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