Gravitational interactions between a protoplanetary disk and its embedded planet is one of the formation mechanisms of gaps and rings found in recent ALMA observations. To quantify the gap properties measured in not only surface density but also rotational velocity profiles, we run twodimensional hydrodynamic simulations of protoplanetary disks by varying three parameters: the mass ratio q of a planet to a central star, the ratio of the disk scale height h p to the orbital radius r p of the planet, and the viscosity parameter α. We find the gap depth δ Σ in the gas surface density depends on a single dimensionless parameter K ≡ q 2 (h p /r p ) −5 α −1 as δ Σ = (1 + 0.046K) −1 , consistent with the previous results of Kanagawa et al. (2015a). The gap depth δ V in the rotational velocity is given by δ V = 0.007(h p /r p )K 1.38 /(1 + 0.06K 1.03 ). The gap width, in both surface density and rotational velocity, has a minimum of about 4.7h p when the planet mass M p is around the disk thermal mass M th , while it increases in a power-law fashion as M p /M th increases or decrease from unity. Such a minimum in the gap width arises because spirals from sub-thermal planets have to propagate before they shock the disk gas and open a gap. We compare our relations for the gap depth and width with the previous results, and discuss their applicability to observations.

The dynamical corotation torque arising from the deformation of horseshoe orbits, along with the vortensity gradient in the background disk, is important for determining the orbital migration rate and direction of low-mass planets. Previous two-dimensional studies have predicted that the dynamical corotation torque is positive, decelerating inward planet migration. In contrast, recent three-dimensional studies have shown that buoyancy resonance makes the dynamical corotation torque negative, accelerating inward migration. In this paper, we study the dependence of the dynamical corotation torque on thermal transport, using three-dimensional simulations. We first show that our results are consistent with previous three-dimensional studies when the disk is fully adiabatic. In more realistic radiative disks, however, radiative diffusion suppresses buoyancy resonance significantly, especially in high-altitude regions, and yields a positive dynamical corotation torque. This alleviates the issue of rapid migration being caused by the negative dynamical corotation torque in adiabatic disks. Our results suggest that radiative diffusion, together with stellar irradiation and accretion heating, are needed to accurately describe the migration of low-mass planets.