The dynamical stability of W Ursae Majoris-type systems
  Theoretical study indicates that a contact binary system would merge into a rapidly rotating single star due to tidal instability when the spin angular momentum of the system is more than a third of its orbital angular momentum. Assuming that W UMa contact binary systems rigorously comply with the Roche geometry and the dynamical stability limit is at a contact degree of about 70%, we obtain that W UMa systems might suffer Darwin's instability when their mass ratios are in a region of about 0.076--0.078 and merge into the fast-rotating stars. This suggests that the W UMa systems with mass ratio $q\leq0.076$ can not be observed. Meanwhile, we find that the observed W UMa systems with a mass ratio of about 0.077, corresponding to a contact degree of about 86% would suffer tidal instability and merge into the single fast-rotating stars. This suggests that the dynamical stability limit for the observed W UMa systems is higher than the theoretical value, implying that the observed systems have probably suffered the loss of angular momentum due to gravitational wave radiation (GR) or magnetic stellar wind (MSW). 
Lifang Li and Fenghui Zhang