Planet Migration and Gap Formation by Tidally Induced Shocks

Gap formation in a gas disk triggered by disk-planet tidal interaction is considered. Density waves launched by the planet are assumed to be damped as a result of their nonlinear evolution leading to shock formation and its subsequent dissipation. As a consequence, wave angular momentum is transferred to the disk, leading to evolution of its surface density. Planetary migration is an important ingredient of the theory; effects of the planet-induced surface density perturbations on the migration speed are considered. A gap is assumed to form when a stationary solution for the surface density profile is no longer possible in the frame of reference migrating with the planet. An analytical limit on the planetary mass necessary to open a gap in an inviscid disk is derived. The critical mass turns out to be smaller than the mass M₁ for which the planetary Hill radius equals the disk scale height by a factor of at least Q^5/7 (Q is the Toomre stability parameter), depending on the strength of the migration feedback. In viscous disks the critical planetary mass could vary from ~0.2M₁ to M₁, depending on the disk viscosity. This implies that a gap could be formed by a planet with mass of 2-15 M_⊕, depending on the disk aspect ratio, viscosity, and the planet's location in the nebula.