H. Frisch1 - M. Sampoorna1,2,3 - K. N. Nagendra1,2
1 - Laboratoire Cassiopée (CNRS, UMR 6202), Observatoire de la Côte d'Azur, BP 4229, 06304 Nice Cedex 4, France
2 - Indian Institute of Astrophysics, Koramangala Layout, Bangalore 560 034, India
3 - JAP, Dept. of Physics, Indian Institute of Science, Bangalore 560 012, India
Received 7 December 2004 /Accepted 4 July 2005
This paper considers the effect of a random magnetic field on Zeeman line transfer, assuming that the scales of fluctuations of the random field are much smaller than photon mean free paths associated to the line formation (micro-turbulent limit). The mean absorption and anomalous dispersion coefficients are calculated for random fields with a given mean value, isotropic or anisotropic Gaussian distributions azimuthally invariant about the direction of the mean field. Following Domke & Pavlov (1979, Ap&SS, 66, 47), the averaging process is carried out in a reference frame defined by the direction of the mean field. The main steps are described in detail. They involve the writing of the Zeeman matrix in the polarization matrix representation of the radiation field and a rotation of the line of sight reference frame. Three types of fluctuations are considered : fluctuations along the direction of the mean field, fluctuations perpendicular to the mean field, and isotropic fluctuations. In each case, the averaging method is described in detail and fairly explicit expressions for the mean coefficients are established, most of which were given in Dolginov & Pavlov (1972, Soviet Ast., 16, 450) or Domke & Pavlov (1979, Ap&SS, 66, 47). They include the effect of a microturbulent velocity field with zero mean and a Gaussian distribution. A detailed numerical investigation of the mean coefficients illustrates the two effects of magnetic field fluctuations: broadening of the -components by fluctuations of the magnetic field intensity, leaving the -components unchanged, and averaging over the angular dependence of the and components. For longitudinal fluctuations only the first effect is at play. For isotropic and perpendicular fluctuations, angular averaging can modify the frequency profiles of the mean coefficients quite drastically with the appearance of an unpolarized central component in the diagonal absorption coefficient, even when the mean field is in direction of the line of sight. A detailed comparison of the effects of the three types of fluctuation coefficients is performed. In general the magnetic field fluctuations induce a broadening of the absorption and anomalous dispersion coefficients together with a decrease of their values. Two different regimes can be distinguished depending on whether the broadening is larger or smaller than the Zeeman shift by the mean magnetic field. For isotropic fluctuations, the mean coefficients can be expressed in terms of generalized Voigt and Faraday-Voigt functions H(n) and F(n) introduced by Dolginov & Pavlov (1972, Soviet Ast., 16, 450). These functions are related to the derivatives of the Voigt and Faraday-Voigt functions. A recursion relation is given in an Appendix for their calculation. A detailed analysis is carried out of the dependence of the mean coefficients on the intensity and direction of the mean magnetic field, on its root mean square fluctuations and on the Landé factor and damping parameter of the line.
Key words: line: formation - polarization - magnetic fields - turbulence - radiative transfer
Observations of the solar magnetic field and numerical simulations of solar magneto-hydrodynamical processes all converge to a magnetic field which is highly variable on all scales, certainly in the horizontal direction and probably also in the vertical one. Solving radiative transfer equations for polarized radiation in a random magnetic field, is thus an important but not a simple problem since one is faced with a transfer equation with stochastic coefficients (Landi Degl'Innocenti 2003; Landi Degl'Innocenti & Landolfi 2004, henceforth LL04). In principle the mean radiation field can be found by numerical averaging over a large number of realizations of the magnetic field and other relevant random physical parameters like velocity and temperature. A more appealing approach is to construct, with chosen magnetic field models, closed form equations or expressions for the mean Stokes parameters. Landi Degl'Innocenti (2003) has given a nice and comprehensive review of the few models that have been proposed.
The problem of obtaining mean Stokes parameters simplifies if one can single out fluctuations with scales much smaller than the photon mean free paths. The radiative transfer equation has the same form as in the deterministic case, except that the coefficients in the equation, in particular the absorption matrix, are replaced by averages over the distribution of the magnetic field vector and other relevant physical parameters. This microturbulent approximation is currently being used for diagnostic purposes in the frame work of the MISMA (Micro Structured Magnetic Atmospheres) hypothesis (Sánchez Almeida et al. 1996; Sánchez Almeida 1997; Sánchez Almeida & Lites 2000) and commonly observed features like Stokes V asymmetries and broad-band circular polarization could be correctly reproduced. In the MISMA modeling the mean Zeeman absorption matrix is actually a weighted sum of two or three absorption matrices, each corresponding to a different constituent of the atmosphere characterized by its physical parameters (filling factor, magnetic field intensity and direction, velocity field, etc.).
The problem simplifies also when the scales of fluctuations is much larger than the photon mean free-paths. The magnetic field can then be taken constant over the line forming region and the transfer equation for polarized radiation is the standard deterministic one. Mean Stokes parameters can be obtained by averaging its solution over the magnetic field distribution. For magnetic fields with a finite correlation length, i.e. comparable to photons mean free paths, the macroturbulent and microtubulent limits are recovered when the correlation scales go to infinity or zero.
The microturbulent limit is certainly a rough approximation to describe the effects of a random magnetic field, but as the small scale limit of more general models, it is interesting to study somewhat systematically the effect of a random magnetic field on the Zeeman absorption matrix. This is the main purpose of this paper. The problem has actually been addressed fairly early by Dolginov & Pavlov (1972, henceforth DP72) and by Domke & Pavlov (1979, henceforth DP79), with anisotropic Gaussian distributions of the magnetic field vector. These two papers have attracted very little attention, although they contain quite a few interesting results showing the drastic effects of isotropic or anisotropic magnetic field distributions with a non zero mean field. More simple distribution have been introduced for diagnostic purposes, in particular in relation with the Hanle effect. For example, following Stenflo (1982), a single-valued magnetic field with isotropic distribution is commonly used to infer turbulent magnetic fields from the linear polarization of Hanle sensitive lines (Stenflo 1994; Faurobert-Scholl 1996, and references therein). A somewhat more sophisticated model is worked out in detail in LL04 for the case of the Zeeman effect. The angular distribution is still isotropic, but the field modulus has a Gaussian distribution with zero mean. The two models predict zero polarization for the Zeeman effect since all the off diagonal elements of the absorption matrix are zero. Recently, measurements of the fractal dimensions of magnetic structures in high-resolution magnetograms and numerical simulations of magneto-convection have suggested that the distribution of the modulus of the magnetic and of the vertical component could be described by stretched exponentials (Cattaneo 1999; Stenflo & Holzreuter 2002; Cattaneo et al. 2003; Janßen et al. 2003). Such distributions are now considered for diagnostic purposes (Socas-Navarro & Sánchez Almeida 2003; Trujillo Bueno et al. 2004). Actually not so much is known on the small scale distribution of the magnetic field vector and on the correlations between the magnetic field and velocity field fluctuations. For isotropic turbulence, symmetry arguments give that they are zero when the magnetic field is treated as a pseudovector (DP79).
Here we concentrate on the effects of Gaussian magnetic field
fluctuations. We believe that a good understanding of the sole action
of a random magnetic field is important before considering more
complex situations with anisotropic random velocity fields and
correlations between velocity field and magnetic field fluctuations,
although they seem to be needed to explain circular polarization
asymmetries. One can find in LL04 (Chap. 9) a simple example showing
the effects of such correlations. So here we assume, as in DP79, that
there is no correlation between the magnetic field and velocity field
fluctuations and that the latter behave like thermal velocity field
fluctuations. They can thus be incorporated in the line Doppler
width. We assume that the medium is permeated by a mean magnetic field
with anisotropic Gaussian fluctuations. We
write the random field distribution function in the form
The distribution written in Eq. (1) is the most general azimuthally symmetric Gaussian distribution. Here we consider three specific types of fluctuations: (i) longitudinal fluctuations in the direction of the mean field, also referred to as 1D fluctuations; they correspond to the case ; (ii) isotropic fluctuations, also referred to as 3D fluctuations; they correspond to (iii); fluctuations perpendicular to the mean field which we refer to as 2D fluctuations; they correspond to the case . In cases (i) and (iii) the fluctuations are anisotropic. They are isotropic by construction in case (ii). In case (i), only the magnitude of is random but in cases (ii) and (iii), both the amplitude and the direction of the magnetic field are random.
For these three types of distribution we give expressions, as explicit as possible, of the mean absorption and anomalous dispersion coefficients. Many of them can be found also in DP72 and DP79 where they are often stated with only a few hints at how they may be obtained. Here we give fairly detailed proofs. Some of them can be easily transposed to non-Gaussian distribution functions. Also we perform a much more extended numerical analysis of the mean coefficients and in particular carry out a detailed comparison of the frequency profiles produced by the longitudinal, perpendicular and isotropic distributions. This comparison is quite useful for building a physical insight into the averaging effects.
This paper is organized as follows. In Sect. 2 we establish a
general expression for the calculation of the mean Zeeman absorption
matrix which holds for any azimuthally invariant magnetic field vector
distributions. In Sects. 3, 4 and 5 we consider in detail the three
specific distributions listed above. Section 6 is devoted to a summary
of the main results and contains also some comments on possible
|Figure 1: Definition of and , the polar and azimuthal angles of the random magnetic field vector , and of and , the corresponding angles for the mean magnetic field .|
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We are interested in the calculation of
In Sect 2.1 we recall the standard expressions of the elements of the Zeeman absorption matrix in the Stokes parameters representation and in Sect. 2.2 we give their expression in the polarization matrix representation. In Sect. 2.3 we explain in detail the transformation of the Ylm and in Sect. 2.4 establish general expressions for the mean coefficients.
We consider for simplicity a normal Zeeman triplet but our results are
easily generalized to the anomalous Zeeman effect (see Sect. 6).
For a normal Zeeman triplet, the line
absorption matrix can be written as (Landi Degl'Innocenti 1976; Rees
1987; Stenflo 1994; LL04)
We introduce a Doppler width
and measure all the
independent variables appearing in
and fq in Doppler
width units. We thus write
We use here Voigt functions which are normalized to unity when
integrated over the dimensionless frequency x, and the associated
Faraday-Voigt functions (a factor
added to the usual definition of H and a factor
usual definition of F). With this definition the Voigt function is
exactly the convolution product of a Lorentzian describing the natural
width of the line and of a Gaussian. The latter can describe pure
thermal Doppler broadening, or a combination of thermal and
microturbulent velocity broadening, provided the velocity field has an
isotropic Maxwellian distribution. What we call here the Doppler width
and denote by
is actually the total broadening
parameter, including the microturbulent velocity field. Thus, with
If the frequency x is measured in units of thermal Doppler width , then become with . The change of variables and and the definition of as in Eq. (9) lead back to Eqs. (6) and (7).
For the calculation of the mean Zeeman propagation matrix, it is
convenient to rewrite the elements as in DP79, namely in the form
The main interest of this formulation, in addition to the fact that
the Ai, (i=0,1,2) depend only on the intensity of the random
magnetic field, is that the functions which contain the angular
dependence can be expressed in terms of spherical harmonics
and Legendre polynomials
which obey simple transformation laws in a rotation of the reference frame.
In terms of these special functions,
We now perform a rotation of the reference frame to obtain the absorption coefficients in a reference frame connected to the mean magnetic field where the averaging process is easily carried out. The initial reference frame is the (x yz) frame, also referred as the LOS reference frame (see Fig. 1). We perform on this reference frame a rotation defined by the Euler angles , and . This rotation is realized by performing a rotation by an angle around the y axis and a rotation by an angle around the initial z-axis. Since the random field is invariant under a rotation about the direction of the mean field, we have taken . Rotational transformations and Euler angles are described in many textbooks (Brink & Satchler 1968; Varshalovich et al. 1988; LL04).
The spherical harmonics Ylm are irreducible tensors of rank
l. They are particular cases of the Wigner
functions corresponding to m=0 or m'=0
(see Appendix A). In a rotation of the reference frame,
defined by the Euler angles ,
transform according to (Varshalovich et al. 1988, p. 141, Eq. (1))
Actually, we need the inverse transformation which will give us the
in terms of the
inverse transformation is (Varshalovich et al. 1988, p. 74, Eq. (13))
To calculate the mean coefficients
to integrate Eq. (12) over .
and the Ai,
(i=0,1,2) are independent of
Eqs. (3) and (11)), only the Ylm have to
be integrated over .
Eq. (14) is integrated over ,
only the term with
m'=0 will remain. For m'=0 the
reduce to Ylm and the Yl0 to Legendre polynomials. Thus
after integration, Eq. (14) reduces to
Using Eq. (15) with l=2, m=0 for
l=1, m=0 for
we obtain the very compact expressions
With the distribution functions considered here (see Eqs. (1) or (3)), the mean coefficients have the same symmetry properties as the non random coefficients, namely and are symmetric with respect to the line center x=0 (they are even functions of x) and is antisymmetric (odd function of x). We stress also that the integrals of and over frequency are not affected by turbulence. Hence if one consider only the integration over , the integral of is zero and the integral of equal to 1/2.
When fluctuations are along the direction of the mean field
the distribution function for the random field can
We introduce the new dimensionless variable y and the parameters
To calculate the mean absorption coefficients it suffices to take the
average of the Ai over
in Eq. (11) since
the random field is along the direction
procedure is equivalent to set
in Eq. (17).
The averaging over the magnetic field distribution amounts to the
convolution product of a Voigt function with a Gaussian coming from
the distribution of the magnetic field modulus. The effect is similar
to a broadening by a Gaussian turbulent velocity field, except that it
does not affect the
term (the -component) since the latter
does not depend on the modulus of the magnetic field. One obtains (see
The broadening of the -components can be described in terms
of a total Doppler width
that combines the effects of
thermal, velocity and magnetic field broadening. It can be written as
When the Zeeman shift by the mean magnetic field
is smaller than the combined Doppler and magnetic broadening
), a situation referred to
as the weak field limit, as in DP72, one has, to the leading order
We now assume that the fluctuations of the magnetic field are
isotropically distributed. This implies that
Eq. (1). The distribution function takes the form
We now calculate the
Eq. (17). Introducing the variable
As shown in Appendix C, the
expressed in terms of the generalized Voigt
and Faraday-Voigt functions H(n) and F(n) defined by
|Figure 2: The H(n) and F(n) functions for several orders n. The damping parameter a = 0. The H(n) are even functions when n is even and odd when n is odd. For the F(n) it is the opposite.|
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have closed form
(i.e. exact) expressions in terms of the H(n) but not
for which only approximate expressions can be given because of
the term with 1/y (see Eq. (34)). The exact expressions for
For , approximate expressions can be constructed in the limiting cases and , which we refer to respectively as the strong mean field and weak mean field limits for reasons explained now. We discuss these two cases separately.
When the mean field intensity
is much larger than the rms
fluctuations, i.e. when
In this case the
by the mean magnetic field is
much larger than the broadening
by the random magnetic field fluctuations. We call
this situation the strong mean field limit but it can also be
viewed as a weak turbulence limit. When
one can neglect
in Eq. (34) and thus obtain
We remark here that if we keep finite but let , we recover the longitudinal turbulence case discussed in the preceding section. This can be checked on Eqs. (38) to (40).
We now consider the case where
This means that the
Zeeman shift by the mean field satisfies
this condition automatically implies that
the mean Zeeman shift is smaller than the combined Doppler and Zeeman
broadening. Thus in this limit, which we refer to as weak mean field
limit, the mean magnetic field is too weak for the
-components to be resolved. The best method to obtain the mean
absorption coefficients is to start from
Eq. (30) and expand the exponentials
in powers of .
Using the change of
variables described in Appendix C with
obtains at the leading order,
For the functions
the expansion in
In this weak field limit the mean value of the absorption coefficient is simply given by since the contribution from , which is of order , can be neglected. Thus is independent of the direction of the mean field. This property holds also when the mean field is constant. The proof given here is an alternative to the standard method which relies on a Taylor series expansion of the Voigt function (Jefferies et al. 1989; Stenflo 1994; LL04).
When the total broadening of the line is controlled by Doppler broadening, i.e. when , one can set . Equations (42) and (43) lead to the standard results and .
When the mean magnetic field is zero, the angular averaging over
in the original variables)
becomes independent of the averaging over the magnitude of the
magnetic field. Because of the isotropy assumption,
and the polarization is zero, namely
diagonal absorption coefficient is given by
equal to the rhs of
Eq. (42). One can verify that our result is identical to the
last equation in Sect. 9.25 of LL04. There
is written in terms of the second order derivative of the Voigt
|Figure 3: Weak mean field limit. Isotropic fluctuations. Absorption coefficients and for a longitudinal mean magnetic field are shown. Mean Zeeman shift ; Voigt parameter a=0. The curve corresponds to a constant magnetic field equal to .|
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|Figure 4: Strong mean field limit. Isotropic fluctuations. Absorption coefficients and for a longitudinal mean magnetic field ( ) are shown. Mean Zeeman shift ; Voigt parameter a=0.The curve corresponds to a constant magnetic field equal to .|
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In Figs. 3 to 5 we show the effects
of an isotropic distribution with a non zero mean field on the
absorption and anomalous dispersion coefficients
We discuss separately the weak and strong field
limits. The results are presented for the damping parameter a=0.
|Figure 5: Strong mean field limit. Isotropic fluctuations. Mean values of and for a transverse mean magnetic field ( ). Same parameters as in Fig. 4 ( ; a=0).|
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In the weak mean field limit, , up to terms of order , , up to terms of order , and which is order of can be neglected. As already mentioned above, is independent of the mean field direction. We show in Fig. 3 the profiles of and for calculated with and . With this choice of parameters, we satisfy the weak mean field condition since stays smaller than unity. As can be observed in Fig. 3a the increase of produces two different effects on . There is a global decrease in amplitude due to the factor in front of the square bracket in Eq. (42) and the appearance of two shoulders created by the increasing contribution of the term with H(2). They are clearly visible for . The position and amplitude of these shoulders can be deduced from the behavior of H(2). Equation (37) shows that the H(n) have maxima at . A rescaling of frequency by the factor , predicts that the position of these shoulders is around and their amplitude around , in agreement with the numerical results. These shoulders are a manifestation of the -components which appear with increasing probability when , i.e. the dispersion of the random magnetic field, increases.
In this limit
are given by
Eq. (16) with
given by the
exact expressions in Eqs. (38), (39) and
given by the approximate relation (40). Thus errors
that can be created by this approximation will all come from
and affect only
and to a lesser extent
using an exact expression. Figures 4 and 5
illustrate the variations of
with the parameter
To satisfy the strong field condition
), we have
smaller than 1.5. The variations of
easy to understand if we expand the sums over q in
Eqs. (38) and (40). This gives
The term containing H(x,a) creates a central component even when the mean field is longitudinal ( ). The existence of this central component, which has no polarization counterpart, was pointed out in DP72. It is created by the averaging of the -component opacity over the isotropic random magnetic field distribution. When , this central component behaves as . It becomes clearly visible when (i.e. ). In Fig. 4 it increases with because we are keeping the product constant. When , this component behaves as . As can be seen in Fig. 5, it is not very sensitive to the value of .
The -components come mainly from the second term in Eq. (44). They vary like for and as for . Thus, an increase in produces a broadening of the components and a decrease in intensity. There is also a shift away from line center more specifically due to the increase of the relative importance of the H(1) terms with respect to the H terms.
The mean coefficients and are given by and with and given in Eqs. (39) and (40). The profiles shown in Figs. 4 and 5 are easy to understand. The dominant contributions come from the terms with , . For , the -components behave essentially as , i.e. as the -components of . Hence their amplitude decreases and their width increases when increases. For , the -components behave as and the central component as , to be compared to for . Hence as observed in Fig. 5, the central component of is more sensitive to the value of than the central component of .
We now discuss the behavior of the mean opacity coefficients when
is of order unity. For
we have exact expressions given in
Eqs. (38) and (39) but there is nothing
Roughly, the weak field limit is valid for
to 0.2 and the strong field limit for
of order unity, neither the weak nor the strong mean field
approximation holds and
must be calculated numerically. For the numerical calculations
it is preferable to return to Eq. (30). The integration
can be carried out explicitly. One obtains, for the mean
The integration over y is performed numerically using a Gauss-Legendre quadrature formula. The integrand varies essentially as e , with the factor e coming from the Bessel function. The maximum of the integrand is around . With 10 to 30 points in the range we can calculate the integrals with a very good accuracy (errors around 10-6). The averaging process increases the overall frequency spread of the mean coefficients. A total band width is adequate to represent the full profiles.
In the following sections we discuss the dependence of on the intensity of the mean field, on its rms fluctuations and on the damping parameter a. A full section is devoted to which has the most complex behavior. Then we discuss the dependence of all the mean coefficients, including the anomalous dispersion coefficients, on the Landé factor for a given random magnetic field. All the calculations have been carried out with a damping parameter a=0, except when we consider the dependence on a.
Equation (45) shows that
central component around x=0 which corresponds to the
-component. It is of the form H(x,a) times a factor which
and on the orientation
of the mean
magnetic field. When
is small, the Bessel functions can be
replaced by their asymptotic expansions around the origin (see
Appendix B) and the central component has the
|Figure 6: Dependence of on the mean magnetic field intensity measured by the parameter . Isotropic fluctuations. The parameters employed are: a=0, . The curves for and 0.1 coincide. The panels a) and b) correspond to the longitudinal ( ) and transverse ( ) cases, respectively.|
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in Fig. 6 for different
values of .
We cover all the regimes of magnetic
splitting from the weak field regime for
to the strong field
These two regimes have been discussed in Sects. 4.1 and 4.2. For
there is a single central peak described by
the H(0) terms in Eq. (42). There is essentially no
contribution from the term with H(2). For
one is in the
intermediate regime described by Eq. (45). There is still a
single peak because the Zeeman shift
smaller than the broadening parameter
one enters in the strong field regime, with well separated
discussed in detail in Sect. 4.2.2. When
is kept finite, the isotropic distribution goes
to the 1D distribution. In the longitudinal case (
the central component goes then to zero and the -components to
while in the transverse case
), they go to H(x,a)/2 and
|Figure 7: Dependence of on the magnetic field dispersion measured by . Isotropic fluctuations. The parameters employed are : a=0, . The panels a) and b) correspond to longitudinal ( ) and transverse ( ) cases respectively. Notice the saturation of the central -component for .|
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Figure 7 shows for a fairly strong mean magnetic splitting and several values of varying from 0 to 6. For we are in the deterministic case with two well separated -components at . Their amplitudes are and for and , respectively. The -component for has an amplitude since a=0. For , we are still in the strong field regime ( ) with -components still roughly at but the peaks have smaller intensity because of the factor in Eq. (44). For , one starts entering into the weak field regime which has been discussed in Sect. 4.2.1 since the corresponding value of is 2/3.
|Figure 8: Dependence of on the damping parameter a in the case. Isotropic fluctuations. Panel a) shows the strong field case ( and ) and panel b) the weak field case ( and ). For large values of a, the and -components decrease in strength.|
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Figure 8 shows for the longitudinal Zeeman effect. Panel (a) is devoted to the strong mean field regime (see also Fig. 4) and panel (b) to the weak field regime (see also Fig. 3). As long as a<10-2, there are no observable effects on the mean value of . The effects of the damping parameter on become noticeable when a>0.1. As expected, the intensity of the and -components decrease and Lorentzian wings appear. When a>0.5, the central component in the strong field case almost disappears. Thus for values of to 0.1, the and -components are insensitive to changes in a and the effects of turbulence discussed in this paper for a=0 survive. In the solar case, this situation will hold except for very strong lines.
|Figure 9: Dependence of on the Zeeman sensitivity (the Landé g factor) introduced through the parameter (see the text for details) Isotropic fluctuations. The parameters employed are a=0, and . Notice the saturation of the -component at the line center. The panels a) and b) correspond to longitudinal ( ) and transverse ( ) cases respectively.|
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|Figure 10: Same as Fig. 9, but for and . The panels a) and b) correspond to and respectively.|
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|Figure 11: Dependence of the magneto-optical coefficients and on the Landé factor. Same model as Fig. 9. Notice the similarity between and as well as and .|
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We now consider the effect of a given random magnetic field on lines with different Zeeman sensitivities. We give and the dispersion , but let the Landé parameter g vary. Thus is constant, but and are varying with g (see Eq. (8)). The mean coefficients have been calculated with and to 6. For this choice of we are in an intermediate field regime and the mean coefficients are given by Eqs. (45), (46) and (47).
Figure 9 shows . For the Zeeman components are not resolved (the same curve is shown in Fig. 6a, ). For , the central peak is quite broad (Full Width at Half Maximum ) because of the superposition of the central -component coming from the first term in Eq. (45) (responsible for the narrow tip) with the two -components given by the two other terms in the same equation. For , the central peak is more narrow (FWHM = 3), because the contribution from the -components is smaller. As can be observed in Eq. (45), the coefficient of is for but only for . We recall that the modified Bessel functions are positive functions. When is large enough, the -component is given by the first term in Eq. (45). It is independent of and its FWHM is around 2. Its amplitude is larger in the transverse than in the longitudinal case since the coefficients of H(x,a) in the integrand are respectively and (I1/2 - I5/2). If it were not for the isotropic distribution, there would be no -component when .
The -components have essentially the same behavior in the longitudinal and transverse case. The positions of the peaks depend little on and can be deduced from the position of the maximum of the integrand in Eq. (45). Ignoring the shifted H functions, keeping only the Bessel function of order 1/2 and the positive exponential in the function (see Eq. (B.1)), we find that the maximum is at . For , we get in fair agreement with the numerical results. The height of the peaks is somewhat larger in the longitudinal than in the transverse case, because the coefficients of the shifted H functions are larger in the first case, as pointed out above.
Figure 10 shows the mean absorption coefficient divided by and divided by (see Eqs. (47) and (46)). The profile of is quite standard. As with the positions of the peaks increase linearly with the Landé factor g and are around . For , the central peak, given by term with H(x,a) is independent of , hence it goes to a constant value when the two -components are sufficiently far away from line center. This constant value will of course depend on .
Finally, in Fig. 11 we have plotted the mean anomalous dispersion coefficients , divided by , and , divided by . They are given by Eqs. (46) and (47) with the Voigt function H(x,a) replaced by the Faraday-Voigt function F(x,a). The coefficient , which has the same symmetry as , keeps more or less the same shape as the Landé factor increases, except for a small broadening long ward of the peaks. This can be explained by considering Eq. (46). The overall shape is controlled by the first term which is independent of . The two other terms are responsible for the broadening of the peaks but since they more or less compensate each other around x=0, they do not affect the central part of the profile.
The coefficient , has the same symmetries as but the opposite sign. Because it involves the difference (see Eq. (47)), it is very sensitive to the value of and hence to the Landé factor. For , one clearly recognizes the shapes of two shifted Dawson integrals with opposite signs in Fig. 11b.
We now assume that the fluctuations of the magnetic field are confined
to a plane perpendicular to the direction of the mean field
Integrating over the longitudinal component in
Eq. (1), we get the distribution function
As shown in DP79, closed form expressions of
can be obtained in terms of the error function when the
damping parameter a=0. For
approximate expressions can be
These different expressions
are easily deduced from Eqs. (52) to (54). We give
them below together with the weak mean field limits for
They will be used to analyze the effects of 2D
Equations (52) and (53) lead to
one has the approximation
The combination of Eqs. (55), (56) and (57) with Eq. (16), yields an expression of for large values of . It contains a term proportional to e-x2, which yields the central component, and terms which are exactly or approximately of the form e which determine the -components.
In the weak mean field limit, i.e. when
we have, to
When the mean field is zero,
are given by the rhs in Eqs. (58) and
(60) which become exact results.
|Figure 12: Dependence of and on the magnetic field distribution in the weak mean field limit for the longitudinal Zeeman effect ( ). The model parameters are a=0, and (hence ). The curves with correspond to a constant magnetic field equal to .|
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|Figure 13: Dependence of on the magnetic field distribution. The model parameters are a=0, , (hence ). The curves with correspond to a constant magnetic field equal to . Panels a) and b) correspond to longitudinal ( ) and tranverse ( ) cases, respectively.|
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|Figure 14: Dependence of and on the magnetic field distribution. Same model parameters as in Fig. 13. Panels a) and b) correspond to transverse ( ) and longitudinal ( ) cases, respectively.|
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We compare in Figs. 12 to
15, the mean absorption coefficients
corresponding to 1D, 2D and 3D
turbulence. Figure 12 corresponds to a weak mean field
limit and the other figures to an intermediate regime, neither weak
nor strong, with
In each figure we also show the absorption
coefficients corresponding to a non-random field equal to the mean
|Figure 15: Dependence of on the magnetic field distribution. The model parameters are a=0, , (hence ). The curves with correspond to a constant magnetic field equal to . Panels a) and b) correspond to longitudinal ( ) and tranverse ( ) cases, respectively.|
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In Fig. 12, is much smaller than the broadening parameter . Hence shows a single central peak. The random fluctuations produce a decrease in the peak intensity and an associated broadening. The decrease in peak intensity is the largest for 3D turbulence and the smallest for 2D turbulence. This can be explained with equations established in the previous sections.
For 1D fluctuations and
we have (see Eq. (27)),
For circular polarization, , with given in Eqs. (26), (43) and (59) for 1D, 3D and 2D turbulence, respectively. The peak intensity is the largest for 2D turbulence and the smallest for isotropic turbulence (see Fig. 12b), exactly as observed for . The frequencies of the peaks are at for zero turbulence, around for 1D turbulence and further away from line center for isotropic turbulence because of the contribution of the term with H(3) (see the discussion in Sect. 4.2.1). For 2D turbulence, numerical simulations show that the maxima are around with not much dependence on the value of . This result is suggested in Sect. 5.1.
We now discuss Figs. 13 and 14 where and . In the non-random case, the two -components of the profile are partially separated when but form a single peak with the -component when . Panel (a) shows that the central frequencies are quite sensitive to the angular distribution of the random field. For 1D turbulence there is a strong broadening of the -components which fill up the depression at line center. For 2D turbulence, the -components are still well marked but have a smaller intensity. As pointed out above, the broadening of the -components is small in the 2D case. For isotropic turbulence, there is also a single broad peak (the same profile is shown in Fig 9, panel (a)). Panel (b), in Fig. 13, corresponds to . We note that 2D and 3D turbulence have essentially the same effects. The decrease in the central peak intensity comes from the angular averaging over . In contrast, the profile is left almost unaffected in the 1D case because the main contribution to the central peak comes from the -component which is insensitive to the fluctuations of the random field intensity.
Figure 14, panels (a) and (b) show and respectively. We see that behaves in much the same way as for . For 1D turbulence, the central peak is not significantly affected for the reason given above. The -components on the other hand suffer some broadening, which goes together with a decrease in intensity. For 2D and 3D turbulence there is a sharp drop in the central peak and also in the -components, but the broadening with 2D turbulence is, as already pointed out, much smaller than with 3D turbulence For , the fluctuations of the magnetic field produce a decrease in the peak intensity, a small shift away from line center and a broadening which has its largest value for 3D and its smallest value for 2D. The strongest effect is produced by isotropic fluctuations. The decrease in the peak intensity can be explained by the factor in Eq. (39).
When the rms fluctuations increase, i.e. when increases, the profiles and keep essentially the same shape but the effects are amplified. All the peaks have a smaller intensity and for 2D and 3D turbulence the -components are moved away further from line center. One also observes a significant decrease in the slope of at line center.
In Figs. 15 we still have , (rms fluctuations equal to the mean field intensity) but and . Hence the -components are well separated as can be observed. Panel (a) of this figure clearly shows the central component created by the averaging of -component over the random directions of the magnetic field for 2D and 3D turbulence. For 2D turbulence, the -components are significantly more intense and more narrow than for 3D turbulence. The central peak on the other hand is shallower. For 1D turbulence, there is no central component but a strong broadening of the -components. The decrease in the intensity of the -components is controlled by the factor (see Eq. (22)). For the transverse case (panel (b)), the -components disappear for 1D turbulence because they are multiplied by but the central peak increases due to the contribution of the broadened -components. This increase of the central peak can also be understood in terms of the constancy of the frequency integral of . For 2D and 3D turbulence, the -components are still well marked but they are somewhat shifted away from line center with the 3D components being broader and shallower than the 2D components. The decrease of the central peak is due to the averaging over the term.
In this paper we have examined the effects of a random magnetic field on the Zeeman line transfer propagation matrix. We have considered a fairly general case where the magnetic field has anisotropic but azimuthally invariant Gaussian fluctuations about a given mean magnetic field which can be set to zero. We have examined in detail three types of random fluctuations : (i) longitudinal fluctuations which take place along the direction of the mean field; referred to as 1D or longitudinal turbulence; (ii) fluctuations which are distributed isotropically around the direction of the mean field referred to as isotropic or 3D turbulence; (iii) fluctuations isotropically distributed in a plane perpendicular to the mean field, referred to as 2D turbulence; the total random field (sum of the fluctuating part and of the mean field) does not lie in this plane unless the mean field is zero. In all three cases, the random field depends on two parameters, the mean field and the dispersion around the mean field (see Eqs. (18), (28), (49)).
First we give a fairly compact and simple expression for the mean coefficients of the propagation matrix. It is valid for any random field invariant in a rotation around the mean field direction (Eq. (16)). This general expression is obtained by taking advantage of the fact that the angular dependence of the Zeeman matrix elements can be written in terms of the spherical harmonics , where and are the polar and azimuthal angles of the random field with respect to direction of the line of sight.
The random fluctuations of the magnetic field have two types of effects. The fluctuations of the magnetic field strength (modulus) produce random Zeeman shifts which lead to a broadening of the -components. It is important to note that the -component is not affected by this phenomenon. The second effect, which occurs only for 2D and 3D turbulence, is the averaging over the angular dependence of the coefficients which affects both the and -components. As a result, the frequency profiles of the mean coefficients can look quite different from the standard profiles created by a constant magnetic field. The physically relevant parameters for the analysis of the mean profiles are the dimensionless parameters , which measures the intensity of the mean magnetic field in units of the rms fluctuations , and , the Zeeman shift by the rms fluctuations. The Zeeman shift by the mean magnetic field is . The broadening by the magnetic field intensity fluctuations combined with the standard Doppler broadening (by thermal and/or microturbulent velocity fluctuations) is described by a parameter . There are two interesting limiting regimes. A weak mean field regime corresponding to , i.e. to a Zeeman shift by the mean magnetic field smaller than the combined Doppler and magnetic broadening. The other interesting limit, referred to as the strong mean field or weak turbulence regime, corresponds to . In this limit, the -components stay well separated in spite of the random field fluctuations, provided stays smaller than . We now briefly summarize the main effects for the three types of fluctuations that we have considered.
For 1D turbulence, the direction of the random magnetic field remains constant and same as the direction ( , ) of the mean magnetic field. The only effect is a broadening and a decrease in intensity by a factor of the -components (see Sect. 3. and Figs. 12 to 15). For the transverse Zeeman effect ( ) and when , a consequence of this broadening is that the central -component can be enhanced by the magnetic field fluctuations while the -components almost entirely disappear (see Fig. 15). When the intensity of the mean magnetic field is zero, the coefficient of circular polarization (and ) are zero but not the mean linear polarization coefficients and . Circular polarization is destroyed by fields of opposite directions but not linear polarization which has a quadratic dependence on the polar angle of the magnetic field.
For isotropic (3D) turbulence, the two effects namely, magnetic broadening of the -components and angular averaging are at work. The dependence of the absorption and anomalous dispersion coefficient profiles on the magnetic field parameters and on the Landé factor is discussed in detail in Sect. 4. One striking effect in the case of the longitudinal Zeeman effect ( ) is the formation in of a central component with no polarization counterpart created by the averaging of . This component is particularly noticeable when (see Figs. 9 and 15). The circular polarization coefficients (and ) can be expressed in terms of generalized Voigt and Faraday-Voigt functions H(n) and F(n). The other mean coefficients can also be expressed in terms of these generalized functions but only in weak mean field and strong mean field regimes. When the mean magnetic field is zero, the random field is strictly isotropic (there is no preferred direction) and both circular and linear and polarization coefficients are zero. The same is true of course for the anomalous dispersion coefficients.
For 2D turbulence the mean profiles resemble the mean profiles for isotropic turbulence. One can observe in particular the formation of a non-polarized central component due to the averaging of over the directions of the random field, but in contrast to isotropic turbulence, there is very little broadening of the -components because the magnitude of the random field is more centered around the magnitude of the mean field. The -components are not only more narrow they are also stronger than with 1D or 3D turbulence (see the figures in Sect. 5). When the mean magnetic field is zero, the mean circular polarization coefficient is zero but not the linear coefficients and . So even if the mean magnetic field is zero, anisotropic turbulence like 1D or 2D turbulence will produce linear polarization.
In this work we have considered for simplicity a normal Zeeman
triplet. In the anomalous Zeeman splitting case, each elementary
must be replaced by a weighted average
of the form
Here we have considered only Gaussian distributions but it is clear that the averaging method and the main effects that we have described will carry over to other types of distributions. Such effects as the broadening of the -components by random Zeeman shifts or the appearance of unpolarized central components due to angular averaging should persist. The assumption that the random fields are azimuthally symmetric plays an important role in the averaging method, but is a fairly realistic assumption for small scale fluctuations. As for correlations between magnetic and velocity fluctuations, they can certainly be incorporated in the averaging method without major difficulties.
For weak lines (optical depth small compared to unity), the opacity
coefficients give a fair approximation to
the observable Stokes parameters and a comparison between
observations and mean coefficient profiles could provide informations
on the statistical properties of the magnetic
field. For example, the intensity of the mean magnetic field could be
obtained with the center-of-gravity method (see e.g. LL04,
p. 640). This method is based on the measurements of the center of
gravity wavelength .
For weak lines, they can be written as
A detailed analysis of Stokes profiles for lines with different Zeeman sensitivity (Landé factors) would be a way to evaluate the dispersion of the random fluctuations. The detection of an unpolarized central component in Stokes I would indicate strong variations in the direction of the magnetic field. However, specific observations at high resolution would be required to verify this fact, because a central unpolarized component may also be produced by a non-magnetic region within the resolution element.
For spectral lines with moderate to large optical depths, radiative transfer effects must be taken into account. The Unno-Rachkovsky solution shows very large differences in the observable Stokes parameters, depending on whether the magnetic field is random or not. This topic will be addressed in subsequent papers where we consider random magnetic fields with a finite correlation length.
M.S. is financially supported by Council of Scientific and Industrial Research (CSIR), through a Junior Research Fellowship (JRF Grant No.: 9/890(01)/2004-EMR-I). This support for the Research work is gratefully acknowledged. She would like to thank the Director, Indian Institute of Astrophysics (IIA) and JAP program, for providing excellent research facilities. Further K.N.N. and M.S. are grateful to the Laboratoire Cassiopée (CNRS), the PNST (CNRS) and the French Ministère de l'Éducation Nationale for financial support during a visit at the Observatoire de la Côte d'Azur where part of this work was completed. H.F. was supported by the Indo-French Center for the Promotion of Advanced Research (IFCPAR 2404-2) and by the Indian Institute of Astrophysics during her visits to Bangalore. She would like to thank Dr. V. Bommier for stimulating discussions. The authors are grateful to the referee for constructive remarks.
The properties that are needed here can be found in Brink & Satchler (1968, Appendix IV), Varshalovich et al. (1988) or in LL04. We reproduce them here for convenience.
The Wigner matrices
are the transformation matrices for
irreducible tensors of rank l in rotations of the reference frame. The
are the Euler angle of the rotation. The
D(l)m m' have an explicit representation:
The Ylm are special cases of
D(l)m m' corresponding to
m'=0 (or m=0):
The functions of order 0, 1 and 2 introduced in
Eqs. (45), (46) and (47) can be obtained by
performing the integration over
in Eq. (30) (see
also Abramovitz & Stegun 1964, p. 443). They may be written as
The mean values are given by the averages, over the magnetic field distributions, of Voigt or Faraday-Voigt functions, multiplied by some polynomials (see Eq. (17)). Explicit expressions for the average values are given in Eq. (22) for 1D turbulence, in Eqs. (38), (39), (40), (41), (42), (43), for 3D turbulence. We show here how to obtain these expressions which for 3D turbulence involve the generalized H(n) (and F(n)) functions introduced in Sect. 4.1 and discussed in Appendix D. Several methods are available to carry out the integration. One can consider the Fourier transforms of the quantities to be averaged. One can write the functions H(0) and F(0) as real and imaginary parts of the function W(0)(z), with z complex (see Appendix D) and then do contour integrations in the complex plane. Here we describe a direct method based on simple changes of variables.
The integrals we want to transform are of the form
First we transform the integral over u. We write
The functions H(n) and F(n) introduced in Eqs. (35)
and (36) of the text are the real and imaginary part
of the function
The W(n) satisfy a recurrence formula which leads
to simple recurrence relations for H(n) and F(n) and thus to a
method of calculation. In the numerator of Eq. (D.1), we write
un = un-1(u-z+z) and immediately obtain
Separating the real and imaginary parts, we find the two recurrence
The recurrence relations take very simple forms when the Voigt
parameter a=0. For H(n), they lead to
Eq. (37) of the text.
For F(n), with
When a is not zero, H(0) and F(0) have been calculated with the algorithm of Hui et al. (1978) which is more accurate than the algorithm of Matta & Reichel (1971), especially for F(0).
We note that the derivatives of the Voigt and Faraday-Voigt functions
can be expressed in terms of the functions H(n) and F(n).