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Astron. Astrophys. 360, 969-990 (2000)

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8. Discussion and conclusions

With the exception of the numerical examples presented in the previous section, where we have assumed the donor star to be on the ZAMS, we have so far not made any explicit assumption about the evolutionary status of the secondary. In fact our stability considerations are valid for any type of donor star as long as it has a sufficiently deep outer convective envelope. In the following we shall therefore briefly address the stability of mass transfer and the peak mass transfer rate in case of instability if the secondary is more or less nuclear evolved. In addition we shall briefly discuss the impact of the donor's metallicity on the stability of mass transfer. A detailed investigation of the irradiation instability in those cases is, however, deferred to a subsequent paper.

8.1. Effects of evolution and metallicity on the stability of mass transfer

Returning to the stability criterion (41) we see that the left-hand side [FORMULA] depends essentially only on the ratio of two time scales, i.e. the thermal time scale of the convective envelope [FORMULA] and the mass transfer time scale [FORMULA] or, respectively, on the time scale on which mass transfer is driven [FORMULA]. Mass transfer is more stable the larger the ratio [FORMULA].

Evolutionary computations such as those shown in Ritter (1994) show that at a given secondary mass [FORMULA] is the shorter the more evolved the star. If magnetic braking according to Verbunt & Zwaan (1981) is invoked for computing the angular momentum loss rate, numerical computations (e.g. those shown in Ritter 1994, or King et al. (1996a) of the mass transfer from a nuclear evolved donor show that [FORMULA] is the longer the more evolved the donor at the onset of mass transfer. Taken together this means that mass transfer is the more unstable (less stable) the more evolved the donor. As an example, evaluating [FORMULA] along the evolutionary tracks shown in Ritter (1994) we find that as long as the secondary has not reached the terminal age main sequence, mass transfer is stable if [FORMULA]. In the sequence starting with a [FORMULA] secondary at the end of central hydrogen burning, systems down to an orbital period [FORMULA] may be unstable.

Main sequence stars of low metallicity are systematically hotter, smaller, more luminous and have a thinner convective envelope than main sequence stars of solar metallicity. Hence [FORMULA] is systematically shorter for low metallicity stars. At the same time [FORMULA] is the shorter the lower the metallicity. Taken together this means that mass transfer is systematically more unstable (less stable) the lower the metallicity. As an example, using results from Stehle et al. (1997), we find that in CVs with a Pop. II donor mass transfer could be unstable for orbital periods as low as [FORMULA].

The case of systems containing a giant donor and in which mass transfer is driven by its nuclear evolution, i.e. [FORMULA], has already been dealt with in some detail by KFKR97. We shall mention here only that such systems are systematically more unstable against irradiation-induced runaway mass transfer than systems with a main sequence donor. There are three reasons for this: First, for a giant [FORMULA] is systematically shorter than for a main sequence star of the same mass (because of the much larger values of R and L), and this despite the fact that [FORMULA] can be quite large. Second, in terms of radius changes due to irradiation, giants react much more strongly than main sequence stars. (Note that in the framework of the homology model presented in Sect. 3 the effective [FORMULA] for giants is -3). Third, the mass transfer time scale [FORMULA]associated with nuclear evolution of a giant is always much longer than [FORMULA] over the entire range of core masses of interest, i.e. [FORMULA]. For details see KFKR97 and Ritter (1999).

8.2. The peak mass transfer rate

Our estimate (72) of the peak mass transfer rate achieved during a mass transfer cycle can be rewritten as


In the case of a fully developed instability it is the second term in (74) which dominates. Therefore, the peak mass transfer rate is essentially determined by the rate of expansion of the donor, i.e.


Thus the lower the thermal inertia of the convective envelope the higher the peak mass transfer rate. As a consequence, very high peak rates can result if the mass of the convective envelope or [FORMULA] is small. If this is the case the long-term evolution of the corresponding systems could be drastically changed. We note e.g. that even if the donor is a main sequence star, the peak mass transfer rate can easily exceed the value required for maintaining stable nuclear burning on the white dwarf, i.e. [FORMULA] (e.g. Fujimoto 1982). With a Pop. I ZAMS donor this limit is reached if [FORMULA] (see Fig. 11), with a donor at the end of central hydrogen burning if [FORMULA] or [FORMULA]. These values are derived from the evolutionary calculations discussed in Ritter (1994).

The consequences of reaching peak mass transfer rates equal to or even in excess of the stable nuclear burning limit of the white dwarf can be far-reaching. First, if steady nuclear burning on the white dwarf is reached, such a system will no longer appear as an ordinary CV but rather look more like a supersoft X-ray source. Because such systems are bright in the EUV only, they become virtually undetectable in our Galaxy: very few such systems have indeed been found (see e.g. Greiner 1996). Second, with nuclear burning on the white dwarf, the nuclear luminosity exceeds the accretion luminosity by a factor of [FORMULA]. With so much more irradiation luminosity available effects other than those discussed in this paper might also become important, e.g. driving of a strong wind from the donor (see van Teeseling & King 1998, King & van Teeseling 1998), and which would change the evolution of such systems altogether. Third, an unavoidable consequence of the very high peak mass transfer rates are very extended low states during which a system is practically detached and thus virtually undetectable. Fourth, from the fact that such systems are barely detectable in both the high and low state and that the transition time between high and low states and vice versa is very short (see KFKR97), one is practically forced to the conclusion that the observed long-period CVs are either stable against the irradiation instability or, for some other reason do not reach peak rates equal to or larger than the stable nuclear burning limit.

8.3. Conclusions

In this paper we have studied the reaction of low-mass stars to anisotropic irradiation and its implications for the long-term evolution of compact binaries. For this we have shown that the case of anisotropic irradiation in close binaries is relevant and that spherically symmetric irradiation is probably not an adequate approximation. We have studied the reaction of low-mass stars to anisotropic irradiation by means of simple homology considerations. We have shown in the framework of this model that if the energy outflow through the surface layers of a low-mass main sequence star is blocked over a fraction [FORMULA] of its surface, it will inflate only modestly, by about a factor [FORMULA] (see Eq. 17) in reaching a new thermal equilibrium, and that the maximum contribution to mass transfer due to thermal relaxation is [FORMULA] times the value one obtains for spherically symmetric irradiation (i.e. [FORMULA]). The duration of the thermal relaxation phase is [FORMULA] (Eqs. 22 and 23), where [FORMULA] is the thermal time scale of the convective envelope. In addition, we have carried out numerical computations of the thermal relaxation process of low-mass stars involving full stellar models and using the modified Stefan-Boltzmann law (9) as an outer boundary condition. The results of these computations (shown in Figs. 2-4) fully confirm results from homology and show that the effects caused by anisotropic irradiation are not only quantitatively but also qualitatively different from those caused by spherically symmetric irradiation.

Next we have carried out a detailed stability analysis. The criterion for stability against irradiation-induced runaway mass transfer in its most general form (arbitary irradiation geometry) is given in Eqs. (41) and (45). One of the remarkable results of this stability investigation is that it is not arbitrarily large irradiation fluxes which destabilize a system most effectively. Rather the most effective irradiating flux is [FORMULA], where [FORMULA] is the surface flux of the unperturbed star.

The reaction of the stellar surface to irradiation is best expressed in terms of a function [FORMULA], where [FORMULA] is the normalized flux. General considerations show that [FORMULA]. For determining [FORMULA] we used a simple, analytic one-zone model for the superadiabatic convection zone. The results of this simple model (Eq. 69) are found to be in good qualitative and satisfactory quantitative agreement with results of computations involving full stellar models (see Fig. 8).

Application of our stability analysis to CVs and LMXBs yields the following results:

CVs which evolve according to the standard evolutionary paradigm, i.e. the model of disrupted magnetic braking, are stable against irradiation-induced runaway mass transfer if the mass of the (ZAMS) donor is [FORMULA]. This holds unless substantial consequential angular momentum losses greatly destabilize the systems. Systems in which the mass of the (ZAMS) secondary is [FORMULA] can be unstable but need not be so, depending on the efficiency of irradiation [FORMULA] (defined in Eq. (24)). Substantial consequential angular momentum losses can however destabilize CVs over essentially the whole range of periods of interest.

CVs with a Pop. II or an evolved secondary are inherently less stable than CVs with a Pop. I ZAMS secondary. The latter are those stars which, for a given mass, have the highest thermal inertia, making the corresponding CVs the most stable ones.

If mass transfer is unstable, we found that it must evolve through a limit cycle in which phases of irradiation-induced mass transfer alternate with phases of small (or no) mass transfer. At the peak of a cycle mass transfer proceeds on a time scale which is roughly [FORMULA] times the thermal time scale of the convective envelope (see Eq. 75). With decreasing mass of the secondary the amplitude of the mass transfer cycles gets smaller and the cycles eventually disappear (after a system has become stable) because the thermal inertia of the secondary (expressed in the functions [FORMULA] and [FORMULA] defined respectively in Eqs. (41) and (45)) increases strongly (see Fig. 10 and Fig. 11). A necessary condition for the maintenance of cycles is that the thermal time scale of the convective envelope has to be much shorter ([FORMULA]) than the time scale on which mass transfer is driven.

LMXBs are very likely to be stable because a) the donor star is strongly shielded from direct irradiation over most of the hemisphere facing the X-ray source, and b) because where this is not the case, [FORMULA], i.e. the sensitivity of the stellar surface to changes in the irradiating flux, is very small.

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Online publication: August 23, 2000