THE EVOLUTION OF CATACLYSMIC VARIABLES AS REVEALED BY THEIR DONOR
STARS
Authors: Knigge Christian , Baraffe Isabelle , Patterson Joseph 
Published: May 2011

ABSTRACT
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
f
GR = 2.47 ± 0.22 below the gap and
f
MB = 0.66 ± 0.05 above (the errors here are
purely statistical, and the standard model corresponds to
f
GR = f
MB = 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
f
GR, 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 P
orb 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
P
orb 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.