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Astron. Astrophys. 332, 395-409 (1998)

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2. Particle acceleration at cluster accretion shocks

2.1. Geometry

Properties of the accretion shocks of clusters of galaxies, the shock radius [FORMULA] and the velocity [FORMULA] of the infalling matter measured in the shock frame, are related to the depth of the gravitational potential of the cluster. Since the cluster gas temperature also depends on the potential, the shock properties can be expressed in terms of the observed temperature [FORMULA]. Using results of one-dimensional simulations of accreting flows onto clusters (Ryu & Kang 1997), Kang et al.  (1997) derived

[EQUATION]

These estimates depend on the assumption of a spherical flow, of a polytropic index of the gas of [FORMULA], and they depend weakly on the assumed Einstein-de Sitter cosmology. Since the real infall pattern to clusters should be aspherical and correlated with the surrounding large-scale structure (Colberg et al. 1997), the shock radii and the shock velocities are expected to differ from these estimates. They might be regarded as average quantities, where from a single cluster can easily deviate. Inserting the X-ray temperature of the Coma cluster of 8.2 keV (Briel et al. 1992) and the redshift of [FORMULA] (Fadda et al. 1996) gives a shock radius of [FORMULA] and a shock velocity of [FORMULA]. If 1253+275 is at the position of this shock, the projected distance to the cluster center of [FORMULA] (Giovannini et al. 1991) implies an angle between line-of-sight and normal of the shock front of [FORMULA]. The true angle is ambiguous ([FORMULA] or [FORMULA]), and depends on the location of the relic on the line-of-sight (foreground or background with respect to the cluster center). A possible configuration is sketched in Fig. 2, where the relic is located on the back side of the cluster accretion shock, since then a physical connection to the nearby radio galaxy NGC 4789 is possible as explained in Sect. 5. The length of the relic is [FORMULA], whereas the thickness cannot be given directly from radio observations. We assume a thickness of [FORMULA], which is smaller than the observed projected thickness due to the action of shock compression in the infall direction. A detailed model of the structure of the relic would allow to deproject the apparent surface. For only moderately large projection angles ([FORMULA]) resulting deprojected surfaces are not much bigger than the observed ones. Since the geometric quantities are only used for order of magnitude calculations of the infalling kinetic energy flux, the shock efficiency, and the diffusion coefficient within the radio plasma, we prefer not to specify a detailed geometrical model. We equate projected to real surface of the relic, which is a weak underestimate. The relic is slightly V-shaped and the radio emission is more sharply edged on the outer side (Giovannini et al. 1991). The V-shape could result from an aspherical shock, but would also be easily explained by an intrinsic shape of the magnetized plasma, since the relic is not seen edge-on. The edges of the observed projected radio structure should differ. The radio plasma seen at the inner edge (left edge in Fig. 2) is more distant to the shock than that at the outer edge (right). The outer edge should be sharper due to the better confined plasma at the shock side, and it should have a lower than average spectral index, since the electrons there are reaccelerated recently. Whereas the inner edge is expected to be smoother and to have a higher than average spectral index. The reason for the latter is that the reaccelerated electrons within the radio plasma seen there, at a position more distant to the shock, had more time for cooling.

[FIGURE] Fig. 2. Sketch of the assumed geometry in the Coma cluster. The accretion flow is indicated by arrows. The shaded region is the intra-cluster medium. The dotted area represents regions of radio plasma. The line-of-sight and the line connecting the relic and the cluster center are lying within the plane of the figure. NGC4789 is placed upstream above the radio relic as a source of relativistic electrons, as discussed in Sect. 5. Its line-of-sight velocity component points away from the observer. All positions could be different due to shifts parallel to the line-of-sight.

2.2. Compression ratio

The radio spectral index [FORMULA] ([FORMULA]) of 1253+275 is [FORMULA] (Giovannini et al. 1991) and is therefore larger than [FORMULA], expected for a strong shock accelerating electrons, with simultaneous synchrotron and inverse-Compton losses. Larger spectral indices are found at the inner edge of the relic, showing evidence for steepening of the electron momentum distribution due to the higher age of the electrons seen there, more distant to the acceleration region of the shock. The radius of the accretion shock structure (assumed to have a spherical surface) is large compared to every intrinsic length scale of the acceleration process. Thus the theory of planar shocks can be applied. The spectral index can be explained by allowing the shock compression ratio R to be lower than the canonical value [FORMULA] of a strong shock of gas with a polytropic index of [FORMULA]. The radio spectral index [FORMULA] of an equilibrium electron population accelerated and cooled at the same time, is related to the shock compression ratio by (e.g. Drury 1983):

[EQUATION]

where a polytropic index of [FORMULA] is assumed. The compression ratio is therefore [FORMULA]. From the theory of shocks (Landau & Lifschitz 1966) the pressure and temperature ratio between the down- and up-stream region (inside and outside the cluster shock front, in the following denoted by [FORMULA] and [FORMULA], etc.) can be derived:

[EQUATION]

For 1253+275 the resulting pressure ratio is [FORMULA]. The temperature ratio [FORMULA] requires that the infalling matter has a temperature of [FORMULA] keV, assuming the post-shock temperature to be roughly half the central cluster temperature. Simulations predict this radial temperature decrease (Navarro et al. 1995), and observations indicate a radially falling temperature profile in clusters (Honda et al. 1996; Markevitch 1996; Markevitch et al. 1996, 1998).

This temperature of the infalling matter is reasonable, since this gas should flow mainly out of sheets and filaments of the cosmological large-scale structure, and therefore was preheated by the accretion shocks at the boundaries of these structures. A simulation of cosmological structure formation by Kang et al. (1996) shows that typical temperatures of filaments are above 0.1 keV. Adiabatic compression and internal shocks within the flows onto clusters might raise these temperatures to a level of [FORMULA] keV, which is needed in order to explain the steepness of the synchrotron spectrum of cluster relic sources caused by weak shocks.

The temperature of the infalling matter gives the sound velocity [FORMULA], which enters into a second estimate of the shock velocity, using the theory of planar shock waves (Landau & Lifschitz 1966):

[EQUATION]

This estimate gives a shock velocity for Coma of [FORMULA], slightly smaller than that following from the theory of Kang et al. (1997). The difference vanishes if a smaller temperature drop of [FORMULA] from the center to the shock radius is used instead of a factor of 2. Comparing both estimates shows self-consistency of our model, but this is not a completely independent test, since both values depend mainly on the same quantity ([FORMULA]).

In the model of acceleration of ultra-high-energy cosmic rays at cluster accretion shocks of Kang et al. (1997) a compression ratio close to [FORMULA] is favored in one particular model to fit the shape of the spectrum of ultra-high-energy cosmic rays using a low and energy independent escape efficiency of cosmic rays from the cluster against the upstream flow. However, a lower compression ratio (and therefore a steeper cosmic ray production spectrum) can be compensated by taking a reasonable energy dependence of the escape efficiency into account. In fact, in Kang et al. (1997) the acceleration and escape efficiency was rather low, with [FORMULA] giving a reasonable fit only to the observed high-energy cosmic ray data. Using the compression ratio of [FORMULA] gives a proton spectrum of [FORMULA] implying for the same flux at a few [FORMULA] eV a combined efficiency which is in the range [FORMULA] (Fig. 5 in Kang et al. 1997) and so consistent with estimates for supernova remnants (e.g. Drury et al. 1989; Völk 1997).

2.3. Shock efficiency

A rough estimate of the gas densities on both sides of the shock sphere can be gained by extrapolating the electron density profile of the Coma cluster, viz. [FORMULA], with the parameters [FORMULA], [FORMULA], and [FORMULA] (Briel et al. 1992). This gives a gas density of the order [FORMULA] at a cluster radius of [FORMULA] Mpc. The density of the infalling matter should therefore be of the order of [FORMULA]. This is consistent with the somewhat lower gas density in the intergalactic space of [FORMULA] far away from clusters, derived e.g. from radio observation of the giant radio galaxy 1358+305 (Parma et al. 1996).

The projected extent of 1253+275 is [FORMULA] (Giovannini et al. 1991), but its real surface S exposed to the accretion flow might be larger. The kinetic power of the infalling matter on the relic

[EQUATION]

is mainly transformed into thermal energy, but some fraction is converted into relativistic particles. [FORMULA] is the velocity of the infalling matter measured in the cluster's inertial frame. The integrated radio power between 10 MHz and 10 GHz of [FORMULA] (Giovannini et al. 1991) is three orders of magnitude lower, and can be easily powered by the dissipative processes in the shock. Assuming a field strength of [FORMULA] G implies that only 10% of the electron radiation losses are synchrotron emission, the rest are inverse-Compton losses by scattering of microwave background photons. The amount of energy loss of the electrons [FORMULA] and the power of the flow [FORMULA] onto the relic gives the necessary minimal efficiency of shock acceleration. The efficiency needs to be higher than [FORMULA] in order to account for the radiative energy requirements and for escaping electrons. This number might be compared to the efficiency of shock acceleration in other astrophysical objects: It is believed that supernova blast waves have efficiencies of [FORMULA] for the acceleration of protons as is necessary in order to explain the galactic cosmic rays below [FORMULA] eV by supernovae (e.g. Drury et al. 1989; Völk 1997). This shows that the assumed cluster efficiency is reasonable.

2.4. Electron spectrum

We apply the theory of plane-parallel shock acceleration, because of the large radius of the shock sphere. We mark upstream quantities with index 1, and downstream quantities with index 2. We use a momentum independent diffusion coefficient, because of the success of this simplification in other circumstances (Biermann 1993, 1996; Wiebel-Sooth et al. 1997). The distribution function of the accelerated electrons cooling by inverse Compton and synchrotron emission is

[EQUATION]

(Webb et al. 1984), where the normalization C can be approximated by being independent of the position x, and [FORMULA] is the spectral index of the injection electron distribution in three-dimensional momentum space. The cutoff momentum is

[EQUATION]

with U denoting the flow velocity in the inertial frame of the shock ([FORMULA] and [FORMULA]), [FORMULA] is the diffusion coefficient, [FORMULA] is the injection momentum into the acceleration mechanism, and a is the ratio between loss and acceleration time scale:

[EQUATION]

The energy loss time scale [FORMULA] of an electron with momentum p is dominated by synchrotron and inverse-Compton losses with:

[EQUATION]

where [FORMULA] is the energy density of the photon field, which is dominated by the microwave background. In Sect. 3.4it is argued that the main fraction of the synchrotron emitting volume belongs to the post-shock region. Thus the electron population, integrated over the synchrotron emitting relic volume [FORMULA] (where S is the surface area, and D is the thickness), can be assumed to be

[EQUATION]

This spectrum has a break at [FORMULA] between the spectral index q and [FORMULA], and a cutoff at [FORMULA]. The break in the momentum spectrum at

[EQUATION]

leads to a break in the radio spectrum at [FORMULA]. Using this relation and solving Eq. 12for the ratio [FORMULA], which is the time the magnetized plasma needed to pass the shock, and therefore the age of the relic as a tracer of the shock structure, we get

[EQUATION]

We use a thickness of [FORMULA], roughly [FORMULA] of the width of the projected structure of 1253+275, due to compression. The age of 1253+275 being a shock-tracer estimated from the kinematics [FORMULA], and that following from Eq. 13: [FORMULA], using the lowest observation frequency of 151 MHz as an upper limit to the break frequency [FORMULA], do not fit exactly. Regarding the approximations of this theory, and the uncertainties of the observational quantities only an order of magnitude accordance can be expected. But since these two estimates of the relic age depend on different observational quantities ([FORMULA] and [FORMULA]), this is a further test of the theory, which could have failed by orders of magnitude. Reasons for a discrepancy might also be hidden in neglected properties of the acceleration mechanism. For instance if the relic has passed the shock already, the diffusion coefficient directly behind the shock [FORMULA] differs from that within the relic [FORMULA], and the ratio [FORMULA] would enter Eqs. 12and 13.

The ratio between break and cutoff momentum of 1253+275

[EQUATION]

must be smaller than 0.18, since the radio spectrum does not exhibit any break or cutoff over a frequency range of about [FORMULA] orders of magnitude. The term in brackets is in the range [FORMULA] and can be neglected for an order of magnitude estimate of the diffusion coefficient behind the shock:

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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