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T"—1 u il 1, '

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Fig. 3.30 Polarization curves of SX Cassiopeiae. This plot, Figs. 4-24 to 4-29 in Terrell (1994), shows mass transfer events of several durations. The upper curves are the U curves, the lower curves the Q curves. Courtesy D. Terrell

Polarization observations of Algol-type binaries are relatively rare for several reasons. Although bright, Algol has low circumstellar activity. The more active systems are too dim for existing telescope/polarimeter combinations. However, polarimetry has seen increased interest in recent years, and the combination of more sensitive polarimeters and the new 8-m class telescopes will be able to collect and detect enough photons to achieve reasonable signal-to-noise ratios. Although there are very few observations to compare to the model at present, the availability of larger telescopes equipped with efficient Polarimeters promises to make polarimetry a very powerful probe of interacting binaries in the next few years.

A desirable polarization model would be based on the solution of the hydro-dynamic flow equations, including viscosity and pressure, providing the electron densities of circumstellar gas elements for many points in space. Today there are no well-developed programs to do the full dynamical circumstellar computations.44 In order to compute Q and U, limb and circumstellar polarization are combined as in Wilson & Liou (1993) . The limb polarization is computed from the WilsonDevinney light curve model, and the circumstellar polarization is computed by the relations given in Brown et al. (1978) assuming optically thin pure electron scattering.

3.8 Modeling Pulse Arrival Times

Binaries that contain an X-ray pulsar are very rich data sources. Besides radial velocity and light curves they also provide pulse arrival times. As in the previous sections, we need to be able to compare observed and theoretical quantities, in this case pulse arrival times in addition to other observables. Therefore, following Wilson & Terrell (1998), we write the pulse arrival time in Heliocentric Julian Date, t , as

with Tref being the arrival time of a reference pulse (which defines pulse phase zero) and At and Atref the light time delays due to orbit crossing for a given pulse and the reference pulse, respectively. S is the number of days in a second of time (1/86400), Pp is the pulse period in seconds, and n is an integer assigned to an observed pulse.

Equation (3.8.1) establishes our model for computing the pulse arrival times. The delays At depend implicitly on the common light curve parameters q, i, a, and e according to

Sadq sin i

where R0 is the radius of the Sun in kilometers, d is the instantaneous separation of the two stars in units of a, 0 is the geometrical phase defined in (3.1.20), and c is the speed of light in kilometers per second.

44 For some specific applications, it certainly could be done - probably within a year or less of development. The real problem is that there are almost no published observations to test the idea (or sufficiently accurate and numerous ones, covering at least several consecutive orbits). So there is little motivation at present to do the calculations.

Wilson & Terrell (1998) proceed as follows to compute At. The pulse ephemeris is used to obtain time. The first two terms on the right-hand side of (3.8.1) are interpreted as time, tp , kept by the pulsar clock and measured in Heliocentric Julian Date, i.e.,

This time, tp, is coupled to the orbital phase, 0, according to ( 2.1.1):

where T0 denotes the reference epoch, P is the orbital period, and 0s is a constant offset which in most cases is simply zero. The next step is to compute the mean anomaly M according to (3.1.34). This gives the true anomaly, u, the geometrical phase, 0, and, finally, the separation, d.

So, the pulse arrival model involves the following set of adjustable parameters:

As noted by Wilson & Terrell (1998), the parameters can be adjusted by differential corrections and analytic derivatives exist for nine parameters.

An analysis by Wilson & Terrell (1994, 1998) of data from a binary containing an X-ray pulsar is briefly outlined in Sect. 7.3.1.

3.9 Self-Consistent Treatment of Parallaxes

As pointed out in Chap. 1 (page 22) the distance D or parallax n of a binary can (in favorable cases) be derived if both light and radial velocity curves are available. If parallax data are available for the binary, for instance, from the Hipparcos mission [cf. Rucinski & Duerbeck (1997)], or if it is a member of an object (e.g., star cluster, galaxy) with known distance, nn > 1, measured values of the parallax might be available. Thus, on the one hand, the parallax is a systemic observable and, on the other hand, it is a model parameter to be estimated by the least-squares analysis.

Let us now consider the parallax n as a parameter in the EB model, and let D denote the distance of the binary connected to n by n = —. (3.9.1)

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