## 5. Concluding remarksWe investigated a number of observed contact binaries in a broad range of periods and mass ratios. Each system was treated individually. Combining observed properties and an approximate treatment of theoretical configurations in thermal equilibrium we derived two arguments against thermal equilibrium. Thermal equilibrium can be excluded (1) when in conflict with the contact condition and (2) when unstable. In each system one of these arguments applies and thermal equilibrium can safely be excluded, provided that Los Alamos opacities are adequate. We did not discuss systems with extremely small mass ratios such as TV Mus, XY Boo, AW UMa. A treatment of these systems is more difficult since the observed value of is very uncertain and since opacities for the secondary's cool interior are needed. We also excluded hotter systems with shallow envelopes since they require a more refined treatment of convection in the outermost mass shells than in the present code. Finally, the discussion concerned only systems in good thermal contact. Taking into account these restrictions we conclude that many (possibly all) observed contact binaries evolve on a thermal time-scale, probably in thermal cycles as proposed by Lucy (1976) and Flannery (1976). This raises the old problem that the semi-detached phase is apparently not observed among shortest-period systems ( days). Note that most systems discussed in this paper belong to this class. We investigated many theoretical equilibrium configurations with realistic combinations of observable properties (period, mass ratio, temperature difference etc.), including configurations violating the contact condition. In all configurations evolutionary effects are important (). Moreover, in a generalised thermal stability problem all configurations turned out to be unstable on a time scale of yr. (In view of the large differences between the observed systems this time interval is remarkably small.) Accordingly, the structure of unevolved real contact binaries is usually incompatible with thermal equilibrium. The structure of evolved real systems is in some cases compatible with thermal equilibrium but usually not with stable equilibrium. For unrealistic combinations of observable properties, however,
stable thermal equilibrium is possible, even in unevolved
configurations. For example, Hazlehurst et al. (1982), Hazlehurst
& Refsdal (1984) and Kähler et al. (1987) found unevolved
stable solutions (Biermann-Thomas models). They are highly unrealistic
since the temperature difference is very large. The stability of these
models is compatible with the present discussion since the entropy
difference is very large and since Eq. (19) with positive values for
More interesting is a stable evolved configuration with a small entropy difference () found by Refsdal & Stabell (1981), using Eq. (19) with . The parameters of this configuration are probably unrealistic as indicated by angular momentum problems. Evolutionary effects are larger than in the configurations discussed in the present paper. (Hydrogen in the primary's centre is exhausted.) These effects have a stabilizing influence. This explains the stability. Equally interesting is a stable evolved configuration found by Robertson & Eggleton (1977). The configuration is in shallow contact and has a small temperature difference and typical values for mass ratio () and period ( days). These examples show that from a purely theoretical viewpoint the existence of contact systems in thermal equilibrium cannot be excluded if major evolutionary effects are invoked. This paper was concerned with a quite different problem. We asked whether real (observed) contact binaries are in thermal equilibrium. As we have seen, this problem can usually be solved if sufficient observational information is available. All systems investigated so far turned out to be in thermal disequilibrium. © European Southern Observatory (ESO) 1997 Online publication: July 8, 1998 |