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

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6. Secular evolution with irradiation

We resume the stability discussion of Sect. 4, but now with the knowledge of the function [FORMULA] which we have gained in Sect. 5. We shall restrict most of the following to the constant flux model (Sect. 4.2.1) and the point source model (Sect. 4.2.2). In addition, we shall assume that all the properties of a binary along a secular evolution without irradiation, in particular the functions [FORMULA] and [FORMULA] defined respectively in Eqs. (41) and (45), are known. Among the compact binaries we specifically discuss cataclysmic variables, and low-mass X-ray binaries. In particular, we wish to examine the following questions: a) which systems are stable (unstable) at 1) turn-on of mass transfer, or 2) during the secular evolution, against irradiation-induced runaway mass transfer, and b) what kind of evolution do systems undergo which are unstable? Because part of these questions have been dealt with extensively by King (1995), KFKR95, KFKR96, KFKR97 and MF98, we shall mainly be concerned with those aspects which have not already been treated in detail in the above papers.

6.1. Cataclysmic variables

In the following we set [FORMULA] and [FORMULA], where [FORMULA] and [FORMULA] are respectively the mass and the radius of the accreting white dwarf. Furthermore we shall restrict our discussion to CVs where the secondary is a low-mass main sequence star. Because this is the case for the vast majority of CVs this is not a very strong restriction. For determining the values of [FORMULA] and [FORMULA] we adopt the standard evolutionary scheme for CVs, i.e. the model of disrupted magnetic braking (e.g. King 1988 for a review) and use results of corresponding model calculations by KR92.

6.1.1. Stability of mass transfer at turn-on

Because [FORMULA] is maximal for [FORMULA], systems are most susceptible to irradiation at turn-on of mass transfer. The relevant stability criterion for the constant flux model is (51) and for the point source model Eq. (58). As we have already pointed out in Sect. 4 these conditions are in fact conditions on the efficiency factor [FORMULA] because everything else is basically fixed. Given the masses of both binary components, i.e. [FORMULA] and [FORMULA], [FORMULA] follows from the mass radius relation of white dwarfs, the secondary's radius [FORMULA] from the mass radius relation of main sequence stars, or from the evolutionary history. The orbital separation a follows from Roche geometry via [FORMULA] and the mass ratio, [FORMULA] and [FORMULA] from the secondary's internal structure, and finally [FORMULA] again from the mass ratio. Thus [FORMULA] (Eq. 45) is uniquely determined by [FORMULA] and [FORMULA]. Given the function [FORMULA], and in particular [FORMULA], condition (51) is one for [FORMULA], and condition (58) one for [FORMULA] only. Taking as an example g from our one-zone model with [FORMULA], [FORMULA] and [FORMULA] for [FORMULA] from Fig. 7, we can plot a line for a given [FORMULA] in the [FORMULA]-[FORMULA] plane along which mass transfer is marginally stable. This is shown in Fig. 9 for various values of [FORMULA]. The figure is to be read as follows: a parameter combination ([FORMULA], [FORMULA]) corresponds to a point in Fig. 9. If that point lies above the line corresponding to a given value of [FORMULA], mass transfer is unstable at turn-on. What we can infer from Fig. 9 is that for typical WD masses [FORMULA], condition (51) is violated for surprisingly small values of [FORMULA], i.e. [FORMULA]. This is the case, in particular, if [FORMULA]. If we take instead of (51) the condition of the point source model (58), s is no longer a parameter. Taking again the same g as above (one-zone model with [FORMULA]) the result is qualitatively the same. The main difference is that the value of [FORMULA] necessary for marginal stability needs to be larger by about a factor of 2.

[FIGURE] Fig. 9. Contour lines in the [FORMULA]-plane along which [FORMULA] for different values of [FORMULA]. Systems above (below) a particular line are unstable (stable) against irradiation-induced runaway mass transfer at the turn-on of mass transfer.

The trends seen in the curves of Fig. 9 are easily explained: they reflect the run of [FORMULA] (cf Eq. 46). The smaller [FORMULA] the less stable mass transfer. Because [FORMULA] is proportional to [FORMULA] and [FORMULA], and [FORMULA] is a steeply decreasing function of [FORMULA] whereas [FORMULA] scales roughly as the relative mass of the convective envelope (cf. Fig. 1, dashed line) and thus decreases strongly with [FORMULA], mass transfer is more likely to be unstable (stable) the higher (lower) [FORMULA] and [FORMULA]. The other factors entering [FORMULA], i.e. [FORMULA], [FORMULA] and [FORMULA] are of comparatively minor importance. The fact that the curves in Fig. 9 are not monotonic results from the steep increase of [FORMULA] with stellar mass near [FORMULA] (cf. Fig. 7).

Because [FORMULA], systems in which mass transfer is stable at turn on will also be stable against irradiation-induced runaway mass transfer for any finite value of [FORMULA], i.e. [FORMULA]. The opposite, however, is not true: not all systems which are unstable at turn-on will be so for the secular mean mass transfer rate.

6.1.2. Stability of secular evolution

The appropriate stability criterion is (41) in its most general form, (50) for the constant flux model and (57) for the point source model. Again, the left-hand side [FORMULA] of these criteria is fully determined by the adopted model of secular evolution, whereas the value of the corresponding right-hand sides still depends on [FORMULA]. As we have stressed in Sect. 4, we can give a sufficient criterion for stability of mass transfer because the function [FORMULA] has a maximum. The corresponding criterion is (43) in its most general form,

[EQUATION]

for the constant flux model, and

[EQUATION]

for the point source model. Conditions (70) and (71) can be read off Fig. 10 where we plot [FORMULA] as a function of secondary mass [FORMULA] (full line) along a standard secular evolution of a CV with [FORMULA] and an initial secondary mass [FORMULA]. The evolutionary data are taken from KR92. Into the same figure we plot Max[FORMULA] [FORMULA] Max[FORMULA] for [FORMULA]) and Max[FORMULA] using results by HR97 for [FORMULA] (dashed lines). As can be seen the values of Max[FORMULA] and Max[FORMULA] are almost independent of secondary mass, i.e. of the orbital period. For CVs above the period gap, i.e. with [FORMULA], [FORMULA] increases very steeply with decreasing secondary mass. This behaviour is mainly due to two factors, first to [FORMULA] which increases strongly with decreasing mass, and second to [FORMULA] which increases strongly towards lower masses mainly because of the mass luminosity relation [FORMULA] of low-mass main sequence stars. So, what [FORMULA] essentially represents is the thermal inertia of the convective envelope. As can be seen from Fig. 10, the intersection of [FORMULA] with Max[FORMULA] or Max[FORMULA] is near [FORMULA]. Because [FORMULA] increases so steeply, small changes in Max[FORMULA] or Max[FORMULA] do not yield a significantly different result.

[FIGURE] Fig. 10. The functions [FORMULA] (Eq. 41) (full line), [FORMULA] and [FORMULA] according to the one-zone model (dashed lines 1 and 2 respectively), and [FORMULA] and [FORMULA] derived from the results of HR97 (dashed lines 3 and 4 respectively) as a function of the secondary's mass along a standard evolution of a CV with [FORMULA] and [FORMULA] calculated by KR92.

When a CV approaches the period gap, i.e. the secondary becomes fully convective at [FORMULA], the system detaches and [FORMULA] because [FORMULA]. When mass transfer resumes [FORMULA] is smaller by typically a factor 10 than immediately above the gap, reflecting the reduced angular momentum loss below the gap and the fact that at turn-on [FORMULA] (cf. 41). [FORMULA] starts increasing again with further decreasing mass because the Kelvin-Helmholtz time [FORMULA] becomes very long as the secondary becomes degenerate.

Because [FORMULA] does not depend strongly on the mass of the white dwarf we can generalize the result found from Fig. 10: In the framework of standard CV evolution, i.e. the model of disrupted magnetic braking, CVs are stable against irradiation-induced runaway mass transfer when the mass of the secondary star is [FORMULA]. If, on the other hand, [FORMULA], a system can be unstable but need not be so, subject to the value of [FORMULA].

We note, however, that by invoking substantial consequential angular momentum loss (CAML), i.e. angular momentum loss which depends explicitly on the mass transfer rate (see King & Kolb 1995 for a discussion of CV evolution with CAML), the mass range over which systems can be unstable is much larger. This has already been pointed out by KFKR96 and confirmed in computations by MF98.

As can be seen in Fig. 10 there is also a slight chance that CVs immediately after turn-on below the period gap are unstable. According to the constant flux model (Eq. 72), some CVs could be unstable, according to the point source model, which is more realistic but still very optimistic, the stability criterion (71) is violated only marginally if at all. Therefore, our conclusion is that CVs below the period gap are very probably stable, at least in the framework of standard CV evolution.

Examining the factors which determine the value of [FORMULA] (Eq. 41) we see that within a given evolutionary model all factors are determined. [FORMULA], [FORMULA] and [FORMULA] depend only on the mass of the secondary and its evolutionary history, [FORMULA] on the adopted rate of angular momentum loss, [FORMULA] and the evolutionary history (via [FORMULA] in (3)), [FORMULA] on the mass ratio and [FORMULA] on the evolutionary history (e.g. Stehle et al. 1996). We see also that making [FORMULA] smaller (in order to get the instability for lower secondary masses) is possible only by either increasing [FORMULA], i.e. lowering the mean mass transfer rate [FORMULA], or by increasing [FORMULA]. The former is practically impossible without upsetting the standard evolutionary paradigm for CVs, i.e. the period gap model which requires that above the period gap ([FORMULA], [FORMULA]) [FORMULA] (e.g. Ritter 1984, King 1988; KR92; Stehle et al. 1996). Increasing [FORMULA], on the other hand, is possible only if a system experiences significant CAML.

6.2. Low-mass X-ray binaries

At first glance one might suspect that in LMXBs irradiation of the secondary represents a much larger threat for the stability of mass transfer than it does in CVs. However, for the following reasons this is very probably not the case: first we note the observational fact that very few of the LMXBs show X-ray eclipses. This has been interpreted as a consequence of the large vertical scale height of the X-ray irradiated accretion disk. This in turn allows the secondary to stay permanently in the disk's shadow. If this is the case, none or at most a small part of the secondary's surface is directly exposed to the accretion light source. Second, indirect illumination of significant parts of the donor (the high latitude regions or part of the back side) is ruled out because this would require a very extended scattering corona indeed, with a typical size of the scattering sphere (at optical depth [FORMULA]) of order or larger than the donor star. As a consequence one would expect X-ray eclipses to occur much more frequently than they are actually observed. Third, heat transport by currents from the hot, illuminated to the cool parts in the X-ray shadow are probably also negligible. This is because in cool stars with a deep convective envelope the superadiabatic convection zone isolates the interior from the surface. It is itself unable to transport significant amounts of heat because of its small heat capacity and the fact that the thermal time scale is much shorter than the time scale of circulation.

Heat transport by currents caused by hydrostatic disequilibrium is, however, of importance in stars with a radiative envelope. Effects of this are e.g. seen in the X-ray binaries HZ Her and V1033 Sco (see e.g. Shahbaz et al. (2000), and references therein).

Because neither indirect llumination nor heat advection can contribute significantly to the blocking of the stellar flux, the integral I on the right-hand side of (41) is much smaller than for a comparable CV, despite the fact that in a LMXB there is potentially much more energy available for irradiating the secondary. The fact that [FORMULA] has a maximum at [FORMULA] is the reason why even large irradiating fluxes do not help. Rather, optimal irradiation is achieved if as large a fraction as possible of the secondary's surface is irradiated with a flux such that [FORMULA] is near its maximum, i.e. if [FORMULA]. This is clearly not the case in LMXBs. Not only is the surface fraction which is directly irradiated small, worse, where the surface of the secondary is directly exposed to the X-ray source, the associated flux is large, i.e. [FORMULA] unless [FORMULA] is assumed to be very small. The latter is very unlikely considering the fact that most of the accretion luminosity emerges in form of rather hard X-rays. Thus, unless the secular evolution of LMXBs is totally unlike that of CVs as far as the nature of the secondary and the typical mass transfer rates are concerned, the value of [FORMULA] is virtually the same as for CVs but, as explained above, the right-hand side of (41) is much smaller than in CVs. Therefore we conclude that very probably LMXBs are stable against this type of irradiation-induced runaway mass transfer. As has been noted by KFKR97, systems in which accretion is intermittent rather than continuous, i.e. transient LMXBs, are even more stable.

6.3. Evolution of unstable systems

Let us now discuss briefly the evolution of systems (i.e. CVs) in which the stationary mass transfer given by (3) is unstable, i.e. systems for which the stability criterion (41) or a special form thereof (Eqs. 50 or 57) is violated. From the fact that [FORMULA] we know that mass transfer is then already unstable at turn-on. Therefore, when mass transfer turns on, the mass transfer rate increases, and because (41) is violated, it does not settle at the secular mean [FORMULA] given by (3). However, because the thermal relaxation caused by irradiation saturates both in amplitude and with time (see our discussion in Sect. 3, in particular Eqs. (20) and (23), and Fig. 4) the mass transfer rate does not run away without bound. Rather, there is an upper limit: with (20) we obtain for the maximum mass transfer rate

[EQUATION]

Because mass transfer can be unstable against irradiation only if [FORMULA], i.e. when the secondary has a relatively thin convective envelope and thus [FORMULA] is large (cf. Fig. 1), the maximum contribution of irradiation to mass transfer can be several times ([FORMULA] times) the thermal time scale mass transfer rate [FORMULA]. Thus, for such systems Max[FORMULA].

After having reached the peak mass transfer rate, mass transfer cannot continue at that rate. Rather it must decrease with time for two reasons: first, thermal relaxation saturates on the time scale given by (23), i.e. [FORMULA] decreases on that time scale. Second, because mass transfer occurs at a rate above the secular mean, the binary system is driven apart, i.e. [FORMULA] [FORMULA] [FORMULA]. Both effects result eventually in the termination of mass transfer. The system becomes slightly detached, the secondary, in the absence of irradiation, shrinks. However, because of the absence of mass transfer the contraction of the system due to angular momentum loss is fast enough to catch up, so that mass transfer resumes and the cycle repeats again. In other words: if a system is unstable at the secular mean mass transfer rate it must undergo a limit cycle in which phases of enhanced, irradiation-driven mass transfer alternate with phases of very low or no mass transfer. The conditions for the occurrence of mass transfer cycles in semi-detached binaries have been investigated in more detail and using more general principles (in the framework of non-linear dynamics) by King (1995), KFKR95, KFKR96 and KFKR97. Their main result is that mass transfer cycles driven by radius variation of the secondary can only occur if [FORMULA] in (1) depends explicitly on the instantaneous mass transfer rate. The only plausible mechanism providing such a dependence is irradiation of the donor star by radiation generated through accretion, i.e. the situation we are studying in this paper. The necessary criterion for the occurrence of mass transfer cycles found by these authors is identical to what we have found here, namely the violation of (41). In the framework of a linear stability analysis of mass transfer, with which we, King (1995), KFKR95, KFKR96 and KFKR97 have been concerned so far, we can not calculate the long-term evolution over time scales [FORMULA] of systems under the irradiation instability. For this the full set of equations describing mass transfer and stellar structure under irradiation have to be solved. Results of such calculations and computational details will be presented in the following section. Because similar computations have already been done by HR97 and MF98, we shall concentrate here on aspects which have not been dealt with in detail by HR97 and MF98, but are, in our opinion, important for better understanding of the evolution under the irradiation instability.

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Online publication: August 23, 2000
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