Magnetic Fields in Galaxy Clusters

1 Introduction 1.3.3 Faraday rotation maps . . . . . . . . . . . . . . . . . . 1.3.4 Reconciling magnetic fields derived from the three methods 7 1.3.5 The RM debate . . . . . . . . . . . . . . . . . . . . . .

1.3.6 Improved methods to determine the magnetic fields from

rotation maps . . . . . . . . . . . . . . . . . . . . . . .

1.4 Simulations of Galaxy Clusters Involving Magnetic Fields . . .

2 Theoretical Considerations 2.1 Friedman equation . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Comoving coordinate transformation . . . . . . . . . . . . . . 2.3 Poisson equation . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 General considerations of magnetohydrodynamics . . . . . . . 2.5 MHD continuity equation . . . . . . . . . . . . . . . . . . . . 2.6 MHD momentum conservation . . . . . . . . . . . . . . . . . . 2.7 Energy conservation . . . . . . . . . . . . . . . . . . . . . . . 2.8 MHD magnetic field evolution . . . . . . . . . . . . . . . . . .

3 Methods 3.1 Principle and Initial Conditions . . . . . . . . . . . . . . . . . 3.2 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Gravitational interaction of gas and dark matter . . . . . . . . 3.4 Adaptive Mesh Refinement . . . . . . . . . . . . . . . . . . . . 3.5 Hydrodynamics and the Riemann problem . . . . . . . . . . . 3.6 Evolution of the magnetic field . . . . . . . . . . . . . . . . . .

4 Results 4.1 Structure formation . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Evolution of the magnetic field . . . . . . . . . . . . . . . . . . 4.3 Properties of the final cluster . . . . . . . . . . . . . . . . . . 4.3.1 Correlation of the density and the magnetic field strength 30 4.3.2 Power spectrum of the magnetic energy . . . . . . . . .

4.3.3 Rotation maps . . . . . . . . . . . . . . . . . . . . . .

5 Summary and Outlook

An adaptive mesh refinement simulation of galaxy cluster formation was performed that included the passive evolution of a magnetic field. It was found that structure formation plays an important role in am- plifying large-scale magnetic fields and that the magnetic properties of the obtained cluster were in good agreement with recent observations. The initial field was amplified by a factor of up to 1000 during the formation of the cluster, and the field strength was seen to be well cor- related with the gas density. We further found a magnetic energy power spectrum that is well described by -5/3 Kolmogorov-type turbulence. Near the accretion shocks on the outskirts of the cluster, the magnetic field is amplified well beyond the value expected from mere compression of gas. Here, shear flows lead to a substantial increase in field strength. Realistic Faraday rotation measures were obtained from the simu- lation data, which was however not resolved well-enough to allow for a more quantitative analysis.

1 Introduction

Recently, magnetic fields in galaxy clusters have come to the attention of the scientific world as the largest-scale magnetic structures measured to the present date. While little is known about the origin of these fields and their evolution throughout cosmic time, the presence of large-scale magnetic fields may have important implications for the processes observed in galaxy clusters. Examples of these implications include inhibition of transport processes - such as heat conduction, spatial mixing of the gas, and the propagation of cosmic rays - or even dynamical importance through the Lorentz force and the additional mag- netic pressure term. It is therefore of great scientific importance to determine the origin, evolution and structure of cluster magnetic fields and their rele- vance in structure formation and in astrophysical phenomena observed today in galaxy clusters.

1.1 Galaxy Clusters: Formation and General Proper-

Clusters of galaxies are the largest gravitationally bound systems in the Uni- verse. They can be recognized in the optical range as groups of galaxies which are located closer together than the average distribution. However, galax- ies represent only around 5% of the total mass of a typical cluster. Most of the baryonic mass, roughly 20% of the total mass of the cluster, is contained in the form of hot ionized gas in the intracluster medium (ICM). The ICM is characterized by high temperatures (in the order of 10 8 K) and electron number densities in the range of 10 −3 cm −3 ; thermal bremsstrahlung emis- sion from the hot gas in the intracluster medium is very intense in the X-ray spectrum, typical luminosities ranging between 10 43 − 10 46 erg/s. By far the largest contribution to the mass of galaxy clusters is the dark matter, which constitutes between 70-80% of the total mass and plays an important role in the development of the clusters. According to the model of hierarchical struc- ture formation, clusters of galaxies are thought to form upon the gravitational merger of smaller units, such as groups and subclusters.

1.2 Theories Regarding the Origin of Cluster Magnetic

Our Universe is permeated by magnetic fields on different length scales and strengths. The most palpable example is our own planet, which has a magnetic field of about 0.5 G at its surface. Moreover, the magnetic activity of our Sun also affects life on Earth. Magnetic fields in the Sun are about 10 G at the poles and can reach up to 2000 G in sunspots. Magnetic fields have been measured in the intraplanetary medium (≃ 50µG), in protostars and neutron stars, in the intragalactic medium (≃ 5µG in the Milky Way). It comes as no surprise

that also the intracluster medium is magnetized. Although the uncertainty in measurements is still significant, the observations are concordant to the existence of fields in the microgauss range having correlation lengths of the order of 10 kpc.

There are several hypotheses regarding the origins of cluster magnetic fields, namely the primordial scenario, the protogalactic scenario and the galac- tic scenario. According to the primordial scenario, the fields originate from the early universe, prior to recombination. Recombination is an epoch in the evolution of the universe (at redshift 1000) when the electrons and protons combined to form atoms, whereby the photons decoupled from matter and the Universe became neutral. The matter in the Universe was reionized at a later stage, possibly after the formation of the first stars. The primordial scenario involves several possible mechanisms for the generation of the magnetic fields prior to recombination. One such mechanism is the Biermann battery effect, which occurs when the gradients of the pressure and the electron number den- sity are not parallel to each other, implying that the system is not in a static equilibrium. To restore the equilibrium, a thermoelectric current is gener- ated, from which the magnetic fields originate. Other mechanisms imply local charge separation occuring during the quark-hadron (QCD) or electroweak (EW) transitions. Estimated values of these seed fields are in the order of 10 −21 G. If the primordial scenario is indeed true, the seed fields should be ob-

servable by means of anisotropies in the cosmic microwave background (CMB) and by their effect on nucleosynthesis. Current measurements of anisotropies of the CMB place upper limits of about 5 nG on the strength of potential seed

The protogalactic scenario predicts the possibility that the most impor- tant growth in large-scale magnetic fields occurred during the reionization era, which corresponds to the early-stage of galaxy formation.

Lastly, the galactic scenario involves the interaction of the magnetized in- tragalactic medium with the ICM in order to generate cluster magnetic fields. Galactic winds and radio jets emitted by active galactic nuclei (AGN) are possible ways in which fields from the galactic medium can be transferred to the ICM. Moreover, the significant concentration of metals in the ICM sug- gests that a large fraction of it is of galactic origin. However, the fields in the ICM have large correlation lengths, comparable to the typical size of a galaxy, therefore a mechanism is required to arrange the galactic-originating fields on ICM correlation scales. One possibility in this sense is to have helical fields, which under the constraint of conservation of helicity tend to order on larger scales for a minimal energy configuration.

1.3 Measuring Cluster Magnetic Fields: Methods and

Apart from the questions posed towards the origin of the cluster magnetic fields, another important scientific research area regards their currently ob- servable properties. There are three different methods which can be employed in order to determine the strengths of magnetic fields in the ICM.

1.3.1 Synchrotron radiation

The first of these methods makes use of synchrotron radiation luminosities. Synchrotron radiation is produced by relativistic electrons gyrating in a mag- netic field due to the Lorentz force. The synchrotron emission spectrum shows a peak at a critical frequency which is proportional to the average magnetic field and the square of the Lorentz factor, γ, while the power emitted in syn- chrotron radiation depends linearly on the square of the average magnetic field and on γ 2 . The energy of the relativistic electrons can be shown to be propor- tional to the synchrotron luminosity and to the −3/2 th power of the magnetic field. The strength of the magnetic field is usually estimated by minimizing the total energy, namely the sum of the energy in relativistic particles and the magnetic energy. This condition is fulfilled when the two different energy contributions are roughly equal, which enables the calculation of the average magnetic field. Furthermore, the degree of polarization of the synchrotron radi- ation gives a measure of the field uniformity (linear polarization corresponding to a uniform magnetic field). From the synchrotron emission method, typical magnetic field strengths in the ICM are computed as 0.4 − 1 µG [3].

1.3.2 Inverse Compton effect

Another method employed to measure the average magnetic field strength in clusters uses Inverse Compton (IC) radiation luminosities in addition to syn- chrotron emission. The inverse Compton effect consists in the scattering of microwave background photons by the relativistic electron population. As a result, the microwave background photons gain momentum from the elec- trons and are turned into X-ray or gamma photons. Given that the IC and synchrotron radiations both originate from the same relativistic electron pop- ulation, one arrives at a proportionality relation: Lsync ∝ u B , where both the

L IC u ph IC and synchrotron radiations can be measured and u ph is the density of the CMB photon field which can be calculated, therefore one can solve for u B , the energy density of the magnetic field. Results of the IC method give average magnetic fields of 0.2 − 1 µG[3].

1.3.3 Faraday rotation maps

The Faraday rotation effect completes the set of methods currently used to measure cluster magnetic fields. The principle of this effect consists in the

fact that a magnetic field induces different refractive indices for left-handed and right-handed circularly polarized beams. Linearly polarized beams can be decomposed into opposite-handed circularly polarized components. Upon passing through a magnetized medium, due to the different refractive indices, a phase difference occurs between the two components, which is equivalent to the rotation of the plane of polarization of the beam by an angle

e 3 λ 2

L

Ψ obs − Ψ int = ∆Ψ =

This is usually written as:

∆Ψ = λ 2 RM

where, comparing the two expressions, the rotation measure RM is defined as:

e 3 L

RM = 2πm 2

In practical units, this is written as

L

RM[rad/m 2 ] = 812

The ”internal” angle of the polarization plane as the beam is emitted, Ψ int , is assumed to be constant over all wavelengths. The angle of polarization at the observer, Ψ obs , is measured as a function of the wavelength and corrected for the rotation effect due to the magnetic field in our own galaxy. Fitting a straight line of the form Ψ obs = f (λ 2 ) to the measurement data yields, according to (2), the rotation measure which contains information about the magnetic field.

In the context of measuring cluster magnetic fields, the rotation measure is determined for each point on a grid of celestial coordinates, thus creating a rotation map of the emitting polarized source as it is seen through the mag- netized cluster medium. These two-dimensional rotation maps are analyzed, revealing information about the strength and structure of the magnetic field. Sources of extended polarized emissions are, for example, radio halos, relics and mini-halos [4].

Rotation maps project the three-dimensional magnetic field profile, com- bined with the electron number density profile, onto a two-dimensional plane by integrating along the line-of-sight. The most common method of disen- tangling the projection-effect from the RM and retrieving information about the average magnetic field in the cluster is to assume that the field varies on a constant scale, Λ c . The cluster is modeled as a set of cells of side-length Λ c inside which the electron number density and magnetic field strength are constant; the orientation of the field according to this model varies randomly from cell to cell. The scale on which the fields vary is usually closely associ- ated with the correlation length calculated from the RM map. Results from RM measurements fall typically in the range of 1-10 µG assuming cell sizes of around 10 kpc.

1.3.4 Reconciling magnetic fields derived from the three methods

Compared to synchrotron and IC measurements, Faraday rotation map analy- ses give magnetic fields which are roughly one order of magnitude larger. Sev- eral arguments can be invoked in order to explain this discrepancy. Firstly, the cluster magnetic field may show a range of coherence scales, and the presence of highly correlated small-scale fluctuations can enhance the rotation measures and thus produce higher estimates of the average field strength. Secondly, an anisotropic pitch-angle distribution would weaken the synchrotron radiation relative to the IC emission, leading to an underestimation of the IC-derived fields. Also, if a large relativistic population is located in the weak-field regions, a large part of the IC emission will come from low magnetic field-strength parts of the cluster.

1.3.5 The RM debate

The scientific debate regarding the calculation of magnetic fields from rotation maps has become increasingly intense as new methods of analyzing RM maps are being developed. As discussed by Rudnick [2], the evidence up to date is insufficient to prove the fact that the rotation measures are due to the ICM and not to a thin thermal skin mixed with the relativistic plasma of the emitting radio source. Moreover, Rudnick cites polarization percentages of 0.9±0.7 and 0.9 ± 1.0 in beams which were used to produce rotation maps, an uncertainty

which is too large for the results to still be considered relevant. Analyzing the rotation map of the Coma cluster by removing sources embedded in the cluster and keeping only the rotation measures of background sources, Rudnick found that this cluster showed, within error limits, a zero averaged absolute value of the rotation measure, even close to the cluster center.

Counterarguments in favor of cluster magnetic field-generated rotation maps are presented in a study by Clarke et. al. [1] who performed a sta- tistical survey of rotation measures as a function of the impact parameter of the emitting source with respect to the cluster center. A clear broadening of the RM distribuion toward small impact parameters (below roughly 1 Mpc) was found, justifying the conclusion that rotation maps are due to the ICM rather than being intrinsic to the emitting source.

1.3.6 Improved methods to determine the magnetic fields from ro-

tation maps

Recently, Enßlin and Vogt ([5], [6]) developed a method of analysing Faraday rotation maps which enabled them to use the RM of Hydra A to calculate the magnetic field of 7 ± 2µG at the cluster center, within the assumptions made about the most likely geometry of the field.

In the first stage, Vogt and Enßlin [6] assume an isotropically distributed