*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
amplitude*and*area 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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