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oa of the primary, with an analogous expression for the secondary. In absolute units we eventually get

Vj(0) =-ajosin 0 sini + rnaAVij(0) + y, j = 1, 2. (3.5.6)

Here a constant velocity, y , is added to account for the velocity of the center-of-mass of the binary system.

More realistic modeling of the proximity effects and their influences on the radial velocity curves might consider the variation of line strength. To measure line strength the line equivalent width is used. On stars with negligible winds, it depends on effective temperature, surface gravity, and aspect angle alone. The formalism discussed above computes the global mean radial velocities based on flux-weighted local velocities, but ideally they should also be weighted by equivalent width. Van Paradijs et al. (1977) seem to be the first to have done so. Van Hamme & Wilson (1994) explored the effects of line strength weighting for a small number of binaries and found that the difference between flux-only and [flux/equivalent width] weighted proximity effects were, at most, a few percent of the velocity amplitude in those examples. Van Hamme & Wilson (1997) extended the approach to eccentric orbits and asynchronous rotation. Again the result is confirmed that for high mass X-ray binaries or binaries with extreme mass ratios the effect of the variation of line-strength is of the order of a few percent. Nevertheless, observers are encouraged to list the lines used to measure the radial velocities in their papers. In order to exploit the line-weighting formalism correctly it is recommended to publish the radial velocities for individual lines when line-strength effects are expected to be significant

3.6 Modeling Line Profiles

Divide et impera (Divide and conquer)

after Philip of Macedon (359-336 B.C.)

Accurate modeling of line profiles enables us to estimate stellar rotation rates as has been discussed in Sect. 2.2.3. In the context of light curve modeling it is natural to use a theory of stellar line broadening for the local profile and a binary star model for the rotational theory (Mukherjee et al. 1996). That strategy avoids the overhead produced by stellar atmosphere theory. The relevant input quantities are the effective temperature, T, the damping constant, r, including both natural and collisional damping, the number of absorbers, Nf, the microturbulent velocity, vtur , the ratio of continuum scattering opacity to total continuum opacity, p, the monochromatic ratio of line opacity to continuum opacity, , at frequency v, and the fraction, 1 - e, of absorbed photons which are scattered. This leads to the auxiliary quantity, lv, defined as

Mukherjee et al. (1996, Sect. 3) obtain the emergent intensity, Iv(0, m), for given m defined in equation (3.2.22):

(p - V3a) (1 - 1) Iv (0,m) = (a + PvM) --• (3.6.2)

The expressions for evaluating a, b, and the probability, pv, that a photon is thermal-ized in an interaction with an atom or ion are given in Mihalas (1978, pp. 312-313). For the continuum intensity, Ic(0, m), Mukherjee et al. obtain

Finally, they obtain the residual intensity, relative to the continuum:

which is then integrated by their light curve program to yield the residual flux. The velocity needed for the calculation of a line profile is AVi as computed in (3.5.4).

Line profiles are generated for each surface element. Figure 3.28 shows how the intrinsic line profile changes as T, r, Nf, and vtur are changed. To get the line profile for the entire star the local line profiles are weighted according to the flux from each of these areas. Figure 3.29 shows an example of profiles computed at

Fig. 3.28 Variation in the intrinsic line profile. This plot, Fig. 1 in Mukherjee et al. (1996), shows the influence of several parameters on the intrinsic line profile (^ = 1): clockwise from lower left, the number of absorbers, effective temperature, microturbulent velocity, and the damping constant. Courtesy J. D. Mukherjee

Fig. 3.28 Variation in the intrinsic line profile. This plot, Fig. 1 in Mukherjee et al. (1996), shows the influence of several parameters on the intrinsic line profile (^ = 1): clockwise from lower left, the number of absorbers, effective temperature, microturbulent velocity, and the damping constant. Courtesy J. D. Mukherjee

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