Edge modes in self-gravitating disc-planet interactions

We study the stability of gaps opened by a giant planet in a self-gravitating protoplanetary disc. We find a linear instability associated with both the self-gravity of the disc and local vortensity maxima which coincide with gap edges. For our models, these edge modes develop and extend to twice the orbital radius of a Saturn mass planet in discs with total masses M-d greater than or similar to 0.06M(*), where M-* is the central stellar mass, corresponding to a Toomre Q less than or similar to 1.5 at twice the planet's orbital radius. The disc models, although massive, are such that they are stable in the absence of the planet. Unlike the previously studied local vortex forming instabilities associated with gap edges in weakly or non-self-gravitating discs with low viscosity, the edge modes we consider are global and exist only in sufficiently massive discs, but for the typical viscosity values adopted for protoplanetary discs. It is shown through analytic modelling and linear calculations that edge modes may be interpreted as a localized disturbance associated with a gap edge inducing activity in the extended disc, through the launching of density waves excited through gravitational potential perturbation at Lindblad resonances. We also perform hydrodynamic simulations in order to investigate the evolution of edge modes in the linear and non-linear regimes in disc-planet systems. The form and growth rates of developing unstable modes are found to be consistent with linear theory. Their dependence on viscosity and gravitational softening is also explored. We also performed a first study of the effect of edge modes on disc-planet torques and the orbital migration of the planet. We found that if edge modes develop, then the average torque on the planet becomes more positive with increasing disc mass. In simulations where the planet was allowed to migrate, although a fast type III migration could be seen that was similar to that seen in non-self-gravitating discs, we found that it was possible for the planet to interact gravitationally with the spiral arms associated with an edge mode and that this could result in the planet being scattered outwards. Thus orbital migration is likely to be complex and non-monotonic in massive discs of the type we consider.