Early stages of gap opening by planets in protoplanetary discs

Annular substructures in protoplanetary discs, ubiquitous in sub-mm observations, can be caused by gravitational coupling between a disc and its embedded planets. Planetary density waves inject angular momentum into the disc leading to gap opening only after travelling some distance and steepening into shocks (in the absence of linear damping); no angular momentum is deposited in the planetary coorbital region, where the wave has not shocked yet. Despite that, simulations show mass evacuation from the coorbital region even in inviscid discs, leading to smooth, double-trough gap profiles. Here we consider the early, time-dependent stages of planetary gap opening in inviscid discs. We find that an often-overlooked contribution to the angular momentum balance caused by the time-variability of the specific angular momentum of the disc fluid (caused, in turn, by the time-variability of the radial pressure support) plays a key role in gap opening. Focusing on the regime of shallow gaps with depths of $\lesssim 20~{{\%}}$, we demonstrate analytically that early gap opening is a self-similar process, with the amplitude of the planet-driven perturbation growing linearly in time and the radial gap profile that can be computed semi-analytically. We show that mass indeed gets evacuated from the coorbital region even in inviscid discs. This evolution pattern holds even in viscous discs over a limited period of time. These results are found to be in excellent agreement with 2D numerical simulations. Our simple gap evolution solutions can be used in studies of dust dynamics near planets and for interpreting protoplanetary disc observations.