deplacements sur le ciel avec grande precision. Le

peuple des pharaons connaissait la precession des equinoxes, et certains indices permettent de penser

qu’ils avaient compris que la Terre est spherique.

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Sumerien Traduction Nom Moderne

GU.AN.NA Taureau celeste Taurus

MASH.TAB.BA Jumeaux Gemini

DUB Tenailles, Pinces Cancer

UR.GULA Lion Leo

AB.SIN Son pere est pecheur Virgo

ZI.BA.AN.NA Destin celeste Libra

GIR.TAB Qui griffe et coupe Scorpio

PA.BIL (Archer) Defenseur Sagittarius

SUHUR.MASH Poisson-chevre Capricorn

GU Seigneur des eaux Aquarius

SIM.MAH Poissons Pisces

KU.MAL Habitant du champ Aries

Table 2. Constellations de la culture Sumerienne

Figure 8. Sceau sumrien montrant la representation cosmique des sumeriens, en haut a gauche :

le Soleil au centre avec les planetes orbitant autour.

En Amerique, les Mayas avaient construit des calendriers, et alignaient souvent leurs batiments et rues

sur des objets celestes. Ils avaient calcule que l’annA c©e dure 365,25 jours, meme s’ils utilisaient une

annee de 365 jours sans annee bisextile. Un de leurs plus grands succes en astronomie est la prediction

precise d’eclipses, jusqu’a des centaines d’annees a l’avance.

Les anciens grecs pensaient que la Terre est ronde, et furent les premiers a mesurer mesurer son

rayon. Ils avaient des competences mahtematiques tres avancees pour l’epoque, et ont meme construit

le mechanisme d’Antikythere en -83, qui servait a calculer les mouvements du Soleil, de la Lune et

des 5 planetes visibles a l’œil nu, et est parfois considere comme le premier ordinateur. Meme si la

majorite d’entre eux pensait que la Terre ´A c©tait le centre du monde, Aristarque croyait deja a un systeme

heliocentrique. Leurs observations des etoiles a conduit a la production de catalogues, le plus celebre

etant celui d’Hipparque.

Sources:

• http://ephemeris.com/history/

17


• http://www.ancient-wisdom.co.uk/astronomy.htm

• https://en.wikipedia.org/wiki/Ancient˙Greek˙astronomy

0.2.2 Motivation de cette these

Meme si l’astronomie est probablement la plus ancienne des sciences naturelles, il reste de nombreuses

questions sans reponse. Comment les planetes, et particulierement les planetes geantes se forment est

l’objet de nombreuses etudes. Nous sommes capables d’observer des systemes exoplanetaires depuis plus

de vingt ans, donc nous savons que notre systeme planetaire n’est pas unique. La premiere exoplanete

fut decouverte en 1992 (Wolszczan & Frail 1992) a partir des irregularites dans les pulsations d’un pulsar

normalement aussi precis qu’une horloge atomique. Ensuite, en 1995, la premiere planete geante a ete

decouverte (Mayor & Queloz 1995) autour d’une etole de type solaire, grace aux oscillations periodiques

de sa vitesse radiale. Les annees suivantes ont conduit a une explosion des decouvertes d’exoplanetes,

aussi bien du sol que de l’espace, grace notamment a des missions comme Corot et Kepler. A la mi-2015,

le nombre d’exoplanetes connues est proche de 2000, et va possiblement franchir ce seuil d’ici la fin de

l’annee. La variete des planetes et des systemes planetaires observes est etonnante. Il y a des systemes

bizarres ou la planete geante est plus proche de son etoile que Mercure de notre Soleil, tandis que dans

d’autres, les planetes sont si proches les unes des autres que la stabilite dynamique parait impossible. Il

y a des planetes geantes decouvertes qui sont plus legeres qu’Uranus, et d’autres qui sont 20 fois plus

massives que Jupiter, voire plus.

Quand une nouvelle etoile nait, le gaz restant autour de la jeune etoile s’effondre en un disque de

gaz et de poussieres. Ce disque est le lieu de formation des planetes (Fig. 9). Les planetes geantes et

telluriques ne se forment pas selon les memes mecanismes. Les planetes geantes se forment en principe

dans les zones externes des systemes planetaires, la ou la temperature est inferieure a la temperature de

sublimation de la glace d’eau. Encore aujourd’hui, nous ne savons pas exactement comment se forment

les planetes geantes ; il y a plusieurs modeles concurrents dans la litterature, et il se pourrait bien qu’il

n’y ait pas une facon unique de former ces planetes.

La theories de formation des planetes geantes la plus populaire parmi la communaute, et dans le cadre

de laquelle s’inscrit cette these, est celle de l’accretion de cœur (Bodenheimer & Pollack 1986) (voir plus

de details dans le chapitre 1). Elle suppose qu’un cœur rocheux se forme en premier qui – une fois sa

masse suffisante – commence a attirer du gaz. Le gaz s’accumule autour du cœur rocheux, et finalement

forme une envelope gazeuse. Mais vers la fin de la formation, quand la masse de l’enveloppe depasse

celle du cœur, la planete croıt de maniere exponentielle (Pollack et al. 1996). Ce taux de croissance est

si grand que notre Jupiter verrait sa masse doubler en 10 000 ans. Ce temps est extremement court, d’un

point de vue astronomique. Nous savons que les disques de gaz et poussiere autour des etoiles jeunes ont

une duree de vie de l’ordre de 3 millions d’annees, ce qui est 300 fois plus long que les 10 000 ans sus-

mentionnes. On s’attend a ce que la planete continue de grandir tant qu’il y a du gaz dans le disque pour

l’alimenter. Autrement dit, si le modele d’accretion de cœur avec cette phase de croissance emballee est

vrai, un disque moyen devrait produire des planetes geantes qui sont plusieurs dizaines a centaines de fois

plus massives que Jupiter. En fait, ou que nous regardions, nous devrions voir ces planetes aux masses

enormes... Mais ce n’est absolument ce que nous voyons dans l’univers. Les observations montrent

que la distribution des masses des exoplanetes, montree sur la Fig. 10, ne presente qu’une minorite de

planetes de plus de 3 masses joviennes.

Il y a clairement un desaccord entre les observations et les modeles theoriques. Quelque chose doit

limiter ou interrompre la derniere phase de croissance exponentielle, et c’est le sujet de ma these. On

18


Figure 9. Vue d’artiste d’un disque de gaz et poussieres entourant une jeune etoile et d’une planete

geante en formation dans la cavite interne. Les planetes geantes sont aussi capables de former un

disque autour d’elles, que nous appelons le disque circum-planetaire (CPD) et qui est l’objet de

mes travaux. Etudier le disque circum-planetaire est tres important pour mieux comprendre les

etapes tardives de la formation des planetes geantes, mais aussi pour comprendre comment les

satellites de ces planetes geantes se forment. Credit image : P. Marenfeld & NOAO/AURA/NSF

pense qu’une jeune etoile forme son propre disque ; de la meme maniere, les planetes geantes a la fin

de leur formation sont capables de former un disque circum-planetaire autour d’elles (Fig. 9). Ce disque

autour des planetes geantes est peut-etre la clef pour limiter la croissance emballee. Comprendre son

role dans le processus de formation planetaire, comprendre quelle quantite de gaz il peut conduire vers la

planete en un temps donne, et quelles sont les proprietes physiques de ce disque peut nous reveler com-

ment et a quelle vitesse se deroule la fin de la formation des planetes geantes. D’autre part, les satellites

reguliers des planetes geantes (tels les satellites galileens de Jupiter) se forment probablement dans ces

disques circum-planetaires donc en les etudiant, nous devrions comprendre mieux l’environnement de

formation de ces satellites.

19


Figure 10. Distribution des masses planetaires de la population des exoplanetes observees. Claire-

ment, la distribution au dela de la masse de Jupiter decroit, et peu de planetes depassent les 3

masses de Jupiter. Figure et donnees de exoplanet.eu .

20

“I love to doubt as well as know.”

Dante Alighieri

Chapter 1

Overview

Contents

1.1 Young Stellar Objects and their Disks . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.1.1 When do planets form? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.1.2 Disk Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.1.3 Structure of the Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.2 Core Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.2.1 Pebble Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.3 Disk Instability Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.4 The missing population of super-giants . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.5 The circumplanetary disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.5.1 What we learned about the subdisk from hydrodynamic simulations so far? . . . 39

1.5.2 What we learned about the subdisk from observations? . . . . . . . . . . . . . . 43

21

Overview Chap.1 Overview

1.1 Young Stellar Objects and their Disks

The dust and gas disks formed around young stars are the birthplaces of planets, therefore it is crucial to

study and characterize these disks in order to understand the conditions in which planets are formed.

Figure 1.1. Summarizing the evolution of the disks around pre-main-sequence stars. These tra-

ditional SED classes were defined by Lada (1987) with the extension by Andre et al. (1993). For

more explanation see text.

Young stars are born in massive, 104 - 106MS un Giant Molecular Clouds. As massive clumps form

in Giant Molecular Clouds, these clumps fragments into smaller cores, which then collapse and star

formation begins. First, the young protostar is surrounded by a massive, accreting envelope. These so-

called Class 0 objects (Andre et al. 1993) are observable mostly in sub-millimeter wavelengths (Figure

1.1). As shown in the figure, the spectral energy distribution (SED) of the object can be fitted by that of

a cold blackbody at this stage. Since the cloud had originally some angular momentum, it rotates, and

slowly an accretion disk forms around the star. Because of its turbulent viscosity, the gas is spiraling

inwards in the disk, feeding the star, while the angular momentum is transported outward in the disk.

Magnetic effects launch powerful polar jets of material, which evacuate part of the mass advected by the

disk. The SED’s of these Class I objects (Lada 1987) can be decomposed into the SED of a black body

22

1.1. Young Stellar Objects and their Disks Chap.1 Overview

plus a large IR excess, which peaks in the far-mid infrared (IR). With time, the accretion decreases, so

the bipolar jets weaken as well. The Class II phase SED (Lada 1987) is characterized by the pre-main-

sequence star’s blackbody emission and mostly near-mid IR excess due to the gas-dust protoplanetary

disk surrounding the young star. The accretion rate of an average T Tauri is of the order of 10−7 −10−8MS un/yr, sometimes with accretion bursts (∼ 10−6MS un/yr), when a large amount of material falls

onto the star in a short amount of time, like in the case of the FU Orionis- and EX Lupi-type objects and

other eruptive young star types. When the accretion drops below ∼ 10−9MS un/yr, the photoevaporation

begins to dominate. First, a large gap starts to form often at about 1 AU (so-called pre-transitional phase),

then large inner cavities can be observed (transitional disks). In the late stages (> 10 Myr) of the disks

the gas is totally dissipated away from the disk by photoevaporation and winds. After that the gas has

been removed only dust grains remains apart from the already formed planets and planetesimals. The

SEDs of the debris disk phase is mostly characterized by stellar blackbody emission, plus a little bit of

excess peaking around 100 microns.

1.1.1 When do planets form?

Thanks to the day by day increasing sample of exoplanets, it is more and more accepted in the community

that every young star has a circumstellar disk around it, and every one of them forms some kind of planets.

The diversity of the observed planetary systems poses a challenge for the planet formation scenarios,

since these models should work in every of the exoplanetary systems. One of the main questions is when

do the planets start to form and how long is their formation timescale.

Inevitably, planets mostly form in the Class I and Class II disk phases. Because these disks are very

bright in infrared wavelengths due to the strong emission of their dust component, it is very difficult

to observe young, forming planets. However, year by year, younger and younger planets are observed.

One of the first stunning cases was LkCa15 (Kraus & Ireland 2012) with its ∼ 2Myr age, and a planetary

candidate of at most 6MJup mass. Another earlier, but more debatable case was the sub-stellar companion

of GQ Lup. It has an age of < 2Myr, but the mass of the companion has a great uncertainty between 1

and 42 MJup (Neuhauser et al. 2005). If this is a planet, rather than a brown dwarf, it must have formed in

the circumstellar disk of GQ Lup within this short 2 Myr timescale. Recently, Quanz et al. (2015) found

HD100546 b via direct imaging. The planet is 5 − 10MJup, and it is only ∼ 1 Myr old, which makes it

the youngest exoplanet found to date. It is inevitable to mention the spectacular ALMA image of HL

Tau, where multiple gaps can be seen, possibly revealing young, forming planets (however, it is not the

only possible explanation; e.g. Zhang et al. 2015, Loren-Aguilar & Bate 2015). The age of the system is

around 100,000 years (with a maximum of 1 Myr, Swift & Welch 2008; Boss et al. 1989), which would

indicate that even in the outer disk, this timescale was adequate to form massive enough planets that can

carve gaps in the midplane’s dust. In conclusion, we have now observational evidence that planets exist

earlier than about 1 Myr, so the formation mechanism has to be fast enough to build giant planets within

this timescale. It is obvious that gaseous giant planets need to form within the disk’s lifetime, and the

latter is constrained to be less than 3-5 Myr (Fedele et al. 2010).

1.1.2 Disk Parameters

In order to estimate the properties of the former protoplanetary disk around the Sun, Weidenschilling

(1977) and Hayashi (1981) came up with the idea of Minimum Mass Solar Nebula (MMSN). The MMSN

is constructed by taking the mass of each planet, increasing it by adding the missing elements in order to

restore a bulk solar composition, and spreading the resulting mass over an annulus ranging from/to the

23


half distance to the neighboring planet. The surface density of the gas, dust and ice components of the

MMSN can be written as follows (Dullemond 2013):

ΣMMS NGas (r) = 1700

(

r

1AU

)−3/2

g cm−2 (1.1)

ΣMMS NDust (r) = 7

(

r

1AU

)−3/2

g cm−2 (1.2)

ΣMMS NIce (r) = 22

(

r

1AU

)−3/2

g cm−2 (1.3)

Note that the ice component is only present beyond the snowline. To extend the surface density to

volume density (since the disk has a thickness), the relationship between the surface and volume densities

is:

Σ(r) =

∫ +∞

−∞ρ(r, z)dz (1.4)

where r is the distance from the star, z is the disk height. The MMSN disk is only an approximation,

the observed real disks are often more massive, sometimes even an order of magnitude more. The other

problem with the MMSN disk is that it has a very steep profile (r−1.5), therefore, the inner disk is very

massive. This is again in contrast with the observed disks, which usually have a flatter profile (Williams

& Cieza 2011). From observations, the power law exponents of the disk surface density profile is around

1.0 (e.g. Andrews et al. 2010). Disks evolving viscously, are also expected to achieve a surface density

profile like 1√r

(in the inner part) and 1r

(in the outer part) as it will be explained later (Sect. 1.1.3). Usually,

a T Tauri (solar type pre-main-sequence stars with < 1.8MS un) disk mass is assumed to be on the order

of 0.01 MS un (e.g. Andrews et al. 2010), with a few orders of magnitude scatter. However, measuring the

mass of the disks is not an easy task. Traditionally, the most common way is to measure the mass of the

dust from continuum observations (the dust continuum emission in the infrared - submillimeter range is

a direct tracer of the total dust opacity and, if a mean grain emissivity is assumed, of the dust column

density (Groves et al. 2015)). Then, the inter-stellar medium’s dust-to-gas ratio of 0.01 is assumed to

estimate the mass of the gas component. However, nowadays it becomes clearer that this value of 0.01

is probably away from reality, and it is probably a different value in every system. We also know that it

changes with the evolution of the disk: in the early stages, the gas component is basically 100% of the

matter in the disk due to the very high temperatures around the time when the star is born, but as the

disk cools, more and more solids condensate out. By the debris disk phase we end up with nearly 100

% dust and no gas. Therefore, the unknown real value of dust-to-gas ratio puts a large error bar on the

disk masses. Note, however that today it is possible to directly measure the gas mass from observations

with very high resolution spectroscopy, but the sample of disks for which this has been done so far is

much smaller than for the dust-based estimation sample. One of the difficulties is that the gas emits only

at specific wavelengths (spectral lines), for which sufficiently high resolution measurements are needed

with the world largest telescopes and best spectrographs.

Another important parameter in disks is the viscosity. There are different origins for the viscosity.

The first one is the molecular viscosity νm ≈ csλ, where λ is the mean free path of the molecules and cs is

the sound speed. However, the molecular viscosity is so low in accretion disks that it could be neglected.

Turbulence, however, should be present in disks, which can lead to effective turbulent viscosity. There

are a few mechanisms which can yield to turbulence in disks, e.g. magneto-rotational instability (e.g.

24


Balbus & Hawley 1991), baroclynic instability (Klahr & Bodenheimer 2003; Klahr 2004) and the vertical

shear instability (Nelson et al. 2013; Stoll & Kley 2014). However, our goal here is to parameterize the

viscosity, instead of studying its cause. In an ideal gas, the viscosity is due to collisions among the

molecules. This molecular viscosity is too small to explain the accretion rates measured on stars, which

are of the order of 10−6 − 10−9MS un/yr, depending on disk mass and stellar age. If the disk is turbulent, a

viscosity can arise from the interaction among the turbulent eddies. Shakura & Sunyaev (1973) reasoned

that, the viscosity having the dimensions of a velocity times a length, the turbulent viscosity should

be proportional to the size and to the rotational velocity of the eddies dominating the turbulent flow.

The largest vortices, which obviously are the dominant ones, cannot have a length scale larger than the

scale-height of the disk (H). Any vortex of a scale l has a rotation with a peripherical velocity Ωl, from

the Keplerian shear formulae. Therefore, a vortex of scale H has a velocity equal to the sound speed

(cs = ΩH). Thus, turbulent viscosity is parameterized by Shakura & Sunyaev (1973) as:

ν = αcsH (1.5)

where α is an adimensional parameter smaller than 1. According to observations, deducing the value

of α from the accretion rate of the star, the typical value is on the order of 10−3 - 10−2 (Dullemond 2013;

Andrews et al. 2010; Hartmann et al. 1998). We will see in Sect. 1.1.3.1 that cs = HΩ, so Eq. 1.5 can be

rewritten as ν = αH2Ω.

The characteristic sizes of disks are a few 10-100 AUs (e.g. Andrews et al. 2010). In general, it can

be said that disk mass correlates with stellar mass, so more massive stars typically have more massive

disks.

1.1.2.1 Disk Temperature

Let us study now the temperature-structure of the disk. First, the heating and cooling processes should

be discussed. Heating is based on two processes: the irradiation by the star and the viscous heating

generated by the differential rotation of the disk (Keplerian shear). The cooling comes from the fact that

the disk is radiating its heat away.

First let us see the case when only viscous heating is present. We first need to find a relationship

between the surface density profile of the disk and the viscosity. For this purpose, consider that the

elementary torque acting on an annulus of the disk of width δr is equal to the derivative of its angular

momentum. Hence:

− 3πd(Σνr1/2)

drδr = δr

d(2πr3/2Σ)

drvr (1.6)

Now, choose the surface density profile like: Σ = ra; and the viscosity like ν = rb then substituting

these power-laws in the equation above, we find the equation −ra+b−0.5 ≈ ra+0.5vr, which implies vr ≈−rb/r = −ν/r. We can assume that the mass transported through the disk is independent of the distance

from the star (an assumption which is correct for an accretional disk far enough from the disk’s outer

edge (Lynden-Bell & Pringle 1974), i.e.: dM/dt = 2πrΣvr = Const. Then, from vr = v0ν/r one can see

that νΣ = Const.

In an annulus of radius r and width of δr the viscous heating can be written like (Bitsch et al. 2013):

Q+visc =9

8ΣνΩ2

K2πrδr (1.7)

25


For the cooling, one can assume that the entire heat is radiated away on the two surfaces of the disk.

Then the cooling in the same annulus is:

Q− = −2σS BT 4e f f

1

κΣ2πrδr (1.8)

Here κ denotes the opacity of the disk. As we will show in Sect. 1.1.3.1 (see Eq. 1.30), the relationship

between the temperature and the scale height of the disk is H ∝√

Tr3. Using this relationship, the scale

height of the disk can be obtained equating Eq. 1.7 and 1.8. This leads to (retaining only the terms

dependent on r and assuming κ=constant):

Ω2r ∝ rH8

r12

1

Σ(1.9)

(

H

r

)8

∝ Ω2r4Σ = rΣ (1.10)

(

H

r

)

∝ (rΣ)1/8 (1.11)

Because Σ ∝ 1/ν, therefore(

Hr

)

∝ (r/ν)1/8. Remembering now that ν ∝ H2Ω one gets:

(

H

r

)

∝(

r5/2

H2

)1/8

=

[

(

r

H

)2

r1/2

]1/8

(1.12)

(

H

r

)5/4

∝ r1/16 (1.13)

(

H

r

)

∝ r1/20 (1.14)

So H ∝ r21/20. Plugging this dependence into the viscosity formula we find:

ν = H2Ω ∝ r21/10 1

r3/2≃ r2

r3/2= r1/2 (1.15)

This means, Σ = 1/ν ∼ 1/√

r.

The other source of the heating is due to the stellar irradiation. The stellar flux is F∗ = L∗/(4πr2). The

irradiation flux is the projection of the stellar flux onto the disk surface, so for the same annulus:

Firr = sin(φ)F∗ ≈sin(φ)L∗

4πr22πrδr (1.16)

where φ is the incidence angle of the stellar radiation on the disk’s surface. Therefore the stellar

heating on the disk’s two surfaces is:

Q+irr =L∗

4πr22 · 2πrδH (1.17)

where δH = sin(θ)δr. Its expression can be derived as follows. The scaleheights of the disk in two

given points r1 and r2 are: H1 = (H/r)1r1 and H2 = (H/r)2r2, the angle θ between a straight line from the

star and the surface of the disk is::

26


sin θ =H2 − H1

δr− H

r=

(H/r)2r2 − (H/r)1r1

δr− H

r(1.18)

from which

δH = rd(H/r)

drδr (1.19)

The cooling of the disk can be written as:

Q− = −2σS BT 4e f f 2πrδr (1.20)

This formula is different from formula 1.8 because the heat is absorbed and re-irradiated at the surface

of the disk, whereas in the viscous heating case the heat is produced near the mid-plane, transported

through the disk to its surface and then irradiated away. This explains why Eq. 1.8 had the 1κΣ

term, that

is not present in formula 1.20.

At equilibrium the heating (Eq. 1.17) should be equal to the cooling (Eq. 1.20). Again, we use that:

H ∝√

Tr3 (1.21)

If we plug this in to Eq. 1.20, the (H/r) profile will result:

H

r∝ r2/7 (1.22)

The equation 1.22 shows that protoplanetary disks dominated by stellar irradiation are flared, i.e. the

scale-height of the disk over the radius is not constant, but H/r ∝ r f . The usual value for f from ob-

servations is around 0.3-0.5 (Chiang & Goldreich 1997) in good agreement with the exponent computed

above. Because Σ ∝ 1/ν and ν =∝ H2Ω, using H ∝ r9/7 implies Σ ∝ r−15/14 ∼ 1/r.

The inner disk is viscous heating dominated, while in the outer disk stellar irradiation is the main

source of heat. The boundary between the two regime is of course depends on the parameters of the disk;

the more massive and viscous is the disk the farther from the star is the boundary between the two heating

regimes (i.e. the radius from which the disk is flared). The consideration of a temperature dependence

opacity κ introduces wiggles in H/r relative to the profiles computed analytically in this section, which

can produce shadowed regions in the disk (Bitsch et al. 2014).

1.1.3 Structure of the Disk

1.1.3.1 Vertical Structure

The vertical disk structure of the disk can be evaluated by solving the hydrostatic equilibrium equation

(Mordasini 2011):

1

ρ

∂P

∂z= −GM∗

r2

(

z

r

)

= −Ω2Kz (1.23)

where p represents the pressure, ρ is the density. Assuming isothermal equation-of-state (EOS) P =

c2sρ and

∂c2s

∂z= 0:

27


c2s

ρ

∂ρ

∂z= c2

s

∂ln ρ

∂z= −Ω2

Kz (1.24)

∂ln ρ

∂z= −Ω2

Kz

c2s

= − z

H2(1.25)

where H is the pressure scaleheight defined as cs/ΩK . After integrating up:

ln ρ = ln ρ0 −z2

2H2(1.26)

So the solution is:

ρ(z) = ρ0 exp− z2

2H2(1.27)

This means the densest part of the disk is the midplane, and the gas is exponentially decreasing as we

go towards the surface. The pressure scaleheight defines an important parameter of the disk, namely the

aspect-ratio, which is H/r.

The scaleheight of the disk is linked to its temperature, which can be derived in the following way.

First, consider the ideal gas law:

P =R

µρT(1.28)

If we plug in this into Eq. 1.23 and recalling that dT/dz = 0 by the assumption of a vertical isothermal

disk we get:

∂ log ρ

∂t= −Ω2

Kzµ

RT(1.29)

Therefore:

H =

(

RTµ

)1/2

ΩK

(1.30)

This means that indeed, H ∝ (Tr3)1/2. Using cs = HΩ we find:

cs =

(

RT

µ

)1/2

(1.31)

1.1.3.2 Radial Structure

In the radial direction, the gravitational force towards the star balanced by the gas pressure and the

centrifugal force, thus the equilibrium can be written as:

v2

r=

GM∗

r2+

1

ρ

∂P

∂r(1.32)

The Eq. 1.32 implies that:

v2

r= Ω2

Kr + c2s

∂ log ρ

∂r(1.33)

28

1.2. Core Accretion Chap.1 Overview

Assuming ρ ∝ r−β, we can rewirte Eq. 1.33:

v2

r= Ω2

Kr − βc2

s

r(1.34)

This is equal to:

Ω2Kr

(

1 −βc2

s

r2Ω2K

)

= Ω2Kr

(

1 − βH2

r2

)

(1.35)

which leads to:

v = vK

[(

1 − βH2

r2

)]

(1.36)

This means that the gas is orbiting sub-Keplerian velocity, it is slower than the dust component.

1.2 Core Accretion

The Core Accretion model by Pollack et al. (1996) and Bodenheimer & Pollack (1986), is the most

accepted giant planet formation scenario. It consist three main phases, as shown on Figure 1.2. The

first phase is the coagulation of solids into a core of ∼ 10 MEarth. This stage ends when the proto-planet

reached the isolation mass, i.e. the planet emptied its feeding zone of planetesimals. When the proto-

planet reached this size, the gravity of the planet is already enough to keep some gas bounded to the

planet, hence the slow envelope accretion – the second phase – begins. When the envelope grows, the

feeding zone of the planet becomes slightly larger as well, therefore a bit more solids are accreted to the

core. As the core and envelope grow, the total gravity grows as well, which leads to the attraction of more

gas. This second stage is the longest, it can last for couple of million years. At every time, the envelope

is in a state of hydrostatic equilibrium, as described by the equations below:

dP

dr= −GM(r)ρ

r2(1.37)

where ρ stands for the density at a particular radius r from the center of the planet, and M(r) is the

integrated mass till this radius r. This equation is valid for r > rcore and M(rcore) = Mcore. The equation

for mass conservation can be written as:

dM(r)

dr= 4πr2ρ (1.38)

while the energy equation is:

dL

dr= 4πr2ρǫ (1.39)

where ǫ represents the heat deposited at radius r by the accretion of planetesimals.

Finally, the equation for the radiative transport:

dT

dr= − 3κLρ

64πσr2T 3(1.40)

here, κ is the opacity, L represents the luminosity, σ is the Stefan-Boltzmann constant and T stands

for the temperature. To close the system of equations, the equation of state is needed to be defined:

29


P =kB

µρT (1.41)

where kB is the Boltzmann constant and µ is the mean molecular weight.

The system of equations above is a closed system that can be solved numerically, once ǫ is known.

Knowing ǫ requires knowledge of the accretion rate of solids and of how the solids sink into the envelope.

The system can be simplified assuming that dL/dr = 0 and L = GMcoreM/rcore. Then, the solution of

the system of equations depends just on two parameters: Mcore and M (or L). Once L is fixed, one finds

that there is a solution M(r) for every value of Mcore up to a limit value Mcrit. Mcrit decreases steeply with

L (Mizuno 1980; Hori & Ikoma 2010; Rafikov 2006). At Mcore = Mcrit, the envelope mass is typically

comparable to the core mass.

When the hydrostatic equilibrium is broken, the accretion proceeds in a runaway (i.e. exponential)

fashion (phase III of Pollack et al. model). In fact, the process is dominated by the gas self-gravity so that

the more massive is the envelope, the faster it accretes. The envelope contracts on the Kelvin-Helmholtz

timescale. This is the approximate time it takes for a planet to radiate away all of its gravitational potential

energy at its current luminosity rate (Ida & Lin 2004):

tKH ≃ 10b

(

Mp

MEarth

)−c (κ

1gcm−2

)

yr. (1.42)

where b and c power-law indexes range between 8-10 and 2.5-3.5, respectively (Ida & Lin 2004)

based on the opacity (κ) considered.

The formation only stops when no more gas can be accreted, e. g. when the left-over gas in the

circumstellar disk cleared away by photoevaporation. The accretion rate is nevertheless limited by the

capability of the circumstellar disk to transport gas towards the planet orbit: so Mplanet < Mdisk. Lubow

& D’Angelo (2006) find that the accretion rate of a Jupiter mass planet can be ∼90% of Mdisk (the latter

measured outside of the planet’s orbit).

Thus, if the runaway accretion phase starts at time t0, the final mass of the planet will be:

∫ ∞

t0

Mdisk exp(−t/τ)dt (1.43)

where Mdisk is the mass flux in the disk at time zero and τ is the decay constant of the mass flux. This

integral (Eq. 1.43) has to be of the order of Jupiter’s mass if one wants the final mass distribution of

giant planets to peak at this mass value. It is unclear if this can be the case. At the beginning of the disk

lifetime, Mdisk is large (a few 10−7 solar masses/yr) and τ is large (about a Myr; Hartmann et al. 1998), so

that the integral gives hundreds of Jupiter masses. Later, when the disk’s accretion rate drops to ∼ 10−9

solar masses per year, photo-evaporation becomes very rapid and τ is equal to a few 105 years only. So,

the value of the integral would be of the order of 1/10 of a Jupiter mass only. Thus, reproducing the giant

planet mass distribution may be problematic.

Another proposed stopping mechanism in the literature for the runaway growth is the gap opening of

giant planets. However, the fact that the planet opens a gap does not prevent accretion, and in this thesis

it will be shown that actually 90% of the gas is accreted through the gap from the polar direction. Some

older studies based on 2D simulations suggested that for very large mass planets (i.e. 5-10 Jupiter-mass),

the accretion is decreased or stopped due to the very large gap width (Kley 1999; Lubow & D’Angelo

2006), but others found that the accretion again enhanced in this planetary mass regime (Kley & Dirksen

2006). In fact, 3D simulations clearly show that the accretion process is very different than what is found

30


Figure 1.2. Core accretion by Pollack et al. (1996). The colors representing the 3 main phases of

core accretion: I - solid core coagulation, II - slow envelope accretion, III - runaway gas accretion.

The pink arrow pointing to the critical core mass, when the core and envelope mass are equal, the

point when the runaway accretion starts.

in 2D, and our own experience is that the accretion does not stop even at 10MJup planets (see Chap. 4).

Therefore, this accretion stopping mechanism due to gap opening is less likely.

Another problem with the core accretion scenario is the overall formation timescale, which is about

10 million years for a Jupiter-mass planet at Jupiter’s location (i.e. at 5.2 AU; see Fig. 1.2). However, the

average lifetime of the gas disk is ∼ 5 million years (Fedele et al. 2010), and for all disks the debris disk

phase (i.e. when almost no gas is left in the system) is definitely reached by 10 million years according to

the recent observations (e.g. Rieke et al. 2005; Su et al. 2006; Ertel et al. 2014; Wyatt et al. 2015; Chen

et al. 2014). Therefore, overall, the core accretion model seems to be too slow to form the majority of

the giant planets. The length of phase II of the core-accretion scenario can be reduced if one assumes

a disk with reduced opacity. Nevertheless, the core accretion process becomes slower with increasing

distance from the star. Hence the large semi-major axis planets (e.g. the directly imaged wide separation

planets, such as in the system of HR8799, for the entire sample see the Figure 1.3) cannot be explained

by this formation scenario, unless these planets migrated outwards significantly from their birth-place.

Even though Piso et al. (2014) argued that the critical core mass, which induce runaway gas accretion,

drops to 5MEarth at 100 AU with realistic equation-of-state and opacities, Rafikov (2011) showed that

this critical core mass cannot be reached over the gaseous disk’s lifetime beyond 40-50 AU. In the Solar

System, the formation of Uranus and Neptune is also problematic (Levison & Stewart 2001).

Finally, the last problem with the Pollack et al. (1996) model is that it is a 1-dimensional model, while

the 3D mechanisms can change the effects and, particularly, the angular momentum of the infalling gas

relative to the planet, which cannot be captured in a 1-D model, as this thesis will show. Since the giant

planets form circumplanetary disk around them, this disk can affect the accretion rate.

On the other hand, the basis of the scenario could be still valid, with some suitable variants. One

of the main supporting factors for the core accretion model is that all giant planet cores but (possibly)

Jupiter’s have massive cores in their interior, of about 10 Earth masses (Guillot 2005). Jupiter might

31


Figure 1.3. Long period exoplanets with semi-major axes over 40 AU – figure and data from

exoplanet.eu

have no core today (Nettelmann et al. 2008). However, there are several tens of Earth masses of “metals”

(molecules heavier than H and He) in Jupiter (Guillot 2005), and it is possible that part or even most

of its primordial core has been eroded and dissolved into the atmosphere (Guillot et al. 2004; Wilson &

Militzer 2012).

From an observational point of view it is crucial to mention how the luminosity of the planet changes

during these stages of formation. Lissauer et al. (2009) and Mordasini et al. (2011) showed that the

proto-planet quickly gains an order of magnitude in luminosity during the first phase of formation, then

the luminosity decreases a bit at the beginning of the second phase, followed by a slow increase as the

planet mass slowly grows as well (Figure 1.4). Then the luminosity very sharply increase again by more

than 2 orders of magnitude in the runaway phase. At that time, the young Jupiter should have reached

a spectacular 10−4LS olar luminosity, which shows that young, forming proto-planets could be observed

especially during the runaway phase. The likelihood of observing a forming planet is even better in the

near-infrared, where the planet is the most luminous compared to its star. Indeed, in the last 7 years, great

efforts have been made and results have been achieved in direct imaging of exoplanets, and even some

of these planets were caught during their formation. A few examples were mentioned before, in Section

1.1.1.

1.2.1 Pebble Accretion

A big improvement of the core-accretion scenario is offered by the pebble accretion model for planetary

cores developed in Lambrechts & Johansen (2012, 2014). They showed that the accretion of cm-sized

pebbles – in contrast to the classical, km sized planetesimals – can enormously speed up the first phase

of the core accretion, namely the solid core accretion phase. According to their analytical calculations

and numerical simulations, the timescale of accretion can be shortened by a factor of 30-1000 at 5 AU,

and 100-10000 at 50 AU. Of course, for this model to work, one needs a significant amount of cm-dm

sized pebbles in the midplane, which is supported by the dust coagulation and drift model developed in

32


Figure 1.4. The top figure represents a formation of a planet via core accretion, similarly to

Figure 1.2. Notice however the reduced timescale relative to that figure, due to a smaller assumed

opacity of the disk. The bottom panel shows the corresponding luminosity of the planet, during its

formation. For more explanation see the text. Figure from Lissauer et al. (2009).

Lambrechts & Johansen (2014). Moreover, ∼1000 km sized embryos needed as seeds. The latter could be

formed by via gravitational instability after that self-gravitating clumps of pebbles are created either by

33

1.3. Disk Instability Model Chap.1 Overview

the streaming instability (Johansen & Youdin 2007) or by concentration in vortices (Johansen & Youdin

2007; Cuzzi et al. 2001, 2008). Also, there are observational evidences for large amount of pebbles in

young circumstellar disks (e.g. Wilner et al. 2005; Zhang et al. 2015), which grew through the process of

dust coagulation and settling in the midplane.

The pebble accretion scenario predicts a large population of ice and gas giant planets, around very

young (∼ 1 Myr) stars. Indeed, a couple of directly imaged planets fit into this picture, and the amount of

planets in this regime grows year by year. One additional advantage of the pebble accretion scenario is

that it can explain why some giant planets end up as ice giants, like Uranus and Neptune, and why some

become massive gas giants like Jupiter or Saturn. In fact, Lambrechts et al. (2014) showed that at a given

core mass, the proto-planet carves a gap in the pebble disk. This stops the further accretion of pebbles

resulting in a drastic drop in core’s luminosity. As shown in Sect. 1.2, a decrease in luminosity decreases

the critical core mass for the runaway gas-accretion phase. Thus, the core of the planet, once pebble

accretion is shut off, becomes suddenly super-critical. This triggers the phase III of the core-accretion

scenario without passing through a long phase II.

In this scenario the ice-giant cores never reached this critical mass to open a gap in the pebble disk.

They have a thin envelope enriched in water vapor by the accreting icy pebbles. According to this model,

the ice giants are predicted to be in the outer disk, where the critical core mass to open a gap in the pebble

disk is very high, while gas giants are supposed to form somewhat closer to the star, but still beyond the

ice-line, which also fits very well for the Solar System case. Note, however, that the pebble accretion

process can provide the solution only for the formation timescale problem of the cores of the planets, but

not to the problem of the final mass distribution of giant planets.

1.3 Disk Instability Model

The major planet formation scenario, alternative to the core-accretion model, is the disk instability model

(Boss 1997). Here the gravitational instability of gas creates spiral arms and clumps in the protoplanetary

disk, which then collapse into proto-planets. Typically, for this scenario to work, one needs massive

disks, and even in that case usually only the outer parts of the disk could become gravitationally unstable.

This model has the advantage over the core accretion model that it needs significantly less time to form

a planet (103 years for a Jupiter-mass planet), especially in the outer disk. Therefore, it is a favored

model to explain planets that orbit far away from their star. However, the weakness of this model is

that it requires special conditions to occur, namely the disk has to become gravitationally unstable. To

determine whether a disk is gravitationally unstable, the Toomre criterion (Toomre 1964) is widely used:

Q =csΩk

GπΣ(1.44)

where cs is the sound speed, G is the gravitational constant, Σ stands for the surface density of the

gas. For the instability to happen, Q < 1 is needed. More precisely, sub-unity values mean that the disk is

unstable with respect to ring-like disturbances. In fact, spiral arms already appear at Q ≤ 1.7 (Durisen et

al. 2007), and it was proved that for isothermal disks, fragmentation happens when Q <1.4 (e.g., Nelson

et al. 1998). One of the reason for the flexibility on the threshold value is that the criterion was derived

for an infinitesimally thin, axis-symmetric disk, and real disks are very different. Therefore, one needs

to keep this in mind when determining the instability of a simulated or observed disk. Often the disk

instability scenario is ruled out to explain an observed young planetary system with the argument that

the disk is not gravitationally unstable – at the present, observed stage. However, one should bear in

34

1.4. The missing population of super-giants Chap.1 Overview

mind that we have no information of what were the disk parameters when the planets formed. Therefore,

it cannot be determined whether the disk could have been unstable originally, and whether this type of

formation mechanism could have created the observed planets in the system. Another constrain for the

disk instability scenario is the cooling timescale of the disk (β = τcoolΩK). The gas clumps need to cool

and contract rapidly, otherwise they are destroyed by the disk’s shear. What is a sufficient cooling time at

a given radius of the disk is highly debated to the literature, but it ranges from a few to a couple dozen of

orbital periods.

The gravitational instability model also presents the problem of the final mass distribution of giant

planets in an even more sever way than the core accretion model. In fact, the giant planets should form

quickly and so they should keep accreting when the disk has still a large accretion rate.

1.4 The missing population of super-giants

Since 1992, when the first exoplanet was discovered around a pulsar (Wolszczan & Frail 1992), and

shortly after a solar-type star (Mayor & Queloz 1995), the planet population increased enormously, to

date, 1855 exoplanets discovered 1. This vast sample allows to study the distribution of parameters of

the planets. If we look at the planetary masses of the giant planets (see Figure 1.5), we see that the large

majority of them has sub-Jupiter masses. The distribution after Jupiter-size sharply declines and only

very few planets exceeds 3 Jupiter-masses. In other words, the super-giants are very rare in the universe.

As we discussed in Sect. 1.2, this is surprising if one considers the runaway phase of the core accretion.

In this stage, the proto-planet accretes gas very fast, limited only by the ability of the disk to supply new

gas to the planet’s vicinity. The accretion rate supported by the disk being large (M ∼ 10−8MS un/yr)

before the photo-evaporation phase, this implies that the majority of the giant planets should have several

Jupiter-masses, in contrast with the observed distribution. There must be either a stopping mechanism for

gas accretion, or a regulator. In this thesis, it will be shown that the circumplanetary-disk formed around

the growing giant planet may, under some conditions, act as a regulator for the gas accretion.

In the recent years, planet population synthesis models (e.g. Mordasini et al. 2009a,b; Alibert et al.

2011; Mordasini et al. 2012a; Fortier et al. 2013; Alibert et al. 2013; Ida & Lin 2008, 2010; Ida et al. 2013)

tried to build populations of planets which using realistic, global formation and evolution simulations.

They set up different initial parameters, and boundary conditions to reproduce the observed distributions

of planets. This task is very difficult, as many aspects regarding the formation and evolution of planets

are uncertain. The most complex models in the recent papers (Fortier et al. 2013; Alibert et al. 2013),

now include improved physical model on the internal structure of forming planets, a detailed description

of the dynamics of the planetesimal disk, taking into account both gas drag and dynamical excitation of

forming planets.

Nevertheless, there are important uncertainties in the theory behind these models, which heavily affect

the planet population synthesis (Mordasini et al. 2015). These are:

• the formation of planetesimals resulting in the initial distribution of solid material in the circum-

stellar disk,

• the mechanism of accretion of the planetary core,

• the opacity which also affects the accretion rate,

1according to exoplanet.eu as of 2015/07/26

35


Figure 1.5. The distribution of planetary masses of the observed exoplanet population. Clearly, the

distribution beyond Jupiter-size sharply declines and only fewer planets exceed above 3 Jupiter-

masses. The distribution of planets more massive than 0.3 Jupiter masses can be considered unbi-

ased (Mayor et al. 2011). Figure and data from exoplanet.eu

• the migration time-scale of embryos, and planets,

• the gas accretion rate in the runaway phase.

This thesis addresses the last point. In the planet population synthesis models, there are two types of enve-

lope accretion rates to calculate. First, for a small core the gas accretion rate is calculated through the in-

ternal structure equations (see Eqs. 1.37-1.40). The accretion rate is depending on the Kelvin-Helmholtz

timescale (see Sect. 1.2). At this stage of the planet formation, the Kelvin-Helmholtz timescale is long,

which means small accretion rate. But as the envelope mass approaches the core mass, the accretion rate

rapidly increases until it reaches a maximal value at which the circumstellar disk can feed the planet with

gas. The envelope contracts and the gas keeps accreting (see also Sect. 1.2). At beginning, when the

planetary envelope accretes from the immediate vicinity of the core, the accretion rate is approximated

as follows (Mordasini et al. 2015):

M ≈ ΣΩR3cap/H (1.45)

where Rcap is the capture radius from where the gas could be captured. After the local reservoir of mass

is depleted and the planet has formed a circumplanetary disk, the accretion is given by(Mordasini et al.

2012a):

36


M = kLub

(

3πνΣ + 6πr∂vΣ

∂r

)

(1.46)

where kLub determines the fraction of the gas reaching the planet through the circumplanetary disk (Lubow

et al. 1999). The first term in the parenthesis is the viscous accretion rate in a protoplanetary disk in

equilibrium. However, the circumplanetary disk is not equilibrium for several reasons. The planet’s

presence as a mass-sink moves out the disk from equilibrium, or the photoevaporation at late stages

disturbs the disk equilibrium as well. Therefore, the second term in the parenthesis accounts for this

non-equilibrium accretion (Mordasini et al. 2012a).

Some other population synthesis studies (e.g. Ida & Lin 2004) use simply the Kelvin-Helmholtz

timescale for the accretion: M = Mplanet/τKH, where the Kelvin-Helmholtz timescale parameterized as

a function of the planetary mass and the opacity: τKH = kM−pκ−q (Ikoma et al. 2000). Obviously, the

choice of the parameters, k, p, q, greatly affects the outcome of the population synthesis models (Miguel

& Brunini 2008), and these parameters are highly unknown. Planet growth is then truncated when some

criteria are met: for instance when the Hill-radius of the planet equals 1.5 H (disk’s scale height) or the

mass of the disk πa2Σ becomes equal to the mass of the planet (Ida & Lin 2005).

Surprisingly, the final mass distribution of the giant planets in the most recent synthesis models (e.g.

Mordasini et al. 2009b, Mordasini et al. 2015) is very good, as one can see comparing Fig. 1.7 to Fig.

1.5 or in the more direct comparison of Fig. 1.6.

Figure 1.6. Figure from Mordasini et al. (2009b) about the mass distribution comparison of the ob-

served and synthesized planets. Panel (A) is a histogram of the projected M sin i of 6075 detectable

synthetic planets (solid line) and the observational comparison sample of 32 planets (dotted line).

Panel (B) is the cumulative distribution function corresponding to (A).

In these models, the runaway phase starts always very late, when the disk’s accretion rate is low. The

distribution P(t0) of times t0 at which runaway growth starts has to be such that the final distribution of

37


planetary masses, which from Eq. 1.43 is:

P(M) = Mdiskτ exp(−t0/τ)P(t0), (1.47)

matches the observed distribution. We believe that the best parameters are chosen in the synthesis mod-

els for this to happen. Indeed, Mordasini et al. (2009b) write “Note that Miguel & Brunini (2008) have

recently shown that the large uncertainties affecting the constants used in parameterized gas accretion

laws as in Ida & Lin (2004) lead to large variations of the predicted final planetary mass distribution”. In

the synthesis model of Ida and Lin, the formation of very massive planets is prevented by the assumption

that gas accretion stops when the Hill radius of the planet exceeds 1.5 disk’s scale heights. Hydrodynam-

ical 3D simulations, however, show that accretion continues well beyond this threshold. If this accretion

cutoff mechanism were removed, the synthesis models of Ida and Lin would produce too many super-

Jupiters (Ida, private communication). Thus we believe that it is worth exploring with state of the art

simulation the gas accretion process for Jupiter mass planets, which is the goal of this thesis. If we found

a mechanism to limit the runaway accretion rate well below the disk’s mass flux, the mass distribution of

giant planets could be explained more simply.

Figure 1.7. Figure from Mordasini et al. (2015) showing the expected mass distribution by the

planet synthesis models. The black line gives the full underlying planet population, while the blue,

red, and green lines are the detectable synthetic planets at a low (10 m/s), high (1 m/s), and very

high (0.1 m/s) radial velocity precisions, respectively. Comparing with the observed distribution

of Fig. 1.5, the observed and predicted distribution of giant planets (M > 100 Earth-masses) agree

quite well.

38

1.5. The circumplanetary disk Chap.1 Overview

1.5 The circumplanetary disk

Because the gas falling towards a planet has some angular momentum relative to the planet, it has to

form a circumplanetary disk. We will see in Sect. 5, however, that if the temperature is high enough due

to the adiabatic compression of gas and limited cooling, the disk may appear as a thick torus or even an

envelope, even for Jupiter-mass planets. We know that giant planets, even the relatively small Uranus

and Neptune, have regular satellites, which presumably formed in a disk. This suggests that in the late

stages of the gaseous circumstellar disk phase, the conditions were adequate for these planets to form a

thin, cold circumplanetary disk.

It is a recurrent question what the mass of the circumplanetary disk could be. This is a very important

parameter both for satellite formation models and for observation campaigns aimed at detecting the CPD.

The mass of the CPD depends on whether the planet formed through core accretion or gravitational

instability (GI). In the GI models, the massive clump eventually collapses into a planet+disk system,

therefore the CPD has a large extent and a large mass. Shabram & Boley (2013) find that a CPD formed

through GI can reach 25% of the planet-mass. On the other hand, if the CPD formed through core

accretion, its mass – according to hydrodynamic simulations and analytical estimates – should only be a

small fraction of the planetary mass, on the order of 1-0.1 %.

We know that in the case of Jupiter and Saturn, the integrated mass of their satellites makes up 2×10−4

of the planetary mass. If one assumes the interstellar medium value for the dust-to-gas ratio, this would

mean a minimum mass for the CPD of 2% of the masses of our two largest gas giants. This is a very

massive disk, but as Canup & Ward (2002) pointed out, this mass has to be processed during the satellite

growth timescale, i.e. it does not have to present at one given snapshot of time. This is because, while

the circumstellar disk is a closed reservoir of mass, the CPD is not. The latter is fed by the circumstellar

disk, and loses mass through the accretion to the planet. Therefore the CPD mass is not constant in time,

but depends on the feeding and mass loss balance and it changes as the circumstellar disk (CSD) evolves,

and as the planet grows. Thus, in principle, the concept of a Minimum Mass circumplanetary disk means

only a time-integral of the CPD mass over the satellite formation timescale. According to Canup & Ward

(2002) the satellite formation timescale was of the order of 105 or even 106 years. This also means a very

light CPD at any given time. In fact, all evidences point to a long formation timescale, at least in the case

of the Galilean satellites. First of all, their large amount of ice suggests that they formed in a cold CPD.

During the accretion phase, when the planet grows rapidly and the CSD feeds the CPD, the environment

is unlikely to be cold according to recent radiative hydrodynamic simulations presented in this thesis and

in e.g. Ayliffe & Bate (2009a), Gressel et al. (2013). Secondly, the outermost Galilean moon, Callisto’s

interior is not differentiated. This also means that it probably formed slowly and late, to avoid accretional

heating and heat production from short-lived radioactive elements. To reach adequate temperatures for

satellite formation, the most likely time of satellite formation is when the planetary accretion has almost

stopped and the CSD gas has mostly been removed. Then, the left-alone, dense CPD can gradually cool

off. How long a CPD like this can last, is a difficult question. The planet and the growing satellites accrete

the leftover material, but the accretion timescale at this stage is difficult to estimate.

1.5.1 What we learned about the subdisk from hydrodynamic simulations so far?

In the last 20 years, as computers evolved, more and more complex hydrodynamic simulations were

made about the evolution of circumstellar-, and circumplanetary disks. First, the simulations were only

low resolution, 2D isothermal simulations, but nowadays 3D, radiative and magneto-hydrodynamical

simulations are also possible. We focus only on the recent years developments here, as they correspond

39


to the most realistic models. We first overview the findings on the CPD itself, then turn to the discussion

of the accretion rate onto the planet.

1.5.1.1 Circumplanetary Disk Qualitative Description

It is known that while low mass planets are capable to form only envelopes around them, larger mass

planets, which can open gaps, can form circumplanetary disks (Paardekooper & Mellema 2008; Ayliffe

& Bate 2009a). Where is this limit is also depends on other factors, such as the opacity. Ayliffe & Bate

(2009a) found that with standard opacities, planets over 100 MEarth are able to form disk, but when they

decreased the opacity, even lower mass planets could form CPD as well (see Fig. 1.8). This thesis will

show that other factors, such as the planetary temperature can also significantly affect the disk forming

capabilities (see in Chap. 5) and the gap-opening ability is perhaps less relevant in this question.

To measure the radial extent of the CPD is a non-obvious task. There is no sharp boundary at the

edge of the disk, the gas density is continuously decreasing from the planet to the bulge region between

the spiral arms, of which the CPD is a subset. The boundary can be determined, for instance, by using

streamlines (e.g. Machida et al. 2010), or by the setting a threshold of the normalized, z-component of

angular momentum of the disk (like in Ayliffe & Bate 2009b). From hydrodynamic simulations, the CPD

radial extent found to be around one-third of the Hill-radius, no matter whether the simulations were

isothermal (Tanigawa et al. 2012), non-isothermal (Machida et al. 2010) or fully radiative with proper

thermal treatment (Ayliffe & Bate 2009b). This result nicely coincides with the analytical estimation of

Quillen & Trilling (1998).

For the CPD mass, there are very diverse values in the literature. For example, the radiative, 2D

simulations of D’Angelo et al. (2003) found 10−5 − 10−6MJ for a Jupiter-mass planet, which is perhaps

one of the lowest values in the literature. While Gressel et al. (2013) argue with a much larger, 10−4 disk-

to-planet ratio. Gressel et al. (2013) also compared their disk masses in isothermal and non-isothermal

simulations, finding that the isothermal disk was more massive. This is contrary to the findings of this

thesis (see Chap. 6), but this discrepancy could be dependent on how the CPD inner and outer boundary

is defined.

All non-isothermal works agree that the temperature in the inner CPD is very high. The temperature

of course depend on the resolution and on the planetary potential treatment, therefore different works

measured a bit different peak temperature. For a 1 Jupiter-mass planet, Ayliffe & Bate (2009b) argue

with 1600 K at the planet surface, if the surface is defined at 0.02 RHill (it is user dependent what the

planetary radius is). They found much larger, 4500 K value with a realistic planetary radius. Of course,

the peak temperature also depends on the viscosity as pointed out by D’Angelo et al. (2003). Their 2D

radiative simulation gave maximum 1500 K with their highest viscosity case (10−16cm2/s) for the Jupiter-

mass planet. The MHD simulations of Gressel et al. (2013) studied a bit smaller mass planets, growing

from 100 MEarth to 150 MEarth, and already at these low-mass cores resulted in peak temperatures over

1500-2000 K. Similarly, the characteristic temperatures in the CPD were 1000-2000 K in the work of

Papaloizou & Nelson (2005). All the above mentioned non-isothermal works agreed that the temperature

profile of the CPD is very steep.

The high temperatures in the CPD naturally means high aspect ratio. Ayliffe & Bate (2009b) measured

0.3-0.6 aspect-ratio for disks around 100, 166 and 333 Mearth planets. This is in agreement with the

findings of D’Angelo et al. (2003). The non-isothermal works of Paardekooper & Mellema (2008),

D’Angelo et al. (2003), and Ayliffe & Bate (2009b) all agree that the circumplanetary material is optically

thick, which can affect the cooling rate.

It is also an interesting question what happens with the planetesimals in the CPD. The very recent

40


work of D’Angelo & Podolak (2015) studied the evolution of planetesimals in protoplanetary disks, com-

bining isothermal simulation with particles that account for icy/rocky bodies of 0.1-100km. The mass

evolution of solids was calculated self-consistently with their temperature, velocity, and position. The

authors found that the larger solids are incapable to enter from the circumstellar disk to the gap region,

therefore, to the CPD as well. Their suggestion is that, after the giant planet becomes massive enough

to open a significant gap, the metallicity/solid body content of the CPD cannot increase a lot anymore.

Similarly, Tanigawa et al. (2014) studied the capture and accretion of 10−2−106 m solid bodies, and high-

lighted that the accretion efficiency peaks at 10 m sized particles. The accretion efficiency for particles

larger than 10 m becomes lower, because gas drag of the CPD’s gas becomes less effective. While for

smaller particles (< 10 m) the efficiency drops because of the strong coupling with the background gas

flow, which prevents particles from accretion.

In fact, Nelson & Gressel (2010) found that magneto-hydrodynamical (MHD) turbulence has a de-

structive effect on the planetesimals in protoplanetary disks, therefore relatively low turbulence level is

required for standard planetesimal accretion onto the planet, which condition can be found in dead-zones.

This highlights that planets likely form in dead zones. Similarly, Fujii et al. (2011), and Fujii et al. (2014)

argue with a low ionization (hence low turbulent viscosity) level in the subdisk; they claim that the mag-

netorotational instability (MRI) in the entire CPD cannot be maintained, therefore, it cannot be the main

driving mechanism of accretion onto the planet. However, Keith & Wardle (2014) and Keith & Wardle

(2015) find that maintaining MRI in the CPD is difficult, but it is possible with adequate parameters

(optimistic surface density and temperature, etc.).

1.5.1.2 Accretion Rate

It has been known for a long time that torques acting on the planet and accretion rates are higher in

2D simulations than in 3D (e.g. Paardekooper & Mellema 2008; Ayliffe & Bate 2009a). But whether

the isothermal simulations or the non-isothermal (i.e. thermal effects included) models result in higher

accretion rates, is not so unequivocal. Paardekooper & Mellema (2008) found reduced accretion rate in

their radiative simulation in comparison with their isothermal model, but they studied a 5 Mearth planet

only. Ayliffe & Bate (2009a) found for a Jupiter-mass planet 3 times lower accretion rate (namely,

2.6×10−5MJ/yr) in their radiative smoothed particle hydrodynamics (SPH) simulation, than in isothermal

setting. Similarly, Gressel et al. (2013) also measured lower accretion rate in non-isothermal (∼ 8 ×10−3Mearth/yr) or MHD simulations, than in the comparison isothermal data, even though their planetary

core was less massive, growing from 100 Mearth to 150 Mearth during the course of the simulation. On the

other hand, Machida et al. (2010) found smaller accretion rate (2.3×10−5MJ/yr) for a 1 MJup planet with

isothermal-like EOS, than with barotropic EOS (4.1 × 10−5MJ/yr). Also, the radiative, 2D simulation

of (D’Angelo et al. 2003) measured comparable accretion rates, than in their previous (D’Angelo et al.

2002), isothermal simulation of a Jupiter-mass planet. So whether the non-isothermal setting undoubtedly

decreases the accretion rate is yet to be answered.

Several factors play a role about the accretion rate. First, where and how it is measured is different

in every work. Some works use reduced density in the cells around the planet, others use a completely

depleted sink area. Furthermore, this accretion area has also different size in the different works. Overall

it can be said that the smaller is the accretion area, the smaller is the accretion rate (Machida et al. 2010;

Ayliffe & Bate 2009a,b, but see also this thesis in Chap. 6).

Furthermore, it is not surprising that the opacity also plays a large role on the accretion rate. Works of

Papaloizou & Nelson (2005) and Ayliffe & Bate (2009a) agree that reduced opacity always mean higher

accretion rate, regardless the planetary core mass considered. However, Ayliffe & Bate (2009a) finds

41


that the opacity plays a smaller role on the accretion rate for planets ≥ 1MJup. Completing 3D hydro

isothermal simulations with 1D planet formation models, Lissauer et al. (2009) have found that Jupiter

could have formed in 3 Myr only if the dust opacity was reduced to 2% of the interstellar medium value,

having a the surface density of 10 g/cm2, and a quite small viscosity (α ∼ 4 × 10−4). They have tried

several values of opacity and viscosity, and with larger viscosity (α ∼ 4 × 10−3) they have ended up with

too massive planets. The evolutionary models of Papaloizou & Nelson (2005) examined gas accretion

onto solid cores of 5M⊕ and 15M⊕ with various dust opacities. They found if the gas is passing through

a circumplanetary disk, indeed the accretion is reduced in contrast with the envelope case. They argued

that the smaller core size would build up a Jupiter-mass planet in 3×108 years, while the larger mass core

would do it in 3 × 106 years. This work also showed that the initial protoplanet mass doubling timescale

is very approximately inversely proportional to the magnitude of the opacity (meaning how much it was

reduced in comparison to the interstellar medium value).

Finally, it is also well known that the accretion rate to the planet is depending on the circumstellar

disk mass. In locally isothermal simulations, the accretion rate linearly scales with the surface density,

i.e. if the circumstellar disk surface density is enhanced by an order of magnitude, the accretion rate will

be an order of magnitude higher. However, the situation is not the same in non-isothermal simulations.

Ayliffe & Bate (2009a) from radiative simulations find that with the increased circumstellar disk surface

density the planet accretes more, but the relation is not linear. They find 2.6 × 10−5MJ/yr accretion rate

for a Jupiter-mass planet in their nominal disk, while this value is five times more (1.3 × 10−4MJ/yr) in

a ten times more massive disk. This accretion rate ratio is fluctuating between 1 and 7 for less massive

planets, when the disk’s surface density is an order of magnitude higher. The exact opposite relationship

was revealed by Paardekooper & Mellema (2008). These authors claim that in their radiative simula-

tion 10 times lower surface density of the circumstellar disk leaded to almost one order of magnitude

enhancement of the accretion rate, although they studied only a very small mass planet with 5Mearth

mass.

Several authors have studied the accretion rate dependence with planetary mass (Ayliffe & Bate

2009a; Kley & Dirksen 2006; D’Angelo et al. 2006; Lubow et al. 1999; Paardekooper & Mellema 2008;

Bate et al. 2003). Overall they found that for small mass cores until (≤ 1MJup) the accretion rate increases

with planetary mass. For larger planetary masses the accretion rate decreases until the point when the

circumstellar disk becomes eccentric due to the large mass planets and the accretion rate again rises

(≥ 2 − 3MJup). This is because the fixed planet in the eccentric gap in these simulations goes through the

circumstellar disk, thus having enhanced gas-accretion episodes, see Fig. 1.10 (Kley & Dirksen 2006;

D’Angelo et al. 2006). The accretion rate mass dependence can be seen on Fig. 1.9 taken from Ayliffe &

Bate (2009a).

Regarding the accretion mechanism to the giant planet we know the following. There is a strong

vertical inflow to the planet and to the circumplanetary disk through the gap (Gressel et al. 2013; Tanigawa

et al. 2012; Paardekooper & Mellema 2008; Ayliffe & Bate 2009b). Once the gas becomes part of the

circumplanetary disk, the viscous forces and the stellar torque will channel the flow down to the planet

(see Chap. 3 and 4). The efficiency of this channeling greatly determines the accretion rate. The gas

which is not accreted leaves the CPD in the midplane region (e.g. Machida et al. 2010; Gressel et al.

2013), id est bringing mass away from the subdisk back to the circumstellar disk (see more in Chap.

4). The vertical influx having a low angular momentum, breaks the rotation of the CPD. But this is not

the only possible mechanism, which affects the rotation. Nelson & Papaloizou (2003) found that the

magnetic field can break the CPD rotation as well, which then affects the accretion rate.

42


1.5.2 What we learned about the subdisk from observations?

There is no resolved CPD observation around a young planet to the date when these lines are written.

So far observations could only reveal extended thermal excess emission around candidate planets. This

extended emission means that the thermal emission is not arising from a point source (like a sole planet),

but from a several Jupiter-radii wide region. This emission can arise from hot circumplanetary material

that is accreting to the planet.

The first extended thermal excess emission from a candidate protoplanet was from LkCa 15 b (Kraus

& Ireland 2012). The two-band observation in K’ and L’ revealed a cold point-like source (that can be the

planet), which seems to be surrounded with hot dust Fig. 1.11. The parent star is very young, only 2 Myrs

old with a still massive circumstellar disk that is in the transitional disk phase. The planetary candidate

is inside a 50 AU wide cavity, orbiting about 11 AU away from its star. The planet has probably a 6 MJup

mass, but it is very uncertain and model dependent.

Figure 1.11. The candidate protoplanet, LkCa 15 b observed by (Kraus & Ireland 2012). The cold

point like source is surrounded by hot dust.

The high-contrast observations on the disk of HD 169142, obtained L’- and J-bands revealed possibly

two planetary candidates with extended thermal emission around them (Reggiani et al. 2014). The parent

star is a 1-10 Myr old Herbig Ae/Be star, with a transitional disk around it. The innermost planetary

candidate is at ∼ 23 AU from the star, it is detected at L’ band but not in J band. This planet is within

the recently resolved inner cavity of the circumstellar disk. If its entire L’ band luminosity comes from

the photosphere of the planet, it would correspond to a very heavy, 28-32 MJup object at the age of the

star. The star is still accreting, which suggests that gas is left in the inner disk cavity. From this reservoir

of gas, the companion could also be accreting. Reggiani et al. (2014) suggest that the planet in this case

can be much less massive, since its luminosity could be enhanced by the accretion process and by a

circumplanetary disk. This assumption is also based on the fact that the gap observed in the inner disk

should be much larger in case the planet is indeed ∼ 30 MJup massive.

Another example about possible ongoing planet formation is in the disk of HD100546 (Quanz et al.

43


2015). This star is also very young (≤ 10Myr) Herbig Ae/Be star with a massive gas and dust disk

extended beyond 300 AU. The planetary candidate orbits around 50 AU from the star and is detected in

M’ and L’ bands. This gives an estimate about the effective temperature of the object that is around

930 K. The extended emitting source has a radius of 7RJup, which also suggests a circumplanetary disk

around the young planetary candidate. The mass of the object is very uncertain, but perhaps around

5-10 MJup. The uncertainty comes from the uncertain stellar age, model dependency, but also from the

extended source, since there is probably a large luminosity contribution by the accreting gas and/or a

circumplanetary disk.

Since brown dwarfs are intermediate in mass objects between stars and planets, their disks can also

tell us something about the circumplanetary disks. A free-floating brown dwarf, OTS 44, with a mass

of 12 MJ has been observed to hold its own disk (Joergens et al. 2013). This is probably the lowest

mass object with a known disk. The subdisk is about 30 Mearth based on radiative transfer modeling,

which means a brown-dwarf-to-disk ratio of 8 × 10−3. The OTS 44 is only ∼2 Myr old and the measured

accretion rate in the subdisk is about 8 × 10−12Msun/yr, which is very high for such a low-mass object.

44


Figure 1.8. Figure from Ayliffe & Bate (2009a) about the densities of the CPD at planetary masses

of 100, 166 and 333 Mearth (left- to right-hand panels) and two different grain opacities. In the top

row, they show the density-map on the midplane with interstellar grain opacities. In the second

row, the density through a vertical slice is plotted, corresponding with the row above. The third row

shows the density on the mid-plane with reduced, 1% grain opacities. Finally, in the bottom row

is the densities through a vertical slice corresponding with the row above. The spatial dimensions

in units of the orbital radius (5.2 AU).45


Figure 1.9. Figure from Ayliffe & Bate (2009a) about the accretion rate planetary mass depen-

dence. The asterisks are isothermal SPH simulations. The diamonds mark the isothermal results

of Bate et al. (2003), and the plus signs those of Lubow et al. (1999), connected with solid lines.

From radiative simulations of Ayliffe & Bate (2009a), results are shown using standard interstellar

grain opacity (dotted), 10% interstellar grain opacity (short dashed), 1% interstellar grain opacity

(dot dash) and 0.1% interstellar grain opacity (long dashed). The inclusion of radiative transfer

substantially lowers the accretion rates of low-mass protoplanets, but Jupiter-mass protoplanets

have similar accretion rates to the locally isothermal result, regardless of the grain opacity. The

analytic approximation of D’Angelo et al. (2003) is shown by the solid curved line.

46


Figure 1.10. Figure from Kley & Dirksen (2006) about a massive giant planet (5.9 MJup) in an

eccentric circumstellar disk. The accretion is episodic, each time the planet on a fixed orbit rushes

through the circumstellar disk, the accretion is greatly enhanced.

47


48

“You really work too hard. If you like hydrodynamics this much, go

to the beach and watch the waves break!”

Frederic Masset, after being tired of the large daily email flux

Chapter 2

Development of modules into the JUPITER

hydro-code

Contents

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.2 The JUPITER code in its isothermal version . . . . . . . . . . . . . . . . . . . . . . 52

2.2.1 Basic description of the JUPITER code . . . . . . . . . . . . . . . . . . . . . . 52

2.3 New Modules for the JUPITER code: energy equation, heating, cooling . . . . . . . 55

2.3.1 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.3.2 The Radiative Module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.3.3 Heating Effects – Stellar Heating . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.3.4 Heating Effects – Viscous heating . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.3.5 Boundary Conditions for Cooling . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.3.6 Equilibrium & Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.4 Parameters of the code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.4.1 The parameter file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.4.2 Predefined variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.4.3 The arguments of the code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.5 Testing the code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.5.1 Testing the Riemann Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.5.2 Testing the Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.5.3 Testing the Radiative Module . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.5.4 The Ultimate Test: the Dimensional Homogeneity Test . . . . . . . . . . . . . . 81

2.6 Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2.6.1 The stability issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

2.6.2 The cold circumplanetary disk problem . . . . . . . . . . . . . . . . . . . . . . 83

49

Chap.2 Code

2.7 Fun-facts about the code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

50

Code Chap.2 Code

”Piled Higher and Deeper” by Jorge Cham– www.phdcomics.com

51

2.1. Introduction Chap.2 Code

2.1 Introduction

Great efforts have been made in the astronomical community to observe planets in their formation to

understand how planets are born, but at the time these lines are written, there are no sufficient observations

to guide us. The only method left to understand the giant planet formation process is with numerical

simulations, such as hydrodynamic simulations. Fortunately, as the computer capabilities evolve very

rapidly, the hydrodynamic codes do too, with almost the same speed. Nowadays, realistic simulations

can be done, which can help observers to plan adequate observations and also adding pieces to the big

puzzle of planet formation theory.

During my thesis I have worked with three different hydrodynamic codes, but my published results

were obtained with one that is the JUPITER code originally developed by Frederic Masset (de Val-Borro

et al. 2006). I used the other two codes for testing and comparison purposes. The three codes which I

will mention in this section are:

• JUPITER – 3D nested mesh, Godunov code; during this thesis energy equation and the radiative

module (radiative transfer with flux limited diffusion, heating, cooling) were added

• FARGOCA – 3D version of the 2D FARGO code developed by Elena Lega in Nice and extended

with radiative modules prior my thesis

• FARGO3D – 3D version of the 2D FARGO code developed by Pablo Benıtez Llambay & Frederic

Masset, including magnetohydrodynamics and which now inherited JUPITER’s radiative module

as well

While I pursued the first doctoral thesis projects with the isothermal version of JUPITER, from my

second year I began to develop a more physically realistic version of it, with the supervision of Frederic

Masset. As the FARGOCA code already contained the radiative modules, we followed the same idea

and extended it to the nested meshes. During the development, it turned out that reaching stability of

JUPITER with these new modules was the hardest part of the job. From now on, when the radiative

module is mentioned, it refers to radiative transfer with flux limited diffusion in a grey approximation,

with a two-temperature approach accounting for the cooling and heating (due to stellar irradiation and

viscous shear) with a realistic opacity table (Bell & Lin 1994). In this chapter the focus will be on the

JUPITER code but the main differences will be mentioned with the FARGO-type codes, since part of

the testing was done through comparisons of the results with these codes. Moreover, the reader can get

a global view on how different hydro-codes work, what are the possibilities, difficulties, advantages of

each one. In this chapter, first the basics of the JUPITER code will be described. Then, a discussion on

the energy equation will follow, along with the radiative module description. After, the parameters of the

code will be discussed and I will finish the chapter with the tests of the new algorithm and the stability

issues.

2.2 The JUPITER code in its isothermal version

2.2.1 Basic description of the JUPITER code

The JUPITER code is a 3-dimensional, hydrodynamic code with nested meshes. The nested meshes are

higher resolution regions in the simulation box, each of them increasing the resolution by a factor of 2

52

2.2. The JUPITER code in its isothermal version Chap.2 Code

in each spatial dimensions. Thanks to these, one or more patch(es) of the simulation box can be studied

with high resolution without wasting computational time by enforcing high resolution everywhere in the

simulation box. This way one can study, for example, an entire circumstellar disk with lower resolution,

while adding nested meshes around the planet to examine the circumplanetary disk (CPD), which was

our goal in this thesis.

There are other methods in the literature to address the same issue. It is important to highlight the dif-

ferences with the JUPITER code and why we chose this algorithm. A different method to study the CPD

is with local box simulations, where only the CPD vicinity - a box - is simulated with high resolution.

But in this case, there is no proper circumstellar disk, therefore one has to deal with complex boundary

conditions in order to simulate the realistic flow from the circumstellar disk to the simulation box. It is

a very difficult task to develop such a realistic boundary condition. Not surprisingly, these simulations –

for example – cannot open proper planetary gaps unlike the global circumstellar disk simulations.

Another way to reach high resolution around the planet is to study the flow via 2D simulations, which

then could also include the circumstellar disk, thus avoiding the boundary condition issue mentioned

earlier. However, this method has its obvious limitations due to the lack of the third dimension. This

thesis will prove that the accretion onto the planet is a truly three-dimensional process, therefore 2D

simulations on the accretion problem are not appropriate.

Finally, the FARGO-type codes are more diffusive than the codes similar to JUPITER; the latter

algorithm is capable of simulating clear shock-fronts, which are present around the planet, e.g. the spiral

wake inside the circumplanetary disk. Thus, our choice fell onto the JUPITER code, since we wanted a

flow as realistic as possible around the CPD, with high resolution, in 3D, within a circumstellar disk, and

this is exactly what this code could offer.

The JUPITER code is parallelized with Message Passing Interface (MPI), i.e. can be run on multiple

CPUs at the same time. The parallelization occurs in each spatial dimension. Therefore, the simulation

box will be automatically divided into smaller boxes corresponding to each CPU-core. It is obviously

better to use many CPUs for the simulations in order to speed up the computation time, but it is necessary

to be careful with their number, since too many cores can in fact slow down the code due to the excessive

communication time among the CPUs. If more cores are used, more communication is needed among

the CPUs to share the information – to avoid mismatch of the flow on the different CPU patches.

Moreover, the different nested meshes also need to communicate with each other, therefore the num-

ber of the refinement levels also slows down the code due to the necessary communication. Hence, it is

wise to test first what is the performance gained with various amounts of CPU-cores, given the resolution

the user wants on the coarse mesh and how many refined mesh will be used later.

As every hydrodynamic code, JUPITER solves the equations of hydrodynamics, which we write

below, and which are respectively the continuity equation (or conservation of mass), the conservation of

momentum, and the energy equation:

∂ρ

∂t+ ∇ · (ρv) = 0 (2.1)

∂(ρv)

∂t+ ∇ · (ρv ⊗ v + PI) = source terms (2.2)

∂(ρE)

∂t+ ∇ · [(ρE + P)v] = source terms (2.3)

where ρ indicates the density, P means pressure, and v is the velocity vector, I represents the unit

tensor. In the last equation, E stands for the energy-density. The energy equation (Eq. 2.3) in its full

53

2.2. The JUPITER code in its isothermal version Chap.2 Code

form with the source terms, which will be discussed in detail in Chapter 2.3.1.

How these equations are solved can be different in the different hydrodynamic codes. Unlike FARGO,

JUPITER is a so-called Godunov code, where the above hyperbolic differential equations are solved

through Riemann-solvers (see below), without the source terms – and the energy equation treated sep-

arately. The energy equation is solved later in the code (see Figure 2.3), along with the addition of the

source terms to the equations 5.1 and 2.2.

There are several Riemann-solvers in the literature, since an entire branch of mathematics studies

and develops newer and newer ones. For simplicity and brevity, here we describe only the few Riemann

solvers of importance for the JUPITER code. More about Riemann solvers can be found in Toro (2009).

Riemann solvers solve the so-called Riemann problem, which is an initial value problem composed

of a conservation equation (like the Euler equations) together with piecewise constant data (like data on

grids) having a single discontinuity. The usefulness of the Riemann solvers is that they solve the Euler

equations in such a way that rarefaction waves and shocks are among the characteristics of the solution.

This is why the Godunov type codes are capable to treat shocks, while finite difference upwind type codes

(such as FARGO) do poorly on this task. Thus, the latter result in more diffusive simulations, making it

more difficult to see fine details.

There are different types of Riemann solvers in JUPITER; in particular there is an exact, iterative

Riemann solver for the adiabatic case, and a nearly exact, two-shock or two-rarefaction Riemann solver

for the isothermal case.

To complete the picture, before the Riemann solver computes the fluxes, there is a so-called predic-

tor step, which predicts the Riemann states for the solvers. This is needed because the hydrodynamic

quantities are not considered uniform within the cells, but they are considered piecewise linear, so their

values need to be interpolated using the appropriate slope and slope limiters. The predictor routine pre-

dicts the hydro quantities after half timestep, at the cell interfaces. The left Riemann state refers to the

set Riemann-states mean the left-value of hydrodynamic quantities on the left side of a given interface,

whereas the right Riemann state refers to the value of hydrodynamic quantities on the right side of this

interface. (see Fig. 2.2). There are three different predictor methods currently present in JUPITER:

• Piece-wise Linear Method (PLM) – second order accuracy in space (i.e. errors are on the order of

the square of the zone size); used in isothermal simulations

• MUSCL-Hancock (MUSCL) – second order accuracy in space; used in adiabatic simulations

• Piece-wise Constant Method (called GFO in the code and in the rest of the thesis) – diffusive, the

error is first order in space, therefore never used for publishable simulations but only for testing

purposes

To avoid problems because of the slopes (e.g. prevent negative densities or spurious oscillations that

would otherwise happen with high order schemes (like MUSCL) due to discontinuities, shocks, or sharp

changes in the solution domain, a Total Variation Diminishing (TVD) slope limiter is used in JUPITER.

The slope limiter that we use is the monotonized central slope limiter (van Leer 1977).

As it was mentioned above, the Riemann-solvers treat the continuity equation (Eq. 5.1) and the

force-less equation for the momenta (Eq. 2.2). Later in the hydro kernel, we apply the source terms. This

is the so-called operator splitting method, in which one decomposes the system of partial differential

equations into simpler subproblems, and treat them separately. Therefore, in JUPITER’s hydro kernel we

later apply the gravitational force on the momenta equations: −ρ∇ · Φ to the right-hand-side of Eq. 2.2.

Moreover, if viscosity were set up, the viscous forces are applied as well on the momenta (∇ · ¯τv – see

54

2.3. New Modules for the JUPITER code: energy equation, heating, cooling Chap.2 Code

Figure 2.2. Figure from Toro (2009) about the Riemann-states.

more in Section 2.3.4). Also, throughout the kernel we also solve the energy equation, described in Sect.

2.3.1. The main steps of the hydro kernel are schematized on Fig. 2.3.

The grid geometries of the code are the following. The coordinate system can be either cartesian,

cylindrical, or spherical. The simulations can be either 2D or 3D, but the radiative transfer module (see

next Section) is only implemented in 3D spherical geometry.

Originally, the code was written with an isothermal EOS. In fact, this means the simulations are

locally isothermal, and the sound-speed is constant at a given radius. One big advantage of locally

isothermal simulations is that they are much faster than the adiabatic or radiative simulations. The first

two papers of this PhD thesis feature locally isothermal simulations. Later, we switched to a more

physically realistic setup.

2.3 New Modules for the JUPITER code: energy equation, heating,

cooling

These new modules took 1.5 years to code, stabilize, and test (roughly 10%-60%-30% breakdown of the

overall time, respectively). Because I worked with Frederic Masset, who resides now in Mexico, I spent

there overall 1.5 months in two segments, and he visited Nice for 3 additional weeks. Otherwise, we

worked from the distance.

2.3.1 Energy Equation

Putting the energy equation into a Godunov, nested mesh code is a very difficult task, especially when

dealing with a Keplerian circumstellar disk. Our original expectation that this task would have been a

matter of a couple of weeks with testing at maximum, quickly got falsified. Not only the programming

part of the job is not trivial, but as we experienced, Godunov type of codes became incredibly unstable

when adding the energy equation; thus, one needs to improvise a series of sparky ideas to stabilize the

55


Figure 2.3. The main steps in the hydro kernel.

56


code. One of the multiple reasons for the instability is that the Godunov codes handle the total energy

instead of the internal energy, therefore one needs to evaluate the internal energy by the end of the

timestep with subtracting the kinetic energy from the total energy. One can easily see that in some parts

of the simulation the kinetic energy is very large in comparison to the internal energy, hence subtracting a

large number from another similarly big number can lead to a very inaccurate (or even negative) internal

energy. Another problem in our simulations is that the coordinate-system is spherical, centered on the

star, while the nested meshes around the planet are almost Cartesians. The shape of the cells, as well

as the resolution, seem to matter a lot when one deals with the energy equation. We dedicate an entire

section on this problem and on its solution, see later Sect. 2.6.1.

The energy equation can be written in different ways depending on what source terms we consider.

For the JUPITER code, the total energy conservation should be written (Mihalas & Mihalas 1978;

Quillen 2014):

∂E

∂t+ ∇ · (Ev) = −ρv · ∇Φ + ∇ · [(−PI + ¯τ) · v] − ∇ · Frad + S (2.4)

where ρ is density, E represents total energy (see below) and v stands for the velocity. Frad is the

radiative flux, P indicates the pressure, Φ is the gravitational potential, I represents the identity matrix,

and ¯τ is the stress tensor (see Sect. 2.3.4). The first term on the right-hand-side is the work done on

the fluid by external forces, while the second term accounts for the advection and viscous heating (see

more in Sect. 2.3.4). In addition, if stellar heating (S ) is set up that is an additional heating source

accounting for the stellar irradiation (see more in Sect. 2.3.3). Note that equation 2.4 does not contain

the thermal conductivity which would go to the right-hand-side, because it is negligible for gases. The

thermal conductivity would matter only for heat transport in solid medium.

Because the total energy consist of the internal energy (ǫ), kinetic energy, and radiation energy (ǫrad,

if any); it can be written as (Mihalas & Mihalas 1978):

E = ǫrad + ǫ + ρv2

2= ǫrad + ρcVT + ρ

v2

2= ǫtot + +ρ

v2

2(2.5)

where cV is the volumic heat capacity, T represents the temperature and ǫtot is the total internal energy

including the radiation energy.

Even though JUPITER uses the total energy equation, the other two codes (FARGOCA and FARGO3D)

compute the equation for the internal energy only. Because I compared the simulations from the various

codes, also to test the new code, it is important to evaluate out the energy equation for the internal energy

as well, as FARGO-type codes use it.

With plugging in Eq. 2.5 into 2.4, one can evaluate out the internal energy equation. First concentrate

on the left-hand-side of Eq. 2.4 and develop it further:

∂ǫtot

∂t+∂(

ρ v2

2

)

∂t+ ∇ · (ǫtotv) + ∇ ·

(

ρv2

2v

)

= (2.6)

∂ǫtot

∂t+ ∇ · (ǫtotv) +

v2

2

∂ρ

∂t+ ρ∂v2/2

∂t+

v2

2v∇ρ + ρv∇ · v2

2+ ρ

v2

2∇ · v = (2.7)

∂ǫtot

∂t+ ∇ · (ǫtotv) +

v2

2

[

∂ρ

∂t+ v∇ρ + ρ∇ · v

]

+ ρ

[

∂v2/2

∂t+ v∇ ·

(

v2

2

)]

(2.8)

Here, the part highlighted with blue is the continuity equation, therefore it vanishes (see Eq. 5.1).

Now, if we consider the entire energy equation with the source terms (right-hand-side) as well:

57


∂ǫtot

∂t+ ∇ · (ǫtotv) − ρv · ∇Φ + v∇ ¯τ − v∇P = −ρv · ∇Φ + ∇ ·

[

(−PI + ¯τ)v]

− ∇ · Frad (2.9)

After simplifying and opening up the square brackets:

∂ǫtot

∂t+ ∇ · (ǫtotv) =

∂ǫrad

∂t+ ∇ · (ǫradv) +

∂ǫ

∂t+ ∇ · (ǫv) = −P∇ · v + ( ¯τ∇) · v − ∇ · Frad + S (2.10)

where (¯τ∇) · v is an operator, which result: (∇ · ¯τ) · v+ τ1∂1v1 + τ2∂1v2 + τ3∂2v1 + τ4∂2v2 if the matrix

components are (τ1τ2τ3τ4) and the vector components are (v1v2).

The choice of the set of equations is obtained by defining the equation-of-state, the gravitational

potential, the radiative flux and providing an additional equation for the rate of change of the radiation

energy. The equation-of-state is:

P = (γ − 1)ǫ (2.11)

The radiation energy equation is obtained in Mihalas & Mihalas (1978) through a complex derivation.

For studying protoplanetary disks, however, we can neglect the advection of radiation energy. Then, the

equation for ǫrad is (Mihalas & Mihalas 1978):

∂ǫrad

∂t= −∇ · Frad + cκPρ

(

B(T )

c− ǫrad

)

(2.12)

Frad = −cλ

ρκR∇ · ǫrad (2.13)

where the radiative flux (Frad) is given by the flux-limited diffusion (Levermore & Pomraning 1981).

Here, λ denotes for the flux limiter that reduces the flux. It approaches to F = 4σT 4/c (σ is the Stefan-

Boltzmann constant, and T represents the temperature) in the optically thin parts, while it approaches

λ = 1/3 in the optically thick parts. Therefore, it accounts for the smooth transition between optically

thick and thin medium. The flux-limited diffusion approximation allows to solve for stable circumstellar

disk models that cover several vertical pressure scale heights. The flux-limiter is defined as in Kley (1989)

and Kley et al. (2009):

λ =

2

3+

√9 + 10R2

for R ≤ 2.0

10

10R+9+

√81 + 180R

for R > 2.0, (2.14)

where

R =1

ρκR

|∇ǫrad|ǫrad

. (2.15)

In Eq. 2.12, c indicates the speed of light, B(T ) is thermal blackbody: 4σT 4 The κP is the Planck

mean opacity, which can be written as (Bitsch et al. 2013):

κP(T, ρ) =

∫

κν,ns(T, ρ)Bν(T )dν∫

Bν(T )dν, (2.16)

where, κν,ns is the frequency (ν) dependent opacity including the effects of scattering. Moreover, κRsymbolizes the Rosseland mean opacity(Bitsch et al. 2013):

58


κR(T, ρ)−1 =

∫

κ−1ν,s(T, ρ)

∂Bν(T )

dTdν

∫

∂Bν(T )

dTdν

. (2.17)

here again, the effects of scattering are included. In the code, we set the Rosseland and Planck

opacities equal to each other, and we used the Bell & Lin (1994) opacities for both. This is because

according to Semenov et al. (2003) they do not differ much and thus we can gain computational time.

As always in numerical codes, one has to balance the running time with the complexity of the physical

models. Therefore, the physical model simplification is often needed to obtain reasonable running time.

Of course, when evaluating out the results, one has to keep in mind the simplifications made and consider

how much these affect the results. We have tested that the assumption on the equality of the two opacities

is not affecting our results much.

The coupling timescale between the thermal component of the radiation energy and the non thermal

one is 1ρcκP

(Bitsch et al. 2013).

When radiative module is active, we have a so-called two-temperature approach, when there is at

least one equation which describes the coupling of the radiation and matter. For this purpose, we solve

together Eq. 2.12 and Eq. 2.4.

Inserting Eq. 2.12 into Eq. 2.10 (remember that we have neglected ∇ · (ǫradv)) we obtain:

∂ǫ

∂t+ ∇ · (ǫv) = Q+ − P∇ · v − ρκPc

[

B(T )

c− ǫrad

]

(2.18)

This is referred as the internal energy equation in FARGO-type codes, like FARGOCA and FARGO3D.

This type of codes can be launched in adiabatic (i.e. only with energy equation but not with radiative

module) without the viscous heating term (Q+ = ¯τ∇ · v) (see Sect. 2.3.4), while JUPITER’s total energy

equation (Eq. 2.4) already includes the viscous heating term, therefore when it is adiabatic, it already

heats due to the viscous shear. The reason why we treated viscous heating in JUPITER this way, because

we want the momenta fluxes (the parenthesis within the bracket of Eq. 2.4) to be consistent with those of

total energy. A mismatch is found to lead to instabilities.

In the JUPITER-code, we solve an equation for Egas = ǫ + ρv2

2which inserting from 2.4 reads:

∂Egas

∂t+ ∇ ·

[(

E + P¯I − ¯τ)

· v]

= −ρv · ∇Φ − ρκPc

(

B(T )

c− ǫrad

)

+ S

∂ǫrad


(

B(T )

c− ǫrad

)

(2.19)

The implementation of the previous set of equations is done following the sketch of Fig. 2.3: E is

advected, the source terms ρv · ∇Φ (and the stellar heating, if considered) are included, then ǫrad and the

temperature is updated in the last step, what we call the radiative module explained in the next section.

2.3.2 The Radiative Module

In the last step of our multistep procedure (see operator splitting in Sect. 2.2.1) we solve:

59


ǫ = E − ρv2

2∂E

∂t= −cκPρ

(

B(T )

c− ǫrad

)

∂ǫrad


(

B(T )

c− ǫrad

)

(2.20)

As this equation shows, we update simultaneously ǫrad and ǫ – i.e. the temperature – and at the end

of the step we update E with the new internal energy.

Now that the stage is set for solving the Eq. 2.20, we switch to the algorithmic description instead

of the physical equations. To solve the two-temperature approach, we followed the method written in

Appendix B of Bitsch et al. (2013), which is based on (Commercon et al. 2011):

The radiation energy equation is given by

ǫn+1rad− ǫn

rad

∆t= 1∆x

(

Dxi+1, j,k

ǫn+1rad,i+1, j,k

−ǫn+1rad,i, j,k

∆x− Dx

i, j,k

ǫn+1rad,i, j,k

−ǫn+1rad,i−1, j,k

∆x

)

+ 1∆y

(

Dy

i, j+1,k

ǫn+1rad,i, j+1,k

−ǫn+1rad,i, j,k

∆y− D

y

i, j,k

ǫn+1rad,i, j,k

−ǫn+1rad,i, j−1,k

∆y

)

+ 1∆z

(

Dzi, j,k+1

ǫn+1rad,i, j,k+1

−ǫn+1rad,i, j,k

∆z− Dz

i, j,k

ǫn+1rad,i, j,k

−ǫn+1rad,i, j,k−1

∆z

)

+ρκP[4σ(T n+1)4 − cǫn+1rad

] , (2.21)

where i,j,k are the cell indices in the 3 spatial directions. The upper indices (n and n + 1) indicate the

time step. Obviously, n + 1 indicates the timestep in the future, the equation has to be solved implicitly,

which is again due to numerical reasons. In an explicit scheme, the time-step would be unbearably small:

δt < (δx)2/(2Di, j,k), where Di, j,k is the diffusion coefficient (see Chapter 19.2 in Numerical Recipes in C

1992. This is the reason we rather use an implicit scheme, where we first guess the new value of ǫrad at

time-step n+1, then we iteratively determine the new value.

The averaged diffusion coefficients are:

Dxi, j,k =

12

(

Di, j,k + Di−1, j,k

)

,

Dy

i, j,k= 1

2

(

Di, j,k + Di, j−1,k

)

,

Dzi, j,k =

12

(

Di, j,k + Di, j,k−1

)

,

which are given by

Di, j,k =λc

ρi, j,kκR,i, j,k. (2.22)

The term (T n+1)4 in Eq. 2.21 is non linear and makes the scheme difficult to invert, yet it is much easier

to solve a linear system implicitly. Assuming that the changes in temperature are small in each time step

(Commercon et al. 2011), we can write

(T n+1)4 = 4(T n)3T n+1 − 3(T n)4 . (2.23)

Now, we need to have T n+1 as a function of T n, ǫnrad

and ǫn+1rad

in order to solve equation. 2.21. We use now

the energy density equation, where we omit advection and compressional heating,

∂ ǫ

∂t= −ρκP(T, P)[B(T ) − cǫrad] + Q+ + S . (2.24)

60


With ǫ = ρcVT , and given that∂ρ

∂t= 0, we get

T n+1 − T n

∆t= −κP

cV

(4σ(T n+1)4 − cǫn+1rad ) +

S

ρcV

+Q+

ρcV

. (2.25)

With our approximation for the temperature (eq. 2.23) we find

T n+1 = η1 + η2ǫn+1rad , (2.26)

where

η1 =T n + 12∆t κP

cVσ(T n)4 + ∆tS

ρcV+∆tQ+

ρcV

1 + 16∆t κPcVσ(T n)3

and (2.27)

η2 =∆t κP

cVc

1 + 16∆t κPcVσ(T n)3

. (2.28)

We can now plug this all into the radiation energy equation and solve for ǫn+1rad

. We arrive at a matrix

equation:

β1,i, j,kǫn+1rad,i+1, j,k + β2,i, j,kǫ

n+1rad,i−1, j,k + β3,i, j,kǫ

n+1rad,i, j+1,k + β4,i, j,kǫ

n+1rad,i, j−1,k

+β5,i, j,kǫn+1rad,i, j,k+1 + β6,i, j,kǫ

n+1rad,i, j,k−1 + Γi, j,kǫ

n+1rad,i, j,k = Ri, j,k ,

The matrix elements are given by:

β1,i, j,k = −∆t

∆x2Dx

i+1, j,k

β2,i, j,k = −∆t

∆x2Dx

i−1, j,k

β3,i, j,k = −∆t

∆y2D

y

i, j+1,k

β4,i, j,k = −∆t

∆y2D

y

i, j−1,k

β5,i, j,k = −∆t

∆z2Dz

i, j,k+1

β6,i, j,k = −∆t

∆z2Dz

i, j,k−1

β1−6 = −(β1 + β2 + β3 + β4 + β5 + β6)

Γi, j,k = (1 + ∆tρκPc − 16∆tρκPσ(T n)3η2) + β1−6

Ri, j,k = 16∆tρκPσ(T n)3η1 − 12∆tρκPσ(T n)4 + ǫnrad

These matrix components are very similar in nature to the one for the one-temperature energy equa-

tion in Kley et al. (2009) and the matrix can be solved with the same matrix solver as in Kley et al. (2009).

We used the Successive Over-Relaxation Method described in Chapter 19.5 in Numerical Recipes in C

(1992). After the iteration, the new radiation energy is known, therefore the new temperature and the new

internal energy as well.

In the algorithm, the new temperature of a given cell is computed through an iterative step, and

jumping on checkerboard in 3D. So every second cell’s temperature is calculated at first (like the black

61


zones on a checkerboard), then the gaps (white spaces) filled up. This is a faster way to calculate the

temperatures, so valuable computer time is gained with this approach. On the other hand, this can be non-

trivial, when one runs it in parallel and has CPU-patches. It has to be defined correctly whether the black

zones in one CPU-patch border stand next to the white zones on the neighboring CPU-patch. We also tried

a method to enhance the speed of this iteration in all three codes (JUPITER, FARGOCA, FARGO3D).

Because the iteration has to start with an initial value, the smart choice of the initial value can reduce the

amount of iterative step needed. It can be assumed that a given cell’s new temperature will likely be the

current temperature plus what it changed in the previous time: T n+1 = T n+ (T n−T n−1)/(dtn−1 ∗dtn). This

gives us a good initial value for the new temperature when the iteration starts.

2.3.3 Heating Effects – Stellar Heating

As the circumstellar disk is illuminated by the star, the surfaces of the disk are heating up. This heating

of course depends on the stellar luminosity. In the parameter file, the effective temperature of the star and

the stellar radius should be given. As it was shown earlier in Sect. 1.1.2, the heating of the outer disk is

dominated by the stellar heating. In the inner disk, the viscous heating is the dominant heating term.

As it was shown in Bitsch et al. (2013), to account for the amount of stellar heating deposited in a

grid cell, one can compute the difference between what flux arrives and what leaves a grid cell:

F⋆e−τi − F⋆e−τi+1 = F⋆e−τi(

1 − e−τi+1

e−τi

)

(2.29)

= F⋆e−τi(

1 − e−(τi+1−τi))

= F⋆e−τi(

1 − e−ρiκi∆r)

, (2.30)

where i + 1 marks the i + 1th grid cell, τ represents the optical depth, which is integrated radially from

the star till the given cell:

τi = Σij=0ρ jκ jδr (2.31)

Since in parallel runs the CPU-patches spread also in azimuth, not only communication is needed to

get the integral, but also, in JUPITER the CPU-patch radial order had to be figured out to perform this

optical depth integration at every restart, since the CPU-patch ordering is different at every restart of the

code. Because, the total flux emitted by the star is given by L⋆ = 4πR2⋆σT 4

⋆, the stellar flux F⋆ per surface

area on a sphere with radius r is thus

F⋆(r) =L⋆

4πr2= σT 4

⋆

(

R⋆

r

)2

. (2.32)

With the front area of a grid cell given by

A = r2∆ϕ(cos θ1 − cos θ2) (2.33)

we can compute the flux received by a single grid cell

F⋆A = R2⋆σT 4

⋆∆ϕ(cos θ1 − cos θ2) = FS⋆ , (2.34)

which then leads to the stellar heating s of a grid cell

s = FS⋆e−τi(


(2.35)

= R2⋆σT 4

⋆∆ϕ(cos θ1 − cos θ2)e−τi(


. (2.36)

62


Because the energy equation (eq. 2.20) is written for energy densities, the division of s by the volume of

the grid cell V = r2∆r∆ϕ(cos θ1 − cos θ2) is needed to get the same units and ultimately to get S = s/V as

it is used in the code:

S = F⋆e−τi1 − e−ρκ⋆∆r

∆r(2.37)

According to Bitsch et al. (2013), this result is dependent on the resolution, but it is only important for the

optically thick regions (ρκ⋆∆r > 1). For optically thin regions the approximation (1−e−ρκ⋆∆r)/(∆r) = ρκ⋆is used. They claim that “in the inner, optically thick regions of the disk, the resolution influences the

repartition of the stellar heating, but it does not influence the global net heating. Resolution studies

indicate that the height of the puffed up inner edge varies by ≈ 3% when lowering the resolution by

33%. As the disk is radially optically thick, the stellar flux is absorbed in the first cells; beyond that the

simulations match perfectly for all tested resolutions.”

In order to illuminate the disk, one has to set up a flared disk right from the beginning of the simu-

lation. This also means that the radial extension of the circumstellar disk has to be large, to have large

enough flared region where the stellar irradiation can have an effect, and keep the disk flared. If the

setup is such that the stellar irradiation cannot illuminate the outer disk, the latter will collapse onto the

midplane. Even though in this thesis none of the projects involves simulations with stellar irradiation, it

is useful to mention the usual parameters used in stellar irradiation simulation, which are:

• flaring index ≈ 0.28

• disk radial extent ≈ 20 AU

• stellar irradiation effect above ≈ 7 degrees opening angle

In the inner disk boundary, where the star would illuminate the cells directly (so called face illu-

mination, which result in a very puffed up inner rim), one should avoid a too large face illumination.

Following an advice of Wilhelm Kley, the FARGOCA team has set up two inner ghost ring for the stellar

irradiation, where most of the irradiation gets absorbed. This is a valid assumption, since for numerical

reasons, we set up the inner radial boundary of the circumstellar disk quite far away from the star, simply

to avoid too small cell-sizes, hence too small time-steps. It is assumed that even though the simulated

circumstellar disk starts from ≈ 0.5 AU from the star, there is a disk component within this radius. So,

adopting the ghosts cells to absorb the stellar irradiation before the inner radial boundary of the disk

is realistic and avoid numerical problems due to face illumination. This only works because one can

demonstrate that F f ace → 0 with r → 0. Since this method worked well in FARGOCA, I have used the

same in JUPITER, and later, FARGO3D inherited the same stellar module. The stellar heating was not

used in any of the simulations presented in this thesis, simply because it is thought to have no important

effect on the circumplanetary disk – given it affects only the surface of the circumstellar disk. Because the

stellar heating requires a radially large, flared circumstellar disk, avoiding it saved computation time for

the circumplanetary disk studies. Nevertheless, it is part of the radiative module we built into JUPITER,

and it was well tested as well.

2.3.4 Heating Effects – Viscous heating

Viscous heating is due to shear. If the kinematic viscosity, ν is not vanishing, one has to include the

heating due to the shear. The stress tensor is:

63


¯τ = 2ρν( ¯D − 1

3(∇~v) ¯I) (2.38)

where ¯D is the strain tensor.

In spherical coordinates the components of the stress tensor can be written as (Tassoul 1978):

τRR = 2ρν(∂vR

∂R− 1

3∇ · ~v)

τϕϕ = 2ρν( 1R

∂vϕ∂ϕ+

vR

R− 1

3∇ · ~v)

τθθ = 2ρν( 1R sinϕ

∂vθ∂θ+

vR

R+

vϕ cot(ϕ)

R− 1

3∇ · ~v)

τRϕ = τϕR = 2ρν[ 12( 1

R

∂vR

∂ϕ+ R ∂

∂R

vϕ

R)]

τθϕ = τϕθ = 2ρν[ 12( 1

R sin(ϕ)

∂vϕ∂θ+

sin(ϕ)

R∂∂ϕ

vθsin(ϕ)

)]

τRθ = τθR = 2ρν[ 12(R ∂∂R

vθR+ 1

R sin(ϕ)

∂vR

∂θ)]

(2.39)

The energy equation (Eq. 2.4) features the divergence of the stress tensor, which was written as f .

The expressions for the components (Tassoul 1978) of f in spherical coordinates are:

fR =1

R sin(ϕ)[

sin(ϕ)

R∂∂R

(R2τRR) + ∂∂ϕ

(τRϕ sin(ϕ)) + ∂τRθ∂θ

] − τϕϕ+τθθR

fϕ =1

R sin(ϕ)[

sin(ϕ)

R∂∂R

(R2τϕR) + ∂∂ϕ

(τϕϕ sin(ϕ)) +∂τϕθ∂θ

] +τRϕR− τθθ cot(ϕ)

R

fθ =1

R sin(ϕ)[

sin(ϕ)

R∂∂R

(R2τθR) + ∂∂ϕ

(τθϕ sin(ϕ)) + ∂τθθ∂θ

] + τRθR+τϕθ cot(ϕ)

R

(2.40)

As mentioned before, in the JUPITER code we are dealing with the viscous heating term inside the

energy equation (see Eq. 2.4), because only this way the code is stable. This means that the viscous heat-

ing is always applied when the adiabatic code is used, even without the radiative module, which would

account for the cooling. For our purposes on the circumstellar disk studies, we always use the radiative

module, hence the cooling as well, so this does not cause any problem. Where the viscous heating is ap-

plied is therefore different in the JUPITER code versus FARGOCA/FARGO3D. In the Fargo-type codes,

the viscous heating is applied in the radiative module, not in the energy equation step, therefore, those

codes can be launched in a pure adiabatic manner without additional heating/cooling terms.

2.3.5 Boundary Conditions for Cooling

In order to cool the disk in a physically realistic manner, we decided to put a simplified radiative transfer

into the code, so that the cooling – and heating – happens using an opacity function of Bell & Lin (1994).

We followed the method described in (Kley et al. 2009), optimized it, and adapted to nested meshes. The

new modules were heavily tested, mostly by comparing the results with the FARGOCA code, which also

features these heating/cooling modules. For the tests see Section 2.5.3.

The cooling is happening through radiation. To control the cooling rate, we set the radiative energy

in the ghost cells above and below the two surfaces circumstellar disk, corresponding to temperatures

cooler than the disk’s. We used 30 K in the case of JUPITER, while FARGOCA uses 3 K, as it represents

the Cosmic Microwave Background (CMB) temperature. These two cooling boundaries above the cir-

cumstellar disk surfaces are quite realistic, since a CSD in nature is also surrounded by space effectively

at CMB temperature.

2.3.6 Equilibrium & Initial Condition

When both heating and cooling are accounted for, the circumstellar disk will find a thermal equilibrium

evolving away from the initial conditions. Through the radiative transfer, the boundary temperature cools

64

2.4. Parameters of the code Chap.2 Code

off the disk, so in comparison with the disk initial opening angle, the disk will have a smaller opening

angle after the – heating/cooling involved – thermal equilibrium is reached. This new equilibrium will

depend on the balance of heating and cooling, which depends on the viscosity, the dust-to-gas ratio,

the circumstellar disk mass, the stellar parameters, etc. To gain simulation time, we used a trick to

quickly reach this new thermal equilibrium. Even if we want to have planet embedded into the disk in

the simulation, we start with a planet-less circumstellar disk. We set up the heating and cooling on the

base mesh. Because the circumstellar disk is azimuthally symmetric before the addition of the planet,

we do not have to use many cells azimuthally. Two cells are enough over the 2 π full disk; this way

the number of cells in the 3D simulation box is very small, which results in a fast run. The simulation

initially runs with only 2 cells in azimuth until a thermal equilibrium is reached, then we divide the 2

cells in azimuth into as many as we want in the simulation. This method saves a lot of computing time.

When the re-binning of the azimuthal coordinate in a larger number of cells is done, we put the planet

into the circumstellar disk, and restart the simulation on the base mesh.

2.4 Parameters of the code

2.4.1 The parameter file

There are parameters which the user can setup in the parameter file and there are others which can be set

in the def.h file containing predefined parameters (see Sect. 2.4.2). The difference between the two, is

that a given compilation contains the parameters setup in def.h file, so if one wants to change any of

the parameters in that file, the code should be recompiled. In the parameter file, whoever, the user can

change any parameter without needing to recompile the code.

Let’s first see the parameters which can be set up in the parameter file:

Table 2.1. Parameters of the Code

Name Type Essential

Pa-

rame-

ter?

Default

Value

Description

DT REAL YES 1. Coarse grain time stepping of the code,

however the CFL condition will divide

this into sub-timesteps, see below the ta-

ble

OUTPUTDIR STRING YES . the path of the output directory

NTOT INT YES total number of timesteps of length DT

of the simulation

NINTERM INT YES 64.0 dump output after this amount of

timesteps

SIZE1 INT YES 64.0 number of cells in the first dimension

SIZE2 INT YES 64.0 number of cells in the second dimension

SIZE3 INT YES 64.0 number of cells in the third dimension

(if any)

Continued on next page

65


Table 2.1 – continued from previous page

Name Type Essential

Pa-

rame-

ter?

Default

Value

Description

GHOSTFILLINGORDER INT NO 1 whether the filling of the ghost of each

level is made by multi-linear interpola-

tion (1) or direct injection (0)

DIM1SPACING STRING NO Arithmetic the spacing between the cells in the first

dimension is arithmetic or logarithmic

DIM2SPACING STRING NO Arithmetic the spacing between the cells in the sec-

ond dimension is arithmetic or logarith-

mic

DIM3SPACING STRING NO Arithmetic the spacing between the cells in the third

dimension is arithmetic or logarithmic

RANGE1LOW REAL NO 0.0 Minimum coordinate of the base mesh

in the 1st dimension

RANGE1HIGH REAL NO 1.0 Maximum coordinate of the base mesh

in the 1st dimension


in the 2nd dimension


in the 2nd dimension


in the 3rd dimension


in the 3rd dimension

COURANTNUMBER REAL NO 0.9 Value of Courant Number, see CFL con-

dition explained below this Table

GAMMA REAL NO 1.6666 the value of the Adiabatic Index

NDIM INT NO 3 Number of dimensions

COORDTYPE STRING NO Cartesian Coordinate system type

COORDPERMUT STRING NO 123 Permutation of the coordinates

DIM1PERIODIC BOOL NO NO Whether the 1st dimension is periodic

DIM2PERIODIC BOOL NO NO Whether the 2nd dimension is periodic

DIM3PERIODIC BOOL NO NO Whether the 3rd dimension is periodic

GRIDFILE STRING NO The name of the file containing the grid

RIEMANNSOLVER STRING NO 2R Desired Riemann solver type (two-

shock, two-rarefaction, automatic /code

decides based on the gradient of the hy-

dro fields in the actual cell boundary/)

ADIABATIC STRING NO NO Isothermal or Adiabatic

METHOD STRING NO MUSCL Method for state prediction (PLM,

MUSCL, GFO)


66



Name Type Essential

Pa-

rame-

ter?

Default

Value

Description

INITCODE STRING NO initial setup of the physical problem,

e.g. kepler3D for a 3D keplerian disk

POTENTIALCODE STRING NO name of the gravitation potential setup

used, e.g. planet3d for a planet’s poten-

tial

KEPLERIAN BOOL NO NO whether an astronomical disk is the

physical problem (the code can solve

of course different physical problems as

well)

OMEGAFRAME REAL NO 0.0 the coordinate system rotation fre-

quency

ASPECTRATIO REAL NO 0.05 initial aspect ratio of the circumstellar

disk at radius=1.0

SIGMASLOPE REAL NO 0.0 The exponent of the surface density

power law profile: 1/rsigmaslope

FLARINGINDEX REAL NO 0.0 initial flaring index for the circumstellar

disk

SIGMA0 REAL NO 6e-4 initial surface density at radius=1.0

MASSTAPER REAL NO 0.0 timescale over which the planet mass

evolves from 0 to the final mass

VISCOSITY REAL NO 0.0 constant kinematic viscosity value

PLANETMASS REAL NO 0.0 mass of the planet

SMOOTHING REAL NO 0.0 smoothing length of the potential ex-

pressed in code units, if one wants the

same value for every mesh, however, it

is better to use level dependent smooth-

ing, which scales with the resolution

SMRATIO REAL NO 2.0 in the case of level dependent smooth-

ing length, this multiplication factor de-

cides how many cell diagonals is the

smoothing length on each level; e.g. the

default value is 2.0 cell diagonal

EXTERNALPOTENTIAL BOOL NO NO in case an additional, external potential

is desired

MERIDIANCORRECTION BOOL NO NO meridian correction in case of nested

meshing

DRIFTVELOCITY REAL NO 0.0 initial drifting radial-velocity of the cir-

cumstellar disk

CS REAL NO 1.0 sound-speed in case of isothermal sim-

ulation


67



Name Type Essential

Pa-

rame-

ter?

Default

Value

Description

INCLINATION REAL NO 0.0 inclination of the disk

SUBCYCLING STRING NO Auto whether level dependent sub-timestep is

calculated through the CFL condition or

not, see more below this Table

NODAMPING STRING NO NO whether there is damping applied at the

simulation box borders, which keeps the

initial condition values for all the hydro

fields

SYMDIMHYDROSTAT STRING NO 000 fixing (yes=1) the hydrostatic

equilibrium in each dimension:

[0||1][0||1][0||1]

CORRDIMHYDROSTAT STRING NO 000 correcting (yes=1) for the hydro-

static equilibrium in each dimension:

[0||1][0||1][0||1]

FLUIDS STRING NO gas in case of multi-fluid simulation, addi-

tional fluid names can be added, e.g.

gas/dust

COUPLING REAL NO 0.0 coupling between dust and gas, if dust

is present

DUSTTOGAS REAL NO 0.01 dust-to-gas ratio

HIGHRESLEVEL INT NO 100 level (number) on which one wants to

switch between the two methods for the

evaluation of the internal energy, see

Sect. 2.6.2

VISCUTOFFLEVEL INT NO 100 turning off the viscosity beyond a given

level

HALFDISK STRING NO YES half disk simulation, or both hemisphere

of the disk is simulated

STELLAR STRING NO NO activating the radiative module

TSTAR REAL NO 5600.0 in case of stellar heating, the star’s tem-

perature

RSTAR REAL NO 3.0 in case of stellar heating, the star’s ra-

dius

STELLDEG REAL NO 7.0 in case of stellar heating, the angle, in

degrees, above which the circumstellar

disk is irradiated

In the following, more in-depth descriptions of some of the important parameters will follow. The

parameters SUBCYCLING and COURANTNUMBER parameterize the Courant-Friedrichs-Lewy con-

68


dition (CFL condition). This is defined in the following manner:

C = ∆t

3∑

i=1

vxi

∆xi

≤ 1.0 (2.41)

where C is the Courant number, i represents the number of dimensions, xi means the spatial variables,

and v indicates the velocity. This condition decides about the timestep, given what is the cell size.

Because we are dealing with several levels and so various cell sizes, the timestep can be different on

each level. On the higher resolution mesh the timestep can be half of the timestep on the previous

level, i.e. two iterations are done on the finer level, while only one iteration is performed on the coarser

mesh. This is called timestep subcycling, which the SUBCYCLING parameter defines. Throughout this

thesis we use an adaptive subcycling procedure, in order to obtain the maximum speed up of the code.

The COURANTNUMBER is a number between 0.0 and 1.0 which controls how small time-step we are

ending up with. The smaller is the Courant number, the slower is the simulation; however, the code is

more accurate in the computations. In this thesis, we use COURANTNUMBER=0.9 or 0.8.

The parameter NODAMPING defines whether we want to use so-called Stockholm-damping at the

simulation box borders. Damping resets the hydro-quantities (density, velocity, energy) to their initial

values. While this technique is widely used to avoid spurious wakes at the simulation box boundaries,

the set-up of some specific boundary conditions and the presence of ghost cells are enough to avoid

problems happening at the borders. In JUPITER, 2 ghost cells mirror the hydro-field values of the two

last active cells. So the boundary, even with reflecting boundary conditions is not a hard wall, like in

some hydro codes. Therefore, I have never experienced wave reflection.

Moreover, I decided not to use damping at all, because it caused only harm in my simulations. First,

in the isothermal simulation the gap did not open around the Jupiter-mass planet’s orbit, because the

damping did reset the initial density in the top layer of the circumstellar disk, resulting in a mass source

refilling the planetary gap. The usage of damping also means that we keep adding new mass into the

simulation, therefore the mass is not conserved any longer. In the radiative simulations, it makes no sense

to use any damping, since the initial conditions of the circumstellar disk evolve into a new hydrostatic

equilibrium due to heating and cooling. Thus reversing the process by enforcing the initial condition of

the circumstellar disk will cause serious problems. Therefore, I decided not to use any damping in my

simulations.

How quickly the planet builds up in the simulation is determined by the parameter MASSTAPER.

This is a number which tells in how many orbits we want to build up the planet. The planet in our

simulations is only a point mass in the corner of 8 cells. It represents just the depth of a potential-

well in the simulation. Through the mass-tapering phase, we deepen the potential well as Mpactual=

Mpfinalsin2

(

t30π2

)

where 30 is the number of orbits. This is the traditional technique used widely in the

hydro-community. One should take care of using long enough mass-tapering to avoid shocking the

circumstellar disk with the sudden appearance of a large-mass planet. I have seen that, especially in the

adiabatic simulation, one should use long mass-tapering even for Earth-sized planets (!) because a too

short mass-tapering changes the results.

The parameters SMRATIO and SMOOTHING are controlling the smoothing of the potential. To

avoid the singularity of the potential-well, we used the traditional technique of smoothing the potential,

which is done in the following way:

Up = −GMp

√

x2d+ y2

d+ z2

d+ rs

2

(2.42)

69


where xd = x − xp, yd = y − yp, and zd = z − zp are the distance-vector components from the planet in

Cartesian coordinates. The smoothing length rs can be used as a constant value in code units on every

level, or better be changed depending on the level we are at. In fact, because we are increasing the

resolution on the nested meshes, the smoothing length could/should be decreased as well. We have set

up a cell-size dependent smoothing, where the cell diagonal divided by 2 sets up the smoothing length

on a given level. The SMRATIO parameter controls how many cell diagonals (divided by 2) we choose.

This way we can set up a level dependent smoothing length, e.g. having 2 cell-size length for level 0 and

level 1, even though the cell size halved on the higher resolution grid. As in the case of mass-tapering,

here a similar method should be used when adding a new level to the simulation. To avoid the instant

change of the potential-well when adding a new level, we have used a cosinusoidal function to slowly

deepen the potential well when each new level was added: rsactual= 0.5

(

rspreviouscos2

(

t−t04

)

+ rsprevious

)

. Again,

this smoothing-tapering should be long enough not to cause any numerical artifact in the simulation and

perturb the flow.

To issue smoothing-tapering, the code should be started with the argument ”-h” (see more in Sect.

2.4.3), id est:

./jupiter -h -s 100 in/parameters.par

2.4.2 Predefined variables

The def.h file contains predefined constants, such as π or the number of ghost cells, which is 2. The ghost

cells ensure the correct communication between levels, CPU-patches and can be used to set up boundary

conditions. In the entire simulation box, there are 2 ghost cells on each side in each spatial dimension.

Similarly, each CPU-patches overlap each other by 2-2 cells in each direction. The overlapping levels

communicate similarly through 2 overlapping ghost zones below the active cells.

We have set up a couple of constants for the radiative module in def.h file. These constants are

mainly in cgs units, because the opacity table used (Bell & Lin 1994) is using values in cgs units as well.

Therefore, each hydro-quantity is transformed from code units to cgs units before the radiative module,

then back again after the radiative module. Here are the definitions for the radiative transfer:

\#define XMSOL 1.989e+33 // Solar Mass

\#define XMH 1.67 e-24 // Hydrogen Mass

\#define AU 1.496e+13 // Astronomical Unit

\#define BOLTZ 1.38e-16 // Boltzmann Constant

\#define CLIGHT 2.99792458e+10 // Light speed

\#define GRAVC 6.67e-08 // Gravitation Constant

\#define ARC 7.56e-15 // Radiation Constant

\#define RGAS 8.314e+07 // Gas Constant

\#define XMSTAR 1.0 // Amount of Solar Masses

\#define RSUN (6.96e+10) /* Sun Radius in cm */

\#define R0 (5.2*AU) // semi-major axis

\#define VOL0 (R0*R0*R0) // volume

\#define XM0 (XMSOL * XMSTAR)

\#define RHO0 (XM0 / VOL0) // density factor between

// code units and cgs

\#define TIME0 (sqrt(VOL0 / GRAVC / (XMSTAR*XMSOL)))

\#define V0 (R0 / TIME0)

70


\#define OPAC0 (R0*R0 / XM0)

\#define XNU0 (R0*V0)

\#define TEMP0 (V0*V0 / RGAS) // temperature factor between

// code units and cgs

\#define P0 (RGAS * RHO0 * TEMP0)

\#define E0 (XM0 * R0*R0/ (TIME0*TIME0))

\#define SLUM (1.0* TIME0 / E0)

\#define C0 (CLIGHT/V0)

\#define A0 (ARC*pow(TEMP0,4.0)/(RHO0*V0*V0))

\#define SIGMARAD (0.25*A0*C0)

\#define BOUNDTEMP 30.0 // cooling temperature

\#define MU 2.30 // mean molecular weight

// for standard solar

// mixture (Kley+ 2009)

\#define CV (1.0/MU/(GAMMA-1.0)) // specific heat coeff (const. volume)

// code units (1.0 in the numerator

// instead of RGAS, latter would lead

// to CV in cgs units)

\#define TSTAR0 (TSTAR/TEMP0) // star temperature

\#define RSTAR0 (RSTAR*RSUN/R0) // radius of the star

\#define FSTAR (SIGMARAD*pow(TSTAR0,4.0)*RSTAR0*RSTAR0) // flux of the star

\#define EPSILON 3.0e-6 // how long the iteration should go in RT

\#define BoundEcode (pow((BOUNDTEMP/TEMP0),4.0)*A0) // energy limit

// corresponding to cooling temperature

2.4.3 The arguments of the code

There are several arguments for JUPITER, if we type ./jupiter we get the following:

Usage : jupiter [-aceghiknptuz] [-j maxlevel] [-o option] [-S number] [-s number]

[-m number] [-r "output# grid# pos"] [-x "output# camera_file"] filename

-a: All CPUs create output directories. Useful if each CPU has a different disk.

-c: monitor CFL condition or coarse ray tracing

-e: monitor kinetic energy.

-g: include ghosts when writing hydro fields to the disk

-j: set maximal level of refinement.

-h: to issue smoothing tapering

-k: computes torque with Stockholm specifications.

-m number: merge output number (to fake mono-CPU)

-n: run disabled. Only read input files, and write first output

-o: redefines a variable (e.g. -o "outputdir=/scratch/test/").

-p: monitor CPU or wall clock time usage.

-r "output# grid# position": refine a specific grid at a given position

-s number: restart run (from mono-CPU output)

-S number: stretch run (from mono-CPU output). Not a real restart ==> output 0

71

2.5. Testing the code Chap.2 Code

-t: monitor torque.

-x "output# camera_file": ray tracing on given output

-z: do not monitor mass and angular momentum

Let us highlight a few important arguments. The code, if running in parallel mode, produces the

output for each CPU, which means we are going to have 100 different density files if 100 cores are used

and the same is true for the other hydro fields. For plotting and other handling reasons, the user might

want a merged version of these files, as if the simulation would have run on 1 core only. After the run is

completed, we have to issue ./jupiter -m 14 in/parameters.par if we want the files of output 14

to be merged.

We can restart the simulation from a previous output after that output was merged. The argument

-s does that for us. Simply, we have to write the output number after -s from which we want to restart

the code, for instance to restart from output 65: ./jupiter -s 65 in/parameters.par. As men-

tioned earlier, the -h argument can be used when smoothing-tapering is needed, i.e. after a new level

addition. Thus, restarting a run from output 65 with smoothing tapering one should type the following:

./jupiter -h -s 65 in/parameters.par.

In the nominal case, the code writes out the hydro fields only in the active domain, without the ghost

cells. But for debugging purposes one might need to write out the ghost cells as well. The argument -g

can be used here with the restriction that the code cannot be restarted from output files created this way.

We mentioned before that the grid levels are added during the simulation, after the flow has reached

a steady state on the last level. Of course, there is a way to add many levels at the same time at the

beginning of the simulation. Then the levels should be set up in the parameter file. But in this thesis, the

physical problem we are studying make it more adequate to add the levels gradually. For example, I wait

to add the first nested mesh until the planetary gap is properly opened on approx. 100-150 orbits. Then,

the level 1 can be added with the -r argument. Here is an example:

./jupiter -r "150 0 -0.2773 0.722 1.461 0.2773 1.2773 1.5707" in/parameters.par

where the first number is the output number where we want to add the new level, the next one is the grid

level to be refined, so for us level 0, because we want to add a finer level, level #1. The next 6 numbers

are the limits of the new mesh in code units: min(azimuth) min(radius) min(co-latitude) max(azimuth)

max(radius) max(co-latitude). This argument defines a new mesh, it does not re-start the code. After this

refinement step, one should restarts with the ./jupiter -h -s 150 in/parameters.par command

and the most likely desired smoothing-tapering.

2.5 Testing the code

During the code-writing, we tested the code several times in numerous ways. It is not possible to describe

all the tests, and show each test results, therefore I only highlight a few, which can convince the reader

about the validity of the algorithm. We did three main groups of global tests: testing the hydro algorithm

(i.e. the Riemann solvers), testing the radiative modules with and without the hydro module, and testing

the communication (between different CPU cores, and between different nested meshes).

2.5.1 Testing the Riemann Solvers

Several known test problems can be found in the literature but the user can of course set up own tests.

The first Riemann problem, which has a known analytical solution, is a shock-tube test, the so-called Sod

72


test in Sect. 4.3.3. in (Toro 2009). It consists of a left rarefaction, a contact discontinuity and a right

shock. The Sod problem exist in 1D and 2D as well. Figure 2.4 shows the density in 1D case after 10

time-step (t=0.1), and the velocity. It is notable that the steps in the density profile are very sharp, which

is a good indication that the code works properly. If the Riemann-solvers were erroneous, one would see

tilted lines in the step function part (i.e. neither horizontal, nor vertical). The reader can compare the

Figures of this work with the Figure 4.7 in (Toro 2009) for the same test.

Figure 2.4. The so-called sod 1D problem. It is a specific kind of Riemann-problem, whose

analytical solution is known. The first row represents the initial conditions, the second is after 10

time-steps (t=0.1). On the left we can see the density, and how sharply the discontinuities occur,

which highlights the accuracy of the algorithm in treating discontinuities.

We can extend the sod problem into 2D. On Figure 2.5, the density initial condition is on the left,

73


while the density map after t=10 (100 time-steps later) is on the right for the two-rarefaction solver. At

the beginning the fluid is at rest (velocity=0). Again, the sharpness of the shocks are remarkable.

Figure 2.5. The sod problem in 2D, testing the two-rarefaction solver. The initial condition of

the density is on the left, while 100 time-step later we get the density map on the right. The

shock-fronts are very clear and sharp.

Different methods can be tested as well. Figure 2.5 shows the case for the PLM method, and we get

an almost identical image for the MUSCL method as well. However, the GFO method, which is much

more diffusive as the other two, results a more blurry, and less proper density map as Figure 2.6 shows.

In 2D we have the opportunity to test the nested meshes as well. In the 2D sod problem, now check-

ing the two-shock solver, I included two nested meshes, which will allow to test whether the mesh-

communication is correct. If this were not the case, one would see discontinuities around the nested

meshes, but Figure 2.7 proves otherwise. On the Figure, on left side the resulting density map is shown

after 50 time-step (t=25) for the PLM method, on the right for the GFO method. Both runs have nested

meshes, discontinuities at the mesh boundaries cannot be seen. We see again that the GFO method gives

a more diffusive result than the PLM or MUSCL methods.

The previous tests were performed in Cartesian coordinate systems. But it is important to test the

different coordinate-systems as well. We set up the 2D sod problem in cylindrical coordinates; and as

Figure 2.8 shows for t=0 (initial condition), t=0.5 and t=1, the performance of the code is very good.

Here, again nested grids were added.

The above test were performed with all different methods (PLM, MUSCL, GFO), in isothermal and

in adiabatic setting, and with all the Riemann-solvers built in the code and their permutations. As it was

shown, the nested meshing, and the different coordinate-systems were tested as well during these basic

Riemann tests.

2.5.2 Testing the Communication

To test the communication, both inter-CPU and inter-level, the developer has quite simple options. How

the code splits the simulation box into different CPUs is code-specific. The JUPITER code cuts the field

in all spatial dimensions into so as to minimize the contact surface of CPUS (individual CPU patches are

74


Figure 2.6. The sod problem in 2D, now with the GFO method. Clearly, the resulting density map

is less sharp than for the PLM and MUSCL methods shown on Fig. 2.5

“as cubic as possible”, Fig. 2.9). Then each CPU performs the computations on the small box attributed

to it. There are several instances in the algorithm when the CPUs need to communicate within each

timestep, sharing their information (hydro quantities) to avoid having any mismatch between the CPU

patches.

To test the CPU communication, one needs to set up a relatively low-resolution simulation of any

given physical problem and run it in sequential (1 core) and in parallel (multi-cores). The resulting hydro

fields then should be compared via plots but also making the subtraction of e.g. the energy field obtained

in the sequential computation and that obtained in the parallel computation. If the difference is below

the numerical accuracy (i.e. 15-17 significant digits for double precision) the test passed. I have set up

several different physical problems (mostly circumstellar disk problems with and without planets), tested

them in 3D with various amount of CPUs. This test needed to be done every time new patches were

added to the code. In case of a mismatch, I commented out parts from the newly added code, until I

nailed down the problem.

The nested meshes can be added as disjoint patches into different parts of the simulation box, but

they can be placed on top of each other, as Figure 2.10 shows. For the inter-level communication, the

available tests are following a similar logic. One can perform a simulation with one nested mesh, and

another one which has only the base mesh but with double resolution, i.e. corresponding to the level 1

resolution of the other simulation on the refined level. Making plots from the hydro fields and making

difference calculations of the hydro fields from the two simulations can highlight problems. In the case

of nested meshes, it is very important to test the case when the coordinate system does not rotate together

with the circumstellar disk. In case of improper inter-level communication, the nested meshes can draw

75


Figure 2.7. The sod problem in 2D, testing the two-shock solver. On the left the PLM method’s

density map, on the right the diffusive GFO method’s. Both cases have two nested meshes, but

discontinuity around the nested grids cannot be seen, which strengthen that the inter-level com-

munications are correct.

Figure 2.8. The sod problem in 2D, in cylindrical coordinates with nested meshes. The density

maps are plotted. The left panel shows the initial condition, the middle is at t=0.5, while the right

shows the t=1 case.

a “tail” in the downstream direction. This is obviously something that should be avoided.

In the tests of communications, the length of the test matters a lot. Small errors can accumulate over

the simulation time; it is therefore necessary to test these issues through long enough runs. A very tricky

issue I learned during these tests is that every iterative routine has to feature inter-CPU communications,

otherwise the iterative loop will provide different values on different CPUs.

76


Figure 2.9. A simulation ran in parallel, showing how the JUPITER code distributes the simula-

tion box among many CPUS. The code splits up the 3D field into smaller boxes, and each CPU

computes one small box.

2.5.3 Testing the Radiative Module

Since the radiative modules are very similar to those built into the FARGOCA code by Elena Lega, I

mostly tested the radiative modules of JUPITER by making exact comparisons between the simulations

made with the two codes. Because FARGOCA has no nested meshing, I tested the radiative modules of

JUPITER with the nested meshes in a similar manner as it was described in the previous Sect. 2.5.2, i.e.

checking that the communication is correct between the nested meshes even when the radiative module

is switched on.

To compare the results of JUPITER with FARGOCA, I had to modify FARGOCA grid system and

initial setups for density/energy so that it would match JUPITER’s. This way I could check the results up

to numerical precision, leading to very accurate testing. Thanks to this very rigorous testing, I discovered

a few small issues in the FARGOCA codes as well. I chose to modify FARGOCA, because it is much-

much easier to do these modifications in that code than in JUPITER. The grids of the codes are defined

differently; here is how I modified FARGOCA’s grid’s radially and co-latitude directions (azimuth is

defined the same way):

/* Original radius array: GlobalRmed[i] = 3.0/4.0*(Radii[i+1]* \

Radii[i+1]*Radii[i+1]*Radii[i+1] \

-Radii[i]*Radii[i]*Radii[i]*Radii[i]);

GlobalRmed[i] = GlobalRmed[i] / (Radii[i+1]*Radii[i+1]*Radii[i+1] \

-Radii[i]*Radii[i]*Radii[i]);*/

77


Figure 2.10. A simulation with 6 levels of refinement. The nested meshes can be put down in

different parts of the simulation box, or as shown here, they can be added on top of each other. In

this thesis, the planet vicinity is studied, therefore the nested meshes are added on each other to

reach the highest possible resolution around the planet.

// Modified by Judit to match Jupiter’s:

GlobalRmed[i] = 2.0/3.0*(Radii[i+1]*Radii[i+1]*Radii[i+1] \

-Radii[i]*Radii[i]*Radii[i]);

GlobalRmed[i] = GlobalRmed[i] / (Radii[i+1]*Radii[i+1] \

-Radii[i]*Radii[i]);

// Co-latitude definitions -- Original:

//Phimed[h] = Phi[h]+0.5*(Phi[h+1]-Phi[h]);

// Modified by judit:

if (.5*( Phi[h]+ Phi[h+1]) < M_PI/2.)

Phimed[h] = asin((cos(Phi[h])-cos( Phi[h+1]))/( Phi[h+1]-Phi[h]));

else

Phimed[h] = PI-asin((cos(Phi[h])-cos( Phi[h+1]))/( Phi[h+1]-Phi[h]));

// modified till here

As the initial density-profiles also differed in the two codes, I changed FARGOCA accordingly:

real Sigma(r,phi)

real r,phi;

78


/*//ORIGINAL:

real cavity=1.0,sigmafactor,rcil,z,height;

if (r < CAVITYRADIUS) cavity = 1.0/CAVITYRATIO; // This is *not* a steady state

// profile, if a cavity is defined. It first needs

// to relax towards steady state, on a viscous time scale

sigmafactor = SIGMA0/ASPECTRATIO;

sigmafactor /= sqrt(2.0*PI);

z = r*cos(phi);

rcil = r;//*sin(phi);

sigmafactor *= pow(rcil,-1.0-SIGMASLOPE-FLARINGINDEX);

height = ASPECTRATIO*pow(rcil,1.+FLARINGINDEX);

//changed to rcil!

return cavity*ScalingFactor*sigmafactor*exp(-(z*z)/(2.0*height*height));

// We keep this density distribution instead of the previous which comes

// from hydrostatic equilibrium since it is less steep and this is preffered

// for the transition to equilibrium in Radiative, especially in stellar case

*/

//Judit’s JUPITER’s version:

real cavity=1.0,sigmafactor,rcil,z,height,fac,w,hm2,init;

FILE *fp;

if (r < CAVITYRADIUS) cavity = 1.0/CAVITYRATIO; // This is *not* a steady state

// profile, if a cavity is defined. It first needs

// to relax towards steady state, on a viscous time scale

sigmafactor = SIGMA0/ASPECTRATIO;

sigmafactor /= sqrt(2.0*PI);

z = r*cos(phi);

rcil = r*sin(phi);

fac=1./sin(phi);

w=(0.5-FLARINGINDEX)*2+1.0+SIGMASLOPE+FLARINGINDEX;

hm2=1./(ASPECTRATIO*ASPECTRATIO*pow(r,2.*FLARINGINDEX));

sigmafactor *= pow(r,-1.0-SIGMASLOPE-FLARINGINDEX);

height = ASPECTRATIO*pow(r,1.+FLARINGINDEX);

init=cavity*ScalingFactor*sigmafactor*pow(fac, w)*pow(fac, -hm2);

if (FLARINGINDEX > 0.0) init=cavity*ScalingFactor*sigmafactor*pow(fac, w)* \

exp(hm2 * (1. - pow (fac, 2.*FLARINGINDEX))/2./FLARINGINDEX);

79


return init;

Finally, the energy (because the velocity initial profiles were the same):

real Fenergy(i,phi)

real phi;

int i;

/* Original:

real energy0,rcil=0.;

if (ADIABATICINDEX == 1.0)

fprintf (stderr, "The adiabatic index must differ from unity to initialize

the gas internal energy. I must exit.\n");

prs_exit (1);

else

rcil = Rmed[i]*sin(phi);

energy0 = Aspectratio[i]*Aspectratio[i]*Sigma(Rmed[i],phi)*\

pow(rcil,-1.0)/(ADIABATICINDEX-1.0);

i,Aspectratio[i],Sigma(Rmed[i],phi),energy0);

return energy0;*/

//Judit’s Jupiter version:

real energy0,rcil=0.;

if (ADIABATICINDEX == 1.0)

fprintf (stderr, "The adiabatic index must differ from unity to

initialize the gas internal energy. I must exit.\n");

prs_exit (1);

else

rcil = Rmed[i];

energy0 = Aspectratio[i]*Aspectratio[i]*Sigma(Rmed[i],phi)*\

pow(rcil,-1.0)/(ADIABATICINDEX-1.0);

Sigma(Rmed[i],phi),energy0);

return energy0;

After these changes, the initial profiles were the same at each grid-point. The first way for testing

the radiative module is to turn off the hydro module, so that the disk evolves under the sole effect of for

cooling. This way one can observe the energy radial profile of the circumstellar disk collapse towards

80


the midplane. Because we do not have heating, eventually the run crashes, as the uppermost layers of

the disk have too low energies, but till 100 orbits it was fine. On Figure 2.11 it is clear that the two

codes give the same profiles even after 100 orbits; the curves corresponding to the different codes are not

distinguishable.

Figure 2.11. The energy profile comparisons with JUPITER and FARGOCA if only cooling is

present and the hydro module is switched off.

This test was performed also with the addition of hydro module, and the addition of stellar heating

as well. I have tested the codes in various resolution and in MPI parallel version too, and even discov-

ered a missing MPI communication inside an iterative loop when computing the new temperatures in

FARGOCA.

Another way of testing the radiative module was to perform the same (low-)resolution simulation

with the two codes, plot different slices of the hydro fields and check the min/max values and the overall

aspect of the images. Of course, there is no way to get the exact same results (meaning the exact same

values for all hydro fields) with two different hydro-codes, especially given that the hydro core algorithm

is completely different, but one would expect a qualitative match even after a 100 orbit timescale. I

performed this test with the FARGO3D code as well, after Frederic Masset put the radiative module from

JUPITER into FARGO3D. I got a reasonable match with all three hydro codes.

2.5.4 The Ultimate Test: the Dimensional Homogeneity Test

One time-consuming task is to check for the dimensional homogeneity of every equation in the algorithm.

Obviously, regardless the unit system we use (SI, cgs, or code units) the results have to be the same apart

from the unit conversion factor. The code units are chosen in such a way that the mass of the star is

1.0, i.e. every mass in the code is in solar masses. Moreover, the gravitational constant is also unity.

The length unit is the radius of the planetary orbit. When preparing the code for the homogeneity test, I

81

2.6. Difficulties Chap.2 Code

browsed through the entire code, and added the G, M, and R units in the physical equations, wherever it

was needed. Then I set up a simulation pair, one run in code units, the second run in SI units. I divided the

resulting fields obtained in the two runs by each other (e.g. density field in simulation 1, divided by the

density field in simulation 2), and checked that their ratio is constant within the numerical accuracy (i.e.

15-17 significant digits for double precision). I performed this task while activating/deactivating every

module, setting up various parameters, checking circumstellar disks with and without planets, checking

for coordinate-system rotation, nested meshing, equation-of-state setups, and so on. This way I found

1-2 small bugs, but after corrections, the code thoroughly passed the homogeneity tests.

2.6 Difficulties

2.6.1 The stability issues

Even though the coding part could have been done on a short timescale, the most significant work on

the code was to make it stable. Roughly speaking, 70% of the 1.5 years period we worked on the code

devoted to code stability and related testing. While the Godunov codes give ”sharper results” by treating

the shocks well, they are a lot more ”touchy” than the more diffusive finite difference codes.

First of all, I experienced that the Godunov code crashes much earlier when dealing with low density

cells (∼ almost vacuum). This means that when the user cools the circumstellar disk through the radiative

module, one has to bear in mind to set up a relatively small opening angle for the disk. In fact, by cooling,

the disk shrinks in scaleheight; therefore, the uppermost cells contain only small amount of mass. The

situation can be solved by setting up a density floor, i.e. setting up a minimal allowed density in each cell.

If the density goes below this threshold value, the code density floor fills up the cell again by changing its

density to the minimum allowed value. The drawback of this method is that the mass of the simulation

is not conserved any longer, because we keep adding new mass in the mass-depleted cells. Furthermore,

the vertical hydrostatic equilibrium of the disk is violated as well. After examining how bad the vertical

hydrostatic equilibrium could be, I decided to:

• set the cooling temperature to 30K instead of 3 K

• try to avoid the use of the density floor by setting up a smaller opening angle.

Even though the 30 K cooling temperature might seem arbitrary at first, it did not cause any harm. It

is still lower that the temperature everywhere in the circumstellar disk, therefore we do not accidentally

heat some parts of the disk. In addition, I tested that this setup does not give observably different result

in the final thermal equilibrium of the disk, it just causes the thermal equilibrium to be achieved slightly

later.

To enhance the stability of the simulation, at one point we even tried to make the code more diffusive

by changing the Riemann-slopes and lowering the Courant-number. However these steps just delayed the

crash and of course did not result in ”sharp” features in the simulations. Finally, we realized that playing

with sequence of operations in the hydro-kernel can help the stability. Of course, there is a necessary order

among some routines, but some others can be swapped. By empirically testing the possible permutations,

we finalized the most stable order, described in Fig. 2.3. Furthermore, the largest step towards the stable

code was to forbid Left/Right states to be less than ∼ 10 − 20% of the current cell state in the predictor

stage.

Moreover, as we encountered other type of instabilities, we realized that the most important quantity

is the minimal temperature in the simulation and the corresponding internal energy. If these quantities

82

2.6. Difficulties Chap.2 Code

go very low, the code crashes or gives unphysical results. Therefore, we set up a temperature floor, and

experienced that after this modification the density floor was not needed any longer. It seems that in

near-vacuum situation the JUPITER code (perhaps all Godunov codes?) does not behave well when the

internal energy is too low.

As we have seen earlier this chapter, the code has various predictor methods. These are each suited

for different physical problems. In the isothermal version of the code only PLM and GFO methods were

existed. GFO is very diffusive, so it was never used for real science simulations. The problem with the

PLM method is that it is not adapted to a steep atmosphere in hydrostatic equilibrium, like we have in the

circumplanetary disk. Therefore, we implemented the MUSCL-Hancock predictor method, which also

helped in the case of radiative simulations on the stability and on the accuracy as well.

It also turned out that the total energy conservative scheme is unstable at too low resolution, but works

well in high resolution. We had designed a work-around meant to stabilize the flow on the base mesh

(which will be part of a forthcoming publication therefore it cannot be detailed here). However, as we

experienced, this work-around, which worked on the base mesh, must not be used on the refined meshes

because it can lead to unphysical solutions (like this cold CPD problem, see next section).

This version does not crash on the first

time step, without MPI and without

nested meshes. Great.

Changeset 174, 06/05/2014 03:52:32

Author: ’judit’

2.6.2 The cold circumplanetary disk problem

Even when we were able to stabilize to code enough to run it for a few hundred orbits without a crash,

we encountered a new, serious problem. As I kept adding the refined meshes around the planet, I realized

that after adding level 3, the CPD almost instantly became unphysically cold, its temperature dropping to

the temperature floor value (see Fig. 2.12).

We remind that the JUPITER code deals with the total energy, the energy equation is also written with

that quantity. This means that one wants to derive out the internal energy to compute temperatures for

the radiative module, the internal energy can be given by ǫ = Etot − Ekin. If we think about it, the kinetic

energy should be very large around the planet. Printing out from the algorithm these values, I confirmed

this hypothesis. The problem was that as we subtract a very large number (Ekin) from another, nearly

equal number (Etot), the remaining number is very inaccurate and even can go negative. To go around this

problem, we have experienced for 1.5 months with the Courant-number, more heating, slope coefficients

(making the code more diffusive), various predictor methods etc. Nothing worked, but eventually we

found the solution. First of all, the place of this subtraction in the global ordering of operations is

important (i.e. where to place in the algorithm the evaluation of internal energy) and that the correct

position of the energy subtraction operation depends on resolution and on the coordinate system. As

we explained in the previous section, the energy conservative scheme turned out to be unstable in too

low resolution (e.g. base mesh), hence we introduced a fix. But this new method, which worked well

for spherical and cylindrical geometries, was not working well on the nearly Cartesian nested meshes

around the planet. This caused the cold CPD problem in very high resolution after a couple of mesh

refinement. So, it turned out that one has to bear in mind the actual geometry of the cells (for instance

83

2.7. Fun-facts about the code Chap.2 Code

Figure 2.12. The unphysical ”cold disk” problem, the CPD became as cold as the temperature

floor on high resolution.

around the planet, if we want to examine the planet vicinity). When we work at high resolution around

the planet (which is basically a cartesian geometry for these cells), we evaluate the internal energy at the

end of the kernel with the conservative energy scheme. Instead, at low resolution in cylindrical/spherical

coordinates we evaluate the internal energy earlier and with a new method. This is also shown in Fig.

2.3.

Therefore a mixed scheme was created, where depending on which level we are, the user can switch

between the internal energy evaluation methods via the “HighResLevel” parameter in the parameter file.

Given the resolutions I used in this thesis, my experience is that for a Jupiter-mass planet one needs

to switch to the high-resolution method at level 2, and in case of larger mass planets with the same

simulation setup, the change should be at level 1. This needs to be experienced out by the user for each

simulation setup (i.e. base mesh resolutions and planet mass used). Because we switch energy evaluation

method between the levels, of course it was checked whether this transition cause any flux mismatch. We

found that the transition between the methods/levels is unnoticeable, the same fluxes of total energy are

shared by the two methods and passed across the resolution jump without any further change needed in

the code.

2.7 Fun-facts about the code

If the reader survived till this point, for reward I summarize here a few fun-facts points about the code

development.

• 195 code versions on SVN Trac

• 3036 new lines of code (of 14358 total)

• 55 new subroutines (from the total of 301)

84

2.7. Fun-facts about the code Chap.2 Code

• ∼ 20% of the code is new

• ∼ 100 years of CPU time in each year for testing and running simulations

• while we rigorously tried to make all the comments in English in the code, accidents did happen

during the coding, therefore some comments were initially in french, spanish, hungarian as well,

reflecting the code writers’ spoken languages

During my Mexican trips, I learned not only code development but also other important life skills.

PhD is a life-experience, it makes us more mature, as they say... Indeed, I learned such life-saving skills,

like what to do when encounter with a scorpion (that is, if you are naive European: spray it with insect

killer to slow it down, then try to trap it with a bowl turned upside-down and ask for help from a local who

will kill it for you – learn how to say that in spanish!). Also I discovered how the world’s most dangerous

spider looks like – which lives in every Mexican home – and therefore, I learned how important is to

shake out clothes before wearing them, and bed sheets before going to bed. It is dubious to me at this

point whether my coding skill-set, or the scorpion trapping capabilities will be more help for me in my

future, but definitely, this PhD equipped me with handful of practical skills, not only abstract theoretical

science.

85

2.8. Conclusion Chap.2 Code

2.8 Conclusion

This Chapter summarized the initial state of JUPITER code and the developments conducted on it during

this thesis project. The JUPITER code is a 3D, nested mesh hydrodynamic code, which is especially

suited to study the vicinity of a planet embedded in a circumstellar disk. The initial nested mesh code

existed in isothermal version developed by Frederic Masset. During the thesis, I co-developed with him

the energy equation, and the radiative transfer into the code. Thanks to the radiative module, now realistic

heating and cooling are taken into account through an opacity table by (Bell & Lin 1994). The heating

includes the viscous heating, the stellar irradiation, while the cooling is through radiation.

Figure 2.13. A 10 Jupiter-mass planet simulation with the fully fledged JUPITER code. The big

image shows the midplane volume density, while the small insert shows the CPD zoomed in with

the nested meshes, from the side view.

To test the new code, I have performed comparison simulations with other two hydro-codes. One of

them is the FARGOCA code, which is a 3D version of the 2D FARGO-code developed by Elena Lega

and extended with radiative modules. The other is the FARGO3D hydro-code, similarly the 3D version

of the 2D FARGO code, which was developed by Pablo Benıtez Llambay and Frederic Masset, including

86


the addition of magnetic field and, after this thesis, the radiative module of JUPITER is also inherited to

it.

Apart from comparison tests of the radiative module, I have deeply tested the inter-CPU and inter-

level communications in JUPITER, moreover, the Riemann-solvers solving the main hydrodynamical

equations, finally, I have tested the dimensional homogeneity of all the physical equations inside the

code.

The largest part of the work was, however, to stabilize the extremely touchy Godunov-type code.

Because the Riemann-solvers lead to very sharp details in the simulations, they turn out to be very fragile.

Therefore we had to come up with several tricks and detours to build a working version of JUPITER. I

performed very detailed, wide-spread testing during this phase as well.

With the addition of the new modules, and the fact that JUPITER treats well the shocks, and have

nested meshes, this hydro code became as one of the bests in the planet/disk community to study both

circumstellar and circumplanetary disks, planet formation and planet-disk interactions. The sharpness

of the simulations provide an unprecedented look into the disk physics with high resolution, without

losing the power of global disk simulations, therefore the feedback between the circumstellar disk and

the planet’s vicinity.

87


88

“We cannot solve problems by using the same kind of thinking we

used when we created them.”

Albert Einstein

Chapter 3

Accretion of Jupiter-mass Planets in the Limit

of Vanishing Viscosity

Contents

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90


3.2.1 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.2.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.3 Structure of the Circumplanetary Disk . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.3.1 The Vertical Structure of the Circumplanetary Disk . . . . . . . . . . . . . . . . 94

3.3.2 Radial Structure of the Circumplanetary Disk . . . . . . . . . . . . . . . . . . . 101

3.3.3 Flow in the Midplane of the Circumplanetary Disk . . . . . . . . . . . . . . . . 102

3.4 Planetary Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.5 Discussion and Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.7 Appendix material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

89

Isothermal Accretion Chap.3 Isothermal Accretion

3.1 Introduction

This chapter describes the first paper of thesis, Szulagyi et al. (2014).

As mentioned in Chapter 1, the most popular giant planet formation theory is the core accretion model

by Bodenheimer & Pollack (1986). As a reminder, there are three main stages of formation in this model.

First, a planetary core is formed and starts attracting the gas within its Bondi-radius. When the core

reaches 12 − 16M⊕ the gas envelope starts to contract quasi-statically while the accretion rate increases

(Pollack et al. 1996). This second stage takes the longest time, of the order of a few million years. The

final, runaway gas accretion phase starts when the envelope and core masses are approximately equal.

This phase should not stop until the planet has opened a deep gap in the gas disk. However, this happens

only when the planet reaches a mass of at least 5-10 Jupiter-masses (Kley 1999; Lubow & D’Angelo

2006). Thus Jovian-mass planets can double their mass on an order of 104 years.

Thus, one should expect to observe a dichotomy in the mass distribution of planets ; planets should

be either smaller than ∼ 30 Earth masses, i.e. those that did not reach the phase of runaway gas accretion,

or larger than a few Jupiter-masses, i.e. those that entered and completed the fast runaway growth phase.

Planets in between these two mass categories should be extremely rare. This is the converse of what is

observed (e.g. Mayor et al. 2011, Fig. 1.5). Thus, there is a need to understand what sets the final mass

of a giant planet (see more in Chap. 1).

An obvious possibility to stop accretion is that the gas disappears while the planet is still growing.

However, the lifetime of gaseous proto-planetary disks is of the order of a few million years (Haisch et al.

2001), which is much longer than the runaway growth timescale (104yr). It seems that the disappearance

of the disk can happen at the right time to stop the runaway growth of the planet at about a Jovian mass.

Another possibility is that a planet cannot accrete more gas than what is delivered to its orbit by viscous

accretion, i.e. it cannot grow faster than the star accretion rate. In general, the accretion rate observed in

proto-planetary disks is on the order of 10−8 − 10−7M⊙/yr. This would allow the accretion of Jupiter’s

atmosphere in 104–105 yr, which is too short relative to the lifetime of the disk. If one requires that

Jupiter takes a million years to accrete its envelope, then its runaway growth needs to be limited by a

stellar accretion rate of 10−9M⊙/yr. But at this very low rate the disk photoevaporates rapidly (i.e. a few

105 years, see e.g. Koepferl et al. 2013; Gorti et al. 2009; Szulagyi et al. 2012). Thus a Jupiter-mass of

gas is unlikely to be accreted by the planet. As discussed in Chapter 1 a very accurate tuning between the

viscous accretion rate, the photoevaporation rate, and the runaway growth seems to be needed to allow

a planet to grow to Jupiter-mass but not beyond this limit. Something must be still missing from the

picture.

What we need is a mechanism that slows down runaway growth, so that it occurs on a timescale

comparable to the disk’s lifetime. A possibility is that the circumplanetary disk acts as a regulator of gas

accretion rate onto the planet. Before the gas is accreted by the planet, it has to pass through the circum-

planetary disk (CPD) because of angular momentum conservation. The actual accretion rate of the planet

then depends on the timescale for angular momentum transport within the CPD. If the circumplanetary

disk has a very low viscosity, then the transport of angular momentum through this disk is inefficient

and gas accretes onto the planet at a slow rate. In this situation the observed mass spectrum of the giant

planets is set by the competition between gas accretion and gas dissipation (Rivier et al. 2012).

There are good reasons to think that the viscosity is very low in the CPD (although see Gressel et al.

2013 for an opposite view). First, the planets are thought to be formed in a dead zone of the circumstellar

disk, where the viscosity is low (Thommes et al. 2008; Martin & Lubow 2011). Second, because the

CPD is shadowed by the circumstellar disk and by the remaining gas in the gap, its irradiation geometry

may be unfavorable for ionization (Turner et al. 2010, 2014). Finally, the large orbital frequencies in the

90

3.2. Setup of the Simulations Chap.3 Isothermal Accretion

CPD make the magnetic Reynolds number too small to derive the magneto-rotational instability (Fujii et

al. 2011; Turner et al. 2014; Fujii et al. 2013).

Motivated by these considerations, in this chapter we study the dynamics of the CPD in detail. Our

simulations are similar to those in Machida et al. (2010) and Tanigawa et al. (2012) with one main

difference. Instead of using a local shearing sheet approximation, we perform global disk simulations.

This better allows us to study the connection between the circumstellar disk and the circumplanetary

disk i.e. opening of a gap, gas flow through the gap etc. Moreover, we investigate in more detail the

accretion rate of a Jovian-mass planet in the limit of vanishing viscosity. To do this, it is not enough to

perform simulations with no prescribed viscosity as in Tanigawa et al. (2012), because every numerical

simulation is affected by numerical viscosity. We need to identify the various accretion mechanisms and

distinguish between those dependent on viscosity and those independent of viscosity (i.e. polar inflow

from the circumstellar disk, loss of angular momentum due to shocks, stellar torque exerted on the CPD,

etc.) and evaluate their magnitude.

A well-known crucial issue for simulating gas accretion onto a planet is the choice the equation-of-

state (EOS). Several works have stressed the need to use an adiabatic EOS – possibly complemented by a

recipe for radiative cooling – in order to study planet accretion (D’Angelo et al. 2003; Klahr & Kley 2006;

Paardekooper & Mellema 2008; Ayliffe & Bate 2009a,b). However, the differences with the isothermal

EOS are believed to be fundamental for small-mass planets (up to Saturn’s mass), but less important for

Jupiter-mass planets. In the latter case the flow of gas is mostly dominated by the planet’s gravity. CPDs

are expected form around Jupiter-mass planets and the differences between isothermal and adiabatic sim-

ulations reported to be limited to the mass of the CPD and its scale height (D’Angelo et al. 2003; Ayliffe

& Bate 2009a; Gressel et al. 2013). Thus, we prefer to use the isothermal EOS, with temperature de-

pendence on stellar distance (hereafter locally isothermal), for multiple reasons. The first is that we wish

to focus our chapter on the role of numerical viscosity and on viscosity-independent transport mecha-

nisms within the CPD, which have never been thoroughly discussed before; these considerations should

be independent of the EOS assumed. Second, radiative simulations imply additional, badly constrained

parameters such as those in the prescription for the opacity laws (e.g. Ayliffe & Bate 2009a; Bitsch et al.

2013). We want to focus the discussion on the objectives stated above without distraction. Third we wish

to make direct comparisons particularly concerning the differences between our global disk simulations

and shearing-sheet studies (Machida et al. 2010; Tanigawa et al. 2012), and the latter have been done

with isothermal EOS. Finally, this chapter is the first in a series of future studies, therefore we wish to

begin with the most simple case and build on it incrementally (see the advancements in Chapters 5 and

chap::CPDmass). Nevertheless, for each result that we present, we will state to what extent we expect it

to be valid or different in a non-isothermal context.

This chapter is structured as follows. In Section 3.2, we describe the setup of our hydrodynamic

simulations. This is followed by the results on the structure of the CPD in Section 3.3. Then, Section

3.4 discusses our findings on the accretion mechanism. Section 3.5 reports discussions and perspectives.

Finally, Section 3.6 summarizes the conclusions of our work.

3.2 Setup of the Simulations

3.2.1 Physical Model

We performed hydrodynamic simulations of an embedded Jupiter-mass planet in a circumstellar disk.

The coordinate system was spherical and centered on the star. The planet was on a fixed circular orbit.

91


The units of the code were the following : the unit mass was the mass of the star (M∗), the length unit

was the radius of the planetary orbit (a), and the unit of time was the planet’s orbital period divided by

2π. Consequently, the gravitational constant (G), the planet’s angular momentum and orbital (angular)

velocity Ω =√

GM∗/a3 are unity. The frame was co-rotating with the planet. Our planet was placed at

coordinates : 0, 1, π2

(azimuth, radius, co-latitude, respectively). In our simulation we used an azimuth

range of −π < θ < π, a radius range of 0.41 < r/a < 2.49, and a co-latitude range of 3 times the pressure

scaleheight : [1.42 < φ < π2]. We assumed symmetry relative to the midplane, therefore only half of the

circumstellar disk was simulated.

The initial surface density is Σ = Σ0(r/a)−1.5 where Σ0 = 6 × 10−4 (in code units). Since our Σ0 is

small, the indirect term is negligible, so that our results scale linearly with Σ0. We chose Σ0 such that with

M∗ = M⊙ (the solar mass) and a = 5.2 AU, our initial surface density profile is very close to Hayashi

(1981)’s Minimum Mass Solar Nebula (MMSN). Because MMSN is proportional to r−3/2, and our Σ0

dimension is M∗/a2, the general relationship between Σ0 and ΣMMS N is : Σ0 = ΣMMS N

√

a jup/a(M∗/M⊙),

therefore, we are using this scaling in the followings.

In our “nominal” simulation the gas was set to be inviscid, i.e. there is no prescribed viscosity in

the fluid equations. We stress, however, that the fluid is nevertheless affected by the numerical viscosity,

whose effects will be quantified by changing the resolution of the numerical grids (see below). For

comparison purposes, we also ran a simulation with an α-prescribed viscosity (Shakura & Sunyaev 1973)

adopting α = 0.004. Hereafter, we will refer to this as our “viscous simulation”. Notice, that α sets a

viscosity that is a function of heliocentric distance (radius). However, since the CPD size is small, the

viscosity in the CPD can be considered uniform.

As discussed in the introduction, the equation-of-state (hereafter, EOS) is locally isothermal : p = c2sρ

with disk aspect-ratio H/r = 0.05, where H = cs/Ω (here, cs is the speed of sound,Ω indicates the angular

velocity, p stands for the pressure and ρ is the volume density). No magnetic field was included in the

computations.

The planetary mass in the simulations was set to 10−3 stellar masses, in order to study planet accretion

at a Jupiter-mass. However, we did not introduce the planet with its full mass from the beginning.

Instead, we prescribed a smooth mass growth of the planet as sin(t/t0), where t0 was 5 planetary orbital

periods. This was done for numerical reasons, so that the gas had the time to adapt to the presence of a

progressively more massive planet. The simulations overall have been ran for 238 planetary orbits.

3.2.2 Numerical Model

For the simulations, we used a three-dimensional nested-grid code, called Jupiter. The Jupiter code

solves the Riemann-problem at every cell boundary (Toro 2009) to ensure the conservation of mass and

the of three components of momentum:

ρt + ∇ · (pv) = 0 (3.1)

(ρv)t + ∇ · (ρv ⊗ v + pI) = 0 (3.2)

where ρ is the density, p the pressure, v the velocity vector, and I indicates the identity matrix. The use

of a Riemann-solver makes Jupiter particularly suited to treat shocks, contrary to the van Leer method

(van Leer 1977). The Riemann-solvers implemented in the Jupiter code are approximated solvers based

on the exact solution : a Two-Shock solver, and a Two-Rarefaction solver (de Val-Borro et al. 2006).

92


Table 3.1. Number of Cells on Different Grid Levels

Level No. of

Cells in

Azimuth

No. of

Cells in

Radius

No. of

Cells

in Co-

latitude

Boundaries of the Levels in Az-

imuth [rad]

Boundaries of the Levels in Ra-

dius [a]

Boundaries of the

Levels in Co-latitude

[rad]

0 628 208 15 [−π, π] [0.41, 2.49] [1.42, π/2]

1 112 112 24 [-0.27735, 0.27735] [0.72264, 1.27735] [1.451041, π/2]

2 112 112 40 [-0.138675, 0.138675] [0.861325, 1.138675] [1.47137, π/2]

3 112 110 56 [-0.0693375, 0.0693375] [0.9306625, 1.0693375] [1.5014588, π/2]

4 112 110 56 [-0.03466875, 0.03466875] [0.96533125,1.03468875] [1.5361276, π/2]

5 112 112 56 [-0.017334375, 0.017334375] [0.98266562, 1.017334375] [1.553462, π/2]

6 112 112 56 [-0.0086671875, 0.00866719] [0.99133281, 1.0086671875] [1.5621291, π/2]

7 112 110 56 [-0.0043335938, 0.00433359] [0.99566641, 1.0043335938] [1.566462737, π/2]

The timestep in the simulation is adapted by the code, in order to satisfy the Courant-Friedrichs-Lewy

condition (CFL condition) for all levels of mesh resolution:

C = ∆t

3∑

i=1

vxi

∆xi

≤ 1.0 (3.3)

where C is the Courant number, i represents the number of dimensions, xi means the spatial variables,

and v indicates the velocity. The timestep at a given level can be the same as the timestep on the higher

resolution level, or it can be twice that timestep, in which case two iterations are performed on the finer

level while one iteration is done on the coarser level. This latter technique is called timestep subcycling.

We use an adaptive subcycling procedure, which will be described in a forthcoming publication, in order

to obtain the maximum speed up of the code (the highest possible ratio of physical time over wall clock

time).

The full viscous stress tensor is implemented in the code in three geometries: Cartesian, cylindrical

and spherical. The spherical implementation, which we use in this work, has been tested thoroughly in a

prior work (Fromang et al. 2011).

We employed a system of 8 nested grids, where at level 0 (i.e. in the coarsest grid) the resolution was

628 × 208 × 15 cells for the directions of azimuth, radius, and co-latitude, respectively. Each additional

grid was added after the gas reached a stationary configuration and each of them was centered on the

planet. The size of the cells in a grid at a given level was 12

in each spatial direction of the cell size of

the next larger grid. Table 3.1 contains the number of cells on each level and the grid boundaries. In the

finest level, the cell length was 7.82 × 10−5a, which is 0.113% of the Hill-radius of the planet, and 87%

of the radius of the present day Jupiter, assuming the planet orbits 5 AU away from the star. The cells

had the same length in every directions (i.e. they were cubes), and the radial spacing in between them

was arithmetic.

On level 0 we used reflecting boundary conditions except in the azimuthal direction where we used

periodic conditions. The communication between the grids at level i and i + 1 (where i = 0, . . . , 7) were

done through ghost cells with multi-linear interpolation.

To test the effects of numerical viscosity, we also ran a simulation with a twice finer resolution (1256×416× 30), which we call hereafter the “high-resolution simulation”. Because the simulation is extremely

slow at this resolution, we did not start it from time zero, but from the output of the nominal resolution

at 238 orbits; we re-binned the gas on the new grids and then ran the code for an additional 10 orbits.

The planet was not modeled, but was treated as a point-mass placed in the corner of four cells on the

93

3.3. Structure of the Circumplanetary Disk Chap.3 Isothermal Accretion

midplane. In order to avoid a singularity, the planetary potential was smoothed as :

Up = −GMp

√

x2d+ y2

d+ z2

d+ rs

2

(3.4)

where xd = x − xp, yd = y − yp, and zd = z − zp are the distance-vector components from the planet in

Cartesian coordinates. The smoothing length rs was set equal to the cell size in levels 0–4. From level 5

on we used 2 cell sizes. Moreover, when introducing levels 6 and 7, we progressively decreased rs from

the value used in the previous level to its desired final value. For example, when introducing level 7, first

the same smoothing length was applied as on level 6 (which is equal to 4 cell sizes on level 7), but then

it was decreased in time with a sinusoidal function until it reached rs = 2 cells. This technique was done

to allow the gas to adapt to a gradual change of the gravitational potential.

Because of the isothermal character of our fluid equations, a huge amount of mass tends to pile up

in the few cells neighboring the point-mass planet. This causes numerical instability. Thus, we applied

a density cut : if the volume density reached 1.42 × 105 (in the code’s units), then the volume density of

that cell was limited to this value (hereafter we refer to this as the “mass-cut”). We keep track of the mass

removed in this operation, from which we compute the planet accretion rate. However, the mass of the

planet that enters in the gravitational potential was not changed. This is because we are interested in the

accretion rate of a Jupiter-mass planet, and not in the growth of the planet itself.

3.3 Structure of the Circumplanetary Disk

In this section we describe our results about the circumplanetary disk structure: its vertical structure, its

radial structure and the radial flow in the midplane. All the results that we present are from our nominal

simulation unless we specify otherwise.

Because the grid at level 0 covers the circumstellar disk globally, the planet can open a gap around its

orbit (see Figure 3.1). This was not the case in the simulations of Machida et al. (2010), and Tanigawa

et al. (2012) because of the shearing-sheet approximation they adopted. Figure 3.1 also shows that the

density wave launched by the planet in the circumstellar disk smoothly joins the CPD and spirals into it

down to the planet (see also Fig. 3.10).

3.3.1 The Vertical Structure of the Circumplanetary Disk

We start by discussing the vertical structure of the CPD. For this purpose, it is convenient to characterize

the CPD based on the z-component (in Cartesian coordinates) of the specific angular momentum with

respect to the planet, normalized to the Keplerian value :

Lz =xdvy − ydvx + (x2

d+ y2

d)Ω

√

GMP

√

x2d+ y2

d

(3.5)

where vx and vy are the velocity-components transformed to Cartesian coordinates in the co-rotating

frame.

Fig. 3.2 represents a vertical slice at azimuth = 0.0 of the Lz distribution in the neighborhood of the

planet, which is located at the center of the upper axis. We see that Lz rapidly drops from ∼ 1 to ∼ 0.5

at a location where the density shows a clear discontinuity (see Fig. 3.3). Therefore, hereafter we define

94


Figure 3.1. Volume density map of our inviscid, low-resolution simulation on the midplane using

data from levels 2-7. The planet is in the middle of the figure. The planet clearly opened a gap and

the spiral density wave launched by the planet connects the circumstellar disk with the CPD. Here

and in the following figures, with ”Azimuth” and ”Radius” we mean the distance from the planet

in the azimuthal and radial direction.

95


the CPD as the region where Lz is larger than 0.65 (see the corresponding contour line in Fig. 3.2). Even

this value is arbitrary; however, given the steep gradient of Lz near the surface of the disk, changing this

threshold would not change significantly the results presented below.

The Fig. 3.2 shows that the gas in the midplane and near the midplane is sub-Keplerian (similarly to

Tanigawa et al. 2012; Uribe et al. 2013). However, notice that near the upper layer of the disk the flow

is slightly super-Keplerian, i.e. in the region bounded by the contour line Lz = 1. This is due to the fact

that the disk is very flared, so that the radial pressure gradient near its surface is positive. In the viscous

simulation, however, this super-Keplerian near-surface layer does not exist. This is due to the higher

viscosity, that limits the vertical shear in the CPD.

The gas located below the CPD is falling toward the CPD with a large vertical velocity, as indicated

by the arrows on Fig. 3.3 (see also Ayliffe & Bate 2009a, and Tanigawa et al. 2012). As pointed out

in Tanigawa et al. (2012) the sharp vertical boundary of the CPD clearly visible in the Lz and density

maps is due to a shock front generated by the vertical influx. As in Tanigawa et al. (2012), we also

notice from Fig. 3.2 that the vertical inflow hits the CPD with a value of Lz that is much smaller than

that characterizing the CPD at the same location. Thus, the vertical inflow slows down the rotation of the

CPD, promoting radial infall at the surface of the CPD.

We find that the vertical influx has also a strong influence on the aspect ratio of the CPD. First of all,

as a reminder, the pressure scale height of the CPD at hydrostatic equilibrium is HCPD ≡ cs/ω, where

ω =√

GMp/d3 is the angular velocity around the planet, and d =

√

x2d+ y2

dindicates the distance from

the planet. The sound speed (cs = 0.05 r−1/2) is almost constant in the CPD in our locally isothermal

simulation. So, we expect the aspect ratio of the CPD to be HCPD/d =cs√GMp

d1/2 ≈ 1.6(d/a)1/2 =

0.16 (d/0.01a)1/2, which is very thick and flared. As we show in Fig. 3.4, the surface of the CPD defined

by zCPD (i.e. the uppermost z-coordinate where Lz ≈ 0.65) is indeed strongly flared, but its aspect ratio

has also a strong dependence on the azimuth relative to the planet. In fact, zCPD/d is changing from

∼ 20% to > 100%. To our knowledge, this wavy surface has not been described yet in the literature. The

wavy surface pattern is due to the dynamical pressure of the vertical mass inflow, which is not uniform

in planetocentric azimuth (see Fig. 3.5); it is maximal along an axis close to the axis of the spiral arm.

Now, we take a Lagrangian approach and consider fluid elements in the CPD orbiting on circles

centered on the planet; because the pressure due to the vertical inflow has two maxima they will feel

a maximum of pressure twice per orbit. But it takes a time tdelay ≈ HCPD/cs for the gas in the CPD to

react to the pressure pulse. In this time the fluid elements rotate by an angle θdelay = ω × tdelay, which

is the angle between the axis marking the minimum height of the CPD and that marking the maximum

pressure. Because ω × tdelay = ω/HCPD/cs = 1, this angle is independent of the distance from the planet

d. The comparison of the Figs. 3.4 and 3.5 clearly shows an angle of order unity (in radians) between the

maximum pressure and the minimum CPD height. A toy-model is presented in the Appendix 3.7 about

how the pressure of the vertical influx leads to the observed structure of the CPD.

Fig. 3.6 shows the vertical density distribution in the CPD from z = 0 to z = zCPD at a given radius

for various values of the azimuth. The mass is conserved along an orbital period, so the integral of each

density curve is the same. On top of the expected equilibrium Gaussian shape, one can notice oscillations

with two knot points where the density does not change with azimuth. This is reminiscent of stationary

waves.

Simulations implementing an adiabatic EOS (e.g. Ayliffe & Bate 2009a, 2012; Gressel et al. 2013)

also find that the vertical inflow is the main feeding mechanism for the CPD. The CPD, however, has a

larger scale height as it is hotter and the boundary between the disk and the vertical flow is less sharp

than in our isothermal simulations. We will come back to this last, important issue in Sect. 3.5.

96


Figure 3.2. A vertical slice of disk passing through the planet’s location, showing in colors the

values of the normalized specific angular momentum Lz of our inviscid, low-resolution simulation.

A value of Lz ≈ 0.65 separates the CPD (see the corresponding contour line) from the environ-

ment. One can see that the gas near the midplane is sub-Keplerian (yellow), while on the surface

layer of the disk it is slightly super-Keplerian (white region bounded by the contour line 1.0). The

blue-violet colors correspond to gas that is falling almost vertically toward the CPD. The blue

circle around the planet symbolizes the 110

of the Hill-radius.

97


Figure 3.3. The same as Fig. 3.2, but showing in colors the volume density of the gas of our nom-

inal simulation; a few velocity vectors schematize the directions of the flow. Notice the vertical

inflow, as well as the accreting flow in the CPD midplane. Again, the orange circle around the

planet symbolizes the 110

of the Hill-radius.

98


Figure 3.4. A color map showing the aspect ratio zCPD/d as a function of azimuth and radius of

our inviscid, low-resolution simulation. The planet is placed at the center of the plot. At any given

distance from the planet, the aspect ratio changes considerably with the planetocentric azimuths.

Thus, the CPD has a “wavy” surface structure.

99


Figure 3.5. The same as Fig. 3.4, but showing in colors the value of ρv2z , representing the ram

pressure exerted by the polar inflow on the CPD surface. It can be seen that the pressure of the

inflow is higher along a diagonal line oriented from top left of the figure to bottom right. Thus,

the CPD is compressed along this line and has a minimum aspect ratio (see Fig. 3.4) along a line

rotated by θdelay relative to the highest pressure line (see text).

100


Figure 3.6. The volume density vs. the z coordinate at d ≈ 0.058 Hill-radii, for various azimuths

relative to the planet. The vertical structure of the CPD changes with azimuth with fixed knots as

a stationary wave. The data are from our inviscid, low-resolution simulation.

3.3.2 Radial Structure of the Circumplanetary Disk

We now move to discuss the orbital motion inside the CPD and the location of its outer radial boundary.

The specific normalized angular momentum Lz on the midplane declines with the distance from the

planet (Ayliffe & Bate 2009a; Tanigawa et al. 2012), but it does not show a steep gradient like the one in

the vertical direction near the surface of the disk. Therefore, previous authors assumed arbitrary limits in

Lz, obtaining different radial extensions for the CPD.

We think that it is more meaningful to look at the orbital motion of the fluid elements, defining the

CPD as the region where the orbits are basically circular. Quasi-circular orbits may have a small value of

Lz if the CPD is strongly sub-Keplerian due to a steep radial pressure gradient, but they are clearly part

of a disk.

In order to visualize easily where the orbits of the disk are quasi-circular, we proceeded as follows.

First, we calculated the semi-major axis a and the eccentricity e in every cell from the cell’s coordinates,

the recorded velocities and the planetary potential; then we plotted the apocenter of the orbit Q = a(1+e)

101


versus the planetocentric radius d of the cell. If, at a given radius every cell, whatever its planetocentric

azimuth, appears to be at the apocenter (Q = d), it obviously means the streamline in the disk is circular,

although sub-Keplerian. We can see on Fig. 3.7 that this is the case up to ∼ 0.48 Hill radii. If we use this

definition for the radial extent of the CPD, then the disk is a bit wider than the previously recorded radial

extensions of ∼ 0.1 − 0.3 Hill radii (Tanigawa et al. 2012; Ayliffe & Bate 2009a).

We remark however that the eccentricity of the streamlines in the disk depends on the viscosity. In

fact, as we will see in Sect. 3.3.3, the streamlines are eccentric if they are shocked at the passage through

the wave generated by the stellar tide. The smaller is the effective viscosity – prescribed or numeric –

in the CPD, the closer to the planet the wave propagates and shocks. Thus, defining the CPD as the

region where streamlines are circular may lead to the uncomfortable situation that the disk may become

vanishingly small in the ideal limit of zero viscosity. In fact, in our viscous simulation, circular orbits

extend up to ∼ 0.55 Hill radii and in the high resolution simulation, which halves the numerical viscosity,

they extend only up to ∼ 0.28 Hill radii. This is an important point that should be kept in mind when

analyzing the results of simulations, regardless if conducted with an isothermal or adiabatic EOS.

Alternatively, we may define the radial extent of the CPD as the largest circle from which stream-

lines wrap around the planet at least once before becoming unbound, in either the forward or backward

integration. If we adopt this definition, the radius of our CPD is approximately 1/2 to 3/4 of the Hill

radius.

On Fig. 3.8 our CPD’s column density (∫

ρdz) profile can be seen. We have less massive CPD than

Tanigawa et al. (2012), probably because our global disk simulation contained a planetary gap in contrary

to the sheering sheet box simulations. Instead the column density at 0.1 Hill radius in our CPD (∼ 100g

cm−2 for Jupiter at 5 AU in a MMSN) is comparable to that in the radiative simulations with reduced

opacity of (Ayliffe & Bate 2009a) and with the most viscous simulation in D’Angelo et al. (2003).

3.3.3 Flow in the Midplane of the Circumplanetary Disk

There is a debate in the literature about the direction of the radial flow on the midplane of the CPD.

Ayliffe & Bate (2009a) found inflow in their simulations, while Tanigawa et al. (2012); Klahr & Kley

(2006); Ayliffe & Bate (2012) found outflow.

We find that the direction of the radial flow on the midplane of the CPD depends strongly on viscosity.

In our viscous simulation the flow is outward, as shown by the streamlines plotted in Fig. 3.9. The outflow

near the midplane – together with inflow in the upper layers – is indeed typical of a three dimensional

viscous-accretion disk (see Urpin 1984; Siemiginowska 1988; Kley & Lin 1992; Rozyczka et al. 1994;

Regev & Gitelman 2002; Takeuchi & Lin 2002).

In our nominal simulation, instead, the net flow is inward. This is due to two reasons: (I) the ef-

fective viscosity is smaller and (II) the flow suffers more pronounced shocks when crossing the spiral

density wave. The latter issue is well visible in Fig. 3.10. The shocks correspond to the points where

the streamlines change abruptly direction. Look in particular at the accreting streamline on the figure.

When it encounters the wave for the first time, the streamline changes abruptly direction relative to the

position of the planet. The streamline now makes a hyperbolic arc around the planet. If unperturbed, it

would leave the planet’s sphere of influence, but it is shocked again when crossing the outer branch of

the density wave at the apex of its trajectory. The shock deviates the motion once again, and reduces its

angular momentum relative to the planet. The streamline now makes a downward arc around the planet

with a large eccentricity, but then it is shocked again and again, every half orbit. Each shock causes a loss

in angular momentum, so that the streamline spirals toward the planet.

The shocks were also visible in Fig. 3.9, but they were less pronounced. The net flow is the result

102


Figure 3.7. Orbital apocenter as a function of planetocentric radius for the cells on the midplane in

the vicinity of the planet. As long as the points lie on a line of slope 1, the streamlines in the disk

are circular. One can read from the figure that the CPD of our inviscid, low-resolution simulation

is quite circular up until ∼ 0.48 Hill radii.

103


Figure 3.8. The column density profile of the CPD with respect to the distance from the planet.

Each curve refers to a different simulation, as labeled.

104


Figure 3.9. Volume density map on the midplane in the vicinity of a Jupiter-mass planet for our

viscous, low-resolution simulation. A few streamlines are also shown with arrows that indicate the

direction of the flow.

105

3.4. Planetary Accretion Chap.3 Isothermal Accretion

of the competition between the viscous stress, which pushes the flow outward, and the shocks, which

cause angular momentum losses. In the viscous simulation the former wins; in our nominal simulation

the latter win. The same competition should occur also for CPDs with adiabatic EOS. Shocks are weaker

in that case (D’Angelo et al. 2003), but in the limit of zero viscosity they should dominate nevertheless.

It is unclear to us why Tanigawa et al. (2012) found outflow in their simulation, which had no pre-

scribed viscosity as in our nominal case. Possibly the numerical viscosity in their simulation was higher

than in ours; or, alternatively, the fact that no gap opened in their simulation changed substantially the

local dynamics.

We stress, however, that the discussion about the direction of the midplane flow in the CPD is mostly

academic. In fact, even in the case of inflow on the midplane, the midplane flow accounts for only 10%

of the total delivered material to the CPD. We derived this percentage through the following procedure.

First, we plotted the azimuthally averaged mass-flux on circles at different planetocentric radii. The

mass-flux is increasing with decreasing distance because mass is continuously added to the CPD from

the vertical direction. Considering a distance of 0.58 Hill radii from the planet, which corresponds to the

largest radius at which all streamlines are accreting (see Fig. 3.10) the mass-flux is 10% of the planet’s

accretion rate (see “mass-cut” accretion rate in Section 3.4 for details). Thus, the remaining 90% of the

accretion has to come from the vertical direction.

3.4 Planetary Accretion

After having analyzed in details the dynamics in the vicinity of the planet, we are now ready to discuss

the planet’s accretion rate.

First, we checked whether we reached a stationary state at the end of the simulation by comparing

the radial mass fluxes (averaged over azimuth and integrated vertically) obtained at different output times

throughout the circumstellar disk and the CPD. Having concluded positively that a quasi stationary state

was reached, we then checked whether the flux of mass onto the planet was a simple consequence of the

mass flux toward the star in the circumstellar disk. We found, as discussed more in details in Chapter 4,

that the flux of gas toward the planet is due to the flux of gas into the gap from both of its sides. Thus,

the planet accretion rate would not be zero even in an equilibrium disk without any net mass flux to the

star. Our disk is indeed very close to an equilibrium disk for α = 0 ; the flux of gas toward the star in our

disk is not significant, and therefore not correlated to the accretion of the planet.

In order to measure the accretion rate in the simulation we measured how much mass was removed

through the mass-cut. In our nominal simulation after reaching a stationary state, we found a large

accretion rate, namely M = 2 × 10−7M∗Ω. Again, the results scale linearly with Σ0 and the relationship

derived in Section 3.2 is Σ0 = ΣMMS N

√

a jup/a(M∗/M⊙). Moreover, Ω =√

GM∗/a3 =

√

M∗M⊙

GM⊙(1AU)3

(1AU)3

a3 .

Plugging in these will lead to M = 5.51 × 10−7M⊙/year × (M∗/M⊙)1/2 × (Σ0/ΣMMSN) × (a/1AU)−1. If the

planet is at 5.2 AU, this corresponds to ∼ 10−4 Jupiter masses per year. We argue that this high accretion

rate is due to numerical viscosity. In fact, in the high resolution simulation, where the numerical viscosity

is halved, the accretion rate is reduced by a factor of two.

Interestingly, in the high resolution simulation, the mass in the CPD is basically the same as in the

nominal simulation (see Fig. 3.8). This is because the polar inflow is also reduced by a factor of two. This

is at first surprising, because in 2D disks at low viscosity the width and depth of a gap is independent of

viscosity (Crida et al. 2006). But 3D gaps behave differently. The detailed analysis of the gas dynamics

in a 3D gap will be the object of another Chapter 4. However, in brief, the dynamics of a gap in a 3D

disk is characterized by an interesting circulation : the gas flows into the gap from the surface of the

106


Figure 3.10. Same as Fig. 3.9, but for the nominal (i.e. inviscid) simulation. One of the stream-

lines shows clear shocks when crossing the spiral density wave, thus the gas flow loses angular

momentum and spirals down to the planet. The planet accretes all the gas flowing between the first

and the third streamlines from the bottom left of the figure and between the first and the second

streamlines from the top right of the panel.

107


circumstellar disk, then precipitates toward the midplane. In doing this, it falls either to the CPD or

gets kicked out by the planet and goes back into the circumstellar disk. The flow into the gap at the

disk’s surface is dominated by the numerical viscosity and therefore changes by a factor of two from the

nominal to the high resolution simulation.

This result shows that it is not possible to assess the accretion rate of a planet in the low-viscosity

limit just by using simulations with no prescribed viscosity. This has to be kept in mind regardless of the

EOS used in the simulations. Instead, we need to identify and quantify the accretion mechanisms that

are independent of viscosity. However, as a reminder, our analysis is based on isothermal simulations; if

a more realistic EOS is used, the quantitative relevance of each mechanisms may change. The analysis

below, therefore, should be regarded as a proof of concept, useful also for future radiative studies, and

not for its quantitative results (see also Sect. 3.5).

Here we envision a scenario in which the circumstellar disk has a layered structure, with a dead zone

near the midplane and an active viscous layer near the surface, in agreement with MRI studies (Gammie

1996). We envision also that the CPD is mostly MRI-inactive, in agreement with (Fujii et al. 2011; Turner

et al. 2014; Fujii et al. 2013), so we investigate the planet accretion rate in the limit of vanishing viscosity

in the CPD.

A first mechanism of accretion, independent of the viscosity in the CPD, is the vertical inflow. We

stress that the vertical inflow is sustained by the flux of gas in the active layer of the circumstellar disk

(see in Chapter 4), so that it should exist also if the planet forms in a dead zone and the CPD is inviscid.

We have seen in Section 3.3.1 that the vertical inflow has a specific angular momentum significantly

smaller than the CPD. Because the contact of the inflow and the CPD happens through a shock, the

inflow subtracts angular momentum from the CPD even in the limit of zero viscosity. Nevertheless, if the

specific angular momentum of the incoming gas is larger than that corresponding to an orbit at the surface

of the planet, the inflow cannot promote accretion onto the planet. Therefore, the mass accreted by the

planet cannot be larger than the mass carried by the inflow of gas with a specific angular momentum

smaller than ( j <√

GMpRp) (Tanigawa et al. 2012). We call this subset of the vertical inflow as a “polar

inflow”.

To estimate the accretion rate due to the polar inflow we proceed as follows. We set the radius of

the planet to be equal to twice the current radius of Jupiter. This is because the planet at the accretion

time was much hotter and therefore its radius was inflated by more or less a factor of two (Guillot et al.

2004). Also, we refer to the viscous simulation. The reason is that, as we said above, the vertical infall

is fed by the gas entering into the gap at the surface of the circumstellar disk, and the latter should be

MRI active. With these settings we find an accretion rate of 4 × 10−9 × M∗Ω = 11.02 × 10−9M⊙/year ×(M∗/M⊙)

1/2 × (Σ0/ΣMMSN) × (a/1AU)−1, i.e. 2 × 10−6 Jupiter masses/yr with the usual scalings and it

should scale linearly with α. This estimate is one order of magnitude smaller than in Tanigawa et al.

(2012), presumably because of the fact that in our simulations the planet opened a gap.

The second accretion process that does not depend on the viscosity in the CPD is the loss of angular

momentum in the CPD due to the torque exerted by the star through the spiral density wave (Martin &

Lubow 2011; Rivier et al. 2012). This torque was already considered in Rivier et al. (2012) in their 2D

simulations. The authors there assumed that in the inviscid case the torque is deposited only in the very

inner part of the CPD. However, as we have seen in section 3.3, the wave shocks and removes angular

momentum also in the outer part of the CPD. The fact that the wave does not seem to shock in the inner

part of the CPD is probably an artifact of numerical viscosity, which increases approaching the planet

and smears out the density contrasts, consequently erasing the wave and its shock front. Because the

simulation does not allow us to resolve where in the CPD the torque is deposited, in order to provide an

upper bound of the planet’s accretion rate promoted by the stellar torque we adopt the following simple

108

3.5. Discussion and Perspective Chap.3 Isothermal Accretion

recipe. We integrate the stellar torque from the planet to the radius where it becomes positive, which is

basically at the edge of the CPD; then we estimate the fraction of the CPD mass accreted per unit time as

the fraction between the integrated stellar torque and the total angular momentum in the disk.

For both the nominal and the high resolution simulation we derive that the stellar torque promotes

the accretion of 3 × 10−3 of the mass of the CPD per planet’s orbital period, i.e. 2.5 × 10−4 of the CPD

mass per year, if the planet is at 5.2 AU around a solar-mass star. Because of the fact that this result is

independent of the numerical resolution, we are confident of its robustness.

The mass of the CPD in our simulation is only 4 × 10−4MJ. However, if the disk could not accrete

onto the planet as fast as in our simulation because of the lack of viscosity, the gas would pile up into the

disk, increasing the CPD mass. How massive the disk can become cannot be studied using isothermal

simulations and will be the object of a future study. In Rivier et al. (2012) it was estimated analytically

that the maximum mass of the CPD is ∼ 10−3MJ; at this mass its vertical pressure gradient becomes large

enough to stop the vertical inflow, so that the mass of the CPD cannot grow further. This estimate is

probably valid only at the order of magnitude level. However, even assuming a CPD mass of 0.01MJ,

the stellar torque would imply an accretion rate of only 2.5 × 10−6MJ/yr, i.e. a mass doubling time of

400,000 years. This timescale is comparable to that of the photoevaporation of the circumstellar disk. If

this result is confirmed in future, more realistic studies, it implies that, if giant planets form toward the

end of the disk’s lifetime, the competition between the planet’s accretion timescale and the disk removal

timescale might explain the wide range of masses observed for giant planets.

3.5 Discussion and Perspective

In this chapter the simulations were all isothermal. Previous studies showed that for small planets (∼10 M⊕) the flow near the planet strongly depends on the equation-of-state (Paardekooper & Mellema

2008, Ayliffe & Bate 2009a, Nelson & Ruffert 2013), but for Jupiter-mass planets the accretion rate in

non-isothermal simulations is close to that in isothermal calculations (Machida et al. 2010). (For more

discussion on this, see Chapter 1.)

We suspect that the Machida et al. (2010) result is due to the large planet’s accretion rate, which is a

consequence of the numerical viscosity. At the level of detail at which we explored the local dynamics

in this chapter, we expect that the equation of state would strongly influence the results at a quantitative

level. In particular, in the limit of vanishing viscosity, the gas should pile up in the CPD, and an adiabatic

equation of state, with flux limited energy transfer is expected to change significantly the final equilibrium

structure of the CPD relative to the isothermal equation-of-state (see our findings with non-isothermal

EOS in Chapters 5 and 6).

The issue of the pileup of material in the CPD is crucial to estimate the planet’s accretion rate in the

inviscid limit. If the mass of the CPD becomes large, the stellar torque can be sufficient to promote a fast

accretion onto the planet. Lubow & Martin (2012) suggested that the disk may become gravitationally

unstable, which would cause FU Orionis-like accretion bursts onto the planet. In fact, works comparing

the results in isothermal and adiabatic simulations, such as D’Angelo et al. (2003) or Gressel et al.

(2013), show that the CPD tends to be less massive if heating and cooling effects are taken into account.

However these works are affected by a large viscosity – prescribed or numeric – which prevents the

pileup of mass in the CPD. It should be investigated as to what actually happens in the ideal inviscid

limit. Of course, neglecting the viscosity in the CPD is only valid if thermal ionization of the gas around

the planet is negligible. Later, in Chapters 5 and 6 we will see that the temperatures are very high in the

circumplanetary material, therefore thermal ionization is probably happening at least in the inner CPD.

109

3.6. Conclusions Chap.3 Isothermal Accretion

Adiabatic simulations like Ayliffe & Bate (2009a) and Gressel et al. (2013) show that the boundary

between the CPD and the vertical flow is less sharp than in our study, suggesting a weakening of the shock

front. If the vertical inflow is diverted before that the CPD becomes gravitationally unstable, and then a

steady state equilibrium can be reached. As in Sect. 3.4 the accretion rate onto the planet will depend on

the mass of this steady state CPD and the stellar torque, but the quantitative estimate will presumably be

different from the one achieved in this chapter. We also notice that adiabatic simulations (e.g. Gressel et

al. 2013) show that the spiral wave launched by the star in the CPD is much less prominent than in the

isothermal case, which would reduce the stellar torque (see for our results on this in Chap. 6).

As the JUPITER code has improved with a radiative module (Chap. 2) during this PhD years, we

carried out newer, more realistic simulations on the problem. For those, see Chapters 5 and 6.

3.6 Conclusions

In this work we studied the dynamics of gas in the vicinity of a Jupiter-mass planet and the properties of

the circumplanetary disk. For this purpose we used the Jupiter code, a 3D nested-grid hydrodynamical

code. We performed locally isothermal simulations with two prescribed α viscosities (α = 0.004 and

α = 0) and, for α = 0, with two different resolutions.

Our results confirm those of Ayliffe & Bate (2009a); Machida et al. (2010); Tanigawa et al. (2012)

concerning the vertical inflow and the CPD vertical structure. We have pointed out, however, that the

CPD upper layer is wavy, i.e. the aspect ratio of the CPD changes with planetocentric azimuth, because

of the pressure of the inhomogeneous vertical inflow. In a reference frame rotating with the gas around

the planet (at a given radius), this pressure exerts a periodic perturbation, leading to the formation of

a stationary wave in the CPD vertical structure. We also were able to reduce the viscosity more than

previous local box simulations; in our inviscid simulation the shocks were more pronounced.

We found that CPD is mostly sub-Keplerian, similarly to Tanigawa et al. (2012), and Uribe et al.

(2013), except in its upper layer, where it can be slightly super-Keplerian because of the significant

flaring of the disk. The radial extent of the disk where the streamlines are quasi-circular depends on

viscosity and, if α = 0, also on numerical resolution. The smaller the effective viscosity is, the smaller

the circular portion of the disk is.

We found that the flow in the CPD midplane is inward if α = 0, in contrast with Tanigawa et al.

(2012), and Ayliffe & Bate (2012). In this case the gas flow in the CPD is crossing the spiral density

wave twice in every orbit, and each crossing leads to the loss of angular momentum due to a shock. Thus

the flow spirals down to the planet. Nevertheless, we showed that the radial inflow of mass through the

outer boundary of the CPD is only 10% of the gas influx coming from the vertical direction. Instead,

in the case of the viscous simulation with α = 0.004, the flow is spiraling outward in the midplane.

Therefore one can conclude, that the viscosity determines the directions of the flow in the CPD.

Our simulation resulted in a high planetary accretion rate, namely M = 1 × 10−4 Jupiter masses per

year for a Jupiter mass planet at 5.2 AU in a MMSN; however we showed that this high rate is due

to numerical viscosity. We identified that the main accretion mechanisms, independent of viscosity, is

the torque exerted by the star onto the CPD. We found that the stellar torque promotes the accretion of

2.5 × 10−4 of the mass of the CPD per year, assuming a planet’s orbital period of 12 years. However, we

cannot provide a reliable estimate of the mass of the CPD with our isothermal simulations, particularly in

the limit of vanishing viscosity, which could lead to a significant pileup of material in the CPD. An order

of magnitude analytic estimate in Rivier et al. (2012) reported a CPD mass of ∼ 10−3MJ. Even assuming

a CPD mass of 0.01MJ, the stellar torque would lead to an accretion rate of only 2.5×10−6MJ/yr. In other

110

3.7. Appendix material Chap.3 Isothermal Accretion

words, a Jupiter would build up in 400,000 years with this accretion rate. This timescale is comparable to

the removal timescale of the circumstellar disk gas (e.g. Koepferl et al. 2013; Gorti et al. 2009; Szulagyi

et al. 2012).

Although future simulations implementing a realistic, non-isothermal equation of state are needed to

achieve a reliable quantitative estimate of the planet’s accretion rate in the limit of vanishing viscosity,

many conceptual results of this work, particularly those on the role of numerical viscosity and viscosity-

independent transport mechanisms in the CPD, should be valid also in a more realistic context.

The main result presented in this chapter is encouraging. The similarity between planet accretion

and disk removal timescales suggests that, if the giant planets form toward the end of the disk’s lifetime,

the competition between the planet’s accretion process and disk’s photoevaporation could explain the

observed, wide range of giant planet masses.

3.7 Appendix material

Here we present a toy-model for the wavy structure of the CPD. Our goal is to schematize with some

basic physical considerations the CPD’s reaction to the pressure of the vertical inflow, which is leading

to this wavy disk surface. This toy-model might help to understand better the process, without giving an

exact, i.e. complex physical model which is not goal of this chapter.

In the reference frame rotating with the fluid elements in the CPD, the periodic excitation by the

vertical inflow’s pressure creates a stationary wave in the vertical structure of the CPD. The solution for

the acoustic wave equation1

c2s

∂2 p

∂t2=∂2 p

∂z2(3.6)

for a stationary wave can be written in the following form :

p(z, t) = 2p0eiνt cos(kz) (3.7)

where t stands for the time, z represents the vertical coordinate, ν indicates the wave frequency, and k

means the wave-number. The term cos(kz) does not involve a phase because the CPD is supposed to

be symmetric with respect to the midplane, so that we have ∂p/∂z = 0 at z = 0. Putting this equation

back into the wave equation we get k = ν/cs = 2π/λ where cs indicates the sound speed and λ is

the wavelength. Because the pressure exerted by the vertical inflow has a frequency that is twice the

planetocentric orbital frequency (ν = 2ω), then at zmax the equation can be written as p(zmax, t) = pzmaxei2ωt.

If we set this equal to Equation 3.7, then we get λ = πcs/ω = πHCPD.

This is precisely what is seen in Fig. 3.6. The figure shows the vertical profile of the volume density

in the CPD, for various values of the azimuth, at a distance d ≈ 0.058 Hill-radii from the planet, where

HCPD/d ≈ 0.1. The z coordinates is normalized by HCPD. The profiles oscillate around the well-

known Gaussian hydrostatic equilibrium profile. One can see two knots, where all curves intersect,

corresponding to the locations in z where the amplitude of the wave is zero, namely corresponding to

cos(kz) = 0 : z = (π/4)HCPD and z = (3π/4)HCPD. The distance between the two knots is λ/2 = π/2HCPD.

The computation of the cumulated mass along the curves shown in Fig. 3.6 reveals that > 97% of the

disk mass is below the knot at 34λ = 3

4πHCPD ≈ 2.4HCPD. Thus, the extreme “waviness” of the surface

of the disk observed in Fig. 3.4 concerns solely an “atmosphere” of the disk accounting only for < 3% of

the disk mass.

111

3.7. Appendix material Chap.3 Isothermal Accretion

112

“Everything is theoretically impossible, until it is done.”

Robert A. Heinlein

Chapter 4

Meridional circulation of gas into gaps opened

by giant planets in three-dimensional

low-viscosity disks

Contents

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.2 Gaps in 3D disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.3 Implications on planet’s accretion from the flow of gas into the gap . . . . . . . . . 120

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

113

Meridional Circulation Chap.4 Meridional circulation

4.1 Introduction

This chapter based on the second paper of this thesis, Morbidelli et al. (2014).

As we described in Chapter 1, understanding what sets the terminal mass of a giant planet in a

runaway gas accretion regime is an open problem in planetary science. Runaway gas accretion is the

third stage of the classical core-accretion scenario for the formation of giant planets (Pollack et al. 1996).

We remind the reader that in stage I, a solid core of 5-10 Earth masses (M⊕) is formed by planetesimal

accretion (or possibly by pebble-accretion; see Lambrechts & Johansen (2012), Morbidelli & Nesvorny

(2012). In stage II, the core starts to capture gas from the protoplanetary disk, forming a primitive

atmosphere in hydrostatic equilibrium; the continuous accretion of planetesimals heats the planet and

prevents the atmosphere from collapsing. In stage III the combined mass of core and atmosphere becomes

large enough (the actual mass-threshold depending on opacity and energy input due to planetesimal

bombardment) that the latter cannot remain in hydrostatic equilibrium anymore; thus the planet enters in

an exponential gas-accretion mode, called runaway.

The accretion timescale in the runaway accretion mode is very fast and, once started, can lead to

a Jupiter-mass planet in a few 104 y (see for instance the hydrodynamical simulations in Kley (1999);

D’Angelo et al. (2003); Klahr & Kley (2006); Ayliffe & Bate (2009a); Tanigawa et al. (2012); Gressel

et al. (2013)) in a proto-planetary disk with mass distribution similar to that of the Minimum Mass Solar

Nebula model (Weidenschilling 1977; Hayashi 1981). This rapid accretion mechanism can explain how

giant planets form. On the other hand, there is no obvious reason for this fast accretion to stop. As its

timescale is much shorter than the proto-planetary disk life time (a few My Haisch et al. (2001)), it is

unlikely that the disk disappears just in the middle of this process, raising the question why Jupiter and

Saturn and many giant extra-solar planets did not grow beyond Jupiter-mass.

It is well known that giant planets open gaps in the protoplanetary disk around their orbits (Lin &

Papaloizou 1986; Bryden et al. 1999). Thus it is natural to expect that the depletion of gas in the planet’s

vicinity slows down the accretion process. Still, all of the hydrodynamical simulations quoted above that

feature the gap-opening process show that the mass-doubling time for a Jupiter-like planet is not longer

than 105 years. We have found the same result in the previous chapter, using 3D simulations.

However, these simulations may have been hampered by the assumption of a prescribed viscosity

throughout the protoplanetary disk, or by significant numerical viscosity in the simulation scheme. It is

expected that planets form in dead zones of the protoplanetary disk (Gammie 1996), where the viscos-

ity is much smaller than in numerical simulations, so that Jovian-mass planets could presumably open

much wider and deeper gaps, with consequent inhibition of further growth (e.g. Thommes et al. (2008);

Matsumura et al. (2009); Ida & Lin (2004).

On this issue, it is worth stressing that there is quite of a confusion on the role of viscosity in gap

opening. In a 2D disk isothermal model, Crida et al. (2006) showed that the width and the depth of a

gap saturates in the limit of vanishing viscosity. This work has been challenged recently by Duffell &

MacFadyen (2013) and Fung et al. (2014), still for 2D disks. For a massive planet, the results of Duffell

and MacFadyen actually agree with those of Crida et al. (2006), because the former group also finds that

the gap has a saturated depth and width in the limit of null viscosity and they demonstrate that this result

is not due to numerical viscosity. The actual disagreement is on the ability of small planets to open gaps

in disks much thicker than their Hill sphere. Fung et al. (2014) address this case just by extrapolation

of formulæ obtained for massive planets in viscous disks, so it is not very compelling. We believe that

gap opening by small planets in Duffell & MacFadyen (2013) is due to the use of an adiabatic equation

of state P = Eint(γ − 1) which, despite adopting a value for the parameter γ very close to unity, is not

equivalent to the isothermal case (see Paardekooper & Mellema (2008). The issue, however, deserves

114

4.2. Gaps in 3D disks Chap.4 Meridional circulation

further scrutiny.

This controversy is nevertheless quite academic, because real disks have a 3 dimensional structure.

Thus, in this chapter we discuss gap opening in 3D disks and we focus on giant planets that are massive

enough to undergo runaway gas accretion, i.e. Jupiter-mass bodies. In Sect. 4.2 we present the structure

of the gaps and the gas circulation in their vicinity, using three-dimensional isothermal simulations. The

gas flows takes the form of a meridional circulation, and explains why the 3D simulations presented in

the previous chapter showed that the flow of gas to the planet happens along the z-direction. We interpret

this result with a simple intuitive model. From this model, we derive in Section 4.3 an estimate for the

flow of gas into a gap in a layered disk (i.e. a disk that is viscous on the surface and “dead” near the

mid-plane), that is in agreement with the numerical results of Gressel et al. (2013). Conclusions and

discussion of the implications for terminal mass problem of giant planet are reported in Sect. 4.4.

4.2 Gaps in 3D disks

In the framework of a study on planet accretion (Chap. 3), we conducted 3D simulations of a Jupiter-

mass planet embedded in an isothermal disk with scale height of 5% and α-prescription of the viscosity

(Shakura & Sunyaev 1973). We adopted α = 4 × 10−3 (viscous simulation) and α = 0 (inviscid sim-

ulation). The latter simulation was conducted with two different resolutions, to change the effective

numerical viscosity of the simulation code. The technical parameters of the simulations are described in

detail in Chap. 3 and the main ones are briefly reported in the caption of Fig. 4.1.

In Chap. 3 we reported that the flux of material into the gap (and therefore the accretion rate of

the planet) decreases by a factor of 2 when the resolution is increased by a factor of 2 (and hence the

numerical viscosity is halved). This suggested that the gap properties do not saturate in the limit of small

viscosity (prescribed or numeric), or at least not in the range of resolution that we have been able to

attain. This observation motivated a deeper investigation, which is the object of the present work.

Fig. 4.1 gives a clear demonstration of how the gap profile opened by a Jupiter-mass planet in 3D

simulations changes with prescribed and numerical viscosity (i.e. with resolution). Each curve shows the

radial profile of the surface density, averaged over the azimuth. The surface density has been computed

by integrating the volume density over the vertical direction. Clearly, here no convergence is achieved

with resolution.

We explain this fact with the following simple model. Consider first the gas in the mid-plane. Its

dynamics has to be similar to that in a 2D disk model, and therefore the planet opens a gap with a given

profile independent of viscosity in the small viscosity limit. In the vertical direction, the disk has to be

in hydrostatic equilibrium. This implies that the volume density ρ scales with z (the distance from the

mid-plane) as

ρ(z) = ρ(0) exp

(

− z2

2H2

)

, (4.1)

where ρ(0) is the mid-plane density and H is the scale-height of the disk. Therefore, at equilibrium the

radial density profile of the gap has to be the same at all altitudes. However, the planet cannot sustain the

same profile at high altitude as on the mid-plane, because its gravitational potential:

Φ(d, z) =GMp√d2 + z2

(4.2)

(where G is the gravitational constant, Mp is the planet’s mass, d is the planet-fluid element distance

projected on the mid-plane) weakens with increasing z. Please notice that z plays the role here of the

115


Figure 4.1. Gap profiles in the simulations from Chap. 3. The perturbed column density distri-

bution is normalized by the original column density. Here “Viscous” means and α parameter of

0.004 in the Shakura & Sunyaev (1973) viscosity prescription and “inviscid” means α = 0; “Low

resolution” means 628x208x15 cells for the directions of azimuth, radius, co-latitude, respectively,

but with mesh refinement in the vicinity of the planet (we use 7 levels of mesh refinements, start-

ing from a box centered on the planet with half-sizes 0.27 × 0.27 × 0.12 in code units resolved in

112 × 112 × 24 cells, and doubling the resolution at each level); “High resolution” means twice

the number of cells in each direction at each level and still α = 0. Notice that the gap becomes

deeper with decreasing viscosity and increasing resolution, i.e. decreasing the effective viscosity

(prescribed or numeric). The simulation has been performed with the code JUPITER written by F.

Masset.

116


smoothing parameter (usually denoted ǫ) used in 2D simulations, and it is well known that the depth of

the gap decreases with increasing smoothing length (see Fig. 4.2)1.

Away from the mid-plane, the gas tends to refill the gap by viscous diffusion because it is not suf-

ficiently repelled by the planetary torque. As soon as the gas penetrates into the gap, however, it has

to fall towards the mid-plane because the relationship (4.1) is no longer satisfied and has to be restored.

Therefore, there is always more gas near the mid-plane in the gap than it would in the ideal case where

vertical motion is disabled. This excess of gas is partially accreted by the planet and partially repelled

away from the planet orbit into the disk, like in the gap opening process. Outside of the gap, then, the

relationship (4.1) is also not fulfilled, because there is an excess of gas near the mid-plane, coming from

within the gap, and a deficit of gas at high altitude, which flowed into the gap. So, the gas has to move

towards the surface of the disk to restore the hydrostatic vertical equilibrium profile (4.1).

In conclusion, there has to be a 4-step meridional circulation in the gas dynamics: (1) the gas flows

into the gap at the top layer of the disk; (2) then it falls towards the disk’s mid-plane; (3) the planet keeps

the gap open by accreting part of this gas and by pushing back into the disk the gas that flowed outside

of the Hill sphere (4) the gas expelled from the gap near the mid-plane rises back to the disk’s surface.

If no gas were permanently trapped in the vicinity of the planet (i.e. no planet growth), this meridional

circulation would basically be a closed loop. Instead, planet accretion makes the flow at step (3) smaller

than that at step (1)

We can observe this meridional loop in the numerical simulations. Fig.4.3 shows the volume density

of the disk in r, φ coordinates and the arrows show the mass transport, both averaged over the azimuth.

Kley et al. (2001); Ayliffe & Bate (2009a); Machida et al. (2010); Tanigawa et al. (2012); Gressel et al.

(2013) and Chap. 3 already showed that gas flows into the gap near the surface of the disk, but Fig.4.3 is

the first clear portrait of the meridional circulation explained above.

Notice that steps 2 to 4 of the meridional circulation occur on a dynamical timescale (i.e. independent

of viscosity), because they are related to pressure and planet’s gravity. Step 1, instead, occurs on a viscous

timescale. If the viscosity is small, the viscous timescale controls the entire timescale of the meridional

circulation. This explains why the flow into the gap and the accretion rate onto the planet scale linearly

with effective viscosity (prescribed or numeric) as found in Chapter 3. Moreover, because step 1 and

step 3 occur on independent timescales, the depth of a gap also increases with decreasing viscosity, as

observed. The relationship, though, is not necessarily linear. Part of the gas falling toward the mid-plane

in the gap ends in the horseshoe libration region and has to diffuse to the separatrices of said region (on a

viscous timescale) before being removed. If all gas followed this fate, the inflow and the removal would

both happen on the same viscous timescale and therefore the gap’s depth would not change with viscosity.

If instead all the gas fell on the separatrices of the horseshoe region, only the inflow would depend on

viscosity and therefore the gap’s depth would be inversely proportional to the viscosity. Reality is in

between the two. In fact, in Fig. 4.1, the gap’s depth increases by a factor ∼ 1.5 as the resolution is

doubled (e.g. the numerical viscosity is halved).

1This shows that neglecting the z-dependence of the potential in the numerical simulations, as done in Zhu et al. (2013),

changes the dynamics of the gas at a qualitative level and can enable small-mass planets to sustain partial gaps.

117


Figure 4.2. Perturbations of the azimuthally averaged surface density profile in a 2D disk, after

5000 orbits in the vicinity of a Jupiter mass planet on fixed orbit. The size of the cells is δr =

0.0167 and δθ = 0.0193. We have checked that this resolution is enough (doubling the number

of cells doesn’t affect the shape of the gap. The viscosity is ν = 10−6.5r2pΩp (independent of

radius). Different colors show results obtained with different values of the smoothing length ǫ of

the planetary potential. This shows the crucial role of the smoothing parameter which, in a 3D

disk, is played by the vertical distance from the midplane.

118


Figure 4.3. Volume density (ing

cm3 ) of the disk plotted in r, φ coordinates, integrated over the

azimuth. The arrows show the mass transport ρ~v, also averaged over the azimuth. This simulation

has been done with the 3D version of FARGO developed in Lega et al. (2014). The viscosity here

is ν = 10−5r2pΩp (independent of radius) and the resolution is 760, 1152, 96 in radius, azimuth and

co-latitude respectively, with radius ranging from rmin = 0.3 to rmax = 4.2 (in units of the orbital

radius of the planet) and co-latitude from 1.42 to 1.72. The Jupiter mass planet is held fixed at

rp = 1, φ = π/2 (midplane). The planet is not allowed to accrete (no mass sink is implemented)

so that, by viscosity, most of the gas that falls onto the circumplanetary disk eventually flows

outside of the planet’s Hill sphere. Consequently, the meridional circulation shown in the figure

has practically reached a steady state as a closed loop.

119

4.3. Implications on planet’s accretion from the flow of gas into the gap Chap.4 Meridional circulation

4.3 Implications on planet’s accretion from the flow of gas into the

gap

The results of the previous section seem to suggest that the flow of gas into the gap and the planet’s

accretion rate have to vanish with viscosity. But remember that what governs the meridional circulation

of gas in the gap’s vicinity is the viscous timescale near the disk’s surface. Protoplanetary disks cannot

be fully inviscid: stars are observed to accrete mass, which implies that angular momentum has to be

transported at least in part of the disk. A popular view is that of a layered disk, i.e. a disk which is

viscous near the surface and inviscid near the mid-plane, due to ionization at high altitude and turbulence

driven by the magneto rotational instability (MRI; Gammie (1996)). This view may have problems (see

Turner et al. (2014) for a review), so that other mechanisms (e.g. the baroclynic instability; Klahr (2004))

may be at work. Possibly these alternative mechanism can make the entire protoplanetary disk viscous,

but at the very least, a viscous layer has to exist near the surface of the disk, as in the MRI view.

We can now make a simple estimate of the gas-flow into a gap, as follows. It is well known that the

radial velocity of the gas in a steady state viscous disk is:

vr =1

Σ j

d

dr[ jνΣ] (4.3)

where ν is the viscosity, j =√

GM⊙r is the specific angular momentum and Σ is the surface density of the

disk, which is not a constant slope anymore, because the profile changed due to the opened gap. From

now on we set the units so that the mass of the star M⊙ and the gravitational constant G are both equal to

1, so that we substitute j with√

r.

Assuming that ν = αH2Ω as in the usual α-model (Shakura & Sunyaev 1973), where Ω is the orbital

frequency of the gas, one has: ν = α(H/r)2 j. Thus, the right-hand side of eq. (4.3) becomes the sum of

two terms. Assuming that H/r is independent of r, we get

vr = α(

H

r

)2(

1 +d lnΣ

d ln r

)

rΩ . (4.4)

According to Crida et al. (2006) (see formula 14 in that paper, with input from formulæ 11 and 13)

in the limit of vanishing viscosity and for a planet with Hill radius Rh ∼ H, the surface density slope is

maximum at a distance ∆ = 2.5Rh and is:

d(logΣ)

dr=

1.3

rp(H/r), (4.5)

where rp is the orbital radius of the planet. As we explained in the previous section, the same slope is

present at every altitude because of the hydrostatic equilibrium relationship in (4.1). At high altitude,

therefore, this is the slope that drives the viscous flow, given that the planet can not sustain the gap

there, due to its reduced potential. From (4.5) the second term of the right-hand side of (4.4) is inversely

proportional to H/r and consequently it dominates over the first term. Thus, we drop the first term from

the formula, getting

vr ∼ α(H/r)rΩ . (4.6)

The flow in the gap is then

M = 2πrvr4Σa (4.7)

120

4.4. Conclusions Chap.4 Meridional circulation

where Σa is the column density of the active layer of the disk, typically 10g/cm2 in the MRI case, if

ionization is due to X-ray penetrating into the disk (Igea & Glassgold 1999). Notice that factor of 4

multiplying Σa, due to the fact that there are two surface layers in a disk and two sides of the gap.

Let’s now take formulæ (4.7) and (4.6) and apply them to a planet with Rh ∼ H at 3.5 AU. Assuming

α = 3 × 10−3, a typical value for the active layer of a MRI disk (Gressel et al. 2011) and H/r = 0.05,

formula (4.6) gives vr = 7.9 × 109 cm/y and M = 1026g/y, i.e. 1.7 × 10−2M⊕/y. If half of this flux is

accreted by the central planet, this value matches the accretion rate of a Saturn-mass planet observed in

the MRI simulations of Gressel et al. (2011).

4.4 Conclusions

In this chapter we have analyzed the dynamics of gas near gaps, opened by planets in three dimensional

proto-planetary disks, with particular emphasis on the small viscosity limit.

We observe a flow of gas into the gap at high altitude in the disk, at a rate dependent on effective

viscosity (prescribed or numeric). We have explained this result with a simple analytic model, that

assumes that the disk is in vertical hydrostatic equilibrium at all radii. Thus the radial profile of the gas

volume density is the same at every altitude in the disk. However, the planet’s potential weakens with

altitude z, so that the planet can not sustain the same gap on the mid-plane and at the surface of the disk.

Consequently, gas flows into the gap, at a viscous rate, at high altitude. Only planets with Hill radius

Rh >> H can sustain the same gap at all altitudes z < H, but this occurs only for very massive planets

(typically many Jupiter masses).

Assuming that proto-planetary disks are layered, as in the MRI description, we have derived from

our model a simple formula for the amount of gas flowing into a gap opened by a planet with Rh ∼ H,

which is in good agreement with the MRI simulations of Gressel et al. (2013). The mass flux is large,

corresponding to a doubling time for the mass of Jupiter of about 50,000 y.

We conclude that gap opening can not be the answer to the terminal mass problem of giant planets,

described in the introduction. A different mechanism is needed in order to slow down the gas accretion

of planets of Saturn to Jupiter mass. The role of the circumplanetary disk is a promising one, as shown

in Rivier et al. (2012) and in Chap. 3, although this needs to be explored further with more realistic

simulations (see Chapters 5 and 6).

121

4.4. Conclusions Chap.4 Meridional circulation

122

“Continuous as the stars that shine

And twinkle on the milky way,

They stretched in never-ending line

Along the margin of a bay”

William Wordsworth: I Wandered Lonely as a Cloud

Chapter 5

Circumplanetary Disk or Circumplanetary

Envelope?

Contents

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.2 Physical Model and Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . 125


5.3.1 Units, Frame and Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3.3 Disk Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3.4 Planetary Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3.5 Simulation sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.4.1 Circumplanetary disk or circumplanetary envelope? . . . . . . . . . . . . . . . . 129

5.4.2 Planet Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.4.3 Velocity and Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

123

5.1. Introduction Chap.5 Circumplanetary Envelope

5.1 Introduction

This chapter based on the third paper of this thesis, which is currently submitted to MNRAS (Szulagyi et

al. 2015).

The importance of studying the circumplanetary disk (CPD) formed around massive giant planets is

twofold: this subdisk regulates the growth of the planet in the last stages (e.g. Lissauer et al. 2009; Rivier

et al. 2012; Szulagyi et al. 2014), and it is the birth-nest for satellites to form (Canup & Ward 2002, 2006;

Mosqueira & Estrada 2003a,b). Currently there is no unambiguous detection of CPD from observations,

although extended thermal emission was detected around the planetary candidates of LkCa15 (Kraus &

Ireland 2012), HD100546 (Quanz et al. 2013, 2015), HD 169142 (Reggiani et al. 2014), and the upper

limit of CPD masses were measured in the system of GSC 6214-210 (Bowler et al. 2015) with the Ata-

cama Large Millimeter Array. Until sufficiently resolved CPDs are observed, hydrodynamic simulations

of these subdisks are the only tool to understand and reveal their characteristics. As computers evolve,

more and more complex (and accurate) physical models are taken into account in the hydrodynamic

simulations as well. But resolving well the circumplanetary disk is challenging even in numerical simu-

lations. One way to do simulations of the CPD is to perform 2D calculations (such as Rivier et al. 2012;

D’Angelo et al. 2003), where sufficiently high resolution can be achievable due to the limitations on 2

spatial directions. However, e.g. Tanigawa et al. (2012); Morbidelli et al. (2014); Szulagyi et al. (2014)

showed that the third dimension really changes the whole picture on the flow of gas in the vicinity of the

planet, thus also on the role of the subdisk. In three-dimensional simulations, another possible way to re-

solve the CPD is to limit the simulation box size. Instead of simulating the entire circumstellar disk, one

can define a box in the vicinity of the planet, where the simulation is performed. These are the so-called

shearing sheet box simulations, such as Machida et al. (2010); Tanigawa et al. (2012). However, this way

the planetary gap is not deep enough, and the CPD is missing the feedback from the circumstellar disk. In

Morbidelli et al. (2014) and in Szulagyi et al. (2014) (Chapters 3 and 4) it is described that the accretion

onto the CPD is a free-fall flow arising from the top layers of the circumstellar disk, which is part of an

entire meridional circulation flow between the CPD and the circumstellar disk. With a shearing sheet

box simulation, this meridional circulation is missed, therefore the accretion and the CPD dynamics are

not correctly addressed. Hence, our approach in this chapter is to do a global disk simulation featuring

the entire circumstellar disk, and using so-called nested meshes, i.e. high resolution grids around the

planet, to zoom onto the planet’s vicinity. We also added a complex radiative module to our JUPITER

code, incorporating the thermal processes (viscous heating, stellar irradiation, and cooling through radi-

ation), assuming a uniform dust-to-gas ratio. The JUPITER code is based on a shock capturing method,

therefore sharp details can be observed in the flow around the planet, which were never shown before.

Previous non-isothermal simulations addressing the subdisk (e.g. Ayliffe & Bate 2009a,b; D’Angelo

et al. 2003; D’Angelo & Podolak 2015; Gressel et al. 2013) have already pointed out that the CPD is hot,

optically thick and it has a steep radial temperature profile. Works of D’Angelo et al. (2003), Ayliffe &

Bate (2009a), and Gressel et al. (2013) also agree that, due to the high temperatures, the spiral density

wake in the CPD is less prominent than in isothermal simulations, suggesting a reduced stellar torque,

which can lead to lower accretion rate (Szulagyi et al. 2014; Rivier et al. 2012). Regarding the measured

accretion rate, the higher temperatures resulted indeed in lower accretion rates in the simulations of

Ayliffe & Bate (2009a), Gressel et al. (2013), and Paardekooper & Mellema (2008), but not in the works

of Machida et al. (2010) and D’Angelo et al. (2003). This can be partially due to the fact that each

author has measured the accretion rate a bit differently; moreover, the thermal models were also different

in complexity. To measure the accretion, or characterize the CPD, it is also important what resolution

the simulations use and what is the applied smoothing technique for the gravitational potential of the

124

5.2. Physical Model and Numerical Methods Chap.5 Circumplanetary Envelope

planet, since these factors highly affect the resulting planet temperatures. Higher resolution means that

the temperatures are more accurate due to the higher iteration precision of the radiative module, but

also that the planetary potential is sampled better and deeper, leading to higher peak temperatures in the

innermost cells around the planet. Therefore, with our new code, we examine here the circumplanetary

material around a 1 MJup planet with maximal resolution of ∼ 80% of Jupiter-diameter. We also compare

our results with a previously existing hydro-code with radiative module, called FARGOCA (Lega et al.

2014).

5.2 Physical Model and Numerical Methods

Our study is based on three-dimensional, grid-based hydrodynamic simulations with the JUPITER code

(Szulagyi et al. 2014; de Val-Borro et al. 2006), originally developed by F. Masset. This code is based

on Godunov’s method and has nested-meshes, which allow to zoom onto the planet’s vicinity with high

resolution. The original code was written with locally isothermal EOS; here we implemented the an en-

ergy equation (so called adiabatic EOS) with simplified radiative transfer module to account for realistic

heating and cooling. The radiative module follows the scheme of Commercon et al. (2011), Bitsch et

al. (2013), and Bitsch et al. (2014). Apart from the equations of the mass-, and momenta-conservations,

the code now solves the energy equation on the total energy. Moreover, the coupling with the radiation

energy (ǫrad) is also taken into account, so the full set of governing equations are:

∂ρ

∂t+ ∇ · (ρv) = 0 (5.1)

∂(ρv)

∂t+ ∇ · (ρv ⊗ v) + ∇P = −ρ∇Φ + ∇ · ¯τ (5.2)

∂E

∂t+ ∇ ·

[(

P¯I − ¯τ)

· v + Ev]

= ρv · ∇Φ − ρκPc

[

B(T )

c− ǫrad

]

(5.3)

∂ǫrad

∂t= −∇ · Frad + ρκPc

[

B(T )

c− ǫrad

]

(5.4)

where ρ is gas density, E is the gas total energy (sum of the internal and kinetic energies), v stands for

the velocity-vector, and P indicates the pressure. Moreover,Φ is the gravitational potential, c indicates the

speed of light, and B(T ) defines the thermal blackbody: 4σT 4 – here σ symbolizes the Stefan-Boltzmann

constant, while T stands for the temperature. The Planck mean opacity, κP, is defined as Eq. 13 in Bitsch

et al. (2013). Eq. 5.3 contains the unit matrix (¯I) and the stress-tensor ( ¯τ), which is defined as:

¯τ = 2ρν

[

¯D − 1

3(∇ · v) ¯I

]

(5.5)

where ν is the kinematic viscosity, and ¯D is the strain tensor. Furthermore, Frad in Eq. 5.3 is defined

as:

Frad = −cλ

ρκR∇ǫrad (5.6)

where λ is the flux limiter taking care of the smooth transition between optically thin and thick

domains. For its definition and usage, see Kley (1989) and Bitsch et al. (2013). The parameter κR

125

5.2. Physical Model and Numerical Methods Chap.5 Circumplanetary Envelope

indicates the Rosseland mean opacity, which is defined as Eq. 15 in Bitsch et al. (2013) and chosen to

be equal to Planck mean opacity. We assume a uniform dust to gas ratio (generally 0.01 but this can be

modified as an input parameter of the code), and use the Bell & Lin (1994) opacity tables.

To close the system of equations, the equation-of-state (hereafter, EOS) also needs to be defined. We

used adiabatic EOS with the adiabatic index (γ) equal to 1.43:

P = (γ − 1)ǫ (5.7)

where ǫ is the total internal energy of the gas, which is ǫ = ρcvT . The JUPITER code is a multi-step,

Godunov-method based code, these steps are the following:

1. Solve for the interface fluxes (Eqs. 5.1–5.3) using exact Riemann solvers, using for the predictor

step a spatially second order accurate scheme of MUSCL-Hancock (Toro 2009).

2. The fluxes are used to update the cell contents, in what is called a “conservative update” (because

the fluxes are shared on interfaces between adjacent cells, the scheme is conservative for the corre-

sponding quantities to platform accuracy). Prior to being used, the fluxes are corrected (augmented)

by the viscous stresses.

3. Source terms (gravitational forces) are applied with finite difference scheme.

4. In the radiative module, Eq. 5.4 is solved and the internal energy is updated through the two-

temperature approach explained in Commercon et al. (2011), and in Bitsch et al. (2013).

The basic hydrodynamic fields in the JUPITER code are the volume density, the 3 components of

the velocity, and the energy. In each cell in the simulation the values of density/energy/velocity are all

centered values, so corresponding to the coordinates of the cell barycenters.

The new parts of JUPITER code (energy equation, radiative module) were heavily tested both sepa-

rately and together with the hydro-kernel as well. This includes testing the adiabatic Riemann-solvers,

the inter-CPU and inter-level communications, the radiative module with comparison tests to other hydro

codes with radiative module, and checking the dimensional homogeneity of all the equations in the entire

code. On details of the testing, see Chap. 2.

For comparison reasons we carried out a simulation with the FARGOCA code (Lega et al. 2014) as

well, which is an improvement of the FARGO code (Masset 2000) in 3D. FARGOCA is finite difference,

staggered mesh code based on an upwind method with van Leer’s slopes. It solves the energy equation

directly for the internal energy:

∂ǫ

∂t+ ∇ · (ǫv) = Q+ − P∇ · v − ρκPc

[

B(T )

c− ǫrad

]

(5.8)

where Q+ = ( ¯τ∇) · v is the viscous heating. While JUPITER is based on a shock-capturing method,

FARGOCA solves the hydrodynamic equation explicitly, thus it is not as well suite as JUPITER to treat

shocks. On the other hand, treating directly the equation for the internal energy alleviates the so-called

high Mach number problem (Ryu et al. 1993; Trac & Pen 2004) faced in JUPITER, where the internal

energy is very small fraction of the total energy. The FARGOCA code does not have nested meshes

that are needed to reach the same high resolution around the planet as with JUPITER. Therefore, we

have implemented a manual nested meshing procedure in FARGOCA as follows. First, for the gap-

opening phase the global disk simulation was performed on the same resolution as JUPITER’s coarsest

mesh. Then, a box around the planet was defined, where the resolution was doubled, and the boundary

126

5.3. Setup of the Simulations Chap.5 Circumplanetary Envelope

Table 5.1. Number of cells on different grid levels

Level N of cells

in azimuth

N of

cells in

radius

N of

cells

in co-

latitude

Boundaries of the levels in az-

imuth [rad]

Boundaries of the levels in ra-

dius [a]

Boundaries of the lev-

els in co-latitude [rad]

0 680 215 20 [−π, π] [0.40005, 2.3845] [1.4416, π/2]

1 120 120 34 [-0.27735, 0.27735] [0.72264, 1.27735] [1.451041, π/2]

2 120 120 62 [-0.138675, 0.138675] [0.861325, 1.138675] [1.47137, π/2]

3 120 120 86 [-0.0693375, 0.0693375] [0.9306625, 1.0693375] [1.5014588, π/2]

4 120 120 86 [-0.03466875, 0.03466875] [0.96533125,1.03468875] [1.5361276, π/2]

5 120 120 86 [-0.017334375, 0.017334375] [0.98266562, 1.017334375] [1.553462, π/2]

6 120 120 86 [-0.0086671875, 0.00866719] [0.99133281, 1.0086671875] [1.5621291, π/2]

7 120 120 86 [-0.0043335938, 0.00433359] [0.99566641, 1.0043335938] [1.566462737, π/2]

conditions for all hydrodynamics variables were set to the values found on the coarser mesh. After the

gas flow has stabilized on a given resolution, the box size around the planet was reduced again, the

resolution was doubled and the same boundary procedure was adopted. This procedure was iterated

until we achieved a resolution corresponding to the second finest level of JUPITER, therefore half of the

resolution in JUPITER’s finest level. There was no possibility to increase the resolution further with the

manual nested mesh technique, i.e. to define a box size inside the circumplanetary disk, because there

is no feedback between the levels, unlike in a real nested meshes with JUPITER. Reducing further the

simulation domain size in FARGOCA would lead to boundary condition problems.

5.3 Setup of the Simulations

5.3.1 Units, Frame and Grid

In our simulations, the coordinate system is spherical – with coordinates of azimuth, radius, co-latitude –

centered onto the star. The mass unit is the mass of the central star (assumed to be solar in numerical appli-

cations), the length unit is the radius of the planetary orbit (rp), while the time unit is such that G, the grav-

itational constant, is one. This implies that the planetary orbital period, 2π/Ω = 2π/√

G(M∗ + Mp)/r3p, is

2π. The frame is co-rotating with the planet, so that the planet is at a fixed position throughout the sim-

ulation. This position is at azimuth=0.0, radius=1.0, co-latitude=π/2. Assuming the planet’s orbit is at

Jupiter’s distance, the length unit in the code is 5.2 AU. The radial limits of the simulation box are 0.4-2.4

code units (2.1-12.5 AU), the azimuthal range is from −π to +π, thus including the entire circumstellar

disk, and the co-latitude range is [1.442, π/2], with the mid-plane being on π/2. This means a 7.4 degree

opening angle for the circumstellar disk. To save computational time, we simulated only the half of the

circumstellar disk, thus assuming symmetry relative to the midplane.

Due to the nested meshes, the resolution changes grid level by grid level. On the coarsest mesh (level

0), the resolution was 680 × 215 × 20, which means dr = 0.009 code units= 0.048 AU. On the next level,

the resolution was double of the previous, and so on, till level 6. In other words, at each level refinement

the resolution doubled in each spatial direction. The highest level of resolution (reached on level 6) was

dr = 1.442 × 10−4code units= 7.498 × 10−4 AU= 0.8 dJup where dJup is the diameter of Jupiter. The

borders of the nested meshes are described in Table 5.1. The simulation began with only the coarsest

mesh, and the successive levels were added in sequence after a quasi-steady state was reached on the

previous level.

127

5.3. Setup of the Simulations Chap.5 Circumplanetary Envelope

5.3.2 Boundary Conditions

At the radial boundaries of the simulation box, we have used reflecting boundary conditions for the radial

velocity. The azimuthal velocity was extrapolated in the 2 ghost cells according to the local Keplerian

velocity. The density and energy values in the ghost cells were set equal to the corresponding values of

their images among the active zones.

At the midplane (colatitude = π/2), a reflecting boundary condition is applied. At the other edge in

colatitude, above the circumstellar disk surface layer, we fix the temperature to 30 Kelvin in the ghost

cells to force the cooling. This accounts for the fact that circumstellar disks are surrounded by the outer

space and are able to radiate away their heat. In the azimuthal direction, periodic boundary conditions

were used.

At the border between nested meshes, the flow should be smooth, therefore the JUPITER code uses a

complex ghost cell communication with multi-linear interpolation. This means that, in each direction, 2

cells overlap with the cells of the previous level beyond the border of the given mesh, and in these ghost

cells the hydrodynamic fields are linearly interpolated from the values available on the previous level.

5.3.3 Disk Physics

The circumstellar disk’s initial surface density was Σ = Σ0( ra)−0.5 with Σ0 = 6.76 × 10−4 code units. This

density was chosen to be close to the Minimum Mass Solar Nebula (MMSN; Hayashi 1981). The initial

disk aspect-ratio was chosen to be H/r = 0.05, where H is the pressure scale-height of the disk, but this

changes as the circumstellar disk cools and therefore contracts a bit towards the midplane. All of our

simulations have a constant viscosity with value 10−5a2Ωp, which corresponds to approximately a value

of α of 0.004 at Jupiter’s orbit in the representation of Shakura & Sunyaev (1973). We remind the reader

that, like every numerical simulation, ours are also affected by the numerical viscosity, in particular close

to the planet where the mesh is locally Cartesian, while the flow has rather a cylindrical symmetry.

Because the opacity table accounts for the dust as well, one needs to define the dust-to-gas ratio in the

simulations. We used the interstellar medium value of 1% dust. The cooling happens through radiation,

therefore one should allow energy to escape through the surface of the circumstellar disk. To minimize

the CPU-time required to reach the initial thermal equilibrium of the circumstellar disk, we run initially

the simulation only with the circumstellar disk (i.e. without a planet). Since the circumstellar disk is

azimuthally symmetric, we defined only 2 cells in azimuth, run the simulation until thermal equilibrium

was reached, we divided the 2 azimuthal cells into the final amount of 680 cells. From this point on we

started to build up the planet increasing its mass over 30 orbits (see below), and run the simulation for

another 120 orbits to reach an equilibrium after the gap opening. Only after this we started to add the

nested meshes.

5.3.4 Planetary Potential

To allow the gas flow to adapt to the presence of our heavy planet, we increased its mass gradually over

30 orbit as

Mp(t) = Mpfinalsin2

(

t

120

)

(5.9)

This meant that the final planet mass was reached after the first 30 orbits of the simulation time. In all

our simulations, Mpfinalis 10−3 code unit, which corresponds to a Jupiter mass planet around a solar mass

star.

128

5.4. Results Chap.5 Circumplanetary Envelope

In the simulations the planet is a point-mass, in the corner of 8 cells (of which only 4 are considered as

active cells and 4 ghost cells due to the symmetry relative to the disk’s midplane). This means that there

is only a potential-well and no physical sphere is modeled for the planet. Hence, no boundary condition

is needed around the planet. However, to avoid the singularity of the gravitational potential, we applied

the traditional smoothing of the potential on a length rs:

Up = −GMp

√

x2d+ y2

d+ z2

d+ rs

2

(5.10)

where xd = x − xp, yd = y − yp, and zd = z − zp are the distance-vector components from the planet

in Cartesian coordinates. The importance of a large enough smoothing length in Godunov codes was

already pointed out by Ormel et al. (2015). Therefore, our smoothing length rs was set equal to three

times of the cell diagonal on levels 0-5, and 6 cell diagonals on level 6. In other words the potential

well was not deepened on level 6 relative to the previous value on level 5. Because the smoothing length

changes on every level to avoid the harsh transition of smoothing length when adding a new level, we

gradually reduced the smoothing length as rs(t) = 0.5(

rspreviouscos2

(

t−t04

)

+ rsprevious

)

, where rspreviousis the

value of rs on the previous level and t0 is the time at which the new level has been introduced. Thus

reaching the new smoothing length – half of the value on the previous, is reached coarser level after 1

orbit.

5.3.5 Simulation sets

We performed altogether 5 simulations, four of them are carried out with the JUPITER code, one is with

the FARGOCA code. In our nominal simulations performed with both codes, the temperatures evolve

freely according to the governing equations (Eqs. 5.3 and 5.4) also on the planet. We refer as “planet”

the set of 32 cells around the point-mass location at [0,1,π/2] (two cells in each coordinate direction).

The side of this cubic region is about 3.6 diameters of Jupiter. This is motivated by the fact that Metis,

the innermost satellite of Jupiter, presently orbits at 1.8 RJup, and that the contraction timescale of Jupiter

is of the order of 1 Myr for this planetary size (Guillot et al. 2004). The nominal simulations resulted in

very high temperature: 13, 000 K on the planet. This is probably an overestimation for several reasons, as

discussed in Section 5.4.2. Therefore, we decided to parameterize the planet’s temperature, by launching

three other simulations with the JUPITER code, where we set a ceiling temperature for the planet at

1000 K, 1500 K and 2000 K in the vicinity of the planet, on the innermost 32 cells. This means that at

the beginning of the simulations we let the temperature on the planet evolve according to the radiative

module, but when the temperature rises above the ceiling temperature, we reset it at the ceiling value,

preventing it to climb further.

5.4 Results

5.4.1 Circumplanetary disk or circumplanetary envelope?

Previous works have suggested that the formation of a circumplanetary disk is linked to the mass of the

planet and the gap-opening process (e.g. Ayliffe & Bate 2009a,b): small planets, which are unable to

open gaps in the circumstellar disks are thought to have some circumplanetary material in the form of an

envelope; instead, planets capable to carve deep enough gaps should form circumplanetary disks. How-

ever, we have found that the situation is more complex. Precisely, if the planetary temperature is high,

129


such as in our nominal simulation, even a gap-opening, Jupiter-mass planet is forming a circumplanetary

envelope instead of a disk (see Fig. 5.1 left column). As long as there is accretion this hot envelope does

not cool off, suggesting that the planet could form a circumplanetary disk only after that the circumstellar

disk has dissipated and accretion has stopped. This way, the envelope would eventually cool, forming a

disk with the same angular momentum as the original envelope.

In Figure 5.1 we show vertical slices at azimuth=0 of the density (first row) and of the temperature

(second row). Each column corresponds to a different simulation: from left to right we show the nominal

case, then the 2000 K, 1500 K, 1000 K fixed planetary temperature cases, respectively. Since we have

small fluctuations between the different output files, we have averaged the fields over the last 3.5 orbits

of the simulations (71 outputs averaged). In the first column we clearly observe a spherical envelope.

The planet temperature here is 13,000 K. However, in all the cases with fixed planetary temperature a

circumplanetary disk develops (columns 2 to 4). Clearly, the planet temperature determines whether an

envelope or a disk forms around the planet. In fact, the higher is the temperature, the more pressure

supported is the disk, and the larger is its scale-height.

Furthermore, plotting the density maps on the midplane revealed that higher temperatures weaken

the trace of the spiral wake in the CPD. The difference is especially striking when comparing our lo-

cally isothermal simulations in Szulagyi et al. (2014) (Chap. 3) with the simulations in this chapter. The

isothermal simulations have the lowest temperature in the circumplanetary region among all the simula-

tions we have performed, therefore the spiral wake is the strongest. This indicates stronger stellar torque,

which probably means higher accretion onto the planet as well. The dependence of the strength of the

spiral wake on the temperature of the simulations was already pointed out in previous works e.g. by

Paardekooper & Mellema (2008) and Ayliffe & Bate (2009a).

On the temperature plots (second row on Fig. 5.1) one can see that in the disk cases, there are two

small regions of bright yellow color (i.e. high temperature), just above and below the central part of the

CPD. We interpret these to be due to a shock between the gas infalling from the vertical direction and the

disk.

We made a simulation with the FARGOCA code as well, which corresponds to the nominal simula-

tion with JUPITER. On Figure 5.2 we show the density map (left) and temperature map (right), which

are quite similar to left column of Fig. 5.1. The two codes with the same initial parameter file gave

qualitatively similar results, namely the planet has a hot circumplanetary envelope (CPE). We recall, that

the simulation made with the FARGOCA code has a resolution, which is half of the simulation made

with JUPITER.

On Figure 5.3, the radial profiles of the CPD/CPE obtained in the JUPITER simulations are compared

for the densities (left) and for the temperatures (right). The density profiles are all very steep. The fixed

central temperature cases almost match each other, their power law index in the outskirts of the CPD is

approximatively -2.6. The nominal simulation’s envelope shows a very different radial profile, which is

due to geometrical reasons (the gas in the envelope has a nearly spherical symmetry). Here, the region

within 0.1RHill contains a larger total mass than in the fixed temperature (disky) cases. However, the

innermost few cells around the planet are always more massive in the fixed central temperature cases

(due to the smaller temperatures, which allow a higher compression of the gas), than in the nominal

simulation. The temperature profiles (right panel on Figure 5.3) shows that the nominal simulation leads

to temperatures higher than those in the fixed central temperature simulations in the whole domain. Even

at 0.5 RHill away from the planet, the nominal case’s envelope is still ∼ 400 K hotter than the fixed

planetary temperature cases. This difference increases up to 7000 K close to the planet. In all cases the

circumplanetary material is optically thick, and the temperatures in the CPD/CPE within ∼10-20% of the

Hill-sphere are over the dust sublimation threshold, so the opacity here is set by the gas opacities. In

130


Figure 5.1. Matrix of figures summarizing the four different simulations (Nominal, planetary

temperature of 2000 K, 1500 K, 1000 K, respectively from left to right). The first row shows the

density maps, while the second shows the temperatures on a vertical slice cut through the planet

along the radial direction. We can see that in the nominal simulation, where the peak temperature

is over 10000 K, a circumplanetary disk cannot form; the circumplanetary material is in a spherical

envelope around the Jupiter-mass planet. Instead, when we fix the planet temperature at 1000 K

– 2000 K, a circumplanetary disk always forms. Therefore, the planetary temperature determines

the formation of a disk or of an envelope even for a gap-opening planet like Jupiter.

Figure 5.2. Simulation of the nominal case made with the FARGOCA code. Left is the density

map around the planet, right the temperature map. The comparison of the results of JUPITER is

very good (see the first column of Fig. 5.1)

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Figure 5.3. Radial profiles of density (left) and temperature (right) at the midplane of the circum-

planetary region for our four simulations with the JUPITER code, and one simulation with the

FARGOCA code (as labelled). Notice that the temperatures of CPDs in the fixed central tempera-

ture simulations reach a maximum at a distance ∼ 10%RHill.

the fixed central temperature cases, the inner part (within a distance of 0.01-0.02 Hill radii) of the CPD

is surprisingly much hotter than the planet. This is because of the cooling action of the planet, whose

temperature is imposed. Instead, far enough from the planet the cooling effects vanish, and the viscous

heating, together with the adiabatic compressional heating can heat the CPD to be hotter than the planet

itself. It is also interesting that the temperatures in the three fixed central temperature cases match beyond

0.02RHill. The power law index of the temperature beyond this distance is around 0.6, so the disks are

flared (a disk with constant aspect ratio would have a temperature proportional to 1/r).

Comparing the nominal simulation of JUPITER and FARGOCA codes, the radial profiles are some-

what different. This is most likely due to the following two reasons. First, the resolution is different in the

two simulations, the FARGOCA simulation has half the resolution of JUPITER’s. This means that the

potential well is more coarsely sampled; it is less deep, resulting in lower peak density and peak temper-

ature. The second reason is that in JUPITER nested meshes are implemented, so that there is feedback

between the different nested mesh levels, in contrary to FARGOCA.

In summary, these findings suggest that even in the case of large mass gas-giants, the gap-opening

capability does not account for whether a circumplanetary envelope or a disk forms around the planet;

but the temperature of the planet is the most critical factor. The characteristics of the circumplanetary

material – density and temperature profiles, rotation, accretion – is strongly dependent on the planetary

temperature we account for.

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5.4.2 Planet Temperature

As we have seen in the previous section, the nominal simulation resulted in very high, 13000 K tempera-

ture on the planet region. This is probably an overestimation for two main reasons.

First, our EOS overestimates temperatures by neglecting dissociation and ionization of hydrogen.

In order to estimate the magnitude of this effect, we compared the changes in specific entropy at the

location of Jupiter in our nominal simulation to those recalculated from our pressures and temperatures

using a more realistic EOS (Saumon et al. 1995). In the nominal simulation, the grid point centered on

the planet was being heated and compressed from 320 K and 0.17 Pa to 13500 K and 1.2 × 105 Pa. This

resulted in the JUPITER code specific entropy decreasing from 0.058 to 0.043 in code units. The specific

entropy calculated directly from the SCvH EOS increased from 13.1 to 21.6 kB/baryon (with kB being the

Boltzmann constant). Because the entropy is defined only to a constant, we did not attempt to match code

entropies and SCvH ones. On the basis that the code entropy did decrease with time in the simulation,

we obtained an upper limit on the temperature by finding the one for which the specific entropy was

13.1 kB/baryon for a central pressure of 1.2× 105 Pa. That upper limit, 3600 K, is significantly lower than

the one obtained in the code. The 1D models of planetary interior structure of Mordasini et al. (2012b)

show similar specific entropies (a pressure of 104 Pa and temperature of 2000 K) at the photosphere of

a proto-Jupiter when it has nearly reached its final mass, detached from the disk and contracted to a

∼ 2 RJup radius. This shows that some caution must be exerted concerning our results in this work or any

hydrodynamic simulation using simplified EOSs at high temperatures.

In spite of this caveat, it is interesting to notice that the entropies that we obtain are always well into

the so called “hot-start” regime, i.e. above 10 kB/baryon. This indicates that Jupiter, and a fortiori any

massive giant planets, should evolve according to the “hot start” scenario rather than according to the

“cold start” one which would involve losses of entropy due to shocks before material is embedded inside

the protoplanet (e.g., Marley et al. 2007; Marleau & Cumming 2014).

The second reason for overestimating the planetary temperature in our simulation is related to the

planet interior structure. One should realize that the gas in the planetary cells in our simulation corre-

sponds to just the last layer of gas accreted by a Jupiter-mass planet and not to the total amount of gas

making the full planet. This layer should be hotter than the whole planet, as one can see with the follow-

ing simple argument based on the virial theorem: |Epot|= 2Ekin. If we assume that the planet is a uniform

sphere of monoatomic hydrogen, we have3GM2

p

5Rp=

3Mpk<T>

mproton, where k is the Stefan-Boltzmann constant.

This gives an average temperature of < T >=GMpmproton

5Rpk. Now the last gas layer falling to the planet will

release a potential energy GMpδM/Rp, which again using the virial theorem will result in a temperature

of 53< T >. This highlights that the temperature we record in the planetary cells in the simulation proba-

bly overestimates of the real temperature of the full planet. In practice, the planet should act as a cooler

for the surrounding infalling gas. Of course this model above is simplistic and it neglects several effects,

e.g. the radiative cooling of the planet, which would lower the temperature below the virial value, and

the additional energy delivered by the accretion of solids, which would increase the temperature.

This uncertainty on what the planet temperature justifies our choice to run several simulations with

parametrized temperatures in the cells near the planet’s position.

5.4.3 Velocity and Angular Momentum

The nominal simulation and the fixed central temperature simulations show completely different nor-

malized angular momentum fields. The normalization of the angular momentum is based on the local

Keplerian velocity, i.e. we divide the angular momentum per unit mass that we measure in each cell of

133


Figure 5.4. Azimuthally averaged normalized angular momentum (z-component) from the av-

erage of 71 outputs over the last 3.5 orbits of the simulations in a non rotating frame. The nor-

malization of the angular momentum was performed based on the local Keplerian velocity. The

coordinates of the plots are cylindrical, planetocentric, so that the planet is in the left-bottom cor-

ner of each figure. The dotted areas symbolize the positive normalized angular momentum values.

Left is the nominal case, with a slightly retrograde inner envelope (within 0.1 RHill mostly negative

angular momentum), and sligthly prograde outer envelope, but overall the rotation of the envelope

is almost stopped. On the right panel, a fixed planet temperature (Tp=2000 K) case is shown with

positive normalized angular momentum values, with a maximum of 80% Keplerian rotation.

the simulation relative to the planet by√

GMpd where d is the radial distance of the cell from the planet

in cylindrical coordinates. The angular momentum is measured in a non rotating frame centred on the

planet Fig. 5.4 shows the z-component of the normalized angular momentum for the nominal case (left)

and for a fixed central temperature case (Tp=2000 K case on the right) through a vertical slice. The values

are azimuthally averaged (so the planet is in the left-bottom corner), and time averaged on 71 output files

over the last 3.5 orbits of the simulations. The dotted areas represent the positive angular momentum val-

ues (i.e. counterclockwise rotation of the gas around the planet). One can see on the right panel that the

circumplanetary disk is sub-Keplerian; it rotates with 80% of the local Keplerian velocity (bright yellow

colors). Above the disk, the gas which – as we will see below – falls towards the midplane, has a very

low angular momentum. This means that as it hits the disk, it slows down the disk rotation. Between the

three fixed central temperature cases there is not a large difference, therefore we show only one example

on Fig. 5.4. However, as the temperature rises, the scaleheight of the CPD is larger, and the rotation of

the disk is slower (lower normalized angular momentum values).

On the left panel of Fig. 5.4, the inner envelope (until 0.1 RHill) has mostly negative normalized

angular momentum. This means, that it rotates to the retrograde direction. The retrograde rotation,

however, is very slow, with a maximum of 4% of the Keplerian rotation speed. Beyond 0.1 RHill, the

envelope within the Hill-radius is rotating prograde, but again very slowly (maximum ∼ 30% of Keplerian

rotation). Overall, it can be said that the rotation of the envelope is almost stalled.

Due to the angular momentum conservation, one could also derive the centrifugal radius

(

Rcent =(J/Menvelope)2

GMp

)

,

where J is the angular momentum of the envelope, whose mass is Menvelope. This radius would correspond

to the one after the envelope collapsed into a ring. According to our computation within ∼ 0.5RHill (i.e.

on the last 3 refined level), this radius is one order of magnitude smaller than Jupiter’s radius, therefore

134


there would be no disk formed after this envelope has collapsed. It is important to highlight though the

limitations of our simulations, e.g. the lack of a rotating planet in the middle, the over-estimated tem-

perature which reduces rotation, etc. Therefore, it is difficult to imagine that Jupiter in our Solar System

went through the same phase. Our envelope would not be able to produce the extended, prograde system

of the Galilean moons. Our fixed temperature simulations show that a colder planet would form a quasi-

Keplerian disk and then, by accreting material from that disk, it would acquire a very rapid rotation. This

raises a problem for explaining the relatively slow rotation of our giant planets. We speculate that the

temperature of Jupiter was below our nominal value, but larger than 2000 K, so that it was surrounded by

a prograde but extremely sub-Keplerian, puffed-up disk.

The velocity fields in the nominal simulation and the fixed central temperature simulations are com-

pared in Figure 5.5. This figure shows the time averaged (71 outputs over 3.5 orbits), azimuthally av-

eraged, mass-weighted, planetocentric radial velocities and vertical velocities in cylindrical coordinates

for the nominal simulation (left column) and for the Tp=2000 K simulation (right column). Since all the

fixed temperature simulations look quite alike, we show here only the Tp=2000 K case for brevity. In the

nominal simulation, we see near the planet (which is placed at the left-bottom corner at 0,0 co-ordinates)

alternate regions of positive and negative velocities in both the radial and vertical directions, which sug-

gest the existence of convective motion. The convective zone is surrounded by a radiative outer layer up

until the edge of the Hill-sphere. Comparing the velocity values to the fixed central temperature simula-

tion’s velocities, the difference is one-two order of magnitude. We also compared the velocity field of the

FARGOCA simulation with JUPITER’s. The same circulation pattern was found.

The radial velocities of the fixed central temperature simulations (see Figure 5.5 bottom-right insert)

show a typical accretion-disk pattern: negative radial velocities on the upper layer of the CPD, so the

flow is rushing toward the planet, and positive values below the upper layer (dotted area on the bottom-

right panel) meaning a receding motion from the planet. The vertical velocities on the upper-right plot

on Fig. 5.5 show the strong vertical influx towards the planet, which shocks on the upper layer of the

CPD above the planet. In fact, the contrast in vertical velocity between the infalling gas and the disk is

about three times higher than the local sound speed. The effect of the very strong shock front was clearly

visible on the temperature plots of Fig. 5.1, where the shock heating highlighted this front in the upper

layers of the CPD, close to the planet. In various slices on non-averaged fields we found that the vertical

influx hits the shock-front so strongly, that some of it is reflected back. The angle of the vertical influx

is not exactly vertical as it hits the CPD, it is slightly tilted, therefore the bounced flow escapes in the

opposite direction at about the same angle. Nevertheless, most of the vertical influx will end up in the

CPD and will be accreted to the planet or leave the disk in the middle regions below the surface layer.

How far from the midplane the vertical influx shocks is also determined by the local pressure, therefore

the planetary temperature. Simulations with hotter planet temperatures have the shock-front further away

from the midplane.

As Szulagyi et al. (2014) and Morbidelli et al. (2014) pointed out, the vertical influx is part of a

larger circulation which connects the circumstellar disk with the circumplanetary disk. Even though

those simulations were isothermal, one can see the same process happening in our radiative simulations

as well. As the circumstellar disk upper layers try to close the gap opened by the planet, gas enters into

the gap, and free-falls onto the planet due to its gravity. This influx hits the CPD and the planet as well.

In fact the CPD is mostly fed by this vertical influx as was pointed out in Szulagyi et al. (2014). Then,

the gas which is not accreted onto the planet leaves the CPD in the outflow near the midplane.

135


Figure 5.5. Azimuthally averaged velocities, from the average of 71 outputs over the last 3.5

orbits of the simulations. The first row shows the vertical velocities in cylindrical, planetocentric

coordinates, the second row corresponds to the radial velocities for the nominal (left column) and

for the fixed planet temperature (Tp=2000 K) simulations. The planet is the left-bottom corner in

each plot. The positive velocity areas are dotted. In the nominal simulation, where an envelope

formed around the Jupiter-mass planet, the sign changes in the velocities indicate a possible inner

convective inner layer (until about 0.15 RHill), surrounded with a radiative zone. On the other hand,

the fixed central temperature simulation formed a disk around the planet; here the radial velocities

show clearly an inflow near the surface layer of the disk (negative vrad), and a radial outflow near

the middle of the CPD (positive vrad, dotted area). The vertical velocities show a strong vertical

influx (negative vz), which shocks on the top of the CPD and above the planet.

136

5.5. Conclusions Chap.5 Circumplanetary Envelope

5.5 Conclusions

In this chapter we have studied the circumplanetary flow around a 1 MJupiter planet with hydrodynamic

simulations in 3D. Thanks to the nested meshes technique, we had an entire circumstellar disk in low

resolution and very high resolution (80% of the Jupiter-diameter) grid around the planet. We also im-

plemented the energy equation and a radiative module into the JUPITER code, which accounts for both

the gas and dust opacities (dust assumed to be 1% of the gas by mass). The heating is due to viscous

heating and adiabatic compression; the cooling is due to radiation and adiabatic expansion. To check

our findings, we made a comparison simulation with the code FARGOCA, which also has the radiative

module following the same logic, but the hydro parts are solved through different mathematical methods.

We performed four simulations with the JUPITER code. In our nominal simulation the temperature

was allowed to evolve according to the energy equations, without further constraints (resulting in a peak

temperature of ∼ 13, 000 K near the planet). In the other three simulations, we enforced a 1000 K, 1500 K

and 2000 K ceiling temperatures in the cells near the planet. This change resulted in a large difference on

the circumplanetary flow between the nominal and fixed temperature cases.

While in the fixed temperature simulations a prograde rotating circumplanetary disk formed, the

nominal case resulted in a very hot spherical envelope, even around this 1 MJupiter planet. Therefore, this

finding suggests that the characteristics (temperature, mass, rotation, accretion, etc.) of circumplanetary

material is mostly determined by the planet’s temperature rather than its mass. Moreover, the ability to

form a circumplanetary disk does not depend simply on the ability of the planet to open a gap in the

circumstellar disk, since even a gap-opening 1 MJupiter planet can form a circumplanetary envelope, like

the low-mass planets, if the planetary temperatures are very high. In any case our results suggest that

giant planets accrete high entropy gas, lending support to the so-called “hot-start” formation scenario

(e.g., Marley et al. 2007; Marleau & Cumming 2014).

We overall found that higher temperatures reduce the disk’s rotation and weaken the trace of the

spiral wake in the subdisk, indicating a reduced stellar torques, hence possible reduced accretion, in

accordance with previous works (e.g.Ayliffe & Bate 2009a, Paardekooper & Mellema 2008). In all of

our simulations the circumplanetary material is optically thick with very steep temperature and density

profiles. Moreover, the nominal case with the circumplanetary envelope has an internal convection layer

of 0.15 RHill, surrounded by a radiative layer extended up to the Hill radius. This envelope has basically

no rotation. In the fixed central temperature simulations, however the subdisk shows moderately sub-

keplerian, prograde rotation and it is fed by a strong, vertical influx arising from the top layers of the

circumstellar disk and the walls of the gap, which then shocks on the CPD surface.

137

5.5. Conclusions Chap.5 Circumplanetary Envelope

138

“Science never solves a problem without creating ten more.”

George Bernard Shaw

Chapter 6

Mass of the Circumplanetary Disk and

Accretion

Contents

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.2 Mass evolution inside the Hill-sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.2.1 Envelope Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.2.2 Disk Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.3 Mass of the Circumplanetary Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

139

Mass of the CPD and Accretion Chap.6 Circumplanetary Disk Mass

6.1 Introduction

In this Chapter we describe the preliminary, unpublished results acquired so far on the accretion rates in

the simulations of Chapter 5. We do not wish to enter into deep discussions, only to describe the findings

we have at this point. More simulations attacking the question of accretion rate are needed to understand

these results and to be able to publish the findings. As the thesis work is limited to 3 years, we had to

draw a line at this point, describing the work done until these lines are written.

The simulations mentioned in this Chapter are explained in Chapter 5, but for a short summary, we

have performed overall 5 simulations. Four simulations with the JUPITER code developed during this

thesis, and one simulation with FARGOCA to compare the results of the nominal simulations. The

nominal simulations were with a Jupiter-mass planet embedded in a MMSN circumstellar disk. The

temperatures were allowed to evolve freely in every cell (including in the cells corresponding to the

planet) according to the radiative module (see Chapter 2). This way we get temperatures over 13,000 K

on the innermost cells around the planet and, instead of a circumstellar disk, an envelope is formed

around the Jupiter-mass planet. Because we think that we probably overestimated the temperatures of

the forming planet (see Chapter 5), in three other simulations with JUPITER, we have fixed the planetary

temperature to maximum of 1000 K, 1500 K and 2000 K, respectively. In these cases, all the time a

circumplanetary disk was formed.

In this Chapter, we first describe the evolution of mass inside the Hill-sphere in the envelope case

in the final phase of the simulation. Here, we also compare the results with those of the corresponding

simulation of FARGOCA. Then, we discuss the simulations where the circumplanetary disks formed.

Finally, we discuss the mass of the subdisks to provide an estimation for observations aiming to detect

the circumplanetary disk the first time.

6.2 Mass evolution inside the Hill-sphere

6.2.1 Envelope Case

In order to derive out accretion rate for the planet, one can define a region around the planet and monitor

the mass enclosed in it during the simulation. In our nominal simulation (see Chapter 5) an envelope

formed around the Jupiter-mass planet filling up the entire Hill-sphere. What is then the radius of the

planet is a tricky question. Therefore, I monitored the enclosed mass within spheres of various radius.

Because the cells are cubic, the sphere is a collection of cubic cells, whose barycenters are within a

desired distance from the planet. If the given cell barycenter is within the radius of the sphere, the entire

mass of the cell is considered (i.e. we do not compute the fraction of the cell’s mass within the sphere).

The smallest sphere was defined based on the smoothing length, i.e. a sphere around the planet whose

radius is the smoothing length: 8.4 × 10−4code units = RHill/82. Next, I kept track of the enclosed mass

within RHill/2. Finally, I of course monitored the mass of the entire Hill-sphere (see Fig. 6.1).

As it was described in the Chapters 3 and 5, the smoothing length – therefore the depth of the

potential-well – was level dependent for levels 0-5 in my simulations. This was done in order to see

correct, fine details in the higher resolution mesh. This can be understood in the following way: the

resolution difference between the coarsest mesh and the finest one is 26 = 64, therefore the smoothing

length should be also decreased by about the same amount. Otherwise, keeping the smoothing length

equal to the value used on the coarsest mesh, the smoothing area would include almost all finer levels,

causing numerical artifact due to the artificially reduced gravity in this region. This way, the flow in the

140

6.2. Mass evolution inside the Hill-sphere Chap.6 Circumplanetary Disk Mass

high-resolution levels would have behaved as if the planet had a mass smaller than 1 Jupiter-mass. The

smoothing of the potential is also smearing out details – e.g. shock-fronts, spiral wake in the CPD –

because of the reduced gravitational potential. When adding the last level (level 6), however, I did not

reduce the smoothing length once again, to avoid possible numerical artifacts due to the change of the

potential and to speed up the computational time. The major numerical artefact I am concerned about

is that changing the smoothing length is an artificial way to enhance accretion. During the smoothing-

tapering procedure, the potential-well is deepened as fast as the user sets that up; this has the conse-

quence of attracting more gas into the innermost cells where we measure the accretion rate. However,

the smoothing-tapering and the smoothing length are user-dependent numerical methods, and therefore

should be avoided when the true accretion rate is to be measured. Obviously, if we impose an accretion

rate (as we do during the smoothing-tapering), this cannot be viewed as the real accretion rate measured

from the simulation, therefore I preferred to leave this out on the finest level.

Figure 6.1. On the last level, the mass evolution of spheres of 1 RHill, 0.5 RHill and within the

smoothing radius (8.4 × 10−4code units = RHill/82).

We can see on Fig. 6.1 that the mass growth of the sphere with radius of the smoothing length (blue

curve) is quickly leveling off. The accretion rate at this flat part is about 6.4 × 10−8MJupyr−1. This is

141


a very low accretion rate, meaning a Jupiter-mass doubling time of 16 million years. Quickly after the

accretion of the innermost cells within the smoothed area have leveled off, the accretion of the entire

Hill-sphere does the same; strongly decreasing from the initial accretion rate of ∼ 4 × 10−5MJupyr−1 to

5.5 × 10−6MJupyr−1 beyond orbit 10 (see red curve on Fig. 6.1). It is also interesting to point out, that

larger volume of sphere accretes more, the envelope seems piling up still.

As we know, the accretion rate is depending on the cooling time. In my simulation where an envelope

formed, the temperatures are very high, peaking above 13,000 K. I made a test about the cooling time, by

simply turning off the hydro module and allow only the radiative module to compute the new temperature.

The observed cooling time was very short (Fig. 6.2), on the order of a fraction of a time step. This of

course means that we need to look for a source of energy, responsible for keeping the temperature high,

so to avoid the accretion of new mass. First, I tested the viscous heating. I set the viscosity to zero,

run the simulation further with both hydro, and radiative modules. Surprisingly, the temperature barely

dropped, which meant that the main heating source is not the viscous heating. Then of course, stellar

irradiation was not used in these simulations, so that cannot be the heating source either. The only heat-

source left was the adiabatic compression (P∇ · v, Eq. 5.8). I created color-maps on this term around the

planet, as shown in Fig. 6.3. Indeed, further tests showed that the main heating source was the adiabatic

compression, overwhelming the viscous heating.

Figure 6.2. Testing the cooling time by eliminating the hydro module and allow only the radiative

module to run further. The cooling this way is extremely fast.

142


Figure 6.3. Color-map of a vertical slice of the adiabatic compression term on the finest level,

P∇ · v. The compressional heating is clearly the largest close to the planet, in the top-middle part

of the figure. The simulation features only the bottom half of the disk below the midplane that is

why the planet is in the top-middle part on this figure.

Due to the very high resolution, just these 15 orbits were achieved during several weeks of computer

time on 100-120 CPUs. This highlights that it is difficult to push this simulation further on the computers

easily available to us today. To understand the accretion rate, more simulations are also needed with

various setups for the accreting planet. How to simulate an “accreting planet” in a realistic way in hydro

simulations is a very complex problem, as it was highlighted by e.g. D’Angelo et al. (2002). Lot of

authors choose to artificially remove some/all of the density in the innermost cells around the planet,

such as in the works of D’Angelo et al. (2002, 2003); Gressel et al. (2013); Ayliffe & Bate (2009a) etc.

We plan to test this and other methods to simulate the accreting planet, but due to the limitations of the

duration of PhD, this work will be done beyond this thesis. The simulation presented here did not use

any artificial mass removal, and thus in that sense corresponding to the so called “non-accreting” models

of D’Angelo et al. (2003) and Paardekooper & Mellema (2008).

6.2.1.1 Comparison with FARGOCA

As we mentioned in Chapter 5, we had a simulation made with the FARGOCA code by Elena Lega with

the same initial parameters than the nominal simulation with JUPITER. But because FARGOCA does not

have nested meshes, the simulation field was gradually and manually reduced to reach similar resolution

than I reached with JUPITER. Finally, due to limitations of the manual nested meshing technique, the

finest resolution of the FARGOCA simulation was half of the resolution of JUPITER’s.

The mass enclosed in various sized spheres around the planet is shown on Fig. 6.4. It can be seen

that the accretion rates are larger than in JUPITER (the mass growing linearly with time at a rate of about

143


3 × 10−5MJupyr−1). However, the simulation is much shorter (only 7.5 orbits, and on this timescale also

the JUPITER simulation was showing accretion at a comparable rate), and there is no feedback between

the nested meshes because of the manual nested meshing technique. This means a continuous mass influx

into the simulation box of FARGOCA that might enhance the accretion rate. The accretion rate of the

smoothed area is almost one order of magnitude higher than in JUPITER on the same resolution. Why

this difference is not entirely clear. It could be that the different history of the simulations is causing it

(e.g. the overall simulation time is different in the JUPITER and FARGOCA simulations, moreover, the

times when a new mesh was added is not the same either). It could also be linked to the fact again that

there is no feedback between the levels in FARGOCA, and the boundaries are damped to the original

hydro fields of the coarsest mesh, causing a continuous mass influx. Moreover the temperatures of the

two simulations are not exactly the same either due to the different energy-equation (i.e. JUPITER solves

for total energy conservation, while FARGOCA solves the internal energy equation).

The entire circumplanetary envelope in FARGOCA seems to be smaller in size than in JUPITER, at

least it is more compact in the inner parts than the envelope in the JUPITER simulation. We see on Fig.

6.4 that the various sized spheres accrete at different rates. The mass doubling times are plotted for the

various spheres on Fig. 6.5. It is clear, the larger is the sphere, the accretion rate is higher, which can

be said about the simulation of JUPITER as well. The fluctuations of the mass curve in the JUPITER

simulation (see Fig. 6.1) were larger than in FARGOCA, which can also be from the fact that FARGOCA

is more diffusive than JUPITER.

It is also interesting to mention that the smoothing function for the potential-well also matters for the

accretion rate and for the simulation in general. JUPITER has epsilon-potential (see e.g. in Chap. 5),

while FARGOCA originally had cubic potential (Kley et al. 2009). Of course, for the above tests, we have

modified FARGOCA to have the same epsilon potential than in JUPITER, with the exact same depths on

the various levels. But we have performed simulations with FARGOCA with the cubic potential as well.

This potential well has a more peaky shape in the cubic potential case, therefore the depth of potential in a

particular level is much deeper than if the potential would have been smoothed with epsilon. This results

in higher densities in the innermost cells, therefore in a higher accretion rate. This test was interesting

for me in realizing how much the accretion rate depends on the smoothing function one uses, and the

smoothing length itself.

6.2.2 Disk Cases

To measure the accretion rate in the disky cases with fixed planetary temperatures was even harder than in

the envelope case. This is due to sinusoidal, very regular variations that can be observed in mass curves

(see Fig. 6.6). Here I plotted again the evolution of the Hill-sphere masses, the masses enclosed into

RHill/2 and the masses enclosed within the smoothing length 8.4×10−4code units = RHill/82 for one fixed

temperature simulation of 2000 K. The other fixed temperature simulations’ curves look similar, but only

the 2000 K could have run sufficiently long enough to measure the accretion. Clearly, this sinusoidal

variations overcast the accretion trend in the Hill-sphere and half-Hill-sphere mass curves, therefore

the accretion rate could not be clearly measured. However, on the smoothed area, we clearly see the

accretion, namely 6.5 × 10−7Mjup/yr. Note that this accretion rate is one order of magnitude higher than

the corresponding accretion rate in the envelope case (see Fig. 6.1).

The sinusoidal variations of the masses are present no matter what sphere we consider. The periodicity

is the same in the various spheres, however the amplitude of the variations are larger as we study a larger

sphere of mass. The period of the variations are about 1 orbit of the planet, but this variations were

observed in previous levels as well, with different periods (less than an orbit) and smaller amplitudes.

144


Figure 6.4. On the last level with FARGOCA, the mass of a given sphere is plotted in time. We

see that larger volume accrete more, so the envelope keeps piling up.

145


Figure 6.5. Mass doubling times for the simulation with FARGOCA on the last level. Spheres

around the planet with various sizes are plotted. Even though the simulations only feature the

bottom half of the disk below the midplane, the values above already multiplied by two to account

for the entire disk.

146

6.3. Mass of the Circumplanetary Disk Chap.6 Circumplanetary Disk Mass

The periodicity in the 1000 K and 2000 K planetary temperature simulations seem to be the same, but

with shifted phase. The origin of these sinusoidal variations of the enclosed masses are not clear to us.

Looking at the distribution of gas near the planet as a function of time, I see that the circumplanetary disk

is periodically truncated to a small radius of fraction of RHill, and the period of these truncation episodes

is the period of oscillation of the mass observed in Fig. 6.6. Moreover, the gap edges seems to fluctuate

a bit as well, showing signs of eccentricity. The cause of these truncation episodes, however, remains to

be understood.

The results with FARGOCA are quite different. Although the FARGOCA simulations were shorter

(only a few orbits) the disk appears to accrete mass at a rate of 1.5 × 10−5Mjup/yr and does not show

the large oscillations seen with JUPITER, because the base mesh is frozen, so there is no feedback with

the gap edges. The difference between the accretion rates of FARGOCA between the envelope and disk

cases are more than an order of magnitude.

Figure 6.6. Left panel: the mass evolution of spheres of 1 RHill, 0.5 RHill and within the smoothing

radius (8.4× 10−4code units = RHill/82) on the last level of refinement of the simulation with fixed

planetary temperature = 2000 K. Right panel: zoom-in of the mass curve of the smallest sphere

with a fitted line to measure accretion rate.

6.3 Mass of the Circumplanetary Disk

To measure the mass of the CPD is interesting for observational campaigns aiming to observe the subdisk

in the future. First, of course the borders of the subdisk should be defined. However, determining the

borders of the CPD is a bit arbitrary. There are fluctuations among the output files damped at various times

on every hydro quantities. After examining the different hydro fields and derived quantities (including

the velocities and normalized angular momentum fields), I have decided to define the CPD boundaries

147


Figure 6.7. On the left panel, the masses of Hill-spheres (thick lines) and CPDs (thin lines)

in the different simulations, as labeled. The last four orbits of each simulation is shown, the

masses reached steady states. On the right panel, I show an example for the density map in the

Tp = 2000 K simulation where the isodensity defining the CPD borders is over-plotted. The radius

of the CPD is ∼ 0.4RHill.

based on isodensity profiles. A density threshold was chosen to be 0.005 in code units (see right panel

of Fig. 6.7), leading to a seemingly nice disk shape. I integrated the mass of the cells which have at

least this density, but which are outside of the smoothed potential region. Even though setting up an

isodensity for the definition of the CPD borders is arbitrary, the position of the outer boundary of the

CPD does not affect much the total CPD mass, as my tests on that revealed. This definition on the

density at ≥ 0.005code units sets the radius of the CPD at about 0.4RHill. This value is in good agreement

what other authors have found between 0.3-0.5 RHill (e.g. Tanigawa et al. 2012; Ayliffe & Bate 2009b;

D’Angelo & Podolak 2015).

Table 6.1 contains the measured (averaged) CPD masses, Hill-sphere masses, and the mass of the

central region within the smoothing length. I measured these quantities over the last 3 orbits of the

simulations (see left panel Fig. 6.7 for the CPD and Hill-masses), when quasi steady state have been

reached (when the masses did not grow any longer). In the fixed temperature simulations, the CPD

masses are roughly equal, ∼ 10−3MJup, while the Hill-sphere masses ∼ 2.6 × 10−3MJup. Comparing the

CPD-mass of the previous isothermal simulations of Szulagyi et al. (2014), the radiative simulation lead

to a one order of magnitude heavier subdisk. This is due to the fact that in the radiative simulation, the

pressure is so large in the vicinity of the planet that the mass cannot fall onto the innermost cells that

easily, as it could in the isothermal simulation. In the isothermal simulation, the central mass became

much heavier, and due to the fast accretion onto the planet, the CPD became anemic. Therefore, the

radiative simulation lead to more realistic estimates for the CPD-mass. It is interesting to point out

that the Hill-sphere versus CPD mass ratio in all fixed temperature simulations is about 2.6 in all three

148


Table 6.1. Masses of CPD, Hill-sphere in our simulations

Central Mass [MJup] CPD mass [MJup] Hill-sphere mass [MJup]

Tp=1000K (1.23 ± 0.09) × 10−4 (9.8 ± 1.9) × 10−4 (2.63 ± 0.22) × 10−3

Tp=1500K (1.62 ± 0.09) × 10−4 (1.03 ± 0.2) × 10−3 (2.75 ± 0.20) × 10−3

Tp=2000K (2.1 ± 0.15) × 10−4 (8.57 ± 0.16) × 10−4 (2.58 ± 0.23) × 10−3

Nominal (8.46 ± 0.02) × 10−4 – (5.48 ± 0.17) × 10−3

cases. Moreover, the mass of the Hill-sphere is almost double in the nominal simulation, than in the fixed

temperature simulations.

149


150

Own Publications on this Thesis Work

Accretion of Jupiter-mass Planets in the Limit of Vanishing Viscosity

Szulagyi, J., Morbidelli, A., Crida, A., & Masset, F.

The Astrophysical Journal, Volume 782, p. 65 (2014)

Chapter 3

Meridional circulation of gas into gaps opened by giant planets in

three-dimensional low-viscosity disks

Morbidelli, A., Szulagyi, J., Crida, A., Lega, E., Bitsch, B., Tanigawa, T., Kanagawa, K.

Icarus, Volume 232, p. 266 (2014)

Chapter 4

Circumplanetary Disk or Circumplanetary Envelope?

Szulagyi, J., Masset, F., Lega, E., Crida, A., Morbidelli, A., Guillot, T.

submitted to Monthly Notices of the Royal Astronomical Society (2015)

Chapter 5

151

Own Publications on this Thesis Work

152

Own Publications Outside this Thesis Work

Planet heating prevents inward migration of planetary cores

Pablo Benıtez-Llambay, Frederic Masset, Gloria Koenigsberger, Judit Szulagyi

Planetary systems are born in the disks of gas, dust and rocky fragments that surround newly formed

stars. Solid content assembles into ever-larger rocky fragments that eventually become planetary em-

bryos. These then continue their growth by accreting leftover material in the disk. Concurrently, tidal

effects in the disk cause a radial drift in the embryo orbits, a process known as migration. Fast inward

migration is predicted by theory for embryos smaller than three to five Earthmasses. With only inward

migration, these embryos can only rarely become giant planets located at Earth’s distance from the Sun

and beyond, in contrast with observations. Here we report that asymmetries in the temperature rise asso-

ciated with accreting infalling material produce a force (which gives rise to an effect that we call “heating

torque”) that counteracts inward migration. This provides a channel for the formation of giant planets

and also explains the strong planet-metallicity correlation found between the incidence of giant planets

and the heavy-element abundance of the host stars.

Nature, Volume 520, Issue 7545, pp. 63-65 (2015)

153


Planet formation signposts: observability of circumplanetary disks

via gas kinematics

Perez, Sebastian, Dunhill, Alex, Casassus, Simon, Roman, Pablo, Szulagyi, Judit, Flores, Christian,

Marino, Sebastian, Montesinos, Matias

The identification of on-going planet formation requires the finest angular resolutions and deepest

sensitivities in observations inspired by state-of-the-art numerical simulations. Hydrodynamic simula-

tions of planet-disk interactions predict the formation of circumplanetary disks (CPDs) around accreting

planetary cores. These CPDs have eluded unequivocal detection – their identification requires predic-

tions in CPD tracers. In this work, we aim to assess the observability of embedded CPDs with ALMA as

features imprinted in the gas kinematics. We use 3D Smooth Particle Hydrodynamic (SPH) simulations

of CPDs around 1 and 5 MJup planets at large stellocentric radii, in locally isothermal and adiabatic disks.

The simulations are then connected with 3D radiative transfer for predictions in CO isotopologues. Ob-

servability is assessed by corrupting with realistic long baseline phase noise extracted from the recent HL

Tau ALMA data. We find that the presence of a CPD produces distinct signposts: 1) compact emission

separated in velocity from the overall circumstellar disk’s Keplerian pattern, 2) a strong impact on the

velocity pattern when the Doppler shifted line emission sweeps across the CPD location, and 3) a local

increase in the velocity dispersion. We test our predictions with a simulation tailored for HD 100546

–which has a reported protoplanet candidate. We find that the CPDs are detectable in all 3 signposts with

ALMA Cycle 3 capabilities for both 1 and 5 MJup protoplanets, when embedded in an isothermal disk.

Accepted to The Astrophysical Journal Letters (2015); arXiv:1505.06808

154


Outwards migration for planets in stellar irradiated 3D discs

Lega, E., Morbidelli, A., Bitsch, B., Crida, A., Szulagyi, J.

For the very first time we present 3D simulations of planets embedded in stellar irradiated discs. It

is well known that thermal effects could reverse the direction of planetary migration from inwards to

outwards, potentially saving planets in the inner, optically thick parts of the protoplanetary disc. When

considering stellar irradiation in addition to viscous friction as a source of heating, the outer disc changes

from a shadowed to a flared structure. Using a suited analytical formula it has been shown that in the

flared part of the disc the migration is inwards; planets can migrate outwards only in shadowed regions

of the disc, because the radial gradient of entropy is stronger there. In order to confirm this result nu-

merically, we have computed the total torque acting on planets held on fixed orbits embedded in stellar

irradiated 3D discs using the hydrodynamical code FARGOCA. We find qualitatively good agreement

between the total torque obtained with numerical simulations and the one predicted by the analytical

formula. For large masses (>20 Earth masses) we find quantitative agreement, and we obtain outwards

migration regions for planets up to 60 Earth masses in the early stages of accretional discs. We find

nevertheless that the agreement with the analytic formula is quite fortuitous because the formula under-

estimates the size of the horseshoe region and therefore overestimates the amount of saturation of the

corotation torque; this error is compensated by imperfect estimates of other terms, most likely for the

cooling rate.

Monthly Notices of the Royal Astronomical Society, Volume 452, p. 1717 (2015)

155


A Resolved Debris Disk around the Candidate Planet-hosting Star

HD 95086

Moor, A., Abraham, P., Kospal, A., Szabo, Gy. M., Apai, D., Balog, Z., Csengeri, T., Grady, C.,

Henning, Th., Juhasz, A., Kiss, Cs., Pascucci, I., Szulagyi, J., Vavrek, R.

Recently, a new planet candidate was discovered on direct images around the young (10-17 Myr)

A-type star HD 95086. The strong infrared excess of the system indicates that, similar to HR8799, beta

Pic, and Fomalhaut, the star harbors a circumstellar disk. Aiming to study the structure and gas content

of the HD 95086 disk, and to investigate its possible interaction with the newly discovered planet, here

we present new optical, infrared, and millimeter observations. We detected no CO emission, excluding

the possibility of an evolved gaseous primordial disk. Simple blackbody modeling of the spectral energy

distribution suggests the presence of two spatially separate dust belts at radial distances of 6 and 64 AU.

Our resolved images obtained with the Herschel Space Observatory reveal a characteristic disk size of

∼6.”0 × 5.”4 (540 × 490 AU) and disk inclination of ∼ 25. Assuming the same inclination for the planet

candidate’s orbit, its reprojected radial distance from the star is 62 AU, very close to the blackbody radius

of the outer cold dust ring. The structure of the planetary system at HD 95086 resembles the one around

HR8799. Both systems harbor a warm inner dust belt and a broad colder outer disk and giant planet(s)

between the two dusty regions. Modeling implies that the candidate planet can dynamically excite the

motion of planetesimals even out to 270 AU via their secular perturbation if its orbital eccentricity is

larger than about 0.4. Our analysis adds a new example to the three known systems where directly

imaged planet(s) and debris disks coexist.

The Astrophysical Journal Letters, Volume 775, Issue 2, article id. L51, 6 pp. (2013)

156


Unveiling new members in five nearby young moving groups

Moor, A., Szabo, Gy. M., Kiss, L. L., Kiss, Cs., Abraham, P., Szulagyi, J., Kospal, A., Szalai, T.

In the past decade many kinematic groups of young stars (<100 Myr) were discovered in the solar

neighbourhood. Since the most interesting period of planet formation overlaps with the age of these

groups, their well dated members are attractive targets for exoplanet searches by direct imaging. We

combined astrometric, photometric and X-ray data, and applied strict selection criteria to explore the

stellar content of five nearby moving groups. We identified more than 100 potential new candidate

members in the beta Pic moving group, and in the Tucana-Horologium, Columba, Carina and Argus

associations. In order to further assess and confirm their membership status, we analysed radial velocity

data and lithium equivalent widths extracted from high-resolution spectra of 54 candidate stars. We

identified 35 new probable/possible young moving group members: four in the beta Pic moving group,

11 in the Columba association, 16 in the Carina association and four in the Argus association. We found

serendipitously a new AB Dor moving group member as well. For four Columba systems Hipparcos-

based parallaxes have already been available and as they are consistent with the predicted kinematic

parallaxes, they can be considered as secure new members.

Monthly Notices of the Royal Astronomical Society, Volume 435, Issue 2, p.1376-1388 (2013)

157


The secondary eclipses of WASP-19b as seen by the ASTEP 400 tele-

scope from Antarctica

Abe, L., Goncalves, I., Agabi, A., Alapini, A., Guillot, T., Mekarnia, D., Rivet, J.-P., Schmider, F.-

X., Crouzet, N., Fortney, J., Pont, F., Barbieri, M., Daban, J.-B., Fantei-Caujolle, Y., Gouvret, C.,

Bresson, Y., Roussel, A., Bonhomme, S., Robini, A., Dugue, M., Bondoux, E., Peron, S., Petit, P.-Y.,

Szulagyi, J., Fruth, T., Erikson, A., Rauer, H., Fressin, F., Valbousquet, F., Blanc, P.-E., Le van Suu,

A., Aigrain, S.

Aims: The Antarctica Search for Transiting ExoPlanets (ASTEP) program was originally aimed at

probing the quality of the Dome C, Antarctica for the discovery and characterization of exoplanets by

photometry. In the first year of operation of the 40 cm ASTEP 400 telescope (austral winter 2010),

we targeted the known transiting planet WASP-19b in order to try to detect its secondary transits in the

visible. This is made possible by the excellent sub-millimagnitude precision of the binned data. Methods:

The WASP-19 system was observed during 24 nights in May 2010. Once brought back from Antarctica,

the data were processed using various methods, and the best results were with an implementation of the

optimal image subtraction (OIS) algorithm. Results: The photometric variability level due to starspots is

about 1.8% (peak-to-peak), in line with the SuperWASP data from 2007 (1.4%) and higher than in 2008

(0.07%). We find a rotation period of WASP-19 of 10.7 ± 0.5 days, in agreement with the SuperWASP

determination of 10.5 ± 0.2 days. Theoretical models show that this can only be explained if tidal

dissipation in the star is weak, i.e. the tidal dissipation factor Q’* ¿ 3×107. Separately, we find evidence

of a secondary eclipse of depth 390 ± 190 ppm with a 2.0 sigma significance, a phase that is consistent

with a circular orbit and a 3% false positive probability. Given the wavelength range of the observations

(420 to 950 nm), the secondary transit depth translates into a day-side brightness temperature of 2690+150−220

K, in line with measurements in the z’ and K bands. The day-side emission observed in the visible could

be due either to thermal emission of an extremely hot day side with very little redistribution of heat to

the night side or to direct reflection of stellar light with a maximum geometrical albedo Ag = 0.27 ±0.13. We also report a low-frequency oscillation in phase at the planet orbital period, but with a lower

limit amplitude that could not be attributed to the planet phase alone and that was possibly contaminated

with residual lightcurve trends. Conclusions: This first evidence of a secondary eclipse in the visible

from the ground demonstrates the high potential of Dome C, Antarctica, for continuous photometric

observations of stars with exoplanets. These continuous observations are required to understand star-

planet interactions and the dynamical properties of exoplanetary atmospheres.

Monthly Notices of the Royal Astronomical Society, Volume 435, Issue 2, p.1376-1388 (2013)

158


Transiting planet candidates with ASTEP400 at Dome C, Antarctica

D. Mekarnia, T. Guillot, I. Goncalves, L. Abe, A. Agabi, J.-P. Rivet, F.-X. Schmider, N. Crouzet,

T. Fruth, M. Barbieri, D.D.R. Bayliss, G. Zhou, E. Aristidi, J. Szulagyi, J.-B. Daban, Y. Fantei-

Caujolle, C. Gouvret, A. Erikson, H. Rauer, F. Bouchy, J. Gerakis and G. Bouchez

ASTEP400, the main instrument of the ASTEP (Antarctica Search for Transiting ExoPlanets) pro-

gramme, is a 40-cm telescope, designed to withstand the harsh conditions in Antarctica, achieving a pho-

tometric accuracy of a fraction of milli-manitude on hourly timescales for planet-hosting southern bright

(V ∼ 12mag) stars. We review the performances of this instrument, describe its operating conditions,

and present results from the analysis of observations obtained during its first three years (2010-2012) of

operation. We observed a total of 22 stellar fields (1 × 1 FoV) during this period. Each field, in which

we measured stars up to magnitude R=18, was observed continuously during ∼ 7 to ∼ 30 days. More than

200000 frames were recorded and 310000 stars analysed. Data were processed using an implementation

of the optimal image subtraction (OIS) photometry algorithm.We found 43 planetary transit candidates

and about 1900 new variables stars inluding 702 eclipsing binaries. Twenty of these candidates were

observed using spectroscopic follow-up. Four of these targets have so far been classified as good planet

candidates for which more accurate well sampled radial velocity observations are required to confirm

their planetary nature and to perform their complete characterization. In addition, from candidates for

which spectroscopic measurements have not been performed, we found 9 targets as suspected good planet

candidates that require future radial velocity follow-up. We present here all of these candidates along with

their detailed properties derived from transit observations as well as from follow-up observations. Our re-

sults demonstrate that extremely stable and precise visible photometry and near-continuous observations

are achievable from the Concordia station at Dome C in Antarctica.

Submitted to Monthly Notices of the Royal Astronomical Society (2015)

159


160

Conclusion

6.4 Conclusion in English

In this thesis the accretion of giant planets and the role of the circumplanetary disk in the accretion

process of a giant planet are discussed. Both isothermal and radiative hydrodynamic, 3D simulations

were performed to study the circumplanetary disk properties and their effects on the planet accretion rate.

The focus was on a Jupiter-mass planet embedded in a Minimum Mass Solar Nebula circumstellar disk.

To perform such simulations, we used the powerful technique of nested meshes, where high resolution

grids are added close to the planet in order to achieve sufficient resolution to study the flow in the planet

vicinity, while simulating the whole circumstellar disk.

Due to the heavy numerical simulations, this thesis is on the boundary between astrophysics, planetary

science and computer science. The main results of the thesis are the followings:

• In Chapter 2 the enormous task of further developing the JUPITER hydrodynamic code was de-

scribed. Together with the original developer of the code, we have added new modules to be able to

achieve realistic temperature estimates. For this task, we added the energy equation, and a radiative

module. The latter deals with cooling due to radiation, heating due to stellar irradiation and viscous

heating. The simplified radiative transfer scheme of this module includes realistic opacities. Even

though the hydrodynamic simulations are made on the circumstellar disk gas component only, the

dust component also plays a role to determine the temperatures through the dust opacities.

• Chapter 3 summarizes our findings with locally isothermal equation-of-state, and vanishing viscos-

ity. The latter assumption was chosen, because planets are thought to form in dead zones, where

the viscosity is negligible. We find that even when there is no viscous angular momentum transport

in the subdisk, the stellar torque drives accretion to the planet. Even without prescribed viscosity,

we found that Jupiter’s mass-doubling time was ∼ 104 years, but we proved that this high accretion

rate is due to resolution-dependent numerical viscosity. To avoid this effect of numerical viscosity,

with semi-analytical methods we derived a realistic accretion rate from stellar torque calculations:

2.5 × 10−6MJ/yr. This accretion rate means that a Jupiter-mass planet would build up in 400,000

years, on a timescale that is comparable to the removal timescale of the circumstellar disk gas due

to photoevaporation. Furthermore, we also showed that the 90% of the accreted gas is coming from

a vertical inflow through the planetary gap, which feeds the subdisk and planet directly as well.

• Chapter 4 describes how the accretion mechanism is happening onto the planet. We have found a

meridional circulation happening between the circumstellar-, and circumplanetary disks. We have

shown that the vertical influx reaching the circumplanetary disk is arising from the top layers of

the circumstellar disk. In these layers, the planetary torque is weaker, hence the viscous effects

161

6.4. Conclusion in English Conclusion

tend to close the gap. As the gas enters the gap, it is pulled by the gravity of the planet, free-falling

directly onto the circumplanetary disk and the planet itself. Inside the subdisk, the gas reaches the

planet flowing in the top layers of the circumplanetary disk. The rest of gas, which is not accreted,

is leaving the subdisk through the mid-plane regions and flowing back to the circumstellar disk. To

maintain the vertical hydrostatic equilibrium of the circumstellar disk, the gas is rising again from

the midplane towards the upper layers of the disk, which feeds the meridional circulation – this is

how the loop closes (see Fig. 6.9).

Figure 6.8. Meridional Circulation schematized with arrows between the circumplanetary disk

(in the center) and the circumstellar disk. The colormap shows the density distribution, with red

colors the higher densities, with blue/black colors the lower densities. The disks are shown through

a vertical slice at the planet’s location.

• In Chapter 5 the radiative simulations, performed with the new version of the code obtained during

this thesis, are discussed. Here we focus on the qualitative description of the circumplanetary

disk and envelope formed around the Jupiter-mass planet. We have shown that even a Jupiter-

mass planet could form a circumplanetary envelope, like the small mass planets, if the planetary

temperature is high enough. In our nominal simulation we let the temperature evolve according to

the radiative module even on the planet, reaching more than 10,000 K. Because the temperatures of

162

6.4. Conclusion in English Conclusion

young, forming, embedded planets are highly unknown, we have carried out simulations were the

planetary temperature was fixed to a maximal value of 1000 K, 1500 K and 2000 K, respectively.

These three simulations all produced circumplanetary disks around the Jupiter-mass planet. This

shows that the planetary temperature, and the planet’s cooling history greatly determines whether a

circumplanetary envelope or a disk forms. Moreover, the planetary temperature has a great impact

on the properties (e.g. scaleheight of the disk, rotational velocity, intensity of the spiral wake in the

disk) of the circumplanetary material.

• Chapter 6 discusses the preliminary results on the accretion rates and on the circumplanetary disk

mass from the radiative simulations of the previous Chapter. We have found very low accretion

rates that we do not fully understand. Clearly, further investigations are needed to characterize

and to understand better the results. We have measured the subdisk mass from the three fixed

temperature simulations, getting ∼ 10−3MJup disk mass in all three cases. We have also shown that

the entire Hill-sphere mass turned out to be 2.6× higher than this value.

Overall, this thesis resulted in 2 peer reviewed publications, and one more submitted. Due to collab-

orations, I have co-authored 7 other publications – including a Nature paper – on planet formation and

migration, exoplanet detection, infrared astronomy of circumstellar disks. This adds up into a total of 10

publications over the 3 years.

163

6.5. Conclusion en franccais Conclusion

6.5 Conclusion en franccais

Dans cette these, l’accretion des planetes geantes et le role du disque circum-planetaire sont discutes.

Des simulations hydrodynamiques 3D isothermes mais aussi radiatives ont ete effectuees pour etudier

les proprietes sur disque circum-planetaire et leurs effets sur le taux d’accretion de la planete. Le cas

d’ecole est une planete de la masse de Jupiter, dans une Nebuleuse Solaire de Masse Minimale. Pour

realiser ces simulations, nous avons utilise la technique efficace des grilles emboitees, ou des grilles a

haute resolution sont ajoutees autour de la planete, pour y atteindre une resolution suffisante pour etudier

le flot du gaz au voisinage de la planete.

Du fait de ces simulations numeriques lourdes, cette these se place a l’interface entre l’astrophysique,

la planetologie, et l’informatique. Les resultats principaux de cette these sont les suivants :

• Dans le chapitre 2 est decrite la tache enorme de developper le code hydrodynamique JUPITER.

En collaboration etroite avec le createur du code, nous avons ajoute de nouveaux modules dans

le but d’obtenir des valeurs de la temperature realistes. Pour cela, nous avons ajoute l’equation

de l’energie et un module radiatif. Ce dernier traite le refroidissement du a l’emission de rayon-

nement, le chauffage du a l’irradiation par l’etoile, et le chauffage visqueux. Le schema simplifie

de transfer radiatif de ce module prend en compte des opacites realistes. Bien que les simula-

tions hydrodynamiques traitent uniquement de la composante gaz du disque circumstellaire, la

composante poussiere a un role tres important pour definir la temperature, via l’opacite ; nous la

prenons en compte en definissant un rapport gaz/poussiere constant et uniforme.

• Le chapitre 3 rassemble nos resultats obtenus avec une equation d’etat localement isotherme, et

une viscosite nulle. Cette derniere hypothese a ete choisie parce que les planete geantes se forment

probablement dans la “zone morte” du disque, ou la viscosite est negligeable. Nous trouvons que

meme en l’absence de transport de moment cinetique dans le sous-disque, le couple exerce par

l’etoile cause de l’accretion sur la planete. Meme sans viscosite imposee dans le code, nous trou-

vons que Jupiter doublerait sa masse en ∼ 104 ans, mais nous avons prouve que ce taux d’accretion

etait en fait cause par la viscosite numerique, qui depend de la resoluion. Pour eviter cet ef-

fet, nous avons calcule semi-analytiquement un taux d’accretion a partir du couple de l’etoile :

2.5 × 10−6MJ/yr. Ce taux d’accretion signifie qu’une planete de la masse de Jupiter se formerait

en 400 000 ans, sur un laps de temps comparable a celui de la dissipation du disque par photo-

evaporation. De plus, nous montrons que 90% du gaz accrete provient d’un afflux vertical dans le

sillon ouvert par la planete, qui alimente le sous-disque et aussi la planete directement.

• Le chapitre 4 le mecanisme d’accretion sur la planete. Nous avons trouve une circulation meridienne

entre les disques circumstellaire et circumplanetaire. Nous avons montre que le flot vertical qui at-

teind the disque circumplanetaire provient des couches superieures du disque circumstellaire. Dans

ces couches eloignees de la planete et qui en sentent donc moins le couple, les effets de la viscosite

tendent a refermer le sillon ouvert par la planete ; quand le gaz penetre a l’interieur du sillon, il

est attire vers le bas par la gravite de la planete et tombe en chute libre directement sur le disque

circum-planetaire et sur la planete elle-meme. Au sein du sous-disque, le gaz atteint la planete via

les couches superieures du disque circumplanetaire. Le reste du gaz, qui n’est pas accrete, quitte le

sous-disque via le plan median, et reflue vers le disque circumstellaire. Pour maintenir l’equilibre

hydrostatique du disque circumstellaire, le gaz remonte du plan median les couches superieures du

disque, ce qui clot la boucle de la circulation meridienne (voir Fig. 6.9).

164


Figure 6.9. La circulation meridienne schematisee avec des fleches, entre le disque circum-

planetaire (au centre) et le disque circumstellaire. Arriere plan : carte de densite selon un plan

de coupe perpendiculaire a celui de l’orbite de la planete, passant par l’etoile et la planete (situee

au centre de l’image).

• Dans le chapitre 5, les simulations avec equation d’etat radiative, realisees avec la nouvelle version

du code obtenue durant cette these, sont pr’esentees. Nous nous focalisons ici sur une description

qualitative du disque et de l’envelope circumplanetaires qui se forment autour de la planete de

masse jovienne. Nous avons montre que meme une planete de la masse de Jupiter peut former une

envelope circumplanetaire, comme els planetes de petite masse, si la temperature de la planete est

assez grande. Dans notre simulation de reference, la temperature evolue librement, determinee par

le module radiatif y compris sur la planete, et y atteint lus de 10 000 K. Comme la temperature

des jeunes planetes en formation est tres incertaine, nous avons aussi mene des simulations ou la

temperature de la planete etait limitee a une valeur maximale de 1000, 1500, et 2000 K. Ces trois

simulations ont produit des disques circumplanetaires autour de la meme planete de la masse de

Jupiter. Ceci montre que la temperature de la planete, et donc l’histoire thermique de la planete

determine pour une grande part si un disque ou une enveloppe se forme autour de la planete. De

plus, la temperature de la planete a une forte influence sur les proprietes de la matiere constituant

le disque circumplanetaire.

165

• Le chapitre 6 presente les resultats preliminaires sur le taux d’accretion et la masse du disque

circumplanetaire, a partir des simulations radiatives du cahpitre precedent. Nous avons trouve des

taux d’accretion tres bas, que nous ne comprenons pas encore tout-a-fait. De plus amples analyses

sont necessaires pour caracteriser et comprendre mieux ces resultats. Nous avons mesure la masse

du sous-disque dans les trois simulations a temperature prescrite, et avons trouve ∼ 10−3MJup dans

tous les cas. Nous avons aussi montre que la masse contenue dans la sphere de Hill entiere est

environ 2,6 fois superieure.

Au final, cette these aura resulte en 3 articles dans des journaux a comite de lecture (dont un soumis en

septembre 2015). Dans le cadre d’autres collaborations, je suis aussi co-auteur de 7 autres articles – dont

un dans Nature – sur la formation et la migration planetaire, la detection d’exoplanetes, ou l’astronomie

infra-rouge de disques circumstellaires. Cela ajoute en un total de 10 publications au cours des 3 ans.


168

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