## 1. IntroductionThe use of spectropolarimetry and Zeeman effect to measure magnetic fields in the sun and stars, requires a transfer theory for polarized light, in the presence of a magnetic field. Such transfer equation was first written in the pioneering paper by Unno (1956), where the effect of anomalous dispersion was not considered yet. Rachkowsky (1962; 1967) was the first to include it, and obtained a general transfer equation for polarized light, in terms of the Stokes parameters I, Q, U and V presented as the
components of vector . This equation
The first particular solution was obtained by Unno, who applied the equation to the case of a Milne-Eddington atmosphere. Rachkowsky (1967), after including anomalous dispersion, solved the RTE for the same homogeneous atmosphere. Several procedures were successful in solving the equation numerically (e.g. Beckers & Schröter 1969, Wittman 1974, Rees, Murphy & Durrant 1989, Landi Degl'Innocenti 1976) even for the general case, by subdividing the atmosphere into numerous layers. The lasts are chosen optically thin in the upper atmosphere and eventually an Unno-Rachkowsky solution is taken for the deepest layer, down to optical depth infinity. Difficulties were encountered as well, depending on the particular method chosen to solve the radiative transfer in each individual sublayer. An universal technique, like the Runge-Kutta method, may not be the best approach for a particular case, mainly because of different scales of variation. The mathematical justifications are often not rigourous and numerical tests are always necessary. The advantage of an eventual analytical solution is obvious. But, up to now, they have been always restricted to homogeneous atmospheric models where only the LTE source function was depth dependent. The constant matrix has been handled with different mathematical techniques: for instance Kjeldseth Moe (1968) and Stenflo(1971; 1994) used -diagonalization, van Ballegooijen(1985) preferred Jones calculus. Now, for the solar case, variations with depth of both the thermodynamical parameters describing the atmosphere and the magnetic field, are not negligible. For instance, Ruiz Cobo & del Toro Iniesta(1992; 1994), and del Toro Iniesta & Ruiz Cobo (1996; 1996) using numerical inversion of the observed Stokes profiles have confirmed the need for inhomogeneous models (see Collados et al. 1994, Westendorp Plaza et al. 1997a,1997b,1997c, and see also del Toro Iniesta & Ruiz Cobo 1996a for a review). All these works raise the interest in analytical methods dealing with non-homogeneous atmospheric models, i.e. with non-constant matrices. At first sight, the RTE for polarized light does not seem more complicated to solve than its scalar equivalent (RTE for pure light intensity, with no polarisation): where as usual, is the absorption
coefficient, A direct extrapolation of this scalar solution would yield: Note that, in this expression, all the mathematical operations involving matrices are well defined. For instance, the exponential of a square matrix is defined as Now, just one difference arises between algebras using matrices and scalars: while two scalars always commute, that is not true in general for two matrices: This difference becomes important when treating for example the derivative of a power of a matrix: it is not true in general that where we have used (and so we will do hereafter) that As a further consequence of the non-commutativity of matrices, we have that in complete contradiction with the scalar case. In short, Eq. (3) is a solution of the polarized RTE, Eq. (1), when the commutation condition, holds. Under this assumption, the previous expressions recover an usual scalar appearance, and therefore we are permitted to write a scalar-like formal solution. It is interesting to note that this condition does not imply a constant absorption matrix. In the following sections we will show how to incorporate variations of with optical depth. Indeed, Landi Degl'Innocenti & Landi Degl'Innocenti (1981; 1985) have already shown how to handle matrices of the form with being a constant matrix. Although not explicitly said in these papers, it is obvious that here satisfies condition (5). For the more general case when the commutation condition (5) does not hold, Landi Degl'Innocenti & Landi Degl'Innocenti (1985) have derived a formal solution for the RTE: where is the with initial condition where represents the identity matrix. Note that when solution (3) applies, the evolution operator takes an explicit form, namely: A general method to solve equations of the form of Eq. (7) for linear operators has already been given by Magnus (1954). In this remarkable paper an exponential expression is proposed: were the exponent is given by an infinite series. Eq. (8) turns out to be Magnus' expression when only the first term in the infinite series is kept (indeed the only non-zero one when condition (5) holds). We now discuss three existing options to solve Eq. (1): -
Constant matrix assumption. Condition (5) is inmediately satisfied and analytical solutions were found (Unno, 1956; Rachkowsky, 1967). -
Multi-layer techniques. The atmosphere is considered as made up by numerous successive layers. A crude assumption on the radiative transfer in each optically thin layer is then advanced, and leads to a procedure of numerical integration expected by intuition to converge to the exact solution. A formal proof of convergence was not given, but the numerical tests were indeed satisfactory. See for instance Rees (1987), Rees et al.(1989), Ruiz Cobo & del Toro Iniesta (1992), del Toro Iniesta & Ruiz Cobo (1996). -
Magnus' solution. By applying linear algebra one can treat the general case, with non commuting matrices (Magnus, 1954).
The constant matrix technique, method (1), is not possible when one
wants to abandon the homogeneous magnetic field and atmosphere
assumptions. Next, poor economy is the main drawback of method (2).
There is some doubt whether one can determine It is striking that Magnus' solution was published two years before the memorial paper by Unno (1956), the first paper on RTE for polarized light, and as yet it has never been mentioned in the astrophysical literature. It is therefore given in Appendix A. The solution given by Magnus is mathematically exact, but it requires the use of Lie algebra, is not economic and can hardly be used in practical computation. It is mentioned here because it confirms the approach of the present paper and complete it. Our general strategy is, first, to "satisfy" condition (5) as far
as possible by extracting from the absorption matrix everything that
commutes with its integral and therefore can easily be integrated
according to Eq. (3), as explained in Sect. 2. In Sect. 3, we
diagonalise the commutative part of the matrix to allow an efficient
integration. Then, in Sect. 4, we treat the residual matrix by an
appropriate approximation and thus obtain a semi-analytical solution
for an optically A few words on the mathematical space where we are working and where the RTE is to be solved, are in order. Magnetometry concerns the 3D real physical space, where the magnetic field can be represented as a 3D "vector" and all physical parameters of the atmosphere determine the coefficients that enter the radiative transfer equation. The last one is much better calculated in another space. Indeed, we have already entered another 4D geometry: the Minkowski space, where the Stokes' 4-vectors are best described. In this geometry, the norm of a vector is given by . It has particular symmetries and is governed by linear algebra. The elementary operations, like absorption and retardation, are presented by matrices for which commutation relations are of particular importance. When condition (5) holds, an exponential solution, scalar like, to a linear equation can easily be derived. Otherwise, we have to turn to Magnus' exponential solution. The main difficulties originate from the fact that only few
variables are In some particular cases, the relations between variables in the Minkowski and real spaces may become simplified. For instance, in absence of absorption of linear polarization, whether in the pure longitudinal magnetic field, or alternatively for particular Zeeman patterns, free of linear polarization. Also for the case when all Zeeman components are separated, simple relations hold as will be discussed in the corresponding sections. © European Southern Observatory (ESO) 1999 Online publication: December 22, 1998 |