THE EVOLUTION OF CATACLYSMIC VARIABLES AS REVEALED BY THEIR DONOR STARS

We present an attempt to reconstruct the complete evolutionary path followed by cataclysmic variables (CVs), based on the observed mass–radius relationship of their donor stars. Along the way, we update the semi-empirical CV donor sequence presented previously by one of us, present a comprehensive review of the connection between CV evolution and the secondary stars in these systems, and reexamine most of the commonly used magnetic braking (MB) recipes, finding that even conceptually similar ones can differ greatly in both magnitude and functional form. The great advantage of using donor radii to infer mass-transfer and angular-momentum-loss (AML) rates is that they sample the longest accessible timescales and are most likely to represent the true secular (evolutionary average) rates. We show explicitly that if CVs exhibit long-term mass-transfer-rate fluctuations, as is often assumed, the expected variability timescales are so long that other tracers of the mass-transfer rate—including white dwarf (WD) temperatures—become unreliable. We carefully explore how much of the radius difference between CV donors and models of isolated main-sequence stars may be due to mechanisms other than mass loss. The tidal and rotational deformation of Roche-lobe-filling stars produces ≃ 4.5% radius inflation below the period gap and ≃ 7.9% above. A comparison of stellar models to mass–radius data for non-interacting stars suggests a real offset of ≃ 1.5% for fully convective stars (i.e., donors below the gap) and ≃ 4.9% for partially radiative ones (donors above the gap). We also show that donor bloating due to irradiation is probably smaller than, and at most comparable to, these effects. After calibrating our models to account for these issues, we fit self-consistent evolution sequences to our compilation of donor masses and radii. In the standard model of CV evolution, AMLs below the period gap are assumed to be driven solely by gravitational radiation (GR), while AMLs above the gap are usually described by an MB law first suggested by Rappaport et al. We adopt simple scaled versions of these AML recipes and find that these are able to match the data quite well. The optimal scaling factors turn out to be fGR = 2.47 ± 0.22 below the gap and fMB = 0.66 ± 0.05 above (the errors here are purely statistical, and the standard model corresponds to fGR = fMB = 1). This revised model describes the mass–radius data significantly better than the standard model. Some of the most important implications and applications of our results are as follows. (1) The revised evolution sequence yields correct locations for the minimum period and the upper edge of the period gap; the standard sequence does not. (2) The observed spectral types of CV donors are compatible with both standard and revised models. (3) A direct comparison of predicted and observed WD temperatures suggests an even higher value for fGR, but this comparison is sensitive to the assumed mean WD mass and the possible existence of mass-transfer-rate fluctuations. (4) The predicted absolute magnitudes of donor stars in the near-infrared form a lower envelope around the observed absolute magnitudes for systems with parallax distances. This is true for all of our sequences, so any of them can be used to set firm lower limits on (or obtain rough estimates of) the distances toward CVs based only on Porb and single epoch near-IR measurements. (5) Both standard and revised sequences predict that short-period CVs should be susceptible to dwarf nova (DN) eruptions, consistent with observations. However, both sequences also predict that the fraction of DNe among long-period CVs should decline with Porb above the period gap. Observations suggest the opposite behavior, and we discuss the possible explanations for this discrepancy. (6) Approximate orbital period distributions constructed from our evolution sequences suggest that the ratio of long-period CVs to short-period, pre-bounce CVs is about 3 × higher for the revised sequence than the standard one. This may resolve a long-standing problem in CV evolution. Tables describing our donor and evolution sequences are provided in electronically readable form.