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Astron. Astrophys. 318, 171-178 (1997)

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2. Theoretical equilibrium configurations

Approximate 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 condition

On 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 [FORMULA] on the surface

[EQUATION]

where [FORMULA] denotes the Roche potential and v is the velocity of the mass motions in a rotating frame.

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

[EQUATION]

where [FORMULA] denotes the surface. This equation gives a generalised surface condition (Eq. 20 in K95). The presence of internal mass motions is expressed by parameters [FORMULA] and [FORMULA] representing respectively the average of the squared velocity on the surface of the component 1 and 2. They are defined by

[EQUATION]

where [FORMULA] with [FORMULA] period is the angular velocity of the system, [FORMULA] is the radius of the component i and the angle brackets denote a spherical average.

In the general case there are closed streamlines on the surfaces of both components and the area [FORMULA] in Eq. (2) must be replaced by [FORMULA], where [FORMULA] is the region enclosed by these streamlines. The generalised surface condition remains an approximation with a modified interpretation of [FORMULA]. This parameter represents the average of the squared velocity on a part of the surface of the component i (at least a belt around the equator).

If [FORMULA] and [FORMULA] are small compared to unity, the departures from Roche geometry are small. In the absence of internal mass motions ([FORMULA]) the generalised surface condition reduces to the Roche equipotential condition.

2.2. Hydrostatic equilibrium

As shown in K95, internal mass motions in the envelope i can approximately be characterized by two parameters [FORMULA], where [FORMULA] expresses the mean kinetic energy of these motions and the coefficient [FORMULA] expresses the correlation of these motions with the orbital motion. The motions are rapid (comparable with the orbital velocity) if [FORMULA] is of order unity, and slow compared with the orbital velocity if

[EQUATION]

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 [FORMULA] since [FORMULA].

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 freedom

We shall consider configurations in good thermal contact. Accordingly, we assume that the equal entropy condition

[EQUATION]

is satisfied, where [FORMULA] denotes the specific entropy in the adiabatic part (more precisely, at the inner border) of the convective envelope k. The mixing-length in units of the pressure scale height will be taken to be [FORMULA].

Let [FORMULA] denote the Roche potential on the surface, and [FORMULA] the potential in the Lagrangian point [FORMULA]. The degree of contact F is defined by

[EQUATION]

and the contact condition is

[EQUATION]

When discussing the degrees of freedom we include also unphysical configurations violating the contact condition.

The physical parameters of a stationary system are mass M, total angular momentum (the sum of orbital and spin angular momentum) J and composition. Concerning the composition, for the hydrogen content in the envelope a standard value [FORMULA] will be adopted. Evolutionary effects in the primary can roughly be taken into account adopting the simple hydrogen profile

[EQUATION]

where [FORMULA] and [FORMULA] is the mass variable. Evolutionary effects in the secondary are much smaller and will be neglected. The composition is then described by the metallicity Z, the hydrogen content in the primary's centre [FORMULA] and the fractional extent of the primary's evolved core [FORMULA]. Unless stated otherwise, [FORMULA] will be assumed.

With these approximations we have the four physical parameters [FORMULA]. Basic observational data are period P and mass ratio q. The angular momentum can be replaced by one of these quantities. Accordingly, if four of the five parameters [FORMULA] are given, the configuration is usually (at least locally) determined uniquely.

2.4. Thermal stability

The 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 curves

For 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 [FORMULA] and the effective temperatures [FORMULA] of the components, their separation A as obtained by Keplers law

[EQUATION]

and the inclination angle i. Normalized radii [FORMULA] of the components and the ratio of temperatures [FORMULA] are defined by [FORMULA]. The separation A will be taken as the unit of length. Let s be the projected separation and B the projected area of the system. The components will be treated as spheres of radii [FORMULA], apart from the neck near the inner Lagrangian point. Two approximations for the neck are shown in Fig. 1. In the first approximation the projected area is overestimated, i.e. we obtain an area [FORMULA]. In the second approximation the area is underestimated ([FORMULA]). We shall take the average

[EQUATION]

Next we make use of the approximations to split the area B into the contributions of the components, writing [FORMULA]. The distribution of surface brightness is then approximately determined by the parameter [FORMULA].

[FIGURE] Fig. 1. Two approximations for the projected area of a contact binary. The area is either overestimated (top) or underestimated (bottom)

It remains to give explicit expressions for these areas. In the case of [FORMULA] we have either an occultation with

[EQUATION]

or a transit with

[EQUATION]

In the case of [FORMULA] we obtain

[EQUATION]

with [FORMULA]. In the subcase of [FORMULA] we find

[EQUATION]

with [FORMULA], and finally

[EQUATION]

In the remaining subcase of [FORMULA] we have

[EQUATION]

with

[EQUATION]

If the primary is in front of the secondary, the final result is

[EQUATION]

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:

[EQUATION]

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.

[FIGURE] Fig. 2. Observed and theoretical light curves of BV Dra (see text)
[FIGURE] Fig. 3. Observed and theoretical light curves of BW Dra (see text)
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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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