2 The magnetic-field and the velocity-field effects on the linear polarization

2.1 The Hanle effect

The Hanle effect results in the modification of the linear polarization parameters (degree and direction of linear polarization) of a spectral line by a local magnetic field vector. From a classical point of view, the excited atom (or ion) acts as a dipole, so the corresponding dipolar electric field is submitted to a precession motion around the magnetic field vector at the Larmor frequency, $\omega$, which is directly related to the magnetic field strength ( ${\displaystyle\omega=\frac{\mu_{\scriptscriptstyle{\rm {B}}}}{\hbar}\,B}$,  $\mu_{\scriptscriptstyle{\rm {B}}}$ is the Bohr magneton and B is the magnetic field strength). In addition, the dipolar electric field strength is damped by the finite lifetime of the upper level of the atomic transition considered. In the case where the magnetic field is strong, the precession motion dominates the damping and, consequently, the electric field vector follows a completely symmetric path, and the re-emitted radiation is depolarized (see cf., Landi Degl'Innocenti 1992: Fig. 8). However, if the damping time and the precession period are approximately of the same order of magnitude, the emitted radiation is linearly polarized with a degree lower than in the case of zero magnetic field. The polarization direction also makes an angle with that for zero magnetic field (for more details, see Mitchell & Zemansky 1934). The modification of the polarization parameters depends on the strength and direction of the magnetic field vector. We should notice that this classical description can explain only the case of the normal Zeeman triplet (i.e., two-level atom $J_{\rm u}=1,\,J_{\rm l}=0$). For more complicated atomic transitions, one must use quantum theory.

The Hanle effect was used successfully to study the magnetic field vector in solar prominences (Leroy 1977; Sahal-Bréchot et al. 1977; Bommier & Sahal-Bréchot 1978; Leroy 1978; Bommier et al. 1981; Sahal-Bréchot 1981; Landi Degl'Innocenti 1982; Athay et al. 1983; Leroy et al. 1983; Leroy et al. 1984; Querfeld et al. 1985; Bommier et al. 1986a,b; Bommier et al. 1994). Bommier & Sahal-Bréchot (1978 & 1982) showed that the sensitivity of a given spectral line to the Hanle effect occurs within the range

$$0.1\le\omega\,\tau\le10,$$ (1)

where $\tau$ is the lifetime of the upper level of the atomic transition. In the visible, hydrogen lines H$\alpha$ and H$\beta$, and the helium line D₃, have an ideal sensitivity to the Hanle effect. This is one of the reasons for the study of the Hanle effect in solar prominences. In the solar corona, the expected order of magnitude for the magnetic field strength is of a few Gauss. Consequently, only lines with short lifetimes are expected to be sensitive to the Hanle effect. Polarization of lithium-like ion lines observed in the far ultraviolet (FUV) are expected to be also sensitive to the effect of the local magnetic field. The measurement and interpretation of the linear polarization parameters of such lines should give us important information on the coronal magnetic field.

2.2 The Doppler redistribution

2.2.1 Case of a directive incident radiation

Figure 2:  Effect of the velocity field of the scattering atoms on the linear polarization parameters of the reemitted radiation. To obtain the modification of the polarization parameters, we need a partially anisotropic incident radiation field. This is the case of the chromosphere-corona transition region radiation field scattered by the coronal ions. The polarization parameters of the reemitted photons carry with them important information on the macroscopic velocity field vector of the coronal ions which can be assimilated to the solar wind velocity field vector. In the present figure, we consider the case of right scattering, the line of sight (PZ) is parallel to the axis (Py).

In the present section, we consider the case of unpolarized radiation beam propagating in the direction $\left(Pz\right)$ (Fig. 1) and which is scattered by moving atoms or ions at P. We observe the scattered photons in the direction of the line of sight $\left(PZ\right)$. We assume that the incident line profile is a Gaussian centered at the frequency $\nu _{\scriptscriptstyle {0}}$ of the corresponding atomic transition. In the atomic frame, an atom (or ion) absorbs the incident radiation and reemits photons at the frequency $\nu _{\scriptscriptstyle {0}}$. This assumes that the atomic sublevels are infinitely sharp and that the absorption profile can be assimilated to a $\delta$-function. Since the natural width of the transition is several orders of magnitude smaller than the (Doppler) width of the incident radiation, this is indeed a very good approximation. In the laboratory frame, the moving atom with a velocity $\vec{V}_{\!\scriptscriptstyle{P}}$ absorbs incident photons at the shifted frequency $\nu=\nu_{\scriptscriptstyle{0}} \left(1+\frac{{V_{\scriptscriptstyle{P_z}}}}{c}\right)$. Thus, the absorption takes place somewhere in the wings of the incident line profile, which means that the absorbed intensity is lower than that of the central zero velocity field frequency of the incident line profile. Consequently, the scattered line is not only shifted (by $\Delta\nu\propto\nu_{\scriptscriptstyle{0}} \frac{V_{\scriptscriptstyle{P_Z}}}{c}$, where $V_{\scriptscriptstyle{P_Z}}$ is the component of the atomic velocity field along the line of sight), but has also its intensity dimmed compared to the incident one (Fig. 1). In this case (perfectly directive incident radiation, i.e., maximum anisotropy), it is found that the atomic velocity field affects only the scattered line intensity but has no effect on its linear polarization parameters, which are the same as those of photons scattered by atoms at rest (here we take into account only radiative excitation). The corresponding formulae are given by Sahal-Bréchot et al. (1998) (formulae 51-55). To get the effect of the velocity field on the linear polarization parameters of the scattered radiation, we need partial anisotropy of the incident radiation.

2.2.2 Case of partially anisotropic incident radiation: qualitative explanation

To explain qualitatively the velocity-field effect on the polarization parameters of the scattered radiation, we consider the case of the coronal ions which scatter the unpolarized and partially anisotropic radiation coming from the underlying chromosphere-corona transition region. In the laboratory frame, the frequency absorption of the incident radiation by the moving ion depends on the incidence direction of the photons, which makes different angles with the ion velocity field as it varies within the radiation cone. Radiation coming from a given direction is more or less dimmed than others as shown by Fig. 2. The angular dependence of the dimming due to the scattering ion motion leads to a modification of the linear polarization parameters of the scattered line. The rotation angle of the direction of linear polarization and the decrease of the polarization degree of the scattered radiation depend on the three components of the velocity vector of the scattering ion. The quantum theory of this phenomenon and the resulting analytical formulae are developed in Sahal-Bréchot et al. (1998).