Non-Periodic Phenomena in Variable Stars
                                              IAU Colloquium, Budapest, 1968


           ON THE REFLECTION EFFECT IN CLOSE BINARIES

                         I. B. PUSTYLNIK
    Physical and Astronomical Institute of the Academy of Sciences, Tartu


  Among various problems of the theory of close binary systems, 
investigations of the effects which are due to the gravitational and 
radiative interactions of the components are of special interest.
  It is inevitable in close binary systems that a part of the radiation of 
either component penetrates into the atmosphere of its mate, suffers 
some changes and will be subsequently reemitted or scattered. This is a 
well known interaction phenomenon, called the reflection effect. At the 
same time, the shape of the reflecting surface is governed by tidal 
perturbations and stellar rotation.
  At present we dispose only of somewhat fragmentary observational data, 
concerning the values of amplitudes and phase functions of the reflected 
light for several dozens of eclipsing variables. As to the physical 
theory of the reflection effect, it is based on a number of simplifying 
assumptions hardly accessible to observational checking. Such a state of 
affairs is due, on one hand, to the fact that the fractionally reflected 
light constitutes a too small part of the whole brightness of the binary 
system to be measured by the direct methods of photometry or 
spectrophotometry. On the other hand, in the outer layers of the 
reflecting star which are responsible for re-emission, deviations from 
LTE and an anisotropy of radiation field can be appreciable.
The present report deals with two different aspects of the problem of 
radiation transfer in a semi-detached binary system, where a B-type star 
of the main sequence combines with secondary subgiant component of F-K 
spectral class. First we discuss the negative O-C values of the 
amplitude of the reflection effect. Another question concerns some 
details of the mechanism of the reflection effect.
  It is generally known that in close binaries of the afore-mentioned
type theoretical estimates of the bolometric amplitude of the reflection effect
appear as a rule to be larger than the values obtained from the analysis of
light curves of eclipsing variables. Sir Arthur Eddington (1927) was the first
to draw attention in his pioneer work to this hardly explicable feature of the
reflection effect. Recently Sobieski (1965) has taken into account the non
greyness of stellar matter and calculated the monochromatic amplitudes for
several well-studied close binaries. His results reaffirmed the presence of the
negative O-C. We still have no comprehensive explanation of this peculiarity.
Recently we examined the following possibility. As long as the reflecting
star, usually subgiant, fills its critical Roche lobe, the gravitational darkening
on its surface must be important. Indeed, let the mass ratio value be equal
to 0.3 (for instance RS Vul or TX UMa). Then the dimensions of critical Roche
lobe are such, that on the top of a tidal bulge the value of gravitational 
acceleration is approximately half as much as in the point diametrically 
opposite to it. If the mechanism of the reflection effect lies in absorption 
with subsequent reemission, rather than scattering, then at the maximum of 
light the top of a tidal bulge will send out in the direction of the observer 
substantially less energy than it would do for the case of a spherical 
reflecting star. Therefore the value of the amplitude of the effect will be 
significantly lower for the distorted star compared to the spherical one.
  A quantitative approach to the problem was outlined in our article in 
"Astrophysics" vol. III, 1. The problem has been reduced to the 
solution of the radiative transfer equation for re-emitting non-grey, 
plane-parallel atmosphere. An irradiation flux of given magnitude and 
spectral distribution is assumed as parallel beam. Next our solution of 
the radiative transfer equation is to be applied to the idealized binary 
system, where the point source represents the irradiating star and the 
reflecting star is identified with its critical Roche lobe. Then the 
reflecting surface is approximated locally by plane-parallel layer. Thus 
the entire irradiated area is divided into elemental zones and the problem 
of computing the brightness of re-emitting surface, as viewed by a distant 
observer, is reduced to the summation of the brightnesses from each visible 
differential zone, allowing for gravitational darkening and fore-shortening 
effects. The numerical calculations required have not yet been performed, 
since a sufficiently powerful computer was not available for the time being.
  It goes without saying that a mere confrontation of the predicted values of 
the bolometric amplitudes with rectification constants for several well-studied 
binaries would essentially contribute to the full understanding in this 
question. But it is worth keeping in mind the low accuracy of determination 
of the reflection effect amplitudes through an analysis of out-of-eclipse 
light variations. It would be even more interesting to study in detail 
the influence of gravitational darkening upon the phase law. We anticipate 
that in presence of the strong gravitational darkening the maximum of the 
reflected light would not fall any more on the phase n, as usually adopted 
in the rectification procedure.
  In this connection the improbably small values of the ellipticities of the
secondary components for the majority of Algol-type binaries may be recalled.
Hosokawa (1957; 1958; 1959) has managed to establish that the systems with
small ellipticities of the secondary components possess also negative O-C
values of the reflection effect amplitudes. Let us examine the system TX
UMa as an example. According to the rectification constants of the light curve
the ellipticity of its secondary subgiant component equals to 0.03. At the
same time the mass of the primary component is three times as much as that
of the secondary and orbital elements indicate that the cooler component
fills it critical Roche lobe. We state the presence of discrepancy of the 
observational data in case of TX UMa. This binary has also large negative O-C.
  It should be noted also that in all theoretical works on the reflection
effect the role of convection has hitherto been neglected. On the other hand,
according to contemporary ideas of stellar evolution the subgiant components
of binaries possess well-developed convective zones. The effective depths of
formation of the Tatters are highly moderate (of order tau = 0.5). If so, then
about a half of the irradiated energy will be absorbed within the convective
zone. Moreover, the convective zone apparently stretches out to the boundary 
of the star here and there, as long as observational evidence exists for mass 
transfer. The problem, whether all this absorbed energy will be re-emitted 
in outer layers or any appreciable amount of it will be swept away by the mass 
loss or even by shock waves, had not ever been studied.
  We proceed now to the brief discussion of the mechanism of the 
reflection effect. It can be shown that Lyman continuum of the B-type 
star is responsible for high electron pressure (~10^2 bars or even more) 
in the reflecting layers. Normal electron pressure for a single star of 
G-K spectral class would be 1 to 10 bars. Lyman continuum photons of 
B-type star ionize atoms of H of its cooler companion and, after 
recombinations, will be transformed into Balmer continuum and L_alpha 
photons. There is some chance to discover this effect through the 
observations in L_alpha of the nearest close binaries with large orbital 
velocities.
  L_alpha photon lives only a split of second in a free state. It will be 
absorbed by a ground state H-atom. The latter will be ionized in its 
turn by photons of the Balmer continuum, as long as these constitute a 
predominant part of irradiative energy. An additional electron density 
depends on the value of the flux in Lyman continuum.
  It would also have been difficult to understand, what induces 
attenuation of the irradiated flux in the reflecting atmosphere if the 
influence of L, continuum had been neglected. Indeed, in a typical 
Algol-type close binary the value of Balmer continuum flux of the 
primary star falling on the surface of its mate exceeds the proper flux 
of the latter. Assuming both stars to be black body radiators of 
definite effective temperature, we are in position to estimate a mean 
number of B, photons falling from outside on each cm^2 of irradiated 
surface per sec and to compare it to the numbers of scatterings or 
absorptions. The latter is proportional to integral alpha_nu B_nu d_nu. 
Calculations indicate that at a normal electron pressure absorption by H^- ions 
and scatterings on H atoms are the most significant opacity sources. The 
former is slightly more effective. At the same time radiation of the 
primary component in Balmer continuum would penetrate quite deeply into 
its companion. Calculations give for the depth 10^9 cm, if the mean 
density equals to 10^-8 g/cm^3. But on the other hand, we expect that 
subjected to irradiation, outer layers of the reflecting star would 
adjust themselves in some way to hamper "strange" radiation. Assuming 
that all free electrons, originated due to L_c continuum, recombine with 
H atoms to form H^- ions, a relatively low ionisation degree of H justifies 
this assumption, we will find that H^- absorption is much more effective than 
Rayleigh scattering. Furthermore, we obtain that the column of matter one or 
two scores of kilometers high at a normal density just will do to absorb 
completely, irradiated energy.
  We hope that our qualitative approach is valid as a satisfactory first 
approximation. Of course, the numerical results must be treated with a 
considerable portion of reservation. A more rigorous approach, allowing 
for the rates of proceeding of the various processes and permitting at 
the same time to work out the equation of equilibrium conditions, is 
highly desirable.

                               REFERENCES

Eddington, A. S., 1927, Mon. Not. R. astr. Soc. 87, 43. 
Hosokawa, Y., 1957, Sendaj astr. Raportoj, 52, 208. 
Hosokawa, Y., 1958, Sendaj astr. Raportoj, 56, 226. 
Hosokawa, Y., 1959, Sendaj astr. Raportoj, 70, 207. 
Sobieski, S., 1965, Astrophys. J. Suppl. Ser. 12, 263.