Analytical solution for the evolution of a binary with stable mass transfer from a giant

Abstract

We derive a simple analytical solution for the evolution of a close binary with nuclear time-scale driven mass transfer from a giant. This solution is based on the well-known fact that the luminosity and the radius of a giant scale to a good approximation as simple power laws of the mass M_c of the degenerate helium core. Comparison with results of numerical calculations by Webbink, Rappaport & Savonije (1983) show the analytical solution and the power law approximation to be quite accurate. The analytical solution presented does also allow (in parametrized form) for non-conservative mass transfer. Furthermore it is shown that the near constancy of the mass transfer rate over most of the mass transfer phase seen in the results by Webbink, Rappaport & Savonije is not a generic feature of this type of evolution but rather a consequence of a particular choice of parameters. The analytical solution also demonstrates that the level of mass transfer is largely set by the core mass of the giant at the onset of mass transfer. Finally we show that the model is selfconsistent and discuss its applicability to low-mass X-ray binaries.