7 Stokes parameters of the reemitted photons

The Stokes parameters of the re-emitted radiation by moving atom with a velocity field ${\textbf{\textit{v}}}_{\scriptscriptstyle{\rm {A}}}$ are given as a function of the density matrix elements of the reemitted photons written in the frame of the line of sight (in number of reemitted photons per unit of time, per unit of volume and per unit of velocity distribution) by the following relations (Sahal-Bréchot et al. 1998, 1986)

$$\begin{array}{lll} {\displaystyle {\cal I}({\textbf{\textit{v}}}_... ...criptstyle{\rm {A}}}, {\textbf{\textit B}}) }&=& {\displaystyle 0}. \end{array}$$ (19)

In the present case, the ${\cal V}({\textbf{\textit{v}}}_{\scriptscriptstyle{\rm {A}}},{\textbf{\textit B}})$ Stokes parameter vanishes because the incident radiation field is considered completely unpolarized.

The general form of the reemitted radiation Stokes parameters (here we take into account only of the linear polarization) is given by

$$\begin{array}{l} \left( \begin{array}{l} {\displaystyle{\cal I}(v... ...al A_I}\\ {\cal A_Q}\\ {\cal A_U} \end{array}\right)\right\}\cdot \end{array}$$ (20)

The quantities ${\cal A_I}$, ${\cal A_Q}$ and ${\cal A_U}$, which contain the effect of the magnetic field, are given in Appendix A.

From the forms of the equations giving the Stokes parameters as a function of the magnetic field and the velocity field, it is clear that the Hanle effect and the Doppler redistribution effect are completely decoupled. Particularly, by cancelling the magnetic field in the terms  ${\cal A_I}$, ${\cal A_Q}$ and ${\cal A_U}$, we obtain the results of Sahal-Bréchot et al. (1998) which give only the velocity-field effect. To obtain the Stokes parameters as a function of the magnetic field alone, we can just cancel the velocity field vector in the dimming term.

We note that we have assumed only the two-level atom approximation and there is no approximation on the incident and the reemitted line profiles. There is also no approximation on the velocity distribution $F(v_{\scriptscriptstyle{\rm {A}}})$ of the scattering atoms. It can be isotropic as in the inner corona (although this point is not really clear) or anisotropic as observed by UVCS/SoHO in the high solar corona.

Following Sahal-Bréchot et al. (1998), to get the distribution in frequency and intensity of the polarization of the scattered radiation, it is necessary to sum the contribution of all the atoms having the velocity ${\textbf{\textit{v}}}_{\scriptscriptstyle{\rm A}_{Z}}$ along the line of sight. This can be performed by an integration over the velocity field distributions ${\textbf{\textit{v}}}_{\scriptscriptstyle{\rm A}_{X}}$ and ${\textbf{\textit{v}}}_{\scriptscriptstyle{\rm A}_{Y}}$ in the plane perpendicular to the line of sight.

The Stokes parameters as a function of the scattered radiation frequency $\nu$ are given (in number of photons per unit of time, per unit of volume, per unit of frequency and per unit of solid angle) by

$$\left( \begin{array}{c} {\cal I}(\nu,{\textbf{\textit B}})\\ {\c... ...d}}v_{\scriptscriptstyle{\rm A}_{X}}{\rm {d}}v_{\scriptscriptstyle{\rm A}_{Y}}.$$ (21)