Particle acceleration in ultra-relativistic obliqu

arXiv:astro-ph/0212329v113Dec2002Particleaccelerationinultra-relativisticobliqueshockwavesA.Melia,b,1andJ.J.QuenbyaaAstrophysicsGroup,BlackettLaboratory,ImperialCollegeofScience,TechnologyandMedicine,PrinceConsortRd.SW72BW,London,UK.bVisitingMaxPlanckInstitutfuerRadioastronomie,AufdemHuegel6953121,Bonn,Germany.AbstractWeperformMonteCarlosimulationsofdiffusiveshockaccelerationathighlyrel-ativisticobliqueshockwaves.HighupstreamflowLorentzgammafactors(Γ)areused,whicharerelevanttomodelsofultrarelativisticparticleshockaccelerationinActiveGalacticNuclei(AGN)centralenginesandrelativisticjetsandGammaRayBurst(GRB)fireballs.Weinvestigatenumericallytheaccelerationpropertiesintherelativisticandultrarelativisticflowregime(Γ∼10−103),suchasangulardistribu-tion,accelerationtimeconstant,particleenergygainversusnumberofcrossingsandspectralshapes.Weperformcalculationsforsub-luminalandsuper-luminalshocks.Forthefirstcase,thedependenceonwhetherornotthescatteringispitchanglediffusionorlargeanglescatteringisstudied.Thelargeanglemodelexhibitsadis-tinctivestructureinthebasicpower-lawspectrumwhichisnotnearlysoobviousforsmallanglescattering.However,bothmodelsyieldsignificant’speed-up’orfasteraccelerationrateswhencomparedwiththeconventional,non-relativisticexpressionforthetimeconstant,oralternativelywiththetimescalerg/cwherergisLarmorradius.TheΓ2energizationforthefirstcrossingcycleandthesignificantlylargeenergygainforsubsequentcrossingsaswellasthehigh’speedup’factorsfound,areimportantinsupportingtheVietriandWaxmanworkonGRBultra-highenergycosmicray,neutrino,andgamma-rayoutput.Secondly,forsuper-luminalshocks,wecalculatetheenergygainforanumberofdifferentinclinationsandthespectralshapesoftheacceleratedparticlesaregiven.Inthisinvestigationweconsideronlylargeanglescattering,partlybecauseofcomputationaltimelimitationsandpartlybecausethismodelprovidesthemostfavourablesituationforacceleration.WeusehighgammaflowswithLorentzfactorsintherange10-40,whicharerelevanttoAGNaccretiondisksandjetultrarelativisticshockconfigurations.Wecloselyfol-lowtheparticle’strajectoryalongthemagneticfieldlinesduringshockcrossingswheretheequivalentofaguidingcentreapproximationisinappropriate,constantlymeasuringitsphasespaceco-ordinatesinthefluidframeswhereE=0.Wefindthatasuper-luminal’shockdrift’mechanismislessefficientinacceleratingparticlestothehighestenergiesobserved,comparedtothefirstorderFermiaccelerationap-plyinginthesub-luminalcase,suggestingthattheformercannotstandasasoleaccelerationmechanismfortheultra-highenergycosmicraysobserved.1IntroductionDetectionof106−1020eVnucleiwithinthesolarsystem,implyingtheirpres-encethroughmuchoftheobservableuniverse,hasstimulatedthedevelopmentofthediffusiveshockaccelerationmodel,wherebyparticlesarerepeatedlyac-celeratedinmultiplecrossingsofashockinterface,duetocollisionswithup-streamanddownstreammagneticscatteringcentres.Thepioneeringworkofthelate70s(Krymsky,1977;Bell,1978a,b;Axfordetal.,1978;BlandfordandOstriker,1978)establishedthebasicmechanismofparticlediffusiveaccelera-tioninnon-relativisticflows.However,sincethen,extensionofthemodeltoobliqueshocks,relativisticplasmaflowsandtheincorporationofnon-lineareffects,haverenderedtheemergenceofaconsensusphysicalpicturemoredif-ficult.Atobliqueshockfrontsthemotionoftheparticlesismorecomplicated,comparedtoaparallelshockconfiguration,astheymayeitherbetransmit-tedorreflectedbytheshocksurface.Apartfromthediffusiveaccelerationmechanism,anobliqueshockcouldalsoaccelerateparticlesintheabsenceofscattering(ArmstrongandDecker,1979),andthismechanismiscalled’shockdrift’acceleration.Providedthatthemotionoftheparticlesisdiffusiveintheobliqueshockconfigurationsithasbeenshown,forananalyticalnon-relativisticflowapproach,thatthesameresultasinparallelshocks,connectsthepower-lawindexofacceleratedparticleswiththecompressionratioandtheaccelerationratewiththediffusioncoefficientsintheshocknormaldirec-tion(e.g.Axfordetal.,1978;Bell,1978a;Drury,1983).Also,followingenergygainintheshockframeonshockcrossingviathedriftapproach,isequivalenttofollowingthegainviaLorentztransformationsfromtheshockframeintoanE=0frame(definedbydeHoffmannandTeller,1950)attheshock,andthenbackintotheshockframe.Ontheotherhand,thethe-oryofobliqueshocksconsidersfeaturesthatarenotinvolvedintheparallelshockcase,inparticulartheabilityoftheshocktoreflectparticlesmeetingaweakertostrongerfieldtransitionandthedependenceoftheaccelerationtimeonthefield-shocknormalangle(e.g.Jokipii,1982).Obliqueshockscanbeclassifiedintosub-luminalandsuper-luminal.Thedifferencebetweenthesetwocategoriesisthat,inthesub-luminalcase,itispossibletofindarelativis-tictransformationtotheframeofreference(deHoffmann-Tellerframe),inwhichtheshockfrontisstationaryandtheelectricfieldiszero(E=0)inbothupstreamanddownstreamregions.Ontheotherhand,super-luminalshock1Correspondingauthor:[e-mail]2frontsdonotadmitatransformationtosuchaframe,astheycorrespondtoshockfrontsinwhichthepointofintersectionofthefrontwithamagneticfieldlinemoveswithaspeedgreaterthanc.KirkandHeavens(1989)employedanumericalsolutiontothetransportequation(inthete