### 12.1 Effects of finite entropy

Results concerning the AM CVn-stars population presented above were obtained assuming a
mass–radius relation for zero-temperature WDs. Evidently, this is a quite crude approximation, having in
mind that in some cases the time span between the emergence of the second white dwarf from the
common envelope and contact may be as short as several Myr [422]. As the first step to more
realistic models, Deloye and coauthors [77] considered the effects of finite entropy of the donors
by using their finite-entropy models for white dwarfs [76]. We illustrate some of these effects
following [77].
The effects of finite entropy become noticeably important for . Isentropic WD with
(i) have larger radii than objects and (ii) the – relations for them are steeper
than for (i.e. in the range of interest they are still negative but have a lower absolute value). By
virtue of Equations (63, 64) this means that for a given orbital period they have higher . This effect is
illustrated in the left panel of Figure 18. (The period–mass relation is not single-valued, since the –
relation has two branches: a branch where the object is thermally supported and a branch where degenerate
electrons provide the dominant pressure support.) The right panel of Figure 18 compares a
model of the population of AM CVn-stars computed under assumptions that the donor white
dwarfs have and a model which takes into account cooling of the prospective donors
between formation and RLOF. The change in the rate of evolution (shown in the left panel)
shifts systems with “realistic” cooling to longer orbital periods as compared to the
population.

Finite entropy of the donors also influences the gravitational waves signals from AM CVn-stars. Again,
by virtue of the requirement of , the systems with donors and hot donors will have a
different for the same combination of component masses, i.e. different radii at the contact
and different relation between chirp mass and . This alters the GW amplitude,
Equation (17.

For instance, if a donor is fully degenerate, it overflows its Roche lobe at and
then evolves to longer . If “realistic” cooling is considered, there are donors that make contact at
up to 25 min. Hotter donors at fixed are more massive, increasing and increasing
. Thus, the contribution of the individual systems to the integrated GW flux from the total ensemble
increases, but their higher rate of evolution decreases the density of population of the sources detectable at
low , since they are lost in the background confusion noise of Galactic WDs. But altogether,
the ensemble of sources detectable by LISA with diminishes by about 10% only.
Note that finite entropy of donors does not significantly affect the properties of the 10,000
systems that are expected to be observed both in electromagnetic spectrum and gravitational
waves.

There are some more subtle effects related to finite entropy for which we refer the reader to the original
paper.