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

## 2. Transformation of matrix

We rewrite the transfer equation as

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

so that the transfer equation reads

Matrix can be decomposed as follows

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

This is the general symmetry of ; the meaning of parameters ,and 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 in the plane Q-U. That is, we introduce a rotation matrix

and its inverse , and we apply them to to obtain

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

where

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

where we note that

so that we can write the transformed transfer equation as

where

with

where we have introduced

The meaning of the new Stokes reference system is as follows: after the rotation, the new generalized Stokes parameters and , projections of vector on axis and in the new reference system, still correspond to linear polarization, but Zeeman linear absorption affects only (absorption along the axis). Faraday rotation may still affect , but with zero absorption. The parameters and are unchanged (the correspoding axis and are not affected by the 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

and its inverse , where

By applying it to the matrix we obtain

where

For the singular case , we adopte , and .

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

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 we can determine the ellipse of polarisation absorbed by the Zeeman effect. We then choose it as axis . The complete new generalized Stokes system follows from transformation matrix . In deriving , stands for the total intensity of the elipse of polarisation and is the rate of circular polarisation. Although easy to calculate, 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 for the component, and for the components, where is the inclination angle of the magnetic field.

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

and calculating the term

so that we can write the transformed transfer equation as

where

with

where we have introduced a new parameter

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

with

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

where has the following aspect:

with

where we have defined

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 . A necessary and sufficient condition for that is a matrix of the form

where is a constant matrix, and is any scalar function of z. Let P,Q and R be the integrals of p,q and r. When calculating the commutator of with its integral , its only a priori non-zero entries are or . It is very easy to see that these expressions vanish when 1) , 2) or 3) p,q and r are all proportional to the same scalar function . While the treatement of cases 1) and 2) is straightforward, the general case 3) needs some discussion: we rewrite p,q r as , and . Matrix keeps the appearance of but with p, q and r substituted by , and . These new variables to be constant is the necessary and sufficient condition for writing an exponential scalar-like solution. A constant matrix implies 7 constant parameters. Matrix contains only three parameters and r, but we need to keep constant only the two ratios and 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, and , angles and , and . 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 and alone in the case of separated Zeeman components. In absence of azimuth variation and .

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

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

and we obtain

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

© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998