The dynamic forces controlling the stellar mass distributions involve the effects of rotation, tides, and noncircular orbits. For an introductory-level discussion of all these effects, see Wilson (1974). Fortunately, tidal forces produce circular orbits and synchronous rotation in many interacting binaries. A detailed and excellent analysis of the tidal evolution in close binary systems is provided by Hut (1981). The orbital period of a synchronous rotator in a circular14 orbit is the same as the rotation period. We will discuss only synchronous rotation in this section.

Another physical simplification reduces the mathematical complexity: Although the stars may be relatively large and considerably distorted, they attract one another nearly as if their entire masses were concentrated into mass points at their centers. Therefore, only two forces need to be considered in the circular orbit and synchronous rotation case:

1. gravitational attractions of two mass points and

2. the centrifugal force due to the rotation of the entire binary system about its center of mass.

Given that both gravitational and centrifugal forces are time-wise constant for coro-tating matter, we can expect to find solutions for static configurations in the coro-tating frame. A somewhat similar problem was solved by the French mathematician E. Roche (1820-1883) in the nineteenth century (Roche, 1849, 1850). The basic

14 Note that eccentric binaries tend to have their angular rotations locked at the orbital angular rate at the periastron (Hut, 1981).

concept for understanding the solutions of that problem is equipotential surfaces (briefly, equipotentials). These are surfaces on which the sum of rotational and gravitational energy per unit mass is constant. On these level surfaces, also called "Roche surfaces," the component of the force vector tangential to these surfaces vanishes, i.e., the local force vector is everywhere normal to them. The Roche surfaces are indeed the static surfaces we are interested in: They corotate with the orbital motion of the binary. Binary component surfaces are now modeled as equipotentials (similarly on Earth, where ocean and lake surfaces follow equipotentials). The force perpendicular to the surface is different for different equipotentials and also varies as a function of location on a particular surface unless this surface is a sphere. In the vicinity of the Earth, equipotentials due to the combined gravitational forces of Earth and Moon are almost spheres. A family of binary system equipotentials projected onto their orbital plane is illustrated in Fig. 1.3. One point in this figure is called the Lagrangian point L1, after the French mathematician J. L. Lagrange (1713-1765). For a corotating test particle at Lp1 the gravitational and rotational forces balance, so the particle feels no force. The Roche surface15 passing through L1p consists of two ovoid surfaces called the Roche lobes of the components. The two ovoids touch at Lp1.

1. detached systems (Fig. 1.4), if neither component fills its Roche lobe;

2. semi-detached systems, if one component fills its Roche lobe, and the other does not; and

3. over-contact systems, if both components exceed their Roche lobes.

Fig. 1.3 Projections of equipotential Roche surfaces. The plot, showing equipotential Roche surfaces projected onto the orbital plane, was produced with Binary Maker 2.0 for a binary system with mass ratio q = 0.5 and Roche potentials = 5,^2 = 3. The outer curve corresponds to the surface passing through Lp. The next inner one passing through Ljp (point where lines cross) represents the Roche lobes for both stars. Both curves depend only on the mass ratio. Finally, the inner near, circular curves are the stars corresponding to the above given potentials and mass ratio

Fig. 1.3 Projections of equipotential Roche surfaces. The plot, showing equipotential Roche surfaces projected onto the orbital plane, was produced with Binary Maker 2.0 for a binary system with mass ratio q = 0.5 and Roche potentials = 5,^2 = 3. The outer curve corresponds to the surface passing through Lp. The next inner one passing through Ljp (point where lines cross) represents the Roche lobes for both stars. Both curves depend only on the mass ratio. Finally, the inner near, circular curves are the stars corresponding to the above given potentials and mass ratio

15 In more general models, including eccentric orbits and asynchronous rotation, the expression Roche surface and Roche lobe will be replaced by critical surface and critical lobe.

The interpretation and physical properties (such as stability) associated with the morphological classes introduced above is discussed in further detail in Sect. 3.1.6. The form of a component is closely related to the contact parameter, or sometimes, fill-out factor, f, which measures the degree of lobe filling [Chap. 3, definition (3.1.101)].

If eccentric orbits and nonsynchronous rotation are considered, then additional configurations besides the one discussed above may occur [see Chap. 3 or Wilson (1979, 1994)].

There is some correspondence between the morphological classification based on the Roche lobes and the phenomenological classification presented in the previous section:

Algol-type light curves ^ semi-detached systems (Fig.1.5), W UMa-type light curves ^ over-contact systems (Fig. 1.6).

Note that the phenomenological classification of the j3 Lyrae-type light curve has no morphological counterpart. Sometimes, j3 Lyrae-type light curves are produced by detached systems, sometimes by semi-detached systems, and sometimes also by systems having marginal over-contact; see the Binary Stars Pictorial Atlas (Terrell et al. 1992), for example. However, there are semi-detached binaries that are not Algols (e.g., cataclysmic variables) and over-contact binaries that are not W UMa's (e.g., over-contact binaries like TUMuscae).

Having this concept and basic understanding of Roche potentials, it is possible to give a physically useful definition of close binaries following Plavec (1968, p. 212): Close binaries are those systems in which a component fills its critical Roche lobe

Fig. 1.6 Roche potential and shape of an over-contact binary. The plot has been produced with Binary Maker 2.0 for the TYBootis parameter set in the examples collection (Bradstreet, 1993). Both components exceed their Roche lobes (0 < f12 < 1, f2 = f1)

at some stage of its evolution. Prior to this evolutionary definition the custom was to define a close binary as one in which the dimensions of at least one component are of the same order of magnitude as the separation.

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