Modeling protoplanetary disk heating by planet-induced spiral shocks

Abstract We investigate the heating of protoplanetary disks caused by shocks associated with spiral density waves induced by an embedded planet. Using two-dimensional hydrodynamical simulations, we explore the dependence of shock heating rates on various disk and planetary parameters. Our results show that the shock heating rates are primarily influenced by the planet’s mass and the disk’s viscosity, while being insensitive to the thermal relaxation rate and the radial profiles of the disk’s surface density and sound speed. We provide universal empirical formulas for the shock heating rates produced by the primary and secondary spiral arms as a function of orbital radius, viscosity parameter $\alpha$, and planet-to-star mass ratio $q$. The obtained formulas are accurate within a factor of a few for a moderately viscous and adiabatic disk with a planet massive enough that its spiral arms are strongly nonlinear. Using these universal relations, we show that shock heating can overwhelm viscous heating when the disk viscosity is low ($\alpha \lesssim 10^{-3}$) and the planet is massive ($q \gtrsim 10^{-3}$). Our empirical relations for the shock heating rates are simple and can be easily implemented into radially one-dimensional models of gas and dust evolution in protoplanetary disks.