Mass Estimates of a Giant Planet in a Protoplanetary Disk from the Gap Structures

Abstract: A giant planet embedded in a protoplanetary disk forms a gap. An analytic relationship among the gap depth, planet mass $M_{p}$, disk aspect ratio $h_p$, and viscosity $\alpha$ has been found recently, and the gap depth can be written in terms of a single parameter $K= (M_{p}/M_{\ast})^2 h_p^{-5} \alpha^{-1}$. We discuss how observed gap features can be used to constrain the disk and/or planet parameters based on the analytic formula for the gap depth. The constraint on the disk aspect ratio is critical in determining the planet mass so the combination of the observations of the temperature and the image can provide a constraint on the planet mass. We apply the formula for the gap depth to observations of HL~Tau and HD~169142. In the case of HL~Tau, we propose that a planet with $\gtrsim 0.3$ is responsible for the observed gap at $30$~AU from the central star based on the estimate that the gap depth is $\lesssim 1/3$. In the case of HD~169142, the planet mass that causes the gap structure recently found by VLA is $\gtrsim 0.4 M_J$. We also argue that the spiral structure, if observed, can be used to estimate the lower limit of the disk aspect ratio and the planet mass.