Roche Lobes in the Second Post-newtonian Approximation

Close binary systems of compact stars, due to the emission of gravitational radiation, may evolve into a phase in which the less massive star transfers mass to its companion. We describe mass transfer by using the model of Roche lobe overflow, in which mass is transferred through the first, or innermost, Lagrange point. Under conditions in which gravity is strong, the shapes of the equipotential surfaces and the Roche lobes are modified compared to the Newtonian case. We present calculations of the Roche lobe utilizing the second order post-Newtonian (2PN) approximation in the ArnowittDeser-Misner gauge. Heretofore, calculations of the Roche lobe geometry beyond the Newtonian case have not been available. Beginning from the general N-body Lagrangian derived by Damour and Schäffer, we develop the Lagrangian for a test particle in the vicinity of two massive compact objects. As an exact result for the transverse-traceless part of the Lagrangian is not available, we devise an approximation that is valid for regions close to the less massive star. We calculate the Roche lobe volumes, and provide a simple fitting formula for the effective Roche lobe radius analogous to that for the Newtonian case furnished by Eggleton. In contrast to the Newtonian case, in which the