On the Saturation of Corotation Resonances: A Numerical Study

The torque exerted by an external potential on a two-dimensional gaseous disk at non-coorbital corotation resonances is studied by means of numerical simulations. The degree of saturation of these resonances is important in determining whether an eccentric giant planet embedded in a protoplanetary disk experiences an eccentricity excitation or damping. Previous analytic studies of the saturation properties of these resonances suffered two important restrictions, since they neglected (1) the possible overlap between neighboring first-order corotation resonances and (2) the fact that first-order corotation resonances share their location with principal Lindblad resonances. We perform calculations restricted to one or two resonances to investigate the properties of two neighboring corotation resonances, as well as the properties of a corotation resonance that overlaps a Lindblad resonance. We find that these properties hardly differ from the case of an isolated corotation resonance, which is found in a first step to agree with the analytical theory. In particular, although the torque of two neighboring corotation resonances may differ from the sum of the torques of the corresponding resonances when considered as isolated, it never exceeds the sum of the fully unsaturated isolated corotation resonances, and it saturates in a fashion similar to an isolated resonance. Similarly, the presence of an underlying Lindblad resonance hardly affects the corotation torque, even if that resonance implies a torque strong enough to significantly redistribute the azimuthally averaged surface density profile, in which case the corotation torque scales with the resulting vortensity gradient. This set of numerical experiments thus essentially validates previous analytic studies. As a side result, we show that corotation libration islands misrepresented by a mesh of too low resolution can lead to a strongly overestimated corotation torque. This may be an explanation of why the eccentricity of embedded Jupiter-sized planets was never observed to undergo an excitation in the numerical simulations performed so far.