Structure and evolution of low-mass W UMa type systems -- II. with   angular momentum loss
  In a preceding paper, using Eggleton's evolution code we have discussed the structure and evolution of low-mass W UMa type contact binaries without angular momentum loss (AML). The models exhibit cyclic behavior about a state of marginal contact on a thermal time-scale. Part of the time of each cycle is spent in contact and part in a semi-detached state. According to observations, W UMa systems suffer AML. We present the models of low-mass contact binaries with AML due to gravitational wave radiation (GR) or magnetic stellar wind (MSW) are presented. We find that gravitational radiation cannot prevent the cyclic evolution of W UMa systems, and the effect of gravitational radiation on the cyclic behavior of contact binary evolution is almost negligible. We also find that the most likely AML mechanism for W UMa systems is magnetic braking, and that magnetic braking effects can increase the period of the cyclic evolution, and shorten the fraction of the time spent in the poor thermal contact state. If W UMa stars do not undergo cyclic evolution, and their angular momentum loss is caused simultaneously by MSW of both components, we find that the value of the parameter, $\lambda$, should be taken a larger value in comparison with those derived from observations of single stars. This indicates that the AML efficiency in W UMa systems may be lowered in comparison with non-contact stars because of the less mass contained in the convective envelopes of the components in W UMa systems. If W UMa systems lose their angular momentum at a constant rate. An angular momentum rate of $\frac{{\rm dln}J}{{\rm d}t}\approx 1.6\times 10^{-9} {\rm yr^{-1}}$ can prevent the cyclic behaviour of the model, and the model can keep in good contact with an essentially constant depth of contact. 
L. Li, Z. Han, and F. Zhang