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 [77Jump To The Next Citation Point] 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 [77Jump To The Next Citation Point].
View Image

Figure 18: Effects of a finite entropy of donors on the properties of AM CVn-stars. The left panel shows the relation between Porb and M˙ along tracks for a system with initial masses of components 0.2 M ⊙ and 0.6 M ⊙ (like in Figure 9View Image). The solid lines show the evolution for donors with Tc = 104 K, 106 K, 5 × 106 K, and 107 K (left to right). The symbols show the positions of models with M2 = 0.01 M ⊙ (triangles), 0.02 M ⊙ (squares), and 0.05M ⊙ (pentagons). The disk stability criteria (for q = 0.05) are shown by the dashed lines (after [406]). The right panel compares the numbers of systems as a function of Porb for the model with a T = 0 WD (dot-dashed line) and the model with “realistic” cooling (solid lines, RWDC). The smooth curves show the percentage of each population laying above a given Porb. (Figures from [77].)

The effects of finite entropy become noticeably important for M ≲ 0.1 M ⊙. Isentropic WD with T > 0 (i) have larger radii than T = 0 objects and (ii) the MR relations for them are steeper than for T = 0 (i.e. in the range of interest they are still negative but have a lower absolute value). By virtue of Equations (63View Equation, 64View Equation) this means that for a given orbital period they have higher M ˙. This effect is illustrated in the left panel of Figure 18View Image. (The period–mass relation is not single-valued, since the MR 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 18View Image compares a model of the population of AM CVn-stars computed under assumptions that the donor white dwarfs have T = 0 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 T = 0 population.

Finite entropy of the donors also influences the gravitational waves signals from AM CVn-stars. Again, by virtue of the requirement of R = R donor L, the systems with T = 0 donors and hot donors will have a different Porb for the same combination of component masses, i.e. different radii at the contact and different relation between chirp mass ℳ and Porb. This alters the GW amplitude, Equation (17View Equation.

For instance, if a 0.2 M ⊙ donor is fully degenerate, it overflows its Roche lobe at Porb ≈ 3.5 min and then evolves to longer P orb. If “realistic” cooling is considered, there are donors that make contact at Porb up to ≈ 25 min. Hotter donors at fixed Porb are more massive, increasing ℳ and increasing h. 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 f, since they are lost in the background confusion noise of Galactic WDs. But altogether, the ensemble of sources detectable by LISA with S∕N > 1 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.

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