7 Discussion

In this paper we have calculated global m=1 modes with low pattern speed corresponding to introducing a finite disc eccentricity. An important aspect was the inclusion of self-gravity which is important for the structure of these modes as well as determining the motion of embedded protoplanets.

We considered disc models that were isolated or contained one or two protoplanets orbiting in an inner cavity. In all cases global modes were found that could be global on scales up to one hundred times the inner cavity radius. The modes could be considered as being of a type that were strongly coupled to the inner protoplanets or essentially free disc modes. In the former case the disc eccentricity could be comparable to that of the protoplanets for up to three times the outer protoplanet orbital semi-major axis with apsidal line antialigned with that of the protoplanet orbits. In the latter case the inner protoplanet orbital eccentricities were small compared to that found in the disc.

We went on to discuss the motion of a protoplanet embedded in an eccentric disc and determined, initially neglecting tidal torques, the equilibrium (non precessing) orbits which maintain apsidal alignment with the disc gas orbits. Equilibrium eccentricities were found to be comparable or possibly even exceed the disc eccentricity. In some cases secular resonance could occur producing particularly large protoplanet eccentricities.

We then formulated the calculation of the response of an eccentric disc to a protoplanet in the earth mass range in order to determine the time rate of change of the eccentricity and orbital migration rate. We found that equilibrium aligned orbits with very similar eccentricity to that of the gas disc may suffer no eccentricity change while undergoing inward migration in general. This was found from the resonant torque calculations but is also expected from direct consideration of the equations governing the tidal response. However, when the non precessing equilibrium aligned orbit has a significantly higher eccentricity than the disc, as can occur generally, but in particular for modes with very small pattern speed, orbital migration may be significantly reduced or reverse from inwards to outwards for the disc models we considered.

Attainment of high eccentricities in this way typically requires the characteristic test particle orbit precession frequency or mode pattern speed to significantly exceed the characteristic orbital circularization rate, a situation more likely for lower mass protoplanets. When tidal circularization dominates, the protoplanet equilibrium eccentricity is reduced while the apsidal line becomes significantly inclined to that of the disc. However, high protoplanet eccentricities could be excited by gravitational interactions between them (Papaloizou & Larwood 2000) and under favourable conditions this effect could act to counter tidal circularization generating significantly higher protoplanet eccentricities than that of the disc. This will be a topic for future investigation.

Although there is some uncertainty in the resonant torque calculations because of the need to sum contributions of different sign, weakening of the tidal interaction is expected on general physical grounds on account of larger protoplanet disc relative velocities at higher eccentricity. This indication of migration reversal at the higher eccentricities was found to be supported by a local dynamical friction calculation applicable in that limit. In this case the interaction near apocentre tends to speed up the protoplanet while the interaction near pericentre tends to slow it down. These effects are of opposite sign but the longer time spent near apocentre results in a net outward migration of the protoplanet for the surface density considered.

Thus the existence of global non circular motions in discs with radial excursions comparable to or exceeding the semi-thickness may have important consequences for the migration of cores in the earth mass range. While processes of the type considered in this paper are unlikely to lead to the very high eccentricities observed for some giant planets, they may be important in controlling migration during planet formation as well as producing modest eccentricities $\sim$0.2.

Acknowledgements

The author thanks the IAP for visitor support and caroline Terquem for valuable and stimulating discussions as well as a carefull reading of a preliminary draft of this paper.