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Cosmicrayaccelerationat relativisticshocks,shear layers,... 8 0 Micha l Ostrowski 0 2 Obserwatorium Astronomiczne, UniwersytetJagiellon´ski n ul. Orla 171, 30-244 Kraków, Poland a (E-mail: [email protected]) J 1 1 Abstract ] h A short discussion of theoretical results on cosmic ray first-order Fermi acceleration at relativistic shock waves is p presented. We point out that the recent results by Niemiec with collaborators change the knowledge about these - o processes in asubstantialway.Inparticular onecannot expectsuchshockstoform particledistributionsextending r toveryhighenergies.Instead,distributionswiththeshockcompressedinjectedcomponentfollowedbyamoreorless t s extendedhighenergytailareusuallycreated.IncreasingtheshockLorentzfactorleadstosteepeninganddecreasing a of the energetic tail. Also, even if a given section of the spectrum preserves the power-law form, the fitted spectral [ indexmay belarger orsmaller thantheclaimed ‘universalindex’σ≈2.2 . 2 AreportedsimplecheckofrealshapesofelectronspectraintheCygAhotspotsprovidesresultsclearlydeviating v fromthestandardexpectationsforsuchshocksmetintheliterature.Thespectrumconsistoftheveryflatlowenergy 9 part(σ≈1.5),uptoelectronenergies∼1GeV,andmuchsteeperpart(σ>3)athigherenergies.Weconcludewith 3 presentation of a short qualitative discussion of the Fermi second-order processes acting in relativistic plasmas. We 3 suggest that such processes can be the main accelerating agent for very high energy particles. In particular its can 1 accelerate electrons toenergies in therange of 1- 103 TeV inrelativistic jets, shocksand radio-source lobes. . 1 0 8 Keywords: cosmicrays,Fermiacceleration, relativisticshock waves, relativisticjets, gammaraybursts 0 : v i 1. Introduction bulentregionsaccompanyingsuchshocksandshear X layers. I will not include here the interesting work r a Relativisticplasmaflowsareobservedinanumber involving collisionless shocks modelling with par- of astrophysicalobjects, ranging from a mildly rel- ticle in cell simulations. This approach uses quite ativisticjetsofthesourceslikeSS433,throughthe- different modelling method in comparison to the Lorentz-factor-of-a-few jets in AGNs and galactic otherworkdiscussedhere,relatinginmostcasesto ‘micro-quasars’, up to highly ultra-relativistic out- thecharacteristicenergyrangeofshockcompressed flowsinsourcesofgammarayburstsorpulsarwinds. thermalplasmaparticles. As nearly all such objects are efficient emitters of Thepresentpaperisamodifiedandupdatedver- nonthermalradiation,whatrequiresexistenceofen- sion of some of my previous reviews of the subject. ergetic particles, our attempts to understand the Also, essentiallythe same slightly shortenedtext is processes generating cosmic rays are essential for provided as my contribution to the HEPRO Work- understandingthefascinatingphenomenaobserved. shopinDublin (September 2007). Below I will discuss the work carried out in order Belowwewillappendthe index ‘1’(‘2’)toquan- tounderstandthecosmicrayaccelerationprocesses tities measured in the plasma rest frame upstream acting at relativistic shocks and within highly tur- (downstream)ofthe shock. Preprintsubmitted toElsevier 2February2008 2. Particle diffusive acceleration at ing to give an overview of the full field I base this non-relativisticshockwaves presentationonmyownandmycollaboratorswork, whichseemstometopresentaconsistentwayofde- Processes of the first-order particle acceleration velopmentandreflectsmyapproachtounderstand- at non-relativistic shock waves were widely dis- ing accelerationprocessesinrelativistic shocks. cussed by a number of authors, let me note still actual reviews by Drury (1983) and Blandford & 3.1. History: acceleration at mildly relativistic Eichler (1987). The most interesting physical fea- shocks ture of the first-order shock acceleration at the non-relativistic shock wave is independence of the test-particle stationary particle energy spectrum Incasesoftheshockvelocityreachingvaluescom- fromthe backgroundconditions nearthe shock,in- parable to the light velocity, the particle distribu- cluding the mean magnetic field configuration and tion at the shock becomes anisotropic. This simple the spectrum of MHD turbulence. The reason is factcomplicatestoagreatextentboththephysical a nearly-isotropic form of the particle momentum pictureandthemathematicaldescriptionofparticle distribution at the shock. If efficient scattering oc- acceleration.Afirstattempttoconsidertheacceler- curs near the shock, this condition also holds for ationprocessattherelativisticshockwaspresented oblique shocks with the shock velocity along the in 1981 by Peacock, and a consistent theory was (upstream) magnetic field UB,1 U1/cosΨ1 v proposedlaterby Kirk& Schneider(1987a).Those ≡ ≪ (Ψ1 -theupstreammagneticfieldinclinationtothe authors considered stationary solutions of the rel- shocknormal).Then,theparticledensityiscontin- ativistic Fokker-Planck equation for particle pitch- uousacrosstheshockandthespectralindexforthe angle diffusion in the parallel shock wave. In the phase-space distributionfunction,α,isgivenexclu- situationwiththegyro-phaseaverageddistribution sively in terms of a single parameter – the shock f(p,µ,z),whichdependsonlyontheuniquespatial compressionratioR: co-ordinate z along the shock velocity, and with µ being the pitch-angle cosine, the equation takes a 3R form: α= . (2.1) R 1 − ∂f Because of the isotropic form of the particle distri- Γ(U +vµ) =C(f)+S , (3.1) bution function, the spatial diffusion equation has ∂z becomeawidelyusedmathematicaltoolfordescrib- where Γ 1/√1 U2 is the flow Lorentz factor, ing particle transportand accelerationprocesses in ≡ − C(f) is the collision operator and S is the source non-relativisticflows. function.Inthe presentedapproach,thespatialco- In realastronomicalobjects some non-linearand ordinatesaremeasuredintheshockrestframe,while time dependent effects, or occurring of additional the particle momentum co-ordinates and the col- energylossesandgainscanmakethe physicsofthe lision operator are given in the respective plasma acceleration more complicated, creating, e.g., non- restframe.Fortheappliedpitch-anglediffusionop- power-low and/or non-stationary particle distribu- erator, C = ∂/∂µ(D ∂f/∂µ), the authors gener- tions. µµ alized the diffusive approach to higher order terms in particle distribution anisotropy and constructed 3. Cosmic ray acceleration at relativistic general solutions at both sides of the shock which shockwaves involved solutions of the eigenvalue problem. By matchingtwosolutionsattheshock,thespectralin- BelowIdescribeworkdoneonmildly-andultra- dex of the resulting power-lawparticle distribution relativistic shock acceleration including important can be found by taking into account a sufficiently recent results of Niemiec et al. With these last re- largenumberofeigenfunctions.Thesameprocedure sults many previous ones occurred to be of histori- yieldstheparticleangulardistributionandthespa- calvalueonly,reflectingspecific individualfeatures tial density distribution. The low-order truncation of the accelerationprocesses.Basing onthese older in this approachcorrespondsto the standarddiffu- worksonecanunderstandtherecentsimulationsre- sionapproximationandtoasomewhatmoregeneral sults in a relatively straightforward way. Attempt- method describedby Peacock. 2 An applicationof this approachto more realistic As stressed by Begelman & Kirk (1990), in rel- conditions – but still for parallel shocks – was pre- ativistic shocks one often finds superluminal con- sentedbyHeavens&Drury(1988),whoinvestigated ditions with UB,1 > c, where the above presented thefluiddynamicsofrelativisticshocks(cf.alsoEl- approach is no longer valid. Then, it is not possi- lison& Reynolds 1991)andused the results to cal- bletoreflectupstreamparticlesfromtheshockand culate spectralindices foracceleratedparticles. totransmitdownstreamparticlesintotheupstream region. In effect, only a single transmission of upstream (or shock injected) particles re-shapes the originaldistributionby shifting particle energies to largervalues,withsuper-adiabaticefficiencydueto anisotropicparticledistributionatthetransmission. Fig.1. Spectralindicesαofparticlesacceleratedatoblique shocksversusshockvelocityprojectedatthemeanmagnetic field,UB,1 (Ostrowski1991a). Theresultsarepresented for the shock compression R = 4. On the right the respective synchrotron spectral index γ is given. The shock velocities U1 are given near the respective curves taken from Kirk & Heavens (1989). The points were taken from simulations computingexplicitlythedetailsofparticle-shockinteractions (Ostrowski 1991a). Asubstantialprogressinunderstandingtheaccel- Fig. 2. Energetic particle density profiles across the rel- erationprocessinthepresenceofhighlyanisotropic ativistic shock with the oblique magnetic field (Ostrowski particle distributions is due to the work of Kirk 1991b). The shock with U1 =0.5, R= 5.11 and ψ1 = 55o & Heavens (1989; see also Ostrowski 1991a and is considered. The curves for growing to the top perturba- Ballard & Heavens 1991), who considered particle tion amplitudes are characterized with the value logκ⊥/κk acceleration at subluminal (UB,1 < c) relativistic (κ⊥/κk is a ratio of the cross-field to the parallel diffusion coefficients) given near each curve. The data are vertically shocks with oblique magnetic fields. They assumed the magnetic momentum conservation, p2/B = shifted for picture clarity. The value Xmax is the distance ⊥ from the shock at which the upstream particle density de- const,atparticleinteractionwiththeshockandap- creases to10−3 part of theshock value. plied the Fokker-Planck equation discussed above to describe particle transport along the field lines outsidetheshock,whileexcludingthepossibilityof 3.2. History: acceleration in the presence of large cross-fielddiffusion.InthecaseswhenUB,1reached amplitude magnetic fieldperturbations relativistic values, they derived very flat energy spectra with γ 0 at UB,1 1 (Fig. 1). In such As the approaches proposed by Kirk & Schnei- ≈ ≈ conditions,theparticledensityinfrontoftheshock der (1987a) and Kirk & Heavens (1989), and the can substantially – even by a few orders of mag- derivations of Begelman & Kirk (1990) are valid nitude – exceed the downstream density (see the only in cases of weakly perturbed magnetic fields, curve denoted ‘-8.9’ at Fig. 2). Creating flat spec- for largeramplitude MHDperturbationsnumerical tra and great density contrasts is possible due to methods have to be used. A series of such simu- the effective repeating reflections of anisotropically lations were performed by numerous authors (e.g. distributed upstream particles from the region of Kirk & Schneider 1987b; Ellison et al. 1990; Os- compressedmagneticfielddownstreamoftheshock. trowski1991a,1993;Ballard&Heavens1992,Naito 3 &Takahara1995,Bednarz&Ostrowski1996).Dif- obliqueshocksofthespectralindexαchangingwith ferentapproachesapplied,includingtheonesinvolv- δB in anon-monotonicway. ing particle momentum pitch-angle diffusion or in- tegratingparticletrajectoriesinanalyticstructures of the perturbed magnetic fields1, allowedfor con- 3.3. History: Energy spectra of cosmic rays accelerated at ultra-relativistic shocks sideringawiderangeofbackgroundconditionsnear theshock.Theresultsobtainedbydifferentauthors canbesummarizedatthefigure(Fig.3)takenfrom The main difficulty in modelling the acceleration Bednarz&Ostrowski(1996).Oneshouldnote,that processes at shocks with large Lorentz factors Γ is essentiallyallderivationsbytheaboveauthorswere the fact that the involvedparticle distributions are performed with assuming scale-free conditions for extremely anisotropic in the shock, with the parti- −1 theaccelerationprocess,resultinginpower-lawdis- cleangulardistributionopeningangles Γ inthe ∼ tributions ofthe acceleratedparticles. upstream plasma rest frame. In the simulations of Bednarz&Ostrowski(1998)aMonteCarlomethod involvingsmallamplitudepitch-anglescatteringwas appliedforparticletransportneartheshockswithΓ intherange3–243.Thesimulationsrevealedanun- expected result, showing convergence, for Γ , → ∞ ofthe resultingpower-lawdistributionstothe’universal’onewiththespectralindexσ 2.2(Fig.4). ≈ Essentially the same result was derived with different methods by many other authors (Gallant & Achterberg1999;Kirketal.2000,Achterbergetal. 2001,Lemoioneetal.2003,Keshet&Waxman2005, Lemoine&Revenu2006,Morlinoetal.2007,etal.), whatcouldsuggestthatintheultrarelativisticlimit theaccelerationprocessbecomesagainsimple,generating cosmic ray spectra essentially independent Fig.3. Relationoftheparticleacceleration timescale Tacc of the considered background conditions. One can versus the particle spectral index α at different magnetic find examples in the literature that some authors field inclinations ψ1 given near the respective curves (Bed- extendthis claimforallrelativisticshocks. narz & Ostrowski 1996). The minimum value of the model parameterκ⊥/κkoccursattheencircledpointofeachcurve andthe waveamplitude monotonously increases alongeach curve up to δB ∼ B, where all curves converge in α. The 16 curveforΨ1=60◦ (UB,1=1)separatethesub-andsuper– λ = -3.44 luminalshockresults.Wedonotdiscussherethepresented 30(cid:176) acceleration times. 12 90(cid:176) Atthefigure(Fig. 3)onecanfindveryflatspec- tra for oblique subluminal shocks if the perturba- σ 20(cid:176) 60(cid:176) 8 tionamplitudesaresmall.Contrarytothatgenera- tionofthepower-lawspectrumispossibleinthesu- 10(cid:176) perluminalshocksonlyinthepresenceoflargeam- 4 plitude turbulence. Then, in contrast to the sublu- 0(cid:176) minal shocks,spectra are extremely steep for small values of δB (not presented at the figure) and α 0 monotonously decreases with increasing magnetic 3 9 27Γ 81 243 field perturbations. A new feature is observed in Fig.4. Thesimulatedspectral indicesσ (σ≡α−2)versus the shock Lorentz factor Γ (Bednarz & Ostrowski 1998). Results for a given ψ1 are joined with dashed lines; the 1 Letusnotethatapplicationbysomeauthorsofpoint-like respective value of ψ1 is given near each curve. Increasing large-anglescattering models inrelativisticshocks does not the turbulence amplitude in a not presented here series of provideaviablephysical representation ofthescattering at simulations led to shifting the resulting curves toward the MHDwaves (Bednarz & Ostrowski 1996). parallel shock, Ψ1=0◦,results. 4 4 ) ) (a) (b) E 3 g( 0.3 (3.08) o 0.3 (3.16) dl 2 N/ 1.0 (3.24) d ( 1 1.0 (3.43) g α = 3.55 o l 0 u = 0.5c u = 0.5c α = 3.69 1 1 ψ1 = 45° δB /B = 3.0 ψ1 = 45° -1 -1 0,1 -5/3 F(k) ~ k F(k) ~ k -2 δB /B = 3.0 0,1 -3 2π /k 2π /k 2π /k 2π /k max min max min ↓ ↓ ↓ ↓ -4 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 log(E/E ) 0 Fig.5. Particlespectraforanobliquemildlyrelativisticshock:theshockvelocityu1=0.5c,themeanmagneticfieldinclination Ψ1 =45◦ and the wave power spectrum are indicated in the respective panels (from Niemiec & Ostrowski 2004). Values of magnetic turbulence amplitude, δB/B, and the indices fitted to the power-law sections of spectra (in parentheses) are given near each result. Ostrowski& Bednarz(2002)reconsideredallthe an additional component of large amplitude short- aboveapproachesto derive particlespectraatrela- wave MHD turbulence was assumed to be produce tivistic shocks and ‘discovered’ that the conditions at the shock. The acceleratedparticle spectra were producingtheuniversalspectralindexwereinsome derived using the Monte Carlo simulations for a wayequivalenttoassuminghighlyturbulentcondi- wide range of shock Lorentz factors - between 2 tions near the shock. Additionally, all these mod- and 30 - and a selection ofthe mean magnetic field els did not introduce any physical scale and thus configurationsandperturbationamplitudes. forced the power-law shape of the resulting spectrum. Do such conditions and the resulting characteristic spectra really exist in astrophysical situa- tions ? 3.4. Towardarealisticdescriptionoftherelativistic shock acceleration Studiesof-asfaraspossible-realisticconditions fortherelativisticshockaccelerationwerepresented in a series of papers by Niemiec et al. (Niemiec & Ostrowski 2004, 2006, Niemiec et al. 2006; see also Lemoine Pelletier 2006). We assumed 3-D static magnetic fieldperturbationsupstreamofthe shock by imposing a large number of sinusoidal waves with different power spectra, F(k). The flat spectrum, F(k) k−1, or the Kolmogorov spectrum, Fig. 6. The particle spectra derived for superluminal rel- F(k) k−5/∝3, were considered in the wide wave- ativistic shocks with Lorentz factors γ1 = 5, 10 and 30 (Niemiec &Ostrowski 2006a). vector∝range (k , k ), with k /k = 105. min max max min The downstream magnetic field structures were Let us take a look at a few characteristic results computed by respective compression of the up- ofthesemodelings.AtFig.5resultsderivedforsub- streamfieldattheshock.Inthelastofabovepapers luminal oblique mildly relativistic shocks are pre- 5 Fig. 7. Particle spectra derived for parallel relativistic shocks with Lorentz factor γ1 = 30 (Niemiec & Ostrowski 2006). For description seeFig.5 andthe originalpaper. sented. One may note that introducing the energy theperturbationsamplitudes.Inparallelshocksthe scales to the modelling, in our units 2π/k and long wave perturbations introduce the acceleration max 2π/k ,leadtodeviationsoftheresultingspectra effectsobservedwithobliquemagneticfields(cf.also min fromthepower-lawshape.Thespectra,asexpected Ostrowski1993).Thusintheultrarelativisticparal- fromthe previousdiscussion,areveryflatfor small lel shocks propagating in highly turbulent medium amplitude turbulence, but steepen at larger ampli- these effects can lead to formation of particle dis- tudes. An interesting feature is seen, that at very tributions with cut-offs at relatively low energies, high particle energies, above the resonance range like in shocks with perpendicular field configura- (E > 2π/k ) the ‘short’ waves weaker influence tions (Fig. 7). If the shock wave generates a large min particletrajectoriesleadingtothehardenergytails amplitudeshort-waveturbulencedownstreamofthe before the cut-offsimposedby the modelling. shock,theaccelerationprocesscanformamoreex- Whenweconsideredobliquesuperluminalshocks tendedpower-lawtail,butathigherparticleenergies the spectra consisted of the shock compressed in- the mean magnetic field or the long wave magnetic jected component and the limited high energy tail field structures start to dominate in shaping par- until a cut-off well within the ‘resonance range’ ticle trajectories and thus the acceleration process, (2π/k , 2π/k ). As illustrated at Fig. 6 the leading to results like the ones described above. In max min tails fast diminish with the growing shock Lorentz anystudiedcasewewerenotabletocreatethescale factor, leaving for largeγ1 the ‘compressedcompo- free conditions in the acceleration process, leading nent’only. tothewiderangepowerlawdistributionofacceler- Thereareafewgeneralobservationsforthefirst- atedparticles.Itwaspossibleonlyinlimitedenergy orderFermiaccelerationprocessesfromtheseseries ranges and the forms of the spectra depended usu- ofmodels.Forparticlesinalowenergyrangeofthe ally alotonthe consideredbackgroundconditions. resonance energies for the considered field pertur- bationstheaccelerationprocessesproceedinanen- semble of different oblique shocks, where each local 3.5. Observational constraints on the shock meanmagneticfieldstructureisformedasasuper- acceleration from Cyg Ahotspots position of the mean magnetic field B0,1 and long wavefieldperturbations.Thus,therecanoccursig- Constraints for the above theoretical derivations nificantdifferencesbetweenspectrageneratedinthe canbeprovidedbypreciseobservationsofenergetic presence of flat and steep turbulence power spec- particle emission from objects harbouring the rela- tra, andthe spectralindices significantly vary with tivistic shocks. Such study performed for hotspots 6 of the Cygnus A radiosource (Stawarz et al. 2007) revealssignificantdeviationsofthederivedelectron 1E12 spectra from the ‘standard’ shock spectra. The re- Hotspot D sulting spectral energy distribution of the hotspot DispresentedatFig.8,showingboththeextended z] 1E11 H synchrotron component and the inverse-compton * (IC) one modelled for optical and X-ray measure- Jy ments. Additionally, the low Spitzer IR points [ 1E10 S provide additional important constraint for the IC spectralcomponent.Inderivationoftherelativistic 1E9 electronsdistributionthesemeasurementsallowfor excluding a possibility of substantial absorption in the low frequency synchrotron spectrum and thus 1E8 1E6 1E8 1E10 1E12 1E14 1E16 1E18 1E20 1E22 1E24 require very flat distribution of low energy elec- [Hz] trons.Thus the intrinsicelectronspectrum(Fig.9) Fig.8. Spectralenergydistributionofemissionfromthehot is composed of a very flat low energy sector, with spot D of Cyg A (Stawarz et al. 2007). One can clearly see the energy spectral index s 1.5, followed above both the synchrotron and the IC components. The Spitzer ≈ a break at E 1 GeV with the steep (s > 3) points in infrared and the optical point allow to exclude ≈ high energy sector.A possible interpretationof the possibility of forming the measured flat low-frequency syn- spectrumconsideredby us wastoassume amildly- chrotron component (dotted lines in synchrotron and IC spectral ranges) dueto someself-absorptionprocesses. relativistic shock acceleration in the jet dominated bytheprotonsbulkkineticenergy.Theprotonsare expected establish the characteristic break energy -2 scale, E 1 GeV, at the shock transition layer. br cooling ∼ The electrons (or pairs) below this energy are ex- effects pected to be accelerated within this layer due to 3 m electron-proton collective interactions. Above Ebr, ) / c -3 either the first order Fermi process acting at the n(e shock creates a steep spectrum, or the accelera- 2 tion process proceeds downstream of the shock in og l -4 the second-order Fermi process. The existing theoretical models do not allow to reject any of these absorption alternatives. effects -5 0 1 2 3 4 5 6 4. Energetic particle acceleration in shear log layers andregionsofrelativisticturbulence 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 high Acceleration processes acting, e.g., in AGN cen- energypartsareexpected tobeintrinsictotheacceleration tral sources and in shocks formed in large scale process,nottheeffectofthedistribution‘aging’downstream jets are not always able to explain the observed of the shock. high energy electrons radiating away from the cen- acceleration,as wellas the process of ‘viscous’par- tre/shock. Among a few proposed possibilities ex- ticleacceleration(cf.thereviewbyBerezhko(1990) plaining these data the relatively natural but still of the work done in early 80-th; Earl et al. 1988, unexploredistheoneinvolvingparticleacceleration Ostrowski 1990, 1998, 2000, Stawarz & Ostrowski within a shear layer transition at the interface be- 2002) can take place. A mean particle energy gain tweenthejetandthesurroundingmedium.Todate per scatteringinthe laterprocessscalesas knowledgeofphysicalconditionswithinsuchlayers isverylimitedandonlyroughestimatesforthecon- 2 <∆E > <∆U > sidered acceleration processes are possible. Within , (4.1) E ∝(cid:18) c (cid:19) the subsonic turbulent layer with a non-vanishing smallvelocitysheartheordinarysecond-orderFermi where < ∆U > is the mean velocity difference be- 7 tween the ‘successive scattering acts’. It is propor- energetictails,providingessentiallythecompressed tional to the mean free path normal to the shear injectedcomponentatlargeΓ.Anotherunexpected layer, λ , times the mean flow velocity gradient in featureisobserveddependenceofthespectrumincli- n thisdirection U~.Withddenotingtheshearlayer nationontheturbulenceamplitudealsoforthepar- n thicknessthis∇gra·dientcanbeestimatedas U~ allel shock waves and formation of cut-offs at such n |∇ · |≈ U/d. Because the acceleration rate in the Fermi II shocks for large Γ. Essentially no conditions stud- process is (V/c)2 (V V is the wave velocity, ied by Niemiec et al. resulted in formation of the A ∝ ≈ V – the Alfvén velocity), the relative importance wide-rangepower-lawparticledistributionwiththe A ofbothprocessesis givenby afactor universal spectralindex 2.2 . Itcanbeofinterest,withthepresentlypublished 2 λnU AUGER results (The Pierre Auger Collaboration . (4.2) (cid:18) d V (cid:19) 2007),thatthemodellingpresentedaboveseemsto exclude the first-order Fermi acceleration at rela- The relative efficiency of the viscous acceleration tivisticshocksaspossiblesourcesforhighestenergy growswithλ andintheformallimitofλ dand n n ≈ particlesregisteredinthis experiment. V c–outsidetheequation(4.2)validityrange–it ≪ In the same time the second-order Fermi pro- dominatesovertheFermiaccelerationtoalargeex- cesses acting in turbulent relativistic plasmas are tent. Becauseacceleratedparticlescanescape from expectedtoplaysignificantroleincosmicrayaccel- the accelerating layer only due to a relatively inef- eration.Thus,evenfacingsubstantialmathematical ficient radial diffusion, the formed particle spectra andphysicaldifficulties,itsdeserveadetailedstudy. areexpectedbeveryflatuptothehighenergycut- Onemaynotethattheaccelerationprocessesaccom- off, but the exactform ofthe spectrum depends on paniedthemagneticfieldreconnectionprocessesare severalunknownphysicalparametersofthebound- analogous to the second-order Fermi acceleration. arylayer(Ostrowski1998,2000). Such processes always accompany the large ampli- For turbulent relativistic plasmas the second- tude MHD turbulence and generate turbulence. A order Fermi acceleration can in principle dominate fewsimpleattemptstoconsidertheseprocesseswere over the viscous process at all particle energies. In recentlypresented(e.g.Virtanen& Vainio2005). the case of electrons the upper energy scale for the When considering the relativistic shock acceler- accelerated particles is provided by the radiation ation one should also note interesting approaches losses.A simple exerciseswith the aboveestimated by Derishev et al. (2003) and Stern (2003), outside acceleration scales and the synchrotron radiation the classical Fermi scheme. They consider the ac- lossscaleyields-forthesourceslikesmallandlarge celeration processes in highly relativistic shocks or scalejets,radiohotspotsorlobes–thehighestelec- jet shear layers (Stern & Poutanen 2006) involving tron energies between 1 TeV and 103 TeV, for the particle-particle or particle-photon interactions at respectiveaccelerationtimescales 103 sin gauss 3 ∼ both sidesofthe shock. fields and up to 10 yrs in µG fields (depending ∼ on the consideredobject). The muchhigher energy limits forprotonsareusuallydeterminedbythe es- cape boundaryconditions,notthe radiativelosses. AcknowledgementsIamgratefultomycollabora- 5. Finalremarks tors Jacek Niemiec andL ukasz Stawarz,whose sig- nificant work forms the main part of this rapport. A recent study of the first order Fermi accelera- ThepresentworkwassupportedbythePolishMin- tion processesat relativistic shocks,taking into ac- istryofScienceandHigherEducationinyears2005- countrealisticassumptionsaboutthe physicalcon- 2008asa researchproject1P03D00329. ditionsneartheshock,revealsafewunexpectedcon- clusions. The modelling of the acceleration process in mostly perpendicular (for relativistic velocities) shocks yields spectra consisting of the compressed injected part appended with a limited high energy References tail. For given upstream conditions increasing the Achterberg, A., Gallant, Y. A., Kirk, J. G., Guth- shock Lorentz factor Γ leads to steepening of such mann,A. 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