In the preceding section we investigated a number of observed contact binaries. All of them turned out to be in thermal disequilibrium. The discussion was based on approximations. Are the results certain, i.e. unaffected by uncertainties in the discussion?
4.1. Arguments against thermal equilibrium
Essentially two arguments against thermal equilibrium have been used. The first argument concerns the structure of theoretical equilibrium configurations with given (observed) properties. Thermal equilibrium can be excluded when in conflict with the contact condition.
If this argument does not apply, i.e. if the contact condition can be satisfied (as in the case of BV Dra, AB And and V508 Oph), we need the second argument which concerns the stability of theoretical configurations with the given properties. Thermal equilibrium can be excluded when unstable. The stability properties depend not only on the structure of the configuration but also on assumptions on thermal perturbations.
4.2. Uncertainties in the structure
Here we ask for the effects of uncertainties in the structure of theoretical equilibrium configurations. We adopted the hydrogen content . In reality X may be somewhat larger. As an approximation we used the equal entropy assumption (5). In reality the entropy difference is small but positive. A third uncertainty concerns the mixing-length .
Potentially most important is the uncertainty in the chemical profile. We used the approximation (8) with . In realistic profiles (e.g. Kippenhahn & Weigert 1990) the evolutionary effects are somewhat more concentrated to the centre. This corresponds to a smaller value of .
Among all theoretical configurations investigated, evolutionary effects are most important ( is smallest) in the last model in Table 1. We compared this model with other models for the given observed properties (including a temperature of 6190 K). They are defined by the values for X, , and . The entropy difference is expressed as entropy per nucleus divided by Boltzmann's constant, and assumed to be constant in the course of a thermal perturbation.
Results are listed in Table 2. All models have the mass . The symbol denotes the e -folding time of the most unstable mode in units of yr. As expected, a decrease in is compensated by a decrease in . The results show that uncertainties in X, and are unimportant for the properties of interest (degree of contact and instability). Uncertainties in the mixing-length have some influence, but the qualitative results are not affected. (It should be noted that the model with is slightly inconsistent. The secondary's convective envelope is very shallow, and in the inner part of this envelope departures from adiabaticity are neglected.)
Table 2. Effects of changes in the hydrogen content, the entropy difference, the mixing-length and the chemical profile (see text)
Uncertainties in the opacity have not been discussed. We used Los Alamos opacities.
4.3. Uncertainties in the stability problem
We adopted the equal entropy condition (5). If this condition is replaced by another equation, the stability properties may change. For example, we may assume that the entropy difference is non-zero and constant. This is of little influence as shown in the preceding subsection. Alternatively, in a configuration satisfying the contact condition and having a positive entropy difference Eq. (5) can be replaced by the equation
where denotes the luminosity (in solar units) transferred from the primary to the secondary and are constants. An equivalent equation was proposed by Hazlehurst & Refsdal (1980). If this equation is adopted, the stability properties depend on the structure (including ) as well as on the coefficients m and n.
Are these uncertainties in the stability problem important? Since we are discussing systems in good thermal contact with deep convective envelopes, the entropy difference is small compared to unity. Since the mass ratios are far from unity, is of order unity. Accordingly, if Eq. (19) is adopted, n must be taken to be positive and K must be sufficiently large. Under these circumstances small changes in are sufficient for the adjustment of , and the stability problem should (approximately) be independent of e stability problem based on the equal entropy assumption. In other words, uncertainties in the definition of the stability problem should be unimportant as long as the entropy difference is bound to be sufficiently small.
To check these arguments we investigated the effects of changes in the stability problem for given observed properties, taken again from the system BV Dra with a temperature of 6190 K. Adopting Eq. (19) we calculated the time scale for different values of and n. Reasonable values for m and n are uncertain, but the case of is certainly unrealistic since the adjustment of entropy between the components is inhibited. Hazlehurst & Refsdal (1980) presented arguments for with a preference for .
Results are listed in Table 3. For the time scale is almost independent of m and n, in accordance with the above arguments. If the entropy difference is larger depends on m and n. An increase in m and a decrease in n have a stabilizing influence. For given positive values of m and n, an increase in has also a stabilizing effect. The uncertainty in is however not large.
Table 3. The e -folding time (in units of yr) of unstable configurations for fixed physical parameters and different values of and n (see text)
This check was applied also to other observed systems, with similar results. Accordingly, uncertainties in the stability analysis are of minor importance since the instability is not affected.
© European Southern Observatory (ESO) 1997
Online publication: July 8, 1998