7. Secular evolution with irradiation: numerical computations
7.1. Computational techniques
To compute the secular evolution of a compact binary with a low-mass donor star we have used the bipolytrope programme described in detail in KR92. With respect to the procedure described in KR92 we have, however, implemented two modifications in order to allow for a proper treatment of irradiation and its consequences for the secular evolution.
Expressions for the photospheric pressure scale height and the factor in terms of stellar parameters of the donor star and binary parameters are also given in Ritter (1988). In the context of this paper the donor star is always a low-mass main sequence star. For such stars both, and are only weak functions of the stellar mass and have typical values and yr-1. The reason for using (73) instead of (3) is that the latter is valid only for stationary mass transfer. However, this is not a good approximation when a system evolves through mass transfer cycles, because such cycles proceed unavoidably through phases of non-stationary mass transfer. In order to simplify the computation of we used fixed values for and , i.e. and yr-1. Furthermore we used (73) also when , i.e. when the donor star overfills its critical Roche volume. (73) is still a reasonable approximation if few , i.e. yr-1 (see e.g. Kolb & Ritter 1990) which is adequate for our purposes.
When computing the mass transfer rate from (73) we use for and , both of which depend on the effective temperature, the value of on the unirradiated part of the star, i.e. . There are two main arguments for this choice: first, in disk-accreting systems the inner Lagrangian point is in the shadow of the disk and the cooling time of gas in the superadiabatic layer flowing from the irradiated parts towards is short compared to the flow time in the shadow region. Second, if we use instead of , and would react practically instantaneously and with a large amplitude to irradiation giving, in turn, rise to a runaway of the mass transfer rate on an extremely short time scale (order of hours). This is obviously not what happens in the systems we do observe.
Second we use either Eq. (52) in the constant flux model or Eq. (59) in the point source model of irradiation in place of the usual Stefan-Boltzmann law as one of the outer boundary conditions for the donor star model. For the calculations presented below we have used the one-zone model described in Sect. 5 rather than results obtained from full stellar models by HR97 described earlier. Specifically, in (52) or (59) was computed by solving (68) in which is determined for an adopted value of from (24) with taken respectively from (48) and (49) in the case of the constant flux model, and from (54) in the case of the point source model. in (68) is taken from the numerical results shown in Fig. 7. Specifically, we have used for . Furthermore, in order to determine we need to specify b, i.e. the temperature exponent of the photospheric opacity law (10). For most of our experiments we have used which is adequate for H- opacity.
The diagnostic quantities and are calculated from (44) and respectively (53) or (60) with the appropriate choice of , i.e. Eqs. (48) or (54), and from (69).
7.2. Computational limitations
By using the bipolytrope approximation for describing the internal structure of the donor star, we can only deal with chemically homogeneous stars which have a radiative core and a convective envelope or are fully convective, i.e. with low-mass zero age main sequence stars. Thus, we are unable to address chemically evolved stars, in particular subgiants. Giants, on the other hand, have been dealt with approximately by KFKR97. We note also that chemically evolved donors among CVs might be more common than hitherto thought (Kolb & Baraffe 2000; Ritter 2000; Baraffe & Kolb 2000) and therefore deserve further study. For our computations, we have adopted Pop. I chemical composition. In the context of the bipolytrope approximation, the chemical composition is of relevance mainly for determining the appropriate gauge functions, i.e. polytropic index of the radiative core and entropy jump h in the surface layers as a function of stellar mass (for details see KR92). Even when adopting the appropriate gauge functions, we know that the value of computed in the bipolytrope approximation is smaller than what one would obtain for a full stellar model if . Because of this, in the bipolytrope approximation binary systems are more likely to be unstable against irradiation than they are in reality if .
We have used the one-zone model described in Sect. 5. As we have discussed there, application of this model is practically restricted to stars with a mass . We have, however, carried out a few calculations of systems with a smaller secondary mass. In those cases we have assumed if and otherwise. This corresponds to what (69) yields in the limit and approximates results of numerical calculations, at least for small fluxes, i.e. , reasonably well.
Our calculations contain a number of free or at best not well-constrained parameters. To mention just the most important ones: in the constant flux model we already make a very simplifying assumption about the function , i.e. the irradiation geometry. This assumption results in the free parameter s. In addition, we have the efficiency factor , a parameter about which we know little beyond the fact that probably . Using our one-zone model (Sect. 5) introduces furthermore the parameters and b, both of which can, however, be fixed reasonably well by comparison with full stellar models. In the point source model we do not need to specify s. But all the other parameters, i.e. , and b remain in the problem. By using numerical results for the functions and we would get rid of the parameters and b. But we would still be left with specifying and . On top of all that we have also a number of input parameters and functions which are already needed for computing a secular evolution without irradiation. Apart from the binary's initial parameters the most important of those are the angular momentum loss rate and parameters arising from assumptions about mass and consequential angular momentum loss from the binary system.
The purpose of the following computations, going beyond the linear stability analysis, is to illustrate the temporal evolution under the irradiation instability and a number of its specific properties which we have uncovered in the foregoing discussion.
As we have demonstrated in the previous section the irradiation instability is more likely of relevance for CVs than for LMXBs. Consequently, in the examples below we have assumed that the accretor is a white dwarf with a mass and a radius according to the mass radius relation (e.g. Nauenberg 1972) of cm. We assume that during the secular evolution the mass of the white dwarf remains constant, i.e. , and that on average the transferred matter leaves the system with the specific orbital angular momentum of the white dwarf. Other parameters characterizing the three examples which we shall discuss subsequently in detail are listed in Table 2.
Table 2. Parameters and model assumptions used for calculating the evolution shown in Figs. 11-13. Parameters common to all three examples are: , cm, , ; irradiation model: one-zone model with ; irradiation geometry: constant flux over surface fraction .
7.3. Numerical results
For the first example of a secular evolution with irradiation, results of which are shown in Fig. 11, we have adopted the constant flux model with , an initial secondary mass and loss of orbital angular momentum on a constant time scale yr derived from the Verbunt & Zwaan (1981) prescription for magnetic braking with . The value was chosen such that on the one hand initially and on the other at all times. The top panel of Fig. 11 shows the run of the mass transfer rate with irradiation (full line) and without (dashed line), the second the evolution of the secondary's radius with irradiation (full line) and without (dashed line). In the third panel we show the run of (full line) and of (dash-dotted line) with time. Finally the bottom panel shows the run of with time. As was to be expected for a system in which (initially) mass transfer evolves through cycles with at least initially large amplitude. The amplitude of the radius variations, on the other hand, are small, a consequence of the small value of . According to (73) we have . We see also that during the mass transfer peaks a significant fraction of the stellar luminosity is blocked. Because has been chosen such that always we always have . What is immediately apparent from this calculation is that though the system evolves through mass transfer cycles, these are damped on a rather short time scale, i.e. a time scale much shorter than . The reason for this is that decreases rapidly with time (mass of the donor). Eventually and mass transfer becomes stable. This is mainly a consequence of the increasing thermal inertia of the convective envelope, i.e. of a decrease of with decreasing donor mass. The damping of the oscillations could only be overcome if at the same time g increases sufficiently strongly with decreasing mass. We shall show below that in a restricted mass range this is indeed possible.
Given the results of the above example we now ask to what extent they are representative, i.e. whether the qualitative behaviour depends strongly on the adopted model parameters or not. With this end in view we have carried out numerous experiments, the results of which we shall now discuss.
Working with the one-zone model (Sect. 5) we ask first how the above results change with the parameter b, i.e. the adopted photospheric opacity. Inspection of Eq. (69) shows that for given values of and increases with b. This means that at least for small fluxes the donor star is more sensitive to irradiation for larger b. The reason for this is easy to understand: the larger b the more pronounced the increase of the optical depth in the superadiabatic layer in response to irradiation, i.e. of the average temperature, and thus the more effective the blocking of the energy outflow from the adiabatic interior. Thus, increasing (decreasing) b above (below) the value we have used in the example shown in Fig. 11 results in more (less) pronounced mass transfer cycles. The time scale on which the mass transfer oscillations are damped remains, however, essentially unaffected by changes of b, reflecting the fact that does not depend on b.
Next we compare the results obtained with the constant flux model (Fig. 11) with those obtained with the point source model. The results of a run with the latter model and parameters identical to those used for producing Fig. 11 (except of s which is not a free parameter in this model) are qualitatively very similar to those found in Fig. 11. The amount of stellar flux blocked during a mass transfer peak is, however, systematically smaller (by about a factor of two) than in the constant flux model. The main reason for this is that the irradiated area on the donor is smaller by about a factor of 1.6 than what we have assumed in the constant flux example, i.e. . Therefore, in order to achieve the same effect as in the constant flux model with , in the point source model needs to be increased by about a factor of two. Otherwise the results obtained from the two models are very similar. Because of this and because the constant flux model is computationally much less demanding we have performed most of our simulations with that model.
Next we examine briefly the dependence on the initial mass of the donor star. From our extensive discussions in Sects. 4 and 6 we know already that below a critical mass which is between about 0.6 and , depending on the adopted models, systems following a standard CV evolution are stable against irradiation.
Adopting the constant flux model and the one-zone model, a system with an initial secondary mass of is still unstable (whereas with the point source model and a more realistic such a system would be stable). Results of an evolution with and and the remaining parameters as in Fig. 11 (cf. Table 2) are shown in Fig. 12. As can be seen this evolution differs in several respects from the one shown in Fig. 11. Initially the amplitude of the mass transfer cycles increases with time. After only a few cycles the peak mass transfer rate is so high that the irradiation effect saturates, i.e. and . Although the one-zone model used here does not apply when approaches , the main effect of saturation can be modelled anyway by setting if . The qualitative behaviour obtained in this way remains the same as if a more realistic and smooth function was used. After an initial phase of increasing amplitudes they later start decreasing and die out very rapidly after about yr. This behaviour can be understood as follows: We have already pointed out above that increasing amplitudes of the cycles can only be expected if with decreasing mass increases fast enough. This is exactly what happens in the evolution shown in Fig. 12. The fast increase of is the result of the fast decrease of when going from to (see Fig. 7). Below , does no longer change much with mass. As a result, the mass transfer cycles are then damped because of the secular increase of . When the system eventually becomes stable, irradiation is still important in blocking the energy outflow. As can be seen from the bottom panel of Fig. 12, is of the order 0.37 after the system has stabilized.
Going to even lower initial secondary masses, mass transfer cycles cannot occur unless either or is much smaller than in a standard CV evolution (cf. our discussion in Sect. 6.1). For illustrating this we show in Fig. 13 the results of a calculation for which we have assumed , , and the much smaller angular momentum loss rate of gravitational radiation. Thus these parameters mimic an unstable system just below the period gap. As we have seen at the end of Sect. 6.1, CVs just below the period gap can be unstable if the constant flux model with is adopted. They are, however, stable if the point source model is used, unless is lowered below the standard value by invoking CAML (see MF98). If we use in addition to the constant flux model also the one-zone model in the limit , at low fluxes the donor star is even more susceptible to irradiation than if a more realistic form of had been used. However, for the purpose of this exercise it does not matter whether CVs below the gap are stable or not. This example was chosen just to demonstrate that with a low enough and depending on the value of a system could evolve through mass transfer cycles as we have concluded from our stability discussion in Sects. 4 and 6. One additional property of this run which is worth mentioning is that the cycles are only rather weakly damped. The reason for this is that a star with a mass is always fully convective and therefore remains constant. Thus the decrease of is mainly due to the slow increase of with time.
© European Southern Observatory (ESO) 2000
Online publication: August 23, 2000