3. Thermal imbalance of observed contact binaries
3.1. Observational properties
From the observational data a few global parameters can be determined. The period is exactly known. The mass ratio and a lower limit for the total mass can be found from radial velocities. The temperature difference and the mean effective temperature can be approximately determined from the colours and spectral types of the components.
If the spectroscopic mass ratio is known, additional properties (e.g. mass M, light ratio , degree of contact F) can be determined in a light curve analysis, but the results are often uncertain. For this reason the light curve will be used directly (i.e. without interpretation in a light curve analysis) as an observational property.
3.2. The temperature difference
In many observed systems the secondary is hotter than the primary. In theoretical configurations with typical mass ratios however the primary is slightly hotter than the secondary, i.e. is positive. For systems in good thermal contact this follows from the equal entropy condition (5) and Lucy's (1967) discussion (in hydrostatic equilibrium the mean effective gravity is slightly larger in the primary than in the secondary). If the thermal contact is poor, the entropy difference is positive and the temperature difference is even larger.
If these theoretical results are reliable, observed systems with a negative temperature difference are far from hydrostatic equilibrium and thus not in thermal equilibrium.
The theoretical temperature difference is however only an approximation. A mixing-length formalism is used to calculate the temperature gradient, and the mixing-length is assumed to be the same in both components. On account of these approximations the argument against thermal equilibrium is not conclusive.
Summarizing, a negative observed temperature difference (and the resulting properties of the light curve) cannot be explained in the present theoretical treatment since in hydrostatic equilibrium the theoretical difference is always positive. Both differences are however small and can be neglected in an approximate discussion.
3.3. Test of thermal imbalance
Consider an observed system with reliable spectroscopic values for q and . We shall ask whether the system can be in thermal equilibrium.
For this purpose we have to check all theoretical equilibrium configurations with the given (observed) values of period and mass ratio. As shown in Sect. 2.3 they can be approximately described by two parameters. For some choice of these parameters, the properties of the theoretical configuration must be compatible with the observations, the contact condition (7) must be satisfied, and the configuration must be thermally stable. Otherwise thermal equilibrium can be excluded within the framework of our approximations.
Taking into account the observable properties as collected in the preceding subsections, the result can be summarized as follows. If an observed system is in thermal equilibrium, there exists a theoretical equilibrium configuration with the observed values of period and mass ratio having the following properties: (I) The mean effective temperature is compatible with the observations. (II) The theoretical light curve is compatible with the observed light curve, with the possible exception of differences between the two minima. Note that the inclination of a theoretical configuration is known from the mass M and the observed value of . The theoretical light curve is therefore also known. (III) The contact condition (7) is satisfied. (IV) The configuration is thermally stable.
For given period and mass ratio, a theoretical equilibrium configuration has two degrees of freedom. They can usually be adjusted to satisfy the conditions (I) and (II). (An exception concerns systems with a very high inclination. Since the mass is approximately known from the observed value of , there is no freedom to satisfy two conditions.) The resulting configuration is determined uniquely if uncertainties in the observational input parameters and the theoretical treatment are ignored. If these uncertainties are taken into account we obtain a set of theoretical configurations with the properties (I) and (II). If any one of these configurations satisfies the conditions (III) and (IV), nothing more can be said. The observed system may or may not be in thermal equilibrium. (The conditions for thermal equilibrium are necessary but by no means sufficient.) However, if configurations satisfying the contact condition do not exist, or if all of them are thermally unstable, the observed system is certain to be in thermal imbalance.
In the following subsections this test will be applied to a number of observed systems. We begin with a detailed discussion of the system BV Dra. For the other systems a short discussion will be sufficient.
3.4. Thermal imbalance of BV Dra
For the system BV Dra (period day) Batten & Lu (1985) determined the parameters and found that the secondary (spectral typ F8) is slightly hotter than the primary (F9). KR86 determined from the colours a mean effective temperature K. Strömgen colours (Ruciski 1983) and the calibration of Philip & Relyea (1979) give K. To cover this range we investigated models with temperatures of 6000, 6100 and 6190 K.
Properties of configurations satisfying the conditions (I) and (II) are listed in Table 1. The contact condition (III) is also satisfied. The theoretical light curves have been fitted to the observed light curve during the transit eclipse, cf. Fig. 4. The temperature difference is positive and too large. This implies that during the occultation eclipse the theoretical light curve is too shallow. Except for these difficulties (which are unavoidable as discussed in Sect. 3.2) the observed properties of BV Dra are reproduced.
Table 1. Theoretical configurations in thermal equilibrium for comparison with BV Dra
Condition (IV) however is violated. The configurations are thermally unstable with an e -folding time of about yr. This was shown in an eigenvalue analysis as well as in evolutionary calculations. Accordingly, within the framework of our approximations (this caveat will be made also in the following subsections) BV Dra is certain to be in thermal imbalance.
3.5. Thermal imbalance of BW Dra
For the system BW Dra ( day) Batten & Lu (1985) determined the parameters . Strömgen colours (Ruciski 1983) and the calibration of Olsen (1984) give a mean effective temperature K. For the primary KR86 derived an effective temperature of about 5900 K. For temperatures in this range, conditions (I) and (II) give evolved () configurations of mass . Since the contact condition is violated (), thermal equilibrium can be excluded. In the generalised (formal) thermal stability problem the models are unstable on a time scale of about yr.
3.6. Thermal imbalance of W UMa
For the system W UMa ( day) Ruciski et al. (1993) determined from 1989 data the parameters , a mean temperature of K and the mass . The light curve (e.g. Linnell 1991) shows that the inclination is very high. Accordingly, there is little freedom in the mass. Using these values of and adopting the metallicity we obtain an evolved () configuration with a mean temperature of 6370 K. The contact condition is violated (). For larger values of Z the temperature is lower but F is still larger. For these reasons thermal equilibrium can be excluded. The configurations are unstable on a time scale of about yr.
3.7. Thermal imbalance of AH Vir
The system AH Vir ( day) has also a very high inclination. Lu & Ruciski (1993) determined the parameters , a mean temperature of 5300 K and a mass of . Adopting this mass and the metallicity we obtain an evolved () configuration which is much too hot ( K) and in severe conflict with the contact condition (). If the metallicity is increased, the temperature decreases but the conflict with the contact condition increases. Accordingly, AH Vir is in thermal imbalance. The configurations are unstable on a time scale of about yr.
3.8. Thermal imbalance of AB And
For the system AB And ( day) Hrivnak (1988) determined the parameters . In a light curve analysis he found . Using these parameters, the present light curve synthesis is not able to reproduce the observed light curve. In the theoretical light curve the difference between the two minima is too large on account of the temperature difference.
The observed light curve shows that the inclination is very high. We calculated models of mass and mean temperatures between 5450 K (the primary's temperature adopted by Hrivnak) and 5800 K. The theoretical light curve is always too shallow. The models are evolved () and satisfy the contact condition (). All of them are thermally unstable with an e -folding time of yr. Accordingly, AB And is in thermal im
3.9. Thermal imbalance of V508 Oph
So far we discussed W-type systems. The A-type system V508 Oph () has parameters similar to those of AB And and also a very high inclination. Lu (1986) and Lapasset & Gómez (1990) found and determined a mean temperature between 5800 and 6000 K. We calculated models with these parameters and the mass . They are evolved () and satisfy the contact condition (). The theoretical light curves are too shallow. All models are unstable on a time scale of yr. We conclude that V508 Oph is in thermal imbalance.
3.10. Thermal imbalance of OO Aql
The A-type system OO Aql ( day) has one of the largest mass ratios. Hrivnak (1989) determined the parameters and a temperature between 5500 and 6000 K. The light curve indicates that the inclination is very high. We investigated models in hydrostatic equilibrium with . Adopting the metallicity we obtain an evolved () model in severe conflict with the contact condition () and with a mean temperature of 4390 K which is much too low. Decreasing Z we obtain models with higher temperature but even lower degree of contact. Thermal equilibrium can therefore be excluded. The configurations are unstable on a time scale of about yr.
© European Southern Observatory (ESO) 1997
Online publication: July 8, 1998