Numerical Calculations of the Linear Response of a Gaseous Disk to a Protoplanet

Abstract

In this paper we present calculations of the linear response of a flat gaseous disk to the presence of an embedded protoplanet. The calculations are numerical and take into account terms neglected by analytic theory and also examine the effects of gradients in surface density and sound speed in the disk. We have also developed a means of rapid calculation of Laplace coefficients for a softened potential (or for a disk of nonzero height). In general, the classical analytic theory is confirmed. However, the magnitude of calculated torque due to the corotation resonance disagrees with previously derived analytical results. The calculated corotation torque is smaller than the analytic value, is much less sensitive to the value of the perturbing potential at corotation, and falls off more rapidly as a function of the azimuthal wavenumber m. It is possible that the rapid variation of the perturbing potential near corotation is responsible for this behavior. We find that the dominant effect of a pressure gradient is to increase the net torque, due to the inward displacement of the Lindblad resonances. The effect countervails the opposing effect of a gradient in disk surface density, in general. We have calculated the inferred orbital evolution timescales for the Earth and outer planets for a variety of model nebulae. In general the orbital evolution timescale for the Earth is on the order of 10 6 years. Timescales for the outer planets are much shorter: 10 3-10 4 years in general. Although the effects of the reaction of the solar nebula to the torques have been neglected in our calculations, the results indicate the importance of the protoplanets for the later stages of nebular evolution.