for and in the coordinate frame of component 2 are used to compute the stellar surfaces and surface normal vectors. That in turn leads to new values of 5*(0, r) and so on.
As computed by Drechsel et al. (1995), in extreme cases such as the one shown in Fig. 3.14 increasing radiation pressure can force the secondary to switch from inner to outer contact configuration. So besides changing the stellar shapes the system configuration can be changed completely due to the shift of the positions of the Lagrangian points and the altered shapes and extents of the Roche lobes. In
scenarios with 51 and S2 of a few percent Drechsel (1997, private communication) reports that the radii of the stars change only by a few percent as well.
Although the modeling of the inner and outer radiation pressure effects of the previous paragraph are based on doubtful assumptions, the computations indicate that radiation pressure can have drastic effects in binary systems and thus require us again to be careful regarding the physical assumptions. Let us therefore summarize the assumptions and their deficiencies and point the reader to the physics to be considered.
Even the modeling of the inner radiation pressure needs to incorporate the gradient of the radiation pressure as done by Howarth (1997), not the radial derivative, because the local physics involves the entire force field. The reason is that a local point on the surface sees only the entire force field, not the separate gravities of the two stars and not the centrifugal force. The outer radiation pressure will lead to horizontal pressure gradients in the surface layers causing instabilities and fluctuations on the surfaces of the stars. Thus the problem is not static. The "potential functions" nrad in the Drechsel et al. membrane formalism are not potential functions in the strict sense because their gradient does not generate the net force field. At best, we can hope that if radiation pressure is sufficiently small the potential is only slightly perturbed and that V nrad approximates the force.
So despite many efforts there is no consistent model for Roche geometry including radiation pressure. If the radiation pressure is negligible as in most stars there is no need to consider it. If it becomes relevant (e.g., in Wolf-Rayet binaries or X-ray binaries) the stars are so hot that the radiation pressure effects become very important and require a dynamical treatment. In these cases, there are additional effects such as radiation-driven colliding stellar winds, as discussed in Sect. 188.8.131.52, that require further modifications of our binary model.
Whereas the original classification of EBs was phenomenological (for types EA, EB, EW see Sect. 1.2.2), based on observed light curves, morphological classification based on equipotentials provided more physical insight. Associated with the concept of equipotentials are "limiting surfaces" or "limiting lobes." A limiting lobe is the volume enclosed by a limiting surface. The usefulness of morphological classifications is that each of the stable configurations is generated by a structural-evolutionary process.
Let us start with the circular orbit and synchronous case. The equipotentials of (3.1.65) are identical with the surfaces of zero relative velocity in the restricted three-body problem [Szebehely (1967), Kopal (1978)]. There exist five Lagrangian points Lp, i = 1,...,5, characterized by the requirement,22 V Q = 0. The Lagrangian point, Lp1, is also called the inner Lagrangian point and is of particular relevance for EB stars because it is critical to the concepts of detached, semidetached, and over-contact binaries. L^ lies between the two stars (see Fig. 3.15), and at that point surfaces of equal potential coalesce in such a manner that the
22 In the more general cases of eccentric orbits or asynchronous rotation we will use the term equilibrium points rather than Lagrangian points.
surfaces passing through L1 are the largest closed equipotentials enveloping the two stars separately. L1 marks the inner Lagrangian surface and the Roche lobes of the components; the relative sizes of the Roche lobes depend directly on the mass ratio such that the star with greater mass has the larger lobe. If one of the stars fills its Roche lobe (semi-detached binary), it may overflow the critical surface, transferring mass to its companion through Lp1. If both stars satisfy this condition, we call the system a contact binary.23 The modified potential at the inner Lagrangian surface is called QI and that at the outer, QO. Note that these quantities depend only on q. The latter potential marks the effective limit of the binary; matter beyond this surface is lost from the binary system through the outer Lagrangian point, L2. When a particle leaves the binary through L2 its energy is too small to escape to infinity. However, it is then no longer forced to corotate with the binary and, for most mass ratios, acquires enough energy by gravitational interaction with the binary to spiral to infinity. If components are in contact, i.e., QI > Q > QO, then Q describes the surface of the common envelope. Such a system is an over-contact binary.
We are now in a position to connect the notions of lobe-filling stars and the values of Roche potential values. If only one component accurately fills its Roche lobe the system is semi-detached. If neither fills its Roche lobe, it is detached.The computation of L1 and of the critical potentials QI and Q O is explained in Appendix E.12. The degree of contact is measured by the contact parameter, f, sometimes called the fill-out factor or parameter :
Note that f = 0 when the component fills its lobe, i.e., Q = QI; and f = 1 when Q = Q O, but when one of the components is within its Roche lobe, the meaning of the contact parameter can be extended: f < 0 for that component.
The fill-out factors need to be computed for each component separately in each component's own reference frame. They are only reasonably defined for circular and synchronous orbits (d = 1 and F1 = F2 = 1). Thus we have
f1:= f (Q1, q), f (Q, q):= qi(q, q) -q0(q, q), Q * QI(Q, q),
where the functions QI(Q, q) and Q0(Q, q) are evaluated as described in Appendix E.12. To compute the fill-out factor of component 2, it is necessary to transform Q2 into the coordinate system of component 2:
23 This configuration is a special case (e = 0, F = 1) of what on 113 page is called a double-contact binary. As is discussed later, a contact binary is not likely to exist. Nevertheless, it is interesting from a mathematical point of view.
The inverse transformation to (3.1.103) is
which enables us to compute f2
The fill-out factor should not be confused with a similar term, fR, also described as the fill-out parameter, introduced by Rucinski (1973):
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