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

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3. Application to the O VI line 103.2 nm of the solar corona

We begin by giving the expression of the polarization matrix of the incident radiation [FORMULA], [FORMULA], [FORMULA] which is that of the same O vi line but emitted by the underlying transition region.

Neglecting inhomogeneities of the solar surface and using the same notations as Paper I, we can write [FORMULA], [FORMULA], [FORMULA] as

[EQUATION]

where [FORMULA] is the incident normalized profile.

[EQUATION]

where [FORMULA] is the limb-brightening coefficient ([FORMULA]) and [FORMULA] is the specific intensity emitted from the center of the disk in the line studied and integrated over the incident line profile.

The incident line profile is a gaussian Doppler profile, with a linewidth [FORMULA]

[EQUATION]

and

[EQUATION]

Then we assume that the atomic velocity distribution can be written as

[EQUATION]

which corresponds to a Maxwellian distribution at a temperature T with an hydrodynamical drift of velocity [FORMULA] [FORMULA] or [FORMULA]. This drift represents the velocity of the solar wind.

We can write, each subdistribution being normalized to 1

[EQUATION]

with

[EQUATION]

and analogous expressions for the two other distributions.

The integrations over [FORMULA] and [FORMULA] can be achieved analytically by means of elementary methods and are not detailed here.

The Stokes parameters [FORMULA], [FORMULA], [FORMULA] of the frequency distribution of intensity and polarization of the reemitted line at the frequency [FORMULA] are given in Appendix A: Eq. (A1). They are expressed in number of photons emitted per steradian, per unit of volume, per unit of time and per unit of interval of frequency.

This equation has to be applied with [FORMULA] (Li -like resonance line [FORMULA]) and [FORMULA] for the other component of the doublet ([FORMULA]).

Without limb-brightening,

[EQUATION]

and

[EQUATION]

After integration over the scattered line-profile, one obtains the global Stokes parameters of the scattered line I, Q, U (in number of photons emitted per steradian, per unit of volume, and per unit of time). They are given in Appendix A: Eq. (A8).

These formulae show that the three first Stokes parameters (intensity and linear polarization parameters) are sensitive to the three components of the vector velocity of the drift. The order of magnitude of the sensitivity to these components will be the object of the next paper (cf. also Sahal-Bréchot et al. 1992b and Sahal-Bréchot & Choucq-Bruston 1994 for preliminary results).

3.1. Asymptotic limit: perfectly directive case

At great distances from the limb (i.e., [FORMULA] in units of solar radius), the results corresponding to a perfectly directive incident radiation (Hyder & Lites 1970; Beckers & Chipman 1974), are recovered. By taking the limit [FORMULA], the integration of Eq. (A1) over [FORMULA] and [FORMULA] becomes analytic. The scattered line has also a gaussian profile, and the frequency dependence is the same for all the Stokes parameters: they are thus characterized by 5 parameters [FORMULA] [FORMULA] [FORMULA] [FORMULA] which are defined in the following.

Writing the intensity of the purely directive incident radiation as

[EQUATION]

with

[EQUATION]

we obtain a scattered intensity in the AZ direction

[EQUATION]

with a width

[EQUATION]

where

[EQUATION]

a shift

[EQUATION]

a dimming

[EQUATION]

which characterizes the dimming of the scattered line intensity with respect to the scattered intensity in the absence of velocity field, a degree of polarization

[EQUATION]

and

[EQUATION]

which corresponds to a direction of polarization perpendicular to the scattering plane ([FORMULA] and [FORMULA]).

In particular, for a [FORMULA] line and for [FORMULA], we obtain the well-known result

[EQUATION]

Therefore, at great distances from the surface of the sun the analysis of the scattered line intensity gives only two components of the velocity field: its projection on the direction of the incident radiation [FORMULA] and its projection on the line-of-sight [FORMULA] through the interpretation of the dimming and of the shift. The other Stokes parameters do not bring any additional information. The third component of the velocity field [FORMULA] - or [FORMULA] -, i.e., the perpendicular to the scattering plane, cannot be determined. Consequently the vectorial diagnostic cannot be performed with this method for a perfectly directive incident radiation.

3.2. Symmetries of the problem

The following discussion is restricted to a scattering atom in the plane of the sky ([FORMULA]). We show that the results for [FORMULA], and for [FORMULA] can be deduced from those of the first quadrant ([FORMULA] and [FORMULA]). The symmetries of the problem are illustrated on Fig. 4.

[FIGURE] Fig. 4. The symmetries and degeneracies of the solution for the diagnostic of the velocity field vector. Oz is the preferred direction of incident radiation

3.2.1. Symmetry due to the change of the direction of propagation of the incident light into the opposite sense

For the scattering atom, this is equivalent to

[EQUATION]

and the line-of-sight is conserved. For [FORMULA], (the present figure is drawn for [FORMULA], this correspond to a symmetry with respect to the line-of-sight AZ.

As the recoil of the atomic motion due to absorption or emission of light is negligible, the atom is not sensitive to the sense of propagation of the light and two opposite directions of propagation have the same effect. Consequently the Stokes parameters of the scattered line will be the same for two velocity field vectors which are symmetrical with respect to the line-of-sight. This is the fundamental degeneracy, which exists in every polarimetric diagnostic, and which has been abundantly discussed in Hanle effect studies (Bommier & Sahal-Bréchot 1978 and further papers); [FORMULA] on Fig. 4. In fact, if the scattering atom was not in the plane of the sky ([FORMULA]), the two corresponding velocity field vectors would not be symmetrical with respect to the line-of-sight.

3.2.2. Symmetry with respect to the preferred direction of the incident radiation

[EQUATION]

The characteristics of the scattered radiation are the same for two opposite lines of sight, the only changes are the sign of the shift d of the scattered line

[EQUATION]

and that of the polarization rotation: I (apart from the sign of the shift) and Q are unchanged and the sign of U is changed

[EQUATION]

this symmetry exists also in Hanle effect studies. [FORMULA] on Fig. 4.

3.2.3. Symmetry with respect to the center of scattering A

[EQUATION]

The only change is the sign of the shift

[EQUATION]

This symmetry does not exists in Hanle effect studies, because [FORMULA] is a real (polar)-vector, whereas the magnetic field [FORMULA] is an axial-vector; [FORMULA] on Fig. 4.

3.2.4. Symmetry with respect to the scattering plane

[EQUATION]

this symmetry is in fact the product of the three preceding symmetries: the result is that I and Q are unchanged, and that the sign of U is changed: the rotation of polarization is changed in its opposite and the shift is unchanged; [FORMULA] on Fig. 4.

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© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998
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