Particle acceleration in ultra-relativistic parall

arXiv:astro-ph/0212328v113Dec2002Particleaccelerationinultra-relativisticparallelshockwavesA.Melia,b,1andJ.J.QuenbyaaAstrophysicsGroup,BlackettLaboratory,ImperialCollegeofScience,TechnologyandMedicine,PrinceConsortRd.SW72BW,London,UK.bVisitingMaxPlanckInstitutfuerRadioastronomie,AufdemHuegel6953121,Bonn,Germany.AbstractMonte-Carlocomputationsforhighlyrelativisticparallelshockparticleaccelerationarepresentedforupstreamflowgammafactors,Γ=(1−V21/c2)−0.5withvaluesbetween5and103.TheresultsshowthatthespectralshapeattheshockdependsonwhetherornottheparticlescatteringissmallanglewithδθΓ−1orlargeangle,whichispossibleifλ2rgΓ2whereλisthescatteringmeanfreepathalongthefieldlineandrgthegyroradius,thesequantitiesbeingmeasuredintheplasmaflowframe.Thelargeanglescatteringcaseexhibitsdistinctivestructuresuperimposedonthebasicpower-lawspectrum,largelyabsentinthepitchanglecase.Also,bothcasesyieldanaccelerationratefasterthanestimatedbytheconventional,non-relativisticexpression,tacc=[c/(V1−V2)][λ1/V1+λ2/V2]where’1’and’2’refertoupstreamanddownstreamandλisthemeanfreepath.AΓ2energyenhancementfactorinthefirstshockcrossingcycleandasignificantenergymultiplicationinthesubsequentshockcyclesarealsoobserved.Theresultsmaybeimportantforourunderstandingoftheproductionofveryhighenergycosmicraysandthehighenergyneutrinoandgamma-rayoutputfromGammaRayBursts(GRB)andActiveGalacticNuclei(AGN).1IntroductionTherearethreedistinctastrophysicalsituationswherethebulkplasmaflowisextremelyrelativisticwithLorentzfactorsΓ=(1−V2/c2)−1/2≥10.ThesearesomeActiveGalacticNuclei(AGN)jetandhotspotsites,GammaRay1Correspondingauthor:[e-mail](GRB)fireballsandpulsarpolarwinds.Withintheseflowsrelativis-ticshocksappearanddiffusiveparticleaccelerationtakesplace.Ineachcase,thereisevidenceforenergeticparticleaccelerationtohighΓfactorsbuttheupperlimittothepossibleenergyattainedbecomeslesscertainwithincreas-ingbulkflowvelocity.ForAGNjets,Γ∼10plausiblyresultsintheknown1019eVcosmicrays(QuenbyandLieu,1989)viadiffusiveshockaccelerationattheterminationshocks.However,themoredirectsynchrotron/self-Comptonmodelingonlyreveals10GeVgamma-rays(e.g.Maraschietal.,1992).ForGRBfireballs,Γ∼100−103appearstoeventuallyproducegamma-raysatleastupto100MeV(Fenimoreetal.,1993)butatsuchflowspeeds,itisnotclearwhetherrepeatedshockcrossingsarepossibleandthepredictionsof1019eVneutrinos(Vietri,1998),whichwouldbedirectevidenceofadiffusivelikeprocesshaveyettobeverified.Similarly,pulsarwindscouldresultinaΓ∼104flowencounteringthenebulaenvelope(Hoshinoetal.,1992),butthereisnoevidenceformorethanTeVaccelerationanywhereinthesystem.HoweverLucekandBell(1994)showthatcombinedmotionacrossanearper-pendicularshockwithanon-uniformfieldandintheelectromagneticfieldofthenebulamayproducemuchhigherenergies.PastworkhassuggestedthatiftheshocksareclosetoparallelsothatanE=0framecanbedefined,diffusiveshockaccelerationisinprinciplepossi-bleandprovidedlargeanglescatteringoccursintheplasmaframe,eachshockcrossingcycleyieldsa∼Γ2energyincrease(QuenbyandLieu,1989).ThisresulthasbeenexploitedbyVietri(1995,1998)modelingGRBfireballparticleshockacceleration.ItisthepurposeofthispapertopresentsimulationresultsonparticleaccelerationwithΓvaluesextendedupto∼103.Hereweconcen-trateontheparallelshockconfigurations(ψ=0o).Ourresults,measuredintheshockframe,willbecomparedwiththestandardspectralpredictionofnon-relativistictheory,wherethemomentumspectrumisn(p)∝p−αwithα=(r+2)/(r−1),forshockcompressionratior=V1/V2,independentofscat-teringdetails,andα=2forstrongunmodifiedshockswithr=4.Analyticallyithasbeenshownthatoveralimitedmomentumrange,thenon-relativistictimescaleforaccelerationisT=[c/(V1−V2)][λ1/V1+λ2/V2]where,’1’and’2’refertoupstreamanddownstreamandλisthescatteringmeanfreepath.TheshockjumpsolutionsofBallardandHeavens(1991)andthereviewofKirkandDuffy(1999)showthat,arangeofshockframervalueslargelylimitedbetween3and4,typicallyariseforoblique,relativisticshocks,whereasr=4istheextremerelativistichydrodynamiclimit.Henceforsimplicityweadoptr=4but,inthecompanionpaperdealingwithobliqueshockswedemonstratetheinsensitivityofourchiefconclusionsontheexactvalue.Forourparame-ters,T=(8/3)(nrg,1/c)where,λ=nrgandrgisLarmorradius.Thefirstrelativisticcorrectionnoticedtothenon-relativistictheory,wasthespectralflatteningseeninparallelshocks(KirkandSchneider,1987a,b),fol-lowedbythediscoveryofaspeed-upintheaccelerationrate(QuenbyandLieu,1989;Ellisonetal.,1990).Theseresultshavesubsequentlybeenex-tendedtonon-parallelandnon-linearsub-luminalshocks(Lieu,etal.,19942andBednarzandOstrowski,1996whouseT=rg/castheunitofaccelerationtime),wherethereislargeanglescatteringorpitchanglediffusion.Mostre-cently,muchinteresthasbeenfocusedondifferencesoccurringbetweensmallangleandlargeanglescatteringmodelsforthetestparticle-turbulencein-teractionsinthefluidframe.GallantandAchterberg(1999)suggestthatforafieldmodelconsistingeitherofrandomlyorientateduniformfieldcellsorauniformfield,thescatteringislimitedtoδθ1/Γintheupstreamfluidframe.Generally,suchamodelyieldsatestparticledifferentialspectralslope∼−2.2(Baring,1999).Furtherm