8 Case of a Gaussian velocity field distribution with a drift velocity field vector

We consider the case of the O VI D₂ coronal line formed in the solar corona by the resonant scattering of the unpolarized and partially anisotropic radiation coming from the underlying transition region. We assume that the incident and reemitted line profiles are both Gaussian with respective linewidths $\Delta\nu_{\scriptscriptstyle{\rm {D_i}}}$ and $\Delta\nu_{\scriptscriptstyle{\rm {D_s}}}$. We obtain a corresponding equation for the incident photons polarization matrix element by replacing, in the Eq. (47) in Sahal-Bréchot et al. (1998), the limb-brightening function $f(\alpha)$ by a function $f(\alpha,\beta)$, for the more general case when the incident radiation field is not cylindrically symmetric around the solar vertical (Pz), which is the case when one observes in the coronal polar holes. The polarization matrix of the incident photons is given by

$$\begin{array}{l} {\displaystyle \varphi\left({\textbf{\textit{v}}... ...iptstyle{\rm {D_i}}}}}\right)^2\right] \, {\cal{M}}(\alpha,\beta)}. \end{array}$$ (22)

We assume also that the atomic velocity distribution is Gaussian combined with a drift velocity field vector. It is given by

$${\displaystyle F({\vec{v}}_{\scriptscriptstyle{\rm {A}}}) = \... ...criptstyle{\rm D}_{\rm s}}}\right)^{\scriptstyle{2}}\right]}.$$ (23)

The macroscopic velocity field vector ${\textbf{\textit{V}}}$ can be assimilated to the solar wind velocity.

The equations giving the Stokes parameters for this case are a generalization of those given in the Appendix A in Sahal-Bréchot et al. (1998) (Eq. (A1) for the frequency dependent Stokes parameters, and Eq. (A8) for the integrated Stokes parameters on the line profile). They can be written in the general form

$$\begin{array}{ll} \left( \begin{array}{l} {\cal I}\\ {\cal Q}\\ ... ...l A_I}\\ {\cal A_Q}\\ {\cal A_U} \end{array}\right) \right\}\cdot \end{array}$$ (24)

The coefficients ${\cal K}_{\rm col}$ and ${\cal K}_{\rm rad}$ are given by Eq. (A1) in the appendix of Sahal-Bréchot et al. (1998) for the frequency dependent Stokes parameters, and by Eq. (A8) in the same paper for the Stokes parameters integrated on the scattered line profile. In both cases, we have

$$\begin{array}{l} {\displaystyle {\cal{K}}_{\rm rad}\propto\frac{1... ...{D_i}}}+ \Delta\nu^2_{\scriptscriptstyle{\rm {D_s}}}} \right]}\cdot \end{array}$$ (25)

The term in the exponential contains the velocity-field effect (it is the dimming term).