The dependence of the structure of planet-opened gaps in protoplanetary disks on radiative cooling

Planets can excite density waves and open annular gas gaps in protoplanetary disks. The depth of gaps is influenced by the evolving angular momentum carried by density waves. While the impact of radiative cooling on the evolution of density waves has been studied, a quantitative correlation to connect gap depth with the cooling timescale is lacking. To address this gap in knowledge, we employ the grid-based code Athena++ to simulate disk-planet interactions, treating cooling as a thermal relaxation process. We establish quantitative dependences of steady-state gap depth (Eq. 36) and width (Eq. 41) on planetary mass, Shakura-Sunyaev viscosity, disk scale height, and thermal relaxation timescale $(\beta)$. We confirm previous results that gap opening is the weakest when thermal relaxation timescale is comparable to local dynamical timescale. Significant variations in gap depth, up to an order of magnitude, are found with different $\beta$. In terms of width, a gap is at its narrowest around $\beta=1$, approximately $10\%$ to $20\%$ narrower compared to the isothermal case. When $\beta\sim100$, it can be $\sim20\%$ wider, and higher viscosity enhances this effect. We derive possible masses of the gas gap-opening planets in AS 209, HD 163296, MWC 480, and HL Tau, accounting for the uncertainties in local thermal relaxation timescale.