A simplified model for the secular dynamics of eccentric discs and applications to planet-disc interactions

Abstract

We develop a simplified model for studying the long-term evolution of giant planets in protoplanetary discs. The model accounts for the eccentricity evolution of the planets and the dynamics of eccentric discs under the influences of secular planet-disc interactions and internal disc pressure, self-gravity, and viscosity. Adopting the ansatz that the disc precesses coherently with aligned apsides, the eccentricity evolution equations of the planet-disc system reduce to a set of linearized ordinary differential equations, which allows for fast computation of the evolution of planet-disc eccentricities over long time-scales. Applying our model to 'giant planet + external disc' systems, we are able to reproduce and explain the secular behaviours found in previously published hydrodynamical simulations. We re-examine the possibility of eccentricity excitation (due to secular resonance) of multiple planets embedded in a dispersing disc, and find that taking into account the dynamics of eccentric discs can significantly affect the evolution of the planets' eccentricities.