Evidence that the maximum electron energy in hotspots of FR II galaxies is not determined by synchrotron cooling
Abstract
It has been suggested that relativistic shocks in extragalactic sources may accelerate the highest energy cosmic rays. The maximum energy to which cosmic rays can be accelerated depends on the structure of magnetic turbulence near the shock but recent theoretical advances indicate that relativistic shocks are probably unable to accelerate particles to energies much larger than a PeV. We study the hotspots of powerful radiogalaxies, where electrons accelerated at the termination shock emit synchrotron radiation. The turnover of the synchrotron spectrum is typically observed between infrared and optical frequencies, indicating that the maximum energy of nonthermal electrons accelerated at the shock is TeV for a canonical magnetic field of 100 G. Based on theoretical considerations we show that this maximum energy cannot be constrained by synchrotron losses as usually assumed, unless the jet density is unreasonably large and most of the jet upstream energy goes to nonthermal particles. We test this result by considering a sample of hotspots observed with high spatial resolution at radio, infrared and optical wavelengths.
keywords:
galaxies: active – galaxies: jets – – acceleration of particles – radiation mechanisms: nonthermal – shock waves1 Introduction
Active Galactic Nuclei (AGN) have been proposed as sources of Ultra High Energy Cosmic Rays (UHECRs). Shocks with different velocities and extents are present in jets of FanaroffRiley (FR) radiogalaxies (Fanaroff & Riley, 1974), where particles can be accelerated via diffusive shock acceleration. In particular, relativistic and mildly relativistic shocks with velocity at the jet termination region might accelerate particles with Larmor radius , where kpc is the jet width. Particles with such a large in a magnetic field G have energy
(1) 
as expected for UHECRs (Lagage & Cesarsky, 1983; Hillas, 1984). In particular, Rachen & Biermann (1993) and Norman et al. (1995) concluded that hotspots of FR II radiogalaxies are plausible sources of UHECRs (see also Nagano & Watson, 2000; Kotera & Olinto, 2011). But, there are two assumptions behind Eq. (1): 1) particles diffuse in the Bohm regime, i.e. the meanfree path is , and 2) the magnetic field persists over distances downstream of the shock.
Protons are the dominant component of UHECRs. Given that ion radiation losses are slow in low density plasmas such as AGN jets, protons can be accelerated up to energies if both assumptions are satisfied. However, there are no hadronic radiative signatures from hotspots and therefore we do not have any observational information about . In consequence, we investigate the validity of assumptions 1) and 2) by modelling the synchrotron emission produced by nonthermal electrons accelerated at the jet reverse shock. The synchrotron turnover at Hz typically observed in hotspots of FR II galaxies (e.g. Meisenheimer & Heavens, 1986; Meisenheimer et al., 1997; Tavecchio et al., 2005; Stawarz et al., 2007; Werner et al., 2012) indicates that the maximum energy of nonthermal electrons is
(2) 
(Ginzburg & Syrovatskii, 1964), much smaller than for reasonable values of the magnetic field. The traditional assumption is that is determined by synchrotron cooling and therefore the diffusion coefficient of particles with such an energy is
(3) 
(e.g. Stage et al., 2006; Kirk & Reville, 2010). Protons with energy also diffuse with and therefore the maximum energy that they can achieve is reduced to 10 TeV instead of 100 EeV as expected from the Hillas constraint in Eq. (1), assuming that persists over distances larger than the synchrotron cooling length of electrons with energy . However, theoretical results of Weibelmediated shocks (e.g. Spitkovsky, 2008a), and observational analysis of the case study of 4C74.26 (Araudo et al., 2015) indicate that the magnetic field is damped in the downstream region of a relativistic shock.
Numerical simulations show that Weibelmediated shocks in relativistic and weakly magnetised plasmas amplify the magnetic field on scale length , where is the Larmor radius of particles being accelerated (Spitkovsky, 2008a, b). The meanfree path for scattering by smallscale turbulence () is larger than the size of the system ( in our case) if is comparable with the plasma skin depth and is the Larmor radius of an EeV proton. These smallscale magnetic fluctuations decay at a distance 100 downstream of the shock, corresponding to 10 cm for a mildly relativistic plasma with density of the order of cm (see Eq. (14)). This rapid decay of the fluctuations inhibits particle acceleration to EeV energies, as was pointed out by Lemoine & Pelletier (2010); Sironi et al. (2013) and Reville & Bell (2014).
Damping of the magnetic field in the downstream region of a relativistic shock was observationally confirmed by modelling the jet termination region of the quasar 4C74.26 (Araudo et al., 2015). The compact synchrotron emission (0.1 kpc) detected in the southern hotspot of this source would require a magnetic field mG to match the size of the emitter with the synchrotron cooling length at the observed frequency of 1.66 GHz. This value of the magnetic field is about 10 times the upper limit imposed by the equipartition condition with nonthermal particles (see more details of the model in Araudo et al. (2016)). Therefore, the compact radio emission delineates the region within which the magnetic field is amplified by plasma instabilities up to 100 G, and it is damped downstream of the shock.
In Araudo et al. (2015) we consider the standard framework to explain the cutoff of the synchrotron spectrum at IR/optical frequencies and we discussed the thickness of the synchrotron emitter in the context of Weibel instabilities. Given that the thickness of the MERLIN radio emitter in the southern hotspot of 4C74.26 is larger than the synchrotron cooling length of electrons, we interpreted this behaviour of being determined by synchrotron cooling, and then, at distances 0.1 kpc downstream of the shock, the magnetic field is damped as a consequence of the small scale of Weibel turbulence. However, 0.1 kpc is much larger than the turbulence decay length predicted by numerical simulations of Weibelmediated shocks in plasmas with densities 10 cm.
In the present work we carry out a deeper study of particle acceleration in the hotspots of FR II radiogalaxies. We revisit the assumption of the synchrotron turnover being determined by synchrotron losses. Given that the scalelength of magnetic fluctuations has to be larger than (see Sect. 3), we show that cannot be determined by synchrotron cooling, as usually assumed, unless the jet density is unreasonably large and most of the jet upstream energy goes to nonthermal particles. We also show that the Weibel instability is not the source of the amplified magnetic field throughout the whole hotspot emission region since not only does it damp too quickly, but also it generates turbulence on a very small scale, insufficient to accelerate particles up to TeV for typical values of the magnetic field. In Sect. 5 we discuss the alternative possibility that the magnetised turbulence is generated by the Non Resonant Hybrid instability (Bell, 2004) which damps less quickly and grows on a larger scale.
In Table 1 we present some of the mathematical relations we use and how the reigning paradigm violates energy conservation even with very conservative assumptions. We consider the sample of hotspots observed with high spatial resolution at radio, infrared (IR) and optical frequencies in Mack et al. (2009). We find that very low values of the magnetic field, and therefore a huge energy density in nonthermal electrons, would be required to explain the flux density at 8.4 GHz if the IR/optical cutoff of the synchrotron spectrum was constrained by synchrotron cooling (see Sect. 3.1). These results invite the revision of previous phenomenological models of the hotspots nonthermal emission.
The results presented in this paper have also important implications for Eq. (1) and the maximum energy that protons can achieve by being accelerated in the jet reverse shock. We conclude that hotspots of FR II radiogalaxies with optical synchrotron cutoff are very poor accelerators of UHECRs.
The paper is organised as follows: In Sect. 2 we introduce the reader to stateoftheart modelling of nonthermal particles in hotspots. In Sect. 3 we revise the assumption that the acceleration process ceases as a consequence of efficient synchrotron losses and show that this standard picture is in disagreement with a limit imposed by plasma physics. In Sect. 4 we show that cannot be explained in the framework of any known instability. In Sect. 5 we explore a possible scenario to constrain the maximum energy of particles accelerated in the jet reverse shock, and in Sect. 6 we present our conclusions. Throughout the paper we use cgs units and the cosmology km s Mpc, and .
Plasma physics  Larmor radius of particles with energy and electric charge in a magnetic field  
Diffusion coefficient and meanfree path of particles in a medium with magnetic turbulence of scale length  
Ion skin depth in a jet with density and mean energy  
is the smallest characteristic plasma scalelength^{a} (see Sect. 3 and Eq. (15))  
Diffusive shock acceleration  Acceleration timescale in a parallel shock with velocity  
The energy density in nonthermal electrons (see Eq. (20)) cannot be greater than the energy budget (see Sect. 2 and Eq. (4))  
Synchrotron radiation  Lorentz factor of electrons emitting synchrotron photons with frequency  
Synchrotron cooling time  
Observations  Hz  Cutoff of the synchrotron spectrum 
Nonthermal electrons’ maximum energy (see Eqs. (2) and (7))  
Reigning paradigm  Synchrotron losses govern where the cutoff is  
Meanfree path of electrons (see Sect. 2.2 and Eq. (8))  
Combining the above   cm  A very large jet density is required to be and (see Sect. 3.1 and Table 2) 
Our conjecture  Condition for particle acceleration in a perpendicular shock with magnetic field  
Maximum energy at which nonthermal protons excite nonresonant turbulence (see Sect. 5) 

In electronpositron plasmas, has to be greater than the electronskin depth , where .
2 Hotspots
The jet termination region in FR II radiogalaxies is characterised by a double shock structure separated by a contact discontinuity, as sketched in Figure 1. Note however that the contact discontinuity is unstable due to the velocity shear and density contrast in both sides of the discontinuity (e.g. Mizuta et al., 2004). Hotspots are the downstream region of the jet reverse shock, where particles accelerated by the shock emit synchrotron radiation.
2.1 Energy budget
The kinetic energy density of relativistic jets with particles of mass and density and moving with bulk Lorentz factor is
(4) 
where corresponds to a jet velocity (Casse & Marcowith, 2005; Steenbrugge & Blundell, 2008) and is the proton mass. Even in the case that we do not know the jet matter composition, we expect that ions (from the jet formation region or from entrainment as the jet propagates) dominate the jet dynamics at the termination region and therefore in Eq. (4). The jet magnetisation parameter is defined as
(5) 
where and is the jet’s magnetic field. The jet (upstream) ram pressure is converted into thermal, nonthermal and magnetic ( erg cm) pressure in the shock downstream region with magnetic field . The magnetic field in the jet downstream region cannot be greater than
(6) 
(Meisenheimer & Heavens, 1986) which corresponds to the extreme case . The jet density is unknown in most cases, but cm is the upperlimit for the primary hotspot in the Western lobe of Cygnus A given the nondetection of radio polarisation (Dreher et al., 1987), and cm is the upper limit in 3C273 (Meisenheimer & Heavens, 1986).
2.2 Model to date
Hotspot (radiotooptical) synchrotron spectra typically show a cutoff at Hz (e.g. Meisenheimer et al., 1997; Tavecchio et al., 2005; Zhang et al., 2010; Werner et al., 2012). The traditional assumption is that the maximum energy of nonthermal electrons accelerated at the jet reverse shock, , where
(7) 
is determined by a competition between synchrotron cooling and acceleration timescales (see Table 1). By equating , the diffusion coefficient is given by Eq. (3) and the meanfree path of the electrons is
(8) 
These electrons radiate half of their energy over a distance
(9) 
downstream of the shock, where we have assumed that the velocity of the shocked plasma is , being the adiabatic shock compression factor. Note however that our results are not sensitive to the exact value of . The condition implies that the size of the acceleration region is .
In some cases, the spectrum is broken at frequency . To avoid misunderstandings between and we show in Fig. 2 two canonical electron and synchrotron spectra: broken (reddashed lines) and unbroken (greensolid lines). (See e.g. Meisenheimer & Heavens, 1986, for a comparison with real spectra.) In sources with enough radiotooptical data to be able to fit the synchrotron spectrum and measure and , the magnetic field is determined by comparing the synchrotron cooling time at with the timescale to be the particles advected a distance from the shock (e.g. Meisenheimer & Heavens, 1986):
(10) 
Therefore, by replacing in Eq. (8), the mean free path of the most energetic electrons accelerated at the shock is
(11) 
The main uncertainty is , that depends on the angle between the jet and the line of sight through the equation
(12) 
where is the observed size at and is the diameter of the source (when hotspots are modelled as cylinders of thickness ). Note that when the jet lies on the plane of the sky, and .
In the seminal paper of Meisenheimer et al. (1989), using observations at optical, near IR, millimetre and radio bands, hotspots are classified into high loss ( GHz) and low loss ( GHz) sources. The latter are characterised by thin emission regions with 0.13 (), 0.06 (), and 0.07 kpc () in the sources 3C20 West, 3C33 South and 3C111 East, respectively. In these low loss sources ^{1}^{1}1Magnetic fields below the equipartition value are also found in hotspots where the Xray emission is also modelled (e.g. Zhang et al., 2010; Werner et al., 2012). Werner et al. (2012) mentioned that this behaviour is in agreement with De Young (2002), who showed that magnetic field amplification by magnetohydrodynamic turbulence to equipartition values requires timescales greater than the dwell time of the plasma in the hotspots, unless special conditions are imposed. and 18 pc, where is the magnetic field in equipartition with nonthermal particles. The thin (disclike) emission regions in these hotspots were suggested to be the result of a drastic change in the downstream flow, producing a rapid decay of the magnetic field.
The detection of diffuse IR and optical synchrotron emission on scales larger than has been interpreted as insitu reacceleration (Fermi II) of the nonthermal electrons (Meisenheimer et al., 1997; Prieto et al., 2002; Brunetti et al., 2003). Later on, Mack et al. (2009) presented high spatial resolution observations at nearIR, optical and radio frequencies of lowpower radio hotspots, finding that Hz in all of them (i.e. low loss sources). The cooling time of electrons emitting synchrotron radiation at these high frequencies in a magnetic field G (in equipartition with nonthermal electrons and protons) is yr and much shorter than the timescales of adiabatic expansion (see Table 5 in Mack et al. (2009)).
In the next section we demonstrate that the maximum energy at which electrons are accelerated cannot be constrained by synchrotron losses, as usually assumed. To demonstrate this, we consider a sample of hotspots that do not show a break in their synchrotron spectra (greensolid lines in Fig. 2), but our arguments are not restricted to these sources.
3 Revising the synchrotron cutoff: when observational astronomy meets plasma physics
The synchrotron turnover at Hz observed in hotspots of FR II radiogalaxies indicates that the maximum energy of nonthermal electrons accelerated at the jet reverse shock is , where and are given by Eqs. (2) and (7), respectively. The Larmor radius of these particles is
(13) 
and the meanfree path is , where is the deflection angle of particles interacting with magnetic inhomogeneities of scale length . Considering the jet as a hydrogen plasma with electron and proton thermal Lorentz factors and , respectively, the ion skin depth downstream of the shock is
(14) 
The ratio
(15) 
shows that the thermal electron Larmor radius is generally larger than (in the “hot electrons/cold protons” scenario) in which case is the smallest characteristic plasma scalelength. Therefore, considering that for suprathermal particles, we find an upperlimit to the meanfree path of the most energetic electrons accelerated at the jet reverse shock:
(16) 
independent of the shock velocity (see Table 1). Therefore, the maximum diffusion coefficient is given by
(17) 
3.1 Is the maximum energy of nonthermal electrons constrained by synchrotron losses?
If is determined by a competition between shock acceleration and synchrotron cooling (i.e. ), the meanfree path of electrons is given by Eq. (8). By comparing with the upperlimit , we find that
(18) 
Equivalently, setting implies a magnetic field , where
(19) 
(Note that the same relationship is found by setting .) In Fig. 3 we plot for the cases of (bluesolid line) and cm (bluedashed line). The small values of would require a very large energy density in nonthermal electrons in order to explain the synchrotron flux measured at radiowavelengths. To demonstrate this, we consider the sample of hotspots observed at radio, IR and optical frequencies by Mack et al. (2009), and with a single radiotooptical spectral index , i.e. no spectral break (see Table 2).
Nonthermal electrons follow a powerlaw energy distribution with and minimum Lorentz factor^{2}^{2}2The value of minimum energy in nonthermal electrons cannot be smaller than the energy of the heated plasma downstream of the shock. By equating , where is the Boltzmann constant and is the temperature of the shocked jet, we find that GeV and therefore has to be greater than 50. assumed to be . The electrons energy density is , where can be determined from the leptonic emission at a particular frequency. Considering the well resolved emission at 8.4 GHz, with luminosity emitted in a (cylindershaped) volume (see Table 2), can be written as
(20) 
The magnetic field is unknown, but we can set the upper and lowerlimits. The former corresponds to the magnetic field in equipartition with nonthermal particles. Setting , where takes into account the contribution of nonthermal protons, we find that
(21) 
We calculate for all the sources in Mack et al. (2009) assuming ; see Table 2 and Fig. 3 (redtriangles down)^{3}^{3}3The equipartition field in Eq. (21) is slightly different from the value in Mack et al. (2009) given that we consider a cylindershaped volume, instead of an spheroid, and we set instead of 1. Note also that Mack et al. (2009) follow the approach of Brunetti et al. (1997) to compute .. Note that G in all the cases, and far greater than (bluesolid line), particularly for those cases with Hz.
3.1.1 Minimum value of B
In the extreme assumption that the nonthermal electron energy density is (see Eq. (4)), the minimum value of the magnetic field required to emit a luminosity at frequency in a volume is
(22) 
We compute for all the sources in Mack et al. (2009); see Table 2 and Fig. 3 (greentriangles up). We can see that (bluesolid line) for those sources with Hz (3C 105S, 3C 195N, 3C 227WE and 3C 403W) whereas for hotspots with Hz (3C 195S, 3C 227E, 3C 445N and 3C 445S). Note however that:

cm is the upper limit found in Cygnus A and 3C475, and therefore we expect values of greater than those plotted in Fig. 3 when the jet density is smaller than cm (). On the other hand, and therefore decreases when smaller values of are considered and the ratio . In particular, the bluedashed line in Fig 3 corresponds to the case of cm and . In such a case, sources 3C 195S, 3C 227E, 3C 445N and 3C 445S move to the regime where . The minimum value of the jet density required to match is listed in Table 2 for all the sources considered in this paper. We can see for instance that the source 3C 195N necessitates cm to satisfy the condition and .

Even when jets in FR galaxies are expected to be perpendicular to the line of sight, a small departure from the plane of the sky (i.e. ) reduces the size of the shock downstream region (see Eq.(12)). In such a case, increases whereas remains constant. Therefore, the situation is even more strongly ruled out when .
In the next section we show that even in the case that the extreme conditions discussed before are assumed, the large value of the diffusion coefficient required for to be determined by synchrotron cooling cannot be explained in any wellestablished theoretical framework.
Source  

[ Hz]  [erg/s]  [kpc]  [G]  [G]  [G]  [cm]  
3C 105S  0.089  1.37  0.75  2.5  1.4210  1205.63  45.27  2.06  1.16  1.9210 
3C 195N  0.110  2.70  0.95  2.9  1.1510  38.12  75.11  7.89  2.30  6.5110 
3C 195S  0.110  5.34  1.00  3.0  1.7110  33.58  91.76  12.45  4.55  3.4210 
3C 227WE  0.086  3.00  0.65  2.3  3.1910  19.26  45.63  1.74  2.55  6.7810 
3C 227E  0.086  11.4  0.75  2.5  7.1410  17.99  62.60  4.12  9.71  3.9610 
3C 403W  0.059  0.29  0.55  2.1  3.9510  167.9  28.46  0.48  0.25  1.9610 
3C 445N  0.056  6.63  0.85  2.7  2.1810  29.36  45.60  2.47  5.65  3.9710 
3C 445S  0.056  8.40  0.80  2.6  5.0410  139.42  35.94  1.38  7.15  1.6010 
4 Particle acceleration and magnetic field amplification
The diffusion coefficient resulting from the assumption that is determined by synchrotron cooling is very large, , as we show in Eq. (3). For comparison, this is 1010 times larger than for TeV particles diffusing through the Galactic interstellar medium. It is even more extreme when compared with during diffusive shock acceleration in supernova remnants where the magnetic field is strongly amplified by the nonresonant hybrid (NRH) instability (Bell, 2004) and structured on the scale of the cosmic ray (CR) Larmor radius.
In nonrelativistic shocks, the condition for the NRH instability to be active is that the upstream magnetic energy density must be less than , where is the efficiency with which the available kinetic energy is given to CR (see Sect. 5 and Appendix A). This condition is easily met in hotspots but it may not apply to relativistic shocks. One possible difference is that magnetic field amplification at relativistic shocks might be driven only by mildly relativistic particles since CR spectra at relativistic shocks are relatively steep with the CR energy density dominated by low energy CR. Fully developed magnetic turbulence on the scale of the GeV Larmor radius would naturally scatter TeV particles with since , as we will see in Sect. 5. However, is not sufficient to explain spectral turnover in the range Hz (implying ) and the NRH instability must be ruled out if we assume that the turnover is due to synchrotron losses.
In ultrarelativistic shocks in weakly magnetised plasmas (), the Weibel instability dominates and generates magnetic field on the small scale of the ion collisionless skin depth . Sironi & Spitkovsky (2011) found that the amplified magnetic field has a scalelength of but the factor 10 may be due to their shock Lorentz factor which increases by when the relativistic ion mass is allowed for (see Eq. (14)). If we assume fully developed Weibel turbulence with CR scattered by randomly orientated magnetic cells on a scale , the diffusion coefficient is given by as shown in Eq. (17). This value of the diffusion coefficient is large but still much smaller than  that would be required to explain the spectral turnover at Hz in a jet with density cm, at least . A further difficulty with a Weibel scenario is that postshock Weibel turbulence decays on a scale of pc (Sironi & Spitkovsky, 2011) which is many orders of magnitude smaller than the size of the hotspot, which is of the order of 10 pc to kpc. We note that the same discrepancy is found in gammaray bursts (e.g Gruzinov & Waxman, 1999; Pe’er & Zhang, 2006), although it is not completely clear at present how the smallscale magnetic turbulence evolves downstream of the shock (see e.g. Sironi et al., 2015).
Fully developed turbulence with a magnetic field of 100 G cannot be responsible for  even if its cell size were as small as , as is shown in Fig. 3 and Eq. (17). There remains the possibility that the magnetic field might consist of a long scalelength component with 100 G with a smallscale perturbation . Magnetic turbulence probably occurs in the Galaxy where CR drift along relatively wellordered magnetic field lines with weak scattering by Alfven waves with amplitude driven by CR drifts at the order of the Alfven speed. Under these conditions the CR current is too weak to drive the NRH instability. The Alfven waves are driven resonantly with a wavelength similar to the CR Larmor radius, and requires fluctuations in the magnetic field as small as in order to reach . Note however that even in the very weak CR drifts in the interstellar medium . For this scenario to hold for hotspots, a valid theory would need to explain how magnetic field could be amplified to 100 G on scales larger than a CR Larmor radius while producing fluctuations on a Larmor scale with only at a relative amplitude smaller than that found in the Galactic interstellar medium.
Although it is impossible to rule out all possibilities, it appears extremely difficult to construct a scenario in which , as required by the supposition that the IR/optical turnover in the synchrotron spectrum is caused by synchrotron radiation losses. We therefore suggest that the cutoff in the spectrum has a different cause, which we now explore in the next section.
5 Nonresonant hybrid instabilities in mildly relativistic shocks
If radiative (synchrotron) losses are not relevant to determining the maximum energy, then this maximum energy must ultimately determined by the ability to scatter particles downstream of the shock. We explore the possibility that the maximum energy achieved by electrons in the jet reverse shock is constrained by magnetic turbulence generated by low energy CRs in perpendicular shocks.
We consider that the amplified hotspot magnetic field is turbulent, and that the largescale background field downstream of the reverse shock is nearly perpendicular to the shock normal because the perpendicular component is compressed and enhanced by a factor of to (i.e. ). In such a case, to accelerate particles up to an energy via a diffusive mechanism, the meanfree path in the shock downstream region, where is a smallscale field, has to be smaller than Larmor radius in (Lemoine & Pelletier, 2010; Reville & Bell, 2014)^{4}^{4}4When the meanfree path of particles in the turbulent field exceeds the Larmor radius in the background compressed field, particles return to helical orbits and diffusion ceases.. The condition is satisfied when the magneticturbulence scalelength is
(23) 
where is the Larmor radius of protons with energy
(24) 
where we take G and G as characteristic values. Note that
(25) 
is greater than in Eq. (14), as required. Note however that this limit, for typical values considered in this paper, cannot be fulfilled by Weibelgenerated turbulence with scale . Therefore, the maximum energy achieved by electrons in the jet reverse shock, , cannot be constrained by Weibel instabilities.
Turbulence on a scale greater than may be excited through the nonresonant hybrid (NRH) instability by the diamagnetic drift of CR on either side of the shock. In the simplest form of the NRH instability (Bell, 2004, 2005), the CR Larmor radius in the unperturbed background field is much greater than the wavelength of field perturbations and therefore the streaming of CRs carrying the electric current is undeflected. The force  acts to expand loops in the magnetic field, and therefore increases. This produces an increment in  and generates a positive feedback loop that drives the NRH instability and amplifies the magnetic field. For the diamagnetic drift in the plane of the shock to amplify the magnetic field (see Appendix A) the NRH growth rate has to be sufficient for the instability to grow through 10 efoldings at the maximum growth rate (Bell, 2004, 2014) in the time the plasma flows through a distance in the downstream region, where is the Larmor radius in the ordered field . That is, the condition must be satisfied (see Appendix A). If the field is strongly amplified, the instability can be expected to saturate when its characteristic scale grows to the Larmor radius of the CR driving the instability. Thus, in Eq. (23) can be expected to match the Larmor radius of the highest energy CR driving the instability. If these CR have an energy , then . From Eq. (24), if then CR with energy correspond to mildly suprathermal protons () in the downstream plasma. It is entirely reasonable that protons with this energy should be present in large numbers downstream of the shock and drive the NRH instability.
In order to check that there is enough energy in protons to excite the nonresonant turbulence, we consider whether the number of efoldings required to amplify the magnetic field up to the saturation value is of the order of 10 (Bell, 2004, 2014). The condition for efficient magnetic field amplification by NRH instabilities is that , as explained above. This condition leads to
(26) 
(see Appendix A), where is the acceleration efficiency and is the jet magnetisation parameter defined in Eq. (5). Given that particles accelerated in relativistic shocks follow a powerlaw energy distribution steeper than the canonical distribution, the CR pressure is dominated by low energy particles. Therefore, the condition for NRH instability growth is that the acceleration efficiency of low energy CR has to be for characteristic values considered in this paper. Such a value of is very reasonable. For comparison, CR acceleration in supernova remnants is usually thought to be in the range 10%50%.
From these estimations we can conclude that NRH instabilities generated by CRs with energies can grow fast enough to amplify the jet magnetic field from 1 to 100 G and accelerate particles up to energies observed in the hotspots of FR II radiogalaxies. The advantage of magnetic turbulence being generated by CR current is that the amplified magnetic field persists over long distances downstream of the shock, and therefore particles accelerated very near the shock can emit synchrotron radiation far downstream. This framework also applies to hotspots with break in the synchrotron spectrum, and we will explore this situation in depth in a following paper.
6 Summary and conclusions
Motivated by the recent realisation of magnetic field damping in the southern hotspot of the radiogalaxy 4C74.26 (Araudo et al., 2015), we have explored in great depth the physical conditions in the hotspots of a larger number of FR II radiogalaxies. In particular, we have investigated the physical mechanism that constraints the maximum energy of particles accelerated at the jet reverse shock.
Based on one observable (the cutoff of the synchrotron spectrum) and one physical requirement () we have found that extreme conditions in the jet plasma would be required for  Hz to be determined by synchrotron cooling, as usually assumed. By equating the acceleration and synchrotron cooling timescales, the mean free path of synchrotron emitting electrons is greater than the maximum value imposed by plasma physics for reasonable values of the magnetic field and jet density (see Eq. (18)). By considering a sample of 8 hotspots observed with high spatial resolution at optical, IR and radio wavelengths (Mack et al., 2009), we show that unreasonably large values of the jet density would be required (see Table 2) to explain the synchrotron flux at 8.4 GHz when (maximum energy of nonthermal electrons) is determined by synchrotron cooling (see Fig. 3). The key steps in our argument are outlined in Table 1.
As mentioned in Sect. 4, the structure of the magnetic field downstream of the shock is not completely understood at the moment. Weibelmediated shocks generate the magnetic field and accelerate particles (e.g. Spitkovsky, 2008b; Martins et al., 2009). However, the characteristic scale of Weibel turbulence cannot account for the cutoff of the synchrotron spectrum observed in hotspots because this scale size is too small., nor the large extent of the hotspot synchrotron emission, much larger than the magnetic decay of predicted by numerical calculations. A viable alternative is that turbulence is generated by the streaming of CRs with energy (see Sect. 5). The amplified magnetic field has a scalelength of the order of the Larmor radius of protons and persists over long distances downstream of the shock, accounting for the extent of the synchrotron emitting hotspot.
In a future work, we will apply our arguments to the very well known sources Cygnus A and 3C445 for which well resolved and multiwavelength data are available (e.g. Orienti et al., 2012; Pyrzas et al., 2015). By modelling the particle acceleration and transport downstream the shock we will be able to determine the details of the magnetic field structure downstream of mildly relativistic shocks.
Acknowledgements
The authors would like to thank the referee for a constructive report, and M.A. Prieto for providing useful information about the sources considered in this paper. A.T.A. thanks M. Perucho, W. Potter and L. Sironi for useful discussions about jet physics. The research leading to this article has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/20072013)/ERC grant agreement no. 247039. We acknowledge support from the UK Science and Technology Facilities Council under grant No. ST/K00106X/1.
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Appendix A Condition for efficient NRH instability in hotspots
As noted in Sect. 5, a condition for effective CR scattering by turbulent magnetic fields amplified by the NRH instability with maximum growth rate is that where is the downstream flow velocity and is the Larmor radius of the CR driving the instability. The perpendicular component of the magnetic field in the jet is compressed by the shock producing a downstream field that is predominantly perpendicular on the large scale. Drift of CR along the shock surface produces a diamagnetic current that can drive the NRH instability. The CR current is perpendicular to both the shock normal and the large scale magnetic field and extends a distance downstream of the shock. The NRH instability must be driven through many efoldings during the time during which a fluid element is subject to the diamagnetic current. In this configuration the NRH growth rate is smaller by a numerical factor of order one than in the case of aligned currents and magnetic field (Bell, 2005). However, is still a good orderofmagnitude measure of the growth rate, where is the density in the shock downstream region. The condition provides a good estimate of the time for strong nonlinear amplification, giving
(27) 
where depends on distance from the shock. From the momentum equation the downstream CR pressure must be balanced by the magnetic force: giving the condition
(28) 
or equivalently
(29) 
where is the CR acceleration efficiency, as quoted in Sect. 5. Equation 29 is thus the condition for efficient NRH instability in jet reverse shocks.