The donor stars of cataclysmic variables

We carefully consider observational and theoretical constraints on the global properties of secondary stars in cataclysmic variable stars (CVs). We then use these constraints to construct and test a complete, semi-empirical donor sequence for CVs with orbital periods P-orb <= 6 h. All key physical and photometric parameters of CV secondaries (along with their spectral types) are given as a function of P-orb along this sequence. This provides a benchmark for observational and theoretical studies of CV donors and evolution. The main observational basis for our donor sequence is an empirical mass-radius relationship for CV secondaries. Patterson and co-workers have recently shown that this can be derived from superhumping and/or eclipsing CVs. We independently revisit all of the key steps in this derivation, including the calibration of the period excess-mass ratio relation for superhumpers and the use of a single representative primary mass for most CVs. We also present an optimal technique for estimating the parameters of the mass-radius relation that simultaneously ensures consistency with the observed locations of the period gap and the period minimum. We present new determinations of these periods, finding P-gap,P-+ = 3.18 +/- 0.04 h (upper edge), P-gap,P-- = 2.15 +/- 0.03 h (lower edge) and P-min = 76.2 +/- 1.0 min (period minimum). We test the donor sequence by comparing observed and predicted spectral types (SpTs) as a function of orbital period. To this end, we update the SpT compilation of Beuermann and co-workers and show explicitly that CV donors have later SpTs than main-sequence (MS) stars at all orbital periods. This extends the conclusion of the earlier study to the short-period regime (P-orb < 3 h). We then compare our donor sequence to the CV data, and find that it does an excellent job of matching the observed SpTs. Thus the empirical mass-radius relation yields just the right amount of radius expansion to account for the later-than-MS spectral types of CV donors. There is remarkably little intrinsic scatter in both the mass-radius and SpT-P-orb relations, which confirms that most CVs follow a unique evolution track. The donor sequence exhibits a fairly sharp drop in temperature, luminosity and optical/infrared flux well before the minimum period. This may help to explain why the detection of brown dwarf secondaries in CVs has proven to be extremely difficult. We finally apply the donor sequence to the problem of distance estimation. Based on a sample of 22 CVs with trigonometric parallaxes and reliable 2MASS data, we show that the donor sequence correctly traces the upper envelope of the observed M-JHK-P-orb distribution. Thus robust lower limits on distances can be obtained from single-epoch infrared observations. However, for our sample, these limits are typically about a factor of 2 below the true distances.