Numerical simulations of disk planet interactions [Elektronische Ressource] / vorgelegt von Gennaro D'Angelo

Sujets

• Physik
• Astronomie

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• Publié par: eberhard_karls_universitat_tubingen
• Publié le: 01 janvier 2003
• Nombre de lectures: 10
• Langue: English
• Poids de l'ouvrage: 6 Mo

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NUMERICAL SIMULATIONS
OF DISK-PLANET INTERACTIONS
DISSERTATION
ZUR ERLANGUNG DES GRADES EINES DOKTORS
DER NATURWISSENSCHAFTEN
¨ ¨DER FAKULTAT FUR MATHEMATIK UND PHYSIK
¨ ¨DER EBERHARD-KARLS-UNIVERSITAT ZU TUBINGEN
VORGELEGT VON
GENNARO D’ANGELO
AUS POZZUOLI (ITALIA)
2003Tag der mundlichen¨ Prufung:¨ 4. Juni 2003
Dekan: Prof. Dr. Herbert Muther¨
1. Berichterstatter: Prof. Dr. Wilhelm Kley
2. Prof. Dr. Klaus WernerNOTICE
Persons attempting to find a motive
in this narrative will be prosecuted;
persons attempting to find a moral in it
will be banished; persons attempting
to find a plot in it will be shot.
MARK TWAIN,
THE ADVENTURES OF HUCKLEBERRY FINN,
1885Thesis advisors Author
Prof. Willy Kley
Prof. Thomas Henning Gennaro D’Angelo
NUMERICAL SIMULATIONS OF DISK-PLANET INTERACTIONS
ABSTRACT
The aim of this dissertation is to study the dynamical interactions occurring between a
forming planet and its surrounding protostellar environment. This task is accomplished by
means of both two- and three-dimensional numerical simulations. In order to render the
proper development of the work, results from such calculations are presented according
to the same temporal order they were achieved.
The first part of my research plan concerned global simulations in three dimensions.
These were intended to investigate the large-scale effects caused by a Jupiter-size body
still in the process of accreting matter from its neighborhood. For the first time, this prob-
lem was tackled in a three-dimensional space. The computations are global in the sense
that they embrace a whole portion of circumstellar disk, extending over a radial distance
interval of eleven astronomical units. For computational reasons, we relied on a local-
isothermal equation of state to describe the thermal properties of disk material. Simu-
lations show that, despite a density gap forms along the orbital path, Jupiter-mass pro-
toplanets still accrete at a rate on the order of 0.01 Earth’s masses per year when they are
embedded in a disk whose mass, inside twenty-six astronomical units, is 0.01 solar masses.
In the same conditions, the migration time scale due to gravitational torques by the disk is
around one hundred thousand years. These outcomes are in good agreement with previ-
ous assessments obtained from two-dimensional calculations of infinitesimally thin disks
as well as from linearized analytical theories of disk-planet interaction.
The global approach is the most rigorous way of treating planets in disks because it
avoids making simplified assumptions on the propagation of the perturbations induced
by the embedded body. Yet, this approach usually prevents from attaining numerical res-
olutions necessary to inquire into the local effects of disk-planet interactions and to handle
those arising from Earth-mass objects. The second part of my work was dedicated to over-
come this restriction by employing a nested-grid technique within the frame of the two-
dimensional approximation. The method allows to perform global simulations of planets
orbiting in disks and, at the same time, to resolve in great detail the dynamics of the flow
inside the Roche lobe of both massive and low-mass planets. Therefore, it was applied to
planetary masses ranging from one Jupiter-mass to one Earth-mass. In each case, the high
resolution supplied by the nested-grid technique permits an evaluation of the torques, re-
sulting from short and very short range gravitational interactions, more reliable than the
one previously estimated with the aid of numerical methods. Likewise, the mass flow onto
the planet is computed in a more accurate fashion. Resulting migration time scales are in
the range from roughly twenty thousand years, for intermediate mass planets, to a million
years, for very low as well as high-mass planets. Growth time scales depend strongly onABSTRACT v
the protoplanet’s mass. Above 64 Earth-masses, this time scale increases as the 4/3-power
of the planet’s mass. Otherwise it raises as the 2/3-power, occasionally yielding short
lengths of time because of the two-dimensional geometry. Circumplanetary disks form
inside of the Roche lobe of Jupiter-size secondaries. Its azimuthally-averaged rotational
velocity is nearly Keplerian, though it becomes sub-Keplerian as the mass of the perturber
is decreased. In contrast, a hydrostatic envelope builds up around a one Earth-mass object.
As a natural evolution, the nested-grid strategy was implemented in three dimen-
sions. In order to evaluate the consequences of the flat geometry on the local flow structure
around planets, simulations were carried out to investigate a range of planetary masses
spanning from 1.5 Earth’s masses to one Jupiter’s mass. Furthermore, in such calculations
protoplanets were modeled as extended structure and their envelopes were taken into ac-
count through physically realistic gravitational potentials of forming planets. Outcomes
show that migration rates are relatively constant when perturbing masses lie above ap-
proximately a tenth of the Jupiter’s mass, as prescribed by Type II migration regime. In a
range between seven and fifteen Earth’s masses, it is found a dependency of the migration
speed on the planetary mass that yields time scales considerably longer than those pre-
dicted by linear analytical theories. Type I migration regime is well reproduced outside of
such mass interval. The growth time scale is minimum around twenty Earth-masses, but
it rapidly increases for both smaller and larger mass values. With respect to accretion and
migration rates, significant differences between two- and three-dimensional calculations
are found in particular for objects with masses smaller than ten Earth-masses. The flow
inside the Roche lobe of the planet is rather complex, generating spiral perturbations in
the disk midplane and vertical shocks in the meridional direction. Recirculation is also
observed in many instances.
The final part of this work was dedicated to the simulation of non-local isothermal
(i.e., radiative) models. Hence, with such calculations the locally isothermal hypothesis
was relaxed and for the first time the full thermo-dynamics evolution of the system could
be modeled. Since the complexity of the problem does not allow a detailed description of
all the energy transport mechanisms, we use a simplified but physically significant form
of the energy equation, by restricting to two-dimensional computations. Different temper-
ature regimes are examined, according to the magnitude of the fluid kinematic viscosity.
The gap structure was found to depend on the viscosity regime, and only cold environ-
ments offer the right conditions for a wide and deep gap to be carved in. The temperature
profile inside the circumplanetary disk falls off as the inverse of the distance from the
planet. Clockwise rotation is established around low-mass non-accreting planets, because
of large pressure gradients. As for migration and accretion, estimates are generally on the
same order of magnitude as those acquired with the aid of local isothermal models. Since
the gap is generally filled in the high-viscosity case, Type I migration regime might extend
to larger planetary masses.CONTENTS
NOTICE ........................................... ii
ABSTRACT .. iv
TABLE OF CONTENTS ................................... vi
LIST OF FIGURES .... ix
LIST OF TABLES ...................................... xi
ACKNOWLEDGMENTS. xiii
DEDICATION ........................................ xv
1E XTRASOLAR PLANETS:O BSERVATIONS,S TATISTICS, AND THEORY 3
1.1 Introduction . . . .................................. 3
1.2 Detection Techniques . . .. 5
1.2.1 Doppler Technique ............................. 5
1.2.2 Astrometric Detections..... 6
1.2.3 Photometric Technique....................... 7
1.2.4 Gravitational Microlensing .. 8
1.2.5 Direct Imaging . . ............................. 8
1.2.6 Circumstellar Disks. 9
1.3 Observations of Extrasolar Planets . ....................... 9
1.4 Statistics of Extrasolar Planets ..... 12
1.4.1 Planetary Mass Function . . . ....................... 12
1.4.2 Orbital Distances . ....... 13
1.4.3 Eccentricities ............................ 14
1.4.4 Properties of the Host Stars .. 15
1.5 Theory of Extrasolar Planet Formation ...................... 16
1.5.1 Interaction between Disks and Embedded Planets...... 18
2G LOBAL SIMULATIONS IN THREE DIMENSIONS 23
2.1 Introduction . . . .................................. 23
2.2 Model Description ..... 23
2.2.1 Basic Equations . . ............................. 24
2.2.2 Numerical Issues .. 25
2.2.3 Initial Setup ................................. 26
2.3 A Test Case.......... 27
2.4 Planet Embedded in a Disk: Standard Model .................. 28
2.4.1 Gravitational Torques .......... 32
viCONTENTS vii
2.5 Varying the Disk Height and the Planetary Mass ................ 34
2.6 Conclusions . . . ....................... 38
3N UMERICAL METHOD AND NESTED-GRID TECHNIQUE 43
3.1 Introduction . . . .................................. 43
3.2 Numerical Method ..... 45
3.3 Nested-Grid Technique . . ............................. 46
3.4 Basic Integration Cycle . .. 48
3.5 Downward Information Transfer . . ....................... 50
3.5.1 Interpolation Formulas in Three Di