Angular momentum losses via GWR may bring detached double degenerates into contact. The mass–radius
relation for degenerate stars has a negative power ( for WDs with a mass exceeding
,
irrespective of their chemical composition and temperature [76
]). Hence, the lower mass WD fills its Roche
lobe first.
In a binary with stable mass transfer the change of the radius of the donor exactly matches the change
of its Roche lobe. This condition combined with an approximation to the size of the Roche lobe valid for
low [196],
For the mass transfer to be stable, the term in the brackets must be positive, i.e.
Violation of this criterion results in mass loss by the donor on a dynamical time scale and, most probably, merger of components. Of course, Equation (65 From the Equations (22, 62
, 64
) it follows that for
the mass loss rate scales as
.
As a result, for all combinations of donor and accretor, the
–
lines form two rather
narrow strips within which they converge with decreasing
. We should note that the time
span between formation of a pair of WDs and contact may be from several Myr to several
Gyr [422
]. This means that the approximation of zero-temperature white dwarfs is not always
valid. Below we discuss the implications of finite entropy of the donors for the population of
AM CVn-stars.
The “theoretical” model of evolution from shorter periods to longer ones is supported by observations
which found that the UV luminosity of AM CVn-stars is increasing as the orbital period gets shorter, since
shorter periods are associated with higher [338].
Note that there is a peculiar difference between white dwarf pairs that merge and pairs that start stable
mass exchange. The pairs that coalesce stop emitting GWs in a relatively small time-scale (of the order of
the period of the last stable orbit, typically a few minutes) [237]. Thus, if we would be lucky to observe a
chirping WD and a sudden disappearance of the signal, this will manifest a merger. However, the chance of
such event is small since the Galactic occurrence rate of mergers of WDsis 10–2 yr–1
only.
Apart from “double-degenerate” and “helium-star” channels for the formation of AM CVn-stars, there
exists the third, “CV”-channel [411, 412, 320]. In this channel, the donor star fills its Roche lobe at the
main-sequence stage or just after its completion. For such donors the chemical inhomogeneity
inhibits complete mixing at typical for initially non-evolved donors. The mixing is
delayed to lower masses and as a result the donors become helium dwarfs with some traces of
hydrogen. After reaching the minimum period they start to evolve to the longer ones. The minimum
of periods for these systems is
5 – 7 min. However, the birth rate of systems that can
penetrate the region occupied by observed AM CVn-stars is much lower than the birth rate in
“double degenerate” and “helium-star” channels and we do not take this channel into account
below.
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