1 Introduction

Processes in the solar atmosphere such as coronal heating and solar wind acceleration have been studied (observationally and theoretically) for a long time. But until now, due to our very limited knowledge of their sources and nature, they still remain among the great challenges of solar physics. The interpretation of these phenomena arises from magnetohydrodynamic theoretical developments, which require the knowledge of vectorial quantities such as the coronal magnetic field and the solar wind velocity field vectors.

In astrophysics, the complete determination of vectorial quantities is more difficult than the determination of thermodynamic scalar quantities (temperature, density, pressure etc.). These in fact can be determined by using spectroscopic methods based on the interpretation of the intensity profile of spectral lines. Common spectropolarimetric methods can only give a partial determination of the vectorial quantities (here, we limit ourselves to resonant scattering): the longitudinal component of the magnetic field via the Zeeman effect, or the line-of-sight component of the scattering atoms velocity field via Doppler shifts of the emitted spectral lines... However, spectropolarimetric methods have the potential to provide a complete determination of vectorial quantities. These methods are based on the interpretation of the Stokes parameters of spectral lines sensitive to the Hanle effect (for the magnetic field) and to the Doppler redistribution (for the velocity field vector). In fact, the Stokes parameters of well selected spectral lines sensitive to the Hanle effect depend on the three components of the magnetic field. Thus, they contain all the information on this vector (strength and direction) as shown by Sahal-Bréchot et al. (1977). Similarly, the Stokes parameters of spectral lines sensitive to the Doppler redistribution of an incident anisotropic and frequency dependent radiation field depend on the strength and direction of the macroscopic velocity-field vector of the scattering atom (Sahal-Bréchot et al. 1998). Consequently, the presence of a magnetic field (velocity field) modifies the polarization of the scattered line which is sensitive to the Hanle effect (Doppler redistribution). Inversely, by measuring the frequency-integrated linear-polarization parameters (degree and direction of linear polarization) of a spectral line sensitive to the magnetic or to the velocity field and another spectral quantity, we can get complete information on the field vector. To determine completely a vector, we need in fact three independent measurements. This is why we need an additional measurement other than the degree and direction of linear polarization. The third quantity may be the electronic density, the Doppler shift, etc.

In the solar corona, the emitted resonance lines from lithium-like ion (O VI, N V, C IV, ...) are excited by isotropic electronic collisions (that do not create polarization in the Zeeman sublevels) and by photoexcitation by the unpolarized and anisotropic radiation coming from the underlying chromosphere-corona transition region. The de-excitation in these lines is by spontaneous emission. The partial anisotropy of the incident radiation from the transition region is at the origin of the atomic alignment in the upper levels. These scattered lines are, consequently, linearly and partially polarized. In addition, they are among the strongest lines emitted in the corona up to large heights above the solar limb (Vial et al. 1980; Kohl et al. 1998; Xing Li et al. 1998; and other papers related to UVCS (UltraViolet Coronagraph Spectrometer, Kohl et al. 1995) aboard the SoHO (the Solar and Heliospheric Observatory, Domingo et al. 1995)).

At low latitudes in the solar corona, these lines have typical widths of a few 10 km s^-1, and they are affected by the Doppler dimming effect due the outflow velocity field of the coronal material (which is also of the order of few 10 km s^-1). The moving atoms (or ions) absorb the incident radiation somewhere in the wings of the incident line profile, at some distance from the zero velocity line center. This means that the absorption takes place away from the maximum of the incident radiation line profile. Consequently, the reemitted line is not only shifted and has its intensity dimmed compared to the incident one, but also its linear polarization parameters (degree and direction of linear polarization) are modified due to the Doppler redistribution effect (Sahal-Bréchot et al. 1998). The linear polarization parameters of the scattered radiation depend on the strength and direction of the macroscopic velocity field vector of the scattering ions, so they carry complete information on the solar wind velocity field vector. Therefore, the measurement of the polarization parameters of such lines (especially the O VI  $\lambda1031.92$ line, hereafter O VI D₂) permits one to determine completely the solar wind velocity field vector. Moreover, the upper levels of the corresponding transitions have short lifetimes. Owing to the smallness of the coronal magnetic field strength, some of these lines are expected to be sensitive to the Hanle effect. The linear polarization profiles of these lines are also expected to contain the complete information on the three components of the coronal magnetic field vector. This makes the measurement of the linear polarization of these coronal lines very important and may be the best method to get information on the solar wind velocity field vector and on the coronal magnetic field vector.

In the second section, we will review briefly the magnetic-field effect (Hanle effect) and the Doppler redistribution effect on the linear polarization parameters of a resonance scattered line. The third section is devoted to the presentation of the physical and astrophysical context of the present work and to the description of the physical system considered to compute the linear polarization of a spectral line due to the resonance scattering of the incident chromosphere-corona transition region radiation by the coronal ions. In our model, we will limit ourself to an elementary scattering medium in a limited volume located on the solar vertical (theoretically, one scattering point). In the fourth section, we will derive the density matrix elements of the unpolarized incident radiation coming from the underlying chromosphere-corona transition region. The fifth section is devoted to the computation of the atomic density matrix elements interacting with the incident radiation and the isotropic electrons of the surrounding medium, in the presence of a local magnetic field. In the computations, we will take into account the effect of the atomic velocity field distribution and the effect of the local magnetic field. The sixth section is devoted to the computation of the density matrix elements of the scattered radiation along the line of sight. In Sect. 7, we use the precedent results to derive the Stokes parameters of a spectral line sensitive to the Hanle effect and to the effect of the drift velocity field of the scattering medium. In the following section, we apply the results of the theory to the case of the O VI D₂ coronal line. We limit ourself for this application to a Maxwellian velocity distribution with a drift velocity field describing the macroscopic motion of the scattering ion. The last section is devoted to the conclusions and the prospects relative to the present work.

Figure 1:  Case of resonance scattering of a purely directive and unpolarized incident radiation in the direction (Pz) by moving atoms with a velocity ${\textbf{\textit{V}}}_{\!\scriptscriptstyle{P}}$ located around the point P. The scattered radiation is observed in the direction (PZ) (the line of sight). The atomic levels being assumed infinitely sharp, in the comoving frame, the atom absorbs and reemits the incident radiation at the eigenfrequency $\nu _{\scriptscriptstyle {0}}$. In the laboratory frame (fixed frame), the atom absorbs the incident photons with the shifted frequency $\nu=\nu_{\scriptscriptstyle{0}} \left(1+\frac{{V_{\scriptscriptstyle{P_z}}}}{c}\right)$. The correspondent intensity $I\left (\nu \right )$ of the incident line profile is lower than the central one $I\left (\nu _{\scriptscriptstyle {0}}\right )$. The scattered line has the frequency $\nu ^\prime$ with a shift $\nu^\prime-\nu_{\scriptscriptstyle{0}}=\nu_{\scriptscriptstyle{0}}\; \frac{{V_{\scriptscriptstyle{P_Z}}}}{c}$, and has also its intensity dimmed compared to that scattered by atoms at rest. The polarization parameters of the scattered radiation are the same as those of a line scattered by atoms at rest. In this figure, we limit ourself to the case of right scattering.