9 Conclusions

We have developed, in the frame of the density matrix formalism, the theory giving the Stokes parameters of a linearly polarized spectral line sensitive simultaneously to the Hanle effect due to a local magnetic field and to the Doppler redistribution due to the macroscopic velocity field of the scattering atoms. In the obtained equations, there is no assumption on the geometry of the magnetic field and velocity field vectors. There is no hypothesis also on the line profiles nor on the velocity distribution of the scattering atoms. The obtained results are general and can be applied to physical problems implicating the magnetic field and/or the velocity field. From the final results, it is possible to obtain the Stokes parameters of a spectral line sensitive only to the Hanle effect or only to the velocity-field effect. This is done by cancelling one or other of these quantities in the final equations. In particular, we get the results obtained by Sahal-Bréchot et al. (1998), by cancelling the Larmor frequency (the magnetic field) in the angular terms ${\cal A_I}$, ${\cal A_Q}$, ${\cal A_U}$. We obtain the Hanle effect alone by cancelling the velocity field in the dimming term.

As an application to the coronal case, we have used a Maxwellian velocity field distribution with a macroscopic velocity field vector for the coronal ions. In future work, we will take into account, in the calculations, the velocity distribution anisotropy observed by UVCS/SoHO (the ion cyclotron effect). The integration along the line of sight will be the subject of a more complete future work.

We will use these results in a further paper to interpret the linear polarization parameters of the O VI D₂ coronal line measured by using observations performed by the SUMER (Solar Ultraviolet Measurements of Emitted Radiation) spectrometer (Wilhelm et al. 1995; Wilhelm et al. 1997; Hassler et al. 1997) on board of SoHO on March 19, 1996 (Raouafi et al. 1999a,b,c,d; Raouafi 2000).

Acknowledgements

The author gratefully acknowledges helpful discussion with S. Sahal-Bréchot and P. Lemaire. The author is also indebted to Roberto Casini for deep reading of the manuscript and for critical and helpful comments.