This potential may be used without significant inconsistencies, if the timescale for nonradial oscillations is much smaller than the orbital period P. In the eccentric orbit case d depends on phase 0 instead of Q (r; q, d) so we also use the notation Q(r; q,0) in the context of eccentric orbits to indicate that the potential and stellar surface depend on phase. We also need the gradient VQ, i.e., the partial derivatives dü

and for the secondary component in the same coordinate system [see Wilson (1979, Eq. 6)]

Note that in the circular-synchronous case, d = 1 and F = 1, (3.1.78) and (3.1.81) give the same expression. The partial derivatives w.r.t. y and z are the same, anyway.

The gradient VQ is needed to compute the normal vector, n, according to (3.1.68) and the local gravity as described in Sect. 3.2.1.

Because the potential (3.1.77) is phase dependent, binary stars moving in elliptic orbits will change their shapes accordingly. The potential formalism is of course only an approximation. What is really needed is an analysis of the response of the stellar surface to varying tidal forces, including nonradial oscillations. Although, a rigorous analysis and computation of the instantaneous volume taking into account that stellar matter is compressible, has not yet been worked out, eccentric binary modeling is often based on the following assumption: The shape of a star varies along the orbit, but it is expected that its volume V remains essentially constant [Avni (1976), Wilson (1979)]. For polytropic stars Hadrava (1986) has formally proven that the contact of the stellar surface with its Roche lobe can occur only at periastron. Therefore the stellar surface may be parametrized by the periastron potential Qp which then yields the periastron volume Vp assumed constant over phase. The phase-dependent potential can then be found from Vp. In what follows we pick up Wilson's (1979) argument. A star's critical lobe size sets an upper limit for its size. In the circular, synchronous case, the maximum size is the Roche lobe (however, if this size is exceeded, we still can have an over-contact binary). In the eccentric case the effective critical lobe size is the one which causes the star to fill its critical lobe exactly at periastron. Hut (1981) shows that rotation will tend to synchronize to the periastron angular rate because of the strong dependence of the tidal force on distance. The periastron-synchronized F is given by Hut (1981, Eq. 44)

Analogous to the Lagrangian point L1 in the circular case with synchronous rotation, the equilibrium point xLp of vanishing effective gravity is, for a given F and periastron separation dp = 1 - e, the solution of the equation dQ

dx 1

The solution of this equation is further discussed in Appendix E.12. The potential Qp corresponding to (xlp, 0, 0) yields Vp of the star by a volume integration. Whereas Q = Q(0) varies along the orbit, the volume V of the star is kept constant, V = Vp.

It should be noted that for e = 0 or F = 1 no over-contact configuration can be stable but now a new configuration enters the stage: double-contact. For a further discussion of binary morphologies we refer to Sect. 3.1.6.

As we have seen, the classical Roche model allows only for gravitational and centrifugal forces. The modifications for eccentric orbits and asynchronous uniform rotation make it possible to analyze a much larger group of binaries. The extended Roche model provides a physically reasonable basis for the description of the geometrical structure and, as we will see in Sect. 3.1.6, evolutionary processes of most systems of intermediate to late spectral type which are not too strongly magnetized. However, in very early spectral-type binaries the interaction between radiation and matter may become important because the radiation pressure increases with the fourth power of the effective temperature. Approaches Including Radiation Pressure

As seen in the literature references below there have been many efforts to extend the Roche model and to include the radiation pressure expected in hot stars. Although these efforts have not yet led to a consistent and commonly accepted model, due to several deficiencies, it seems appropriate to discuss them briefly, to comment on their deficiencies, to point the reader to the problems involved in including radiation pressure, and hopefully, to raise further interest in the subject.

In very hot stars the radiation pressure is due to the interaction between electromagnetic radiation and matter and can be important. The radiation pressure decreases the effect of gravity, depends on the momentum transfer associated with absorbed or scattered photons, and is a complicated function of the local conditions. Because a large fraction of the momentum transfer is due to absorption in prominent ultraviolet resonance lines, the problem is related to the radiative acceleration of stellar winds [see, for instance, Castor et al. (1975) or Hearn (1987)].

Dynamically, radiation pressure leads to complicated situations. Stars with radiative envelopes have solutions at depth that are insensitive to surface boundary conditions. Thus, controlled by the optical depths, not too far below the surface, the state variables including the total radiation pressure [cf. Mihalas (1978, Eq. 1-46, p. 17)]

PR = IaT4, a = 4- = 7.5647 ■ 10-15 erg ■ cm-3K-4 (3.1.84)

3 c will be constant on the standard Roche equipotential surfaces. Accordingly, to get the shape of the photosphere, we integrate the structure equations inward along normals to these potential surfaces and determines the starting height by requiring asymptotically the constancy of state variables on equipotential surfaces. The solution thus obtained necessarily has horizontal pressure gradients in the surface layers of nonspherical stars (Kippenhahn and Weigert, 1989), but they become vanishingly small in deep layers. These gradients will give rise to "geostrovarPhic winds" analogous to the Earth's jet stream. Because the depth of the photospheres of hot stars on the main sequence is about 1% of the radius, this is the order of magnitude of the deviations from Roche geometry that we could expect in best cases (Lucy 1997). For very hot stars and certainly for WR components, the photosphere is formed in the star's radiatively driven wind and large deviations from Roche geometry will arise as the problem becomes nonstatic.

In a binary system the radiation pressure influences not only the shape of the surface by the gravitational force field but also deforms the companion's surface directly (this might be called the outer radiation pressure effect).

Despite the physical effects and complexity mentioned above some early and simple attempts to include radiation pressure have been made by Schuerman (1972), Kondo & McCluskey (1976), Vanbeveren (1977, 1978), Djurasevic (1986), and Zhou and Leung (1987). These approaches have in common that they use a modified force field, and they consider only the inner radiation pressure effect due the radiation of the star itself. They replaced the potential GM1/r1 of the hotter component (and if necessary also that of the secondary accordingly) by


r FG

where G is the gravitational constant, and FG is the force due to gravity

G Mi

Assuming that 5 is constant, the potential (3.1.59) in the binary system is now supposed to be

- ^rad(x, y, z) = G(1 - 51)-1 + G(1 - 52)—2 + — r2ffl. (3.1.87)

R1 R2 2

Note that FR accounts only for the interaction of stellar matter with the star's own radiation field and is derived as follows. At first, the radiation pressure, PR, acting on a unit surface element is given by

where y is the angle between the surface normal and the incident radiation, d&> is the solid angle element, and Iv is the monochromatic intensity in the frequency interval dv around v .If p denotes the mass density, force and radiation pressure are coupled by

Equation (3.1.89) is true if PR includes all radiation pressure contribution from both stars. However, in the papers above, using the monochromatic average opacity kv of the envelope and absorption coefficient kv = Kv/p per unit mass, a plane parallel radiative or spherically symmetric transfer equation is assumed, and PR is replaced by the inner radiation pressure. Then V PR is replaced by the radial derivative of PR and the radiation force per unit mass follows as

Fr =---— = — Kv Iv cos y d«dv = -—- kv Iv dv, (3.1.90)

P 9r cp Jo Ja 4nr2c Jo where c = 2.9979 ■ 108 m/s is the speed oflight, and kv is an average absorption coefficient of the envelope per unit mass. Thus, using (3.1.86), we finally get a constant expression for á

The assumption VPR = 9PR/9r is true only for spherical stars. For nonspherical stars, gravitation and flux-proportional inner radiation pressure do not vary with the inverse square of the distance, r. Nevertheless, based on this approach, the shape of equipotentials under the influence of the inner radiation pressure has been investigated by several authors: Djurasevic (1986), Zhou & Leung (1987), Drechsel et al. (1995), and Niedsielska (1997). Figure 3.13 (courtesy Drechsel) shows the meridional intersections of equipotential surfaces of a binary system with mass ratio, q = 1, for different á1 values. The top part shows the shrinking of a fixed equipotential surface = 3.75) with increasing á1; the bottom part demonstrates the influence of increasing inner radiation pressure on the extent of the Roche lobe of the primary.

Fig. 3.13 Inner radiation pressure effects [Fig. 1 in Drechsel et al. (1995)]. Courtesy H. Drechsel

Fig. 3.13 Inner radiation pressure effects [Fig. 1 in Drechsel et al. (1995)]. Courtesy H. Drechsel

This approach, although used by many authors, has not been without criticism. Howarth (1997) shows that the inner radiation pressure does not change the stellar figure at all. His arguments are based on radiative equilibrium andvon Zeipel's law (see page 117). For a lobe-filling star the gravity at the inner Lagrangian point,

L1, is zero and thus according to von Zeipel's law, the temperature (and hence the inner radiation pressure) is also zero, and thus cannot change the location of L1. His mathematical argumentation is: According to von Zeipel's law the flux vector F is proportional to the gradient of the gravitational potential [see Eq. (3.2.10)]. The radiative and gravitational acceleration of star 1 are antiparallel and coupled by arad = -5g. (3.1.92)

This leads to the effective surface gravity acceleration geff = g + arad = (1 - 5)g. (3.1.93)

According to (3.1.58), g is the (negative) gradient of the potential &, so it is also possible to represent geff as the gradient of the effective potential

The potential ^eff differs from the modified potential ^rad; ^eff follows from ^ by simple scaling. On page 97 we derived the dimensionless potential fl from ^ by dividing it by GM1. Note that if we divide ^eff by (1 - 5)GM1 we get the same dimensionless potential fl. That clearly tells us that the inner radiation pressure does not change the shape of the components.

The description of the radiation pressure also needs to consider the incoming radiation of the companion (outer radiation pressure effect). Even under mild conditions it is no longer possible to derive an analytical expression describing the equipotential surface. Drechsel et al. (1995) treat the photosphere as a deformable membrane subject to the radiation of the companion and compute iteratively its shape. However, if inward integrations were made, enormous unbalanced pressure gradients would be found in deep layers. Nevertheless, because it is the first time that the inner radiation effect and the radiation pressure of the companion are considered separately, we briefly sketch their approach coded into a light curve program.21

To account for the consequences of irradiation of the companion, Drechsel et al. (1995) introduced two functions = (0, y, r),

depending on the local coordinates of a surface point on component j . These functions vary according to

21 They used a circular orbit version of the Wilson-Devinney program. It is also the first time that the outer radiation pressure has been coded into a light curve program.

and take their maximum values at the intersection points of the lines connecting both mass centers with the stellar surfaces = 0°) and the minimum values at the stellar horizons (# = 90°).

For a given surface, the integration of the incident flux, F(0, p, r), on a unit area element located at (0, p, r), is very similar to that used to compute the reflection effect. The computation of 5* (0, p, r) is performed by the formula

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