## 4. DiscussionIn 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 equilibriumEssentially 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 structureHere we ask for the effects of uncertainties in the structure of
theoretical equilibrium configurations. We adopted the hydrogen
content . In reality 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 Results are listed in Table 2. All models have the mass
. The symbol denotes the
Uncertainties in the opacity have not been discussed. We used Los Alamos opacities. ## 4.3. Uncertainties in the stability problemWe 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 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, 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 Results are listed in Table 3. For the
time scale is almost independent of
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 |