2. Computing the mass transfer rate
In the context of this paper, compact binaries are either cataclysmic variables (CVs) or LMXBs, i.e. systems consisting of a low-mass star (the secondary) with a mass which fills its critical Roche lobe and transfers matter to a compact companion (the primary), either a white dwarf (in CVs), or a neutron star or a black hole (in LMXBs). The secular evolution of such systems is a consequence of mass loss from the secondary which, in turn, is driven either by the secondary's nuclear evolution or by loss of orbital angular momentum and possibly other mechanisms such as irradiation on which this paper focuses. In the standard picture of the secular evolution see (e.g. King 1988; Kolb & Ritter 1992, hereafter KR92; Ritter 1996) the nature of the compact star is of no importance, i.e. the star is considered as a point mass (of mass ). The nature of the compact component is, however, of importance for accretion phenomena occurring in such systems, e.g. dwarf nova and classical nova outbursts in CVs and X-ray bursts in LMXBs, and when illumination of the secondary by radiation generated through accretion is considered, as we shall do in the following. Thus, calculating the standard secular evolution of such a binary reduces to calculating the evolution of a low-mass star under mass loss, where the mass loss rate derives from the boundary conditions imposed by the fact that the star is in a binary. In the simplest case one obtains the mass loss rate by requiring that the radius of the secondary is always exactly equal to its critical Roche radius . By decomposing the temporal change of and as (see e.g. Ritter 1988, 1996)
the mass radius exponent of the critical Roche radius, where the subscript * indicates that for evaluating this quantity one has to specify how mass and angular momentum are redistributed in the system. J is the orbital angular momentum, the rate of change of due to thermal relaxation and the one due to nuclear evolution. The virtue of (3) is that it shows immediately how secular evolution works: If the binary is dynamically stable against mass transfer; i.e. if , then mass transfer must be driven by some mechanism. This can either be the secondary's expansion due to nuclear evolution or the shrinkage of the orbit due to loss of orbital angular momentum. In the standard evolution of low-mass binaries, mass transfer is usually not driven by thermal relaxation. However, as we shall see below, this is not necessarily true if irradiation of the secondary is taken into account.
Attempts to bring the observed population of millisecond pulsars in line with the death rate of their presumed progenitors, i.e. the LMXBs, have resulted in the speculation that the secular evolution of LMXBs might be drastically different from, and their lifetime much shorter than that of CVs (e.g. Kulkarni and Narayan 1988). A possible reason for this is seen in the fact that the donor in a LMXB is exposed to a high flux of hard X-ray radiation emitted by the accreting compact star. In fact, Podsiadlowski (1991) has shown that irradiating a low-mass main-sequence star () spherically symmetrically with a flux erg cm-2 s-1 results in an expansion of the star on a thermal time scale and in its gradual transformation into a fully radiative star. It is this expansion which can drive mass transfer on a thermal time scale and this was thought to shorten the lifetime of a LMXB. On the formal level the effect of irradiation is taken into account by replacing (1) by
Here, is the thermal relaxation term arising from mass loss alone and the thermal relaxation term caused by irradiation (at constant mass). The latter term arises because irradiating the donor star means that its surface boundary conditions are changed and that it tries to adjust to them on a thermal time scale. With (6) instead of (1) we have now for the mass transfer rate
From (7) it is clearly seen that if , irradiation amplifies mass transfer. However, there are limits to what irradiation can do because the time scale and the amplitude of the effect are limited. The secondary's expansion due to irradiation saturates at the latest when it has become fully radiative, where it is larger by than a star of the same mass in thermal equilibrium without irradiation. As we shall show below the time scale on which the irradiated star initially expands is of the order of or less than the thermal time scale of the convective envelope . As a result, we can expect a maximum contribution from irradiation to at the onset of irradiation (i.e. of mass transfer) which is of the order
However, with ongoing irradiation not only does decrease roughly exponentially on the time scale , but other effects also tend to decrease . With the star's transformation from a mainly convective to a fully radiative structure, increases from to a large positive value. Thus increases, with the result that decreases. In addition, even the driving angular momentum loss is affected: for both braking mechanisms discussed in the literature, i.e. magnetic braking (e.g. Verbunt & Zwaan 1981; Mestel & Spruit 1987) and gravitational radiation (e.g. Kraft et al. 1962), . Thus decreases if the secondary inflates. Worse, magnetic braking which is thought to be coupled to the presence of a convective envelope might cease altogether once the star has become fully radiative. In this case the system might no longer be able to sustain the mass transfer necessary to keep the secondary in its fully radiative state. As a consequence, it is therefore possible that such systems evolve through a limit cycle in which short phases (of duration ) with irradiation-enhanced mass transfer alternate with long detached phases during which the oversized secondary shrinks as it approaches thermal equilibrium, again on the thermal time scale, whereas the system contracts on the much longer time scale of angular momentum loss.
A number of theoretical studies along these lines, under the assumption of spherically symmetric irradiation, have been performed (e.g. by Podsiadlowski 1991; Harpaz & Rappaport 1991; Frank et al. 1992; D'Antona & Ergma 1993; Hameury et al. 1993; Vilhu et al. 1994), all more or less confirming the behaviour outlined above, including cyclic evolution (Hameury et al. 1993; Vilhu et al. 1994). It should be stressed once more that the validity of these studies, in which the effect of irradiation is maximized, rests on the validity of the assumption that spherically symmetric irradiation is an adequate approximation. In fact, Gontikakis & Hameury (1993) and Hameury et al. (1993) have examined whether the spherically symmetric approximation is adequate and find that it is not. Worse, when taking into account the anisotropy of the irradiation they find that the long-term evolution differs significantly from the spherically symmetric case. Now, anisotropic irradiation is a rather difficult 3-dimensional problem involving hydrostatic disequilibrium to some extent, and with it circulations which can transport heat from the hot to the cool side of the star. Because of this one might dismiss simple theoretical arguments such as those given by Gontikakis & Hameury (1993) and Hameury et al. (1993). There is, however, a much more compelling argument supporting the case of anisotropic irradiation. This derives from the observations of a number of close but detached binary systems in which a low-mass companion, exposed to an intense radiation field emerging from its hot (degenerate) companion, shows a bright illuminated and an essentially undisturbed cool hemisphere. A compilation of the systems in question is given in Table 1, where we list the object's name, its association with a known planetary nebula, its orbital period (in days), the effective temperature of the irradiating white dwarf, the amplitude (in magnitudes) of the "reflection effect", and relevant references. These systems all show a pronounced "reflection effect"in their light curves which, in turn, is explained by the anisotropic temperature distribution on the irradiated companion. Among them are 7 binary central stars of planetary nebulae. The systems listed in Table 1 demonstrate that a cool star, exposed to strong irradiation from a hot companion, can live without problems with a hot and a cool hemisphere. Moreover equal effective temperature over the whole surface is not established, at least not over atime scale yr associated with the age of central stars of planetary nebulae, despite the fact that the irradiated star need not even rotate nearly synchronously. Obviously, heat transport from the hot side to the cool side by means of circulations is sufficiently ineffective that the large difference in effective temperatures can be maintained over long times. Thus the case of anisotropic irradiation has to be taken seriously and deserves a more detailed study. This is the objective of this paper.
Table 1. Detached close binaries showing a reflection effect.
© European Southern Observatory (ESO) 2000
Online publication: August 23, 2000