  It has been suggested that long-period giant planets, such as HD 95086b and
HR 8799bcde, may have formed through gravitational instability of
protoplanetary discs. However, self-gravitating disc-satellite interaction can
lead to the formation of a gravitationally unstable gap. Such an instability
significantly affects the orbital migration of gap-opening perturbers in
massive discs. We use 2D hydrodynamical simulations to examine the role of
planet mass on the gravitational stability of gaps and its impact on orbital
migration. We consider giant planets with planet-to-star mass ratio q=0.0003 to
q=0.003, in a self-gravitating disc with disc-to-star mass ratio M_d/M_*=0.08,
aspect ratio h=0.05, and Keplerian Toomre parameter Q = 1.5 at 2.5 times the
planet's initial orbital radius. Fixed-orbit simulations show that all planet
masses we consider open gravitationally unstable gaps, but the instability is
stronger and develops sooner with increasing planet mass. The disc-on-planet
torques typically become more positive with increasing planet mass. In
freely-migrating simulations, we observe faster outward migration with
increasing planet mass, but only for planet masses capable of opening unstable
gaps early on. For q=0.0003, the planet undergoes rapid inward type III
migration before it can open a gap. For q=0.0013 we find it is possible to
balance the tendency for inward migration by the positive torques due to an
unstable gap, but only for a few 10's of orbital periods. We find the unstable
outer gap edge can trigger outward type III migration, sending the planet to
twice its initial orbital radius on dynamical timescales. This triggered type
III migration may be a way of bringing giant planets to large orbital
distances. On the other hand, our results also imply gap-opening is not a
sufficient condition for the in situ formation of giant planets on wide orbits
through disc fragmentation.
