aa r X i v : . [ a s t r o - ph ] J a n Cosmic ray acceleration at relativistic shocks, shear layers, ...
Micha l Ostrowski
Obserwatorium Astronomiczne, Uniwersytet Jagiello´nskiul. Orla 171, 30-244 Krak´ow, Poland(E-mail: [email protected])
Abstract
A short discussion of theoretical results on cosmic ray first-order Fermi acceleration at relativistic shock waves ispresented. We point out that the recent results by Niemiec with collaborators change the knowledge about theseprocesses in a substantial way. In particular one can not expect such shocks to form particle distributions extendingto very high energies. Instead, distributions with the shock compressed injected component followed by a more or lessextended high energy tail are usually created. Increasing the shock Lorentz factor leads to steepening and decreasingof the energetic tail. Also, even if a given section of the spectrum preserves the power-law form, the fitted spectralindex may be larger or smaller than the claimed ‘universal index’ σ ≈ . σ ≈ . ∼ σ >
3) at higher energies. We conclude withpresentation of a short qualitative discussion of the Fermi second-order processes acting in relativistic plasmas. Wesuggest that such processes can be the main accelerating agent for very high energy particles. In particular its canaccelerate electrons to energies in the range of 1 - 10 TeV in relativistic jets, shocks and radio-source lobes.
Key words: cosmic rays, Fermi acceleration, relativistic shock waves, relativistic jets, gamma ray bursts
1. Introduction
Relativistic plasma flows are observed in a numberof astrophysical objects, ranging from a mildly rel-ativistic jets of the sources like SS433, through the-Lorentz-factor-of-a-few jets in AGNs and galactic‘micro-quasars’, up to highly ultra-relativistic out-flows in sources of gamma ray bursts or pulsar winds.As nearly all such objects are efficient emitters ofnonthermal radiation, what requires existence of en-ergetic particles, our attempts to understand theprocesses generating cosmic rays are essential forunderstanding the fascinating phenomena observed.Below I will discuss the work carried out in orderto understand the cosmic ray acceleration processesacting at relativistic shocks and within highly tur- bulent regions accompanying such shocks and shearlayers. I will not include here the interesting workinvolving collisionless shocks modelling with par-ticle in cell simulations. This approach uses quitedifferent modelling method in comparison to theother work discussed here, relating in most cases tothe characteristic energy range of shock compressedthermal plasma particles.The present paper is a modified and updated ver-sion of some of my previous reviews of the subject.Also, essentially the same slightly shortened text isprovided as my contribution to the HEPRO Work-shop in Dublin (September 2007).Below we will append the index ‘1’ (‘2’) to quan-tities measured in the plasma rest frame upstream(downstream) of the shock.
Preprint submitted to Elsevier 30 October 2018 . Particle diffusive acceleration atnon-relativistic shock waves
Processes of the first-order particle accelerationat non-relativistic shock waves were widely dis-cussed by a number of authors, let me note stillactual reviews by Drury (1983) and Blandford &Eichler (1987). The most interesting physical fea-ture of the first-order shock acceleration at thenon-relativistic shock wave is independence of the test-particle stationary particle energy spectrumfrom the background conditions near the shock, in-cluding the mean magnetic field configuration andthe spectrum of MHD turbulence. The reason isa nearly-isotropic form of the particle momentumdistribution at the shock. If efficient scattering oc-curs near the shock, this condition also holds foroblique shocks with the shock velocity along the(upstream) magnetic field U B, ≡ U / cos Ψ ≪ v (Ψ - the upstream magnetic field inclination to theshock normal). Then, the particle density is contin-uous across the shock and the spectral index for the phase-space distribution function, α , is given exclu-sively in terms of a single parameter – the shockcompression ratio R : α = 3 RR − . (2 .
3. Cosmic ray acceleration at relativisticshock waves
Below I describe work done on mildly- and ultra-relativistic shock acceleration including importantrecent results of Niemiec et al. With these last re-sults many previous ones occurred to be of histori-cal value only, reflecting specific individual featuresof the acceleration processes. Basing on these olderworks one can understand the recent simulations re-sults in a relatively straightforward way. Attempt- ing to give an overview of the full field I base thispresentation on my own and my collaborators work,which seems to me to present a consistent way of de-velopment and reflects my approach to understand-ing acceleration processes in relativistic shocks.3.1.
History: acceleration at mildly relativisticshocks
In cases of the shock velocity reaching values com-parable to the light velocity, the particle distribu-tion at the shock becomes anisotropic . This simplefact complicates to a great extent both the physicalpicture and the mathematical description of particleacceleration. A first attempt to consider the acceler-ation process at the relativistic shock was presentedin 1981 by Peacock, and a consistent theory wasproposed later by Kirk & Schneider (1987a). Thoseauthors considered stationary solutions of the rel-ativistic Fokker-Planck equation for particle pitch-angle diffusion in the parallel shock wave. In thesituation with the gyro-phase averaged distribution f ( p, µ, z ), which depends only on the unique spatialco-ordinate z along the shock velocity, and with µ being the pitch-angle cosine, the equation takes aform: Γ( U + vµ ) ∂f∂z = C ( f ) + S , (3 . ≡ / √ − U is the flow Lorentz factor, C ( f ) is the collision operator and S is the sourcefunction. In the presented approach, the spatial co-ordinates are measured in the shock rest frame, whilethe particle momentum co-ordinates and the col-lision operator are given in the respective plasmarest frame. For the applied pitch-angle diffusion op-erator, C = ∂/∂µ ( D µµ ∂f /∂µ ), the authors gener-alized the diffusive approach to higher order termsin particle distribution anisotropy and constructedgeneral solutions at both sides of the shock whichinvolved solutions of the eigenvalue problem. Bymatching two solutions at the shock, the spectral in-dex of the resulting power-law particle distributioncan be found by taking into account a sufficientlylarge number of eigenfunctions. The same procedureyields the particle angular distribution and the spa-tial density distribution. The low-order truncationin this approach corresponds to the standard diffu-sion approximation and to a somewhat more generalmethod described by Peacock.2n application of this approach to more realisticconditions – but still for parallel shocks – was pre-sented by Heavens & Drury (1988), who investigatedthe fluid dynamics of relativistic shocks (cf. also El-lison & Reynolds 1991) and used the results to cal-culate spectral indices for accelerated particles. Fig. 1. Spectral indices α of particles accelerated at obliqueshocks versus shock velocity projected at the mean magneticfield, U B, (Ostrowski 1991a). The results are presented forthe shock compression R = 4. On the right the respectivesynchrotron spectral index γ is given. The shock velocities U are given near the respective curves taken from Kirk& Heavens (1989). The points were taken from simulationscomputing explicitly the details of particle-shock interactions(Ostrowski 1991a). A substantial progress in understanding the accel-eration process in the presence of highly anisotropicparticle distributions is due to the work of Kirk& Heavens (1989; see also Ostrowski 1991a andBallard & Heavens 1991), who considered particleacceleration at subluminal ( U B, < c ) relativisticshocks with oblique magnetic fields. They assumedthe magnetic momentum conservation, p ⊥ /B = const , at particle interaction with the shock and ap-plied the Fokker-Planck equation discussed aboveto describe particle transport along the field linesoutside the shock, while excluding the possibility ofcross-field diffusion. In the cases when U B, reachedrelativistic values, they derived very flat energyspectra with γ ≈ U B, ≈ superluminal con-ditions with U B, > c , where the above presentedapproach is no longer valid. Then, it is not possi-ble to reflect upstream particles from the shock andto transmit downstream particles into the upstreamregion. In effect, only a single transmission of up-stream (or shock injected) particles re-shapes theoriginal distribution by shifting particle energies tolarger values, with super-adiabatic efficiency due toanisotropic particle distribution at the transmission. Fig. 2. Energetic particle density profiles across the rel-ativistic shock with the oblique magnetic field (Ostrowski1991b). The shock with U = 0 . R = 5 .
11 and ψ = 55 o is considered. The curves for growing to the top perturba-tion amplitudes are characterized with the value log κ ⊥ /κ k ( κ ⊥ /κ k is a ratio of the cross-field to the parallel diffusioncoefficients) given near each curve. The data are verticallyshifted for picture clarity. The value X max is the distancefrom the shock at which the upstream particle density de-creases to 10 − part of the shock value. History: acceleration in the presence of largeamplitude magnetic field perturbations
As the approaches proposed by Kirk & Schnei-der (1987a) and Kirk & Heavens (1989), and thederivations of Begelman & Kirk (1990) are validonly in cases of weakly perturbed magnetic fields,for larger amplitude MHD perturbations numericalmethods have to be used. A series of such simu-lations were performed by numerous authors (e.g.Kirk & Schneider 1987b; Ellison et al. 1990; Os-trowski 1991a, 1993; Ballard & Heavens 1992, Naito3 Takahara 1995, Bednarz & Ostrowski 1996). Dif-ferent approaches applied, including the ones involv-ing particle momentum pitch-angle diffusion or in-tegrating particle trajectories in analytic structuresof the perturbed magnetic fields , allowed for con-sidering a wide range of background conditions nearthe shock. The results obtained by different authorscan be summarized at the figure (Fig. 3) taken fromBednarz & Ostrowski (1996). One should note, thatessentially all derivations by the above authors wereperformed with assuming scale-free conditions forthe acceleration process, resulting in power-law dis-tributions of the accelerated particles. Fig. 3. Relation of the particle acceleration time scale T acc versus the particle spectral index α at different magneticfield inclinations ψ given near the respective curves (Bed-narz & Ostrowski 1996). The minimum value of the modelparameter κ ⊥ /κ k occurs at the encircled point of each curveand the wave amplitude monotonously increases along eachcurve up to δB ∼ B , where all curves converge in α . Thecurve for Ψ = 60 ◦ ( U B, = 1) separate the sub- and super–luminal shock results. We do not discuss here the presentedacceleration times. At the figure (Fig. 3) one can find very flat spec-tra for oblique subluminal shocks if the perturba-tion amplitudes are small. Contrary to that genera-tion of the power-law spectrum is possible in the su-perluminal shocks only in the presence of large am-plitude turbulence. Then, in contrast to the sublu-minal shocks, spectra are extremely steep for smallvalues of δB (not presented at the figure) and α monotonously decreases with increasing magneticfield perturbations. A new feature is observed in Let us note that application by some authors of point-likelarge-angle scattering models in relativistic shocks does notprovide a viable physical representation of the scattering atMHD waves (Bednarz & Ostrowski 1996). oblique shocks of the spectral index α changing with δB in a non-monotonic way.3.3. History: Energy spectra of cosmic raysaccelerated at ultra-relativistic shocks
The main difficulty in modelling the accelerationprocesses at shocks with large Lorentz factors Γ isthe fact that the involved particle distributions areextremely anisotropic in the shock, with the parti-cle angular distribution opening angles ∼ Γ − in theupstream plasma rest frame. In the simulations ofBednarz & Ostrowski (1998) a Monte Carlo methodinvolving small amplitude pitch-angle scattering wasapplied for particle transport near the shocks with Γin the range 3 – 243. The simulations revealed an un-expected result, showing convergence, for Γ → ∞ ,of the resulting power-law distributions to the ’uni-versal’ one with the spectral index σ ≈ . σ λ = -3.44 Γ Fig. 4. The simulated spectral indices σ ( σ ≡ α −
2) versusthe shock Lorentz factor Γ (Bednarz & Ostrowski 1998).Results for a given ψ are joined with dashed lines; therespective value of ψ is given near each curve. Increasingthe turbulence amplitude in a not presented here series ofsimulations led to shifting the resulting curves toward theparallel shock, Ψ = 0 ◦ , results. (a) ) δ B /B = 3.02 π /k min π /k max ↓ ↓ u = 0.5c ψ = 45 ° F(k) ~ k -1 α = ) l og ( d N / d l og ( E )) (b) ) δ B /B = 3.02 π /k min π /k max ↓ ↓ u = 0.5c ψ = 45 ° F(k) ~ k -5/3 α = ) log ( E/E ) Fig. 5. Particle spectra for an oblique mildly relativistic shock: the shock velocity u = 0 . c , the mean magnetic field inclinationΨ = 45 ◦ and the wave power spectrum are indicated in the respective panels (from Niemiec & Ostrowski 2004). Values ofmagnetic turbulence amplitude, δB/B , and the indices fitted to the power-law sections of spectra (in parentheses) are givennear each result. Ostrowski & Bednarz (2002) reconsidered all theabove approaches to derive particle spectra at rela-tivistic shocks and ‘discovered’ that the conditionsproducing the universal spectral index were in someway equivalent to assuming highly turbulent condi-tions near the shock. Additionally, all these mod-els did not introduce any physical scale and thusforced the power-law shape of the resulting spec-trum. Do such conditions and the resulting charac-teristic spectra really exist in astrophysical situa-tions ?3.4.
Toward a realistic description of the relativisticshock acceleration
Studies of - as far as possible - realistic conditionsfor the relativistic shock acceleration were presentedin a series of papers by Niemiec et al. (Niemiec &Ostrowski 2004, 2006, Niemiec et al. 2006; see alsoLemoine Pelletier 2006). We assumed 3-D staticmagnetic field perturbations upstream of the shockby imposing a large number of sinusoidal waveswith different power spectra, F ( k ). The flat spec-trum, F ( k ) ∝ k − , or the Kolmogorov spectrum, F ( k ) ∝ k − / , were considered in the wide wave-vector range ( k min , k max ), with k max /k min = 10 .The downstream magnetic field structures werecomputed by respective compression of the up-stream field at the shock. In the last of above papers an additional component of large amplitude short-wave MHD turbulence was assumed to be produceat the shock. The accelerated particle spectra werederived using the Monte Carlo simulations for awide range of shock Lorentz factors - between 2and 30 - and a selection of the mean magnetic fieldconfigurations and perturbation amplitudes. Fig. 6. The particle spectra derived for superluminal rel-ativistic shocks with Lorentz factors γ = 5, 10 and 30(Niemiec & Ostrowski 2006a). Let us take a look at a few characteristic resultsof these modelings. At Fig. 5 results derived for sub-luminal oblique mildly relativistic shocks are pre-5 ig. 7. Particle spectra derived for parallel relativistic shocks with Lorentz factor γ = 30 (Niemiec & Ostrowski 2006). Fordescription see Fig. 5 and the original paper. sented. One may note that introducing the energyscales to the modelling, in our units 2 π/k max and2 π/k min , lead to deviations of the resulting spectrafrom the power-law shape. The spectra, as expectedfrom the previous discussion, are very flat for smallamplitude turbulence, but steepen at larger ampli-tudes. An interesting feature is seen, that at veryhigh particle energies, above the resonance range( E > π/k min ) the ‘short’ waves weaker influenceparticle trajectories leading to the hard energy tailsbefore the cut-offs imposed by the modelling.When we considered oblique superluminal shocksthe spectra consisted of the shock compressed in-jected component and the limited high energy tailuntil a cut-off well within the ‘resonance range’(2 π/k max , 2 π/k min ). As illustrated at Fig. 6 thetails fast diminish with the growing shock Lorentzfactor, leaving for large γ the ‘compressed compo-nent’ only.There are a few general observations for the first-order Fermi acceleration processes from these seriesof models. For particles in a low energy range of theresonance energies for the considered field pertur-bations the acceleration processes proceed in an en-semble of different oblique shocks, where each local mean magnetic field structure is formed as a super-position of the mean magnetic field B , and longwave field perturbations. Thus, there can occur sig-nificant differences between spectra generated in thepresence of flat and steep turbulence power spec-tra, and the spectral indices significantly vary with the perturbations amplitudes. In parallel shocks thelong wave perturbations introduce the accelerationeffects observed with oblique magnetic fields (cf. alsoOstrowski 1993). Thus in the ultrarelativistic paral-lel shocks propagating in highly turbulent mediumthese effects can lead to formation of particle dis-tributions with cut-offs at relatively low energies,like in shocks with perpendicular field configura-tions (Fig. 7). If the shock wave generates a largeamplitude short-wave turbulence downstream of theshock, the acceleration process can form a more ex-tended power-law tail, but at higher particle energiesthe mean magnetic field or the long wave magneticfield structures start to dominate in shaping par-ticle trajectories and thus the acceleration process,leading to results like the ones described above. Inany studied case we were not able to create the scalefree conditions in the acceleration process, leadingto the wide range power law distribution of acceler-ated particles. It was possible only in limited energyranges and the forms of the spectra depended usu-ally a lot on the considered background conditions.3.5. Observational constraints on the shockacceleration from Cyg A hotspots
Constraints for the above theoretical derivationscan be provided by precise observations of energeticparticle emission from objects harbouring the rela-tivistic shocks. Such study performed for hotspots6f the Cygnus A radiosource (Stawarz et al. 2007)reveals significant deviations of the derived electronspectra from the ‘standard’ shock spectra. The re-sulting spectral energy distribution of the hotspotD is presented at Fig. 8, showing both the extendedsynchrotron component and the inverse-compton(IC) one modelled for optical and X-ray measure-ments. Additionally, the low Spitzer IR pointsprovide additional important constraint for the ICspectral component. In derivation of the relativisticelectrons distribution these measurements allow forexcluding a possibility of substantial absorption inthe low frequency synchrotron spectrum and thusrequire very flat distribution of low energy elec-trons. Thus the intrinsic electron spectrum (Fig. 9)is composed of a very flat low energy sector, withthe energy spectral index s ≈ .
5, followed abovea break at E ≈ s > E br ∼ E br ,either the first order Fermi process acting at theshock creates a steep spectrum, or the accelera-tion process proceeds downstream of the shock inthe second-order Fermi process. The existing the-oretical models do not allow to reject any of thesealternatives.
4. Energetic particle acceleration in shearlayers and regions of relativistic turbulence
Acceleration processes acting, e.g., in AGN cen-tral sources and in shocks formed in large scalejets are not always able to explain the observedhigh energy electrons radiating away from the cen-tre/shock. Among a few proposed possibilities ex-plaining these data the relatively natural but stillunexplored is the one involving particle accelerationwithin a shear layer transition at the interface be-tween the jet and the surrounding medium. To dateknowledge of physical conditions within such layersis very limited and only rough estimates for the con-sidered acceleration processes are possible. Withinthe subsonic turbulent layer with a non-vanishingsmall velocity shear the ordinary second-order Fermi S [ Jy * H z ] [Hz] Hotspot D Fig. 8. Spectral energy distribution of emission from the hotspot D of Cyg A (Stawarz et al. 2007). One can clearly seeboth the synchrotron and the IC components. The Spitzerpoints in infrared and the optical point allow to excludepossibility of forming the measured flat low-frequency syn-chrotron component (dotted lines in synchrotron and ICspectral ranges) due to some self-absorption processes. l og n e () / c m log absorptioneffects coolingeffects Fig. 9. Relativistic electron spectra in Cyg A hot-spots(Stawarz et al. 2007; here γ is an electron Lorentz factor).Different spectral indices of the low energy and the highenergy parts are expected to be intrinsic to the accelerationprocess, not the effect of the distribution ‘aging’ downstreamof the shock. acceleration, as well as the process of ‘viscous’ par-ticle acceleration (cf. the review by Berezhko (1990)of the work done in early 80-th; Earl et al. 1988,Ostrowski 1990, 1998, 2000, Stawarz & Ostrowski2002) can take place. A mean particle energy gainper scattering in the later process scales as < ∆ E >E ∝ (cid:18) < ∆ U >c (cid:19) , (4 . < ∆ U > is the mean velocity difference be-7ween the ‘successive scattering acts’. It is propor-tional to the mean free path normal to the shearlayer, λ n , times the mean flow velocity gradient inthis direction ∇ n · ~U . With d denoting the shear layerthickness this gradient can be estimated as |∇ n · ~U | ≈ U/d . Because the acceleration rate in the Fermi IIprocess is ∝ ( V /c ) ( V ≈ V A is the wave velocity, V A – the Alfv´en velocity), the relative importanceof both processes is given by a factor (cid:18) λ n d UV (cid:19) . (4 . λ n and in the formal limit of λ n ≈ d and V ≪ c – outside the equation (4.2) validity range – itdominates over the Fermi acceleration to a large ex-tent. Because accelerated particles can escape fromthe accelerating layer only due to a relatively inef-ficient radial diffusion, the formed particle spectraare expected be very flat up to the high energy cut-off, but the exact form of the spectrum depends onseveral unknown physical parameters of the bound-ary layer (Ostrowski 1998, 2000).For turbulent relativistic plasmas the second-order Fermi acceleration can in principle dominateover the viscous process at all particle energies. Inthe case of electrons the upper energy scale for theaccelerated particles is provided by the radiationlosses. A simple exercises with the above estimatedacceleration scales and the synchrotron radiationloss scale yields - for the sources like small and largescale jets, radio hot spots or lobes – the highest elec-tron energies between 1 TeV and 10 TeV, for therespective acceleration time scales ∼ s in gaussfields and up to ∼ yrs in µ G fields (dependingon the considered object). The much higher energylimits for protons are usually determined by the es-cape boundary conditions, not the radiative losses.
5. Final remarks
A recent study of the first order Fermi accelera-tion processes at relativistic shocks, taking into ac-count realistic assumptions about the physical con-ditions near the shock, reveals a few unexpected con-clusions. The modelling of the acceleration processin mostly perpendicular (for relativistic velocities)shocks yields spectra consisting of the compressedinjected part appended with a limited high energytail. For given upstream conditions increasing theshock Lorentz factor Γ leads to steepening of such energetic tails, providing essentially the compressedinjected component at large Γ. Another unexpectedfeature is observed dependence of the spectrum incli-nation on the turbulence amplitude also for the par-allel shock waves and formation of cut-offs at suchshocks for large Γ. Essentially no conditions stud-ied by Niemiec et al. resulted in formation of thewide-range power-law particle distribution with theuniversal spectral index 2.2 .It can be of interest, with the presently publishedAUGER results (The Pierre Auger Collaboration2007), that the modelling presented above seems toexclude the first-order Fermi acceleration at rela-tivistic shocks as possible sources for highest energyparticles registered in this experiment.In the same time the second-order Fermi pro-cesses acting in turbulent relativistic plasmas areexpected to play significant role in cosmic ray accel-eration. Thus, even facing substantial mathematicaland physical difficulties, its deserve a detailed study.One may note that the acceleration processes accom-panied the magnetic field reconnection processes areanalogous to the second-order Fermi acceleration.Such processes always accompany the large ampli-tude MHD turbulence and generate turbulence. Afew simple attempts to consider these processes wererecently presented (e.g. Virtanen & Vainio 2005).When considering the relativistic shock acceler-ation one should also note interesting approachesby Derishev et al. (2003) and Stern (2003), outsidethe classical Fermi scheme. They consider the ac-celeration processes in highly relativistic shocks orjet shear layers (Stern & Poutanen 2006) involvingparticle-particle or particle-photon interactions atboth sides of the shock.Acknowledgements I am grateful to my collabora-tors Jacek Niemiec and Lukasz Stawarz, whose sig-nificant work forms the main part of this rapport.The present work was supported by the Polish Min-istry of Science and Higher Education in years 2005-2008 as a research project 1 P03D 003 29.ReferencesAchterberg, A., Gallant, Y. A., Kirk, J. G., Guth-mann, A. W. 2001,
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