Mon.Not.R.Astron.Soc.000,000–000(0000) Printed5January2012 (MNLATEXstylefilev2.2) Radiative signature of magnetic fields in internal shocks P. Mimica1⋆ and M. A. Aloy1 1DepartamentodeAstronom´ıayAstrof´ısica,UniversidaddeValencia,46100,Burjassot,Spain 2 1 0 2 5January2012 n a ABSTRACT J Commonmodelsofblazarsandgamma-rayburstsassumethattheplasmaunderlyingtheob- 4 servedphenomenologyismagnetizedtosomeextent.Withinthiscontext,radiativesignatures ofdissipationofkineticandconversionofmagneticenergyin internalshocksofrelativistic ] E magnetizedoutflowsarestudied.We modelinternalshocksasbeingcausedbycollisionsof H homogeneousplasmashells.Wecomputetheflowstateaftertheshellinteractionbysolving Riemannproblemsatthecontactsurfacebetweenthecollidingshells,andthencomputethe . h emissionfromtheresultingshocks.Undertheassumptionofa constantflowluminositywe p findthatthereisacleardifferencebetweenthemodelswherebothshellsareweaklymagne- o- tized(σ<10−2)andthosewhere,atleast,oneshellhasaσ>10−2.Weobtainthattheradiative r efficienc∼yislargestformodelsinwhich,regardlessofthe∼ordering,oneshellisweaklyand t theotherstronglymagnetized.Substantialdifferencesbetweenweaklyandstronglymagne- s a tizedshellcollisionsareobservedintheinverse-Comptonpartofthespectrum,aswellasin [ the optical, X-ray and 1GeV light curves. We propose a way to distinguish observationally between weakly magnetized from magnetized internal shocks by comparing the maximum 2 v frequency of the inverse-Compton and synchrotron part of the spectrum to the ratio of the 8 inverse-Comptonandsynchrotronfluence.Finally,ourresultssuggestthatLBLblazarsmay 0 correspondto barely magnetizedflows, while HBL blazarscould correspondto moderately 1 magnetizedones.Indeed,bycomparingwithactualblazarobservationsweconcludethatthe 6 magnetizationoftypicalblazarsisσ<0.01fortheinternalshockmodeltobevalidinthese 1. sources. ∼ 1 Keywords: BLLacertaeobjects:general–Magnetohydrodynamics(MHD)–Shockwaves 1 –radiationmechanisms:non-hermal–radiativetransfer–gammarays:bursts 1 : v i X r 1 INTRODUCTION is commonly assumed when modeling shell interactions analyt- a ically (e.g., Kobayashietal. 1997; Daigne&Mochkovitch 1998; Highly variable radiation flux has been observed in the rela- Spadaetal.2001; Bosˇnjaketal.2009). Particularly,theinfluence tivistic outflows of blazars and gamma-ray bursts (GRBs). Even of the magnetic field (if present) has been shown to significantly though theradiationenergy andtimescalesaredifferent for both alter the dynamics (Mimicaetal. 2007). In spite of these efforts, classes of objects (γ-rays on a millisecond timescale for GRBs westilldonotknown withcertaintywhether theflow,whoseen- versus X-rays on a timescale of hours for blazars) the underly- ergyisbeingdissipated,issignificantlymagnetized,orwhetherit ing physics responsible for the energy dissipation might be very isonlythekineticenergywhichultimatelypowerstheemission. similar.The internal shock scenario (Rees&Meszaros 1994) has been used to explain the variability of blazars (e.g., Spadaetal. Inapreviouswork(Mimica&Aloy2010,MA10 inthefol- 2001;Mimicaetal.2004)andGRBs(e.g.,Kobayashietal.1997; lowing)wehavestudiedthedynamicefficiency,i.e.theefficiency Daigne&Mochkovitch1998;Bosˇnjaketal.2009).Inthisscenario ofconversionofkinetictothermaland/ormagneticenergyininter- inhomogeneities in a relativistic outflow cause parts of the fluid nalshocks.Wefoundthatthedynamicefficiencyisactuallyhigher tocollideand produce shockswaveswhichdissipateenergy. The iftheshellsaremoderatelymagnetized(σ 0.1,seethenextsec- ≈ shellcollisionsareoftenidealizedascollisionsofdenseshells.In tionfor thedefinitionofσ)than ifbothareunmagnetized. How- recentyearsone-andtwo-dimensionalrelativistichydrodynamics ever, wedid not compute the radiativeefficiency of such interac- (RHD,Kinoetal.2004;Mimicaetal.2004,2005)andrelativistic tions, but instead used the dynamic efficiency as an upper bound magneto-hydrodynamics(RMHD,Mimicaetal.2007)simulations ofit.RecentlyBo¨ttcher&Dermer(2010,BD10inthefollowing), oftheshellcollisionshavebeenperformedandhaveshowedthat Joshi&Bo¨ttcher(2011)andChenetal.(2011)havepresentedso- thedynamicsofshellinteractionismuchmorecomplexthanwhat phisticated models for the detailed computation of the emission frominternalshocks.Whilethesemodelsassumeasimplehydro- dynamicevolution,theyemployatime-dependent radiativetrans- ⋆ E-mail:Petar.Mimica@uv.es fer scheme which involves the synchrotron and synchrotron self- 2 P.Mimica andM. A. Aloy Compton(SSC)processesaswellasthecontributionComptonized theparticularsoftheinitialstatesthesewavescaneitherbeshocks externalradiation(externalinverseCompton-EIC),allthewhile orrarefactions.Welabeltheleft-goingwavewithRStodenotea takingintoaccounttheradiativelossesoftheemittingnon-thermal reverseshock,andwithRRincaseareverserarefactionhappens. particles.WehaveadaptedthemethodofBD10anduseittoper- Similarly,welabel theright-going wave withFSor FR todiffer- formaparametricstudy,addressedtoinferthemagnetizationofthe entiatethecasesinwhichaforwardshockoraforwardrarefaction flowfromthelightcurvesandspectraofinternalshocksinmagne- occurs,respectively.WewilluseasubscriptS torefertotheprop- tizedplasma. erties of the shocked fluid in general, and the subscripts FS and Theorganizationofthispaperisasfollows:Section2briefly RS whendistinguishingbetweenthefrontandreverseshockedflu- summarizes the model of MA10 which isused to study theshell ids.Finally,wewillusethesubscript0forpropertiesoftheinitial collisiondynamics,andinSections3and4wedescribethenumer- statesingeneral,andthesubscriptsLandRwhenweneedtodis- icalmethodweemploytocomputethenon-thermalradiation.We tinguish between left and right initial states. Because we assume discusstheradiativeefficiencyinSection5andpresentthespectra thattheflowluminosityisthesameforbothinitialstates,using(1) and light curves in Section 6. A global parameter study is elab- wedeterminethenumberdensityintheshellstobe orated in Section 7. We close the paper with a discussion of our L resultsandgiveourconclusions(Sec.8). n = , (3) L,R πR2m c3 Γ2 (1+ǫ+χ+σ ) Γ 1 Γ 2 p h L,R L,R − L,Ri q − −L,R whereΓ =(1+∆g)Γ . 2 SHELLCOLLISIONDYNAMICS L R TheRiemannsolverprovidesuswiththebulkvelocityofthe AswasdiscussedindetailintheSection2ofMA10,ouraimisto shockedfluidβc(anditsLorentzfactorΓ=(1 β2) 1/2),andveloc- − − model a largenumber of shell collisions withvarying properties. itiesβ candβ coftheFSandRS,respectively(providedthey FS RS Therefore, we employ a simplifiedmodel for a singleshell colli- exist).Thevelocityoftheinitial(unshocked)statesintheCDrest sion, based ontheexact solutionof theRiemann problem. When frameis describing the initial states of the Riemann problem we will use β β subscriptsLandRtodenoteleft(faster)andright(slower)shells, β′0= 10−ββ . (4) respectively. − 0 TheshockvelocitiesintheframeoftheCDcanbecomputedas WeassumeacylindricaloutflowwitharadiusR.Mimicaetal. (2004)showthatthejetlateralexpansioninthiscaseisnegligible. β β Forsimplicity,andbeingconsistentwithpreviousworkinthefield β′S = 1S −ββ , (5) (e.g.,BD10,Joshi&Bo¨ttcher2011),wealsoignoretheshelllon- − S whereprimedenotesquantitiesintheCDrestframe.Inthisframe gitudinalexpansionaftertheshockscrosstheshell(seealsoSec- theshockcrossestheshellatatime tion3.4).Followingtheequation 9of MA10wedefinethelumi- nosityas tc′ross,S = c∆βr0′ , (6) L:=πR2ρc3 Γ2(1+ǫ+χ+σ) Γ √1 Γ 2, (1) ′S − − − (cid:12) (cid:12) wherecisthehspeedoflightinvacuuim,ρisthefluidrest-massden- where∆r0′(cid:12)(cid:12)ist(cid:12)(cid:12)heshellwidthintheCDframe, sity,ǫisthespecificinternalenergy,χ:= p/(ρc2)istheinitialratio β β betweenthethermalpressureandtherest-massenergydensity,and ∆r0′ =Γ∆rβ − βS . (7) σ := B2/(4πρΓ2c2)isthemagnetization parameter.Here Bisthe 0− S strength of the large-scale magnetic field, which is perpendicular tothedirectionofpropagationofthefluidmovingwithvelocityv 3 NON-THERMALPARTICLES andacorresponding LorentzfactorΓ := 1/ 1 (v/c)2.Thespe- − cific internal energy is related to the pressupre and to the density Inthissectionweshowthepropertiesofnon-thermalparticlesand throughtheequationofstate.WeusetheTManalyticapproxima- theiremission.Wefirstdiscussthemodelforthemagneticfieldand tiontotheSyngeequationofstate(deBerredo-Peixotoetal.2005; non-thermalparticles,andthenoutlinethemethodusedtocompute Mignoneetal.2005)andobtain: theiremission. 3 p 9 p 2 1/2 ǫ:= + +1 1. (2) 2ρc2 4 ρc2!  − 3.1 Magneticfield WeassumethatL = L andχ = χ .Furthermore,asinMA10, L R L R AsinMimica&Aloy(2010)andBD10,weassumethatthereex- weassumeΓ := (1+∆g)Γ .ThisleavesuswithR,σ , σ and L R L R ists a stochastic magnetic field, which is created in situ by the ∆g as parameters, because all other quantities can be determined shocksarisinginthecollisionoftheshells.Welabelthisfieldby usingequations(1)and(2).Tothese,weaddanadditionalparam- B , and by definition itsstrength isa fraction ǫ of the internal eter∆r,whichistheinitialwidthoftheshellsintheLAB-frame. S,st B energy density of the shocked shell u (obtained, in our case, by S WhileitdoesnotinfluencethesolutionoftheRiemannproblem, theexactRiemannsolver): it provides the physical scale necessary for the calculation of the observedemission. B = 8πǫ u . (8) S,st B S Oncetheinitialstatesareconstructed,wecomputetheexact p Sinceweallowforarbitrarilymagnetizedshells,thereisalso solutionoftheRiemannproblemusingthesolverofRomeroetal. anordered(macroscopic)magneticfieldcomponent B ,which (2005). The initial discontinuity between left and right states de- S,mac isadirectoutputoftheexactRiemannsolver.Thetotalmagnetic composesintoacontactdiscontinuity(CD),andaleft-goinganda rightwave(intheframeinwhichtheCDisatrest).Dependingon fieldisthenBS := B2S,st+B2S,mac. q Radiativesignaturesof magneticfields 3 3.2 Injectionspectrumofnon-thermalparticles 3.3 Particleinjectioncut-offs We assume that a fraction of electrons in the unshocked shell is As was done in Mimicaetal. (2010), we assume that the upper accelerated to high energies at the shock front. Following Sec- cutoff for the electron injection is obtained by assuming that the tion3ofBD10,weassumethatafractionǫ ofthedissipatedki- acceleration time scale isproportional to the gyration timescale. e neticenergyisusedtoaccelerateelectrons.Weassumethatsome ThenthemaximumLorentzfactorisobtainedbyequatingthistime particle acceleration mechanism operates at shocks, such that a scaletothecoolingtimescale, fraction of the electrons in the unshocked shell is accelerated to 3m2c4 1/2 high energies in the vicinity of the shock front. As it is com- γ = e . (13) 2 4πa e3B ! monly done (Bykov&Meszaros 1996; Daigne&Mochkovitch acc S 1998;Mimicaetal.2004,BD10),weassumethatafractionǫ of where a > 1 is the acceleration efficiency parameter (BD10). e acc the dissipated kinetic energy is used to accelerate electrons. The Thelowercut-offisobtainedbyassuming,incompleteanalogyto widthoftheaccelerationzone∆r isparametrizedasamultiple Eq. 10, that the number of accelerated electrons is related to the a′cc ∆ of the proton Larmor radius inthe shocked fluid. Therefore, numberofelectronspassingthroughtheshockfront, acc theaccelerationatagivenpointintheshockedfluidlastsforatime dN ∆ta′cc=∆ra′cc/(β′Sc).Wehave dti′nj,0 =ζeπR2n0Γ′0β′Sc, (14) ′ Γ m c2 whereζ isthefractionofelectronsacceleratedintothepower-law ∆ra′cc=∆acc ′0eBp , (9) distributeion.FromEqs.14,10and11weget S aFtacrkcoeemsleptrhalaitsicoeenxarpsergVeisao′scncioi=ns,πwRe2∆corma′ccp.uTtehetheenveorgluyminejewchtieorneraactceeilnertoatitohne Rγγγ1γ122 ddγγγγ1−−qq = ζǫeeΓ′0nu0mS ec2 . (15) R Sincewearedealingwithpotentiallyhighlymagnetizedfluids,the dEdit′nj,0 =πR2ǫeuS∆∆rta′cc , (10) wcoencdaitnionnotγu2se≫thγe1ecqaunantiootnbseucahssausmEeqd.(1s3eeoEfqB.D1130).,Tanhdertehfeorreefowree ′ a′cc computeγ fromEq.15numericallyusinganiterativeprocedure. andweassumethattheenergyspectrumoftheinjectedrelativistic 1 particlesisapower-lawintheelectronLorentzfactorγ, 3.4 Evolutionoftheparticledistribution dn dt ′idnjγ =Q0γ−qH(γ; γ1,γ2), (11) In this work we assume that particles cool via synchrotron and ′ external-Compton processes. We ignore the adiabatic cooling in wheren′inj isthenumber densityoftheinjectedelectrons, Q0 isa thisworksinceweareprimarilyinterestedincollisionsofmagne- normalizationfactorandγ1andγ2arethelowerandupperinjection tizedshells,wheretheelectronsarefast-cooling.Theconsequence cutoffs (computed below), all measured in the shocked fluid rest of not accounting for the adiabatic loses of the particle distribu- frame. The step function is defined as usual by H(x;a,b) = 1 if tionisthatourmethodoverestimatestheemissionaftertheshocks a6x6band0otherwise. crosstheshells.Nevertheless,mostofthefeaturesthatmakesub- Acautionarynoteshouldbeaddedhereregardingthefactthat stantivedifferencesbetweenthedynamicstriggeredbymagnetized wechoosethatthespectralenergydistributionoftheinjectedparti- andnon-magnetizedshellshappenintheearlylightcurveand,thus, clesisapurepower-laweveninthehigh-σregime.Boththeoretical neglectingtheexpansionoftheshellsplasmadoesnotchangethe arguments(e.g.,Kirk&Heavens1989)andrecentparticle-in-cell qualitativeconclusionsofthispaper. (PIC)simulations(e.g.,Sironi&Spitkovsky2009)haveshownthat TheradiativelossesforanelectronwithaLorentzfactorγcan particleaccelerationisnotveryefficientinthepresenceofastrong bewrittenas mpaargtinceleticinfijeecldtiopnarsaplelecltrtuomthmesighhotckinvfroolnvte.tThheepmreosdeinficceaotifonthsetothtehre- γ˙ =−43cσTu′Bm+cu2′extγ2, (16) malpopulation.Recently,acalculationbyGiannios&Spitkovsky e (2009)showshowthespectrumoftheGRBpromptemissionmight whereu′B=B2S/8πΓ2andu′extaretheenergydensityofthemagnetic lookinsuchacase:abumpatthespectralmaximumandalower fieldandtheexternalradiationfield(seeSec.4.2)intheshocked contribution at ultra-high energies. However, current PICsimula- fluidframe,respectively. Once theenergy losseshave been spec- tionshavenotbeenrunforsufficientlylongtimetoachieveastable ified, we use the semi-analytic solver of Mimicaetal. (2004) to situation.Thus,thefractionoftheenergywhichgoesintothermal computetheparticledistributionatanytimeafterthestartofthein- electrons (parameter δ of Giannios&Spitkovsky 2009) is still to jectionatshock.Moreprecisely,weusethesolutionforthecontin- bedetermined.Inthisparticularstudywesetthisfractiontozero uousinjectionandparticlecooling(Eq.19ofMimicaetal.2004) andthusavoidintroducinganotherfreeparameterinourmodels. for the time ∆ta′cc since the beginning of the shock acceleration. IntegratingthedistributionEq.11inLorentzfactorandequat- After that time the shock acceleration ends and we approximate ingtheresulttoEq.10dividedbyV (inordertoobtaintheenergy theresultingparticledistributionby apiecewise power-law func- a′cc densityinjectionrateintotheaccelerationregion)wecancompute tion. Then we employ Eq. 17 of Mimicaetal. (2004) on each of thenormalizationfactorfortheelectroninjection, thepower-lawsegmentstocomputethesubsequentevolution. q 2 − ifq,2 Q0 = dVEa′ic′ncjm,0/ecd2t′ × γ121−/ql−nγγ22−2q ifq=2 . (12) 4WeNasOsuNm-TeHthEatRtMheAoLbsRerAveDr’IsAlTinIeOoNfsightmakesanangleθwith  γ1! thejetaxis,whichisalsothedirectionofpropagationofthefluid. 4 P.Mimica andM. A. Aloy Wedenoteby xandtthepositionandtimeintheobserverframe, andRSif x < 0(i.e.inEqs.12,13, 15and8thevalues for the ′ andbyx andt thelocationandtimeintheCDframe.Weassume correspondingshockedfluidshouldbeused). ′ ′ thattheCDislocatedat x = 0forallt.Letµbeµ := cosθ,and Consideringthatν = ν(1+z)/ ,andusingEq.22,wecan ′ ′ ′ D definethetimeatwhichanobserverseestheradiationemittedfrom computethefluxintheobserverframeobtaining(BD10, Dermer xattimetas 2008) T =t xµ/c. (17) x′max(T) − 4πR2 Thenthetimeasafunctionofthetimeofobservationandposition νFν(T)= Dd2 Z dx′ν′j′ν′[ta′(T,x′)], (25) L canbewrittenas x′min(T) whered istheluminositydistance.Weperformtheintegrationin t′=D(T/(1+z)+Γ(µ−β)x′/c), (18) Eq.25nLumerically. where := [Γ(1 βµ)] 1 istheDoppler factor andzisthered- The total emissivity is assumed to be the result of combin- − D − shift.LorentztransformationshavebeenappliedtoEq.17toobtain ingthreeemissionprocesses:(1)synchrotronradiation,(2)inverse Eq.18. Comptonwithanexternalradiationfield(EIC),andthesynchrotron Animportantquantityisthetimeelapsedaftertheshockhas self-Compton (SSC) up-scattering. These emission processes are passedagivenx.Fromsuchvalueonecancalculatetheageofthe consideredinmoredetailinthenextsections. ′ electrondistributionfunctionatthatposition,whichturnstobethe timesincetheshockaccelerationhasbegun.Thus,theagecanbe 4.1 Synchrotronemission definedas x Wecomputethesynchrotronemissionforeachpower-lawsegment ta′ :=t′− β′c. (19) of the electron distribution (see Sec. 3.4) separately. In order to ′S speedupthecalculationweusetheinterpolationmethoddescribed AmoreusefulexpressioninvolvesT.UsingEqs.19and18weget, in Mimicaetal. (2009, section 4) and, in more detail, in Mimica fortheFS (2004,sections2.1.3and4.3.1). T x 1 β µ ta′,FS =D"1+z − cΓ′ β−FS −FSβ #, (20) 4.2 EICemission andfortheRS Following BD10, we assume that the external radiation field is T x 1 β µ monochromaticandisotropicintheAGNframe.Wedenotethefre- ta′,RS =D"1+z − cΓ′ β−RS −RSβ #. (21) quencyandtheradiationfieldenergydensityinthisframebyνext and u , respectively. Transforming into the shocked fluid frame WenotethatEq.20hastobeusedwhen x > 0,whileEq.21is ext ′ weget validwhen x < 0. Ift 6 0,thentheshockhasnot crossedthat ′ a′ positionyetand,consequently,thatplacedoesnotcontributetothe ν′ext = Γνext (26) emissionyet. u′ext = Γ2uext TheobservedluminosityintheCDrestframeis Analogously to the computation of the synchrotron emission (Sec.4.1),wecomputetheEICemissivityforeachpower-lawseg- x′max(T) ment separately. We use Eq. 2.94 of Mimica (2004), but replac- ν′L′ν′(T)=πR2 Z dx′ν′j′ν′[ta′(T,x′)], (22) ing I(ν0)/ν0 by cu′ext/ν′ext and with an additional cut-off (approx- x′min(T) imating the Klein-Nishina decline of the Compton cross-section) wherethelowerandupperlimitsdependon(1)whethertheshock such that the emissivity is zero for hν > m2ec4/(hν′ext) (see also, exists,and(2)whetherithascrossedtheshell1.IftheRSdoesnot Aloy&Mimica2008).Valuesofνextanduextusedinthisworkcan exist,thenx =0;otherwiseitis befoundinTable1. ′min ΓcT β β x′min(T)=max 1+z1RSβ− µ,−∆rL′!, (23) 4.3 SSCemission − RS where∆r isthefastershellwidthintheCDframe.Analogously, Analogously to Sec. 4.2, we use the Eq. 2.94 of Mimica (2004). L′ forx =0theFSisnon-existent;otherwiseitis However, in the case of SSC the incoming intensity of the syn- ′max chrotron radiation depends on x and T. For a point on the shell ′ x′max(T)=min 1Γc+Tz1βFSβ−βµ,∆rR′!, (24) axisthe(angleaveraged)intensityatfrequencyν0canbewrittenas − FS π L(θ′) 1 t∆orR′pebrefionrgmthtehesloinwteegrrsahlelilnwEiqd.th2i2n,thje(CtD) sfrhaomuled.Wbeepcoominptuotuetdthfaotr, I0,ν0(T,x′)= 2Z dθ′ Z ds j′ν0,syn(t′(T)−s/c,x′+scosθ′), (27) theparticledistributionevolvedusin′νg′ va′aluesfortheFSif x > 0, 0 0 ′ whereL(θ)isthelengthofthesegmentindirectionθ fromwhich ′ ′ synchrotronemissionhashadtimetoarriveto x atatimet,and ′ ′ t(T)iscomputedusingEq.18.Thesynchrotronemissivity j 1 Sincetheassumedgeometryinthismodeliscylindrical,theeffectsofthe ′ ′ν0,syn canberewrittenintermsofT usingEqs.19,and20(or21), high-latitudeemissionareignored.Thismeansthatallthepeaksandbreaks inthelight curves aresharperthanwouldbeincaseaconical geometry s woshvaeeslrlesass(tsismuemeatSeedet.chtAeionrnaot6teh.2ae)tr.wcohnicsheqthueenlicgehtocfutrhveeadsesculimneesdagfeteormshetorcyksiscrthoassttwhee j′ν0,js′νy0n,s(cid:18)ytn′("Tta′)(T−,xc′,)x−′+css 1co+scθo′(cid:19)s=θ′1β−βββS!#, (28) S − Radiativesignaturesof magneticfields 5 We point out that, alternatively or simultaneously, magnetic dis- Parameter value sipation can provide a source for the emission in internal shocks ΓR 10 (e.g., Gianniosetal. 2009; Nalewajkoetal. 2011, and references ∆g 1 therein),whichwearenotconsideringhere.Thus,theradiativeeffi- σL [10−6,101] ciencywecomputeinthispaperisonlyalowerboundtotheactual σR [10−6,101] radiativeefficiencyofthebinarycollisionofrelativisticmagnetized ǫB 10−3 shells.AsisshownintheAppendixA,wecanusethedefinitionof ǫe 10−1 thedynamicefficiencyinspiredbytherecentworkofNarayanetal. ζe 10−2 (2011).ItsadvantageistheLorentz-invariance,whichenablesusto ∆acc 10 compareittotheradiativeefficiencyofourmodel. aRacc 3 1100166cm Following MA10, but using the definitions for the different ∆r 6×1013cm energycomponentsEˆK,EˆT andEˆM oftheAppendix(Eq.A2),we q × 2.6 denotebyEˆ0thetotalenergyintheunshockedshells, L 5 1048ergs 1 uνeexxtt 5××101−0414eHrgzcm−−3 Eˆ0 :=EˆK(ΓR(1+∆g),nLmp,1)+EˆT(ΓR(1+∆g),nLmp,χnLmpc2,1) z 0.5 +EˆM(ΓR(1+∆g),nLmp,σL,1)+EˆK(ΓR,nRmp,1) (29) θ 5o +Eˆ (Γ ,n m ,χn m c2,1)+Eˆ (Γ ,n m ,σ ,1). T R R p R p M R R p R Table1.Blazarmodelparametersusedinthiswork.NotethatσLandσR whereχisthepressure-to-densityratioofthecoldinitialshellsand canvarycontinuouslyintheindicatedrange. is set to 10 4. We also define the width of the shocked shells in − termsoftheirinitialwidth, β β FromEq.28wecanseethat L(θ′)canbecomputed byrequiring ζW := β| − βS| (30) thatthefollowingconditionbesatisfiedforeachθ′, 0− S sothat06ζ 612.Thedynamicthermalefficiencyforthefaster s 1 ββ W ta′(T,x′)− c 1+cosθ′ β− βS!>0. shellisdefinedas S − 1 If this condition is not satisfied, it means that the shock has not ǫT,L := Eˆ × (31) psyansscehdrotthreonpoeimntisxs′i+onsfcroosmθ′tahtattimpoeint′t(Tto)−cosn/tcribyuette,it.oe.tthheeriencisomno- EˆT(ΓS,L,0nS,Lmp,pS,L,ζW,L)−EˆT(ΓR(1+∆g),nLmp,χnLmpc2,1) h i ingintensity.Inaddition,wealsorequirethatL(θ)6R.Finally,it andanalogousdefinitionscanbewrittenforǫ ,ǫ andǫ (see ′ M,L T,R M,R shouldnotbeforgottenthatwhen x +scosθ > 0theemissivity Eqs. 13, 14, 16 and 17 of MA10). The total (Lorentz invariant) ′ ′ offluidshockedbytheFSshouldbeused,andtheonecorrespond- dynamicthermalandmagneticefficiencyis ingtotheshockedfluidbytheRSotherwise.Also,ifeitherofthe ǫ =ǫ +ǫ , (32) shocksisnotpresent,thereisnocontributionfromthecorrespond- T T,L T,R ingregion. ǫM =ǫM,L+ǫM,R. (33) Inpracticethenumericalcostofadirectevaluationofdouble Fromtheseequationsitcanbeseenthattheradiativeefficiencycan integral in Eq. 27 is prohibitive if we take into account that this beatmostǫ (ǫ +ǫ ).Moreformally,wecanwritetheradiative intensity has to be evaluated for each x in Eq. 25. To overcome e T M ′ efficiencyas(neglectingadiabaticcooling) this problem we approximate Eq. 27 by discretizing the angular integralinanon-uniform θ-intervals.Thechoiceofnon-uniform ǫ :=ǫ f (ǫ +ǫ ), (34) ′ rad e rad T M intervalsismotivatedbythefactthatmostofthecontributionofthe where f := ǫ /(ǫ +ǫ ). It should be noted that Eq. 34 refers incomingradiationcomesfromanglesclosetoµ= β ,sothatwe rad T T M − S tothe“bolometric”emission,i.e.itincludesallfrequenciesforthe concentratemostofthebinsclosetothatangle.Numericaltesting wholedurationoftheshellinteraction.SinceEarth-basedobserva- showsthatusing13binsprovidesanacceptabletradeoffbetween tionshavealimitedspectralandtemporalcoverage,theEq.34is theaccuracyandthecomputationalrequirements. only an upper limit for radiative efficiencies inferred from actual observations. Figure 1shows that the radiativeefficiency isnot a one-to-one map of the total dynamic efficiency. In particular, we 5 RADIATIVEEFFICIENCY notethat f dropstounder10%intheregion(σ >10,σ >10). rad L R Furthermore,thereisaregionof maximal dynamic efficiencyfor In this section we compare the radiative efficiency of the inter- σ 0.2 and σ > 1, where the radiative efficiency from internal shocks with their corresponding dynamic efficiency. We use R ≈ L nal shocks, which can only tap thethermal energy in theregions thekinematicparametersfromMA10intheblazarregime,while downstream shocks, is not maximum. Nevertheless, for small-to- theparametersusedtocomputeemissionareguidedbythevalues moderatevalues ofthemagnetizations of bothshells(lowerright fromBM10(seeTable1forthecompletelist).Allparametersof quadrantofFig.1),theradiativeefficiencyisagoodproxyofthe ourmodelsarefixedexceptforσ andσ ,whichcanvaryinthe L R totaldynamicalefficiency. rangeindicatedbytheTable1.Intherestofthepaper wedistin- guishmodelsbythevalueofthemagnetizationofeachshell,e.g., amodelwithσL =0.1andσR =1isdenotedbythepair(0.1,1). 2 NotethatcomparingEqs.7and30weseethat∆r0′ = ΓζW∆r,whichis As can be seen from Eq. 10, in our model only the thermal justaLorentztransformation oftheshocked shellwidthfromlabtoCD energy can be injected into the non-thermal particle population. frame. 6 P.Mimica andM. A. Aloy 1013 1012 ] z H y 1011 J F [ν tsoytna.l ν 1010 SSC EIC 109 108 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 ν [Hz] Figure2.Averagedspectra(timeintegrationinterval:0-100ks)formod- els(10 6,10 6),(10 2,10 2)and(1,0.1)(full,dashed,anddot-dot-dashed − − − − lines,respectively).Blackcoloredlinesshowthetotalspectrum,whilethe red,blueandgreenlinesshowthecontributionduetosynchrotron,SSCand EICemission,respectively. Figure 1. Contours: frad (see Eq. 34) for different values of (σL,σR). The contours indicate the following values of frad (per cent): 1,5,10,20,30,40,50,60,70,80,90, and100.Inthe region oftheparam- eter space above the dashed line there is no FS,while the RS is always presentfortheconsideredparametrization(seealsoFig.1ofMA10).Filled contours:totaldynamicefficiencyǫT +ǫMinpercent. mediatemagnetizationcase.Furthermore,ICspectralcomponents arenotablyweakerthanthesynchrotrononeforlargemagnetiza- 6 SPECTRAANDLIGHTCURVESOFMAGNETIZED tion,beingtheICspectralpeaklocatedat4.2 1018Hz. × INTERNALSHOCKSINBLAZARS AscanbeseenfromFig.2,thesynchrotronemissionfromall threemodels isof comparable intensity, whiletheIC emissionis Our aim is to produce synthetic spectra and light curves from muchweakerinthestronglymagnetizedmodel(1,0.1).Thereason our numerical models of the interaction of two relativistic, mag- forsuchalargedifferenceinthehigh-energyemissionbetweenthe netized shells. With this purpose, we chose three models from magnetizedandthenon-magnetizedmodelsliesinthelowernum- our parameter space, which are representative of different condi- berdensityofemittingelectrons(Eq.3)andinthehighermagnetic tions that can be encountered in blazar jets. The first model cor- fieldofthemagnetizedmodel.Themagnetizedmodelhasamuch responds to a regime of very low magnetization of both shells lowernumber densityduetoitsrelativelyhighσ,inbothFSand (σ ,σ )=(10 6,10 6).Thesecondandthirdmodelscorrespondto L R − − RSemittingregions,whichmeansthattherearelessscatterersfor intermediate (10 2,10 2) and moderate/high shell magnetizations − − theSSCandEICprocesses.Ahighmagneticfieldalsomeansare- (1,0.1). ductionoftheupperinjectioncut-off(Eq.13),whichinturnmeans thattheseedsynchrotronphotonsintheSSCprocessarebeingup- scatteredtolowerfrequencies,explainingthesmallcontributionof 6.1 Averagespectra theSSCcomponent tothespectrum.TheEICcomponent’supper Thespectrumoftheweaklymagnetizedmodel(10 6,10 6),(Fig.2; cut-offsaredeterminedbytheKlein-Nishinadecline(seeSec.4.2) − − full lines) reproduces the typical double-peaked spectrum of andnotbytheupperinjectioncut-off,whichexplainswhytheEIC blazars.Thesynchrotronemission(Fig.2;solidredline)peaksat spectralpeaksofthemodelsareinasimilarfrequencyrange. 4.6 1013Hz,whiletheinverse-Compton(IC)emission,dominated Figure3showsthecontributionsfromtheFSandRS(redand × bytheSSCcomponent, peaksat6.7 1021Hz(Fig.2;solidblue bluelines,respectively)tothetotalspectrum(blacklines).Except × line). In this case, the IC spectral component is clearly dominat- closetothelocalminimabetweenthetwospectralpeaks, theRS ing the overall spectrum. At intermediate magnetizations (Fig. 2; contributionisdominantintheaveragespectraofthemodelswith dashed lines) the synchrotron emission peaks at higher frequen- lowandintermediatemagnetizations,(10 6,10 6)and(10 2,10 2) − − − − ciesthaninthepreviouscase,namely,at7.5 1014Hz,whilethe respectively.Inthevicinityoftheaforementionedminima(located × IC emission peaks at 6.0 1020Hz, as one would expect, since intheX-raysrange),theFScontributiontendstobroadenthewidth × a larger magnetic field increases the synchrotron peak frequency oftheminimaandtosoftenthespectralslope.Atthemoderateto (e.g., Mimica2004).Fortheseshellmagnetizations,theSSCalso highmagnetizationsofthemodel(1,0.1)theFSisdominantexcept dominatesthehighenergyemission,butnowboththeSSCandEIC intherange1015 1016Hz,where theFSandtheRShave com- − componentsaresignificantlyweakerthaninthemodel(10 6,10 6). parable contributions. The reason is that the faster shell, through − − Interestingly, moderate to high magnetizations (Fig. 2; dot-dot- which the RS propagates, is substantially more magnetized than dashed lines)reducethepeak frequency(1.9 1014Hz)andsub- theslowershell,sothattheRShaslessparticlestoacceleratethan × stantiallyflattenthesynchrotronspectralcomponentw.r.t.theinter- theFS. Radiativesignaturesof magneticfields 7 1013 total SSC 1013 FS syn. RS total 1012 1012 ] z ] H z Jy 1011 y H 1011 [ J F ν [ν ν 1010 νF 1010 109 109 108 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 ν [Hz] 102 103 104 105 t [s] Figure3.SameasFig.2,butdistinguishingthecontributionsoftheFS(red lines)andoftheRS(bluelines)tothetotalspectrum(blacklines). Figure5.X-raylightcurvesforthemodels(10−6,10−6),(10−2,10−2)and (1,0.1),shownwithfull,dashedanddot-dashedlines,respectively.Theto- tallightcurveisshowninblack,whilethesynchrotronandSSCcontribu- tionsareshowninredandblue,respectively. 1014 R-band 1-10 keV 100 MeV 1013 1GeV 1014 SSC EIC ] total Hz 1012 1013 y [Jν1011 Hz] 1012 F ν y J 1010 [ν1011 F ν 109 1010 102 103 104 105 109 t [s] Figure 4. Multi-wavelength light curves for the models (10−6,10−6), 102 103 104 105 (10 2,10 2)and(1,0.1),shownwithfull,dashedanddot-dot-dashedlines, t [s] − − respectively. TheR-band(5 1014Hz),X-rayband(1 10keV),aswell as0.1GeVand1GeVlightc×urvesareshowninblack,re−d,greenandblue, Figure6.1GeVlightcurvesforthemodels(10−6,10−6),(10−2,10−2)and respectively. (1,0.1),shownwithfull,dashedanddot-dot-dashedlines,respectively.The totallightcurveisshowninblack,whiletheEICandSSCcontributionsare showninredandblue,respectively. 6.2 Lightcurves Themulti-wavelengthlightcurvesofthemodelspresentedinthe 3.Atintermediatemagnetizations(10 2,10 2)thefirstsharpdrop − − previoussectionaredisplayedinFig.4.Wehavepickedupseveral isobservableaswell,thoughherethereisaweaklate-timeoptical characteristic bands to analyse the data (R-band, X-ray, 0.1GeV emissionbetween104and105secondsduetotheSSCprocess. and1GeV). TheemissionintheX-raybandisabitmoreinvolved,andto Comparing the R-band light curves of the three models we perform a proper analysis we show in Fig. 5 both, the total light can seethat models (10 2,10 2) and (1,0.1) exhibit properties of − − curve(blacklines),andtheindividualcontributionstoitofthesyn- thefast-coolingelectronsemittingsynchrotronradiation,whilethe chrotronandSSCprocesses(redandbluelines,respectively)4.Ex- model (10 6,10 6) shows the opposite, slow-cooling behavior. In − − ceptatveryearlytimes,theemissionisdominatedbytheSSCpro- thelattercasethemaximumof theR-band lightcurveisreached cess in all cases. The synchrotron emission in this band happens when the shocks cross the shells, and afterward the emission de- cays as the particlescool down (no new particlesare accelerated afterbothshockscrosstheshell).Inthecaseofthemodel(1,0.1) 3 Notethat,sincethehigh-latitude emissionisignoredduetocylindrical onecanclearlynoticetwosuddendropsinemissionaround 4ks, geometry,thedropsaretoosharpandwouldbesmootherwereaconicaljet whichcorrespondtothemomentswhenfirsttheRS,andlater,the geometryassumed. FScrosstheirrespectiveshells.Thealmostverticaldropsinemis- 4 WedonotshowtheEIClightcurvebecauseitscontributionisnegligible sionareindicativeofaveryefficientelectronsynchrotron-cooling atthesefrequencies. 8 P.Mimica andM. A. Aloy tion5.Figure7showsthefluenceasafunctionofσ andσ .Wecan L R seethat,asexpected,thefluenceroughlyfollows f (seeSect.5). R Theregionwithmostluminousinternalshocks(upper leftcorner of Fig. 7) happens for a moderately-to-strongly magnetized slow shellandaweaklymagnetizedfastshell,wherebytheFSdoesnot exist.Theemissionweakensasthemagnetizationofthefastshell increases, withtheexception of theregion wherethefast shellis stronglymagnetizedbuttheslowshellisweaklymagnetized(lower right corner of Fig.7).Weconclude that,as wasindicated inthe Section 4.4 of MA10, a large difference in the magnetization of the shells yields stronger dissipation and more luminous internal shock(s)thanwhenbothshellsareweaklymagnetized. Figure 8 shows the spectral maxima of the synchrotron, ν , and the inverse Compton emission, ν (left and right max,syn max,IC panels,respectively).FromFig.2,onecouldanticipateatrendwe confirmhere,withtheparametricspacecoverage,namely,thatthe ICemissionismoresensitivetochangesinthemagnetizationthan the synchrotron emission. This statement reflects itself in Fig. 8 throughthefactthattherangeofvariationofν islargerthan max,IC thatofν .Thus,theICspectralpeakbecomesabetterproxy max,syn of themagnetization of theshellsthanthesynchrotron peak. Ex- cept at small shell magnetizations, the IC emission happens in a fast-cooling regime, and the dependence of ν with the mag- max,IC Figure7.Contoursofthelogarithmofthetime(0 100ks)andfrequency (1012 1025Hz) integrated flux (i.e., fluence) as−a function of the shell netizationissimilartothatof frad.Complementary, atsmallshell magne−tization σL andσR.Notethat, different fromFig.1,the regionof mf ag(nceotmizpatairoentsh,ethleowmearphoaflfνomfaxF,siygns,.re1saenmdb8le-slevfte)r.ymuchtothatof ultra-high magnetizations (σL,R > 10)isnotincluded inthisfigure.The rad Theleftpanelofthefigure9showstheratiooftheICtosyn- reasonbeingthatthecomputationoftheintegratedfluxwithsuchextreme magneticfieldsrequiresadiscretizationofthetwo-dimensionalintegralin chrotronfluence.Thetrendisquitesimilartothatoftheintegrated Eq.27inaverylargenumberofintervals,makingthecalculationnumeri- flux shown in Fig. 7. When both shells are strongly magnetized callyimpractical.Nevertheless,thetrendsatsuchhighmagnetizationscan (σ>0.1)theICemissiondropssignificantly.Intheregionσ <10 3 L − beeasilyextrapolatedfromthevaluesdisplayedinthefigure. the∼ratio isbetween a unity and 60, witha similar behav∼ior in the region (σ >0.1,σ <0.1).≃The region of low radiative effi- L R ciency around σ∼ 0.01∼appears as a dark vertical band in the plot(σ <10 4,wLh≃erebothsynchrotronandICprocessesprovide inanefficientfast-coolingregime,whichcanbeinferredfromthe asimilaRr∼fluen−ce.Therightpanel ofFig.9showstheratioofthe fastdropofthesynchrotroncomponents between4and9ks.The frequencies of the IC and synchrotron spectral maxima shown in factthatincreasingmagneticfieldsmakethatparticlescoolfaster, Fig.8.Fromanobservational point ofview,thefluencemightbe explainsthatthenon-magnetizedmodelpeaksmuchlater( 60ks) ≃ muchmorerobustandsignificantthanthepeakICandsynchrotron inthisbandthantheothertwo(moremagnetized)models. frequencies,whichcanbedifficulttomeasure.However,forthera- At energies of 1GeV, there is only emission from IC pro- tioν /ν ,thelowerrightandupperleftcornersoftheplotdisplay IC syn cesses (Fig. 6). The model with the smaller magnetization dis- noticeablydifferentvalues. Thus,onecanusethefrequencyratio plays a clearly dominant EIC emission at early times, while in togetherwiththefluenceratioinordertobreakthedegenerationin theother two models EICdominates thelater times. Inthe mod- thefluenceratiowhenoneoftheshellisverymagnetizedandthe els(10 2,10 2)and (1,0.1) EIC,similartothesynchrotron emis- − − otherisnotmagnetized(seeSec.8.2forfurtherdiscussionofthis sionintheX-rayband, sinks veryquickly before8ks, indicating point). that the electrons are in a fast-cooling regime. In the latter models,becauseofthedelaysassociatedtothephysical lengthofthe emittingregion,theSSCcontributionpeaksveryearlyanddecays exponentiallybeforethesharpdropoftheEICemission(thisispar- 8 DISCUSSIONANDSUMMARY ticularlythecaseofthemostmagnetizedmodel,inwhichtheSSC componentdoesnotsignificantlycontributetothelightcurveafter Wehaveextendedthestudyofthedynamicefficiencyperformedin 400s). In contrast, the EICemission of the model (10 6,10 6) MA10bycomputingthemulti-wavelength,time-dependent emis- − − ≃shows a much more prominent peak and a shallower decay from sionfrominternalshocks.Inthissectionwediscussandsummarize themaximum(at 9ks),bothfeaturesbeingcharacteristicsfrom ourfindings. ≃ electronsinaslow-coolingregime. 5 All two-dimensional plots in this section have been produced using a logarithmicallyspacedgridof30 30intheσL σRparameterspace.For × × eachofthepointswecomputedlightcurveson96logarithmicallyspaced 7 GLOBALPARAMETERSTUDY frequencies (between 1012 Hz and 3 1025 Hz) for 120 logarithmically spacedpointsintime(between2and1×05seconds).Afinercoverageofthe Inthefollowingwepresenttheresultsoftheglobalparameterstudy σL σR parameter spacewasnotpractical duetotheprohibitively high × ofthedependenceoftheemittedradiationontheshellmagnetiza- memoryandcomputationaltimerequirementsontheavailablemachines. Radiativesignaturesof magneticfields 9 Figure8.Contoursofthefrequencyofthespectralmaximaofthesynchrotron(leftpanel)andoftheinverseCompton(rightpanel)emissionasafunctionof σLandσR. Figure9.Leftpanel:contoursofthelogarithmoftheratiooftheICandsynchrotronfluenceasafunctionoftheshellmagnetizationσLandσR.Rightpanel: sameasleftpanel,butfortheratioofthefrequencyofthespectralmaximaofthesynchrotronandinverseComptonemission. 8.1 Emissionmechanismsandmagnetization coolingelectronsonlyfortheweaklymagnetizedmodel,whilefor shellswithσ>0.01electronsarefast-cooling. In Section 6 we show the average spectra and multi-wavelength ∼ lightcurvesofthreetypicalmodelsfromtheparameterspacecon- TheSSCemissiondominatesintheX-raybandandhigherfre- sideredinthispaper.Synchrotronemissiondominatesforν<1017 quencies(Fig.2).However,atearlytimesthesynchrotronemission Hz,and israther independent of theshellmagnetization (F∼ig.2). dominatesinX-rays(Fig.5),whileinγ-raysthesituationismore The RS dominates synchrotron emission for weakly magnetized complex. For the weakly magnetized model (slow-cooling elec- shells,whileinthecaseofstronglymagnetizedshellstheFSand trons), EIC dominates the early emission, whilein themoderate- RShavecomparablecontributions.R-bandlightcurves(blacklines to-highly magnetized models EIC dominates the late-time emis- onFig.4)show thatthesynchrotron emissionisduetotheslow- sion. The reason for this is that, in the magnetized models, the 10 P.Mimica andM. A.Aloy high-energytailoftheelectrondistributiondisappearsveryquickly, Thepreviousanalysissuggeststhatwiththecombinedinfor- so that the incoming synchrotron photons cannot be up-scattered mationofthefluenceand peak-frequency ratios,onecould tryto intothe1GeVrange. Intheweaklymagnetizedmodelsthereare figureout,byusingobservationaldata,whichistheroughmagne- enoughslow-coolingelectronsatsufficientlyhighenergiesforthe tization of the shells of plasma whose interaction yields flares in SSCtodominateoverEICatlatertimes. blazars.Toservesuchapurpose,wedisplayinFig.10ourmodels Finally,fromFig.2weseethattheICemissionisweakerthe ina2D parameter space whose horizontal andvertical directions moremagnetizedtheshells.Thisisduetotherequirementofour areν /ν andF /F ,respectively.Wenoticethatthecomputed IC syn IC syn modelthattheshellluminosity(Eq.1)beconstantregardlessofσ. modelsaredistributedinabroadregionwhich,nevertheless,shows FromEq.3weseethatforσ 1thenumberdensityintheshells arelativelytightcorrelationbetweenF /F andν /ν .Inthe ≫ IC syn IC syn behavesas σ 1.SincetheICemissiondependsonthenumberof leftpanelofFig.10,wedisplayourmodelsinthreedifferentcolors − ≈ electrons(EIClinearlyandSSCquadratically), itisclearthatthe accordingtothemagnetizationoftheleftshell.Thesamehasbeen ICemissionmussnecessarilydropforlargeσ.Fromtheanalysis doneintherightpanel,butfortherightshell. ofthethreerepresentativemodels,weconcludethattheshellmag- Based on the degree of variation of magnetization between netizationimprintstwomainfeaturesontheemissionpropertiesof thefastandtheslowshells,wehavedividedtheparameterspacein blazars. On the one hand, the magnetization changes the ratio of threebroadregions(labeledwithromannumeralsI,IIandIIIin integrated flux below the synchrotron peak to the integrated flux Fig.10),wheretheshellshavethefollowingcharacteristics: below theIC-dominatedpartof thespectrum.Ontheother hand, I: moderately magnetized fast shell colliding with a weakly the magnetization determines whether electrons are slow-cooling magnetizedslowshell,orweaklymagnetizedfastshellinter- (for weakly magnetized shells) or fast-cooling (moderate-to-high actingwithastronglymagnetizedslowone; magnetization). II: strongly magnetized fast and moderately magnetized slow shells; 8.2 Globaltrends III: stronglymagnetizedfastandslowshells. Weperformedaglobal parameter study(Section7)toinvestigate ThefirstthingtonoteisthatformodelsintheregionIIIboththe thedependenceofsomeobservationalquantitiesontheshellmag- ICemissionanditsfrequencymaximumarelowercomparedtothe netization. As discussed in Section 7, the integrated flux (Fig. 7) rest of themodels. This leads us tothe conclusion that when the followsthetrendalreadyshownbytheradiativeefficiency(Fig.1). flow is strongly magnetized and the magnetization does not vary However, theintegrated flux and the spectral maxima are quanti- substantiallytheICsignatureisexpectedtoberelativelyweak.Fur- tiesdependentontheparticularvalueswehavetakeninourmodel, thermore,regionIIshowsthatinthecaseof alargervariationin specifically,onthephysicalsizeoftheshellsandtheirbulkLorentz magnetization (i.e.,weakly magnetized slow shell) the frequency factors, as well as on the source redshift. On the other hand, in maximumremainslow,buttheICsignaturebecomessubstantially MA10weshowthatthedynamicefficiencyisveryweaklydepen- higher. Finally, in the region I we see that when the variation in dent on the shell bulk Lorentz factor, i.e. it only depends on the magnetizationismoreextreme(i.e.acollisionofaweaklyanda shellmagnetizationforafixed∆g.Inordertoeliminatethedepen- stronglymagnetizedshells)wegetaverystrongICsignatureand denceonabsolutequantitiesinFig.9weshowIC-to-synchrotron itsfrequencymaximumisshiftedtomuchhigherenergies. flux ratio, as well as the ratio of the frequency of IC to the synchrotron spectral maxima. The shape of the contours on the left 8.3 Radiativeanddynamicefficiency panelofFig.9doesnotexactlyfollowtheoneinFig.1:thereisa muchstrongerdependenceonσLthanonσRinmostofthescanned Asdiscussed in Section 5, the radiative efficiency ǫefrad(ǫT +ǫM) parameterspace.Nonetheless,inthelowerhalfoftheplots,thebe- does not have a one-to-one correspondence to the dynamic effi- haviorofbothFIC/Fsyn and fRissimilar.Forexample,ifwekeep ciency (ǫT + ǫM). While the latter peaks in the region σL ≈ 1 σdiRatcivoensetffianctieanncdyeisqulaarlgteor1th0a−n69a0n%d vfaorryσσLR<, w0e.0n1o,ttehetnhaittdtheecaryas- (aσndL<σR10−≈4,σ0.R2,>t1h0e).foTrhmeesramreeacchaensbietscmonacxliumdeudmfrionmthteheretigmioen- toalocalminimum,andsuccessivelygrowsagaintoreachvalues andf∼requency-i∼ntegratedfluxshowninFig.7.Forpurposesofthe inexcess of 90% (σL >1). Comparatively, at small values of σR restofthisdiscussionwewilluse frad asaproxyfortheradiative theratioF /F iscl∼osetoitsmaximumfor0.01<σ >1,and efficiency. IC syn L touches aminimuminthesameinterval as fR. The∼uppe∼r half of Wenotethat intheregion ofmaximum frad theFSdoes not Figs. 9 (left) and 1 does not show the same qualitative behavior. exist.However,weseeanother regionofhigh frad intheopposite ThereasonforthisdiscrepancyisthatRSdominatestheemission, cornerofFig.1theefficiencyisquitehighaswell.Consistentwith and thus the overall radiative properties are more sensitive tothe thediscussionintheprevioussubsection andwithwhatisshown magnetizationofthefastshellthroughwhichitpropagates. inregionIinFig.10,weconclude thattheradiativeefficiencyis Interestingly, thereisacertaindegreeof degeneration inthe maximalwhenthevariationinmagnetizationbetweenthecolliding values both of the radiative efficiency and of the F /F ratio shellsislarge. IC syn considering the regionswhere oneof the twoshellsisvery magnetized and the other is basically non-magnetized (i.e.,the upper 8.4 Conclusionsandfuturework leftandlowerrightcornersofFigs.9and1).Inbothcases,thera- diativeefficiencyandthefluenceratioareclosetotheirrespective Under the assumption of a constant flow luminosity we find that maximumvalues.However,wecandistinguishbetweenthecaseof thereisacleardifferencebetweenthemodelswherebothshellsare high-σ /low-σ andthecaseoflow-σ /high-σ bylookingatthe weaklymagnetized(σ<10 2)andthosewhere,atleast,oneshell L R L R − ratioofpeakfrequenciesν /ν (rightpanelofFig.9).Anotice- hasaσ>10 2.Weobt∼ainthattheradiativeefficiencyislargestin IC syn − ablysmaller ν /ν ratiocorresponds totheformer case thanto thosemo∼delswhere,regardlessoftheordering,thereisalargevari- IC syn thelater. ation in the magnetization of the interacting shells. Furthermore,

Radiative signature of magnetic fields in internal shocks

Similar Radiative signature of magnetic fields in internal shocks