A&A 460, 425-438 (2006)
DOI: 10.1051/0004-6361:20065964
F. Govoni1 - M. Murgia1,2 - L. Feretti2 - G. Giovannini2,3 - K. Dolag4 - G. B. Taylor5,6
1 - INAF - Osservatorio Astronomico di Cagliari,
Loc. Poggio dei Pini, Strada 54, 09012 Capoterra (CA), Italy
2 -
INAF - Istituto di Radioastronomia,
via Gobetti 101, 40129 Bologna, Italy
3 -
Dipartimento di Astronomia,
Univ. Bologna, via Ranzani 1, 40127 Bologna, Italy
4 -
Max-Planck-Institut for Astrophysik, Garching, Germany
5 -
Department of Physics and Astronomy, University of New Mexico,
Albuquerque, NM 87131, USA
6 -
National Radio Astronomy Observatory, Socorro, NM 87801, USA
Received 4 July 2006 / Accepted 3 August 2006
Abstract
Aims. The goal of this work is to constrain the strength and structure of the magnetic field in the nearby cluster of galaxies A2255. At radio wavelengths A2255 is characterized by the presence of a polarized radio halo at the cluster center, a relic source at the cluster periphery, and several embedded radio galaxies. The polarized radio emission from all these sources is modified by Faraday rotation as it traverses the magnetized intra-cluster medium. The distribution of Faraday rotation can be used to probe the magnetic field strength and topology in the cluster.
Methods. For this purpose, we performed Very Large Array observations at 3.6 and 6 cm of four polarized radio galaxies embedded in A2255, obtaining detailed rotation measure images for three of them. We analyzed these data together with the very deep radio halo image recently obtained by us. We simulated random 3-dimensional magnetic field models characterized by different power spectra and produced synthetic rotation measure and radio halo images. By comparing the simulations with the data we are able to determine the strength and the power spectrum of the intra-cluster magnetic field fluctuations which best reproduce the observations.
Results. The data require a steepening of the power spectrum spectral index from n=2, at the cluster center, up to n=4, at the cluster periphery and the presence of filamentary structures on large scales. The average magnetic field strength at the cluster center is 2.5
.
The field strength declines from the cluster center outward with an average magnetic field strength calculated over 1 Mpc3 of
1.2
G.
Key words: globular clusters: general - galaxies: clusters: individual: A2255 - magnetic fields - polarization - cosmology: large-scale structure of Universe
The existence of magnetic fields associated with the intracluster medium in clusters of galaxies is now well established through different methods of analysis (see e.g. the review by Govoni & Feretti 2004, and references therein). The strongest evidence for the presence of cluster magnetic fields comes from radio observations. Magnetic fields are revealed through the synchrotron emission of cluster-wide diffuse sources, and from studies of the rotation measure (RM) of polarized radio galaxies. Other techniques, not discussed in this paper, include the study of inverse Compton hard X-ray emission (e.g. Fusco-Femiano et al. 2004; Rephaeli et al. 2006), cold fronts (e.g. Vikhlinin & Markevitch 2002) and magneto hydrodynamic simulations (e.g. Dolag et al. 2002).
Direct evidence for the presence of relativistic electrons and magnetic
fields in clusters of galaxies comes from the detection,
in an ever increasing number of
galaxy clusters, of large-scale, diffuse, steep-spectrum synchrotron
sources known as "radio halos''
or "relics'' depending on their morphology and location
(e.g. Giovannini & Feretti 2002).
Under the assumption that radio sources are in a minimum energy condition,
it is possible to derive a zero-order estimate of the magnetic field
strength averaged over the entire source volume.
Typical equipartition magnetic fields, estimated
in clusters with wide diffuse synchrotron emission, are 0.1-1 G.
However, the equipartition estimate is critically dependent on the low
energy cut-off of the relativistic electrons for steep spectrum
sources such as radio halos. Since this quantity is not known
the equipartition estimates of the magnetic field strength in these
radio sources should be used with caution.
Table 1: Pointed VLA observations of radio galaxies in the A2255 cluster of galaxies.
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Figure 1:
Radio contours of the A2255 cluster of galaxies obtained at 20 cm (Govoni et al. 2005) overlaid on the ROSAT X-ray image.
The white symbols + and ![]() ![]() ![]() ![]() |
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Figure 2: Radio contours at 3.6 cm of four cluster radio galaxies overlaid on the DSS2 red plate (J1712.4+6401, J1713.3+6347, J1713.5+6402 and J1715.1+6402, from left to right). The radio contours are: 0.04, 0.08, 0.16, 0.32, 0.64, 1.28, 2.56, and 5.12 mJy/beam. See Table 2 for more information (e.g. resolution, sensitivity, peak brightness) of these observations. |
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The presence of a magnetized plasma between an observer and a radio source
changes the properties of the polarized emission from the
radio source.
Therefore a complementary set of information on clusters
magnetic fields along the line-of-sight can be determined, in conjunction
with X-ray observations of the hot gas, through the analysis of the
RM of radio sources.
Many high quality RM images of extended radio galaxies
are now available in the literature (see e.g. the review by Carilli & Taylor 2002, and
references therein). These data
are consistent with magnetic fields of a few G
throughout the clusters.
In addition, stronger fields exist in the inner regions of
strong cooling core clusters.
In a few cases it has been possible to study the cluster magnetic field
in more detail by sampling several
extended radio galaxies located in the same cluster of galaxies
(e.g. Feretti et al. 1999; Taylor et al. 2001; Govoni et al. 2001a).
It is worth noting that the RM observed toward radio galaxies may not be entirely representative of the cluster magnetic field if the RM is locally enhanced by compression of the intracluster medium due to relative motions. There are, however, several statistical arguments against this interpretation. In particular the statistical RM investigation of point sources (Clarke et al. 2001; Clarke 2004) shows a clear broadening of the RM distribution toward small projected distances from the cluster center indicating that most of the RM contribution comes from the intracluster medium. This study included background sources, which showed similar enhancements as the embedded sources.
Recent work (e.g. Enßlin & Vogt 2003; Murgia et al. 2004) showed that detailed RM images of radio galaxies can be used to infer not only the cluster magnetic field strength, but also the cluster magnetic field power spectrum. Moreover, Murgia et al. (2004) pointed out that morphology and polarization information of radio halos may provide important constraints on the power spectrum of the magnetic field fluctuations on large scales. In particular, their simulations showed that if the intracluster magnetic field fluctuates up to scales of some hundred kpc, then steep magnetic field power spectra may give rise to detectable polarized emission in radio halos.
A2255 is a nearby (z=0.0806, Struble & Rood 1999),
rich cluster with signs of
undergoing a merger event (e.g. Yuan et al. 2005;
Sakelliou & Ponman 2006).
It is a suitable target to study the intracluster magnetic field
because it is characterized by the presence of
a diffuse radio halo source at the cluster center,
a relic source at the cluster periphery, and several
embedded radio galaxies (Jaffe & Rudnick 1979;
Harris et al. 1980; Burns et al. 1995; Feretti et al. 1997).
Govoni et al. (2005) found, for the first time
in a cluster, that the radio halo of A2255 shows
filaments of strong polarized emission (20-40%).
The distribution of the polarization angles in the filaments
indicates that the cluster magnetic field
fluctuates up to scales of
400 kpc in size.
In the rest of the cluster they did not detect significant diffuse polarized emission
except in the brighter regions of the relic (
15-30%).
Here we present multi frequency (3.6 and 6 cm) Very Large Array (VLA) observations of four polarized radio galaxies in the A2255 cluster. We apply the numerical approach proposed by Murgia et al. (2004), of analyzing the polarization properties of both radio galaxies and halo, to
investigate the cluster magnetic field strength and power spectrum.
The paper is organized as follows. In Sect. 2 we discuss details about the radio observations and the data reductions. In Sect. 3 we present the total intensity and polarization properties of the four radio galaxies at 3.6 and 6 cm. In Sect. 4 we present the RM images, discuss the results, and consider the presence of a cluster magnetic field. In Sect. 5, by following the same approach presented in Murgia et al. (2004), we introduce the 3D multi-scale magnetic field modeling used to determine the intra-cluster magnetic field strength and structure. In Sects. 6 and 7 we show the results obtained with a constant and variable magnetic field power spectrum slope, respectively. In Sect. 8, we draw conclusions from this study.
Throughout this paper we assume a CDM cosmology with H0 = 71 km s-1 Mpc-1,
= 0.3, and
= 0.7.
At the distance of A2255, 1
corresponds to 1.5 kpc.
The four radio galaxies J1712.4+6401, J1713.3+6347, J1713.5+6402 and J1715.1+6402 have been investigated with multi-frequency, high-resolution, VLA observations. These sources have been selected on the basis of their high flux density (>80 mJy at 20 cm), extension, and the presence of polarized emission in the NVSS (Condon et al. 1998). The details of the observations are provided in Table 1. The sources were observed at two frequencies (4535/4885 MHz) within the 6 cm band and at two frequencies (8085/8465 MHz) within the 3.6 cm band, both in the B and C configurations. All observations were made with a bandwidth of 50 MHz.
The (u,v) data at the same frequencies but from different configurations were first handled separately and then combined. The source 1331+305 (3C286) was used as the primary flux density calibrator, and as an absolute reference for the electric vector polarization angle. The phase calibrator was the nearby point source 1642+689, observed at intervals of about 30 min. Calibration and imaging were performed with the NRAO Astronomical Image Processing System (AIPS), following standard procedures. Self-calibration was applied to remove residual phase variations.
Total intensity images I have been produced by averaging the two frequencies in the same band
while U and Q images have been obtained for each frequency separately.
Images of polarized intensity
P=(Q2+U2)1/2,
fractional polarization FPOL=P/I and position angle of polarization
were derived from the I, Q and U images.
Figure 1 shows the contour image of A2255 (Govoni et al. 2005)
at 20 cm overlaid onto the ROSAT PSPC image of the cluster.
The X-ray image is in the band 0.5-2 keV and has been
obtained from the ROSAT public archive
by binning the photon event table in pixels of 15''
and by smoothing the image with a Gaussian of
.
The centroid of the X-ray emission is approximately at
RA(J2000) =
,
Dec(J2000) = 64
03
54
(Feretti et al. 1997).
The X-ray peak is shifted to the West with respect to this position.
The large field of view of the radio image shows the location
of the cluster radio galaxies with respect to the gas density distribution.
For clarity the first radio contour is at 25
.
Therefore we see only the radio galaxies
of the cluster while the low brightness diffuse emissions
(radio halo and relic) are not visible here.
The high resolution (FWHM
2
2
)
images obtained at 6 cm for the four radio galaxies analyzed in this work are inset in Fig. 1.
The sources are located at different distances from the cluster center
which allows,
through the study of their RM (presented in Sect. 4),
the cluster magnetic fields to be estimated along different lines-of-sight.
Figure 2 shows the high resolution (FWHM
1
1
)
observations in the 3.6 cm band for the four sources overlaid on the DSS2
red plate
.
Every source has a clear optical
counterpart and all are classified as cluster radio galaxies
on the basis of the optical spectroscopy analysis
(Miller & Owen 2003; Yuan et al. 2003).
The relevant parameters of the radio images at high resolution are listed in Table 2.
In the following we give a brief description of the individual sources.
In the radio images presented in Figs. 3-6
we restored the maps at different frequencies with the same
beam (2
2
).
The relevant parameters of these total and polarization intensity images are listed
in Table 3. The intensity and polarized flux densities were
obtained, after the primary beam correction,
by integrating in the same area the I and P surface brightness, respectively,
down to the noise level.
In Figs. 3-6 contours represent total intensity
while vectors represent the orientation of the projected
E-field and are proportional in length to the fractional polarization.
In the fractional polarization images, pixels with error greater than
were blanked.
The fractional polarization values given in the following subsections are calculated in the
regions where the error is less than
.
Table 2: Total intensity images at high resolution (see Figs. 1 and 2).
Table 3:
Total and polarization intensity radio images restored with a beam of 2
(see Figs. 3-6).
The source has a narrow-angle tail structure
and it is located in projection quite near to the cluster center (see Fig. 1).
The images show an unresolved core, in the position
RA(J2000) = 1712
23
,
Dec(J2000) = 64
01
57
,
South-West of the cluster center, and a long tail elongated North-East in the direction of the cluster X-ray centroid. The maximum projected angular size of the source at 6 cm is
about 70
(
105 kpc) but it appears much more extended
in our previous observation at 20 cm (2.5
).
At 3.6 cm only the brightest emission of the tail is detected.
Figure 3 shows the polarization images of the source
with an angular resolution of 2''
2''.
The source is strongly polarized both at 6 cm and 3.6 cm with
a mean fractional polarization of about 14% and 12.5% respectively.
In the core the fraction polarization is 2% at 6 cm and 4% at 3.6 cm
and increases along the tail.
This narrow-angle tail radio galaxy is located
in the southern part of the cluster far from the center (see Fig. 1).
From the compact component, in the position
RA(J2000) = 1713
16
,
Dec(J2000) = 63
47
37
two jets emanate and then bend to the north. The maximum projected angular size at 6 cm is about 90
(
135 kpc) but the source appears much more extended at 20 cm (
3
).
At 3.6 cm only the brightest emission of the jets have been detected.
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Figure 3:
Source J1712.4+6401: Left: total intensity contours and polarization vectors at 6 cm (4535 MHz). Right: total intensity contours and polarization vectors at 3.6 cm (8465 MHz).
The angular resolution is 2.0'' ![]() ![]() |
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Figure 4:
Source J1713.3+6347: Left: total intensity contours and polarization vectors at 6 cm (4535 MHz). Right: total intensity contours and polarization vectors at 3.6 cm (8465 MHz).
The angular resolution is 2.0'' ![]() ![]() |
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Figure 5:
Source J1713.5+6402: Left: total intensity contours and polarization vectors at 6 cm (4535 MHz). Right: total intensity contours and polarization vectors at 3.6 cm (8465 MHz).
The angular resolution is 2.0'' ![]() ![]() |
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Figure 4 shows the polarization images of the jets
with an angular resolution of 2''
2''.
The sensitivity of the observations reveal significant polarization
only in the inner parts of the jets up to a distance of about 15
(
20 kpc)
from the core. The mean fractional polarization for the source
is about 11% at 6 cm and 14% at 3.6 cm.
In the core the fraction polarization is 2% at 6 cm and 0.4% at 3.6 cm.
The jet on the east shows a lower fractional polarization
(10% and 12% at 6 cm and 3.6 cm respectively)
than the jet on the west (16% and 23% at 6 cm and 3.6 cm respectively).
The source has a total extension of about 35
(
50 kpc)
and a double structure with an unresolved core in
the position RA(J2000) = 17
13
29
,
Dec(J2000) = 64
02
49
(see Fig. 1).
Figure 5 shows the polarization images of the source
at 6 cm and 3.6 cm with an angular resolution of 2''
2''.
The mean fractional polarization is
11% at 6 cm
and
13% at 3.6 cm. The fraction polarization of the core is 2% at 6 cm and 3% at 3.6 cm
while the lobes are polarized at similar levels.
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Figure 6:
Source J1715.1+6402: Left: total intensity contours and polarization vectors at 6 cm (4535 MHz). Right: total intensity contours and polarization vectors at 3.6 cm (8465 MHz).
The angular resolution is 2.0'' ![]() ![]() |
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This wide-angle tail source, located in the periphery of the cluster (see Fig. 1),
is very extended with a size of about 3.5
at 20 cm.
The maximum projected angular size at 6 cm is about 200
(
300 kpc) but the low surface brightness lobes are only marginally
detected at 6 cm and completely missed at 3.6 cm
where only the first 30
(45 kpc) of the jets are visible.
An unresolved component is located in the position:
RA(2000) = 1715
09
,
Dec(2000) = 64
02
54
.
Twin low brightness oppositely directed jets emanate from the core to the north-east
and south-west. After 40
(60 kpc) the north-east jet
bends to the west toward the X-ray centroid while the other jet remains straight.
Figure 5 shows the polarization images of the
central part of the source with an angular resolution of 2''
2''.
We detect polarization only in the core and in the inner arcseconds of the jets.
In the core the fraction polarization is 0.6% at 6 cm and 2% at 3.6 cm
while in both jets it is about 14% at 6 cm.
The polarization in the northern jet at 3.6 cm is below the sensitivity of
the present observation while in the southern jets it is 12%.
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Figure 7: Images of the rotation measure computed using the polarization angle maps at the frequencies 4535, 4885, 8085 and 8465 MHz with a resolution of 2''. Contours refer to the total intensity images at 3.6 cm. Contour levels are: 0.06, 0.15, 0.5, 1, and 2 mJy/beam. |
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Polarized radiation from cluster and background radio galaxies
may be rotated by the Faraday effect if magnetic fields are present
in the intra-cluster medium. In this case, the observed polarization angle (
)
at a frequency
is connected to the intrinsic polarization angle (
)
through:
Table 4: Rotation measure.
Figure 7 shows the total intensity contours at 3.6 cm
overlaid on the RM images of the four cluster galaxies.
The RM values range from about -300 rad/m2 up to 250 rad/m2and reveal patchy structures with RM fluctuations down to scales of a few kpc. Figure 8 shows the histograms of the RM distribution for the four sources.
Due to the very low number of reliable RM pixels in J1715.1+6402
this source will not be considered in the following statistical analysis.
Table 4 reports, for the other three sources sorted by distance from the X-ray
centroid, the mean value
the
root mean square
and the maximum absolute value
of the RM distribution. These data are not corrected for the Galactic contribution
which is likely negligible. A2255
in galactic coordinates is located at lon = 93.97
and
lat = +34.95
and based on the average RM for extragalactic
sources published by Simard-Normandin et al. (1981),
the RM Galactic contribution in the region occupied by A2255 is expected to be
about -6 rad/m2.
Our data suffer from two kinds of uncertainty.
The first uncertainty is related to the fit
and takes into account the presence of errors in the measurements.
In the RM images all the pixels with an error in the fit greater
than 40 rad/m2 were blanked. This ensures that at all
frequencies we always consider pixels with an error
in polarization angle lower than 10
(see Eq. (1)).
The mean fit error calculated in the images
is about 20 rad/m2 for J1713.5+6402 and about 30 rad/m2 for J1712.4+6401
and J1713.3+6347. The second source of error is the statistical error.
Ideally one would determine the parameters of the RM distribution
by investigating an area as large as possible.
The morphology of sources and the presence of blanked pixels in the RM images introduce a further uncertainty in this determination. This uncertainty is particularly strong in sources
where the reliable RM pixels are present in a small
number of observing beams N (see Table 4). The statistical error has been
calculated to be about 10 and 5 rad/m2 for the
and
respectively in J1713.5+6402 and
about 15 and 10 rad/m2 for the
and
respectively in the other two sources.
As found in other studies (Feretti et al. 1999;
Taylor et al. 2001; Govoni et al. 2001a) aimed to study the cluster magnetic
field by sampling the RM of extended radio galaxies located in the same
cluster of galaxies, the most striking result is the trend of
the RM values with distance from the cluster center.
The innermost source, J1712.4+6401, has the highest
and
.
The source J1713.5+6402, located at about 1 core radius from the cluster center,
shows considerably lower RM values. Finally, the peripheral source J1713.3+6347
shows a
consistent with its uncertainty but still has a significant
.
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Figure 8: Histograms of the rotation measure images for all significant pixels. |
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The RM results are in agreement with the interpretation that the external Faraday screen is the same for all the sources, i.e. the decreasing radial profile of the RM data may be due to the intracluster medium of A2255, whose differential contribution depends on how much magneto-ionized medium is crossed by the polarized emission. Thus the data are consistent with the existence of a magnetic field associated with the intracluster medium.
The RM structures on small scales can be explained by the fact that the cluster magnetic field fluctuates on scales smaller than the size of the sources. On the other hand, the RM distribution with a non-zero mean indicate that the magnetic field fluctuates also on scales larger than the radio sources. Therefore to study in detail the cluster magnetic field properties it seems necessary to consider cluster magnetic fields models where both small and large scale coexist.
An analysis of the rotation measure of radio sources sampling different lines-of-sight across the cluster, together with an X-ray observation of the intracluster gas, can be used to derive information on the strength and structure of the cluster magnetic field. Complementary information on the cluster magnetic field can be derived by analyzing the radio halo emission. In particular, for a given distribution of the relativistic electrons in a cluster, different magnetic field models will generate very different total intensity and polarization brightness distributions for the radio halo emission. For those clusters containing a radio halo, a realistic magnetic field modeling should be able to explain both the small scale fluctuations seen in the RM images of cluster radio galaxies, and the morphology and polarization of the large scale diffuse emission.
The FARADAY tool, described in Murgia et al. (2004), permits an investigation of cluster magnetic fields by comparing the observations with simulated RM and radio halo images, obtained by considering 3-dimensional multi-scale cluster magnetic field models. We applied this method to A2255 with the aim of finding the optimal magnetic field strength and structure capable of describing both the high resolution RM images presented in this work and the large scale polarized features visible in the radio halo of A2255 (Govoni et al. 2005).
There are a number of quantities that are critical in our modeling, these are: the gas density distribution of the thermal electrons, the characterization of the magnetic field fluctuations and radial scaling, the energy density distribution of the synchrotron electrons.
For the distribution of the thermal electron gas density in A2255
we assumed a standard -model profile:
We considered different power law cluster magnetic field power spectrum models:
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(4) |
We adopted a magnetic field distribution decreasing from the cluster center according to:
Provided the distribution of the thermal electrons and the magnetic field model described above we simulated the RM by integrating Eq. (2) along the line-of-sight.
To simulate the radio halo emission
at each point of the computation grid we calculated the
synchrotron emissivity by convolving the emission spectrum
of a single relativistic electron with the particle energy distribution
of an isotropic population of
relativistic electrons whose distribution follows:
The energy density of the relativistic electrons is:
In our model the magnetic field energy density
and
are in equipartition at every
point in the cluster, therefore both energy densities have the same radial decrease.
We adopted an electron energy spectral index
and a Lorentz factor for the
high-energy cutoff of the electron distribution
104. The resulting radio halo emission spectrum is
roughly consistent with the radio spectral index map (not shown, Govoni et al. in preparation)
obtained by combining our radio halo image at 1.4 GHz (Govoni et al. 2005) with
the image at 327 MHz (Feretti et al. 1997).
With
we have
.
The low-energy cutoff
and
,
are adjusted to guarantee the observed halo brightness and
.
In practice we fix
= 10 and let
vary.
Below we present the simulated RM and radio halo images corresponding to two different variants of the above model. As a first approximation, we keep the power spectrum spectral index n constant throughout the entire cluster (Sect. 6). However, we find that this model does not reproduce the radio halo polarization level observed in A2255. To describe both the RM profiles and the radio halo polarization the data require a steepening of the power spectrum spectral index from the cluster center to the periphery and the presence of filamentary structures on large scales (i.e. a non-Gaussian magnetic field). In Sect. 7 we will show that a model with these characteristics can better account for both the observed RM statistic and the radio halo polarization.
Table 5: Gas density and magnetic field parameters adopted in the simulations.
The computational grid, gas density, and magnetic field model
parameters of the simulations are provided in Table 5.
A grid size of 10243 pixels with a cellsize of 2 kpc
allows us to simulate a cluster field of view of
2048
2048 kpc2. The simulated
magnetic field is periodic at the grid boundaries. Thus, to extend our RM simulations up to the source J1713.3+6347, the computational grid has been
replicated realizing a field of view of 4096
4096 kpc2
(i.e. up to about one virial radius).
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Figure 9:
Simulated RM images for different values of the magnetic field power
spectrum spectral index n. The three power spectra are normalized to have the
same total magnetic field energy which is distributed over the range of spatial scales
from 4 kpc up to 512 kpc.
In these simulations the average field at the cluster center is
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In Fig. 9 we show the central 2
2 Mpc2 region of the simulated RM images
of A2255 for three different values of the magnetic field power spectrum spectral index, n = 2, 3 and 4.
To obtain such images we performed the integration of
Eq. (2) from the cluster center up to 4 Mpc
along the line-of-sight using the magnetic field model described
in the previous section.
The three magnetic field power spectra are normalized to generate the same
average magnetic field energy density.
The average magnetic field strength at the cluster
center is
G and
its energy density decreases from the cluster center following the gas energy profile. The central
magnetic field strength of this set of simulations provides the best fit to the observed RM radial
profiles (see below).
It is evident from Fig. 9 that the same cluster
magnetic field energy
density generates different magnetic field configurations with
correspondingly different RM structures, for different values of the power
spectrum spectral index.
RM images of the whole cluster, such as those presented in
Fig. 9, cannot be observed in reality.
However it is possible to derive the RM from limited regions of the cluster
by observing radio sources located at different projected distances from
the center as described in Sect. 4.
The comparison of the observations with the simulations
can constrain the magnetic field strength and power spectrum.
In such a comparison the details of the three-dimensional structure of the radio sources are
neglected, and the entire observed Faraday rotation is assumed to
occur in the intracluster medium. The three radio galaxies considered
in this work are cluster members. For what concerns their location
along the line-of-sight, we followed two approaches.
As first approximation, we supposed that the three galaxies
lie on a plane perpendicular to the line-of-sight at the distance of the cluster center.
The simulations shown in Fig. 9 reproduce such a situation.
As an improvement, we extracted the Faraday depth of the simulated RM images randomly from the
3 dimensional distribution of bright member galaxies observed in A2255. We determined
this trend by fitting with a King model the spatial galaxy distribution of the
brightest spectroscopically confirmed member galaxies taken from Yuan et al. (2003).
The fit, shown in Fig. 10, has been extended only to those members
characterized by a red magnitude brighter than
since radio galaxies usually populate this optical luminosity range (e.g. Govoni et al. 2000).
To compare directly the simulations with the observations
we added to the simulated RM images a noise of 30 rad/m2
(corresponding to the typical fit error in our data)
and an offset of -6 rad/m2
(representing the RM galactic contribution calculated in
the direction of A2255).
Therefore, in the peripheral regions of the simulated RM images,
where the contribution of the cluster Faraday screen is negligible,
we have
30 rad/m2 and
rad/m2.
These are the lower limits of our simulations.
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Figure 10:
Spatial distribution of member galaxies in A2255 taken from Yuan et al. (2003).
The solid line represents the fit of the King model
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Figure 11:
Comparison of
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In Fig. 11 we compare the observed
and
with the expectation of the simulations
(for n=2,3,4 and
=
G, see below).
The plotted data are those presented in Table 4. The error bars
represent only the statistical errors discussed in Sect. 4 and
do not account for fitting errors since those are already included in the simulated RM images.
Since the observed
and
are calculated in limited regions over
the radio galaxies, we have to sample the corresponding
simulated values in regions of equivalent size.
The expected radial trends have been obtained
by covering the simulated RM images with a grid of rectangular boxes
of 50
50 kpc2 in size.
These boxes reproduce the regions covered by the RM images of the radio galaxies
in A2255. Inside each box we calculated the values
of
and
exactly as it would be done if they were
real radio sources. Due to the random nature of the simulated magnetic field, the values of
and
at a given radial distance vary from box to box.
However, the numerical simulations allow us to quantify this
statistical variance.
We calculated the mean and the dispersion of
and
values from all
the boxes located at the same projected
distance from the cluster center.
The dark lines show the mean while the dark gray regions
show the dispersion of the profiles calculated in the simulations shown in Fig. 9.
The light gray regions represent the increase of the scatter of simulated RM in the case in which
the location of the boxes along the line-of-sight is extracted randomly
from the 3 dimensional distribution of galaxies in A2255.
The horizontal dashed lines represent the lower limits of
and
caused by the presence of fitting errors and the RM galactic contribution.
Both the values of
and
depend linearly
on the cluster magnetic field strength.
The best fit of the simulated
radial profiles with
respect to the
values observed in A2255 was obtained by considering a central magnetic field
of about 2
G. It reproduces quite well the observed
for each of the
considered values of the magnetic field spectrum spectral index (n = 2, 3, 4), as shown
in the left panel of Fig. 11.
The comparison between simulated and observed
(Fig. 11, right panels) allows us to constrain the value of the best
index of the power spectrum. In order to quantify the relative goodness of the fit
of the
profiles we calculated the sum
![]() |
Figure 12:
Simulated radio halo brightness (contours) and fractional polarization (color)
for the three different magnetic field spectral indices n = 2, 3 and 4.
For a direct comparison with the data (Fig. 15; bottom right panel)
these images have been convolved with a beam of 25
![]() |
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A flat spectral index power spectrum, n=2, is ruled out since it predicts
too low
for all three sources. A steep spectral index power spectrum, n=4,
fits the outer source but predicts too much
for the two central sources.
The model with n=3 agrees better with the data.
![]() |
Figure 13: Left: magnetic field power spectra as a function of the distance from the cluster center for the variable magnetic field power spectrum slope model. The dashed and dotted reference lines have a slope of 2 and 4, respectively. Right: radial profile of the magnetic field strength. The solid line represents the average magnetic field strength while the gray region indicates the root mean square of the magnetic field fluctuations. |
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We simulated the expected total intensity and polarization brightness
distribution at 1.4 GHz, for a magnetic field strength
2
as found in the RM data.
In Fig. 12 we show the simulated radio halo brightness
(contours) and fractional polarization (color)
for the three different magnetic field spectral indices n = 2, 3 and 4.
For a direct comparison with the data (see Fig. 15; bottom right panel)
these images have been convolved with a beam of 25
(37.5 kpc)
and the lowest contour level has been chosen to match
the dynamic range of 10 as in the observations.
As in the case of the simulated RM images, the same cluster magnetic field energy density generates different magnetic field configurations with correspondingly different halo morphologies, for different values of the power spectrum spectral index, n. However, in this case these differences are even more evident given that the radio halo intensity depends upon the square of the magnetic field fluctuations.
Flat power spectrum indexes (n=2) give rise to a regular and smooth radio halo morphology. Increasing the spectral index, n, and thus the power on large scales, the radio halo becomes increasingly irregular. For n=4 the radio halo emission is confined to a few bright filaments. Again, the simulated radio halo corresponding to n=3 seems to better reproduce the observed morphology.
The internal depolarization of the radio halo emission is stronger where the RM is higher and where the magnetic field is tangled on small scales. Thus, as one would expect, we found that the degree of fractional polarization increases with increasing values of n and decreases towards the cluster center (Murgia et al. 2004). In any case, one can observe significant polarized emission only from the outer layers of the halo observed along the line of sight paths that do not cross the cluster center. The maximum fractional polarization attained in this set of simulations at the outer halo edges ranges from 5%, for n=2, up to 9%, for n=4. These values are definitely lower than the values observed in the radio halo of A2255.
Therefore, our results seem to indicate that a single magnetic field power spectrum slope cannot account for both the observed RM and radio halo images. Even n=3, which gives a good results for the RM profiles and radio halo morphology, yields insufficient polarization levels.
![]() |
Figure 14:
Top: simulated RM image as appears for a variable value of the magnetic
field power spectrum spectral index n=2-4. In this simulation the average field at the cluster center is
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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The results of the previous section suggest that the magnetic field
power spectrum slope may change as a function of distance from the
cluster center. The data require that much of the magnetic field energy
density at the cluster center should be concentrated on small scales
in order to account
for the observed RM dispersion. On the other hand, to explain the highly
polarized structures observed in the radio halo it is necessary that
most of the magnetic field energy density is on large scales at the cluster periphery.
This scenario is also consistent with the relatively large
of the most external radio galaxy
J1713.3+6347 (see Fig. 11).
We thus considered a magnetic field configuration whose power spectrum gradually steepens with increasing distance from the cluster center (see Fig. 13).
After a series of simulations, we found that the best global magnetic field is composed by the sum of:
![]() |
Figure 15:
Top: simulated ( left panel) radio halo brightness for a variable magnetic field spectral index n = 2-4. For a direct comparison with the data ( right panel)
this image has been convolved with a beam of 25
![]() |
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Non-Gaussian fields are characterized by a lower filling factor with respect to Gaussian random fields since magnetic field energy
is more concentrated in filaments. The non-Gaussianity is indeed a necessary feature of the
outer field since it allow us to enhance the field strength in the polarized filaments
boosting their prominence. Moreover, the presence of large voids in the magnetic field
ensures that the
over the two central radio galaxies is not exceeded as would happen
for a Gaussian field with a similar power spectrum slope (see bottom left panel of Fig. 11).
The resulting cluster RM images and profiles, obtained with the same procedure described in
Sect. 6.1, are shown in Fig. 14.
The average magnetic field strength at the cluster center is 2.5
.
The RM image shows that the small scale field fluctuations are stronger at the cluster center while the large scale magnetic field
dominates at the periphery. The RM profiles shows that this model is able to reproduce both the
and the
of all three sources better than a constant magnetic field slope model can.
In Fig. 15, we present the observed radio halo total intensity and fractional polarization images (right panels) along with their simulated counterparts (left panels) generated by considering the magnetic field model outlined above. By comparing the total intensity images we are able to reproduce both the brightness level, the size, and the overall morphology of the halo. More importantly, the fractional polarization of the simulated filaments reaches values as high as 10%-20%. These values, although still somewhat lower than those observed, (the average polarization percentage in filaments is 30%), are substantially higher than those presented in the previous section and clearly represent an improvement over the model based on a constant slope n.
Note that with the same magnetic field strength we are able to explain both the RM data and the
radio luminosity of the radio halo while keeping relativistic particle and field energy densities
in equipartition. However, the equipartition magnetic field of the simulated radio halo
calculated using standard formulas (Pacholczyk 1970)
results in about 0.3
over a 1 Mpc3 volume (the same result obviously holds also for the observed radio halo since both have roughly the same size and
luminosity). With such a magnetic field strength, which is lower than the average magnetic field of the simulations over the same volume
,
it would be virtually impossible to fit the RM data, since the expected
and
profiles will be one order of magnitude lower
than those shown in Fig. 14.
This problem is usually addressed in the literature as a discrepancy between the magnetic field
strength estimated from the RM and that estimated from the minimum energy argument.
In our simulation this problem is not present since most of the relativistic particles
are at low energies and are undetectable at observable radio wavelengths.
We do not claim that we have solved the aforementioned discrepancy. But we note that
the classical equipartition estimate is critically dependent on the low energy cut-off of the
electron energy spectrum for steep spectrum sources such as radio halos.
Thus, until a precise knowledge of the low energy spectrum of the
synchrotron electrons in radio halos can be reached, the equipartition
estimates of the magnetic field strength in these radio sources should be
used with caution.
In conclusion, we find that a magnetic field power spectrum whose slope steepen from the cluster center outwards provides a good description of the RM radial profiles and is able to reproduce both the luminosity and the fractional polarization of the radio halo, thus reconciling the magnetic field strength estimated obtained using these two complementary methods of analysis.
The fractional polarization of the observed radio halo is still higher than the simulated one. However, we do not exclude that more sophisticated magnetic field models, e.g. non-Gaussian random fields with ad-hoc correlated phases, can better reproduce the observed filamentary polarized structures.
We checked that the magnetic field radial profile adopted in our simulations,
in particular the value of ,
does not strongly affect the results.
We present new VLA observations at 3.6 and 6 cm for four polarized radio galaxies embedded in A2255, obtaining detailed RM images for three of them. The dispersion and the mean of the RM decrease with increasing distance from the cluster center. We analyze these data, together with the very deep radio halo image recently obtained by Govoni et al. (2005). Using the numerical approach described in Murgia et al. (2004), we simulate random 3-dimensional magnetic field models characterized by different power spectra and produce synthetic RM and radio halo images. By comparing the simulations with the data we investigate the strength and the power spectrum of the intra-cluster magnetic field fluctuations.
We find that a magnetic field power spectrum with a single spectral index is not suitable to reproduce both the rotation measure distribution of radio galaxies, and the radio halo polarization. The data require a steepening of the power spectrum spectral index from n=2, at the cluster center, up to n=4 at the cluster periphery and the presence of filamentary structures on large scale. This result seems to indicate that the magnetic field at the cluster center would be more dominated by small-scale structures, while at the periphery most of the magnetic field energy density is concentrated on the large-scale structures. A magnetic field with power spectrum steepening with radius is likely to reflect a complex behaviour of turbulence and motions in clusters. Subramanian et al. (2006) discuss the evolving turbulence due to dynamo action, and find that turbulence can coexist with ordered filamentary gas structure. Since turbulence is likely driven by a merger event, it may be expected that this driving is stronger in the cluster central region. In this case, turbulence in the peripheral regions can be in a more advanced stage of decay than at the cluster center. Since smaller scales decay first, and the integral scale of decaying turbulence increases with time, larger-scale magnetic field structures may be more easily present in the outer cluster region.
The average magnetic field strength at the cluster center is 2.5
.
The field strength declines from the cluster center outward and the average magnetic field
strength calculated over 1 Mpc3 is about 1.2
G.
In conclusion, we find that the above magnetic field model provides a good description of the RM radial profiles and is able to reproduce both the luminosity and the fractional polarization of the radio halo, thus reconciling the magnetic field strength estimated using two complementary methods of analysis.
Acknowledgements
We are grateful to Dan Harris and Anvar Shukurov for discussions and suggestions. This research has made use of the NASA/IPAC Extragalactic Data Base (NED) which is operated by the JPL, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research was partially supported by PRIN-INAF 2005.