Low-mass planets in nearly inviscid disks: numerical treatment

Abstract

Context. Embedded planets disturb the density structure of the ambient disk, and gravitational back-reaction possibly will induce a change in the planet´s orbital elements. Low-mass planets only have a weak impact on the disk, so their wake´s torque can be treated in linear theory. Larger planets will begin to open up a gap in the disk through nonlinear interaction. 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. Aims: We study the process of planet-disk interaction for low-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. Methods: 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. Results: With respect to the wake structure and the torque density acting on the planet, we demonstrate that the FARGO-algorithm yields correct a correct and stable evolution for the planet-disk problem, 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 H of the disk. By comparing the torque densities of two- and three-dimensional 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.