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Astron. Astrophys. 345, 986-998 (1999)

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Appendix A: line profile parameters

The line shift is defined as [FORMULA], where [FORMULA] is the wavelength of the red, respectively blue [FORMULA]-component peak. The line broadening is the difference between the centre-of-gravity wavelengths of the blue and red [FORMULA]-components:

[EQUATION]

The function [FORMULA] stands for V, Q, or U and [FORMULA] for the unsigned wavelength relative to line-centre. We isolate the effects of the wave by removing the width of the reference profile (i.e., the corresponding Stokes profile calculated in the absence of the wave) according to [FORMULA]. The unsigned [FORMULA]-component amplitudes are [FORMULA] (where [FORMULA] and [FORMULA] indicate the blue and red [FORMULA]-components, respectively). In order to stress the variations the total amplitude is normalized to the amplitude [FORMULA] of the reference profile: [FORMULA]. The relative amplitudeandarea asymmetry are defined as

[EQUATION]

respectively. Here, [FORMULA] and [FORMULA] are the unsigned areas of the blue and red [FORMULA]-components, respectively.

Appendix B: analytical considerations based on a Milne-Eddington atmosphere

In this Appendix we use analytical solutions of the polarized radiative transfer equations, including the magneto-optical effects, describing a Zeeman-split line in a Milne-Eddington atmosphere to explicate the dependence of the V, Q and U profiles on [FORMULA] and [FORMULA]. The solutions (due originally to Rachkovsky, 1967) are taken from, e.g., Arena & Landi degl'Innocenti (1982) and read

[EQUATION]

Here [FORMULA], while the [FORMULA] (with [FORMULA], V, Q, or U) are defined as

[EQUATION]

The definitions of the [FORMULA] are obtained if in Eqs. (B5)-(B7) [FORMULA] is replaced by [FORMULA]. Each [FORMULA] (with [FORMULA]) is basically a Voigt function and [FORMULA] a Faraday function (e.g. Landi degl'Innocenti 1976), but their precise functional form does not play a role for the present purpose. If we introduce [FORMULA] and [FORMULA], as well as the similarly defined quantities [FORMULA] and [FORMULA], then Eqs. (B1)-(B3) read

[EQUATION]

Note that since [FORMULA] and [FORMULA] are independent of [FORMULA] and [FORMULA], so are [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. [FORMULA] and s in (B8)-(B10) still depend on [FORMULA], but all [FORMULA] dependences are explicitly written in the [FORMULA] and [FORMULA] terms. In Eqs. (B8) to (B10) the terms proportional to [FORMULA] generally give the bulk of the signal and the terms proportional to [FORMULA] are due exclusively to the magneto-optical effects. Except very close to the limb [FORMULA] does not cause a change in sign of any terms. The azimuth [FORMULA], in contrast, has opposite signs in the left and right halves of the flux tube (as seen from the vantage point of the observer). Eq. (B8) then predicts that the Stokes Q amplitude has the same sign for both [FORMULA] and [FORMULA]: [FORMULA] is dominated by [FORMULA] while the magneto-optical effects introduce a weaker dependence on [FORMULA]. Stokes U, however, has the reversed dependence on [FORMULA] according to Eq. (B8) the bulk of the signal is proportional to [FORMULA] and only the magneto-optical effects give a term proportional to [FORMULA]. Finally, Stokes V remains independent of [FORMULA] in this approximation.

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999
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