4. Stability against irradiation-induced mass transfer
Let us now examine the situation in which a low-mass star transfers mass to a compact companion (of mass and radius ) and, in turn, is irradiated (directly or indirectly) by the accretion light source. Because irradiation can enhance mass transfer and more irradiation can give rise to even higher mass transfer, we must examine under which conditions such a situation is stable against irradiation-induced runaway mass transfer.
4.1. Arbitrary irradiation geometry
In this subsection we wish to keep the discussion as general as possible. Therefore, we do not specify the irradiation geometry. Specific models which do just that will be presented in Sect. (4.2).
Here is the mass transfer rate, a the orbital separation, a dimensionless function of the position on the secondary's surface (characterized by polar coordinates ) which describes the irradiation geometry, and a dimensionless efficiency factor. only if the accretion luminosity is radiated isotropically from the compact star and if all the energy which is radiated into the solid angle subtended by the donor as seen from the accretor is absorbed below the photosphere. In a real situation will be considerably less than unity for several reasons. The most important of these are: 1) the accretion luminosity will, in general, not be emitted isotropically, e.g. if accretion occurs via a disk which radiates predominantly perpendicular to the orbital plane and casts a shadow onto the donor, or if the accretor is strongly magnetized and accretes mainly near the magnetic poles, as e.g. in AM Her systems. 2) energy emitted in certain spectral ranges, as e.g. EUV radiation and soft X-rays, is unlikely to reach the photosphere of the donor (in the case of EUV and soft X-ray radiation because of the high column density of neutral hydrogen). 3) part of the incident flux will be scattered away before penetrating into the photosphere. 4) not all of the transferred mass needs to be accreted by the compact star. Part of it may leave the system before releasing much potential energy, e.g. via a wind from the outer regions of the accretion disk.
From this it is clear that computing a reliable value for is a formidable if not nearly impossible task. Therefore, we treat as a free parameter and our goal must be to arrive at conclusions which are as independent of as possible.
For deriving the criterion for adiabatic stability against irradiation-induced mass transfer we follow Ritter (1988) (see also Ritter 1996). The only difference is that we use now Eq. (6) instead of (1) (the latter corresponding to Eq. (12) in Ritter's (1988) paper). Accordingly the criterion for adiabatic stability becomes
The dimensionless number which is defined by (25) measures how sensitively the stellar radius changes in response to irradiation produced by mass transfer. Ignoring the influence of irradiation, i.e. setting , gives and (25) reduces to the usual criterion for adiabatic stability.
We compute the derivative in (25) in the framework of the bipolytrope model (e.g. KR92) from which we obtain (cf Eq. 20)
For calculating we shall make use of what we shall refer to as the weak irradiation assumption. Making this assumption is tantamount to assuming that at any point lateral heat transport is negligible compared to radial transport. Lateral heat transport occurs in the form of radiative diffusion and advection because of non-vanishing lateral temperature gradients and , and departures from strict hydrostatic equilibrium. We shall show in the Appendix that the weak irradiation assumption can be justified in those cases we are interested in and that we may safely neglect lateral heat transport. Accordingly, energy conservation requires that at any point the stellar flux, i.e. the energy lost by the star from its interior per unit time and unit surface area is
Before working out in (25), we shall first examine the reaction of the stellar surface to irradiation in rather general terms. For that it is convenient to introduce the dimensionless irradiating flux
where is the effective temperature and the stellar flux in the absence of irradiation. We note that and that we expect , i.e. that for very high irradiating fluxes energy outflow from the stellar interior is totally blocked. Next we introduce the function
The second factor in (36) describes the incremental blocking of the energy loss from the interior with changing irradiating flux . The maximum of arises because vanishes for large . This, in turn, is a consequence of the fact that as long as the star keeps a negative temperature gradient in its subphotospheric layer, i.e. the superadiabatic convection zone, irradiation can not block more than the total flux .
As we shall see below, the fact that has a maximum is very important for the stability discussion. In fact, we shall see in the next Sect. 5 that for realistic situations the maximum of is smaller by about a factor of two than the strict upper limit given by (35).
where we note that the first factor in the integral is equal to . With
is the mass loss time scale. Although the relations (40) and (41) do not show an explicit dependence on they nevertheless depend on it via x. However, because of the fact that has a maximum, the integral on the right-hand side of (40) and (41) must have a maximum that is smaller than Max. Hence we can state that systems which fulfill the condition
which is independent of , are definitely stable against irradiation-induced runaway mass transfer.
The reason for rewriting the stability criterion (40) in the form of (41) or (43) is that in the latter conditions does not depend on irradiation but only on the internal structure of the donor star (via , , , ) and on the secular evolution model (via , and ). On the other hand, all the information about the irradiation model is contained in the expression on the right-hand side.
As above, we have separated in the conditions (45) and (46) factors which do not depend on irradiation (collected in ) from those which do (on the right-hand side). Like , depends only on the internal structure of the donor star and the secular evolution model. In essence, Eqs. (45) and (46) are conditions on the value of in the sense that for a given irradiation model, i.e. given and , must not exceed a certain value if mass transfer is to be stable.
Furthermore, Eqs. (45) and (46) show in particular that because is maximal for (see Eq. 34), i.e. for , systems with an unirradiated donor star are the most susceptible to irradiation. In other words, if a system is stable at the turn-on of mass transfer it will remain so later, unless secular effects diminish the value of .
4.2. Specific irradiation models
In the following we describe two specific irradiation models, a rather simple one, hereafter referred to as the constant flux or constant temperature model, and a more realistic one, hereafter referred to as the point source model.
4.2.1. The constant flux model
Note that the parameter s, introduced above, and which we have introduced in Sect. 3 are not the same quantity. However, s and are related and the corresponding relation will be given below.
where now is the colatitude with respect to the substellar point and the azimuth around the axis joining the two stars. If the star is assumed to be spherical, the colatitude of the "terminator"of the irradiated part of the surface and s are related via
Because of (48) not only is the irradiating normal flux constant but also the effective temperature on the irradiated part (hence the name constant temperature model). (48) inserted in (28) yields the luminosity of the star
4.2.2. The point source model
In this model, which has already been discussed in some detail by KFKR96, we assume the donor star to be irradiated by a point source at the location of the compact star. For simplicity we assume the secondary to be spherical. The chosen geometry is axisymmetric with respect to the axis joining the two stars. Denoting again by the colatitude of a point on the surface of the irradiated star with respect to the substellar point (for a sketch of the geometry see Fig. 5), in (24) becomes
is the secondary's radius in units of the orbital separation a. Because the secondary fills its critical Roche volume, is a function only of the mass ratio . The terminator of the irradiated part of the star is at the colatitude
© European Southern Observatory (ESO) 2000
Online publication: August 23, 2000