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

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4. Limits of the polarization degree

Neglecting limb darkening (i.e., setting [FORMULA]), we can integrate [FORMULA] [FORMULA] [FORMULA] [FORMULA] over the angle [FORMULA]. For the polarization degree we have, in the two limiting cases [FORMULA] and [FORMULA]

[EQUATION]

where

[EQUATION]

[EQUATION]

In particular, for [FORMULA] (the atom is at the limb), one gets

[EQUATION]

On the other hand,

[EQUATION]

where, setting

[EQUATION]

[EQUATION]

[EQUATION]

Again, for [FORMULA], one gets

[EQUATION]

4.1. Limiting values of [FORMULA] with limb darkening

It is well established that, as a first order approximation, limb darkening can be described by a linear equation of the form

[EQUATION]

where [FORMULA] is defined in Fig. 2.1 and where u is a parameter given by [FORMULA] in the continuum surrounding H [FORMULA] (see for instance Allen 1973 ).

From elementary geometry we find

[EQUATION]

[EQUATION]

where

[EQUATION]

We therefore must evaluate integrals of the form

[EQUATION]

i.e., integrals such as

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The integrals [FORMULA] and [FORMULA] can be found in Gradshteyn and Ryzhik (1965 ). One gets

[EQUATION]

[EQUATION]

For the continuum, we use Eq. (26), and we split the integrals relative to [FORMULA] and [FORMULA] to obtain

[EQUATION]

[EQUATION]

Also

[EQUATION]

[EQUATION]

Setting [FORMULA] we obtain for [FORMULA]

[EQUATION]

For [FORMULA] (atom at the limb), we have

[EQUATION]

[EQUATION]

[EQUATION]

so that

[EQUATION]

4.2. The rotation of the plane of polarization

To 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

[EQUATION]

[EQUATION]

[EQUATION]

one obtains

[EQUATION]

[EQUATION]

The integrals over the two variables [FORMULA] can be performed by using a numerical Gauss-Legendre integration procedure with 32 points. The results of the integration depend on h, the height of the atom above the limb, the magnitude and direction of the velocity flow [FORMULA] the temperature [FORMULA] (and therefore [FORMULA]) of the scattering region, and the law of limb darkening specified by the function [FORMULA]. Fig. 3, obtained for [FORMULA] K , [FORMULA] km, [FORMULA], shows the rotation angle versus the line shift [FORMULA] for various directions and moduli of the ensemble velocity [FORMULA]. The Stokes parameter profiles are integrated in frequency. The rotation angle goes to zero when the modulus of the ensemble velocity goes to zero. This diagram is useful for experimental purposes since both the shift of the scattered profile and the rotation of the polarization plane are observable quantities.

[FIGURE] Fig. 3. Plot of the angle of rotation of polarization [FORMULA], as a function of the line shift d for [FORMULA] and [FORMULA] varying from 0 to [FORMULA]. The plots are performed for 5 values of the modulus V. Other parameters are specified in the text. Note that the rotation angle goes to zero when the velocity V goes to zero.

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 [FORMULA] km/s and then decreases. When limb darkening is taken into account the polarization degree is larger.

[FIGURE] Fig. 4. The polarization degree is plotted as a function of V for three different altitudes. Collisional rates and limb darkening are neglected.
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© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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