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

2. Basic equations

All the equations needed to calculate the Hanle effect produced by a depth-dependent magnetic field, for a line formed with complete frequency redistribution, are given in this section. They generalize equations obtained in FS91 for the case of a magnetic field with a uniform (constant with depth) azimuthal angle. Many of the equations to be given here have already been published, but are spread out in several articles, not all of them with easy access (such as Faurobert-Scholl 1993). Hence the presentation here of a complete set of equations.

2.1. Polarized line radiative transfer equation

In the presence of a weak magnetic field, the radiative transfer equation for the Stokes vector may be written as

where is the scalar absorption profile function (Landi Degl'Innocenti 1985). All the sign conventions, and the symbols for the physical quantities have the same meaning as in FS91 : is the frequency averaged line optical depth, x is the frequency separation from line center, measured in Doppler width units, is the propagation direction of the ray where is the co-latitude and is the azimuth of the ray. The positive optical depth is measured in the direction opposite to the vertical axis (see Fig. 1). For lines formed with complete frequency redistribution, the vector source function is independent of frequency and may be written as

where is a given primary source term and is the Hanle phase matrix. In this paper all matrices are denoted with italic letters accentuated with a hat. As in FS91 and Landi Degl'Innocenti (1985), we neglect depolarizing collisions. The main results of this paper are independent of this simplifying assumption. Depolarizing collisions can be introduced by making a vector instead of scalar (Landi Degl'Innocenti et al. 1990).

Fig. 1. Geometry specifying the direction of the magnetic field and of the line-of-sight . Angles and are the co-latitudes of and , respectively. The azimuthal angles and are measured starting from the x -axis in the anti-clockwise direction in the xy -plane

The vector magnetic field is characterized by its strength B and by the angles and defined as shown in Fig. 1. The full Hanle phase matrix is a matrix which couples together the three Stokes parameters I, Q and U but does not couple them to the Stokes parameter V. Here we are interested in the Hanle effect on the linear polarization of spectral lines. We may thus consider only the three-component Stokes vector and source vector . The primary source term is assumed to be of thermal origin. Hence it is unpolarized and we may write it as where , with the Planck function at the line center frequency. An explicit analytical expression of the Hanle phase matrice was first given by Landi Degl' Innocenti & Landi Degl' Innocenti (1988).

2.2. The azimuthal Fourier expansion method

In a 1D medium, in the absence of magnetic field and of incident collimated radiation, the radiation field is axially symmetric, i.e. it does not depend on the azimuth . This is no longer true when a magnetic field is present. It is well known that the non-axisymmetric transfer problem of Rayleigh scattering polarization can be simplified by expanding the azimuthal angle dependence of the specific intensity and source vector (see Chandrasekhar 1960 p. 250). This method is generalized for the Hanle scattering problem in FS91, where the azimuthal angle dependence of and is expanded in a Fourier series with respect to the azimuthal angle difference , where is assumed to be depth-independent. Here we present a more general formulation, where and are expanded in Fourier series with respect to . It can thus be used in cases where varies with depth.

Because of its -periodicity with respect to the variable , the specific intensity vector may be expanded as

where

The Hanle phase matrix may be expanded in a two-dimensional Fourier expansion with respect to and . In FS91 and Faurobert-Scholl (1993) it was shown that this expansion is limited to terms of order 2. Namely,

Explicit expressions of the Fourier coefficients were also given, however with some misprints.

2.3. Fourier coefficients of the Hanle phase matrix

The Fourier components may be written as linear combinations of matrices which depend only on the angular variables µ and µ' with scalar coefficients which depend only on the magnetic field variables and B. Namely,

In the particular case ,

where is the isotropic matrix all the elements of which are zero except the element (1,1) which is unity. The coefficients are complex scalars whereas the matrices have real elements.

A remarkable property of the Hanle phase matrix is that it has a diadic representation. It is of the same nature as the diadic representation for Rayleigh scattering introduced by Domke (1971; see also Ivanov 1995). The matrices can be factorized as tensor products of two vectors depending on µ and µ' respectively and the isotropic matrix as the tensor product of two constant vectors. Furthermore, only six vectors are necessary to construct the Hanle phase matrix. We indeed have

and

The index i depends on k and m and the index j on l, m. Table 1 shows how to obtain i and j for positive values of k and l. For example . The symmetry relations and provide the for negative values of k and/or l.

Table 1. Values of the indices i and j for Eq. (9)

The vectors are given by :

The parameter W is the standard atomic depolarization factor which is equal to unity for a transition , . Note that all the are proportional to W.

The coefficients , , may be written as

The notation stands for complex conjuguate.

All the coefficients , can be expressed in terms of the :
For and all values of k :

For and all values of l :

For ( is the product of k by l) :

For :

The coefficients satisfy symmetry relations. For :

For ,

The symmetry relations satisfied by the , which for simplicity are henceforth denoted by , are also shown in Table 2. The depend thus only on 15 different coefficients which are given in Eq. (39).

Table 2. Symmetries of the coefficients

2.4. Fourier expansion of the Stokes source vector S

Substituting the azimuthal Fourier expansions of and in Eq. (2), we can perform analytically the azimuthal integration over . This yields the azimuthal Fourier expansion of the source vector. As the Hanle phase matrix has no Fourier component of order higher than 2, the same property holds for the source vector. The complex Fourier components of the vector , defined as in Eq. (4), are given by

where is the Kronecker symbol. Since we have an axially symmetric primary source term, only the Fourier component has an inhomogeneous term. In the following we prefer to deal with real quantities (as in FS91). Boldface calligraphic uppercase letters accentuated with a tilde are used to denote the complex Fourier components. For the real Fourier components, we change the calligraphic font to an italic font and replace the tilde by an horizontal line. Boldface calligraphic uppercase letters are also used for the Stokes vector and associated source function.

The real Fourier components are given by

The Fourier expansion of may then be written as

Each vector is a three-component vector. However for symmetry reasons, the azimuthal average of Stokes U vanishes. Thus , which is the azimuthal average of , has only two components.

2.5. Factorization of the Fourier source vector

Following FS91, we introduce a new vector defined by

It is a 14-component vector since is a 2-component vector, while the other are 3-component vectors. A 14-component vector denoted by is constructed in a similar fashion with the components of the real Fourier expansion coefficients of . It satisfies the radiative transfer equation :

Using Eqs. (7)-(28) and (30), it is straightforward, although lengthy, to show that can be written in the factorized form

A key property of (32) is that and are six-component vectors. Hereafter all six-components vectors are denoted with boldface italic uppercase letters. The matrices and vectors appearing in this equation are defined in the following sub-sections.

2.5.1. The irreducible mean intensity J

The vector is defined by

It is directly related to the six irreducible tensors introduced by Landi Degl'Innocenti et al. (1990) in their density matrix formalism of the Hanle effect (see also Landi Degl'Innocenti 1984). The difference between the vector of this paper and the vector introduced in Paper I and in FS91 is that has no factor and does not include the primary source term (see Eq. (14) in Paper I). It is a six-component vector while is a two-component vector. In analogy with Paper I, we denote the first two components of by and , whereas the other components are denoted by and .

2.5.2. The matrices and

is a matrix. In symbolic notation, it may be written as

To obtain the explicit expression it suffices to replace the line vectors by their three components, except for and for which only the first two components are being used (the vectors are given in Eqs. (10) to (13)). Because of the block structure of , the first two components of and of the vector depend only on the azimuthal average of (i.e. on , the third and fourth components depend only on the Fourier components whereas the fifth and sixth components, , depend only on .

is a matrix which is the transpose of . In symbolic notation, it may be written as

Here the are three-component column vectors, except for and which, as above, are two-component vectors. Clearly, is made of three blocks. The first block, which contains the first two elements of the first row, is identical to the matrix of Paper I.

2.5.3. Primary source term

The second term in the r.h.s. of Eq. (32) is a primary source term. It comes from the first term in the r.h.s. of Eq. (27). It is easy to see that

and hence that . Being able to write the primary source term in this factorized form is necessary to arrive at the reduced problem described in Sect.  3. If the primary source term in Eq. (2) is not isotropic and unpolarized this factorization may not hold. A simple method for overcoming this difficulty is to write the Stokes vector as

where is the solution of the problem with the internal source but no scattering term. A similar decomposition is used in Ivanov (1995) for Rayleigh scattering and in Ivanov et al. (1997) for resonance polarization. In the transfer equation for the diffuse radiation field , the primary source term is then of the required form. When there are no internal primary sources but an external non-axisymmetric incident radiation field , the same technique applies. It is now the directly transmitted field created by the incident radiation which should be subtracted from the total field.

2.5.4. The matrix

The matrix depends only on the magnetic field strength B and its co-latitude . Except for the first row and the first column, it is almost the matrix of the coefficients , . Taking into account the symmetries of the shown in Table 2, may be written as

where

where

The dimensionless parameter , which depends on the intensity of the magnetic field, is given by

where is the Larmor frequency of the electron in the magnetic field, is the Landé factor of the upper level and A the destruction rate of the upper level alignment. It is the sum of the radiative, inelastic and depolarizing collision rates (see e.g. Bommier 1996, Eq. (32)). We note here that the matrix differs from the one in FS91. The elements to have opposite signs.

2.5.5. The matrix

The matrix may be written as

where

The matrix comes from the factor in Eq. (27) and from . This factor yields a rotation matrix which becomes when it is commuted with . The matrix is a unitary matrix. It satisfies .

© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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