2019ApJ...884..142G

Gravitational interactions between a protoplanetary disk and its embedded planet are 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 two-dimensional 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 that the gap depth δ_Σ in the gas surface density depends on a single dimensionless parameter K≡q²(h_p/r_p)^–5α^–1 as δ_Σ=(1+0.046K)^–1, consistent with the previous results of Kanagawa et al. 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 decreases from unity. This 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.