Cosmic Ray Acceleration at Relativistic Shock Wave

arXiv:astro-ph/9610189v123Oct1996CosmicRayAccelerationatRelativisticShockWavesM.OstrowskiObserwatoriumAstronomiczne,UniwersytetJagiello´nski,Krak´ow,Poland(E-mail:[e-mail])AbstractTheoryofthefirst-orderFermiaccelerationofcosmicrayparticlesatrelativisticshockwavesisreviewed.Weconsidershockswithparallelandoblique,sub-andsuper-luminalmagneticfieldconfigurationsandwithfinite-amplitudemagneticfieldperturbations.Aroleofobliquefieldconfigurationsandfieldperturbationsinformingthecosmicrayenergyspectrumandchangingtheaccelerationtimescaleisdiscussed.1.Particleaccelerationatnon-relativisticshockwavesProcessesoffirst-orderparticleaccelerationatnon-relativisticshockwaveswerewidelydiscussedbyanumberofauthorsduringthelasttwodecades(forreview,see,e.g.Drury(1983),Blandford&Eichler(1987),Berezkoetal.(1988),Jones&Ellison(1991)).Below,weremindthebasicphysicalpictureandsomemostoftheimportantresultsobtainedwithinthistheory,tobelatercomparedwiththeresultsobtainedforrelativisticshocks.Thepreferredbyussimpledescriptionoftheaccelerationprocessconsistsinconsideringtwoplasmarestframes,theupstreamframeandthedownstreamone.Weuseindices‘1’or‘2’toindicatequantitiesmeasuredintheupstreamorthedownstreamframerespectively.Ifoneneglectsthesecond-orderFermiacceleration,theparticleenergyisaconstantofmotioninanyplasmarestframeandenergychangesoccurwhentheparticlemomentumisLorentz-transformedateachcrossingoftheshock.Inthecaseofparallelshock,withthemeanmagneticfieldparalleltotheshocknormal,theaccelerationofanindividualparticleisduetoaconsecutiveshockcrossingsbythediffusivelywanderingparticle.Eachupstream-downstream-upstreamdiffusiveloopresultsinasmallincrementofparticlemomentum,Δp∝U1/v,wherevistheparticlevelocityandU1istheshockvelocityintheupstreamframe,U1≪v≈c.Inobliqueshocks,theparticlehelicaltrajectoriescancrosstheshocksurfaceanumberoftimesatanyindividualshocktransitionorreflection.Themostinterestingfeatureofthefirst-orderFermiaccelerationatanon-relativisticplane-parallelshockwaveisindependenceofthetest-particlestationaryparticleenergyspectrumfromthebackgroundconditionsneartheshock,includingthemeanmagneticfieldconfigurationandthespectrumofMHDturbulence.Themainreasonbehindthatisanearly-isotropicformoftheparticlemomentumdis-tributionattheshock.Ifasufficientamountofscatteringoccursneartheshock,thisconditionalwaysholdsfortheshockvelocityalongtheupstreammagneticfieldUB,1≡U1/cosΨ1≪v(Ψ1-theupstreammagneticfieldinclinationtotheshocknormal).Independentlyofthefieldinclinationattheshock,theparticleden-sityiscontinuousacrossitandthespectralindexforthephase-spacedistributionfunction,α,isdefinedexclusivelyinthetermsoftheshockcompressionratioR:α=3RR−1.(1.1)Becauseoftheisotropicformoftheparticledistributionfunction,thespatialdiffu-sionequationhasbecomeawidelyusedmathematicaltoolfordescribingparticletransportandaccelerationprocessesinnon-relativisticflows.ThecharacteristicaccelerationtimescaleattheparallelshockisTacc=3U1−U2κ1U1+κ2U2,(1.2)whereκi≡κk,iistherespectiveparticlespatialdiffusioncoefficientalongthemagneticfield,asdiscussedbye.g.Lagage&Cesarsky(1983).Ostrowski(1988,seealsoBednarz&Ostrowski1996)derivedananalogousexpressionforshockswithobliquemagneticfieldsandsmallamplitudemagneticfieldperturbations.Foranegligiblecross-fielddiffusionandforUB,1≪citcanbewritteninessentiallythesameformastheonegiveninEqu.(1.2),withallquantitiestakenasthenormal(n)oneswithrespecttotheshock(κn,iforκi(i=1,2)).Usuallyκnκk.Therefore,asconfirmedbymeasurementsintheheliosphere,theobliqueshocksmaybemuchmorerapidacceleratorsascomparedtotheparallelshocks.2.Cosmicrayaccelerationatrelativisticshockwaves2.1TheFokker-PlanckdescriptionoftheaccelerationprocessInthecaseoftheshockvelocity(oritsprojectionUB,1)reachingvaluescomparabletothelightvelocity,theparticledistributionattheshockbecomesanisotropic.Thisfactcomplicatestoagreatextentboththephysicalpictureandthemathematicaldescriptionofparticleacceleration.Thefirstattempttoconsidertheaccelerationprocessattherelativisticshockwaspresentedin1981byPeacock(seealsoWebb1985);however,noconsistenttheorywasproposeduntilapaperofKirk&Schneider(1987a;seealsoKirk1988)appeared.Thoseauthorsconsid-eredthestationarysolutionsoftherelativisticFokker-Planckequationforparticlepitch-anglediffusionforthecaseofparallelshockwave.Inthesituationwiththegyro-phaseaverageddistributionf(p,μ,z),whichdependsonlyontheuniquespa-tialco-ordinatezalongtheshockvelocity,andwithμbeingthepitch-anglecosine,theequationtakestheformΓ(U+vμ)∂f∂z=C(f)+S,(2.1)whereΓ≡1/√1−U2istheflowLorentzfactor,C(f)isthecollisionoperatorandSisthesourcefunction.Inthepresentedapproach,thespatialco-ordinatesaremeasuredintheshockrestframe,whiletheparticlemomentumco-ordinatesandthecollisionoperatoraregivenintherespectiveplasmarestframe.Fortheappliedpitch-anglediffusionoperator,C=∂/∂μ(Dμ,μ∂f/∂μ),theygeneralisedthediffusiveapproachtohigherordertermsinparticledistributionanisotropyandconstructedgeneralsolutionsatbothsidesoftheshockwhichinvolvedsolutionsoftheeigenvalueproblem.Bymatchingtwosolutionsattheshock,thespec-tralindexoftheresultingpower-lawpar