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Astron. Astrophys. 333, 1130-1142 (1998)

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5. The effect of collisions

The presence of free electrons and protons in the plasma of the spicule leads us to consider the effect of collisions between the scatterer and the surrounding particles.

These implies that collisional rates have to be added in the statistical equilibrium equations for [FORMULA] (Eq. (A1)). As an example we give the equations for the evolution of the populations [FORMULA] and [FORMULA]

[EQUATION]

[EQUATION]

In these equations we have neglected the collisional rates due to inelastic collisions, with [FORMULA], since they are very small compared to those due to the elastic ones, with [FORMULA], (see Tables 5 and 6). The factor [FORMULA] is defined by Sahal et al. (1996)

[EQUATION]


[TABLE]

Table 5. Inelastic transition [FORMULA]. Values of the inelastic collisional coefficient [FORMULA] for various temperatures and densities.



[TABLE]

Table 6. Values of the elastic collisional cefficients involved in the transitions for [FORMULA] K and [FORMULA] cm-3. [FORMULA] refers to collisions with electrons and [FORMULA] with protons. [FORMULA].


In order to find the solution for [FORMULA], we solve symbolically the equations in the [FORMULA] subsystem with the collisional rates taken into account. This is possible through Mathematica. The Stokes parameters Q and U are proportional to the tensors [FORMULA], [FORMULA] and for these tensors we are led to the following approximate result

[EQUATION]

[EQUATION]

For electronic densities less than [FORMULA] cm-3, one can just consider relaxation in the [FORMULA] subsystem and write the populations as

[EQUATION]

[EQUATION]

For higher densities one has to consider efficient inelastic collisional rates from the [FORMULA] level which is by far the most populated. The relevant collisional rates, for a temperature [FORMULA] K, are

[EQUATION]

[EQUATION]

One can show that:

[EQUATION]

with

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

This expression reduces to the preceding one for [FORMULA] (Eq. 15) when [FORMULA] that is [FORMULA] [FORMULA] [FORMULA] [FORMULA] 1.

We can introduce [FORMULA], [FORMULA] the Stokes parameters when collisions are considered. For the angle of rotation of the polarization plane, we find

[EQUATION]

The rotation angle [FORMULA] is unaffected by collisions due to the fact that the Stokes parameters Q and U are equally affected by collisions. Isotropic collisions depolarize, which is an effect well identified in the literature. This is shown in Fig. 5 displaying the polarization degree at the same altitude [FORMULA] [FORMULA] km and for three different densities [FORMULA], [FORMULA], [FORMULA] cm-3. The central density corresponds to a model of spicules given in the literature (Heritschi & Mouradian 1992 ). At this density the polarization degree is reduced by a factor 2.

[FIGURE] Fig. 5. The polarization degree is plotted as a function of V for three values of the electron density. The atom is at an altitude of [FORMULA]  km above the limb. Limb darkening is taken into account.
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© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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