An analytic approach to the secular evolution of cataclysmic variables

We investigate the reaction of low-mass stars to mass loss within the context of the evolution of cataclysmic variables (CVs). Based on homology we derive a first-order differential equation for the radius reaction of a low-mass star upon mass loss. The solution of the differential equation yields the stellar radius as a function of the mass-transfer time-scale tau(M) = \M/M\ and the stellar mass M. The main property of the differential equation is the convergence of solutions which differ only in the initial conditions. The solutions converge on an e-folding time-scale tau(per). A linearized analysis yields tau(per) less than or equal to 1/20 tau(KH) (with tau(KH) the Kelvin-Hehmholtz time of the star). Applying our model to CV evolution, we furthermore show that after a short turn-on phase the CV evolution is independent of its initial conditions, which explains the similarity of secular evolution tracks previously found by Paczynski & Sienkiewicz and Kolb & Bitter in numerical studies. This is why the detached phase (M=0) in the life of a single CV can still be seen in the CV population as a period gap. An analytic solution of the linearized differential equation near thermal equilibrium is applied to the period flag subsequent to the turn-on of mass transfer after the detached phase of CV evolution.