In orbits with significant eccentricity, the separation d between the components varies significantly, leading to variable deformation; in the case of asynchronous rotation expressed by F (for definition of F, see page 100) the axes depend on rd := r0/d according to

\ /1 - 1(2 - 4q)^2rd + |(1 + q)F^ \ b = rJi - 1(2 + 5q)A2rd + 1(1 + q)F2A2^3 . (3.1.54)

The error in the length of the semi-major axes in this ellipsoid approximation is of the order of qrJ. Thus, for extremely close stars with r0 « 1 d the error is about 6% for q = 1. For r0 < 2d the error is smaller than 3%. The approximations up to O(r®) are not valid for the modeling of close (especially contact and over-contact) binary systems. The surface shapes are not correctly represented by triaxial ellipsoids in such cases, and the use of ellipsoidal models to derive photometric mass ratios is inappropriate.

Let us summarize: The Wood model is most useful for sufficiently detached systems, for which the surface distortions are adequately approximated by triaxial ellipsoids. It is certainly better for the analysis of these systems than any model based on spherical stars or rectification. However, for an adequate treatment of severely distorted components, only a model based on equipotential surfaces will suffice.

3.1.5 Roche Geometry and Equipotential Surfaces

Auspicium melioris aeui (An omen of a better age)

The Roche model is based on the following assumptions about mass distribution and orbits.

First, both components are assumed to act gravitationally as point masses (surrounded by essentially massless envelopes). This allows a relatively simple analytical representation of the potential. Theories of stellar structure show that in most cases the approximation of the potential as of two point sources plus a centrifugal potential is sufficient.

Second, it is implicitly assumed that periods of free nonradial oscillations are negligible when compared with the orbital period P, so that the shape of the components is determined by the instantaneous force field. This fact becomes very important for modeling eccentric orbit binaries. The timescale of these oscillations is of the order of the hydrostatic timescale, which for solar type stars is about 15min. Surfaces of constant potential are assumed to be surfaces of constant density. In particular, this is true for the stellar surface, viz., the visible photosphere. For fixed mass ratio, rotation rates, etc., the stellar surface is parametrized by only one quantity: The potential energy of that surface.

As the basic assumption of the applicability of the Roche model is that the stars must be in hydrostatic equilibrium, strictly speaking, Roche potentials are only valid for components moving in circular orbits and rotating synchronously. The solution of the nonsynchronous problem was first presented by Plavec (1958) and, in an apparently independent work, by Limber (1963). A generalization of Roche potentials to treat eccentric orbits was investigated by Avni (1976). The asynchronous and eccentric solutions were first properly combined by Wilson (1979). In the following subsections, the equations for circular and eccentric orbits are presented separately. Circular Orbits and Synchronous Rotation

Consider point masses moving in circular orbits around their center of mass. Assume an orthogonal right-handed coordinate frame (see Fig. 3.1) with origin at point 1, corotating with the system so that component 2 lies always on the (positive) x-axis and has the vector coordinates r2 = (1, 0, 0)T. The z -axis is parallel to the normal vector of the orbital plane. Component masses are labeled Mi and M2, and S denotes the center of mass. In this environment, a test particle of unit mass in the atmosphere of component 1 experiences a gravitational plus a centrifugal force. The total force F acting on the test particle is given by

R1 R2

where G = 6.673 ■ 10-11 m3kg-1s-2 is the gravity constant, and Rj denotes the distance of the point r = (x, y, z)T from the center of component j. r0( is the vector rQ( = Mr - (xc, 0, 0)T, M := diag(1, 1, 0), (3.1.56)

originating in (xc, 0, 0)T and pointing to (x, y, 0)T, xc is the position of the center of mass on the x-axis, viz.,

M2 q M2

where d is the separation of the components centers, and q denotes the mass ratio.

The force F per unit mass (this is the surface gravity acceleration g) can be computed as the gradient

of the potential (Kopal 1959)

M1 M2 o2 2 - ^(x, y, z) = GR- + GR- + — r2(, (3.1.59)


2 = (x - Xc)2 + y2 = (x2 + y2) - 2xxc + xc (3.1.60)

is the perpendicular distance of the particle from the orbital rotation axis which is parallel to the vector (xc, 0, 1)T. Whereas the first and second right-hand side terms of (3.1.59) are the gravitational potentials of M1 and M2, the third term is the centrifugal potential due to the rotation of the frame of reference.

The relation (3.1.58) expresses that we have a conservative force field, energy is conserved, and the integral

along any closed path vanishes.

Let us now transform the potential into a more convenient form. Under the assumptions that the stars revolve in circular orbits and the axial rotation is synchronized with the orbital revolution, the angular18 velocity o can be replaced according to Kepler's law by

Substituting (3.1.62) in (3.1.59) and using spherical polar coordinates (3.1.1)

leads to a replacement of the physical potential U by the normalized or modified Roche potential Q

V d GM1

taking the form

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