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We have given a brief overview of particle acceleration in astrophysical plasmas in general, and acceleration of electrons in the ICM in particular. We have pointed out the crucial role plasma waves and turbulence play in all acceleration mechanisms and outlined the equations that describe the generation, cascade and damping of these waves and the coupling of these processes to the particle kinetics and energising of the plasma and acceleration in both relativistic and nonrelativistic regimes.

We have applied these ideas to the ICM of clusters of galaxies with the aim of production of electron spectra which can explain the claimed hard X-ray emission either as non-thermal bremsstrahlung emission by nonrelativistic electrons or as inverse Compton emission via scattering of CMB photons by a population of relativistic electrons. It is shown that the first possibility which can come about by accelerating background electrons into a non-thermal tail is not a viable mechanism, as was pointed earlier in P01. The primary reason for this difficulty is due to the short Coulomb collision and loss timescales. Quite generally, it can be stated that at low rates of acceleration one obtains a hotter plasma and an insignificant non-thermal tail. Discernible tails can be obtained at higher rates of acceleration but only for short periods of time. For periods on the order of a billion year such rates will also cause excessive heating and will lead to runaway conditions where most of the electrons are accelerated to relativistic energies, at which they are no longer bound to the cluster, unless there exists a strong scattering agent.

This leads us to the model where hard X-rays are produced by the inverse Compton process and relativistic electrons. Moreover, even if the hard X-ray radiation turns out to be not present, or one finds a way to circumvent the above difficulties, we still require the presence of relativistic electrons to explain the radio emission. These electrons must be injected into the ICM by some other means. They can come from galaxies, specially when they are undergoing an active nuclear (or AGN) phase. Or they may be due to interactions of cosmic ray protons with thermal protons and the resultant pion decays. We have shown that just injection may not be sufficient, because for reasonable injected spectra the transport effects in the ICM modify the spectrum such that the effective radiating spectrum is inconsistent with what is required. Thus, a re-acceleration in the ICM is necessary and turbulence and merger shocks may be the agent of this acceleration. In this case, it also appears that a steady state scenario, like the hadronic mechanism described above, will in general give a flatter than required spectrum unless the electrons escape the ICM unhindered. This requirement is not reasonable because the expected tangled magnetic field will increase this time. But, more importantly, the presence of turbulence necessary for re-acceleration will result in a short mean free path and a much longer escape time.

A more attractive scenario is if the injection of electrons and/or the production of turbulence is episodic. For example for a short lived electron injected phase (from say an AGN) but a longer period of presence of turbulence one can determine the spectral evolution of the electrons subject to acceleration and losses. We have shown that for some periods of time lasting several times the acceleration timescales one can obtain electron spectra consistent with what is required by observations. The same will be true for a hadronic source if there is a short period of production of turbulence. In either case we are dealing with periods on the order of several hundreds of million years to a billion years, which is comparable with timescales expected from merging of Mpc size clusters with velocities of several thousands of km s-1 which are theoretically reasonable and agree with observations (see e.g. Bradac et al. 2006).

Acknowledgements The authors thank ISSI (Bern) for support of the team "Non-virialized X-ray components in clusters of galaxies". A.M. B. acknowledges the RBRF grant 06-02-16844 and a support from RAS Programs.

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