2009MNRAS.394..501A

When the total angular momentum of a binary system is at a critical (minimum) value, a tidal instability occurs (Darwin's instability), eventually forcing the stars to merge into a single, rapidly rotating object. The instability sets in at some critical separation which in the case of contact binaries corresponds to a minimum mass ratio depending on dimensionless gyration radius k₁. If one considers n = 3 polytrope (fully radiative primary with Γ₁= 4/3), k²₁= 0.075 and q_min~ 0.085-0.095. There appears to be, however, some W UMa-type binaries with q values very close, if not below these theoretical limits, implying that primary in these systems is probably more centrally condensed. We try to solve the discrepancy between theory and observations by considering rotating polytropes. We show by deriving and solving a modified Lane-Emden equation for n = 3 polytrope that including the effects of rotation does increase the central concentration and could reduce q_min to as low as 0.070-0.074, more consistent with the observed population.