SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


Astron. Astrophys. 357, 351-358 (2000)

Previous Section Next Section Title Page Table of Contents

Appendix A: Stokes V asymmetry in a canopy configuration with vertical magnetic field

In 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, [FORMULA] the value of t at the interface of the canopy with the field-free atmosphere below, [FORMULA] and [FORMULA] the intensities of the emerging left- and right-circularly polarized radiation, respectively, [FORMULA] and [FORMULA] the absorption coefficients of the respective polarized radiation, [FORMULA] the ratio of the opacity of polarized radiation to that of the continuous radiation and [FORMULA] the outward directed radiation intensity at the interface. Assuming LTE with [FORMULA] the Planck function, the intensity of the polarized light emerging from the atmosphere for a given wavelength is then represented by

[EQUATION]

with

[EQUATION]

where

[EQUATION]

The subscript p refers to left- or right-circularly polarized light, i.e. it represents either L or R. Strictly speaking, [FORMULA] and [FORMULA] 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 [FORMULA] and [FORMULA], where [FORMULA] 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:

[EQUATION]

[EQUATION]

The Stokes V parameters at both wavelengths are [FORMULA] and [FORMULA]. Because of the symmetry of the Zeeman effect [FORMULA] and [FORMULA] and therefore [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. Hence

[EQUATION]

Denoting by [FORMULA] the intensity ratio [FORMULA] we obtain

[EQUATION]

and

[EQUATION]

For the Stokes V amplitude asymmetry, defined as [FORMULA], we thus obtain

[EQUATION]

Assuming the magnetic field to point downwards, [FORMULA] is negative, hence [FORMULA] and [FORMULA]. Let us further assume that the Zeeman shift is large compared to the width of the spectral line. Then we have [FORMULA] and [FORMULA] where [FORMULA] denotes the line absorption coefficient. Thus we obtain:

[EQUATION]

with

[EQUATION]

where [FORMULA] denotes the line optical depth and [FORMULA] the outward directed continuous intensity at the interface. Since [FORMULA] increases with decreasing [FORMULA] and [FORMULA] decreases with [FORMULA], the asymmetry increases with decreasing [FORMULA] as long as the fraction in (A.2) is dominated by [FORMULA]. To assess the role of [FORMULA] let us consider its numerator :

[EQUATION]

which is positive for large and negative for small values of [FORMULA]. Since the denominator of [FORMULA] is always positive the sign of [FORMULA] is equal to the sign of the numerator. Thus the asymmetry as function of [FORMULA] is smaller than [FORMULA] for large values of [FORMULA]; it becomes equal to [FORMULA] at some intermediate depth whence [FORMULA] becomes larger than [FORMULA]. The limit of R(t) for vanishing [FORMULA] is:

[EQUATION]

hence

[EQUATION]

If Zeeman and Doppler shift are equal, [FORMULA] represents the relative intensity of the spectral line at its center. Further, in a standard model of the quiet solar atmosphere [FORMULA], hence

[EQUATION]

Fig. A.1 shows [FORMULA] and [FORMULA] as function of [FORMULA] 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 [FORMULA] and [FORMULA] become equal is at about [FORMULA] while for the FeI  6302 line the crossing occurs much deeper in the atmosphere, close to [FORMULA]. In both cases the prediction of (A.3) is exactly realized: for the line FeI  5250.2 [FORMULA], [FORMULA], hence [FORMULA]; as can be seen from Fig. A.1 the limit of the amplitude asymmetry is 82 %. Similarly for the line FeI  6302, [FORMULA], [FORMULA], hence [FORMULA]; Fig. A.2 shows that the limit of the asymmetry is 70 %.

[FIGURE] Fig. A.1. Stokes V amplitude asymmetry of FeI  5250 emerging from a quiet sun canopy configuration with 1500 G and 3.3 km s-1 as function of [FORMULA] of canopy interface. Also plotted are [FORMULA], [FORMULA] and the asymmetry obtained by the evaluation of (A.2).

[FIGURE] Fig. A.2. The same quantities as in Fig. A.1, for the line FeI  6302.

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000
helpdesk.link@springer.de