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In this paper we attempt to constrain the acceleration models based on the observations described in the papers by Durret et al. (2008), Rephaeli et al. (2008) and Ferrari et al. (2008) - Chapters 4-6, this volume, and the required spectrum of the accelerated electrons shown in Fig. 6 of Petrosian et al. 2008 - Chapter 10, this volume. However, before addressing these details we first compare various acceleration processes and stress the importance of plasma waves or turbulence (PWT) as an agent of scattering and acceleration, and then describe the basic scenario and equations for treatment of these processes. As pointed out below there is growing evidence that PWT plays an important role in acceleration of particles in general, and in clusters of galaxies in particular. The two most commonly used acceleration mechanisms are the following.

1.1. Electric field acceleration

Electric fields parallel to magnetic fields can accelerate charged particles and can arise as a result of magnetic field reconnection in a current sheet or other situations. For fields less than the so-called Dreicer field, defined as ED = kT / (e lambdaCoul), where

Equation 1 (1)

is the collision (electron-electron or proton-proton) mean free path 1, the rate of acceleration is less than the rate of collision losses and only a small fraction of the particles can be accelerated into a non-thermal tail of energy E < LeED. For the ICM ED ~ 10-14 V cm-1 and L ~ 1024 cm so that sub-Dreicer fields can only accelerate particles up to 100's of keV, which is far below the 10's of GeV electrons required by observations. Super-Dreicer fields, which seem to be present in many simulations of reconnection (Drake 2006, Cassak et al. 2006, Zenitani & Hoshino 2005), accelerate particles at a rate that is faster than the collision or thermalisation time tautherm. This can lead to a runaway and an unstable electron distribution which, as shown theoretically, by laboratory experiments and by the above mentioned simulations, most probably will give rise to PWT (Boris et al. 1970, Holman 1985).

In summary the electric fields arising as a result of reconnection cannot be the sole agent of acceleration in the ICM, because there are no large scale magnetically dominated cosmological flows, but it may locally produce an unstable particle momentum distribution which will produce PWT that can then accelerate particles.

1.2. Fermi acceleration

Nowadays this process has been divided into two kinds. In the original Fermi process particles of velocity v moving along magnetic field lines (strength B) with a pitch angle cosµ undergo random scattering by moving agents with a velocity u. Because the head (energy gaining) collisions are more probable than trailing (energy losing) collisions, on average, the particles gain energy at a rate proportional to (u / v)2 Dµµ, where Dµµ is the pitch angle diffusion rate. This, known as a second order Fermi process is what we shall call stochastic acceleration. In general, the most likely agent for scattering is PWT. An alternative process is what is commonly referred to as a first order Fermi process, where the actual acceleration occurs when particles cross a shock or any region of converging flow. Upon crossing the shock the fractional gain of momentum delta p / p propto ush / v. Ever since the 1970's, when several authors demonstrated that a very simple version of this process leads to a power law spectrum that agrees approximately with observations of the cosmic rays, shock acceleration is commonly invoked in space and astrophysical plasmas. However, this simple model, though very elegant, has some shortcomings specially when applied to electron acceleration in non-thermal radiating sources. Moreover, some of the features that make this scenario for acceleration of cosmic rays attractive are not present in most radiating sources where one needs efficient acceleration of electrons to relativistic energies from a low energy reservoir.

The original, test particle theory of diffusive shock acceleration (DSA), although very elegant and independent of geometry and other details (e.g. Blandford & Ostriker 1978) required several conditions such as injection of seed particles and of course turbulence. A great deal of work has gone into addressing these aspects of the problem and there has been a great deal of progress. It is clear that nonlinear effects (see e.g. Drury 1983, Blandford & Eichler 1987, Jones & Ellison 1991, Malkov & Drury 2001) and losses (specially for electrons) play an important role and modify the resultant spectra and efficiency of acceleration. Another important point is the source of the turbulence or the scattering agents. A common practice is to assume Bohm diffusion (see e.g. Ellison et al. 2005). Second order acceleration effects could modify the particle spectra accelerated by shocks (see e.g. Schlickeiser et al. 1993, Bykov et al. 2000). Although there are indications that turbulence may be generated by the shocks and the accelerated particle upstream, many details (e.g. the nature and spectrum of the turbulence) need to be addressed more quantitatively. There has been progress on the understanding of generation of the magnetic field and turbulence on strong shocks (Bell & Lucek 2001, Amato & Blasi 2006, Vladimirov et al. 2006) as required in recent observations of supernova remnants (see e.g. Völk et al. 2005). There is also some evidence for these processes from observations of heliospheric shocks (see e.g. Kennel et al. 1986, Ellison et al. 1990). Basic features of particle acceleration by cosmological shocks were discussed by Bykov et al. 2008a - Chapter 7, this volume, so we will concentrate here on the stochastic acceleration perspective.

1.3. Stochastic acceleration

The PWT needed for scattering can also accelerate particles stochastically with a rate DEE / E2, where DEE is the energy diffusion coefficient, so that shocks may not be always necessary. In low beta plasmas, betap = 2(vs / vA)2 < 1, where the Alfven velocity vA = (B2 / 4pi rho)1/2, the sound velocity vs = (kT / m)1/2, rho = nm is the mass density and n is the number density of the gas, and for relativistic particles the PWT-particle interactions are dominated by Alfvenic turbulence, in which case the rate of energy gain DEE / E2 = (vA / v)2 Dµµ << Dµµ, so that the first order Fermi process is more efficient. However, at low energies and/or in very strongly magnetised plasmas, where vA can exceed c, the speed of light 2, the acceleration rate may exceed the scattering rate (see Pryadko & Petrosian 1997), in which case low energy electrons are accelerated more efficiently by PWT than by shocks. 3

Irrespective of which process dominates the particle acceleration, it is clear that PWT has a role in all of them. Thus, understanding of the production of PWT and its interaction with particles is extremely important. Moreover, turbulence is expected to be present in most astrophysical plasmas including the ICM and in and around merger or accretion shocks, because the ordinary and magnetic Reynolds numbers are large. Indeed turbulence may be the most efficient channel of energy dissipation. In recent years there has been a substantial progress in the understanding of MHD turbulence (Goldreich & Sridhar 1995, Goldreich & Sridhar 1997, Lithwick & Goldreich 2003, Cho & Lazarian 2002, Cho & Lazarian 2006). These provide new tools for a more quantitative investigation of turbulence and the role it plays in many astrophysical sources.

1 The proton-proton or ion-ion mean free path will be slightly smaller because of the larger value of the Coulomb logarithm lnLambda ~ 40 in the ICM. Back.

2 Note that the Alfven group velocity vg = c [vA2 / (vA2 + c2)]1/2 is always less than c. Back.

3 In practice, i.e. mathematically, there is little difference between the two mechanisms (Jones 1994), and the acceleration by turbulence and shocks can be combined (see below). Back.

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