5. The effect of collisions
These implies that collisional rates have to be added in the statistical equilibrium equations for (Eq. (A1)). As an example we give the equations for the evolution of the populations and
In these equations we have neglected the collisional rates due to inelastic collisions, with , since they are very small compared to those due to the elastic ones, with , (see Tables 5 and 6). The factor is defined by Sahal et al. (1996)
Table 5. Inelastic transition . Values of the inelastic collisional coefficient for various temperatures and densities.
Table 6. Values of the elastic collisional cefficients involved in the transitions for K and cm-3. refers to collisions with electrons and with protons. .
In order to find the solution for , we solve symbolically the equations in the subsystem with the collisional rates taken into account. This is possible through Mathematica. The Stokes parameters Q and U are proportional to the tensors , and for these tensors we are led to the following approximate result
For electronic densities less than cm-3, one can just consider relaxation in the subsystem and write the populations as
For higher densities one has to consider efficient inelastic collisional rates from the level which is by far the most populated. The relevant collisional rates, for a temperature K, are
One can show that:
This expression reduces to the preceding one for (Eq. 15) when that is 1.
We can introduce , the Stokes parameters when collisions are considered. For the angle of rotation of the polarization plane, we find
The rotation angle 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 km and for three different densities , , 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.
© European Southern Observatory (ESO) 1998
Online publication: April 28, 1998