  We investigate the existence and properties of equipotential surfaces and
Lagrangian points in non-synchronous, eccentric binary star and planetary
systems under the assumption of quasi-static equilibrium. We adopt a binary
potential that accounts for non-synchronous rotation and eccentric orbits, and
calculate the positions of the Lagrangian points as functions of the mass
ratio, the degree of asynchronism, the orbital eccentricity, and the position
of the stars or planets in their relative orbit. We find that the geometry of
the equipotential surfaces may facilitate non-conservative mass transfer in
non-synchronous, eccentric binary star and planetary systems, especially if the
component stars or planets are rotating super-synchronously at the periastron
of their relative orbit. We also calculate the volume-equivalent radius of the
Roche lobe as a function of the four parameters mentioned above. Contrary to
common practice, we find that replacing the radius of a circular orbit in the
fitting formula of Eggleton (1983) with the instantaneous distance between the
components of eccentric binary or planetary systems does not always lead to a
good approximation to the volume-equivalent radius of the Roche-lobe. We
therefore provide generalized analytic fitting formulae for the
volume-equivalent Roche lobe radius appropriate for non-synchronous, eccentric
binary star and planetary systems. These formulae are accurate to better than
1% throughout the relevant 2-dimensional parameter space that covers a dynamic
range of 16 and 6 orders of magnitude in the two dimensions.
