## 2. Theoretical equilibrium configurationsApproximate structure equations for contact systems in thermal equilibrium have been derived in K95, allowing for the effects of large-scale mass motions in the common turbulent envelope. ## 2.1. The surface conditionOn account of these mass motions the Roche equipotential condition is replaced by a more general surface condition. The underlying assumptions require some comments. Neglecting turbulent viscosity in the Navier-Stokes equations for steady motions we obtained Bernoulli's equation for a stream line on the surface where denotes the Roche potential and
If all streamlines meet in stagnation points, as in the velocity field proposed by Webbink (1977), the constant is the same for all streamlines and we find where denotes the surface. This equation gives a generalised surface condition (Eq. 20 in K95). The presence of internal mass motions is expressed by parameters and representing respectively the average of the squared velocity on the surface of the component 1 and 2. They are defined by where with period is
the angular velocity of the system, is the
radius of the component In the general case there are closed streamlines on the surfaces of
both components and the area in Eq. (2) must be
replaced by , where is
the region enclosed by these streamlines. The generalised surface
condition remains an approximation with a modified interpretation of
. This parameter represents the average of the
squared velocity on a part of the surface of the
component If and are small compared to unity, the departures from Roche geometry are small. In the absence of internal mass motions () the generalised surface condition reduces to the Roche equipotential condition. ## 2.2. Hydrostatic equilibriumAs shown in K95, internal mass motions in the
envelope If this inequality is satisfied, the effects of internal mass motions are small. The effects on the surface condition are of the second order in since . The spectral line broadening of typical contact binaries is consistent with the assumption that the internal mass motions are slow compared with the orbital velocity (cf. Ruciski et al. 1993 and references therein). Rapid motions are almost certainly incompatible with the observed broadening functions. For this reason the inequality (4) will be treated as an observational constraint. Using this constraint we conclude that the effects of internal mass motions are very small. In other words, thermal equilibrium implies approximate hydrostatic equilibrium. In the further discussion hydrostatic equilibrium will be assumed. ## 2.3. The degrees of freedomWe shall consider configurations in good thermal contact. Accordingly, we assume that the equal entropy condition is satisfied, where denotes the specific
entropy in the adiabatic part (more precisely, at the inner border) of
the convective envelope Let denote the Roche potential on the
surface, and the potential in the Lagrangian
point . The degree of contact When discussing the degrees of freedom we include also unphysical configurations violating the contact condition. The physical parameters of a stationary system are mass where and is the mass
variable. Evolutionary effects in the secondary are much smaller and
will be neglected. The composition is then described by the
metallicity With these approximations we have the four physical parameters
. Basic observational data are period ## 2.4. Thermal stabilityThe thermal stability problem concerns perturbations in the distribution of entropy. The perturbed system is supposed to be in hydrostatic equilibrium and uniform rotation and to satisfy the equal entropy condition. Mass and total angular momentum and the composition of each mass element are kept fixed. With these definitions the thermal stability problem is uniquely defined. Note that (on account of perturbations in the mass ratio) the chemical profile of a perturbed system is not described by Eq. (8). The stability can be tested either in a stability analysis (i.e. in a discussion of the eigenvalues of the stability problem) or studying the evolution, starting with a small perturbation. This stability problem is formally applicable also to systems violating the contact condition. (Physically, equal entropy requires contact, but this point is unimportant in a formal stability discussion.) Other thermal stability problems can be obtained on replacing the equal entropy condition by another equation. The connection between different thermal stability problems will be discussed in Sect. 4. ## 2.5. Theoretical light curvesFor comparison with the observations we shall need theoretical light curves. Here we present a simple light curve synthesis without reference to Roche geometry. Input parameters are the spherically averaged radii
and the effective temperatures
of the components, their separation and the inclination angle Next we make use of the approximations to split the area
It remains to give explicit expressions for these areas. In the case of we have either an occultation with or a transit with In the case of we obtain with . In the subcase of we find with , and finally In the remaining subcase of we have with If the primary is in front of the secondary, the final result is If the secondary is in front of the primary, the indices 1 and 2 must be interchanged. The systems BV Dra and BW Dra can be used to test this simple light curve synthesis. Light curves of BV Dra (Yamasaki 1979) and BW Dra (Ruciski & Kaluzny 1982) and spectroscopic observations of both systems (Batten & Lu 1985) were used by Kaluzny & Ruciski (1986, hereafter KR86) to derive the following parameters: Theoretical light curves with these parameters as obtained by the present simple procedure are shown in Figs. 2 and 3, together with the observed light curves. The agreement is rather good for BV Dra (the theoretical light curve is somewhat too shallow) and very good for BW Dra. A comparison with the theoretical light curves of KR86 shows that the present results are only slightly inferior. Accordingly, effects neglected in the present light curve synthesis are of minor importance.
© European Southern Observatory (ESO) 1997 Online publication: July 8, 1998 |