## Ellipsoidal Models

Autre temps, autres mÅ“urs (Other times, other customs)

The models by Wood (1971, 1972) assume that the forms of the components can be described by triaxial ellipsoids with semi-axes aj, bj, and Cj, with the major axes along the line of centers at periastron. The orbit is allowed to be eccentric. Usually it is assumed that tidal forces in close binaries require the orbital angular momentum vector and the rotation axes of the stars to be parallel. Furthermore, axial and mean orbital rotation are usually synchronized.

The orbital parameters are the same as in the spherical model with addition of eccentricity e and argument of the periastron In addition, we have six geometric parameters, the semi-axes aj, bj, and cj of the ellipsoids. Instead of these parameters the Wood model alternatively also uses the following six dimensionless parameters: a = ai/A, k = a2/ai, the ellipticities sj = bj /aj in the orbital plane, and relative deviations c j /b j

perpendicular to the orbital plane and normalized w.r.t. ej. As outlined in Sect. 2.8, the component eclipsed at the deeper minimum is defined as star 1. Thus the ratio k = a2/a1 can be larger than 1 if star 2 is the larger one. For k > 1 the primary minimum is an occultation;16 for k < 1 it is a transit.11

For triaxial ellipsoids with semi-axes a, b, and c the direction cosines (nx, ny, nz), of the surface normal, n, are given by

16 Occultation is an eclipse of the smaller star by the larger one.

17 Transit is the passage of the smaller star in front of the larger star.

The distance r from a surface point, r, r = (ka,bb,vcf , (3.1.44)

to the center is given by r = V (la)2 + (bb)1 + (vc)2. (3.1.45) Finally, by exploiting l2 + b2 + v2 = 1, (3.1.46) the differential surface element (3.1.5) takes the form da = r2 sin 0d0= Dr3 sin 0d0d<P. (3.1.47)

cos p

The basis for Wood's ellipsoidal model is provided by Chandrasekhar's (1933a) investigations of equilibrium figures in close binary systems, where the companion is assumed to be a point source, and the star itself is described by a polytropic stellar model. In such models (Chandrasekhar, 1939, p. 43) the density p varies with the radial coordinate 0 according to p ~ 0n where n is the polytropic index. Chandrasekhar analyzed the distortion of such polytropes under the influence of rotation and tides. This leads to an expansion of the potential in spherical harmonics up to order 4, or equivalently, terms of O (r0) are neglected. The radius vector r from the center of mass to a surface point in direction (l, b, v) is r = r0