A&A 453, 1095-1109 (2006)
DOI: 10.1051/0004-6361:20054532
H. Frisch^{1} - M. Sampoorna^{1,2,}^{} - K. N. Nagendra^{1,2}
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
Received 17 November 2005 / Accepted 27 March 2006
Abstract
Context. The Zeeman effect produced by a turbulent magnetic field or a random distribution of flux tubes is usually treated in the microturbulent or macroturbulent limits where the Zeeman propagation matrix or the Stokes parameters, respectively, are averaged over the probability distribution function of the magnetic field when computing polarized line profiles.
Aims. To overcome these simplifying assumptions we consider the Zeeman effect from a random magnetic field which has a finite correlation length that can be varied from zero to infinity and thus made comparable to the photon mean free-path.
Methods. The vector magnetic field is modeled by a Kubo-Anderson process, a piecewise constant Markov process characterized by a correlation length and a probability distribution function for the random values of the magnetic field. The micro and macro turbulent limits are recovered when the correlation goes to zero or infinity.
Results. An integral equation is constructed for the mean propagation operator and explicit expressions are obtained for the mean values and second-order moments of the Stokes parameters at the surface of a Milne-Eddington type atmosphere. The expression given by Landi Degl'Innocenti (1994) for the mean Stokes parameters is recovered. Mean values and rms fluctuations around the mean values are calculated numerically for a random magnetic field with isotropic Gaussian fluctuations. The effects of a finite correlation length are discussed in detail. Various extensions of the Milne-Eddington and magnetic field model are considered and the corresponding integral equations for the mean propagation operator are given.
Conclusions. The rms fluctuations of the Stokes parameters are shown to be very sensitive to the correlation length of the magnetic field. It is suggested to use them as a diagnostic tool to determine the scale of unresolved features in the solar atmosphere.
Key words: line: formation - polarization - magnetic fields - methods: analytical - Sun: atmosphere
The Zeeman effect has been used in Astrophysics for more than a century to measure magnetic fields in the Sun, stars and other objects. The very first analyses of the Zeeman effect were carried out with uniform magnetic fields. Together with a higher quality of data, appeared multi-components models (Stenflo 1994), each component having a different but uniform, or slowly varying, magnetic field. For these models, the observable Stokes parameters are given by a conveniently weighted average of the Stokes parameters of each component. Prompted by measurements of asymmetrical Stokes V profiles, multi-components models of another type were introduced under the name of MISMA (Sánchez Almeida et al. 1996). In this model, each component is optically thin and the Zeeman propagation matrix is replaced by an average over the various components. These two types of models can be made quite sophisticated. With the terminology used for random velocity fields broadening, one can say that the first model is of the macroturbulent type, since the averaging is over the radiation field, whereas the second type of model is of the microturbulent type since the averaging is done locally over the propagation matrix.
These two types of models may be insufficient to encompass the complexity of the solar atmosphere which shows inhomogeneities, undoubtedly related to the magnetic field structure, down to scales at the limit of the resolution power of present day telescopes. For example there is an active discussion on the fine structure of sunspot penumbrae. It seems accepted that penumbral magnetic fields have a more or less horizontal component in the form of flux tubes embedded in a more vertical background. However the diameter of these flux tubes and their spatial distribution is still a matter of controversy, the number quoted in the literature varying from 1-15 km to 100 km (Sánchez Almeida 1998; Martinez Pillet 2000; Borrero et al. 2005). In addition, because of very large kinetic and magnetic Reynolds numbers prevailing in the solar atmosphere (Childress & Gilbert 1995), turbulent magnetic and velocity fields have spectra extending over a wide range of wave-numbers. We were thus strongly motivated to consider the Zeeman effect in a medium where the magnetic field is random with a correlation length, i.e. characteristic scale of variation, comparable to radiative transfer characteristic scales. The importance of this problem has been stressed again recently (Landi Degl'Innocenti 2003; Landi Degl'Innocenti & Landolfi 2004, henceforth LL04).
The general regime, neither macro nor microturbulent, leads to polarized radiative transfer equations with random coefficients. Only a few papers have been devoted to this subject in the past (see however, Faulstich 1980; Landi Degl'Innocenti 1994, henceforth L94). Recently this field seems to be receiving some renewed interest (Caroll & Staude 2003, 2005; Silant'ev 2005). Similar problems, somewhat simpler though, have been solved in the seventies for the transfer of unpolarized radiation in the presence of a turbulent velocity with a finite correlation length (see Mihalas 1978, for a list of references). Turbulent velocity field models introduced were less or more sophisticated. The simplest one, is the Kubo-Anderson process (henceforth KAP). For radiative transfer problems, it was employed in the context of turbulent velocity fields for LTE lines (Auvergne et al. 1973) and non-LTE lines (Frisch & Frisch 1976; Froeschlé & Frisch 1980), and also in the context of random magnetic fields for the Zeeman (L94) and Hanle (Frisch 2006) effects. Actually the KAP was introduced for nuclear magnetic resonance (Anderson 1954; Kubo 1954). It was also employed to model the electric field in the stochastic Stark effect (Brissaud & Frisch 1971; Frisch & Brissaud 1971). The name Kubo-Anderson process was introduced in Auvergne et al. (1973).
The idea of the KAP is to describe the atmosphere in a number of "eddies'' having lengths distributed according to a Poisson distribution with given density. It is assumed that in each eddy the magnetic field and other random parameters, such as the velocity or temperature, are constant and their values drawn at random from a probability distribution function. The mean polarized radiation field is obtained by averaging over this distribution and the distribution of the length of the eddies. A KAP is thus characterized by a correlation length and a distribution function for the values of the random variables. The correlation length and the distribution function can be selected independently. This model is fairly simple but has the correct micro and macroturbulent limits corresponding to a correlation length which is zero or infinite. As we show here, when associated to a simple atmospheric model like the Milne-Eddington model, it yields a convolution-type integral equation for the mean propagation operator from which one can deduce explicit expressions for the mean and rms fluctuations of the Stokes parameters at the surface of the atmosphere, and also for the cross-correlations between Stokes parameters. In L94, only the mean Stokes parameters at the surface are considered. It is quite clear that having explicit expressions is very useful for exploring finite correlation length effects.
In this paper the main focus is on the effects of random magnetic fields with a finite correlation length. For a full description of say, turbulent eddies or random distribution of flux tubes in a sunspot penumbrae, it is necessary to incorporate all the other relevant atmospheric parameters which typically should be described by the same type of random process as the magnetic field, in particular the same correlation length. When the magnetic field is described by a KAP, incorporating other random parameters, in particular a velocity field, also described by a KAP with the same correlation length as the magnetic field is no additional work as we explain in the Remark at the end of Sect. 2.6.
In Sect. 2 we define the random magnetic field model, establish a convolution-type integral equation for the mean propagation operator, solve it exactly for its Laplace transform and give an explicit expression for the mean value of the Stokes parameters at the surface of the atmosphere. The latter is used in Sect. 3 to study numerically the sensitivity of the mean Stokes parameters to the correlation length of a random magnetic field with isotropic Gaussian fluctuations. In Sect. 4 we establish an explicit expression for the second-order moments of the Stokes parameters and study numerically the dispersion of the Stokes parameters about their mean values. The second-order moments give also access to the mean cross-correlations between Stokes parameters. In Sect. 5 we introduce various extensions of the Milne-Eddington and magnetic field model and establish the corresponding integral equations for the mean propagation operator. A summary of the main results is presented in Sect. 6.
We consider a line formed in LTE in semi-infinite one-dimensional
medium and assume that the source function is a linear function of depth.
The radiative transfer equation for the Stokes vector
for rays propagating outwards along
the normal to the surface may be written as
Following the usual procedure, we define the evolution operator
,
as the linear operator which transforms
into
when the source term
in Eq. (1) vanishes (Landi Degl'Innocenti 1987; see also
the appendix A).
Since photons propagate from positive s (inside) to s=0 (surface), we always take s'>s. The formal solution of the transfer equation at s=0 may be written as
Assuming that the magnetic field is a KAP implies that is piecewise constant, jumping at randomly chosen points between random values. The jumping point s_{i} are uniformly and independently distributed in with a Poisson distribution of density independent of s. In each interval s_{i-1}< s < s_{i}, the magnetic field takes a constant value . The are random variables with a probability distribution function independent of s. Hence a KAP is fully characterized by a probability distribution function and a correlation length here defined as . We recall that for a Poisson distribution of density , the probability of having r jumps in an interval of length L is . Since is a KAP, any element of the Zeeman propagation matrix is also a KAP.
The absence of memory of the Poisson process implies that a KAP is a Markov process (see the definition after Eq. (7)). The Markov property and the fact that is piecewise constant are the two properties which allow us to obtain an integral equation for the mean propagation operator. In addition, because and are chosen independent of s, the KAP is a stationary process (unconditioned statistical properties are invariant under space translations). As a consequence, the integral equation for the mean propagation operator is of the convolution type (see Eq. (10)). Examples of integral equations, which are not of the convolution type because the stationarity assumption has been relaxed, are given in Sect. 5.
The mean value can be calculated by summing the contributions from realizations having N=0, N=1, N=2, etc. jumping points (e.g. Brissaud & Frisch 1971). This technique yields the mean value as sum of a series. The latter is equivalent to a Neumann series expansion of the convolution-type integral equation (see Eq. (10)). Following Brissaud & Frisch (1974; see also Auvergne et al. 1973) we show how to establish a integral equation for directly. A summation method is used in Sect. 4 to calculate the second-order moments of the Stokes parameters.
When the propagation matrix is independent of space, the propagation operator
is an exponential and depends only on the difference s-s'. Henceforth referred to as
the "static'' evolution operator and denoted by
(
stands for static), it may be written as
First we consider all the realizations without jumping point between 0 and s. For each realization
is constant in the interval [0,s] and the propagation operator is given by its static value. The probability that there is no jump in the
interval [0,s] is
.
Thus, the contribution to the mean
propagation operator from the realizations with no jump is given by:
Next we assume that there are one or several jumping points between 0 and s and denote by t the last jumping point before s. For a Poisson distribution, the probability distribution of s-t is the same as the probability distribution of the intervals between successive jumps. Hence the probability that t falls within the small interval is given by the usual Poisson formula .
The mean of the propagation operator, when there is at least one jump, is obtained by integrating its conditional mean, knowing
that the last jump falls in the small interval ,
weighted
by the probability of the conditioning event. The integral is over all
possible values of s', that is from 0 to s. (Note that the
probability that the KAP has its last jump in the small interval
is proportional to ,
but the conditional
probability is, to leading order, independent of .) The
mean of the propagation operator for the case with at least one jump
may thus be written as
Two key properties are now used: (i) the Markov property of the KAP, which guarantees that, after conditioning, the "past'' (0<t<s') and the "future'' (s'<t<s) are independent and (ii) the semi-group property (see Appendix A).
Using (i) and (ii), and the fact that the propagation operator in the
interval [s',s] is just the static one, we have
Adding the contributions from Eqs. (6) and (7), we obtain a closed convolution-type integral equation for the mean propagation operator:
Equation (10) can be solved explicitly by introducing
the Laplace transforms,
In principle, by performing an inverse Laplace transform on the r.h.s. of Eq. (15) we can obtain the mean propagation operator . Actually in our applications, only the Laplace transform is needed.
Returning to Eq. (4), we see that the integral in the r.h.s. is the Laplace transform of
for p=0
(see Eq. (11)). The mean value of the Stokes vector at the surface can thus be written as
Equation (16), combined with Eqs. (17) and (13), yields an explicit expression for the mean value of the Stokes vector at the surface. The sole averaging which has to be performed is the averaging over in Eq. (13).
As mentioned above, this expression has first been obtained
in L94, with a stochastic magnetic field model identical to
ours, even if it is not referred to as a KAP. The proof, which is
very elegant, starts from Eq. (4). The
integral over
is first
replaced by a sum from i=1 to
over all the intervals
[s_{i-1},s_{i}]. Elementary algebra shows that each term in the sum is of the form
The macroturbulent limit corresponds to a correlation length
going to
infinity. In this case the magnetic field is independent of optical depth but its value is
random with a probability distribution function
.
Setting
in
Eq. (17) we obtain for the macroturbulent limit,
The propagation matrix will usually contain a contribution from the
background continuum opacity which we assume here to be unpolarized.
The propagation matrix is then of the form
At the surface, the Stokes vector in the continuum is given by
Equation (16) shows that the magnetic field effects
are contained in
.
This suggests to introduce
The mean residual Stokes vector can also be written as
In the macroturbulent and microturbulent limits,
Eq. (29) reduces to
The expressions given here for the residual Stokes vector are similar
to the expressions given in Auvergne et al. (1973) for the broadening
by a turbulent velocity field. The only difference is that the line
absorption coefficient is
now a matrix instead of a simple scalar. From a numerical point of view,
it is more convenient to work with the residual Stokes vector than
with the Stokes vector itself because the averaging is done on
quantities which go to zero at large frequencies.
Remark
In the proof given above we have assumed for
simplicity that randomness in
,
and thus in ,
comes only from the magnetic field. If randomness comes from other
physical parameters and provided they are described with the same type
of random process as the magnetic field, in particular the same
correaltion length, all the theoretical results
given here will still hold, but the averaging over
must
be replaced by an averaging over a joint probability distribution function
,
where the
are
scalar or vector random parameters. This remark holds
also for the results in Sect. 4 on the
second-order moments.
Figure 1: Definition of , , and , the inclinations and longitudes of the random magnetic field vector , and of its mean . The angles and are defined with respect to the direction z along the line of sight. | |
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In this section we use Eq. (29) to study the dependence of on the correlation length of a random magnetic field with isotropic Gaussian fluctuations. We assume that the velocity field is microturbulent. Its effects can thus be incorporated in the definition of the profile and of the Doppler width. This assumption allows us to clearly identify the effects of the random magnetic field. The function is defined in Sect. 3.1 and numerical results are presented in Sect. 3.2.
To calculate
we must perform the averaging
over
of the r.h.s. in Eq. (30) where
.
For a random magnetic field with isotropic Gaussian
fluctuations,
Figure 2: Variation of the full width at half maximum of the emergent Stokes I profile with the jump frequency for various values of the line strength . The model parameters employed are ; ; ; and a=0. | |
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Figure 3: Dependence of the mean Stokes parameters on the correlation length for a weak mean magnetic field and strong turbulence (f=10). The model parameters are: , , . The mean field is in the direction of LOS. The full lines show the macro () and micro limits. The line types are: dotted (); dashed (); dot-dashed (). The long-dashed lines correspond to the Unno-Rachkovsky (UR) solution calculated with . | |
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Figure 4: Same as Fig. 3 but for moderate turbulence (f=1). The model parameters are: , , . | |
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Figure 5: Same as Fig. 4, but with the mean field perpendicular to the direction of the LOS. | |
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The effects of the random magnetic field are controlled by two parameters:
The numerical method for averaging over is described in Paper I, where it is applied to the calculation of . Although the expressions here are somewhat more complicated, the same technique can be applied. The averaging involves a triple integration over the variables , and . The y-integration requires some care. It is performed using a Gauss-Legendre quadrature formula with 10 to 30 points in a range . We have choosen for y_{0}<1 and for y_{0}>1. The mean residual Stokes parameters are calculated in a frequency-bandwidth with . All the calculations reported here are performed with a damping parameter a=0. In Paper I it is shown that the elements of are not very sensitive to the value of a, unless it becomes larger than 0.1.
Equation (29) shows that involves the parameter and the ratio . When is small, and a fortiori , Eq. (28) (or (30)) shows that can be neglected compared to the identity matrix. Hence, for small values of , . Therefore for weak lines, the Stokes parameters depend only on (through ) in the region of the line formation. For lines sensitive to the value of , the microturbulent regime is reached when , i.e. when the correlation length has a line optical depth smaller than unity. These remarks are illustrated in Fig. 2 which shows , the full frequency width at half-maximum of (the mean value of Stokes I), for different choices of and . We have assumed and = 1, in order to have for Stokes I a single well defined peak allowing for an unambiguous definition of . Figure 2 clearly shows that the dependence on increases with and that the microturbulent regime, indicated by the fact that reaches a constant value, sets in at roughly .
Figure 6: Dependence of the mean Stokes parameters on the correlation length of the magnetic field for a strong mean field and weak turbulence (f=1/3). The model parameters are: , , . The mean field is in the direction of the LOS. The line types have the same meaning as in Fig. 3. | |
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Figure 7: Same as Fig. 6, but with the mean field perpendicular to the direction of the LOS. | |
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Numerical results illustrating the dependence of , and , the Stokes I, Q and V components of , are shown in Figs. 3 to 9 for different values of (10 and 100) and different magnetic field parameters. To simplify the notation, without risk of confusion, we have omitted the subscript "KA'' and the value for the components of . In all the figures , which means that the random magnetic field broadening is of the same order as the broadening by the combined thermal and turbulent velocities. Hence the strength of the fluctuations is always . For comparison we also show the Unno-Rachkovsky solution calculated with the mean field , henceforth referred to as the mean Unno-Rachkovsky solution and denoted UR. The relative variation between the micro and macroturbulent limits are evaluated by considering the ratio where the subscript X stands for I, Q or V.
(i) Behavior of . All Figs. 3 to 9 clearly show that the profiles corresponding to a finite value of lie, as expected, between the microturbulent and macroturbulent limits, with the microturbulent profiles being at all frequencies broader than the macroturbulent ones, especially around the frequencies corresponding to the -components. When (Figs. 3 to 7), the relative variations, measured with , are between 10% and 20% at line center and also in the -components, when the latter are well separated. The main trend at line center is an increase of with . The value of f seems to be essentially irrelevant. In Fig. 6, where the mean field is longitudinal, shows an unpolarized -component created by the angular averaging of the factor in the -component of the absorption coefficient (see Paper I). The strength of this component is very sensitive to the angular distribution of the magnetic field fluctuations.
When (Figs. 8 and 9), deviates strongly from the UR solution. When is longitudinal (Fig. 8), a peak appears at line center and its value is almost independent of the correlation length. As shown by the Unno-Rachkovsky solution the central component behaves essentially as , with the mean value of the absorption coefficient. At line center, when the magnetic field is random, becomes much larger than its deterministic counterpart calculated with . Hence when is fairly large, the value of the central peak may approach unity. When is in the transverse direction, one observes drastic changes between the macroturbulent and microturbulent limits which can also be explained in terms of the behavior of .
(ii) Behavior of . A striking feature (see Figs. 3, 4, 6, 8) is the strong deviation from the UR solution for strong and moderate turbulence (see Figs. 3 and 4 with f=10 and f=1) while for weak turbulence, stays very close to the UR solution (see Figs. 6 and 8 with f=1/3). The relative variations between the micro and macroturbulent limits seems to be largely independent of the value of f. They are always smaller than 10% and in general smaller than the variation of at line center, except for the case of Fig. 3 where they are both of the same order and slightly less than 10%. It thus seems that , can be calculated with the microturbulent limit, with reasonable confidence, ignoring the correlation length of the magnetic field.
Figure 8: Dependence of the mean Stokes parameters on the correlation length of the magnetic field for a strong line: . The other model parameters are and . They are the same as in Fig. 6 and correspond to a weak turbulence case (f=1/3). The mean field is in the direction of the LOS. The line types have the same meaning as in Fig. 3. | |
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Figure 9: Same as Fig. 8, but with the mean field perpendicular to the direction of the LOS. | |
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(iii) Behavior of . Figures 5, 7, 9 show a strong deviation from the UR solution which decreases when the strength of the turbulent fluctuations decreases. For , at line center reaches 75% when f=1 but decreases to 20% when f=1/3. For this value of , one can observe that the line center is more sensitive to the correlation length than the -components. For and although f=1/3 only (see Fig. 9), is very sensitive to the correlation length, at line center and also in the wings. At line center, is bounded by the macro and microturbulent limits, but in the -components the behavior is not so simple because the position of the peaks moves away from the line center when increases. The maximum depth of the -components stays however always above the macroturbulent value. Finally we remark that for weak fluctuations (f=1/3), will depart more from the UR solution than (compare Figs. 6 and 7).
All the figures shown in this section confirm the remark that microturbulence is reached when .
We now examine the fluctuations of the Stokes parameters around their
mean values. For each Stokes parameter, we consider the square of
the dispersion,
Second-order moments are investigated in Brissaud & Frisch (1974) for systems of linear stochastic equations, but only for homogeneous systems or systems with a white noise inhomogeneous term. Here we show that explicit expressions for second-order moments can also be obtained for inhomogeneous systems with a constant inhomogeneous term. Our method is inspired by Brissaud & Frisch (1974).
When the source function
varies linearly with optical depth, one can easily obtain a vector transfer equation with a constant inhomogeneous term. It suffices to introduce
the new unknown vector
In Sect. 4.1, we use Eq. (38) to establish a transfer equation for the tensor product and solve it for . In Sect. 4.2 we establish an explicit expression for by a summation method and use it in Sect. 4.3 to illustrate the dependence of the dispersion on the correlation length and strength of the magnetic field fluctuations.
To calculate the dispersions
,
we need only
,
however the latter cannot be calculated
independently of the other
.
We therefore introduce the tensor product
It follows from Eq. (37) that
satisfies the transfer equation
The Green's function (or propagation operator)
associated to Eq. (44) satisfies
In terms of
,
the solution of Eq. (44) at the surface may be written as
In this section we show that the average of
over all the realizations of the KAP can be written in the form
Proof
Taking the average of Eq. (50), we see that
We now consider an interval [0,s'], and examine all
the realizations of the KAP. We characterize them by the number of
jumping points N in the interval [0,s']. We stress that s' varies
from s to ,
while s varies from 0 to .
In
Sect. 2.3 we have already introduced the
elements needed here, namely that the probability to have no jump in
an interval of length L is
and that the probability
to have a jump in a small interval
around s_{i} is
.
The proof is based on the remarks that
and
satisfy a semi-group property and that they can be replaced by their static values if there is no jumping points between s and s'.
For N = 0, we have no jump in [0,s'] hence no jump in [0,s] and [s,s'], so we can replace
and
by
and
,
respectively. We can thus write
For N = 1, we have one jump, say at a point s_{1}, within an interval
,
which can lie in either one of the intervals [0,s] or [s,s']. We consider
the two cases separately.
Case (a):
0<s_{1}<s<s'
First we use the semi-group property to write
The integrations over s_{1}, s and s' can be carried out explicitly
in terms of the Laplace transforms
and
.
The integral over s' is already
a Laplace transform. Changing the order of integration, the integral
can be transformed into
.
We thus obtain
Case (b):
0<s<s_{1}<s'
Since s_{1} is to the right of s, we now write
Summing all the contributions from N=0 to infinity, we find the result given in Eq. (54) for the matrices and . The central term corresponds to the interval [s_{-},s_{+}], the term to its right contains the contributions of all the intervals to the right of s_{+} and the term to its left the contributions of all the intervals between 0 and s_{-}.
We can now write an explicit expression for
.
Since
we have assumed that the line and continuum source
functions are unpolarized,
and
.
Hence, only the
first column in the matrix
will contribute to
.
For Stokes I and V we thus have
In the microturbulent and macroturbulent limits, the expressions for
the dispersion of the Stokes parameters are simpler. In the
microturbulent limit, the dispersion is simply zero since all the
coefficients in the transfer equation are replaced by their mean
values. One is actually dealing with a deterministic problem. In the
macroturbulent limit the second order moments can be deduced from the
Unno-Rachkovsky solution which leads to
We checked the result given in Eqs. (52) to (56) by applying our summation method to a scalar transfer equation where the propagation matrix is replaced by an absorption coefficient K. For this scalar problem, the second-order moment can also be calculated with a method introduced by Bourret et al. (1973) which relies on the introduction of new quadratic dependent variables, chosen in such a way that they satisfy a homogeneous system of linear stochastic equations. This method, restricted to scalar problems, has been applied by Auvergne et al. (1973) for the broadening of spectral lines by a turbulent velocity field.
Once the problem of calculating the second-order moments of has been reduced to the calculation of the mean value of the r.h.s. in Eq. (50), it is very likely that methods somewhat different from the summation method presented here can be set up. In particular L94 method should work, although it could be algebraically somewhat cumbersome since it does not make direct use of the Laplace transform of the evolution operator.
To calculate the dispersion of the Stokes parameters we must evaluate the elements of the matrix . The averages over (see Eq. (54)) are performed with Gauss-Legendre quadratures. The integration over the magnetic field intensity can be carried out with the same grid points as for the calculation of the mean Stokes parameters (see Sect. 3.2). The angular integrations over the polar angles and require more refined grids. Typically one needs around 30 points to calculate the dispersion while 10 or less are enough for the mean values. We note also that the width of the frequency domain must be significantly increased.
In the macroturbulent limit, the calculation of the dispersion is much
simpler since
We note also that all the results obtained for the second-order moments of the Stokes parameters hold for the residual Stokes parameters, provided we divide them by B_{1}^{2} (see Eq. (52)).
Figure 10 shows , ( ) for the four Stokes parameters and different values of (we use the subscript the same convention as in Sect. 3.2). The magnetic field parameters are and as in Figs. 4 and 5 and hence correspond to a case of moderately strong fluctuations (f=1). The direction of the mean field is and . Comparing with Figs. 4 and 5 where and or , we see that has become much smaller, as expected, and has become almost insensitive to the value of (on the scale of Fig. 10). Of course, has also become somewhat smaller and remains almost independent of . For , the dependence on does seem to depend on the direction of the mean field.
Figure 10: Mean Stokes parameters with dispersion for moderately strong fluctuations of the magnetic field. The model parameters are: , , . The direction of the mean magnetic field is defined by and . The full lines show the mean profiles and the discontinuous lines the mean values plus and minus the square root of the dispersion. The line types are: dotted (); dot-dashed (); triple-dot-dashed (). In this figure stands for . | |
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Figure 11: Variation of the dispersion with the strength of the magnetic field fluctuations. The dispersion is shown only for the macroturbulent limit . The line strength and the mean magnetic field parameters are the same as in Fig. 10. The curves are labeled with the value of f. | |
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In contrast with the mean values, we see that the are very sensitive to the value of . They have their largest values in the macroturbulent limit () and go to zero in the microturbulent limit. In the macroturbulent limit, the dispersion is quite large compared to the mean value. For the mean Stokes profiles, we have seen that the microturbulent limit is essentially reached when . Figure 10 shows that the dispersion has still a significant value when . This makes the dispersion much more sensitive to the characteristic scale of the random magnetic field.
Figure 11 shows the macroturbulent limit of calculated with Eq. (70) for (as in Fig. 10) and different values of f varying between 0.5 and 4. For Stokes I, Q and U the dispersion has maxima at line center and at the frequencies corresponding to the inflexion points in the Stokes I profile. For Stokes V, the dispersion is zero at line center for symmetry reason, and has its maximum at the inflexion points of I also. The minima of and correspond to the zero-crossing frequencies in the mean Stokes profiles.
Starting from a case of weak fluctuations (f=0.5), we observe that the dispersion increases with f, as expected, until say f=2. For larger values of f, we observe a decrease of the peak value in the wings of , and , associated to a significant broadening which reflects the fact that the stronger the fluctuations of the magnetic field, the further out from line center can they be felt. At line center and keep increasing with f, even beyond f=2. Numerical experiments, not presented, here indicate that and saturate to a value around 0.25 but that this phenomenon is related to the choice of . When the random magnetic field has a fixed direction and varies in intensity only, the values of and at line center will decrease after going through a maximum. For Stokes I, the dispersion has a fairly complicated behavior, specially around the line center. The initial increase is also followed by some kind of saturation, but again, this is related to the choice of . In the wings, the behavior is essentially the same as for the other Stokes parameters. A more detailed analysis of the dispersion is deferred to a subsequent paper.
When some of the assumptions that were introduced to obtain explicit expressions for the mean Stokes parameters are dropped, it may still be possible to write an integral equation for the mean propagation operator. In some cases this equation can still be solved explicitly by a Laplace transform method, but in general a numerical solution is required. A few examples are given below.
It follows from the solution of Eq. (1) (see
Eq. (A.11)) that the mean value
of the Stokes vector at the surface can be written as
Let us consider an example presented in LL04 (p. 419) in which the continuum
and line source functions are different and both have exponential
terms. The transfer equation is now of the form
When contains an exponential, it does not seem possible to transform the original transfer equation into a new equation with a homogeneous source term and obtain with the method described in Sect. 4 an explicit expression for the dispersion around the mean Stokes parameters.
We now assume that the line and continuum source functions and the ratio
(introduced in
Sect. 2.6) can vary with optical
depth, but not the Zeeman propagation matrix
.
This
implies that the Doppler width is taken constant. There is no hope to
obtain an exact result for the mean Stokes parameters, however
an expression given in Pecker & Schatzman (1959) for the difference
I_{c}(0) -I(0), in the case of non-polarized transfer, could be a good starting point for their numerical calculation.
For the polarized case, the expression given in the above reference becomes
The mean value still satisfies Eq. (10) but the static propagation operator, as shown by Eq. (78), is now a function of s and s'.
In the preceding sections, it has been assumed that ,
the
density of the Poisson distribution, is independent of the optical
depth s along the line of sight. If we let
vary with depth,
the Poisson process becomes a non-homogeneous Poisson
process^{}. The probability that no jumps occur between s and s' is
.
Equation (10) becomes
Some other generalizations can still lead to convolution equations for the mean evolution operator. For example, if depends on the modulus of the random magnetic field or if the random magnetic field consists of several fields with different characteristic scales. Such generalizations have been considered for the statistical Stark effect (Brissaud & Frisch 1971).
The results given in the previous sections hold for an outward directed ray normal to the surface of the atmosphere. They can easily be extended to the case of a ray making an angle with the vertical. It suffices to project on to the line of sight the quantities which describe the variations of the model along the normal to the atmosphere, such as the source function, absorption coefficients, and correlation length.
For the example treated in Sect. 5.1,
will be given by Eq. (75) with B_{1} changed to
and
and
changed to
and
,
where
.
For the linear source function
treated in Sect. 2.6,
the usual residual Stokes vector
will be given by Eq. (27) multiplied by
.
For the example treated in
Sect. 5.2, w(s) and
become
For the calculation of , the correlation length should also be projected along the line of sight, which means transforming into . Thus in Eqs. (27)-(29), should be changed to . This is also the change made in LL04 (see Eq. (9.280), p. 500), where , the mean length of the eddies measured in the vertical direction, becomes along the line of sight. As a consequence, the more inclined with respect to the vertical are the rays, the closer is one to a macroturbulent type of averaging. This is consistent with a picture of random fluctuations organized in turbulent layers. Now, even in a plane parallel atmosphere, one may want to have a more or less isotropic distribution of turbulent eddies. This can be achieved by keeping the same value of (i.e. same correlation length) in all directions.
This paper presents the first detailed investigation of the Zeeman effect created by a random magnetic field with a finite correlation length. The goal of this work is to overcome usual treatments whereby the correlation length of the magnetic field is either much smaller, or much larger, than a photon mean free-path, i.e. the microturbulent and macroturbulent limits. The random magnetic field is described by a Kubo-Anderson process which takes constant but random values on intervals of random length distributed according to a Poisson distribution of density . The random magnetic field is thus characterized by a mean correlation length defined here as and the probability distribution function of the random values taken by the magnetic field. The micro and macroturbulent limits are recovered when the correlation length goes to zero or infinity.
The Kubo-Anderson process has been associated to a Milne-Eddington atmospheric model with a linear source function. This combination has allowed us to construct explicit expressions that were used to study numerically the mean Stokes parameters and their dispersion at the surface of the atmosphere. The main theoretical results concern the construction of:
Numerical investigations have been carried out for a probability distribution function describing a random magnetic field with mean value and isotropic Gaussian fluctuations with dispersion . We have assumed a microturbulent velocity with a Gaussian distribution which is equivalent to incorporating an additional thermal broadening into the Doppler width of the line. In agreement with the Milne-Eddington model, the ratio of the line to continuum opacity has been taken constant. For weak lines ( order of unity or less), the Stokes parameters are essentially given by the profiles of the absorption coefficients and hence depend only on . For stronger lines, sensitive to the correlation length of the magnetic field, the mean Stokes parameters lie between the micro and macroturbulent limits. This is strictly true for Stokes I, because it is a positive quantity, and at line center for Stokes Q and U. It is a bit more complicated for Stokes V and the -components of Stokes Q and U, because the position of the peaks depend on the correlation length. The microturbulent limit is reached when the correlation length is around unity in the line optical depth unit, i.e. when .
The numerical calculations have been performed for (a few cases with have also been considered) for different values of the mean magnetic field , dispersion and correlation length . The dispersion and mean field have been combined to construct a dimensionless parameter which measures the relative strength of the magnetic field fluctuations. The assumption that the magnetic field fluctuations are isotropic influences some of the results but not the general trends which are summarized here.
Concerning the mean values, we have found that:
In addition to the magnetic field, a whole set of other atmospheric random parameters (velocities, temperatures, densities, ) are needed to properly describe a distribution of flux tubes or magnetohydrodynamic turbulence. These additional parameters should typically be described by the same type of random processes as the magnetic field, in particular the same correlation length. In this case all the theoretical results given here will hold, provided is replaced by a joint distribution function , where the stand for the other random parameters. If the random parameters have different correlation lengths, a KAP-type of modeling can still be set up. An example can be found in the case of the stochastic Stark effect (Brissaud & Frisch 1971). A composite KAP is introduced to handle simultaneously the ion and electron electric fields with their quite different characteristic lengths due to the large mass difference between the two types of particles.
Acknowledgements
M.S. is financially supported by the Council of Scientific and Industrial Research (CSIR), through a Junior Research Fellowship (JRF Grant No: 9/890(01)/2004-EMR-I), which is gratefully acknowledged. M.S. is also grateful to the Indo-French Sandwich Thesis Program for making possible a visit to the Observatoire de la Côte d'Azur. 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 visits 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 is grateful to J. Sánchez Almeida for stimulating discussions. The authors have also benefitted from constructive remarks from a referee.
For the benefit of the reader we recall here some of the main
properties of the radiative transport operator for polarized transfer
(Landi Degl'Innocenti 1987). The homogeneous transfer equation
associated to Eq. (1) of the text may be written as
The Green's function, also called evolution or transport or
propagation operator, is here defined by
and
When the propagation matrix is a constant, the evolution operator
is given by