The influence of heating of subphotospheric layers by an external irradiation
flux is entirely described by a modification of the boundary condition, as
all of the absorbed X-ray flux is thermalized before being re-emitted. In the
plane parallel approximation, the standard Stefan-Boltzmann law 
where L is the surface luminosity of the
secondary star in the absence of illumination (note that L need not be
equal to the nuclear luminosity), 
its radius, and 
is
the effective temperature of the unilluminated star has to be replaced by
(Ritter et al. 1996b):
where 
is the normalized
illuminating flux, and
Here 
is the effective temperature of a stellar surface element
subject to the illuminating flux x.
The function G(x), in the range 0 - 1, describes the blocking of the
intrinsic stellar luminosity as a result of illumination. 
includes only the fraction of the flux that is deposited below the
photosphere; as mentioned above, energy deposition in optically thin regions
does not result in modifications of the internal structure of the star.
In the case of low-mass secondaries which have a convective envelope, the entropy deep in the envelope remains constant over the whole surface of the star, and varies slowly with time, on the Kelvin-Helmholtz time of the whole envelope. By contrast, the superadiabatic and radiative outer layers have a very short thermal time, and are in thermal equilibrium. Moreover, Gontikakis & Hameury (1993) have shown that the flux, in the plane parallel approximation, does not vary with depth in the convective zone. The problem of determining the reaction of the star to illumination thus reduces to finding the structure of an initially convective layer in thermal equilibrium when one changes the outer boundary condition while keeping the entropy at its base constant. This is easily done solving the\
Figure 1: Function G(x) for secondaries on the main sequence, with a mass
0.8 a), 0.6 b), 0.4 c) and 0.2 
d)
standard stellar structure
equations, which, in the plane parallel approximation write:
Figure 2: Functions 
and x g(x) for secondaries on the
main sequence with masses 0.8 a), 0.6 b), 0.4 c) and
0.2 
d)
where P is the pressure, T the temperature, 
the
temperature gradient calculated using the mixing length approximation in the
convective zone, 
the column density, and g the surface gravity,
assumed to be constant. The energy flux F is also assumed to be constant
throughout the layer. The layer is integrated down to a depth 
of 
g cm
, which is deep enough that the departure from
adiabaticity is negligible; our results have been found to be independent of
the particular value of 
. The surface boundary conditions
are standard:
In the absence of illumination, these equations are integrated for a given
g and 
, and give the entropy 
deep in the convective zone.
In the presence of an irradiating flux, one adds the condition 
at
the base of this layer. The set of Eq. (3) are integrated using the
method, equation of state and opacities described in Hameury (1991). This
gives F, and thus G as a function of g, 
, and x.
Figure 1 (click here) shows G(x) for main sequence stars of various masses. A
significant difference can be seen between very low mass secondaries (less
than 0.4 
) and more massive ones. This difference is due to the low
value of the energy flux that has to be carried through the convective zone,
and hence to a small deviation to adiabaticity, even in the outer
superadiabatic layers. For those low-mass stars, G(x) is not very
different from the relation 
that one would obtain
assuming that there is no superadiabatic region, so that the temperature
gradient and thus the surface temperature is fixed.
The stability analysis of mass transfer involves the derivative of the
function G(x) (Ritter et al. 1996c), 
. More precisely,
the criterion for stability against irradiation-induced mass transfer is:
where 
is the adiabatic mass radius exponent of the
secondary, 
the Kelvin-Helmholtz time, 
the mass transfer time scale, 
a dimensioneless function which scales
roughly as the inverse of the mass of the secondary convective envelope, 
the fraction of the secondary exposed to illumination, and
is the mass radius exponent of the Roche radius. This
criterion is deduced from the assumption that the mass-radius exponent of the
secondary under the sole influence of illumination must be less than
; Eq. (5) means that, for instability to
occur, one must satisfy both conditions of sufficient illuminating fluxes
and sensitivity of the secondary response to irradiation, i.e. large value of
g(x).
C does
not depend on the very detailed structure of the illuminated star, and is
easily accessible in the bi-polytropic approximation. The response of the
secondary is entirely contained in the function x g(x), which is plotted in
Fig. 2 (click here). The small discontinuities are due to linear interpolations in
determining the opacity. A remarkable characteristic of this function is
that (1) g(x) is a monotonously decreasing function, which is less than
unity, and (2) that x g(x) has a maximum of the order of 0.5, whatever the
secondary mass, although the position of this maximum does depend on it.
Thus, there will be no cycles if C > s (note that this is a sufficient but
not necessary condition for stability).
It must be stressed that this procedure is valid only for stars which have a convective envelope, even though the whole star can be far out of thermal equilibrium. This would in fact be the main limitation in following the evolution of systems in which the effect of irradiation is so strong that at some point the star becomes fully radiative. This was a natural outcome of models assuming spherically symmetric illumination, but Hameury et al. (1991) have shown that this is no longer the case when one accounts for the asymmetry of irradiation.
These calculations can easily be generalized to stars which are not on the
main sequence, provided they still have a convective envelope. Given the
effective temperature 
and surface gravity g, the set of
Eq. (3) can be integrated in the absence of illumination and provide
the unperturbed outer structure; the same procedure as previously is then
applied. Table 1 (available only in electronic form at the CDS via anonymous
ftp 130.79.128.5) gives 
(Col. 4) as a function of 
(Col. 1), 
(Col.
2) and 
(Col. 3).
Figure 3: Evolution of a cataclysmic variable using either a two
hemispheres model with standard boundary conditions (top panel), or a
spherically symmetric model with condition (9)
If one assumes that the irradiation source is a point located at the distance
a from the center of the secondary, Eq. (1) can be easily integrated over
the whole stellar surface, which gives, following the notations of Ritter et
al. (1996b):
where 
is the colatitude with respect to the substellar point of a
point on the secondary surface, 
, with 
the mass
ratio, and the normalized irradiation flux x is now given by:
with
where 
is an efficiency factor accounting for the albedo of the
secondary, the fraction of energy deposited in optically thin regions or the
anisotropy of emission at the surface of the primary, and
Equation (6) can thus be written in the form
We have integrated numerically Eq. (6), and the results can be fitted within
1% by:
Table 2 (available only in electronic form at the CDS via anonymous ftp
130.79.128.5) gives the coefficients 
,
, and 
as a function of 
and 
. For
arbitrary values of 
and 
, a linear interpolation
of y (not of the coefficients 
, 
and 
) can be done.
The stability criterion now involves the integral of the function x g(x);
mass transfer will be stable if:
We have calculated this maximum using the values of G(x) as determined
above, and found that this maximum is again almost independent of the
secondary mass, and is smaller by about a factor 2 than the value 2s found
assuming uniform illumination over a fraction s of the secondary, being
equal to (0.2 - 0.3) 
.
We have compared evolutionary calculations using either a spherically
symmetric code with a boundary condition given by (10), or the code described
in Hameury et al. (1993) in which both hemispheres with a coupling term are
modeled. The results are given in Fig. 3 (click here), for a cataclysmic variable with
= 1 
and 
= 0.8 
; the white dwarf radius is 
cm, and the efficiency 
is 0.1 (spherically symmetric
model) and 0.09 (two hemispheres model). Both are very similar, i.e. the mass
transfer rate exhibits damped oscillations; the different values of 
were chosen so as to obtain the same initial maximum value of the mass
transfer rate. The differences (slightly shorter damping time and 
) are
mainly due to the fact that in the two hemispheres model, the illuminating
flux is assumed to be constant over the heated regions, whereas the boundary
condition (10) includes an angular variation of this flux, which has a
strongly non-linear effect.
For X-ray luminosities close to the Eddington limit, we predict that the
illumination effect is small, as x g(x) is small for large values of x,
and as a significant fraction of the secondary surface might be shielded from
illumination by the accretion disc (Ritter et al. 1996b). However, Eq. (10)
might significantly underestimate the effect of heating, as the heat flux in
the illuminated regions can be negative, and be of the same order of
magnitude as the intrinsic stellar flux. The unilluminated layers have then
to re-radiate this additional flux, which may, in some cases, be sufficient
to also block the intrinsic stellar flux. This is equivalent to having 
in these regions which then become very sensitive to illumination.
This effect, responsible for the short outbursts of mass transfer obtained
by Hameury et al. (1993), is indeed not accounted for by the boundary
condition (10), but depends sensitively on the strength of the coupling
between both hemisphere, i.e. on the circulation timescale which is poorly
known.
