Modeling protoplanetary disk heating by planet-induced spiral shocks.

This article investigates heating in protoplanetary disks caused by shocks from spiral density waves induced by an embedded planet, using two-dimensional hydrodynamical simulations. The study quantifies shock heating rates by measuring specific entropy jumps (denoted as δ) across primary and secondary spiral arms and finds that these rates follow universal scaling relations dependent primarily on the planet-to-star mass ratio (q), the disk's viscosity parameter (α), and orbital radius, while being largely insensitive to the disk's surface density profile, sound speed distribution, and thermal relaxation timescale. Empirical formulas for δ are provided, accurate within a factor of two for moderately viscous (α ≈ 10⁻³ to 10⁻³.⁵) and adiabatic disks with sufficiently massive planets (q ≳ 10⁻⁴), enabling predictions of when shock heating can dominate over viscous heating—specifically for low-viscosity disks (α ≲ 10⁻³) with massive planets (q ≳ 10⁻³). The authors note limitations due to the two-dimensional nature of the simulations and the restricted radial domain, suggesting that future three-dimensional and extended-domain studies are needed to fully capture vertical effects and inner disk shock heating.