The radio afterglow of Swift J1644+57 reveals a powerful jet with fast core and slow sheath

1.6 MB·English

by  P. Mimica

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed13April2015 (MNLATEXstylefilev2.2)
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The radio afterglow of Swift J1644 57 reveals a powerful jet with
fast core and slow sheath
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P. Mimica1⋆, D. Giannios2, B. D. Metzger3 and M. A. Aloy1
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2 1DepartamentodeAstronom´ıayAstrof´ısica,UniversidaddeValencia,46100,Burjassot,Spain
2DepartmentofPhysicsandAstronomy,PurdueUniversity,525NorthwesternAvenue,WestLafayette,IN47907,USA
r
p 3ColumbiaAstrophysicsLaboratory,ColumbiaUniversity,NewYork,NY,10027,USA
A

0
13April2015
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]
E ABSTRACT
H
We model the non-thermaltransient Swift J1644+57as resulting from a relativistic jet
.
h poweredbytheaccretionofatidally-disruptedstarontoasuper-massiveblackhole.Accom-
p panyingsynchrotronradioemissionisproducedbytheshockinteractionbetweenthejetand
- thedensecircumnuclearmedium,similartoagamma-rayburstafterglow.Anopenmystery,
o
however,istheoriginofthelate-timeradiore-brightening,whichoccurredwellafterthepeak
r
t ofthejettedX-rayemission.Here,we systematicallyexploreseveralproposedexplanations
s
forthisbehaviourbymeansofmulti-dimensionalhydrodynamicsimulationscoupledtoaself-
a
[ consistentradiativetransfercalculationofthesynchrotronemission.Ourmainconclusionis
that the radio afterglow of Swift J1644+57 is not naturally explained by a jet with a one-
2
dimensionaltop-hatangularstructure.However,amorecomplexangularstructurecomprised
v
ofanultra-relativisticcore(LorentzfactorΓ∼10)surroundedbyaslower(Γ∼2)sheathpro-
1
videsareasonablefittothedata.Suchageometrycouldresultfromtheradialstructureofthe
6
3 super-Eddingtonaccretionfloworastheresultofjetprecession.Thetotalkineticenergyof
0 theejectathatweinferof∼few1053ergrequiresahighlyefficientjetlaunchingmechanism.
0 Ourjetmodelprovidingthebestfittothelightcurveoftheon-axiseventSwift J1644+57is
. usedtopredicttheradiolightcurvesforoff-axisviewingangles.Implicationsforthepresence
1
ofrelativisticjetsfromTDEsdetectedviatheirthermaldiskemission,aswellastheprospects
0
5 fordetectingorphanTDE afterglowswith upcomingwide-fieldradiosurveysandresolving
1 thejetstructurewithlongbaselineinterferometry,arediscussed.
:
v Keywords: radiationmechanisms:non-thermal– galaxies:active galaxies:jets galaxies:
i nuclei–hydrodynamics–radiativetransfer
X
r
a
1 INTRODUCTION timescalesareexpectedtoresultfromcoherent processessuchas
‘giant pulses’ from Galactic pulsars (e.g. Jessneretal. 2005) and
Radioastronomyisundergoing arevolutioninthestudyof time-
cyclotronmaseremissionfrombrowndwarfsandplanets(Berger
domain phenomena due to the advent of sensitive wide-field ar-
2002;Hallinanetal.2008).Brighterextra-galacticanalogsofsuch
rays. At meter wavelengths these include LOw Frequency AR-
events have not yet been detected (although see Thorntonetal.
ray (LOFAR; vanHaarlemetal. 2013), the Murchison Widefield
2013;Spitleretal.2014),butmayaccompanyrareviolentevents,
Array (MWA; Lonsdaleetal. 2009) and the Long Wavelength
suchasgiantmagnetarflares(Lyubarsky2014)andthemergerof
Array (LWA; Ellingsonetal. 2009). At centimetre wavelengths
binaryneutronstars(Hansen&Lyutikov2001).
(GHz frequencies) a new generation of wide-field facilities is
alsobeingdeveloped,suchasApertif/WesterborkSynthesisRadio At centimeter wavelengths and on longer timescales, in-
Telescope(WSRT;Oosterlooetal.2009),MeerKAT (Boothetal. coherent synchrotron sources may instead dominate the tran-
2009), and the Australian Square Kilometer Array Pathfinder sient sky. These generally result from the shock interaction be-
(ASKAP;Johnstonetal.2008).However, despitetheoverallma- tween rapidly expanding matter from an energetic explosion and
turityof radio astronomy, surprisingly littleisknown about what dense ambient gas. Known examples include radio supernovae
astrophysicalsourceswilldominatethetransientsky. (Weileretal. 2002), off-axis (‘orphan’) afterglows of gamma-
The brightest sources at meter-wavelengths on short ray bursts (Totani&Panaitescu 2002), and other relativistic out-
flows from the core collapse of massive stars (e.g., SN 2009bb;
Soderbergetal. 2010). The transient Swift J164449.3+573451
⋆ E-mail:[email protected] (hereafterSwiftJ1644+57)istheprototypeforanewtypeofrel-
2 P.Mimica, D.Giannios,B. D. Metzgerand M.A. Aloy
ativistictransientthatwillalsocontributesignificantlytothevari-
ableradiosky(Giannios&Metzger2011).
SwiftJ1644+57 was characterised by powerful non- 10-9
thermal X-ray emission (Bloometal. 2011; Levanetal. 2011; 101 -1s]
BSwurirfotwJ1s6e4t4a+l.5720a1n1d; Zaapuodseirtieornetcaoli.nc2i0d1e1n)t.wTihthe ltohnegnudculreautisonofoaf mJy] -2cm
previouslyquiescentgalaxyleadtotheconclusionthatitwaspow- f [ν 10-10 erg
e(SreMdBbHy)rfaoplildowacincgretthioentiodnaltoditshreupcteionntraolfsausptearrm(aas‘stiivdealbdliascrkuphtioolne 100 114...489 GGGHHHzzz F [X
15.4 GHz
event’,orTDE;Carter&Luminet1982;Rees1988),althoughal- 2443..46 GGHHzz 10-11
ternativeexplanationshavebeenproposed(e.g.Quataert&Kasen 0.3-10 keV
2012).TherapidX-rayvariabilitysuggested anorigininternalto 10-1100 101 102
a jet that was relativistically beaming its radiation along our line time [days]
of sight, similar to the blazar geometry of normal active galactic
nuclei(Bloometal.2011).SwiftJ1644+57wasalsocharacterised Figure 1. Radio light curves of SwiftJ1644+57 at observing frequen-
byluminoussynchrotronradioemission,thatbrightenedgradually cies1.4,1.8,4.9,15.4,24.4and43.6GHzfromBergeretal.(2012)and
over the course of several months (Zaudereretal. 2011). Unlike Zaudereretal.(2013).AlsoshownaretheSwiftXRTobservations inthe
0.3−3 keV band (brown crosses, right axis). Note the broad secondary
the rapidly varying X-ray emission, the radio emission resulted
maximumintheradio occurring ∼ 6months after trigger, wellafter the
from the shock interaction between the TDE jet and the dense
X-raymaximum.
external gas surrounding the SMBH (Giannios&Metzger 2011;
Bloometal. 2011), similar to GRB afterglows. A second jetted
TDE,SwiftJ2058+05, withsimilarX-rayandradiopropertiesto risetoSwiftJ1644+57, inorder toidentify which(if any) of the
SwiftJ1644+57wasreportedbyCenkoetal.(2012). proposedmodelsbestreproducetheradiodata.
JettedTDEsinprincipleofferauniqueopportunitytowitness This paper is organised as follows. In §2 we review the ob-
thebirthof anAGN,thusproviding anatural laboratorytostudy servationsofSwiftJ1644+57andsummarisepreviousworkonits
the physics of jet production across a wide range of mass feed- interpretation.§3describesourmodelforthejetandthetechnical
ingrates.ObtainingabetterunderstandingofSwiftJ1644+57and detailsofscenariosthatweconsider.In§4wedescribeourresults.
J2058+05 would thus have far-reaching consequences for topics §5describesapplicationsandimplicationsofourresults,includ-
such as the physics of relativistic jet formation and super Ed- ingtheinferredphysicalstructureofTDEjetspoweredbysuper-
dington accretion,theconditions (e.g.distributionof accretingor Eddingtonaccretionandtheprospectsfordetectingorphanjetted
outflowing gas) in nominally quiescent galactic nuclei, and pos- TDEswithfutureradiosurveys.In§6wesummariseourresults.
sibly even the astrophysical origin of ultra-high energy cosmic
rays (Farrar&Gruzinov 2009; Farrar&Piran2014). Despitethis
promise, theory has yet to fully exploit the wealth of data avail- 2 AJETTEDTIDALDISRUPTIONEVENT
ableforSwiftJ1644+57.Accurate,multi-dimensionalmodelsfor
2.1 ObservationsofSwiftJ1644+57
TDEjetsareneededtoquantifyhowsimilareventswouldappear
toobserversoffthejetaxis.Suchinformationisnecessarytocon- The two distinct components in the SED of SwiftJ1644+57 in-
strainthepresenceofoff-axisjetsinTDEsdetectedviatheirquasi- dicated different origins for the X-ray and radio emission
isotropic thermal emission (Boweretal. 2013; vanVelzenetal. (Bloometal. 2011, Burrowsetal. 2011). Rapid variability of the
2013)ortoassesshowjettedTDEswillcontributetofutureradio X-raysplacedtheiroriginatsmallradiiclosetotheSMBH,likely
transientsurveys(Giannios&Metzger2011;Frailetal.2012). fromasourceinternaltothejetitself.Constraintsonthebrightness
Perhaps the greatest mystery associated with temperatureandvariabilityoftheradioemissioninsteadplaceits
SwiftJ1644+57 is the unexpected flattening or rebrightening originatlargerradii,wheretherelativisticoutflowbegantointer-
of the high frequency radio light curves on a timescale of a actwiththeCNMsurroundingtheblackhole(Giannios&Metzger
few months after its initial rise (Bergeretal. 2012; Fig. 1). This 2011).Figure1showstheobservedradioandX-raylightcurvesof
behavior contradicts expectations of one-dimensional blast wave SwiftJ1644+57.
models that assume energy input proportional to the observed Afterseveral days ofpeak activity,theX-rayluminosityde-
X-rayemission,andwhichinsteadpredictthatthehighfrequency creasedasapowerlawLx ∝ t−α intimewithα ∼ 5/3,consistent
emissionshoulddecayrapidlyoncethejetenergyistransferedto withthepredicteddeclineinthefall-backrate M˙ ofthedisrupted
theblast (Metzgeretal. 2012). Thisbehavior suggests that either star(Rees1988).Unlesscoincidental,theapparentfactthatLx∝M˙
thetemporal or angular structureof thejetismorecomplex than impliesthatboththejetpowerLj trackedtheinstantaneousaccre-
assumed in the basic models usually applied to GRB jets (e.g., tionrateandthattheradiativeefficiencyandthebulkLorentzfactor
Bergeretal. 2012; Wangetal. 2014), or that some key physical oftheoutflowremainedrelativelyconstantthroughouttheevent1.
processhasbeenneglected(e.g.,Kumaretal.2013). AmodelfortheradioemissionofSwiftJ1644+57usingthe
first three weeks of data was developed by Metzgeretal. (2012,
In this paper we model the hydrodynamical interaction be-
hereafterMGM12),undertheassumptionthattheX-rayluminos-
tween a TDE jet and the circumnuclear medium (CNM) using
itytrackedtheinstantaneouskineticpowerofthejet.Thissudden
oneandtwo-dimensionalrelativistichydrodynamical simulations.
Assumingthat non-thermal electronsareacceleratedattheshock
fronts,weexploitthesimulationresultstoperformaradiativetrans- 1 Substantial temporal variation of the bulk Lorentz factor would cause
fer calculation for thesynchrotron radio emission. Our goal isto equally greatchanges inthebeamingdegree ofthejetemission,making
provideacomprehensivestudyofthestructureofthejetthatgave hardtounderstandwhyLx∝M˙ for∼1yearaftertheTDE.
SwiftJ1644+57:powerfuljetwith fastcoreandslowsheath 3
injectionofenergyintotheCNMproducesaforward-reverseshock environment.Thelatterwasassumedtoincludeboththeextended
structure,whichactstodeceleratetheejecta.Theevolutionischar- hydrostaticatmospherewhichisproducedbyreturningstellarde-
acterisedbytwostages:(1)anearlyphaseduringwhichthereverse bris(Loeb&Ulmer1997;Guillochonetal.2014)onsmallradial
shockcrossesbackthroughtherelativisticallyexpandingejectathat scales (. 10−3 pc) and the CNM of the nuclear cluster on larger
wasreleasedovertheinitialfew-dayperiodofpeakactivity;(2)a scales. The cocoon created by shocked CNM plays an important
laterphaseduringwhichtherelativisticblastwaveundergoesself- role in the model of DeColleetal. (2012) in collimating the jet
similarexpansion(Blandford&McKee1976). and,potentially,impactingthelate-timeradioemission.Thejetis
Astrikingfeatureoftheearly-timeradiolightcurvewasthe onlyself-collimatedbyitsenvironment,however,ifitskineticlu-
presenceofanachromaticbreakatt ∼ 10days(Fig.1).MGM12 minosityislessthanacriticalvaluewhichweestimatetobe∼1044
showed that this break could result from the transition between ergs−1usingtheformalismofBrombergetal.(2011),i.e.wellbe-
stage (1) and (2), and that the temporal slopes of the pre- and low the peak (beaming-corrected) luminosity of SwiftJ1644+57,
post-break light curve requires that the CNM density n scale &1045−46ergs−1.
cnm
with radius as ∝ r−β with β ∼ 2. MGM12 also constrained the Radiorebrighteningcouldinprinciplealsoresultfromthejet
initial (unshocked) Lorentz factor Γ ∼ 10 and opening angle encounteringanabruptchangeintheradialslopeoftheCNMden-
j
θ ∼ 1/Γ ∼ 0.1 to values remarkably similar to those of blazar sityprofile.Suchabreakmightbeexpectedbecauseonsmallradial
j j
jets(e.g.Pushkarevetal.2009).TheestimatedvalueoftheCNM scalesthedensityprofileisexpectedtoapproachthatofaBondiac-
density∼fewcm−3atradii∼1018cmfromtheSMBHwassome- cretionflowontotheSMBH(n ∼ r−3/2),whileonlargerscales
cnm
whatlowerthanthatmeasuredonasimilarradialscalefromSgA* outside the SMBH sphere of influence the density profile should
inourowngalaxy(Bergeretal.2012).Thereverseshockindirectly flatten.However,paststudiesofGRBafterglowsshowthatchanges
shapedtheearlylightcurvesofSwiftJ1644+57throughitseffect intheexternal density profileproduce onlyminor changes inthe
onthehydrodynamics,buttheobservedsynchrotronemissionwas synchrotron light curve behavior (e.g. Mimica&Giannios 2011;
probablydominatedbyelectronsacceleratedattheforwardshock Gatetal.2013,andreferencestherein)andrarelyrebrightening.
(MGM12)2. Kumaretal. (2013) emphasise that electrons accelerated at
Bergeretal. (2012) presented updated radio light curves of the forward shock are subject to Compton cooling by the X-rays
SwiftJ1644+57, which showed the surprising radio rebrighten- fromthejet.Accordingtotheirmodel,theapparent excessinthe
ing mentioned earlier (Fig. 1). This behavior deviated signifi- late radio emission is in fact a deficit in the early-time emission,
cantly from that predicted by MGM12 for a blast-wave evolv- resulting from the synchrotron flux being suppressed due to the
ing with constant energy, thus demanding that an additional lower effective cooling frequency of X-ray cooled electrons. Al-
source of energy contribute to the forward shock at late times thoughX-raycoolingwillcertainlyplayaroleinshapingtheearly
(e.g.BarniolDuran&Piran2013).Althoughatemporaryrebright- radio emission, it becomes less efficient as the jet spreads later-
ening can be created by the self-absorption frequency passing ally, since a smaller fraction of the freshly accelerated electrons
throughtheobserversfrequencyband(MGM12),suchatransition (onlythosewithintheopeningangleoftheoriginaljet)arecooled.
cannotexplaintheobservedlightcurves. Kumaretal.(2013)alsoconsidertheeffectofX-raycoolingonop-
ticallythinemission.Asweshow here,itseffectisnegligiblefor
theradiobandsatwhichtheobservedemissionisselfabsorbed.
2.2 Proposedexplanationsfortheradiorebrightening A final possibility, which we advocate in this work, is that
Aftertheone-zonejetmodelfailedtoreproducethelateradioex- the late radio brightening indicates that the jet responsible for
cessobservedforSwiftJ1644+57,severalauthorsproposedexpla- SwiftJ1644+57possessedamorecomplexangularstructurethan
the top hat structure adopted in most previous models. A natural
nationsforthisapparentincreaseintheenergyoftheforwardshock
(BarniolDuran&Piran2013).Webrieflyreviewthesepossibilities alternative is that the jet possesses an ultra-relativistic (Γj ∼ 10)
here. coreresponsibleforthehardX-rayemission,whichissurrounded
by a wider-angle, mildly relativistic (Γ ∼ 2) sheath. Core emis-
Bergeretal.(2012)proposedthattheradiorebrighteningcan j
be explained as the result of slower material catching up and in- siondominatestheearly-timeemission, whilethemoreenergetic
sheathdominatesthelateradioemission(e.g.Tchekhovskoyetal.
jectingenergytothedeceleratingforwardshockatlatetimes(e.g.,
2014; Wangetal. 2014; see, however, Liuetal. 2015). This sce-
Granot&Kumar2006).Inthismodeltheslowermaterialisejected
aftertheinitialultrarelativisticjetstageanditcontains∼20times nario allows both components of the jet to be ejected simultane-
more energy than the Γ ∼ 10 ejecta. The decline of the X-ray ouslyduringthefirstweeksfollowingtheTDE,buttheemissionis
j
initiallydominatedbythatproducedattheexternalshocksdriven
lightcurvewasbelievedtoreflectthedecreaseinthejetpowercor-
bythefastcore.Suchajetstructureisphysicallymotivatedifthe
responding tothepredictedevolutionoftherateofmassfallback
slow ejecta results from an initial phase of precession of the fast
totheSMBH,indicatingthatthebeamingofthejetemission(and
jet (Stone&Loeb 2012; Tchekhovskoyetal. 2014) or simply as
thereforethejetLorentzfactor)didnotvarysubstantiallyoverthat
the result of the separate fast and slow outflow components ex-
period.Itisthusnotclearinthisscenariowhatprocessrevivedthe
source,resultingintherequiredslowerandmorepowerfulsecond pected toaccompany super-Eddington accretion (Mineshigeetal.
2005; Coughlin&Begelman 2014; Jiangetal. 2014). A similar
ejectionevent.Ontheotherhand,theenergetic,slowerejectamay
fastcore/slowsheathconfigurationismotivatedbyblazarobserva-
havebeenejectedsimultaneouslywiththerelativisticcomponent,
e.g.,encasingit(seebelow). tions(Ghisellinietal.2005),indicating that such astructure may
beagenericfeatureofrelativisticjets.
DeColleetal. (2012) performed 2D hydrodynamic simula-
tions of the interaction between the TDE jet and its surrounding TherichdatasetavailableforSwiftJ1644+57,nowspanning
over threeyears, makes it possible to construct amuch more de-
tailedmodelfortheafterglow,whichexplorestheeffectsoflateen-
2 With thepossible exception oftheradio emission during thefirst∼10 ergyinjection,complexangularstructure,andX-raycoolingofthe
daysaftertrigger;seebelow. radiatingelectronsdescribedabove.However,mostmodelstodate
4 P.Mimica, D.Giannios,B. D. Metzgerand M.A. Aloy
include only one dimensional hydrodynamics. Multi-dimensional wherev isthejetvelocity.TheexternalCNMisalsoinitialisedtoa
j
effects,suchasangularstructureandjetspreadingorpressurecon- coldtemperature,withapower-lawradialdensityprofilen ∝r−k
cnm
finementbytheshockedexternalmedium,becomeimportantasthe withk∼1−2.Finally,thejettoexternalmediumeffectiveinertia
jetdeceleratestomildlyrelativisticspeeds.Indeed, sincebynow, densityratioη =ρ hΓ2/(ρ h Γ2 )issettobeconstantinall
R j j j cnm cnm cnm
∼ 4yearsafterthetrigger,eventhefastestcomponents ofthejet oursimulationsandroughlyequalto1000.
have decelerated to Γ . 2, modeling the jet as part of a spheri-
cal flow is clearly inadequate. Multidimensional models are also
necessary to properly address the emission observed by off-axis 3.1 1DSimulationDetails
viewers,whichpeaksoonafterthejetbecomesmildlyrelativistic.
A numerical grid with radial resolution ∆r = 1.1×1013 cm ex-
Belowwedescribeaseriesofoneandtwo-dimensional hydrody-
tendsfromtheinnerboundaryR=r =1016cm.Thelattervalue
namic simulations that explore systematically the above physical j,0
istakentobesufficientlysmallsothatoursimulationsadequately
effects in order to more accurately model the radio afterglow of
followtheinitialjet-CNMinteractionsincludingthereverseshock
SwiftJ1644+57.
crossingofthejet.Thejetevolutionisfollowedforatotalduration
of∼6.7×108s.Atotalof1600snapshotsofthejetstatearesaved
atregularintervals(every8.3×104s)duringthefirst1.3×108s,af-
3 MODELDESCRIPTIONANDNUMERICAL terwardthetimeintervalbetweenthesnapshotsisprogressivelyin-
PROCEDURE creased,resultinginanadditional785snapshotsbeingstoredover
theremaining∼ 5.4×108s.Thecollectionofsavedsnapshotsis
The numerical code MRGENESIS (Mimicaetal. 2007, 2009) is usedasaninputfortheSPEVcode(Mimicaetal.2009)inorderto
usedtoperform1Dand2Drelativistichydrodynamicsimulations
computethesynchrotronradiolightcurves.AppendixAprovides
in spherical symmetry. MRGENESIS is a high-resolution, shock-
additionaltechnicaldetailsonthelightcurvecalculationinSPEV.
capturingschemewhichusesMPI/OpenMPforhybridparalleliza-
tionandHDF5librariesforparallelinputandoutput.
TheTDEjetresponsibleforSwiftJ1644+57ismodeledasa
3.2 2Dsimulationdetails
relativisticoutflowwithatemporallyconstantLorentzfactorΓ and
j
akineticluminosityL thatvariesintimewithaformmotivatedby 2Dmodelsaresimulatedusingthesameinitialandboundarycon-
j
theobserved(isotropic)X-raylightcurve, ditionsasintheequivalent 1Dcase. Theradialandangular reso-
lution are ∆r = 4×1014 cm and ∆θ = π/600 rad, respectively.
L if t6t
j,0 j We output the snapshots at exactly the same evolutionary times
wLjh(te)re=t= 5L×j,01 0tt5j!s−5∼/31wifeekt>isttjh,eapproximateduration oft(h1e) catsrohsilboviusneeton1arDt’to3oc+mtah1sae’e-kdce(io§mmth3epe.n1usS)ti.aoPtInEitoaVnlshacrloaayulcc-loudtmrlaabpcteilioenrxngeismtpyfraeoorabkfsleitebdhmliest,h,wias.etoin.r∆ckthreiesaSmnSPdPEaEVi∆nVθm,conaunrose-tt
j MRGENESIS.
epochofpeakX-rayactivity(e.g.,Burrowsetal.2011).Thepeak
luminosityL isrelatedtothetotalisotropicenergyofthejetac-
j,0
∞
cordingtoE = L (t)dt≈5L t/2.Motivatedbytheisotropic
iso 0 j j,0 j 3.3 Non-thermalparticlesandemission
energyradiatedinRthesoftX-rayband∼ 3×1053erg(Levanetal.
2011;Burrowsetal.2011),werequirethatE &3×1053erg.The Energeticelectronsareinjectedbehindtheforwardshockwithan
iso
true (beaming-corrected) energy of the jet is E = E f−1, where energyspectrum f(γ) ∝ γ−p,asmeanttomimictheeffectsofe.g.
f = (1−cosθ) is the beaming fraction. In our moisodebl for fast, diffusive shock acceleration. The forward shock is identified us-
fb j
narrowjetsweassumeaninitialjetLorentzfactorΓ = 10andjet ing the algorithm described in Sec. 3.1. of Cuesta-Mart´ınezetal.
j
half-openingangleθ =1/Γ =0.1.Wealsoconsiderlowervalues (2015).ParticleaccelerationattheRSisignored,motivatedbythe
j j
ofΓ =2forθ <θ =1/Γ =0.5inthecaseofawider,slower lackofevidenceforasubstantialRSemissioncomponentfromthe
j,s j,s j,s
jet,ortwo-zonemodelswithbothfastandslowcomponents. observedradiolightcurvesofSwiftJ1644+57(MGM12;although
Theinnerradiusofthenumericalgrid,R = r ,corresponds theRSmaybecontributingtothelow-frequencyemissionduring
j,0
tothelocationwherethejetisinjected.Theinitialsizeofthera- the first week post trigger–see below). Fractions ǫe and ǫB of the
dial grid for 1D simulations is r = 3.2×1017cm, but this is post-shockthermalenergyareplacedintorelativisticelectronsand
out,0
periodicallyenlargedasthejetpropagatestolargerradii.Thisen- the magnetic field, respectively. Optionally, we allow that only a
largementoccursinsuchawayastokeepthenumericalresolution fractionζeofavailableelectronsisacceleratedatshocks(weadopt
per unit length constant in the radial direction, i.e. new uniform ζe = 1exceptwhereindicatedinSection4.2andinAppendixB).
zonesareaddedtothegrideachtimeitisextended. For2Dsim- Weadopt p=2.2,ǫe =0.1andǫB =10−3 asfiducialvaluesofthe
ulationstheangularcoordinateθhasafixedrange[0,π/2],while microphysical parameters, similar tothose used tofit GRB after-
thejetisinjectedatR = r acrosstheangularrangeθ 6 θ.The glows(e.g.,Mimicaetal.2010).
j,0 j
ratioofthejetpressuretorestmassdensityisP(t)/ρ(t)c2 =0.01, Injectedparticlesloseenergytosynchrotroncoolingandadi-
j j
sothatitissufficientlysmallfortheinitialthermalpressureofthe abaticlosses,andinsomecaseswealsoincludeinverseCompton
jet material to be negligible and, hence, the specific jet enthalpy, (IC)coolingbyX-rayphotons(ICcoolingoccursintheThomson
h = 1+ε/c2 +P(t)/ρ(t)c2 ≃ 1(where ε is thespecific inter- regimeforparametersofinterest).TheelectronLorentzfactorsγ
j j j j j
naljetenergy).Thejetdensityρ(t)attheinner boundary isthen evolveaccordingto
j
relatedtothejetluminosityandLorentzfactoraccordingto dγ 1dlnρ 4 cσ
= γ− T u′ +u′ γ2 (3)
ρ (t)= Lj(t) , (2) dt′ 3 dt′ 3mec2 (cid:0) B rad(cid:1)
j 4πr2 hΓ2vc2 wheret′,ρ,u′ and u′ arethetime,fluiddensity, magneticfield
j,0 j j j B rad
SwiftJ1644+57:powerfuljetwith fastcoreandslowsheath 5
andexternalradiationenergydensity,respectively,inthecomoving
frame of the fluid. The value of u′ is only nonzero for models 10
rad
includingICcooling(§4.1.2). ]
3
-m n
c P
2 102 Γv
c
4 RESULTS p
m c
[ v/
Exploring each of the proposed scenarios for the afterglow of P 5Γ
SwiftJ1644+57(§2.2)acrosstheirfullparameterspacewouldre- ],
quire at least tens of numerical simulations, resulting in a non- -3m 101
thermalemissioncomputationaltimethatiscurrentlynotfeasible c
fortwodimensional(axisymmetric)simulations.Wethusadoptthe n [ T = 9.6 x 106 s
strategyoffirstperformingalargenumberofexploratoryrunsun-
dertheassumptionofsphericalsymmetryfortheblastwaveevo-
lution(§4.1,§4.2).Resulting1Dlightcurvesarethencomparedto 10 20 0
theobservations,allowingustoidentifythemostpromisingmod- r [1016 cm]
els,alongwiththejet/CNMparametersthatbestreproducetheob-
servations. These most promising scenarios are then explored in
2
greatdetailbyrepeatingthemusingafulltwodimensionalcalcu-
]
lation(§4.3). -3m n
c P
2 101 Γv
c
4.1 One-component1Dmodels p
m c
Webeginbyconsideringlightcurvesproducedbyone-component, [ v/
P 1Γ
sθpj.hNeroitcealtlhyatsaylmthmouegtrhicthjeetsinwitiiathljaentoapsesnuimngedan(figxleedθj)doopeesnninogtfaancgtoler -3m],
intothehydrodynamicevolutionintheonedimensionalmodel,its
vreagluioenmduosetssftialcltboersinpteocitfiheedebmeicsasuiosnetchaelcaunlgautiloanr3s.izeoftheemitting n [c 100 T = 4.29 x 107 s
0
110 120
4.1.1 Fast,narrowjet
16
r [10 cm]
Wefirstconsiderafast,narrowjetwithanisotropicenergyE =
iso
1053ergexpandingintoastratifiedexternalmediumwiththeradial Figure2.Upper panel:Numberdensityn(fullline), thermalpressure P
densityprofilen = n (r/1018cm)−k,wheren = 1.5cm−3 and (dashedline) andfour-velocity Γv/c(greyline) ofthefast, narrow jetat
cnm 18 18
kisvariedbetween1,3/2,and2.ThejetLorentzfactorandhalf- thetimeT ∼ 9.6×106 s.Thereverseandforwardshocksarevisibleas
openingangleareassumedtobeΓ =10andθ =0.1,respectively. discontinuities on the right half of the plot. The density peak marks the
j j location of the contact discontinuity in both the top and bottom panels.
Figure2showstwosnapshotsofthe1Djetevolution,corre-
Lowerpanel:Sameasupperpanel,butforalaterstageinthejetevolution
sponding to an early time in the jet evolution, shortly before the
(T ∼4.3×107s).Bythistimethereverseshockhascrossedtheportionof
RScrossestheejectainjectedatconstantluminosity(t∼107s;top
thejetshown,whiletheregionnearthefrontofthejetisgraduallyentering
panel), and alater time(t ∼ 4.3×107s; bottom panel). At early aself-similarevolution.
timesboththeforwardandreverseshocksarestillclosetothehead
ofthejet.Behindthedensejethead,whichisproducedduringthe
normalizationoftheCMNdensityn forafixedvalueofη .Be-
initialphaseof constant jetluminosity, thedensitydecreases toa 18 R
minimumat∼1.5×1017cm.Atyetsmallerradii,thedensitythen causechangingEiso/n18doesmakeadifferenceinthecalculatedra-
againincreases∝ r−2 (eq.[2])approaching thepointof jetinjec- dioemission,weexploitthisscalingfreedomtoperformalimited
scanoftheparameter spacewithouthavingtoperformadditional
tion.Inthemorephysical2Dcasethislow-densitytailleadstothe
simulations.
collapseofthejetchannelbythecocoon(§4.3.1).
Figure3showsourresultsforthesyntheticradiolightcurves
Therelativistichydrodynamical equationsareinvarianttoan
incomparisontotheobservationsofSwiftJ1644+57(Fig.1).We
overallchangeinthenormalizationofthedensityandlengthscale
setn =5cm−3,whichsetsE =3.33×1053erg.Eachpanelcor-
(Schecketal.2002).Equation(2)showsthatanincreaseinthejet 18 ISO
respondstoadifferentradiofrequency,withdifferentvaluesofthe
luminosity L (and hence E ) is equivalent to an increase in the
j iso
assumedCNMdensityindexkshownwithdifferentcolours.Thick
andthinlinesshowlightcurveswithandwithoutsynchrotronself-
absorption taken intoaccount. Asexpected, theearly-timeobser-
3 ContrastthiswithGRBafterglows,inwhichtheshockedjetmaterialhas vationscanbefitreasonablywellusingther−3/2profile(MGM12),
Γdsuoshpeθpsjrneo>sstian1ffgeptcrhtieotrhoebtosoebtrhsveeerdvjeefltdubflxruebxay.kH,aiof.awec.,etovtrehreo,fffio∼nri(TtΓeDsohEθpjej)en2tisnw,gΓitshahnθrgejls<∼epe1oc,ftthtthoeertehbjaeytt ebmutistshieonlaistes-ttriomnegleymaibsssoiorbnedisactl<∼ea2r0lyGuHnzdderuprrinogdutcheedfi.rTsth1e0r−ad1i5o
fromanotherwiseequivalentsphericaloutflow(Bloometal.2011).Models days,andthepeakateachfrequencycorrespondstothetransition
forSwiftJ1644+57thatassumeasphericallysymmetricflowmaystrongly fromopticallythicktoopticallythinregime.Furthermore,ateach
overestimatetheafterglowemissionoftheactualbeamedevent. frequency we can see the achromatic break in the optically thin
6 P.Mimica, D.Giannios,B. D. Metzgerand M.A. Aloy
3
10
obs.
mJy] 110001 orrr---b132/2s. mJy] 110021 nn1188 == 51 0c0m c-m3-3
[ν [ν 100
f -1 f
10 -1
1.4 GHz 1.8 GHz 10 1.4 GHz 1.8 GHz
3
10
1
y] 10 y] 102
J J
m 100 m 101
[ν [ν 100
f -1 f
10 -1
4.9 GHz 15.4 GHz 10 4.9 GHz 15.4 GHz
3
10
1
y] 10 y] 102
J J
m 100 m 101
[ν [ν 100
f -1 f
10 24.4 GHz 43.6 GHz 10-1 24.4 GHz 43.6 GHz
101 102 101 102 101 102 101 102
time [days] time [days] time [days] time [days]
Figure3.Radiolightcurvescalculated forourfiducial“fast,narrow”jet Figure4.SameasFigure3,comparingradiolightcurvesfromajetwith
model (§4.1.1), with Eiso = 3.33×1053erg, n18 = 5 cm−3, ǫe = 0.2, higherdensityandenergy(Eiso =6.67×1054erg;dashedline)thatbetter
ǫB=2×10−3.Thickandthinlinesshowthelightcurveswithandwithout fitsthelate-timeradiodatatotheEiso=3.33×1053ergmodelfromFigure3
synchrotronself-absorptionrespectively.Differentpanelsshowingdifferent (fullline)thatbetterfitstheearlydata.Inbothcasesweassumeanexternal
observingfrequencies,withlinestylesshowingdifferentassumptionsabout densityprofilencnm∝r−3/2.
thepower-lawslopeoftheradialdensityprofileoftheCNM,ncnm(r)∝r−k,
wherek=1(fulllines),3/2(dashed),1(grey).Shownforcomparisonwith
circlesaretheradioobservationsofSwiftJ1644+57(Fig.1). overall normalization, L = 0, 5×1047ergs−1, and 1048ergs−1.
x,0
The observer time t is related to the simulation time t used in
obs
equation(1)accordingto
lightcurves(thinlines).Asdiscussedin§2,thisbreakmarksthe
jetdecelerationtime. rcosθ
t =t− , (6)
Figure4showsthelightcurvesforacaseinwhichn18 isin- obs c
creasedto100cm−3 (andhenceE to6.67×1054 erg).Although
iso whererandθaretheradiusandtheanglewithrespecttotheline
the late radio emission is now reasonably well fit, the early-time
ofsightoftheemittingparticleattimet.
emission is somewhat overproduced. However, such narrow jets
Figure 5 shows the radio light curves including IC cooling,
sufferfromlatteraljetspeadingeffectsatratherearlystagesintheir calculatedassumingapowerfuljet(E =6.67×1054erg)andpa-
iso
evolutionthatarenotprobedby1Dmodels.InSect.4.3,weexplore rameters (k = 1.5, ǫ = 0.1, ǫ = 0.002) as necessary to match
e B
asimilar,fast/narrowjetmodelwith2Dsimulationsdemonstrating
the late radio peak. IC cooling does noticeably influence the op-
that,asaresultoflatteralspreading,thejetemissionisgreatlyun- tically thin synchrotron emission. However, the effect on the ob-
derpredictsobservationsontimescaleslongerthan∼ 1month.We
served, optically-thickemissionisalmost negligible, evenfor the
havenotbeenabletofindany1-component jetmodel capableof
highest assumed valueof L .Whentheemissionisstronglyab-
x,0
explainingthelateradiorebrightening.
sorbedtheobserveronlyseesthenewlyinjectedparticlescloseto
theexternalshock,whiletheparticlesthathavecooledarealready
4.1.2 X-raycoolinginthefastjet locatedfurtherbehindandtheirsynchrotronemissiondoesnotes-
cape.WethusconcludethatICcooling,withinanarrow-jetmodel,
Kumaretal. (2013) propose that the flat radio light curve in cannotexplainthelateradiorebrightening.
SwiftJ1644+57 results from an early-time deficit of the syn-
chrotronemissioncausedbyICcoolingbyX-rayphotonsfromthe
jet.Wetestthisideabyrunningjetsimulationsincludingradiative 4.1.3 Slow,widejet
lossesinequation(3)accordingto
To contrast with the fast, narrow jet model (Sec. 4.1.1) we also
L (t )Γ
u′rad = 4xπorb2sc , (4) consideraslow,widejetwithΓj =2andθj =0.5(Sec.4.1.1).Fig-
ure6showsourresultsfortheslowjetlightcurves(dashedlines),
whereΓisthebulkLorentzfactoroftheemittingparticlesandLx calculated for an external density n18 = 50 cm−3 and jet energy
istheX-rayluminosity,whichscalesinthesamewayasthethejet E =3.33×1054erg,incomparisontotheequivalentfastjetcase
iso
power(eq.[1]),i.e.accordingto (fulllines).Theslowjetcanexplainthelatetimepeak,butitunder-
t −5/3 predicts the early-time emission observed from SwiftJ1644+57.
L (t )=L max 1, obs , (5) This is because a slow jet coasts for a long time before appre-
x obs x,0 t !
j ciably slowing down, since it must first sweep up ∼ 1/Γ of its
j
whereagaint = 5×105sandweconsider severalvaluesforthe ownrest mass. Duringthisextended coastingphase, theemitting
j
SwiftJ1644+57:powerfuljetwith fastcoreandslowsheath 7
3 3
10 10
obs. obs.
L = 0 fast
mJy] 110021 LLXxX == 51x01480 4e7r ge rsg- 1s-1 mJy] 110021 slow
[ν 100 [ν 100
f f
-1 -1
10 1.4 GHz 1.8 GHz 10 1.4 GHz 1.8 GHz
3 3
10 10
y] 102 y] 102
J J
m 101 m 101
[ν 100 [ν 100
f f
-1 -1
10 4.9 GHz 15.4 GHz 10 4.9 GHz 15.4 GHz
3 3
10 10
y] 102 y] 102
J J
m 101 m 101
[ν 100 [ν 100
f f
10-1 24.4 GHz 43.6 GHz 10-1 24.4 GHz 43.6 GHz
101 102 101 102 101 102 101 102
time [days] time [days] time [days] time [days]
Figure5.RadiolightcurvesincludingtheeffectsofX-raycooling(§4.1.2), Figure6.Comparingradiolightcurvesfromaslow-widejet(Γj=2, θj=
calculated forajetwithEiso = 6.67×1054ergexpandingintoak = 1.5 0.5;dashed lines) tothe fast-narrow jet (Γj = 10, θj = 0.1;full lines).
mediumfordifferentvaluesoftheX-rayluminosity Lx,0 (eq.[5]).While Thickandthinlinesshowmodelsincludingandneglectingself-absorption,
X-raycoolingcansuppressthesynchrotronemissionforanopticallythin respectively. Bothjets have Eiso = 3.33×1054 ergandpropagate intoa
medium(thinlines),itseffectontheactualemissionisnegligiblebecause ncnm ∼ r−3/2externalmediumwithn18 = 50cm−3.Weassumeǫe = 0.1
thelatterisselfabsorbed(thicklines). andǫB=0.001.
3
area increases and so does the luminosity of the source at bands 10
wfahsterbeeftohreeermeaiscshiionngiistsseplefa-akb,swoirtbhedth.eTpheearkestiumlteindgeltiegrhmtcinuerdvebryiseeis- Jy] 1021 oεεBBb ==s. 00..000012
m 10
ther the thick-thin transition or the onset of the jet deceleration,
by itself inconsistent with the relatively flat light curve observed f [ν 100
forSwiftJ1644+57.Thecomparisonbetweenthelightcurvespro- 10-1 1.4 GHz 1.8 GHz
ducedbyafast-narrowjetandaslow-widejet(Fig.6)issuggestive
3
10
ofthefactthatdifferentpartsofauniquejet,movingatdifferentan-
glesandwithdifferentLorentzfactors,couldaccountforboththe y] 102
J
earlyandthelatetimelightcurveofSwiftJ1644+57. m 101
[ν 100
f
-1
4.2 Two-component1Dmodels 10 4.9 GHz 15.4 GHz
3
10
Theone-component jetmodelsconsideredthusfarcanreproduce
either the early or late time light curves well (Fig. 6), but none y] 102
J
acceptably reproduce the entire emission. We are thus motivated m 101
to consider a two-component jet, comprised of a fast core (Γj = [ν 100
10) with a narrow half-opening angle of θ = 0.1 rad, which is f
j 10-1 24.4 GHz 43.6 GHz
surrounded byaslower sheath(Γ = 2) from0.1to0.5rad. Both
components of the jet are assumed to possess an equal isotropic 101 102 101 102
energy E = 4×1054 erg.Weemployanexternal mediumwith time [days] time [days]
iso
an ∝ r−1.5 profileandn = 60cm−3.Asinpreviouscaseswe
cnm 18
assumeǫ = 0.1,however weallowǫ tovarymodestlyaboutits Figure7.Lightcurvesofatwo-component1Djetmodel(§4.2)including
fiducialvealue10−3soastoquantitativeBlyminimizethedifferences afast core(Γj = 10;θj = 0.1)surrounded byaslower sheath (Γj = 2;
amongobservationaldataandoursyntheticmodels. θj=0.5,eachwiththesameisotropicenergyEiso=4×1054erg.Different
linestylesshowdifferentchoicesoftheassumedpost-shockmagneticfield
modeFliwguitrhe7ǫsh=ow1s0−th3eprreosvuidltessoaforeuars2o-ncaobmleporenpernets1eDntamtioodneol.fTthhee strengthǫB=10−3(fullline)and2×10−3(dashedline).
B
well-sampledν>∼15GHzlightcurve,butitunderpredictstheflux
at low frequencies ν . 5GHzand early timest . 10 days. This
excess at early times in the observational data appears to require
either asignificant reduction of theoptical depth at timest . 10
days,oranadditionalemittingregionnotincludedinthecalcula-
8 P.Mimica, D.Giannios,B. D. Metzgerand M.A. Aloy
tions,andquitelikely,acombinationofbothfactors.Mimicaetal. T=0.96×107s[111days]
30.00
(2009) showed that during theearly stages of the jet-CNMinter- 101
action the reverse shock that decelerates the jet material can be 103
particularlypowerful.Indeed,fortheadoptedparameters,therate m] 20.00 100
ocofmdipsasripabatlieotnootfheenoebrsgeyrvaetdthjeetrdevuerarstieonsh(o∼ck1p0edaakyssoinnathteimceassceaolef nth102 16y[c10 10.00 10−1Γβ
SwiftJ1644+57).However,forthefiducialparametersadoptedin 101
10−2
thepresentpaper(includingthoseoftheCNM),theemissionofthe 0.00
−25.00 0.00 25.00
reverseshockisstronglyabsorbedbelow∼10GHzand,hence,the
x[1016cm]
reverseshockcontributionalonecannotaccountforthefluxdeficit T=4.3×107s[497days]
ofourmodelsatearlytimes4.Thelackofadominantreverseshock
120.00 101
componentatearlytimesispotentiallysupportedbytheupperlim-
102
iwetvseeoerk,fs.thaa2ftt−eirf1t0wh%eeBocAnoTnthstierdi4egr.g8eaGrs(HmWzailleilrnesreeamvraapluoeeltaarolif.za2tht0ieo1n2ζ)eo.nWpatiermanmeosetcetae,lrhe,ostwhoef- nth 110001 16y[cm]10 6900..0000 11000−1Γβ
synchrotronself-absorptionatboththeforwardandreverseshocks 10−1 30.00
issubstantiallyreduced,andevencountingonlytheforwardshock 10−2
0.00
emission,thefluxdeficitatearlytimescanbeameliorated(seeAp- −100.00 −50.00 0.00 50.00 100.00
x[1016cm]
pendixBfordetails). T=11×107s[1276days]
Themodelwithǫ =0.002,thoughslightlyoverpredictingthe
B 101 300.00 101
observed emission, nevertheless provides a good candidate to set
theinputforourfull2Dsimulationsdescribednext.Itisdesirable 100 m] 200.00 100
tfoulsll2igDhtclyasoevtehreprjeedtiwctiltlheexepmeriisesniocenlwatiethratlhsep1reDadminogd.eTlshiinscleeaindsthtoe nth10−1 16y[c10 10−1Γβ
100.00
earlier deceleration of the jet in comparison to spherical models, 10−2
resultinginlowerfluxatbandswheretheemissionisopticallythin 10−2
(seenextSection). 0.00 −250.00 0.00 250.00
x[1016cm]
T=63×107s[7263days]
100 100
1200.00
10−1
m] 900.00
nth10−2 16[c10 600.00 10−1Γβ
y
10−3 300.00
10−2
0.00
−750.00 0.00 750.00
x[1016cm]
Figure8.Snapshotsofthehydrodynamicevolutionofthetwo-component
4.3 2Dsimulations jet.Thetimeintheuppertwopanelscorrespondstothetimesofthesnap-
shotsinFig.2andFig.9.Ineachpanelthelefthalfshowsthe(comoving)
Motivatedbytheimprovementsinaccommodatingthelightcurves
numberdensitynthofthethermalfluid,whiletherighthalfshowsthefluid
whenconsideringone-dimensionaltwocomponentmodels(Fig.7), four-velocity Γβ := Γv/c. Since bothfast(Γf = 10)andslow (Γs = 2)
we perform 2D simulations of two component jet models with jetsareinjectedwiththesameluminositypersolidangle,nthinthe(more
E = 4×1054 erg. The jet is injected at the inner grid bound- relativistic) fastjetisinitially ∼ Γ2/Γ2 lowerthanintheslowerjet.This
iso s f
ary(r =2×1016cm)intheangularrange[0,0.5]rad.Afastcore canbeseenintheuppertwopanels,wherethefastjetisseenasadark-
j,0
(Γ =10)isinjectedbetween0and0.1rad,whileaslowersheath colouredchannel closetothesymmetryaxis.Notethatthedensitycolor
f
(Γ =2)isinjectedbetween0.1and0.5rad.Theexternalmedium scalechangesfrompaneltopanel.Thesphericalboundariesontheleftside
s
intowhichthejetispropagating hasar−3/2 profileand n = 60 ofsomepanels denotetheedgeofaradialgridwhichisperiodically ex-
18 tended,see§3.
cm−3.ThejetluminositychangesintimeaccordingtoEq.1.
Theradialresolutionofthesimulationsis∆r = 4×1014 cm
(Sec.3.2),buttheradialgridisregularlyextendedasthejetprop- 4.3.1 Hydrodynamicevolution
agatesintotheexternalmedium.Thisisachievedbystoppingthe
Oursimulationstartsfromsufficientlyshortdistancefromthecen-
simulationoncethejetheadnearstheouterradialedgeofthecur-
tralenginesothatwecanfollowallthestagesofthejetdeceleration
rentgrid,extendingtheradialgridandinitialisingwiththeexternal
asthejetsweepsthroughtheCNM.Initiallyboththeslowandfast
medium.Theangularresolutionis∆θ =π/600radandwealways
jetsareoverdensewithrespecttoexternalmediumandpropagate
simulatethefullrange[0,π/2]using300points.
almost ballisticallyalongtheir initial opening angle (toppanel in
Inordertocomputethelightcurvesthesnapshotsofthegrid
stateneedtobesavedperiodically(Sec.3.1).Thetimeatwhichthe
snapshotsarewrittenisexactlythesameasin1Dsimulations,thus 4 Besidesthefactthatitistechnicallyverychallengingtoaccuratelyde-
enablingustodirectlycompare1Dand2Dmodelsbothinterms tectthereverseshockinthehydrodynamicsimulations,thereverseshock
oftheirhydrodynamical evolution(Sec.4.3.1),aswellasintheir emissionisstronglysuppressedfortheparametersathand,whichjustifies
observationalsignatures(Sec.4.3.2). ourchoiceofnotincludingitinthecurrentmodels.
SwiftJ1644+57:powerfuljetwith fastcoreandslowsheath 9
slower shock it decreases almost to the slow jet value (Γ = 2).
s
10 In order to estimate the shock front velocities, it is important to
n (1D)
3] P (1D) considerthatthefastjetislaunchedtoexpandradiallyatanangle
-m Γv (1D) θj =0.1and,thus,withavelocitycomponentperpendiculartothe
2c c 102 nP ((22DD)) rjeatmaxpirsesvs⊥ur∼eθpje(r1pe−nΓdi−fc2u)1la/2r=to0th.1ec.axBiaslawnecicnogncthluedfeastthaatndthseloswhojcekt
m p Γnv ( 2(2DD-s)) c propagation speed towards the axis is very slow, while the effect
P [ P (2D-s) 5Γv/ of theinitial(thermal) pressure imbalance between both jetsgets
-3m], 101 Γv (2D-s) rneadrurocwedretodlaayreelrastiuvrerolyunthdiinnglatyheerbaeraomunodftthheeffaassttjjeett-w(sheeit,ees.hga.,dtehse-
intherightsideofthesecondpanelofFig.9).
c
n [ T = 9.6 x 106 s Having explained the differences with previous jet stability
studies caused by the pressure imbalance, we now use the re-
sults of those works to analyze fast/slow jet interface properties.
0 Hardee&Hughes(2003)showedthatpressurematched,cylindri-
10 20
16 caljetswithafast-spineandaslow-sheatharemorestablethanjets
r [10 cm]
withoutslow sheaths. Inour 2Dmodel theroleof thesheath can
beplayedbytheslowjet.Hardee&Hughes(2003)showthatthe
10
decreaseinthegrowthrateofKelvin-Helmholtzinstabilitiesispro-
-3m] 101 T = 4.29 x 107 s portionaltothevelocityshear∆v.Inourcase,∆v=vfj−vsj ≃0.13c
(v andv arethespeedsofinjectionofthefastandoftheslowjets,
c fj sj
2 respectively).Theyalsofindthatthepresenceofanexternalmov-
c
p ingsheardampensasymmetricmodesandraisesthegrowthrateof
m c thefundamentalpinchmode.Inthisregard,wenotethat,initially
P [ 5Γv/ (T . 100days)thefastjetdoesnotdevelopanyrecollimationin-
], 100 duced by the pinching of the slow jet. Later (T & 100days), the
-3m bowshockandthereverseshockproducedbytheslowjetpinchthe
c beamofthefastjetonlyveryclosetothejethead(y∼2.8×1017cm;
n [ toppanelofFig.8).Thispinchingspeedsupthebeamofthefast
jetwhich,asaresult,propagatesabitfasterthanapure1D“jet”
withthesameproperties.Thiseffectisvisibleinthetoppanelof
110 120 0 Fig.9,whereprofilesalongtheaxisofthe2Djetmodelarecom-
r [1016 cm] paredwithanequivalent1Djetmodel.Thereweobservethatthe
forwardshock of the1D jetmodel lagsbehind that of the2D jet
Figure9.SameasFig.2,butinadditionshowingthevaluesattheaxisof (locatedatr ≃2.9×1017cm).AlsoinFig.9,wedisplaytheaxial
thetwo-component2Djetsimulation(thinlines),anda2Dsimulationofa profilesofa2Djetwiththesamepropertiesasthefastandnarrow
single-componentfastjet(brownlines).Forbettercomparison,resultsofa jet, but without a broader slow jet flanking it (“top-hat” 2D jet).
1Dsimulationwiththesame∆rasthe2Doneareshown(seeSec.3.2). Without the stabilisinginfluence on thedynamics of thebroader,
slowerjet,thetop-hat2Djetclearlylagsbehind1Djetmodel.At
latertimes(bottompanelof Fig.9),thesituationisreversed,and
Fig.8).Atthisstage,bothfastandslowjetsmoveradiallyandare the2Deffects(transferoftransversalmomentum)makesthetwo-
slowed down at the reverse shock (located at distance ∼ 1018cm componentjetbowshocktopropagateatsmallerspeed.
at the second panel in Fig. 8). The reverse shock strongly com- Atthispointtherampressureofthefastjetis∼3timeslarger
pressesand heatsthe jetmaterial.Theshock front can beclearly thanthatoftheslowjetandtheformer“breaks-out”ofthelatter’s
seen in the right panels of Fig. 8 where the unshocked CNM is bowshock,ascanbeobservedfromthethirdpanelofFig.8.Inthe
at rest (black area). Throughout this work, we assume that parti- subsequentevolution,theinitiallyoblatestructureoftheoutermost
cleacceleratedat theexternal shock areresponsible for theradio bowshockofthefastjetbecomesincreasinglyprolate.AfterT ∼
emissionofSwiftJ1644+57. 20yr(bottompanelofFig.8),thelongitudinalsizesofthecavity
Thereisafundamentalfeatureofthestructuredjetdynamics blownbythefastjetandtheslowjetcavitybecomecomparable.
thatdeservesspecialattention:thejetisremarkablystablethrough- Another important difference we find between a top-hat 2D
outmostofthesimulatedevolution.Ananalyticalstudyofthesta- jetmodelandourtwo-component jetmodelistheradialwidthof
bilityofourconfigurationisrathercomplex(andbeyondthescope thebow-shock-to-reverseshockregion.Astimeevolvesthisradial
of thispaper), thereasonbeing thatthefast jetisunderpressured widthbecomesshorterinthetop-hatjet,sincethereisnodynam-
withrespect totheslow one. Thisbreaksoutacommonassump- icalcollimationexertedbythebroadcomponent. Forinstance, at
tioninthestudyofrelativisticjets,namelythatthejetispressure- T ∼500days(bottompanelofFig.9andsecondpanelofFig.8),
matchedwiththesurroundingmatter.Asaresultofthis,theslow/ theradialwidthoftheformerregionisof∼5×1016cmforthetop-
fastjetinterfacedevelopsaRiemannstructureconsistingonapair hat2Djetmodel,whileitisabouttwiceaslargeinourreference
ofshocksseparatedbyacontactdiscontinuity. Fortheconditions sheared2Djet.Thisdifferenceincreaseswithtime,particularlyaf-
we study, both shocks move towards the jet axis, though the one tertheconstantinjectionphasesfinishes.
shocking the fast jet isfaster (and weaker) than the one crossing Atlatertimestheinjectionluminosityinbothjetsdecreases.
thebeam of theslow jet.Acrossthe faster shock, the jetLorentz Thetenuousjetfindsitselfsurroundedbyahotcocoonofshocked
factorslightlydecreasesfromΓ = 10toΓ ∼ 9,whileacrossthe material. The pressure of the surrounding cocoon causes the jet
f
10 P.Mimica,D. Giannios,B. D. MetzgerandM. A. Aloy
103 Atlowfrequenciestheemissionisabsorbedand1Dand2D
y] 102 jceotnsichaalvley.vIenrythseim2Dilatrwloig-chotmcuprovneesnatssilmonuglaatisonthsethjeeteimsiesxspioanndfliantg-
mJ 101 o1Dbs heirgvha trieosn.s tensearlierasaresultofthejetdecelerationandexpansionalong
[ν 100 212DDD tloownwoe rcceoosmm.pp.. the azimuthal direction. In the 1D simulations such expansion is
f
-1 notallowedandthejetdecelerationtakesplacemoreslowly.
10 1.4 GHz 1.8 GHz
At higher frequencies the 2D two-component jet emission
3
10 peaksearlierandatlowerluminosity.The1Dmodeloverpredicts
y] 102 theopticallythinemissionbyfactorof∼3−5,takingintoaccount
mJ 101 thatour 1Dtreatment alreadytakesintoaccount thattheopening
[ν 100 angle of the jet is finite when calculating the emission. Models
f thatassume1D,sphericallyemittingblastoverpredicttheresulting
-1
10 4.9 GHz 15.4 GHz emission by alargefactor (∼ 100) for timescales∼ months after
3 theTDE.
10
Atearlytimes,whenallmodelsradiateinanopticallythick
Jy] 102 regime, the differences among 1D and 2D models are negligible,
m 101 andallofthemfallatroughlythesamedistancetotheobservations.
[ν 100 However,theone-component 2Djetfailsbadlytoaccountforthe
f
10-1 24.4 GHz 43.6 GHz observed flux after its light curve peak, when the flux decreases
muchfaster thaneitherthetwocomponent 2Dmodel or thanthe
101 102 101 102 1Dsingle-component jetmodelsshowninFig.3.Forbrevity,we
time [days] time [days]
donotshowherethelightcurvesobtainedwithasinglecomponent
2Dslow-widejet,buttheeffectsonthelightcurveareverysimilar
Figure10.SameasFig.7,butshowingthelightcurvesproducedby1D
to the ones shown in Fig. 6 and discussed in Sect. 4.1.3 for 1D
simulations(fulllines,correspondingtodashedlinesinFig.7)and2Dsim-
ulationsofatwo-componentjet(redlines)forǫe=0.1,ǫB=0.002,aswell models.Consideringthata2Dhydrodynamical modelingismore
asa2Dsimulationofone-componentfastandnarrowjet(greylines).Both realisticthanany1Dmodeling,wefindthattheordersofmagnitude
1Dand2DsimulationsassumeEiso = 4×1054 erg,anexternalmedium fluxdeficitinsinglecomponent 2Dfast-narrowjetmodelsatlate
withr−3/2 profileandn18 = 60cm−3.Thefastandnarrowjet(Γf = 10) times,togetherwiththefluxdeficitofsinglecomponent2Dslow-
spanstheangle from0to0.1rad, whiletheslowandwidejet(Γs = 2) widejetmodesatearlytimesisanunequivocalproofthatasingle
extendsfrom0.1to0.5rad. componentjetcannotsimultaneouslyexplainbothearly-andlate-
timeemission,andarguesinfavorofatwo-layerjetmodel.
channel tocollapse(thirdpanelinFig.8).Atlatertimesverylit-
tlenewinjectionofenergyandmomentumtakesplace.Theblast
wavethatboundsthestructureofthecomplexcavitywillrelaxinto 4.3.3 Originofnon-thermalemission
a spherical Sedov solution. While we do not follow its evolution
As remarked in § 3.3, SPEV models non-thermal particle evolu-
untilthislatestage,theresultingemission(ofinteresthere)isprac-
tion by injecting them behind relativistic shock fronts. Figure 11
ticallysphericallysymmetricbytheendofthesimulation(seenext
shows the particle distribution and density for the four hydrody-
Section).
namicsnapshotsdiscussedinFig.8.
Even though there is substantial dissipation at the reverse
shocks that can contribute to the emission the first ∼10days, we
4.3.2 Lightcurves
only model particles accelerated at forward/external shocks. The
Figure10showsacomparisonbetweenlightcurvescomputedfrom reason is threefold: (1) the emission from the reverse shock is
1Dand2Dsimulationswiththesameinitialconditions,aswellas stronglydimmedduetothestrongsynchrotronself-absorptionre-
alightcurvecomputedfroma2Dsimulationofanone-component sultingfromtheadoptedmicrophysicalanddynamicalparameters
jet (fast and narrow component only). Since the 2D simulations ofourmodels;(2)wefocusonthemonthtoyeartimescaleevolu-
areperformedwithlowerresolutionintheradialdirectionincom- tionwherethereverseshockemissionisexpectedtobelesspromi-
parison to the 1D ones, we have also performed 1D simulations nent; and (3) computing the reverse shock particle evolution and
withreducedresolution(tomatchthe2Dradialgrid,dottedline). emissionisnumericallyverydemanding.
Though,fromadynamicalpointofviewthedifferencesinchang- At early times (top panel in Fig. 11) the particles are con-
ingresolutionarenegligible,theinitialspeedofpropagationofthe finedtoarelativelysmallvolumebehindtheshockfrontsandcon-
forward (bow) shock of the jet isslightly different (∼ 10−3c; see tributetotheobservedemission.Atlatertimes(bottomtwopanels
Mimicaetal. 2009), which slightly affects the flux arrival times inFig.11)asignificantfractionofparticlesstreamawayfromthe
and the optical thickness of the overall ejecta. One can see that shockfrontsincethedistancebetweentheforwardshockandthe
thereducedresolutionaffectsthepredictedopticallythinemission contactdiscontinuityseparatingtheshocked jetmaterialfromthe
atearlytimes(dottedandfullthincurvesinFig.10).However,the shocked CNMalsogrows.Itshouldremarkedthatwestopinjec-
emission at these stages is optically thick and the low resolution tion when γ of the electrons becomes less than ∼ 2. We have
min
has negligible effect on the emission (see thicklines). We, there- checked that our results do not depend on the exact value of the
fore,expect thattheresolutionof the2D simulationsissufficient cutoff.Fort>∼10yearstheblastwaveentersthe“deepNewtonian”
to capture the hydrodynamic evolution of the jet well enough to regime,whereγ ∼1forallfreshlyinjectelectronsandtheparti-
min
computetheresultingemission(seeAppendixAformoredetails cleaccelerationspectrumismodified(seeSironi&Giannios2013
onothernumericalparametersinfluencingtheemission.) fordetails).