9 Evolution of Interacting Double-Degenerate Systems

Angular momentum losses via GWR may bring detached double degenerates into contact. The mass–radius relation for degenerate stars has a negative power (≃ − 1∕3 for WDs with a mass exceeding 0.1 M ⊙, irrespective of their chemical composition and temperature [76Jump To The Next Citation Point]). 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 q [196],

( )1 ∕3 R ≈ 0.4622 a ---m---- , (62 ) L M + m
where m and M are, respectively, the masses of the prospective donor and accretor, renders two relations important for the study of compact binaries.
  1. It provides a relation between the orbital period of the binary Porb and the mass M2 and radius R2 of the donor:
    ( R )3∕2( 0.1M )1 ∕2 Porb ≃ 101 s ----2--- ------⊙ . (63) 0.01R ⊙ M2
  2. It allows to derive the rate of mass transfer for a semidetached binary in which mass transfer is driven by angular momentum losses:
    ( ) ( ) m˙ ˙J ζ (m ) 5 m −1 m- = J- × --2-- + 6-− M-- , (64) GWR
    where ζ(m ) = d ln r∕d ln m.

For the mass transfer to be stable, the term in the brackets must be positive, i.e. 

m-- 5- ζ-(m-) M < 6 + 2 . (65 )
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 (65View Equation) clearly oversimplifies the conditions for stable mass exchange. A rigorous treatment has to include a consideration of tidal effects, angular momentum exchange, and possible super-Eddington M˙ immediately after RLOF by the donor and associated common-envelope formation, possible ignition of accreted helium [446138252122]. However, the study of these effects is still in the embryonic state and usually the evolution of IDDs or other binaries with WD donors is calculated applying MR relations, without considering detailed models of WDs (or low-mass helium stars in the appropriate formation channel); see, e.g., [301436100417Jump To The Next Citation Point341395433Jump To The Next Citation Point422Jump To The Next Citation Point286Jump To The Next Citation Point472Jump To The Next Citation Point287Jump To The Next Citation Point]. Figure 9View Image shows examples of the evolution for systems with a helium degenerate donor or a low-mass “semidegenerate” helium star donor and a carbon-oxygen accretor with initial masses that are currently thought to be typical for progenitors of AM CVn systems – 0.2 and 0.6M ⊙, respectively. For Figure 9View Image the mass–radius relation for zero-temperature white dwarfs from the article by Verbunt and Rappaport [433] is used; for the low-mass He-star mass–radius relation approximating results of evolutionary computations by Tutukov and Fedorova [409] were used: R ≈ 0.043 m −0.062 (in solar units). The same equations are applied for obtaining the model of the population of AM CVn-stars discussed in Sections 10, 11, and 12. The mass of the donor in the system may be a discriminator between the formation channels. For instance, a large mass of the donor (0.18M ⊙ ) found for the prototype of the class, AM CVn itself [355] favours the helium channel for this system.
View Image

Figure 9: Mass-loss rate vs. orbital period for “typical” AM CVn-stars: an interacting double degenerate system with initial masses of donor and accretor 0.2M ⊙ and 0.6 M ⊙ (red line) and a “semidegenerate” low-mass helium star donor plus white dwarf accretor of the same initial masses (blue line). Black dots on the red curve mark positions of the system at log T (yr) = 5,6,7,8,9,10 from Roche-lobe overflow; on the blue curve they mark logT (yr) = 5,6, 7,8. Green lines mark lower and upper limits of the disk instability region according to Tsugawa and Osaki [406Jump To The Next Citation Point]. Below the magenta circles q < 0.02 and conventional evolutionary computations may be not adequate for description of mass-transfer process (see the text).

From the Equations (22View Equation, 62View Equation, 64View Equation) it follows that for m ≪ M the mass loss rate scales as 1∕3 M. As a result, for all combinations of donor and accretor, the Pm˙ lines form two rather narrow strips within which they converge with decreasing m. We should note that the time span between formation of a pair of WDs and contact may be from several Myr to several Gyr [422Jump To The Next Citation Point]. 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 ˙ M [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 [411412320]. 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 M ≃ 0.3 M ⊙ 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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