arXiv:astro-ph/0112429v1 18 Dec 2001 Astronomy & Astrophysics manuscript no. (will be inserted by hand later) Nested-grid calculations of disk-planet interaction Gennaro D’Angelo 1,2 , Thomas Henning 1 , and Wilhelm Kley 2 1 Astrophysikalisches Institut und Universit¨ats-Sternwarte, Schillerg¨aßchen 2-3, D-07745 Jena, Germany 2 Computational Physics, Auf der Morgenstelle 10, D-72076 T¨ ubingen, Germany Received —; accepted — Abstract. We study the evolution of embedded protoplanets in a protostellar disk using very high resolution nested-grid computations. This method allows us to perform global simulations of planets orbiting in disks and, at the same time, to resolve in detail the dynamics of the flow inside the Roche lobe of the planet. The primary interest of this work lies in the analysis of the gravitational torque balance acting on the planet. For this purpose we study planets of different masses, ranging from one Earth-mass up to one Jupiter-mass, assuming typical parameters of the protostellar disk. The high resolution of the method allows a precise determination of the mass flow onto the planet and the resulting torques. The obtained migration time scales are in the range from few times 10 4 years, for intermediate mass planets, to 10 6 years, for very low and high mass planets. Typical growth time scales depend strongly on the planetary mass, ranging from a few hundred years, in the case of Earth-type planets, to several ten thousand years, in the case of Jupiter-type planets. Key words. accretion, accretion disks – hydrodynamics – methods: Numerical – planetary systems 1. Introduction During the past five years radial velocity studies have allowed the detection of planetary companions around other main-sequence stars. Until now about sixty so-called “extrasolar planets” have been discov- ered, which orbit their stars within a distance of a few AU. A recent catalog of extrasolar planets, in- cluding their orbital characteristics, is provided by Butler et al. (2001) and up-to-date versions can be found at http://www.obspm.fr/encycl/encycl.html and http://exoplanets.org/, maintained by Jean Schneider and the Department of Astronomy at UC Berkeley, respectively. In contrast to the solar system, these new planets dis- play quite different orbital properties that challenge the accepted formation scenario for solar planets. The ma- jor differences are their high minimum masses (up to 17 Jupiter-masses), their proximity to the central star (a frac- tion of the Sun-Mercury distance) and their high eccen- tricities (up to 0.7). One of the main problems to deal with is the very close distance of massive planets to their parent star. The formation of Jupiter-type planets at these locations is, on theoretical grounds, very unlikely. First of all, from purely geometrical arguments, the matter reservoir of the sur- rounding disk is too little so that a planet could hardly Send offprint requests to : G. D’Angelo, e-mail: gennaro@astro.uni-jena.de accrete its mass. Second, the temperatures within the disk are too high for a rocky core to condense easily. For these reasons it is generally believed that planets have formed from disk material further out, at distances of several AU from the star, and have then migrated to their present positions. This radial motion of the planet through the disk is primarily caused by gravitational torques acting on the planet. The presence of the planet in the disk dis- turbs the disk gravitationally, creating spiral density wave perturbations, which emanate from the planet through the disk. Hence, the disk is no longer axisymmetric which re- sults in a net torque on the planet. The sign and magni- tude of the vertical component of the torque determines the direction and efficiency of the radial migration. While initial fully non-linear hydrodynamical numer- ical computations of embedded planets assumed a fixed circular orbit of the planet (Kley 1999; Bryden et al. 1999; Lubow et al. 1999), more recent simulations took into account the back reaction of the disk and allowed for a change in the parameters of the planetary orbit (Kley 2000; Nelson et al. 2000). For a Jupiter-mass planet and typical parameter values for the disk, the obtained orbital decay time is about 10 5 years, which agrees reasonably well with previous estimates based on analytic linear the- ories (Goldreich & Tremaine 1980; Ward 1997). The majority of the computations, performed so far, have used a single grid which resolves the Roche lobe of a Jupiter-mass planet only with very few grid cells. Recently, Cieciel¸ ag et al. (2000a, 2000b) used an Adaptive