A&A 401, 835-848 (2003)
DOI: 10.1051/0004-6361:20030172
T. A. Enßlin - C. Vogt
Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str.1, Postfach 1317, 85741 Garching, Germany
Received 30 July 2002 / Accepted 7 February 2003
Abstract
The autocorrelation function and similarly the Fourier-power spectrum
of a rotation measure (RM) map of an extended background radio source
can be used to measure components of the magnetic autocorrelation and
power-spectrum tensor within a foreground Faraday screen. It is
possible to reconstruct the full non-helical part of this tensor in
the case of an isotropic magnetic field distribution statistics. The
helical part is only accessible with additional information; e.g. the
knowledge that the fields are force-free. The magnetic field strength,
energy spectrum and autocorrelation length
can be obtained
from the non-helical part alone. We demonstrate that
can
differ substantially from
,
the observationally easily
accessible autocorrelation length of an RM map. In typical
astrophysical situation
.
Any RM study, which does not take this distinction
into account, likely underestimates the magnetic field strength.
For power-law magnetic power spectra, and for patchy magnetic
field configurations the central RM autocorrelation function is shown
to have characteristic asymptotic shapes. Ways to constrain the volume
filling factor of a patchy field distribution are discussed.
We discuss strategies to analyse observational data, taking into
account - with the help of a window function - the limited extent of
the polarised radio source, the spatial distribution of the electron
density and average magnetic energy density in the screen, and
allowing for noise reducing data weighting. We briefly discuss the
effects of possible observational artefacts, and strategies to avoid
them.
Key words: magnetic fields - radiation mechanism: non-thermal - galaxies: active - intergalactic medium - galaxies: clusters: general - radio continuum: general
The interstellar and intergalactic plasma is magnetised. The origin of the magnetic fields is partly a mystery, yet it allows fascinating insights into dynamical processes in the Universe. Magnetic fields are an important constituent of cosmic plasma in so far as they couple the often collisionless charged particles by the Lorentz-force. They are able to inhibit transport processes like heat conduction, spatial mixing of gas, and propagation of cosmic rays. They are essential for the acceleration of cosmic rays. They mediate forces through their tension and pressure, giving the plasma additional macroscopic degrees of freedom in terms of Alfvénic and magnetosonic waves. They allow distant cosmic ray electron populations to be observed by magneto-curvature (synchrotron) radiation.
Observational studies of spiral galaxies have revealed highly organised magnetic field configurations, often in alignment with the optical spiral arms. These magnetic fields are believed to be generated and shaped by the dynamo action of the differentially rotating galaxy disks from some initial weak seed fields.
The seed fields could have many origins, ranging from outflows from stars and active galactic nuclei over battery effects in shock waves, in ionisation fronts, and in neutral gas-plasma interactions, up to being primordially generated in high energy processes like phase transitions or inflation during the very early Universe.
In order to learn more about the magnetic field origin, less processed plasma has to be studied. There is the possibility that the magnetic fields outside galaxies, in galaxy clusters and - if existing - even in the wider intergalactic space carry more information on the field's origin. In clusters, magnetic fields with a much lower degree of ordering, compared to the organised fields in spiral galaxies, have been detected. However, they may be highly processed by turbulent gas flows driven by galaxy cluster mergers, which may mask their origin. Regardless, cluster magnetic fields are an interesting laboratory to study magneto-hydrodynamical (MHD) turbulence, and are of great importance to understand thermal and non-thermal phenomena in the intra-cluster medium.
Despite their obvious importance for many astrophysical questions, and despite many observational efforts to measure their properties, our knowledge of galactic and intergalactic magnetic fields is still poor. For an overview on the present observational and theoretical knowledge the excellent review articles by Rees (1987), Wielebinski & Krause (1993), Kronberg (1994), Beck et al. (1996), Kulsrud (1999), Beck (2001), Grasso & Rubinstein (2001), Carilli & Taylor (2002), and Widrow (2002) should be consulted.
One way to probe magnetic fields is to use the Faraday rotation effect. Linearly polarised radio emission experiences a rotation of the polarisation plane when it transverses a plasma with a non-zero magnetic field component along its propagation direction. If the Faraday active medium is external to the source, a wavelength-square dependence of the polarisation angle measured can be observed and used to obtain the RM, which is the proportionality constant of this dependence. Such situations are realised in nature in cases where a polarised radio galaxy is located behind the magnetised medium of a galaxy, or behind or embedded in a galaxy cluster.
The focus of this work is on the analysis of RM maps of Faraday screens, in which the fields are statistically isotropically distributed. This should be approximately fulfilled in galaxy clusters, but not in the highly organised spiral galaxies. However, our analysis should also give some insight into the statistics of RMmaps of galaxies, since many of the results do not strictly require perfect isotropy.
Magnetic fields in galaxy clusters are known to exist due to detection of cluster wide synchrotron emission (Willson 1970), and detection of their Faraday rotation effect. Although the association of the RM with the intra-cluster medium is not unambiguous, since it could also be produced in a magnetised plasma skin of the observed radio galaxy (Bicknell et al. 1990), there are arguments in favour of such an interpretation: (i) the asymmetric depolarisation of double radio lobes embedded in galaxy clusters can be understood as resulting from a difference in the Faraday depth of the two lobes (Laing 1988; Garrington et al. 1988). (ii) A recent RMstudy by Clarke et al. (2001) of point sources located mostly behind (but 40% inside) galaxy clusters show a larger dispersion in RM values than a reference sample without a galaxy cluster intersecting the line-of-sight. (iii) The cluster-wide radio halos observed in some clusters of galaxies (e.g. Feretti 1999) show synchrotron emission of relativistic electrons within magnetic fields. The cluster fields strength should be within an order of magnitude of their Faraday rotation estimates for the radio emitting electrons to have a reasonable energy density (compared to the thermal one) (e.g. as can be read off Fig. 1 in Enßlin & Biermann 1998).
Typical RM values of galaxy clusters are of the order of a few 100 rad/m^{2}, being consistent with field strengths of a few G which are well below equipartition with the thermal cluster gas. However, in cooling flow clusters extreme RM values of a few 1000 rad/m^{2} were detected (see Carilli & Taylor 2002), indicating possibly substantial magnetic pressure support of the intra-cluster gas there.
Although the magnetic fields of galaxy clusters are less ordered than these of spiral galaxies, the presence of coherent structures is suggested by high resolution Faraday maps, which exhibit sometimes RMbands (e.g. Dreher et al. 1987; Taylor & Perley 1993; Taylor et al. 2001; Eilek & Owen 2002). Such bands may be caused by shear-amplification of originally small-scale magnetic fields, as seen in numerical MHD simulations of galaxy cluster formation (Dolag et al. 1999). They are likely embedded within a magnetic power-spectrum which extends over several orders of magnitude in wavevector space. Similar to hydrodynamical turbulence, a broad energy injection range is followed by a power-law spectrum at larger wavevectors. For attempts to measure the magnetic power spectrum from cluster simulations and radio maps see Dolag et al. (2002) and Govoni et al. (2002) respectively^{}.
Another area of application of the theory developed here can be to measure the properties of an hypothetical large-scale magnetic field outside clusters of galaxies, which could be of primordial origin. A pioneering feasibility study in this direction was done by Kolatt (1998), who already outlined several of the ideas investigated in this work. He proposed to probe the cosmological magnetic fields by using catalogues of RM measurements of distant radio galaxies and to measure spatial RM correlations between them in order to measure the magnetic power spectrum, as we propose to do for extended Faraday rotation maps. An RM search for fields in the Lyman- forest by Oren & Wolfe (1995) found at most a marginal detection. If they exist, primordial magnetic fields may be detectable by Faraday rotation of the cosmic microwave background (CMB) polarisation during and shortly after the epoch of recombination, as Kosowsky & Loeb (1996) proposed.
Ohno et al. (2002) proposed to use the CMB polarisation even for RMstudies of nearby galaxy clusters. Should this speculative proposal become technically feasible, a lot of detailed information on intra-cluster magnetic fields could be obtained.
If magnetic fields are sampled in a sufficiently large volume, they can hopefully be regarded to be statistically homogeneous and statistically isotropic. This means that any statistical average of a quantity depending on the magnetic fields does not depend on the exact location, shape, orientation and size of the used sampling volume.
The quantity we are interested in this paper is the autocorrelation (or two-point-correlation) function (more exactly: tensor) of the magnetic fields. The information contained in the autocorrelation function is equivalent to the information stored in the power-spectrum, as stated by the Wiener-Khinchin Theorem (WKT). We therefore present two equivalent approaches, one based in real space, and one based in Fourier space. The advantage of this redundancy is that some quantities are easier accessible in one, and others in the other space. Further, this allows to crosscheck computer algorithms based on this work by comparing results gained by the different approaches.
The observable we can use to access the magnetic fields is Faraday rotation maps of extended polarised radio sources located behind a Faraday screen. Since an RM map shows basically the line-of-sight projected magnetic field distribution, the RM autocorrelation function is mainly given by the projected magnetic field autocorrelation function. Therefore measuring the RM autocorrelation allows to measure the magnetic autocorrelation, and thus provides a tool to estimate magnetic field strength and correlation length.
The situation is a bit more complicated than described above, due to the vector nature of the magnetic fields. This implies that there is an autocorrelation tensor instead of a function, which contains nine numbers corresponding to the correlations of the different magnetic components against each other, which in general can all be different. The RM autocorrelation function contains only information about one of these values, the autocorrelations of the magnetic field component parallel to the line-of-sight. However, in many instances the important symmetric part of the tensor can be reconstructed and using this information the magnetic field strength and correlation length can be obtained. This is possible due to three observations:
The same magnetic power spectrum can have very different realizations, since all the phase information is lost in measuring the power spectrum (for an instructive visualisation of this see Maron & Goldreich 2001). Since the presented approach relies on the power spectrum only, it is not important if the magnetic fields are highly organised in structures like flux-ropes, or magnetic sheets, or if they are relatively featureless random-phase fields, as long as their power spectrum is the same.
The autocorrelation analysis is fully applicable in all such situations, as long as the fields are sampled with sufficient statistics. The fact that this analysis is insensitive to different realizations of the same power spectrum indicates that the method is not able to extract all the information which may be in the map. Additional information is stored in higher order correlation functions, and such can in principle be used to make statements about whether the fields are ordered or purely random (chaotic). The information on the magnetic field strength (, which is the value at origin of the autocorrelation function), and correlation length (an integral over the autocorrelation function) does only depend on the autocorrelation function and not on the higher order correlations.
The presented analysis relies on having a statistically isotropic sample of magnetic fields, whereas MHD turbulence seems to be locally inhomogeneous, which means that small scale fluctuations are anisotropic with respect to the local mean field. However, whenever the observing window is much larger than the correlation length of the local mean field the autocorrelation tensor should be isotropic due to averaging over an isotropic distribution of locally anisotropic subvolumes. This works if not a preferred direction is superposed by other physics, e.g. a galaxy cluster wide orientation of field lines along a preferred axis. However, even this case can in principle be treated by co-adding the RM signal from a sample of clusters, for which a random distribution of such hypothetical axes can be assumed. In any case, it is likely that magnetic anisotropy also manifests itself in the Faraday rotation maps, since the projection connecting magnetic field configurations and RM maps will conserve anisotropy in most cases, except alignments by chance of the direction of anisotropy and the line-of-sight. The presence of anisotropy can therefore be tested, which is discussed later in great detail.
Since there are cases where already an inspection by eye seems to reveal the existence of magnetic structures like flux ropes or magnetic sheets, we briefly discuss their appearance in the autocorrelation and the area filling statistics of RM maps. As already stated, the presence of such structures does not limit our analysis, as long as they are sufficiently sampled. Otherwise, one has to replace e.g. the isotropy assumption by a suitable generalisation. In many cases this will allow an analysis similar to the one proposed in this paper. We leave this for future work and applications where this might be required. A criteria to detect anisotropy statistically is given in this work.
In Sect. 2 the autocorrelation functions of magnetic fields and their RM maps are introduced, and their interrelation investigated. In Sect. 3 the same is done in Fourier space, which has not only technical advantages, but also provides insight into phenomena such as turbulence. Faraday map signatures of magnetic structures like flux-ropes are briefly discussed in Sect. 4. Possible pitfalls due to observational artefacts are investigated in Sect. 5. The conclusions in Sect. 6 summarise our main findings, and give references to the important results and formulae in detail.
The Faraday rotation for a line-of-sight parallel to the z-axis and
displaced by
from it, which starts at a polarised radio source
at
and
ends at the observer located at infinity is given by
(1) |
The focus of this work is on the statistical expectation of the
two-point, or autocorrelation function of Faraday rotation maps. This
is defined by
(2) |
The RM signal from different subvolumes of the Faraday screen will
differ due to electron density and typical magnetic field strength
variations within the source. Such global variations can be regarded
as variations of a window function
,
which mediates the
relation between the observed RM signal and an underlying (rescaled)
magnetic field, which is virtually homogeneous in a statistical
sense. To be more specific, we choose a typical position
within the screen (e.g. the centre of a galaxy cluster), and define
and
.
We then define the window function
by
The expectation of the observed RM correlations are
In the following we ignore the influence of the window function in the discussion, since for sufficiently large windows it only affects a_{1}. We therefore write for and keep in mind that our measured field strength B_{0} is estimated for a volume close to the reference location . At other locations, the average magnetic energy density is given by . This approach assumes implicitly that typical length scales are the same throughout the Faraday screen. For sufficiently extended screens, this assumption can be tested by comparing results from different and separately analysed regions of the RM map.
The magnetic autocorrelation tensor for homogeneous isotropic
turbulence, as assumed throughout the rest of this paper, can be
written as
(10) |
(12) |
Now, an observational program to measure magnetic fields is obvious:
from a high quality Faraday rotation map of a homogeneous, (hopefully)
isotropic medium of known geometry and electron density (e.g. derived
from X-ray maps) the RM autocorrelation has to be calculated
(Eq. (7)). From this an Abel integration (Eqs. (14)
or (15)) leads to the magnetic autocorrelation function,
which gives
at its origin. Formally,
For isotropic magnetic turbulence statements about integrals of w(r)and
can be made. If correlations are short ranged,
in the sense that
for
,
e.g. because of a finite size of the magnetised volume, then
Eq. (12) implies
For example, the frequently used magnetic cell-model, in which cells of length-scale are filled with a from cell-to-cell randomly oriented but internally homogeneous magnetic field, does not fulfil . As a consequence it gives positive autocorrelation volumes and surfaces of the order and , respectively. Therefore it should be possible to exclude such an oversimplified model observationally.
Other integral quantities of the turbulent magnetic field to look at
are the existing non-zero magnetic and RM autocorrelation lengths.
The magnetic autocorrelation length can be defined as
An observationally easily accessible length scale is the Faraday
rotation autocorrelation length:
Having now defined two characteristic length-scales of the fields and their RM maps, suitable criteria testing the observed autocorrelation volume and area for statistical completeness can be formulated: and . We note that for strictly positive and therefore unphysical autocorrelation functions one would expect and statistically.
This can be turned around. For a sufficiently large RM map of a
Faraday rotation screen with isotropic magnetic fields any homogeneous
RM contribution from additional magnetised foregrounds can relatively
accurately be measured. If there is a weak screen-intrinsic
homogeneous magnetic field component, its z-component is
(22) |
Now, all the necessary tools are introduced to test if the window
function
was based on a sensible model for the average
magnetic energy density profile
and the proper geometry of
the radio source within the Faraday screen
(see
Eq. (4)). Models can eventually be excluded a-posteriori
on the basis of
(24) |
We note, that this method of model testing can be regarded as a refined Laing-Garrington effect (Laing 1988; Garrington et al. 1988): the more distant radio cocoon of a radio galaxy in a galaxy cluster is usually more depolarised than the nearer radio cocoon due to the statistically larger Faraday depth. This is observed whenever the observational resolution is not able to resolve the RMstructures. Here, we assume that the observational resolution is sufficient to resolve the RM structures, so that a different depth of some part of the radio source observed, or a different average magnetic energy profile leads to a different statistical Faraday depth . Since this can be tested by suitable statistics, e.g. the simple statistic proposed here, incorrect models can be identified.
It may be hard in an individual case to disentangle the effect of changing the total depth of the used polarised radio source if it is embedded in the Faraday screen, and the effect of changing , since these two parameters can be quite degenerate. However, there may be situations in which the geometry is sufficiently constrained because of additional knowledge of the source position, or statistical arguments can be used if a sufficiently large sample of similar systems were observed.
We use the following convention for the Fourier
transformation of a n-dimensional function
:
= | (26) | ||
= | (27) |
= | (29) | ||
= | |||
= | (30) |
(31) |
Figure 1: Left: for power-law spectra of magnetic fluctuations within the range k_{1} = 1 to k_{2} = 10^{4}with normalisation and different slopes s as labelled. For comparison, the thin line shows the RM correlation function expected for a single-scale power spectrum, as it can arise for isotropic linear force-free fields ( with , and ; see Sect. 3.4). Right: in logarithmic units for the same power-law spectra. The unbent lines are the asymptotic spectra given by the first two terms in Eq. (44). | |
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We recall that the power spectrum
of a function
is given by the absolute-square of its Fourier transformation
.
The WKT states that the Fourier
transformation of an autocorrelation function
,
estimated within a window with volume V_{n} (as in
Eq. (3)), gives the (windowed) power spectrum of this
function, and vice versa:
(33) |
Thus, the 3-d magnetic power spectrum (the Fourier transformed
magnetic autocorrelation function
)
can be directly connected
to the one-dimensional magnetic energy spectrum in the case of
isotropic turbulence:
Also the correlation lengths can be expressed in terms of
:
The isotropic magnetic autocorrelation function can be expressed
as
(41) |
(42) |
In many cases the small-scale magnetic energy spectrum is a power-law,
say
(e.g. s=5/3 for Kolmogorov-like
turbulence, as expected if the magnetic fields were shaped by a mostly
hydrodynamical turbulence) or
for
k_{1}<k<k_{2} (with
). For the behaviour of
on such scales Eq. (43) can be
written as
(45) | |||
(46) | |||
(47) |
The shape of the RM correlation function close to the origin allows a direct read-off of the type of magnetic turbulence. A top-down scenario, where most of the energy resides on large scales (s>1) and the smaller scales are populated by a turbulent cascade as in the Kolmogorov-, Kraichnan-, and Goldreich-Sridhar-phenomenologies, leads to a flat cusp at the origin, and a convex shape near to it. A bottom-up magnetic turbulence scenario, where the fields originate on small scales and are enlarged by shear flows or other inverse cascade actions (s<1), leads to a sharp cusp at the origin, and a concave slope next to it. A spectral energy distribution with as much energy on small as on large scales (s= 1) leads to a linear cusp at the origin. The behaviour of for these three cases is illustrated in Fig. 1.
Faraday rotation maps do not contain information about the helical part of the autocorrelation tensor. Therefore, additional information is required in order to be able to measure the helical correlations. For example any relation between the helical and non-helical components would be sufficient.
In order to give an example for such additional information, we discuss the case of force-free fields (FFFs). The condition for FFFs reads where can in general be a function of position. For simplicity, we restrict to be spatially constant. Such so-called linear FFFs lead to a very simple structure of the components of : for all components vanish, and for one gets , leaving the magnetic energy density (or the helicity) as the only remaining free parameter for a given characteristic wave-vector . FFFs are therefore also called maximally helical fields.
From the fact that for a linear FFF only one spherical shell in
wave-vector space is populated with magnetic power, the spatial
autocorrelation function is easily obtained as
(49) |
(50) |
Here, we discuss the effect of a finite window function on magnetic
field estimates, in order to possibly correct for the bias made with
the robust weighting scheme introduced in Sect. 2.1. Taking a
finite window function
into account, Eq. (32)
becomes
(52) |
In any situation in which there is substantial magnetic power on scales comparable or larger than the window size Eq. (51) can be used to estimate the response of the observation to the magnetic power on a given scale p by inserting . Ideally, this is then used within a matched-filter analysis or as the response matrix in a maximum-likelihood reconstruction of the underlying power-spectra. The computation of the response matrix relating input power and measured signal can be cumbersome since in general a 2- or 3-dimensional integral (depending if one uses the delta function) has to be evaluated for each matrix element. Therefore we restrict our discussion here to three highly symmetric, idealised cases, and an approximative treatment of a more realistic configuration, which should give a feeling for the general behaviour.
Figure 2: Response in to a single-scale magnetic power signal at wavenumber p for different radii R of a cylindrical window function. | |
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A cylindrical window: suppose a circular radio source with
radius R is seen through a very deep Faraday screen, so that the
depth L_{z} can be approximated to be infinite long. The window
function
leads to
(53) |
(54) |
Figure 3: Response in to a single-scale magnetic power signal at wavenumber p for a sheet-like window seen edge-on with diameter L_{x} = 10/p for different wavevector orientations with respect to the sheet normal. | |
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A sheet-like window seen edge-on: suppose a very elongated radio
source is seen through a deep Faraday screen, so that the window
function is approximated by
.
This
gives
(55) |
(56) |
(57) |
Figure 4: Response in to a single-scale magnetic power signal at wavenumber p for a sheet-like window seen face-on for different diameters L_{z}. | |
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A sheet-like window seen face-on: suppose the window is an
infinitely extended homogeneous layer of thickness L_{z} in z-direction,
e.g. a magnetised skin layer of a large radio source, so that
,
and
(58) |
The response function
(59) |
The observed magnetic energy density estimated with the help of
Eq. (38) can be shown to be related to the real
magnetic power spectrum via
(60) |
D(k) | = | (61) | |
(62) |
In principle, it is possible to correct for any bias, if the window function is reliably known and if the statistical sampling is sufficiently good even on the large scales so that dividing the observed magnetic power spectrum by the weighting function gives sensible results (and not just amplifies the noise).
Approximative treatment of a realistic window: in a realistic
situation, often a relatively small sized radio galaxy is seen through
a deep Faraday screen. In such a case the depth can again be
approximated to be infinite for the purpose of the Fourier-space
window:
(63) |
(64) |
(65) |
(66) |
= | (67) | ||
= | (68) |
Since isotropy of the magnetic field statistics is a crucial ingredient of the proposed analysis, it is important to test if indications of anisotropy are present, and to see how anisotropy can affect the results.
Anisotropy can manifest itself in two different ways: (a) The Fourier space magnetic power distribution can be anisotropic, by being not only a function of k but a full function of , and (b) the magnetic power tensor itself can be anisotropic. Certainly both flavours of anisotropy can be present simultaneously. However, their effects can be well separated, so that we discuss them one by one.
Before doing so, we note that the relation
(a) Anisotropic power spectrum: Eq. (69) shows that an anisotropic power spectrum can be detected, since it leads very likely to an anisotropic RM power map if the anisotropy is not aligned with the z-direction by chance. Since the latter can not be excluded, it is hard to prove isotropy. On the other hand, a perfect alignment of the line-of-sight and the anisotropy axis is not very likely. By studying a number of independent Faraday screens, an anisotropic power spectrum can be ruled out on a statistical basis. Furthermore, by co-adding the signals of several systems, statistical isotropy can be enforced, even if an individual system is anisotropic. However, in order to be able to co-add different observations, the window functions have to be well understood. Especially the scaling of the average magnetic field energy density with location within the Faraday screen and from screen-to-screen should be known. Since this is still poorly known, it is worth to check if indications of anisotropy are present in every dataset itself.
A good way to check for indications of anisotropy is by eye inspection
of maps of
or equivalently
or by
comparing profiles which were calculated using different angular
slices. A more quantitative estimate of apparent anisotropy can be
obtained by the use of multipole moments. Since the dipole moment
vanishes due to mirror symmetries in
and
,
the first non-trivial multipole is the quadrupole
moment:
(70) |
(71) |
(b) Anisotropic tensor: in the anisotropic case, the only constraint on the magnetic autocorrelation tensor is due to the divergence-freeness of the magnetic fields. This translates for the z-components into , which leaves the observable absolutely unconstrained since k_{z} = 0. can therefore be an arbitrary function of . However, if it is not circularly symmetric, this can be detected with the methods described above.
In order to have a working example of an anisotropic part
of the magnetic tensor we assume that a preferred
direction
exists, so that
(72) |
Furthermore, since any anisotropy of the magnetic power tensor should have a physical cause, e.g. a large-scale shear flow in the Faraday active medium, an accompanying anisotropic power spectrum is very likely, which can principally be detected by the methods described above (a).
Finally, if anisotropy turns out to be inherently present in Faraday screens, one might replace Eqs. (8) and (28) by a more complex, anisotropic model in order to be able to extract information from individual screens. In that case this work may help as a guideline for such a more elaborate analysis.
The possibility exists that the magnetic fields of a Faraday screen
consist of several distinct magnetic structures like flux ropes,
magnetic tori etc. If the positions and orientations of the structures
can be regarded as statistically independent the magnetic
autocorrelation function can be written as
(73) |
(74) |
For a magnetic structure, which consists of a mostly constant magnetic
field B_{s} within the volume V_{s}, and negligible field strength
elsewhere, the autocorrelation function is asymptotically for small
r
W_{s}(r) = B_{s}^{2} V_{s} ( 1 - r/l_{s} ), | (75) |
In order to calculate the RM autocorrelation of such a Faraday screen,
we use as a toy model w(r) from Eq. (76) as long as
,
and otherwise w(r)=0. Equation (17) would then
requires
,
but the actual choice is
only important for numerical values of constants of proportionality,
and not for the qualitative shape of the RM autocorrelation function
at the origin. Integrating Eq. (13) leads to an asymptotic
expansion of the form
We summarise that a Faraday screen built from structural elements with internally constant magnetic fields, and only a single characteristic length-scale leads to a flat central autocorrelation function, with at most a logarithmic cusp of the form given by Eq. (77).
Although there exist characteristic shapes of the RM autocorrelation function in the case of a patchy magnetised Faraday screen, as demonstrated in Sect. 4.1, the presence of such patches cannot be deduced from alone. Since the phase information is missing, the special form of the cusp arising from magnetic structures as given by Eq. (77) can not be distinguished from a complete random phase turbulence with steep power-law like spectra with spectral index , as can be seen from comparison with Eq. (48).
In order to measure the patchiness of the magnetic field distribution in galaxy clusters Clarke et al. (2001) used the area filling factor of the line-of-sight of extended radio sources which do not show any RM due to the Faraday screen. For their sources, they concluded that %.
If the magnetic fields are in flux-rope like structures, with typical
length
and diameter ,
the cross section of a flux-rope
to be intersected by a line-of-sight is of the order
.
Their volume filling factor is
.
If their locations can be regarded as
being uncorrelated, the number K of flux ropes intersected by a
line-of-sight of length
is Poisson-distributed:
with
.
From that it follows by inserting K=0 that
(78) |
Another constraint for the magnetic filling factor can be obtained
from energetic arguments. The magnetic field energy density in
magnetised regions can be expected to be below the environmental
thermal energy density
,
since otherwise a magnetic
structure would expand until it reaches pressure equilibrium. Since
the autocorrelation analysis of RM maps is able to provide the volume
averaged magnetic field energy density
,
the magnetic
volume filling factor can be constrained to be
(79) |
The finite size of a synthesised beam of a radio interferometer should smear out RM structures below the beam size, and therefore can lead to a smooth behaviour of the measured RMautocorrelation function at the origin, even if the true autocorrelation function has a cusp there. Substantial changes of the RM on the scale of the beam can lead to beam depolarisation, due to the differentially rotated polarisation vectors within the beam area (Conway & Strom 1985; Laing 1988; Garrington et al. 1988). Since beam depolarisation is in principal detectable by its frequency dependence, the presence of sub-beam structure can be noticed, even if not resolved (Tribble 1991; Melrose & Macquart 1998). The magnetic power spectrum derived from a beam smeared RM map should cut-off at large .
Instrumental noise can be correlated on several scales, since radio
interferometers sample the sky in Fourier space, where each antenna
pair baseline measures a different -vector. It is difficult
to understand to which extend noise on a telescope antenna baseline
pair will produce correlated noise in the RM map, since several
independent polarisation maps at different frequencies are combined in
the map making process. We therefore discuss only the case of
spatially uncorrelated noise, as it may result from a pixel-by-pixel
RM fitting routine. This adds to the RM autocorrelation function
(80) |
(81) |
If it turns out that for an RM map with an inhomogeneous noise map (if provided by an RM map construction software) the noise affects the small-scale power spectrum too severely, one can try to reduce this by down-weighting noisy regions with a suitable choice of the data weighting function which was introduced in Sect. 2.1 for this purpose.
An RM map is often derived by fitting the wavelength-square behaviour of the measured polarisation angles. Since the polarisation angle is only determined up to an ambiguity of (where n is an integer), there is the risk of getting a fitted RM value which is off by from the true one. m is an integer, and is a constant depending on the used wavelength range from to .
This can lead to artifical jumps in RM maps, which will affect the RMautocorrelation function and therefore any derived magnetic power
spectrum. In order to get a feeling for this we model the possible
error by an additional component in the derived RM map:
(82) |
(83) |
Comparing this with Eq. (48) shows that the artificial power induced by the -ambiguity mimics a turbulence energy spectrum with slope s=1, which would have equal power on all scales. A steep magnetic power spectrum can therefore possibly be masked by such artefacts.
Fortunately, for a given observation the value of is known and one can search an RM map for the occurrence of steps by over a short distance (not necessarily one pixel) in order to detect such artefacts.
We have investigated the statistics of Faraday rotation maps on the level of the autocorrelation function and the power spectrum. We proposed ways to analyse extended Faraday maps in order to reconstruct the magnetic autocorrelation tensor (Eqs. (6) and (34)) from which quantities like the average field strength, the magnetic energy spectrum, and their autocorrelation length can be obtained (Eq. (20)). We showed that under the assumption of isotropy of the observed magnetic field ensemble the symmetric part of the magnetic autocorrelation tensor (Eqs. (8) and (28)) can be reconstructed. This makes use of the condition and the additional assumption (which can be tested a-posteriori) that the gradient scale of the electron density (e.g. the core radius of a galaxy cluster) is much larger than the typical field length-scale. The anti-symmetric or helical part of the magnetic correlation tensor can only be measured if additional information is available, e.g. in the case of force-free fields (Sect. 3.4).
The assumption of isotropy of the magnetic field statistics should be justified in cases where a sufficiently large volume of the screen is probed. In principle, it can also be tested by searching for non-circular distortions of the 2-dimensional autocorrelation function (Sect. 3.6).
A further test for statistical isotropy and sufficient sampling of the field fluctuations is the fact that if these conditions are given in a finite Faraday screen (which cannot maintain infinitely long correlations) the rotation measure (RM) autocorrelation area (Eq. (18)) has to vanish. This means that there is a balance between the positively and negatively valued areas of the autocorrelation function. In practice, one would require it to be much smaller than the RM autocorrelation length squared. We note that e.g. the popular magnetic cell-model, in which cells are filled by from cell-to-cell independently oriented and internally homogeneous magnetic fields, does not have these properties, since it violates the required condition.
Our approach is meant to be applied directly to real data. Effects of incomplete information, due to the limited extent of polarised radio sources, are properly treated in form of a window function (Eqs. (4), (5), and (51)). This window function contains additional information on the screen geometry, and allows for proper bookkeeping of data weighting, in case of noisy data being analysed. Since the window function also requires some working hypothesis about the average magnetic energy density profile of the Faraday screen, ways to test it a posteriori are sketched (Sect. 2.4).
The most efficient way to analyse Faraday rotation maps leads through Fourier space (Sect. 3). The Fourier transformation of a map gives direct insight into the magnetic energy spectrum (Eqs. (35) and (37)), which fully specifies the magnetic autocorrelation function (Eq. (32)). Many quantities of interest can be obtained from it, such as the average field strength (Eq. (38)), the correlation lengths (Eq. (39)), and the bias resulting from the used window function (Sect. 3.5).
The Fourier domain formulation gives also important insight into the real space behaviour of the RM autocorrelation function (Sect. 3.3): a power-law magnetic energy spectrum leads to a cusp at the origin of this function, where the shape of the cusp is determined by the power law index s. A steep spectrum s>1 (as e.g. expected for turbulent cascades) leads to a flat cusp, whereas a flat spectrum s<1 gives a pronounced sharp cusp. The limiting case s=1 with equal power on all scales leads to a linear (decreasing) behaviour of the RM autocorrelation function close to the origin.
Such cusps of the RM autocorrelation (or power-law spectra in Fourier space) can be signatures of turbulent cascades, but they can also occur in other situations. We demonstrated that a Faraday screen which is composed of finite magnetic structures of roughly constant field strength lead to a flat cusp, too (Sect. 4.1). We show ways to constrain the magnetic volume filling factor for such magnetic field models (Sect. 4.2). Observational artefacts, like noise or jumps in the measured RM values due to the so called -ambiguity, are able to produce sharp cusps (Sect. 5). We therefore stress the importance to check maps for such distortions and describe ways to do this.
A very important result of this work is that the magnetic autocorrelation length (Eq. (20)) is in general not identical to the autocorrelation length of the RM fluctuations (Eq. (21)). is much more strongly weighted towards the large-scale part of the magnetic power spectrum than (Eq. (39)). In typical astrophysical situations a broad spectral energy distribution can be expected so that will be much smaller than . Since enters the classical formulae to estimate magnetic field strength from Faraday measurements (see Eq. (40)), but is sometimes used instead, we expect that several published magnetic field estimates from Faraday rotation maps are actually underestimates.
We hope that our work aids and stimulates further observational and theoretical work on the exciting field of Faraday rotation measurements of cosmic magnetic fields in order to give us deeper insight in their fascinating origins and roles in the Universe.
Acknowledgements
We thank Kandaswamy Subramanian for many discussions and important suggestions. We acknowledge further stimulating discussions on RM maps with Matthias Bartelmann, Tracy E. Clarke, Klaus Dolag, Luigina Feretti, Gabriele Giovannini, Federica Govoni, Philipp P. Kronberg, and Robert Laing. We thank Matthias Bartelmann and Tracy Clarke for comments on the manuscript. TAE thanks for the hospitality at the Istituto di Radioastronomia at CRN in Bologna in April 2000, where several of the discussions took place. This work was done in the framework of the EC Research and Training Network The Physics of the Intergalactic Medium.