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Note that r is a dimensionless quantity. This follows from the definition d = 1 . If d is known in physical units, then r scales accordingly.

If we want to compute the force in physical units, it follows from (3.1.64)

Note that once we know the potential and its gradient we can compute the normal vector, n, at each surface point by

The negative sign in (3.1.68) ensures that the normal vector points inward. The explicit formulas to compute Vfl in the most general case are provided on page 101.

3.1.5.2 Circular Orbits and Asynchronous Rotation

Already by the early 1950s there was well-established evidence for asynchronous rotation in many close binaries; cf. Struve (1950). Some of the more interesting Algols have rapidly rotating primaries [Van Hamme & Wilson (1990), Wilson (1994)]. Fast rotation strongly deforms a star as is demonstrated for RZ Scuti and RW Persei in the Pictorial Atlas (Terrell et al. 1992). Wilson (1994) also discusses slow or subsynchronous rotation which is pertinent to the study of common envelope evolution.

To model fast and slow rotation binaries, it is necessary to extend the concept of Roche surfaces to asynchronous rotation. It is assumed that the stars rotate uni-formly,20 so that star 1 rotates with angular velocity vector mi . We further simplify the dynamics by neglecting minor rotation-induced changes in the mass distribution. We use m to refer to the angular velocity vector of orbital rotation. The acceleration of a mass element in the rotating frame with center in that star (Fig. 3.11) was derived by Limber (1963) and has the form d2ri 1 ( Mi M2

+ m x (m x r01) + m1 x (M1xr1) - 2m^ ( —1 ) , (3.1.69)

20 An asynchronous theory by Peraiah (1969, 1970) includes even nonuniform rotation. But it seems that it has not been applied to real observations or incorporated into a general light curve program.

where r1 is the radius vector from the center of star 1 to the point of interest, r01 is the vector pointing from the center of mass to the center of star 1, p is the stellar density, and p is the gas pressure. Note the different sign convention for the potentials. The first term is the force due to pressure gradients in the stars, the second term represents gravity, the last term is the Coriolis force, and the other terms are centrifugal force and an offset from the center of mass.

If (3.1.69) is transformed to the corotating frame of the orbit with center in star 1 defined in Sect. 3.1.5.1, following Limber (1963), all other terms can be expressed by means of an effective potential

^eff := G — + G —+ rn r01X1 + 1 &?r&, (3.1.70)

where r& denotes the distance between the point of interest and the rotation axis of star 1. A special case arises when & and & are parallel to each other, i.e., & x & = 0. In this case, the effective potential takes the form tf-eff = gM + gM + 1 &2r2&+1 ¿y2r2i + &&r&, & := & - &. (3.1.71)

Note that in the limit & = 0, the effective potential ^eff is identical to the potential ^ in (3.1.59) describing the synchronous case. Here we will consider only the case that & and are parallel. For that case, Fig. 3.12 shows the x, y-plane and the quantities r0&, r01, r&1, and x.

If, following Limber (1963), we now assume that the mass motions in star 1 with respect to the rotating frame are negligible, i.e., r1', r1, and as a consequence the Coriolis forces are small compared to all other terms in (3.1.69), we end up with

Thus, under this assumption, according to (3.1.72), surfaces of constant pressure and constant density are identical with theequipotential surfaces of ^eff. Thus, from now

Fig. 3.12 Nonsynchronous rotation. Definition of geometrical quantities in the orbital plane

on, it is sufficient to concentrate on theeffective potential ^eff- With the geometrical relations r( = x2 + y2, rf)m = (x - xc)2 + y2 = (x2 + y2) - 2xxc + xc, (3.1.73)

and the definition of the rotational parameter F (the ratio of angular rotation rate to the mean orbital revolution rate a)

a the term involving the centrifugal potential takes the form

1 (2r2a+ 2a2r((i + ((< = 2a2 [r2«+ (F-1)2r( + 2(F - 1)r( ]

Combining (3.1.71) and (3.1.75) and proceeding as in Sect. 3.1.5.1 yields

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