  Finite eccentricities in mass-transferring eccentric binary systems can be
explained by taking into account mass-loss and mass-transfer processes that
often occur in these systems. These processes can be treated as perturbations
to the general two-body problem. The time-evolution equations for the
semi-major axis and the eccentricity derived from perturbative methods are in
general phase-dependent. The osculating semi-major axis and eccentricity change
over the orbital timescale and they are not easy to implement in binary
evolution codes like MESA. However, the secular orbital element evolution
equations can be simplified averaging over the rapidly varying true anomalies.
In this paper, we derive the secular time-evolution equations for the
semi-major axis and the eccentricity for various mass-loss/transfer processes
using either the adiabatic approximation or the assumption of delta-function
mass-loss/transfer at periastron. We begin with the cases of isotropic and
anisotropic wind mass-loss. We continue with conservative and non-conservative
non-isotropic mass ejection/accretion (including RLOF) for both point-masses
and extended bodies. We conclude with the case of phase-dependent mass
accretion. Comparison of the derived equations with similar work in the
literature is included and explanation of the existing discrepancies is
provided.
