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 \leq 0.076 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 ) .