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Astron. Astrophys. 342, 201-211 (1999)

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2. Transformation of matrix [FORMULA]

We rewrite the transfer equation as

[EQUATION]

And, [FORMULA] being invertible, we can define the source function vector, either LTE or not,

[EQUATION]

so that the transfer equation reads

[EQUATION]

Matrix [FORMULA] can be decomposed as follows

[EQUATION]

Where [FORMULA] and [FORMULA] are the usual scalar absorption coefficients: the selective (at line center) and the continuum one, respectively; [FORMULA] is the azimuth angle of the magnetic field, relative to a fixed reference system, and [FORMULA] is the [FORMULA] identity matrix.

This is the general symmetry of [FORMULA]; the meaning of parameters [FORMULA],and [FORMULA] can be found by comparing expression (10) with the corresponding ones in Landi Degl'Innocenti & Landi Degl'Innocenti(1981; 1985), Rees (1987) or Kawakami(1983).

We can simplify this matrix by rotating it an angle [FORMULA] in the plane Q-U. That is, we introduce a rotation matrix

[EQUATION]

and its inverse [FORMULA], and we apply them to [FORMULA] to obtain

[EQUATION]

Applying this transformation to Eq. (9) we obtain:

[EQUATION]

where

[EQUATION]

[EQUATION]

The left hand side of the transformed transfer equation (12) is equal to

[EQUATION]

where we note that

[EQUATION]

so that we can write the transformed transfer equation as

[EQUATION]

where

[EQUATION]

with

[EQUATION]

where we have introduced

[EQUATION]

The meaning of the new Stokes reference system is as follows: after the [FORMULA] rotation, the new generalized Stokes parameters [FORMULA] and [FORMULA], projections of vector [FORMULA] on axis [FORMULA] and [FORMULA] in the new reference system, still correspond to linear polarization, but Zeeman linear absorption affects [FORMULA] only (absorption along the [FORMULA] axis). Faraday rotation may still affect [FORMULA], but with zero absorption. The parameters [FORMULA] and [FORMULA] are unchanged (the correspoding axis [FORMULA] and [FORMULA] are not affected by the [FORMULA] rotation). In the real space, the meaning of this rotation is that the reference for the usual definition of the Stokes parameters is taken parallel to the magnetic field for Q. These new axes rotate with the field.

A second simplification is obtained by the use of a new rotation, given by

[EQUATION]

and its inverse [FORMULA], where

[EQUATION]

By applying it to the matrix [FORMULA] we obtain

[EQUATION]

where

[EQUATION]

For the singular case [FORMULA], we adopte [FORMULA], [FORMULA] and [FORMULA].

The meaning of the new Stokes reference system is as follows: after the [FORMULA] rotation, the new generalized Stokes parameters [FORMULA] and [FORMULA], projections of [FORMULA] on axis [FORMULA] and [FORMULA] correspond to elliptic polarizations, Zeeman elliptic absorption affects [FORMULA] only (absorption along axis [FORMULA]). Faraday rotation still affects [FORMULA] (and [FORMULA]) but with zero absorption. Parameters [FORMULA] and [FORMULA] are unchanged (the correspoding axis [FORMULA] and [FORMULA] are not affected by the [FORMULA] rotation). Note that [FORMULA] is wavelength dependent and therefore the rotation in the Minkowski space is not constant with [FORMULA]!

In the real space, the meaning of this rotation is not any longer as simple as before. However, note that the most general state of polarisation is elliptic!! At each [FORMULA] we can determine the ellipse of polarisation absorbed by the Zeeman effect. We then choose it as axis [FORMULA]. The complete new generalized Stokes system follows from transformation matrix [FORMULA]. In deriving [FORMULA], [FORMULA] stands for the total intensity of the elipse of polarisation and [FORMULA] is the rate of circular polarisation. Although easy to calculate, [FORMULA] has no simple meaning in terms of the magnetic field, except for the case of a strong field when all the Zeeman components are completely separated. Then [FORMULA] for the [FORMULA] component, and [FORMULA] for the [FORMULA] components, where [FORMULA] is the inclination angle of the magnetic field.

We repeat here all the steps made for first rotation [FORMULA], defining the new transformed Stokes and emission vectors

[EQUATION]

and calculating the term

[EQUATION]

so that we can write the transformed transfer equation as

[EQUATION]

where

[EQUATION]

with

[EQUATION]

where we have introduced a new parameter

[EQUATION]

This parameter s, a new non-zero entry in [FORMULA], makes it a little more complicate than before. A new transformation is necessary if we want to obtain a simpler matrix like [FORMULA]. The way for this simplification is a third rotation [FORMULA] given by

[EQUATION]

with

[EQUATION]

The same mathematical steps of the two previous rotations are repeated for [FORMULA]. We pass directly to the final expression for the transfer equation:

[EQUATION]

where [FORMULA] has the following aspect:

[EQUATION]

with

[EQUATION]

where we have defined

[EQUATION]

By now the matrix, and consequently the RTE, has been simplified in a general way, without any assumption nor constraint. To proceed to an exponential solution for the transfer equation, we need to ensure that commutation condition (5) holds for [FORMULA]. A necessary and sufficient condition for that is a matrix [FORMULA] of the form

[EQUATION]

where [FORMULA] is a constant matrix, and [FORMULA] is any scalar function of z. Let P,Q and R be the integrals of p,q and r. When calculating the commutator of [FORMULA] with its integral [FORMULA], its only a priori non-zero entries are [FORMULA] or [FORMULA]. It is very easy to see that these expressions vanish when 1) [FORMULA], 2) [FORMULA] or 3) p,q and r are all proportional to the same scalar function [FORMULA]. While the treatement of cases 1) and 2) is straightforward, the general case 3) needs some discussion: we rewrite p,q r as [FORMULA], [FORMULA] and [FORMULA]. Matrix [FORMULA] keeps the appearance of [FORMULA] but with p, q and r substituted by [FORMULA],[FORMULA] and [FORMULA]. These new variables to be constant is the necessary and sufficient condition for writing an exponential scalar-like solution. A constant [FORMULA] matrix implies 7 constant parameters. Matrix [FORMULA] contains only three parameters [FORMULA] and r, but we need to keep constant only the two ratios [FORMULA] and [FORMULA] to satisfy condition (5). After transformation, only two variables are requested to be constant,i.e., two constraints instead of seven originally: 5 degrees of freedom have been earned. Reviewing the transformation process, those 5 degrees of freedom may be used to treat analytically gradients with depth in azimuth, [FORMULA] and [FORMULA], angles [FORMULA] and [FORMULA], and [FORMULA]. Details about how to do it and its application to a numerical code, are left for a forthcoming paper.

For the sake of demonstration, we discuss the variation of angles [FORMULA] and [FORMULA] alone in the case of separated Zeeman components. In absence of azimuth variation [FORMULA] and [FORMULA].

For the [FORMULA] Zeeman component, polarisation is purely linear, [FORMULA], and we adopt [FORMULA] [FORMULA]: the integration is straightforward (case 1). For the [FORMULA] Zeeman components, polarisation is elliptic. Since [FORMULA], we adopt [FORMULA] and [FORMULA].

[EQUATION]

Absorption in each [FORMULA] component is proportional to [FORMULA] and also [FORMULA], [FORMULA] being the only variable depending on depth. We now suggest [FORMULA] to match the depth variation of q, r and [FORMULA] (case 3) with [FORMULA] To keep the same depth variation for [FORMULA], we impose a variation of [FORMULA] such that

[EQUATION]

and we obtain

[EQUATION]

In general, it will be impossible to interpret [FORMULA] in such a simple way.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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