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

Appendix A: line profile parameters

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

The function stands for V, Q, or U and 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 . The unsigned -component amplitudes are (where and indicate the blue and red -components, respectively). In order to stress the variations the total amplitude is normalized to the amplitude of the reference profile: . The relative amplitudeandarea asymmetry are defined as

respectively. Here, and are the unsigned areas of the blue and red -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 and . The solutions (due originally to Rachkovsky, 1967) are taken from, e.g., Arena & Landi degl'Innocenti (1982) and read

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

The definitions of the are obtained if in Eqs. (B5)-(B7) is replaced by . Each (with ) is basically a Voigt function and 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 and , as well as the similarly defined quantities and , then Eqs. (B1)-(B3) read

Note that since and are independent of and , so are , , and . and s in (B8)-(B10) still depend on , but all dependences are explicitly written in the and terms. In Eqs. (B8) to (B10) the terms proportional to generally give the bulk of the signal and the terms proportional to are due exclusively to the magneto-optical effects. Except very close to the limb does not cause a change in sign of any terms. The azimuth , 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 and : is dominated by while the magneto-optical effects introduce a weaker dependence on . Stokes U, however, has the reversed dependence on according to Eq. (B8) the bulk of the signal is proportional to and only the magneto-optical effects give a term proportional to . Finally, Stokes V remains independent of in this approximation.

© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999
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