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

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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 [FORMULA] and radius [FORMULA]) 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).

The component normal to the stellar surface of the irradiating flux generated by accretion can be written as

[EQUATION]

Here [FORMULA] is the mass transfer rate, a the orbital separation, [FORMULA] a dimensionless function of the position on the secondary's surface (characterized by polar coordinates [FORMULA]) which describes the irradiation geometry, and [FORMULA] a dimensionless efficiency factor. [FORMULA] 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 [FORMULA] 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 [FORMULA] is a formidable if not nearly impossible task. Therefore, we treat [FORMULA] as a free parameter and our goal must be to arrive at conclusions which are as independent of [FORMULA] 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

[EQUATION]

The dimensionless number [FORMULA] 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 [FORMULA], gives [FORMULA] and (25) reduces to the usual criterion for adiabatic stability.

We compute the derivative [FORMULA] in (25) in the framework of the bipolytrope model (e.g. KR92) from which we obtain (cf Eq. 20)

[EQUATION]

For calculating [FORMULA] 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 [FORMULA] 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 [FORMULA] and [FORMULA], 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 [FORMULA] the stellar flux, i.e. the energy lost by the star from its interior per unit time and unit surface area is

[EQUATION]

where [FORMULA] is the effective temperature of the surface element in question. With (27) the stellar luminosity, i.e. the energy loss per unit time from the interior becomes

[EQUATION]

Before working out [FORMULA] 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

[EQUATION]

and the dimensionless stellar flux

[EQUATION]

where [FORMULA] is the effective temperature and [FORMULA] the stellar flux in the absence of irradiation. We note that [FORMULA] and that we expect [FORMULA], i.e. that for very high irradiating fluxes energy outflow from the stellar interior is totally blocked. Next we introduce the function

[EQUATION]

which has the following notable properties: First

[EQUATION]

Second, for positive albedos

[EQUATION]

Third, if F is a monotonically decreasing function of x (and there is no physical reason why this should not be so), then

[EQUATION]

From (32) and (34) we can immediately prove through integration of [FORMULA] by parts that

[EQUATION]

The reason why [FORMULA] attains a maximum can be understood as follows: Rewriting [FORMULA] in dimensional form (using (29) and (31)) we see that

[EQUATION]

The second factor in (36) describes the incremental blocking of the energy loss from the interior with changing irradiating flux [FORMULA]. The maximum of [FORMULA] arises because [FORMULA] vanishes for large [FORMULA]. This, in turn, is a consequence of the fact that as long as the star keeps a negative temperature gradient [FORMULA] in its subphotospheric layer, i.e. the superadiabatic convection zone, irradiation can not block more than the total flux [FORMULA].

As we shall see below, the fact that [FORMULA] 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 [FORMULA] is smaller by about a factor of two than the strict upper limit given by (35).

We can now return to Eq. (28) and compute [FORMULA]. This can be written as

[EQUATION]

where we note that the first factor in the integral is equal to [FORMULA]. With

[EQUATION]

[EQUATION]

from (24) we have (using (38a))

[EQUATION]

where [FORMULA] is the luminosity of the star in thermal equilibrium without irradiation. Combining now Eqs. (25), (26) and (39) we can rewrite the stability criterion as

[EQUATION]

or

[EQUATION]

where

[EQUATION]

is the mass loss time scale. Although the relations (40) and (41) do not show an explicit dependence on [FORMULA] they nevertheless depend on it via x. However, because of the fact that [FORMULA] has a maximum, the integral on the right-hand side of (40) and (41) must have a maximum that is smaller than Max[FORMULA]. Hence we can state that systems which fulfill the condition

[EQUATION]

which is independent of [FORMULA], 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 [FORMULA] does not depend on irradiation but only on the internal structure of the donor star (via [FORMULA], [FORMULA], [FORMULA], [FORMULA]) and on the secular evolution model (via [FORMULA], [FORMULA] and [FORMULA]). On the other hand, all the information about the irradiation model is contained in the expression on the right-hand side.

For discussing the stability of mass transfer in the limit of very small irradiating fluxes, i.e. [FORMULA], we must use (38b) instead of (38a) in (39). This results in the following stability criterion:

[EQUATION]

or

[EQUATION]

Because of (34) we arrive at a necessary and sufficient condition for stability

[EQUATION]

As above, we have separated in the conditions (45) and (46) factors which do not depend on irradiation (collected in [FORMULA]) from those which do (on the right-hand side). Like [FORMULA], [FORMULA] 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 [FORMULA] in the sense that for a given irradiation model, i.e. given [FORMULA] and [FORMULA], [FORMULA] must not exceed a certain value if mass transfer is to be stable.

Furthermore, Eqs. (45) and (46) show in particular that because [FORMULA] is maximal for [FORMULA] (see Eq. 34), i.e. for [FORMULA], 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 [FORMULA].

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

In this model we assume a fraction [FORMULA] of the stellar surface to be irradiated by a constant average normal flux

[EQUATION]

Note that the parameter s, introduced above, and [FORMULA] which we have introduced in Sect. 3 are not the same quantity. However, s and [FORMULA] are related and the corresponding relation will be given below.

Comparison of (47) with (24) shows that we may write [FORMULA] as follows:

[EQUATION]

where now [FORMULA] is the colatitude with respect to the substellar point and [FORMULA] the azimuth around the axis joining the two stars. If the star is assumed to be spherical, the colatitude [FORMULA] of the "terminator"of the irradiated part of the surface and s are related via

[EQUATION]

With (48) and (49), and [FORMULA], the stability conditions (41) and (45) become respectively

[EQUATION]

and

[EQUATION]

These are the results presented earlier in Ritter et al. (1995, 1996a).

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

[EQUATION]

Comparing this modified Stefan-Boltzmann law with (9) yields the relation between s and [FORMULA]:

[EQUATION]

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 [FORMULA] 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), [FORMULA] in (24) becomes

[EQUATION]

where

[EQUATION]

is the secondary's radius in units of the orbital separation a. Because the secondary fills its critical Roche volume, [FORMULA] is a function only of the mass ratio [FORMULA]. The terminator of the irradiated part of the star is at the colatitude

[EQUATION]

[FIGURE] Fig. 5. Sketch of the geometry of the point source model involving a spherical secondary irradiated by a point source at a distance a. Note that [FORMULA] is the irradiating flux normal to the stellar surface.

Inserting (54) in (41) or (45), the stability criteria become respectively

[EQUATION]

and

[EQUATION]

With (54) inserted in (28), the luminosity of the star is

[EQUATION]

Comparing (59) with (9) we find that the effective fraction of the stellar surface over which the energy outflow is blocked is

[EQUATION]

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Online publication: August 23, 2000
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