Radiative signature of magnetic fields in internal shocks

3.9 MB·English

by  P. Mimica

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:[email protected] 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 inter-
nal 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 mod-
els,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 syn-
chrotron 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 mag-
netized 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,