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

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1. Introduction

The 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,

[EQUATION]

in terms of the Stokes parameters I, Q, U and V presented as the components of vector [FORMULA]. This equation resembles the transfer equation for the scalar case when polarization is ignored. The scalar intensity is replaced by the Stokes vector [FORMULA]. The emision term is now a four component vector [FORMULA], and the scalar absorption coefficient becomes a [FORMULA] matrix [FORMULA] which describes absorption (including anomalous dispersion) in the presence of Zeeman effect. The variable z parameterizes the light path. The transformation from a single equation to a system of four equations (abridged in a vectorial form) changes drastically the nature of the problem and no general explicit analytical solution has been proposed so far.

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 [FORMULA] matrix has been handled with different mathematical techniques: for instance Kjeldseth Moe (1968) and Stenflo(1971; 1994) used [FORMULA]-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 [FORMULA] 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):

[EQUATION]

where as usual, [FORMULA] is the absorption coefficient, I and J are the scalar intensity and emission function respectively and z the geometrical path. This equation has an explicit formal solution, for the layer [FORMULA]:

[EQUATION]

A direct extrapolation of this scalar solution would yield:

[EQUATION]

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

[EQUATION]

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:

[EQUATION]

This difference becomes important when treating for example the derivative of a power of a matrix: it is not true in general that

[EQUATION]

where we have used (and so we will do hereafter) that

[EQUATION]

As a further consequence of the non-commutativity of matrices, we have that

[EQUATION]

in complete contradiction with the scalar case.

In short, Eq. (3) is a solution of the polarized RTE, Eq. (1), when the commutation condition,

[EQUATION]

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 [FORMULA] with optical depth. Indeed, Landi Degl'Innocenti & Landi Degl'Innocenti (1981; 1985) have already shown how to handle matrices of the form

[EQUATION]

with [FORMULA] being a constant matrix. Although not explicitly said in these papers, it is obvious that here [FORMULA] 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:

[EQUATION]

where [FORMULA] is the evolution operator , a new [FORMULA] matrix which obeys the homogeneous equation

[EQUATION]

with initial condition

[EQUATION]

where [FORMULA] represents the [FORMULA] identity matrix. Note that when solution (3) applies, the evolution operator takes an explicit form, namely:

[EQUATION]

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:

[EQUATION]

were the exponent [FORMULA] 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):

  1. Constant matrix assumption. Condition (5) is inmediately satisfied and analytical solutions were found (Unno, 1956; Rachkowsky, 1967).

  2. 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).

  3. 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 a priori the number of layers necessary for a desired precision. Last but not least, a more analytical insight than what a pure numerical method can give is always desired as well. Moreover, our ultimate purpose is magnetometry of the sun or stars. We want to go beyond the first method, the most used, at present, but limited to a constant absorption matrix and therefore also constant field. Still we must admit that actual observations will allow us to determine only "little" more than a homogeneous atmosphere model, say, at most the magnetic fields at two or three levels in the atmosphere. It is therefore not "economic" to calculate more than a few layers in the atmosphere.

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 finite layer with arbitrary depth variations. Eventually we can then borrow the techniques from the multi-layer approach and apply our semi-analytical solution to a few layer model to improve the computation.

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 [FORMULA] is given by [FORMULA]. 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 explicitlycommon to the "two spaces". Typically scalar variables like [FORMULA] (see Sect. 2. for their definitions) will appear in both spaces in similar ways. However, rotations of the Stokes reference system will not. Exception is the azimuth rotation. The angle of rotation of the azimuth of the magnetic field in the "real 3D space" corresponds to a rotation in the Minkowski space, but with a double amount. Naturally, when a constant atmosphere is selected in the real space, the corresponding matrix in the Minkowski space will be constant as well. On the other hand some rotation in the Minkowski space may be much easier to handle. For instance, one may find convenient to use generalized Stokes vectors expressed in terms of elliptic states of polarization. Transformations from one set of Stokes reprensentation to another are expressed simply as rotations in the Minkowski space. Except in some limiting cases it is not possible to translate these angles in terms of angles in the physical space. At the same time, the highly non linear relations between magnetic field and the entries of the absorption matrix cannot in general be simplified. Thus, while the RTE can be solved for a given depth variation of the absorption matrix, we cannot, in general, recover analytically the corresponding variation of the magnetic field. We anticipate that numerical methods can overcome this difficulty and profite from the analytical solution in the Minkowski space to treat the depth variations of the magnetic fields and improve both the economy and the precision of the calculations. These considerations apply as well to all other atmospheric conditions, like temperature, pressure, velocity etc.

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.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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