Embedded planets disturb the density structure of the ambient disk and gravitational back-reaction will induce possibly a change in the planets orbital elements. The accurate determination of the forces acting on the planet requires careful numerical
analysis. Recently, the validity of the often used fast orbital advection algorithm (FARGO) has been put into question, and special numerical resolution and stability requirements have been suggested. In this paper we study the process of planet-disk interaction for small mass planets of a few Earth masses, and reanalyze the numerical requirements to obtain converged and stable results. One focus lies on the applicability of the FARGO-algorithm. Additionally, we study the difference of two and three-dimensional simulations, compare global with local setups, as well as isothermal and adiabatic conditions. We study the influence of the planet on the disk through two- and three-dimensional hydrodynamical simulations. To strengthen our conclusions we perform a detailed numerical comparison where several upwind and Riemann-solver based codes are used with and without the FARGO-algorithm. With respect to the wake structure and the torque density acting on the planet we demonstrate that the FARGO-algorithm yields correct results, and that at a fraction of the regular cpu-time. We find that the resolution requirements for achieving convergent results in unshocked regions are rather modest and depend on the pressure scale height of the disk. By comparing the torque densities of 2D and 3D simulations we show that a suitable vertical averaging procedure for the force gives an excellent agreement between the two. We show that isothermal and adiabatic runs can differ considerably, even for adiabatic indices very close to unity.
We study the development of finite eccentricity in accretion disks in close binary systems using a two-dimensional grid-based numerical scheme. We perform detailed parameter studies to explore the dependence on viscosity, disk aspect ratio, the inclu
sion of a mass-transfer stream and the role of the boundary conditions. We consider mass ratios 0.05<q<0.3 appropriate to superoutbursting cataclysmic binary systems. Instability to the formation of a precessing eccentric disk that attains a quasi-steady state with mean eccentricity in the range 0.3-0.5 occurs readily. The shortest growth times are ~15 binary orbits for the largest viscosities and the instability mechanism is for the most part consistent with the mode-coupling mechanism associated with the 3:1 resonance proposed by Lubow. However, the results are sensitive to the treatment of the inner boundary and to the incorporation of the mass-transfer stream. In the presence of a stream we found a critical viscosity below which the disk remains circular. Incorporation of a mass-transfer stream tends to impart stability for small enough viscosity (or, equivalently, mass-transfer rate through the disk) and does assist in obtaining a prograde precession rate that is in agreement with observations. For the larger q the location of the 3:1 resonance is pushed outwards towards the Roche lobe where higher-order mode couplings and nonlinearity occur. It is likely that three-dimensional simulations that properly resolve the disks vertical structure are required to make significant progress in this case.
Over 30 planetary systems have been discovered to reside in binary stars. For small separations gravitational perturbation of the secondary star has a strong influence on the planet formation process. It truncates the protoplanetary disk, may shorten
s its lifetime, and stirs up the embedded planetesimals. Due to its small semi-major axis (18.5 AU) and large eccentricity (e=0.35) the binary $gamma$ Cephei represents a particularly challenging example. In the present study we model the orbital evolution and growth of embedded protoplanetary cores of about 30 earth masses in the putative protoplanetary disk surrounding the primary star in the $gamma$ Cep system. We assume coplanarity of the disk, binary and planet and perform two-dimensional hydrodynamic simulations of embedded cores in a protoplanetary disk. The presence of the eccentric secondary star perturbs the disk periodically and generates strong spiral arms at periapse which propagate toward the disk centre. The disk also becomes slightly eccentric (with e_d = 0.1-0.15), and displays a slow retrograde precession in the inertial frame. For all initial separations (2.5 to 3.5 AU) we find inward migration of the cores. For initial semi-major axes (a_p gsim 2.7), we find a strong increase in the planetary eccentricity despite the presence of inward migration. Only cores which are initially far from the disk outer edge have a bounded orbital eccentricity which converges, roughly to the value of the planet observed in the $gamma$ Cep system. We have shown that under the condition protoplanetary cores can form at around 2.5 AU, it is possible to evolve and grow such a core to form a planet with final outcome similar to that observed.
