Detached systems have Q > Q1 and fR < 1, lobe-filling components have fR = 1, and over-contact systems are described by 1 < fR < 2. Similar to fi and f2, for computing f1R and f2R we have to apply the coordinate transformations described above.

Another definition being used in Binary Maker 3.0 is BM | Q!/Q - 1 , Q > Q1

which is properly normalized for detached systems between — 1<fBM < 0 as well.

For circular orbits and synchronous rotation the Roche potential approach led to the morphological types of detached, semi-detached, and over-contact binaries. The names and the full set of categories were used first by Kopal (1954) but the term "over-contact" dates back to Kuiper (1941) who already understood the relevant principles. Unfortunate or not, many authors have used the adjective contact, rather than over-contact, for binaries with common envelopes. However, in a more general context (namely eccentric orbits, asynchronous rotation) a consistent approach is possible only if we have a concept in which the word contact has a meaning in the sense "in contact with a critical surface." To stress again, the word contact refers to contact with a limiting surface (not necessarily with the other component). In that sense, the term "semi-contact" would be a more accurate usage than "semidetached." For the case of circular orbits and synchronous rotation, the "degree of contact" can be quantitatively described by the term contact parameter, or sometimes, fill-out factor, f, defined according to (3.1.101). It measures the degree to which a component fills its Roche lobe: 0 if the potential matches that of the inner, and 1 if it matches the outer Lagrangian surface.

For circular orbits and synchronous rotation the limiting surfaces are the inner Lagrangian surface (the Roche lobes) and the outer Lagrangian surface. These simple scenarios already explain many observed configurations and enable us to link them to evolutionary states taking into account evolutionary expansion, gravitational radiation, mass loss and exchange, and magnetic braking. The first three morphological types are the following:

1. Detached binary systems, where both components are within their lobes. The fill-out factor of each component is negative. If the components are small compared with their Roche lobes, their shapes closely approximate spheres. Whereas morphology and evolutionary state are related for semi-detached and over-contact binaries, such a connection does not exist in detached systems.

2. Semi-detached systems, with one component within its critical lobe, whereas the other exactly fills its lobe (f = 0 for this component). This morphological type includes Algols, cataclysmic variables, and some X-ray binaries,24 in which one component is highly evolved and in which mass transfer occurs. Algol itself may serve as an example of the Algol type. In general, the lobe-filling star can lose matter through the inner Lagrangian point.

3. Over-contact systems or common envelope binaries, where each component has a surface larger than its Roche lobe. Mechanical equilibrium requires that the surfaces match in potential. That is, the common surface must coincide with a single equipotential above the Roche lobes (0 < f1>2 < 1,and f1 = f2). Configurations are limited by the outer Lagrangian surface. This morphological type explains W UMa stars very well. Whereas in all over-contact binaries the more massive star is larger than its companion, Binnendijk (1965, 1970) defined two subclasses of W UMas observationally: W-type and A-type W UMa stars on the basis of the larger star being cooler or hotter than the other, respectively; i.e., the primary minimum being an occultation or a transit. So we have A-type contact systems, in which the more massive star has the greater surface brightness and the W-type systems, in which the more massive and larger star has less surface brightness. Our present interpretation [Lucy (1976), Flannery (1976), Robertson & Eggleton (1977), Wilson (1978), and Lucy & Wilson (1979)] is that W-types are formed by slightly over-contact binaries with moderate mass ratio such as 0.4-0.6 and with components close to the zero age main sequence (ZAMS), and that A-types are somewhat evolved - on the main sequence but not on the ZAMS. Configurations with one component larger than its critical lobe while the other is not do not have closed surface equipotentials and are not expected to exist for more than a few orbits. However, the early (extremely brief) stages of common envelope evolution specifically involve exactly this configuration [cf. Webbink (1992), Taam & Bodenheimer (1992), or Iben and Livio (1994)]. Binaries in which both components exactly fill their Roche lobe (f1 = f2 = 0; the true contact system as we might use the term) could in principle exist. But no mechanism is known by which they could come into existence and they are not expected to be stable against small perturbations. Small effects caused by evolutionary changes lead to either the semi-detached or over-contact scenario.

24 The basic model of X-ray binaries is a close binary system with a "normal" star (main sequence or giant, in exceptional cases also a degenerate star) filling its Roche lobe and transferring matter to the compact object, a neutron star, or a black hole (Krautter 1997).

4. Double-contact system (Wilson, 1979), where each component fills its lobe (see Fig. 3.16) exactly, and at least one rotates supersynchronously. For asynchronous rotation (F = 1) or eccentric orbits (e = 0), over-contact binaries can no longer exist. The extreme case is a centrifugally limited binary or a double-contact system, where two components fill their limiting lobes but do not touch each other. P Lyrae and V356 Sagittarii are likely candidates. What is the astrophysical meaning of double-contact binaries? It is observed that some Algol primaries (among those with primaries well within their Roche lobes) rotate much faster than synchronously, some even close to or approximately at the centrifugal limit. An underlying physical process to account for that fast rotation is spin-up by the accretion process. As described in Wilson (1994), gas transferred from the contact component arrives with considerable angular momentum and converts orbital to rotational angular momentum. The outer envelope of the primary component now spins-up. Rather than the star expanding to reach the lobe, the limiting lobe contracts to meet the star. The secondary component rotates synchronously and already fills its limiting lobe (the ordinary Roche lobe).

Fig. 3.16 The double-contact binary RZ Scuti. This figure, reproduced from the Pictorial Atlas (Terrell et al. 1992, p. 342) and provided by Dirk Terrell, shows the shape of RZ Scuti at phases 0, 0.05, 0.5. Courtesy D. Terrell

In summary, the following stable configurations can occur:

• detached: both components are smaller than the critical or limiting lobe;

• semi-detached: one component is smaller than the critical lobe, while the other fills its critical lobe at periastron;

• double-contact: each component exactly fills its critical lobe (again, at perias-tron); and

• over-contact for F = 1 and e = 0 only: common envelope binary.

In Chap. 4, we show and discuss how a priori knowledge about the configuration of a binary system can be used as a constraint. The Wilson-Devinney program, for example, implements such explicit constraints by different modes of operation.

3.2 Modeling Stellar Radiative Properties

Ignorantia legis neminem excusat (Ignorance of the law excuses none)

The computation of the flux emitted from the binary components requires the integration of local quantities over the surfaces. In the Roche potential models, especially in the circular orbit and synchronous rotation case, the stellar surface S' is defined as the set of all points rs on the equipotential surface (3.1.65) specified by n0,

For each star, in the circular orbit and synchronous rotation case, the surface is parametrized by only two quantities: q and n0. For fixed q, the larger n0 the smaller the star, and vice versa. The surface defined by the set S' of vectors or points rs has the surface area S = fS, da. Scaling with R2 gives the real surface measure. The differential surface element da in spherical coordinates was given in (3.1.5), repeated here for convenience,

cos P

Corresponding to the equipotential condition n(rs, q) = no, rs = (rs,0,^), (3.2.3)

it is possible to define the function25

rs : [-n, n] x [0, 2n] ^ IR+, (0, <p) ^ rs(0, <p), (3.2.4)

which gives the distance of a surface point rs to the center of the star. The computation of the surface area follows as

S = S(n0) = f da = f f —r2(0,<p)sin0d0d<p (3.2.5)

and the volume is

25 In order to define a function we explicitly define the domain of its argument, here 0 and p 2n p n p rs (0,p)

In the eccentric orbit case, there is no potential in the strict sense as discussed in Sect. However, as discussed on page 102, there is good physical reasoning that the volume remainsconstant along the orbit. Thus, the shapes of the stars vary with phase but we require that the volume remains constant. At first, V = V (flp) is computed according to (3.2.6) where flp is the potential at periastron. Note that flp plays the role flo played in the circular orbit case. Then, for a given phase ^, the corresponding Roche potential fl$ is computed. fl$ is derived from the requirement that it yields a stellar surface which leads to the correct volume, i.e., V(fl$) = V. Therefore, to compute fl# the following iterative procedure based on Wilson (1979):

(ß(rs; q, 0) = ß^) ^ rs(k)(0, p) ^ V (ßf ) (3.2.7)

is applied until, after a number of iterations, k = 0,..., K,

a predefined tolerance is achieved. The result is again a radial function rs (0, y) defining the stellar photosphere.

The computation of the flux emitted by the stellar photosphere is based on several assumptions about the underlying photospheric physics. These include the choice of a model atmosphere and several physical effects:

• gravity brightening;

• reflection effect; and

• blackbody radiation, gray atmosphere, or a model atmosphere.

In addition, special physical effects such as dark or bright spots on the star surfaces might be included.

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