Formation of a disc gap induced by a planet: Effect of the deviation from Keplerian disc rotation

The gap formation induced by a giant planet is important in the evolution of the planet and the protoplanetary disc. We examine the gap formation by a planet with a new formulation of one-dimensional viscous discs which takes into account the deviation from Keplerian disc rotation due to the steep gradient of the surface density. This formulation enables us to naturally include the Rayleigh stable condition for the disc rotation. It is found that the derivation from Keplerian disc rotation promotes the radial angular momentum transfer and makes the gap shallower than in the Keplerian case. For deep gaps, this shallowing effect becomes significant due to the Rayleigh condition. In our model, we also take into account the propagation of the density waves excited by the planet, which widens the range of the angular momentum deposition to the disc. The effect of the wave propagation makes the gap wider and shallower than the case with instantaneous wave damping. With these shallowing effects, our one-dimensional gap model is consistent with the recent hydrodynamic simulations.Comment: 15 pages, 13 figures, accepted for publication in MNRA

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

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.Comment: 16 pages, 5 figures, accepted for publication in The Astrophysical Journal Letter