Eccentricity Evolution for Planets in Gaseous Disks

Abstract

At least several percent of solar-type stars possess giant planets. Surprisingly, most move on orbits of substantial eccentricity. We investigate the hypothesis that interactions between a giant planet and the disk from which it forms promote eccentricity growth. These interactions are concentrated at discrete Lindblad and corotation resonances. Interactions at principal Lindblad resonances cause the planet's orbit to migrate and open a gap in the disk if the planet is sufficiently massive. Those at first-order Lindblad and corotation resonances change the planet's orbital eccentricity. Eccentricity is excited by interactions at external Lindblad resonances that are located on the opposite side of corotation from the planet, and damped by co-orbital Lindblad resonances that overlap the planet's orbit. If the planet clears a gap in the disk, the rate of eccentricity damping by co-orbital Lindblad resonances is reduced. Density gradients associated with the gap activate eccentricity damping by corotation resonances at a rate that initially marginally exceeds that of eccentricity excitation by external Lindblad resonances. But the corotation torque may be reduced as the result of the trapping of fluid in libration around potential maxima. This nonlinear saturation can tip the balance in favor of eccentricity excitation. A minimal initial eccentricity of the order of 1% is required to overcome viscous diffusion, which acts to unsaturate corotation resonances by reestablishing the large-scale density gradient. Thus, eccentricity growth is a finite-amplitude instability. Formally, the apsidal resonance, which is a special kind of co-orbital Lindblad resonance that exists in pressure-dominated disks, appears to damp eccentricity faster than external Lindblad resonances can excite it. However, the wavelength of the apsidal wave in a pressure-dominated disk is so long that it does not propagate. A self-gravity-dominated disk does not have an apsidal resonance. Nevertheless, apsidal waves are excited at gap edges. Although these propagate, their long wavelengths suggest that they are likely to be reflected at disk edges to form standing waves. Viscous damping of standing waves results in eccentricity damping, but at level far below that which traveling waves would produce. Although the level of eccentricity damping due to apsidal waves is reduced to a modest level in both pressure- and self-gravity-dominated disks, whether it drops well below that of Lindblad resonances depends sensitively on the disk's thickness and planet's mass. However, our analysis shows that with reasonable parameters, planet-disk interactions can promote eccentricity growth.