Potential softening and eccentricity dynamics in nearly Keplerian disks

In many astrophysical problems involving disks (gaseous or particulate) orbiting a dominant central mass, the gravitational potential of the disk plays an important dynamical role. The dynamics driven by disks can usually be studied using secular approximation. This is often done using softened gravity to avoid singularities arising in the calculation of the orbit-averaged potential — disturbing function — of a razor-thin disk using classical Laplace-Lagrange theory. We explore the performance of several softening formalisms proposed in the literature in reproducing the expected eccentricity dynamics in the disk potential. We identify softening models that, in the limit of zero softening, give results converging to the expected behaviour exactly, approximately, or not converging at all. We also develop a general framework for computing the secular disturbing function given an arbitrary softening prescription. Our results suggest that numerical treatments of the secular disk dynamics — i.e. representing the disk as a collection of N gravitationally interacting softened rings/annuli — are rather demanding, requiring a fine numerical sampling. We find that this requirement is further aggravated for disks with sharps edges, such as planetary/debris rings.