Numerical Simulations of Disk-Planet Interactions

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Dokumentart: PhDThesis
Date: 2003
Language: English
Faculty: 7 Mathematisch-Naturwissenschaftliche Fakultät
Department: Sonstige - Mathematik und Physik
Advisor: Kley, Wilhelm
Day of Oral Examination: 2003-06-04
DDC Classifikation: 530 - Physics
Keywords: Akkretionsscheiben , Hydrodynamik , Planetenentstehung
Other Keywords:
Accretion Disks , Hydrodynamics , Planetary Systems
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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 problem 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. Simulations show that, despite a density gap forms along the orbital path, Jupiter-mass protoplanets 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 previous 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 resolutions 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 overcome 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, resulting 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 on 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 dimensions. 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 account through physically realistic gravitational potentials of forming planets. Outcomes show that migration rates are relatively constant when perturbing masses lie above approximately 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 predicted 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 temperature 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 environments 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.

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