Emergence of vortices at the edges of planet-driven gaps in protoplanetary discs

Young planets embedded in protoplanetary discs (PPDs) excite spiral density waves, which propagate shock and deposit angular momentum in the disc. This results in gap opening around the planetary orbit, even for low (sub-thermal) mass planets, provided that the effective viscosity in the disc is low. The edges of these planet-induced gaps are known to be prone to emergence of observable vortices via the Rossby wave instability (RWI). We study time-scales for the development of vortices driven by low-mass planets in inviscid discs. We employ a recently developed semi-analytical theory of vortensity production by the planet-driven shock to predict vortensity evolution near the planet, from which we derive the radial profile of the planet-induced gap as a function of time (this procedure can have multiple other uses, e.g. to study dust trapping, suppression of pebble accretion, etc.). We then analyse the linear stability of the gap edges against the RWI, obtaining the time-scales for the first appearance of unstable modes and (later) fully developed vortices at gap edges. We present useful formulae for these time-scales as functions of planetary and disc parameters and provide their physical justification. We also thoroughly test our semi-analytical framework against high-resolution 2D hydrodynamic simulations, confirming the accuracy of our theoretical predictions. We discuss ways in which our semi-analytical framework can be extended to incorporate additional physics, e.g. planetary accretion, migration, and non-zero disc viscosity. Our results can be used to interpret observations of PPDs and to predict emergence of vortices in simulations.