Astron. Astrophys. 340, 579-592 (1998)

1. Introduction

Accurate knowledge of the physical conditions which prevail in the solar corona and in the solar wind acceleration region is crucial for understanding the physical mechanisms which are at the origin of coronal heating, mass and energy transport and acceleration of the solar wind. In particular, determination of both magnetic and matter velocity field vectors is essential since their coupling must be taken into account in magnetohydrodynamical modelling.

Determination of scalar quantities, such as temperatures and densities can be achieved through usual spectroscopic methods based on interpretation of the frequency dependence of the intensity of suitable lines or of the continuum of the observed radiation field. Yet determination of vectorial quantities is more complex: in fact, the complete information on strengths and directions is contained in the polarization parameters of the received radiation. Intensity transports a part of the information only. Consequently, for achieving a complete vector diagnostic, polarimetric measurements are required, and theoretical methods of spectropolarimetric diagnostics leading to all the Stokes parameters of adequate lines must be developed.

During the twenty past years methods of determination of vector magnetic fields in astrophysics have been developed (Bommier 1977; Bommier & Sahal-Bréchot 1978, and further papers), leading to interpretation of the linear polarization parameters of lines affected by the Hanle effect in terms of magnetic field vectors in solar prominences (Leroy et al. 1983, 1984; Bommier et al. 1981, 1986a, 1986b; Bommier et al. 1994; Bommier & Leroy 1997). This has been carried out with the help of the quantum theory of the matter-radiation interaction within the density matrix formalism (Fano 1957; Cohen-Tannoudji 1962, 1968, 1977; Cohen-Tannoudji et al. 1988). The entirely new vectorial data which have been obtained for solar prominences have changed our knowledge on their magnetic structure; they have shown the interest of having developed a method capable of determining the complete vector field and not only its projection on the line-of-sight as a result of conventional Zeeman studies. Likewise it would be certainly interesting to also develop a method of determination of the matter velocity field vector. Usual methods based on the interpretation of the Doppler shift of spectral lines yield only the component along the line-of-sight.

The first attempts for determining one more component of the matter velocity field of the solar acceleration region have been carried out by Gabriel (1971) and Beckers & Chipman (1974) and then by Kohl & Withbroe (1982) and Withbroe et al. (1982a, 1982b). Following Gabriel et al. (1971) who observed high in the corona during an eclipse the Ly line of hydrogen formed by resonance scattering of the incident Ly chromospheric radiation, they remarked that a number of other interesting lines of the transition region should also be observed high in the corona: they should be partially formed by resonance scattering of the incident transition region radiation. In particular, Li -like ion lines should be expected, and the O vi 103.2 nm line should be one of the most intense: in fact, these ions have broad abundance curves and thus a sufficient amount of ions can remain at coronal temperatures, allowing their detection. Besides, these ions may also be "frozen in" within several solar radii (Bame et al. 1974; Withbroe et al. 1982a, 1982b; Kohl & Withbroe 1982), which enhances the probability of finding them high in the corona. The EUV spectrometer Sumer of the spatial Solar and Heliospheric Observatory Soho of ESA-NASA offers a new and important opportunity for such observations (Wilhelm et al. 1995, 1997; Lemaire et al. 1997; Hassler et al. 1997).

The interest of detecting these resonance scattered lines lies in the fact that they should be affected by the velocity field of these coronal ions: qualitatively, the moving ion absorbs the incident radiation somewhere in the incident line wing because of the Doppler effect, and the absorbed intensity is smaller. This leads to a decrease of the scattered line intensity, which is called the Doppler dimming effect (Hyder & Lites 1970). Thus the intensity of the scattered line will be sensitive to the projection of the velocity field on the direction from the scattering ion towards the region of incident radiation (the vertical to the surface of the sun in average). The scattered line being also shifted by the Doppler effect (leading to the determination of the projection of the velocity field on the line-of-sight), Kohl & Withbroe (1982) and Withbroe et al. (1982a, 1982b) suggested that the interpretation of the Doppler shift associated to that of the Doppler dimming should offer an opportunity of increasing our knowing on the velocity field of the solar wind acceleration region. They based their analysis on the basic theory of resonance scattering for an incident radiation perfectly directive, the moving scattering atoms having an anisotropic Maxwell distribution of velocities with a hydrodynamical ensemble velocity. The diagnostic can then give two informations (cf. Fig. 1): the velocity field component along the direction of the incident radiation, and that along the line-of-sight . However, the component perpendicular to the scattering plane is not attained and the complete vector diagnostic cannot be achieved.

Fig. 1. Resonance scattering in the perfectly directive case: A is the scattering atom, with velocity . Az is the direction of incident radiation line. Its intensity is frequency-dependent. AZis the line-of-sight. The atom absorbs and reemits the incident light at the eigenfrequency in its atomic frame at rest. Due to the Doppler effect, in the laboratory frame, the atom absorbs the frequency and reemits the line in the AZ direction at the frequency . Thus the scattered line is shifted and dimmed through . The shift is sensitive to and the dimming to . The scattered line is linearly polarized along the perpendicular to the scattering plane but the degree of polarization does not depend on the atomic velocity: for a scattering at right angles, it is equal to 1 for a normal Zeeman triplet and to 0.428 for a line.

The aim of the present paper is to show that the complete information on the three components of the matter velocity field vector is contained in the three first Stokes parameters (I, Q, U: intensity and linear polarization) of the coronal scattered line sensitive to the Doppler dimming effect, provided that the complete geometry of the scattering would be taken into account (i.e., the incident radiation is partially and not perfectly directive, cf. Fig. 2).

Fig. 2. The direction of polarization of the reemitted line: OAz is the preferred direction of incident radiation. (a ) Without velocity field, the direction of polarization of the reemitted line is most often perpendicular to the scattering plane zAZ (i.e. along Ax (or AX)). It can be in the scattering plane (along AY) in the case of a strong limb brightening and a scattering atom very close to the limb. (b ) In the presence of a velocity field, the polarization degree and polarization direction can be modified (rotation of angle ).

In Sect. 2 we will focus on the formalism which is general and can be applied to a variety of astrophysical problems where anisotropies of resonance scattering occur.

In Sect. 3 we will apply the formalism to the two-level atom and we will give the expression of the Stokes parameters of a coronal line formed by resonance scattering of the same line originating from the transition region and having a Doppler profile. Other line-broadening mechanisms will be neglected. Doppler redistribution of radiation will be only considered.

Quantitative results will be presented in the next paper on the example of the O vi 103.2 nm line which should be one of the most intense lines that are sensitive to the Doppler dimming effect in the solar wind acceleration region (cf. also Sahal-Bréchot et al. 1992b; Sahal-Bréchot & Choucq-Bruston 1994 where preliminary results have already been given).

© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998
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