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We now address the problem of particle acceleration in clusters of galaxies. The current information on the ICM does not allow us to treat the problem as outlined above by solving the coupled kinetic equations. In what follows we make reasonable assumptions about the turbulence and the particle diffusion coefficients, and then solve the particle kinetic equation to determine N(E, t). We first consider the apparently simple scenario of acceleration of the background thermal particles. Based on some general arguments, Petrosian (2001, P01 hereafter) showed that this is not a viable mechanism. Here we carry out a more accurate calculation and show that this indeed is the case. This leads us to consider the transport and acceleration of high energy particles injected into the ICM by other processes.

3.1. Acceleration of background particles

The source particles to be accelerated are the ICM hot electrons subject to diffusion in energy space by turbulence and Coulomb collisions, acceleration by turbulence or shocks, and energy losses due to Coulomb collisions 8. We start with an ICM of kT = 8 keV, n = 10-3 cm-3 and assume a continuous injection of turbulence so that its density remains constant resulting in a time independent diffusion and acceleration rate. The results described below is from a recent paper by Petrosian & East (2007, PE07 hereafter). Following this paper we assume a simple but generic energy dependence of these coefficients. Specifically we assume a simple acceleration rate or timescale

Equation 12 (12)

Fig. 2 shows a few examples of these time scales along with the effective Coulomb (plus IC and synchrotron) loss times as described in Fig. 3 of Petrosian et al. 2008 - Chapter 10, this volume.

Figure 2

Figure 2.Acceleration and loss timescales for ICM conditions based on the model described in the text. We use the effective Coulomb loss rate given by Petrosian et al. 2008 - Chapter 10, this volume, and the IC plus synchrotron losses for a CMB temperature of TCMB = 3 K and an ICM magnetic field of B = 1 µG. We also use the simple acceleration scenario of Eq. 12 for Ec = 0.2mec2 ( ~ 100 keV) and for the three specified values of p and times tau0 / tauCoul (from Petrosian & East 2007).

We then use Eq. 10 to obtain the time evolution of the particle spectra. After each time step we use the resultant spectrum to update the Coulomb coefficients as described by Petrosian et al. 2008 - Chapter 10, this volume. At each step the electron spectrum can be divided into a quasi-thermal and a 'non-thermal' component. A best fit Maxwellian distribution to the quasi-thermal part is obtained, and we determine a temperature and the fraction of the thermal electrons. The remainder is labelled as the non-thermal tail. (For more details see PE07). The left and middle panels of Fig. 3 show two spectral evolutions for two different values of acceleration time tau0 / tauCoul = 0.013 and 2.4, respectively, and for Ec = 25 keV and p = 1. The last spectrum in each case is for time t = tau0, corresponding to an equal energy input for all cases. The initial and final temperatures, the fraction of particles in the quasi-thermal component Nth, and the ratio of non-thermal to thermal energies Rnonth are shown for each panel. The general feature of these results is that the turbulence causes both acceleration and heating in the sense that the spectra at low energies resemble a thermal distribution but also have a substantial deviation from this quasi-thermal distribution at high energies which can be fitted by a power law over a finite energy range. The distribution is broad and continuous, and as time progresses it becomes broader and shifts to higher energies; the temperature increases and the non-thermal 'tail' becomes more prominent. There is very little of a non-thermal tail for tau0 > tauCoul and most of the turbulent energy goes into heating (middle panel). Note that this also means that for a steady state case where the rate of energy gained from turbulence is equal to radiative energy loss rate (in this case thermal Bremsstrahlung, with time scale >> tauCoul) there will be an insignificant non-thermal component. There is no distinct non-thermal tail except at unreasonably high acceleration rate (left panel). Even here there is significant heating (almost doubling of the temperature) within a short time ( ~ 3 × 105 yr). At such rates of acceleration most particles will end up at energies much larger than the initial kT and in a broad non-thermal distribution. We have also calculated spectra for different values of the cutoff energy Ec and index p. As expected for larger (smaller) values of Ec and smaller (higher) values of p the fraction of non-thermal particles is lower (higher).

Figure 3

Figure 3. Upper left panel: Evolution with time of electron spectra in the presence of a constant level of turbulence that accelerates electrons according to Eq. 12 with tau0 / tauCoul = 0.013, Ec = 0.2 (~ 100 keV) and p = 1. For the last spectrum obtained for time t = tau0, the low end of the spectrum is fitted to a thermal component (dashed curve). The residual 'non-thermal' part is shown by the dotted curve. We also give the initial and final values of the temperature, the fraction of electrons in the thermal component Nth, and the ratio of energy of the non-thermal component to the thermal components Rnonth. Upper right panel: Same as above except for tau0 / tauCoul = 2.4. Note that now there is only heating and not much of acceleration. Lower panel: Evolution with time (in units of tau0) of electron spectral parameters, T(t) / T0, Nth, Rnonth and the power-law index delta for indicated values of tau0 / tauCoul and for p = 0 and Ec = 100 keV. Note that for models with the same value of p at t = tau0 roughly the same amount of energy has been input into the ICM (from Petrosian & East 2007).

The evolution in time of the temperature (in units of its initial value), the fraction of the electrons in the 'non-thermal' component, the energy ratio Rnonth as well as an index delta = -d lnN(E) / d lnE for the non-thermal component are shown in the right panel of Fig. 3. All the characteristics described above are more clearly evident in this panel and similar ones for p = -1 and +1. In all cases the temperature increases by more than a factor of 2. This factor is smaller at higher rates of acceleration. In addition, high acceleration rates produce flatter non-thermal tails (smaller delta) and a larger fraction of non-thermal particles (smaller Nth) and energy (Rnonth).

It should be noted that the general aspects of the above behaviour are dictated by the Coulomb collisions and are fairly insensitive to the details of the acceleration mechanism which can affect the spectral evolution somewhat quantitatively but not its qualitative aspects. At low acceleration rates one gets mainly heating and at high acceleration rate a prominent non-thermal tail is present but there is also substantial heating within one acceleration timescale which for such cases is very short. Clearly in a steady state situation there will be an insignificant non-thermal component. These findings support qualitatively findings by P01 and do not support the presence of distinct non-thermal tails advocated by Blasi (2000) and Dogiel et al. (2007), but agree qualitatively with the more rigorous analysis of Wolfe & Melia (2006). For further results, discussions and comparison with earlier works see PE07.

We therefore conclude that the acceleration of background electrons stochastically or otherwise and non-thermal bremsstrahlung are not a viable mechanism for production of non-thermal hard X-ray excesses observed in some clusters of galaxies.

3.2. Acceleration of injected particles

The natural way to overcome the above difficulties is to assume that the radio and the hard X-ray radiation are produced by relativistic electrons injected in the ICM, the first via synchrotron and the second via the inverse Compton scattering of CMB photons. The energy loss rate of relativistic electrons can be approximated by (see P01)

Equation 13 (13)


Equation 14 (14)

are twice the loss time and the energy where the total loss curve reaches its maximum 9 (see Fig. 4). Here r0 = e2 / (me c2 ) = 2.82 × 10-13 cm is the classical electron radius, uph (due to the CMB) and B2 / 8pi are photon (primarily CMB) and magnetic field energy densities. For the ICM B ~ µG, n = 10-3 cm-3 and the Coulomb logarithm lnLambda = 40 so that tauloss = 6.3 × 109 yr and Ep = 235 me c2.

Figure 4

Figure 4. Comparison of the energy dependence of the total loss time (radiative plus Coulomb; thick solid line) with timescales for scattering tauscat (lower dashed line), acceleration tauac (thin solid lines), crossing time taucross ~ R / cbeta (long dashed lines) and escape time Tesc ~ taucross2 / tauscat (upper dashed line) of electrons. At low energies the scattering and escape times are dominated by Coulomb scattering but at high energies turbulence scattering becomes more important. The acceleration is constant at low energies (A(E) propto E) but increases at high energies, corresponding to an acceleration rate A(E) --> a constant at high E or to a turbulence spectral index q = 1. A chaotic magnetic field with scale of 10 kpc will increase the crossing time to the primed curve. The arrows show the maximum scattering times for which the escape times are equal or longer than the total loss times. The critical energies where the acceleration time is equal to the escape time, the Coulomb and inverse Compton loss times are shown.

The electrons are scattered and gain energy if there is some turbulence in the ICM. The turbulence should be such that it resonates with the injected relativistic electrons and not the background thermal nonrelativistic electrons for the reasons described in the previous section. Relativistic electrons will interact mainly with low wavevector waves in the inertial range where W(k) propto k-q with the index q ~ 5/3 or 3/2 for a Kolmogorov or Kraichnan cascade. There will be little interaction with nonrelativistic background electrons if the turbulence spectrum is cut off above some maximum wave vector kmax whose value depends on viscosity and magnetic field. The coefficients of the transport equation (Eq. 10) can then be approximated by

Equation 15 (15)

For a stochastic acceleration model at relativistic energies a = 2, but if in addition to scattering by PWT there are other agents of acceleration (e.g. shocks) then the coefficient a will be larger than 2. In this model the escape time is determined by the crossing time Tcross ~ R/c and the scattering time tauscat ~ Dµµ-1. We can then write Tesc ~ Tcross(1 + Tcross / tauscat). Some examples of these are shown in Fig. 4. However, the escape time is also affected by the geometry of the magnetic field (e.g. the degree of its entanglement). For this reason we have kept the form of the escape time to be more general. In addition to these relations we also need the spectrum and rate of injection to obtain the spectrum of radiating electrons. Clearly there are several possibilities. We divided it into two categories: steady state and time dependent. In each case we first consider only the effects of losses, which means D = 0 in the above expressions, and then the effects of both acceleration and losses.

3.2.1. Steady state cases

By steady state we mean variation timescales of order or larger than the Hubble time which is also longer than the maximum loss time tauloss / 2. Given a particle injection rate dot{Q} = dot{Q}0 f(E) (with integf(E)dE = 1) steady state is possible if Tesc = dot{Q}0 / integ N(E)E-s dE.

In the absence of acceleration (D = 0) Eq. 10 can be solved analytically. For the examples of escape times given in Fig. 4 (Tesc > tauloss) one gets the simple cooling spectra N = (dot{Q} tauloss / Ep) integeinfty f(E)dE / (1 + (E / Ep)2), which gives a spectral index break at Ep from index p0 - 1 below to p0 + 1 above Ep, for an injected power law f(E) propto E-p0. For p0 = 2 this will give a high energy power law in rough agreement with the observations but with two caveats. The first is that the spectrum of the injected particles must be cutoff below E ~ 100 mec2 to avoid excessive heating and the second is that this scenario cannot produce the broken power law or exponential cutoff we need to explain the radio spectrum of Coma (see Fig. 6 and the discussion in Petrosian et al. 2008 - Chapter 10, this volume). A break is possible only if the escape time is shorter than tau0 in which case the solution of the kinetic equation for a power law injected spectrum (p0 > 1 and s > -1) leads to the broken power law

Equation 16 (16)

where Ecr = Ep((s + 1) (Tesc / tauloss)-1/(s+1). Thus, for p0 ~ 3 and s = 0 and Tesc appeq 0.02tauloss we obtain a spectrum with a break at Ecr ~ 104, in agreement with the radio data (Rephaeli 1979 model). However, this also means that a large fraction of the E<Ep electrons escape from the ICM, or more accurately from the turbulent confining region, with a flux of Fesc(E) propto N(E) / Tesc(E). Such a short escape time means a scattering time which is only ten times shorter than the crossing time and a mean free path of about ~ 0.1R ~ 100 kpc. This is in disagreement with the Faraday rotation observations which imply a tangled magnetic field equivalent to a ten times smaller mean free path. The case for a long escape time was first put forth by Jaffe (1977).

Thus it appears that in addition to injection of relativistic electrons we also need a steady presence or injection of PWT to further scatter and accelerate the electrons. The final spectrum of electrons will depend on the acceleration rate and its energy dependence. In general, when the acceleration is dominant one expects a power law spectrum. Spectral breaks appear at critical energies when this rate becomes equal to and smaller than other rates such as the loss or escape rates (see Fig. 4). In the energy range where the losses can be ignored electrons injected at energy E0(f(E) = delta(E - E0)) one expects a power law above (and below, which we are not interested in) this energy. In the realistic case of long Tesc (and/or when the direct acceleration rate is larger than the rate of stochastic acceleration (i.e. a >> 1) then spectral index of the electrons will be equal to -q + 1 requiring a turbulence spectral index of q = 4 which is much larger than expected values of 5/3 or 3/2 (see Park & Petrosian 1995). This spectrum will become steeper (usually cut off exponentially) above the energy where the loss time becomes equal to the acceleration time tauac = E / A(E) or at Ecr = (Ep a D tauloss)1/(3 - q). Steeper spectra below this energy are possible only for shorter Tesc. The left panel of Fig. 5 shows the dependence of the spectra on Tesc for q = 2 and s = 0 (acceleration and escape times independent of E). The spectral index just above E0 is p = (9/4 + 2tauac / Tesc)1/2 - 1.5. In the limit when Tesc --> infty the distribution approaches a relativistic Maxwellian distribution N propto E2e - E / Ecr. For a cut-off energy Ecr ~ 104 this requires an acceleration time of ~ 108 yr and for a spectral index of p=3 below this energy we need Tesc ~ tauac/18 ~ 5 × 106 yr which is comparable to the unhindered crossing time. This is too short. As shown in Fig. 4 any scattering mean free path (or magnetic field variation scale) less than the cluster size will automatically give a longer escape time and a flatter than required spectrum. For further detail on all aspects of this case see Park & Petrosian (1995), P01 and Liu et al. (2006).

Figure 5

Figure 5. Left panel: The steady state electron spectra injected at energy E0 and subject to continuous acceleration by turbulence with spectral index q = 2 for different values of the ratio Tesc / tauloss. Note that tau0 in the label is the same as tauloss in the text. For very high values of this ratio we get a relativistic Maxwellian distribution (from Liu et al. 2006). Right panel: Evolution with time of a power law injected spectrum (top line) subject to Coulomb and inverse Compton (plus synchrotron) losses as given by Eq. 17. Solid lines for b=2 (t / tau0 = 10-n; n = 3, 2, 1, 0), and dashed lines for b = 60 (t / tau0 = 10-2.18 + n/3; n = 0, 1, 2, 3, 4). The heavy portions show the energy range of the electrons needed for production of radio and hard X-rays.

In summary there are several major difficulties with the steady state model.

3.2.2. Time dependent models

We are therefore led to consider time dependent scenarios with time variation shorter than the Hubble time. The time dependence may arise from the episodic nature of the injection process (e.g. varying AGN activity) and/or from episodic nature of turbulence generation process (see e.g. Cassano & Brunetti 2005). In this case we need solutions of the time dependent equation (Eq. 10). We start with the generic model of a prompt single-epoch injection of electrons with Q(E, t) = Q(E) delta(t - t0). More complex temporal behaviour can be obtained by the convolution of the injection time profile with the solutions described below. The results presented below are from P01. Similar treatments of the following cases can be found in Brunetti et al. (2001) and Brunetti & Lazarian (2007).

It is clear that if there is no re-acceleration, electrons will lose energy first at highest and lowest energies due to inverse Compton and Coulomb losses, respectively. Particles will be peeled away from an initial power law with the low and high energy cut-offs moving gradually toward the peak energy Ep. A more varied and complex set of spectra can be obtained if we add the effects of diffusion and acceleration. Simple analytic solutions for the time dependent case are possible only for special cases. Most of the complexity arises because of the diffusion term which plays a vital role in shaping the spectrum for a narrow injection spectrum. For some examples see Park & Petrosian (1996). Here we limit our discussion to a broad initial electron spectrum in which case the effects of this term can be ignored until such features are developed. Thus, if we set D(E) = 0, which is a particularly good approximation when a >> 1, and for the purpose of demonstration if we again consider the simple case of constant acceleration time (q = 2 and A(E) = a DE), then the solution of Eq. 10 gives

Equation 17 (17)

where delta2 = 1 - b2 / 4, b = a D tau0 Ep2 = tauloss / tauac and T± = 1 ± btan(delta t / tauloss) / (2delta). Note that b = 0 correspond to the case of no acceleration described above. This solution is valid for b2 < 4. For b2 > 4 we are dealing with an imaginary value for delta so that tangents and cosines become hyperbolic functions with delta2 = b2 / 4 - 1. For delta = 0 or b = 2 this expression reduces to

Equation 18 (18)

The right panel of Fig. 5 shows the evolution of an initial power law spectrum subjected to weak acceleration (b = 2, solid lines) and a fairly strong rate of acceleration (b = 60, dashed line). As expected with acceleration, one can push the electron spectra to higher levels and extend it to higher energies. At low rates of acceleration the spectrum evolves toward the generic case of a flat low energy part with a fairly steep cutoff above Ep. At higher rates, and for some periods of time comparable to tauac, the cut off energy Ecr will be greater than Ep and there will be a power law portion below it. 10 As evident from this figure there are periods of time when in the relevant energy range (thick solid lines) the spectra resemble what is needed for describing the radio and hard X-ray observations from Coma described in Fig. 6 of Petrosian et al. 2008 - Chapter 10, this volume.

In summary, it appears that a steady state model has difficulties and that the most likely scenario is episodic injection of relativistic particles and/or turbulence and shocks which will re-accelerate the existing or injected relativistic electrons into a spectral shape consistent with observations. However these spectra are short lived, lasting for periods of less than a billion years.

8 In our numerical results we do include synchrotron, IC and Bremsstrahlung losses. But these have an insignificant effect in the case of nonrelativistic electrons under investigation here. Back.

9 We ignore the Bremsstrahlung loss and the weak dependence on E of Coulomb losses at nonrelativistic energies. We can also ignore the energy diffusion rate due to Coulomb scattering. Back.

10 At even later times than shown here on gets a large pile up at the cut off energy (see P01). This latter feature is of course artificial because we have neglected the diffusion term which will smooth out such features (see Brunetti & Lazarian 2007). Back.

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