Shifting of the resonance location for planets embedded in circumstellar disks

Abstract

Context. In the early evolution of a planetary system, a pair of planets may be captured in a mean motion resonance while still embedded in their nesting circumstellar disk. Aims. The goal is to estimate the direction and amount of shift in the semimajor axis of the resonance location due to the disk gravity as a function of the gas density and mass of the planets. The stability of the resonance lock when the disk dissipates is also tested. Methods. The orbital evolution of a large number of systems is numerically integrated within a three-body problem in which the disk potential is computed as a series of expansion. This is a good approximation, at least over a limited amount of time. Results. Two different resonances are studied: the 2:1 and the 3:2. In both cases the shift is inwards, even if by a different amount, when the planets are massive and carve a gap in the disk. For super-Earths, the shift is instead outwards. Different disk densities, Σ, are considered and the resonance shift depends almost linearly on Σ. The gas dissipation leads to destabilization of a significant number of resonant systems, in particular if it is fast. Conclusions. The presence of a massive circumstellar disk may significantly affect the resonant behavior of a pair of planets by shifting the resonant location and by decreasing the size of the stability region. The disk dissipation may explain some systems found close to a resonance but not locked in it.