Properties of Density and Velocity Gaps Induced by a Planet in a Protoplanetary Disk

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 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 $\alpha$. We find the gap depth $\delta_\Sigma$ in the gas surface density depends on a single dimensionless parameter $K\equiv q^2(h_p/r_p)^{-5}\alpha^{-1}$ as $\delta_\Sigma = (1 + 0.046K)^{-1}$, consistent with the previous results of Kanagawa et al. (2015a). The gap depth $\delta_V$ in the rotational velocity is given by $\delta_V= 0.007 (h_p/r_p) K^{1.38}/(1 +0.06K^{1.03})$. The gap width, in both surface density and rotational velocity, has a minimum of about $4.7 h_p$ when the planet mass $M_p$ is around the disk thermal mass $M_{\text{th}}$, while it increases in a power-law fashion as $M_p/M_{\text{th}}$ 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.