## 4. Limits of the polarization degreeNeglecting limb darkening (i.e., setting ), we can integrate over the angle . For the polarization degree we have, in the two limiting cases and where In particular, for (the atom is at the limb), one gets On the other hand, where, setting Again, for , one gets ## 4.1. Limiting values of with limb darkeningIt is well established that, as a first order approximation, limb darkening can be described by a linear equation of the form where is defined in Fig. 2.1 and where
From elementary geometry we find where We therefore must evaluate integrals of the form i.e., integrals such as The integrals and can be found in Gradshteyn and Ryzhik (1965 ). One gets For the continuum, we use Eq. (26), and we split the integrals relative to and to obtain Also Setting we obtain for For (atom at the limb), we have so that ## 4.2. The rotation of the plane of polarizationTo obtain the profiles of the Stokes parameters of the scattered radiation, one has to go back to calculate the integrals appearing in Eqs. (28). By setting one obtains The integrals over the two variables can be
performed by using a numerical Gauss-Legendre integration procedure
with 32 points. The results of the integration depend on
We show in Fig. 4 the polarization degree for two altitudes above the limb, and for the atom at the limb. The degree of polarization rises to a maximum value for km/s and then decreases. When limb darkening is taken into account the polarization degree is larger.
© European Southern Observatory (ESO) 1998 Online publication: April 28, 1998 |