\ = 4481.2 Â (Mg II)

Fig. 3.29 Line profiles and Rossiter effect. This plot, Fig. 4 in Mukherjee et al. (1996), shows the Rossiter effect on the line profiles in S Cancri. The phase of observation is 0.184. The orbital shifts corresponding to phases 0.05 and 0.184 differ by too small an amount to show significantly. Courtesy J. D. Mukherjee several phase shifts (0.01, 0.02, 0.03, 0.05) for S Cnc. The phase of the observations is 0.184. The orbital shifts corresponding to phases 0.05 and 0.184 differ by too small an amount to show up significantly. In order to compare the computed line profiles with observed ones they are convolved with the instrumental profile. The instrumental profile is usually represented by a Gaussian given in terms of full width at half maximum. The last step in getting the final line profile is to take phase smearing into account by averaging computed line profiles from the beginning and end of the observation interval. In case the interval is sufficiently small the profile can be calculated at the phase corresponding to the middle of the interval.

3.7 Modeling Polarization Curves

Embarras de choix (Embarrassment of choice)

As an example of how additional observables may be incorporated into a light curve model we can consider circumstellar polarization based on scattering in optically thin stellar envelopes.

Chandrasekhar (1946a, b) stimulated interest in polarized radiation from binary stars when he showed that eclipses would break the disk symmetry and allow limb polarization to be observed. In a binary system with a nonspherically symmetric cir-cumstellar envelope, various processes (such as scattering, reflection, and Zeeman splitting of spectral lines in the presence of a magnetic field) can produce a small contribution to the intrinsic linear and/or circular polarization in the total light from the system. The type and degree of polarization will vary over the binary period and will depend on the polarizing mechanism (Thomson scattering, Rayleigh scattering, and others), the distribution of scattering material, and the aspect of the system as seen by an observer.

In principle, polarization data provide a chance to derive the inclination i and thus masses of binary stars in the case of noneclipsing spectroscopic binaries because in this case i is not available from either light curves or radial velocity curves. However, the main polarization mechanism(s) need to be identified and the model must be rather good.

In a binary star system, two main sources of polarization have to be considered: Photospheric polarization, namely the limb polarization effect predicted by Chandrasekhar (1946a, b) and circumstellar polarization . The additional observables are the Stokes quantities

where 0p is the position angle (measured conventionally counterclockwise from North) of maximum signal as an analyzer is rotated, and p _ Fmax Fmin (3 7 2)

Net photospheric polarization should be zero for a centrally symmetric star face even if there is significant polarization at the limb, as expected when electron scattering is important. Eclipses break the symmetry and can lead to net observable polarization, called the limb eclipse effect. Although many attempts have been made to detect the limb eclipse effect, its unambiguous detection remains elusive. Kemp et al. (1983) claimed to have detected it in Algol, but this claim is not universally accepted. Wilson & Liou (1993)presented an analysis of the Kemp et al. (1983) observations and argued that their model, based on the Wilson-Devinney light curve model, showed that the eclipse effect caused a significant part of the rapid variation of the Stokes quantities during primary eclipse.

Wilson & Liou (1993) used the relations outlined in Brown et al. (1978) to compute polarization arising fromcircumstellar matter. Later, Terrell and Wilson (unpublished) combined a gas flow program [Terrell (1994), Terrell & Wilson (1993)] with the Wilson-Liou program to compute observable polarization curves. As can be seen in Fig. 3.30,the polarization curves are very sensitive to the location of and physical conditions in the circumstellar gas. In these simulations, the ionization scheme was relatively simple, but the results show that polarimetry can be a very effective tool. The ionization is computed either from theSaha equation (telling us the relative populations of two adjacent stages of ionization)

N: ne or the Boltzmann-Saha equation (giving the number of atoms available for a transition and so to produce a given spectral line)


The Saha and Boltzmann-Saha equations involve the following symbols: Nis the relative number of atoms in any state of excitation, s, of a stage of ionization i; N, the sum populations of all ionization states; ne, the electron density; x: , the ion-ization potential between adjacent stages of ionization; and some proportionality constants, B and C, which include several atomic constants.

1 2 Phase

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