## 2. Transformation of matrixWe 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 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 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 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 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
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 |