Astron. Astrophys. 357, 351-358 (2000) Appendix A: Stokes V asymmetry in a canopy configuration with vertical magnetic fieldIn order to understand the behaviour of the Stokes V asymmetry of a photospheric spectral line in a canopy configuration as function of interface height we shall consider the case of a purely vertical magnetic field in the canopy region because then the radiation transport equations of right- and left-circularly polarized light decouple and become identical to the transport equation of unpolarized light (Landi Degl'Innocenti 1992) and may easily be solved. A `normal' Zeeman effect is assumed. Let t be the optical depth of the continuous radiation at the frequency of the spectral line, the value of t at the interface of the canopy with the field-free atmosphere below, and the intensities of the emerging left- and right-circularly polarized radiation, respectively, and the absorption coefficients of the respective polarized radiation, the ratio of the opacity of polarized radiation to that of the continuous radiation and the outward directed radiation intensity at the interface. Assuming LTE with the Planck function, the intensity of the polarized light emerging from the atmosphere for a given wavelength is then represented by with where The subscript p refers to left- or right-circularly polarized light, i.e. it represents either L or R. Strictly speaking, and should also have the index p; we omit it for better readability. To derive Stokes V and its asymmetry we shall consider two wavelength points and , where is the wavelength at center of the spectral line. Denoting by the superscripts - and + these wavelength positions we have the four intensities of the polarized radiation emerging from the atmosphere: The Stokes V parameters at both wavelengths are and . Because of the symmetry of the Zeeman effect and and therefore , , and . Hence Denoting by the intensity ratio we obtain and For the Stokes V amplitude asymmetry, defined as , we thus obtain Assuming the magnetic field to point downwards, is negative, hence and . Let us further assume that the Zeeman shift is large compared to the width of the spectral line. Then we have and where denotes the line absorption coefficient. Thus we obtain: with where denotes the line optical depth and the outward directed continuous intensity at the interface. Since increases with decreasing and decreases with , the asymmetry increases with decreasing as long as the fraction in (A.2) is dominated by . To assess the role of let us consider its numerator : which is positive for large and negative for small values of . Since the denominator of is always positive the sign of is equal to the sign of the numerator. Thus the asymmetry as function of is smaller than for large values of ; it becomes equal to at some intermediate depth whence becomes larger than . The limit of R(t) for vanishing is: If Zeeman and Doppler shift are equal, represents the relative intensity of the spectral line at its center. Further, in a standard model of the quiet solar atmosphere , hence Fig. A.1 shows and
as function of
for the line FeI
5250.2 and a model of the quiet solar atmosphere, Fig. A.2
the same for the line FeI 6302. For the
FeI 5250 line the point where
and
become equal is at about
while for the FeI
6302 line the crossing occurs much deeper in the atmosphere,
close to . In both cases the
prediction of (A.3) is exactly realized: for the line
FeI 5250.2 ,
, hence
; as can be seen from Fig. A.1
the limit of the amplitude asymmetry is 82 %. Similarly for the line
FeI 6302, ,
, hence
; Fig. A.2 shows that the limit
of the asymmetry is 70 %.
© European Southern Observatory (ESO) 2000 Online publication: May 3, 2000 |