Astron. Astrophys. 332, 610-628 (1998)

3. The irreducible transfer equation

The transfer equation (31) for the vector is simpler than the original transfer equation (1) for the Stokes vector , because the real Fourier components do not depend on the azimuth. However the source term is still a function of two variables : the optical depth , and the co-latitude of the ray. The factorization of , given in Eq. (32), suggests to introduce a new radiation field , and a new source function , that depends only on the optical depth, defined by :

We shall refer to the six-component vectors and as the "irreducible radiation field" and "irreducible source vector". Eq. (32) shows that

where

When , the matrix becomes an unit matrix and so does . The vector has already been defined in Eq. (36). Introducing Eq. (45) into Eq. (33), we can rewrite as

The irreducible radiation field satisfies the transfer equation

Left multiplying this equation on both sides by and commuting the matrix multiplication with the derivative with respect to , we indeed recover the transfer equation for .

We can now establish a vector integral equation for the irreducible source vector . It will be the basis for the iterative method presented in Sect.  4. Following the standard method, we first write the formal solution of Eq. (50). Using then Eqs. (47) and (49), we obtain :

where T is the optical thickness of the medium and the matrix defined in Eq. (48), the other three arguments being dropped for convenience.

The matrix is defined by

It is a matrix which may be written as,

The first block is identical to the kernel matrix for axisymmetric resonance polarization problems. The kernels and were introduced by Landi Degl'Innocenti et al. (1990). We recall that is normalized to unity and to zero. All the kernels , and have the same normalization, viz.,

The and their primitives , defined by

are shown in Fig. 2 for the case and positive values of (remember that they are even functions of ). The and their primitives are positive except for and . They decrease algebraically to zero at large optical depths and increase logarithmically as . The properties of the propagating kernels and and of the mixing kernel have been discussed at length in Paper I. The kernels and play a similar role as .

Fig. 2. The Hanle scattering kernels (upper panel) and their primitives (lower panel) in lin-log scales for Voigt profile with a damping parameter and . The normalization of is given by

To end this section we briefly comment on Eq. (51). It looks very much like the vector integral equation for resonance polarization in zero magnetic field considered in Paper I. However the true kernel of this integral equation is the product . When depends on optical depth, the integral equation is not of the Wiener-Hopf type since the kernel is not a displacement kernel. When is a constant matrix, the Wiener-Hopf character is maintained. However, in contrast with resonance polarization in zero magnetic field, the kernel is not a symmetric matrix and hence the transport operator is not self-adjoint. We stress also that in this equation all the components of are coupled inspite of the fact that the matrix has a very simple structure (see Eq. (53)).

For completeness we give below the analytical expressions of all the non-zero elements of :

The are the usual exponential integral functions.

We note here for further use that also satisfies an integral equation. Combining Eqs. (51) with (47) we readily obtain

where

For a non-polarized primary source of thermal origin, with

is also given by Eq. (63) with in place of .

Thanks to the transformations carried out in the preceding sections, the calculation of the Stokes vector has been reduced to the solution of the vector transfer equation (50) where the vector source function depends only on optical depth and satisfies the integral equation (51). We solve it in the next section by an operator perturbation method.

© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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