Potential softening and eccentricity dynamics in razor-thin, nearly-Keplerian discs

Abstract

In many astrophysical problems involving discs (gaseous or particulate) orbiting a dominant central mass, gravitational potential of the disc plays an important dynamical role. Its impact on the motion of external objects, as well as on the dynamics of the disc itself, can usually be studied using secular approximation. This is often done using softened gravity to avoid singularities arising in calculation of the orbit-averaged potential --- disturbing function --- of a razor-thin disc using classical Laplace-Lagrange theory. We explore the performance of several softening formalisms proposed in the literature in reproducing the correct eccentricity dynamics in the disc potential. We identify softening models that, in the limit of zero softening, give results converging to the expected behavior exactly, approximately or not converging at all. We also develop a general framework for computing secular disturbing function given an arbitrary softening prescription for a rather general form of the interaction potential. Our results demonstrate that numerical treatments of the secular disc dynamics, representing the disc as a collection of $N$ gravitationally interacting annuli, are rather demanding: for a given value of the (dimensionless) softening parameter, $\varsigma\ll 1$, accurate representation of eccentricity dynamics requires $N \sim C\varsigma^{-\chi}\gg 1$, with $C\sim O(10)$, $1.5\lesssim \chi\lesssim 2$. In discs with sharp edges a very small value of the softening parameter $\varsigma$ ($\lesssim 10^{-3}$) is required to correctly reproduce eccentricity dynamics near the disc boundaries; this finding is relevant for modelling planetary rings.