Table Of Content

An RMHD study of transition between prompt and afterglow GRB phases 8 0 0 2 n a J 9 PetarMimica∗ DepartmentforAstronomyandAstrophysics,UniversityofValencia ] h E-mail: [email protected] p - Dimitrios Giannios o Max-PlanckInstituteforAstrophysics r t E-mail: [email protected] s a [ Miguel-AngelAloy 1 DepartmentforAstronomyandAstrophysics,UniversityofValencia v E-mail: [email protected] 5 2 3 WestudytheafterglowphasesofaGRBthroughrelativisticmagnetohydrodynamicsimulations. 1 Theevolutionofarelativisticshellpropagatingintoahomogeneousexternalmediumisfollowed. . 1 Wefocusontheeffectofthemagnetizationoftheejectaontheinitialphasesoftheejecta-external 0 8 medium interaction. In particular we are studying the condition for the existence of a reverse 0 shock into the ejecta, the timescale for the transfer of the energy from the shell to the shocked : v mediumandtheresultingmultiwavelengthlightcurves. Tothisend,wehavedevelopedanovel i X schemetoincludenon-thermalprocesseseswhichiscoupledtotherelativisticmagnetohydrody- r a namiccodeMRGENESISinordertocomputethenon-thermalsynchrotronradiation. Supernovae:lightsinthedarkness October3-5,2007 Maó(Menorca) ∗Speaker. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ AnRMHDstudyoftransitionbetweenpromptandafterglowGRBphases PetarMimica 1. Introduction Magneticfieldsmayplayanimportantroleintherelativisticflowofagamma-rayburst(GRB), but the extent to which they are important remains uncertain. Looking at the process which pro- ducesarelativisticGRBoutflow,twoalternativesareusuallyconsidered. Ontheonehand,neutrino annihilationmaybeaprocesswhichleadstotheformationoffireballdominatedbythermalenergy. Heremagneticfieldsaredynamicallyunimportant. Alternatively,powerfulenoughmagneticfields can efficiently extract the rotational energy from the central engine and launch a Poynting-flux dominated flow. Inordertoachieverelativisticvelocities,GRBshavetobelaunchedwithhighenergy-to-mass ratio. In the fireball model, most of the energy is assumed to be thermal [17, 30]. This can be be result of neutrino-antineutrino annihilation in the polar region of an accreting object [35, 3, 4, 6]. Theacceleration ofthefireballisduetotheinternal pressure gradient, whereby thermal energy of the fireball is converted into kinetic energy of the baryons. At the end of the acceleration phase fasterpartsoftheflowmaycollidewiththesloweronesproducingtheinternalshockswhichpower the GRBprompt emission [31, 11, 27, 28]. Processes such astwo-stream instability inthe shocks might amplify weak magnetic fieldspresent inthe flow[24]. These fieldsare expected toaccount for less than 1% of the energy of the flow. After the internal shock phase the flow expands and cools asitenters the afterglow phase. In this case veryweakly magnetized flowis expected atthe onsetoftheafterglow. Magnetic fields with appropriate topology can efficiently extract rotational energy from the central engine, be it an accretion disk [8], rotating black hole [9] or a millisecond magnetar [34] launching a Poynting-flux dominated flow (PDF). The acceleration of the PDF depends on the field geometry and the dissipation processes. Magnetic dissipation can convert Poynting flux into kineticenergy [12,14]andalsopowertheGRBpromptemission[23,13,15]. Differentstudiesof MHD jet acceleration show that magnetic energy is not completely converted into kinetic energy at the end of the acceleration phase. At larger radii we expect magnetic energy of the flow to be comparable to the kinetic energy of the baryons [14] or even much larger [23, 33]. It should be notedherethatthemagnetization oftheflowmightbedecreased bythepair-loading causedbythe n n¯-annihilation nearthecentralengine[22,5]. FireballandPDFmodelsrespectivelypredictweaklyandstronglymagnetizedflowattheonset oftheafterglowphase. TheinitialphasesoftheinteractionoftheGRBflowwiththe(circumburst) externalmediumdependonthestrengthofthemagneticfieldsintheflow. Aparticularlypromising probe of the magnetization of the GRB flow can come from understanding the early afterglow emission[20,36,16]. Inthispaperweoutlinethestatusoftheongoingstudyoftheinteraction ofmagnetizedejecta with external medium. In Sec. 2 we describe in more detail the current understanding of ejecta- medium interaction. Sec. 3 gives an overview of numerical methods and a plan for numerical simulations. SummaryisgiveninSec.4. 2 AnRMHDstudyoftransitionbetweenpromptandafterglowGRBphases PetarMimica 2. Ejecta-medium interaction We consider a homogeneous shell expanding into an external medium of constant density1. At large distances from the central engine (typically R ≈1015−1017 cm) substantial interaction 0 begins, whereby ejecta begins to decelerate due to the accumulation of the external material. We assume that at these distances the flow has already been accelerated and collimated. The internal dissipation mechanism, presumably responsible for the prompt emission (e.g. internal shocks, magnetic dissipation) is also expected to take place at a shorter distance from the central engine withrespecttothatoftheafterglow phases. Aftertheinternaldissipation isover,theflowexpands and cools. Since we are interested in the afterglow phases, the shell is assumed to be cold. We denote the shell Lorentz factor by g ≫ 1 and its width by D . In a radially expanding outflow 0 0 the magnetic field component perpendicular to the direction of motion drops as r−1 while the component in the direction of motion drops as r−2, so that we expect the magnetic field to be dominated bytheperpendicular component. Wedefinethemagnetization parameteras E B2 s := P = 0 , (2.1) 0 E 4pg r c2 K 0 0 whereE andE arePoyntingandkineticenergiesintheshell,r andB itsdensityandmagnetic P K 0 0 field measured in the central engine frame. With this definition a fireball corresponds to s ≪1 0 whileaPDFhass ≥1. cisthespeedoflight. Thetotalenergyoftheshellis 0 E =4p R2D (g r c2+B2/4p )=E (1+s ). (2.2) 0 0 0 0 0 K 0 FromEqs.2.1and2.2wecanseethats parametrizesthefractionofthetotalenergyinthekinetic 0 (1/(1+s ))andmagnetic(s /(1+s ))form. 0 0 0 Wefirstdiscusstheejecta-medium interaction fornon-magnetized ejecta, andthenturntothe arbitrarily magnetized case. We also focus on the conditions for the existence of a reverse shock intoejectaofarbitrary magnetization. 2.1 Thecaseofs ≪1 0 The evolution of the interface between the cold unmagnetized shell and the external medium is well understood. It was studied in detail analytically [32, 26] as well as using one-dimensional [19]andtwo-dimensional [10,18,25]numerical simulations. At the interface between the shell and the ambient medium two shocks form, the forward shock propagating into the external medium, and the reverse shock propagating into the shell. Shocked shell andexternal medium areseparated bythecontact discontinuity. Theforward shock is always ultra-relativistic, while the strength ofthe reverse shock depends onthe density contrast betweentheshellandtheexternalmediumandthebulkLorentzfactorg . Wedistinguish between 0 relativistic andNewtonianreverseshocks[32]. Thecriticalparameter is x :=l1/2D −1/2g −4/3, (2.3) 0 0 where l = (3E/4p n m c2)1/3 is the Sedov length, n the external medium number density, and e p e m the proton mass. In the Newtonian case (x ≫1) the shock is non-relativistic in the shell rest p 1Similaranalysiscanbeperformedforthewindprofilewherethedensityoftheexternalmediumscalesasr−2. 3 AnRMHDstudyoftransitionbetweenpromptandafterglowGRBphases PetarMimica frame and does not decelerate the ejecta much, rather the ejecta decelerate once they accumulate a mass g −1 times their own mass from the external medium. In the relativistic case (x ≪1) the 0 shock crosses the ejecta quickly and slows them down considerably. After the shock crosses the ejecta, there isa phase where shocks and rarefaction waves cross theejecta, passing the energy to theforwardshock. Atlaterstagestheevolutionoftheejectaonlydependsontheirtotalenergyand theexternalmediumdensity [7]. 2.2 Thecaseofarbitrarys 0 Thedynamicsofmagnetizedejectahasnotbeenstudiedasthoroughlyasthatofunmagnetized ejecta. A qualitative difference to the unmagnetized case is that later phases of the evolution are influenced bytheinternal evolution ofthemagnetized shell. Theinitial phase oftheevolution has recently been studied [37] by solving the ideal MHD shock conditions for arbitrarily magnetized ejecta withtoroidal field. Inparticular, thedynamics ofshock crossing hasbeenstudied assuming that there is a reverse shock. In that case, the reverse shock crosses the shell faster the higher the magnetization is. However,aswehaverecently shown[16],itisnotalwaysthecasethatareverse shockforms. 2.3 Conditionsfortheexistenceofareverse shock Cold, non-magnetized ejecta are always crossed by areverse shock upon interacting with the external medium. This is the case since the sound speed of the ejecta is low and does not allow for fast transfer of the information of the interaction with the external medium throughout their volume. Ontheother hand, inaflowthatisstrongly magnetized andsub-fast magnetosonic (asin the Lyutikov & Blandford 2003 model [23]) there is no reverse shock forming. The flow adjusts gradually to the changes ofthe pressure inthe contact discontinuity that separates the magnetized flowfromtheshockedexternalmedium. In[16]wegeneralizetoarbitrarilymagnetizedejectaand derivethecondition fortheformation ofareverseshock. Afteradetailedtreatmentofthisproblemwearrivetothefollowingconditionfortheformation ofareverseshock[16] 1 x < , (2.4) (4s )1/3 0 whichcanberewrittenintermsofshellparametersas2 s <0.6n1/2D 3/2g 4 E−1/2. (2.5) 0 0 12 2.5 53 Fig. 1 (taken from [16]) shows the division of x −s parameter space in two regions, one 0 whereareverseshockformsandanother whereitsformationissuppressed. Wenotethatfromthe conditions inEqs.2.4and2.5itfollowsthatevenformildlymagnetizedshellsareverseshockcan be suppressed. This indicates that the paucity of the observed optical flashes in GRB afterglows (associated with the reverse shock emission) may be caused by the suppression of the shock in manyGRBs. 2WeusetheconventionthatA=Ax10x. 4 AnRMHDstudyoftransitionbetweenpromptandafterglowGRBphases PetarMimica 10 No Reverse Shock Newtonian Reverse Shock 1 x Relativistic Reverse Shock 0.1 0.001 0.01 0.1 1 10 100 s o Figure 1: Existence of a reverse shock in the x −s parameter space. The dashed black line delimits 0 regions where a reverse shock formsfrom the region where there is no reverse shock, ignoring the radial shellspreading. Thesolidblacklineshowsthedelimitationwhentheshellspreadingistakenintoaccount. See[16]fordetails. 3. Numerical simulations In this section we describe the reasons and motivation for performing numerical simulations ofshell-ejecta interactions: 1. Verification of the analytic approach: Results of the analytic work described in Sec. 2, especially in Sec. 2.3, need to be verified by means of numerical simulations. Wenote that the line dividing regions of formation and suppression of the reverse shock in Fig. 1, given by Eq. 2.4 is approximate, and it is necessary to perform numerical simulations for models whoseinitialparameters lieinthevicinityoftheline. 2. Dynamicsofshockpropagation: Wewanttousenumericalmodelstostudytheinfluenceof the magnetization onthepropagation ofthereverse shock through the shell. OnFigs.2and 3 we show snapshots of two simulations, one with the unmagnetized (s =0), and another 0 with the magnetized (s =1)shell interacting with the external medium, both shown atthe 0 sameevolutionary time. Itcanbeseenthatinthemagnetized case(Fig.3)thereverseshock haspenetratedtheshelldeeperthanintheunmagnetizedcase(Fig.2). Ourgoalistoperform a parametric study where we study the propagation of shocks and rarefactions through the shellfordifferentcombinations ofx ands . 0 5 AnRMHDstudyoftransitionbetweenpromptandafterglowGRBphases PetarMimica Figure 2: Snapshot from relativistic hydrodynamicalsimulation of a spherical, non-magnetized (s =0) shellthatdeceleratesinteractingwithexternalmediumofdensity10cm−3. Thetotalenergyoftheshellis E =1051erg,itsinitialwidthD =1014cmandbulkLorentzg ≃50.P∗(dashedline),r (solidline)stand 0 0 forthegaspressureand(labframe)densityrespectively(inarbitraryunits). Withincreasingradius,onecan clearlyseethereverseshock,contactdiscontinuityandforwardshocklocatedatr∼1.483·1016cm. 3. Longtermevolutionandenergycontent: Oneoftheimportant questions thatsimulations can answer is the timescale of the transfer of energy from the shell to the shocked external medium. Long-term numerical calculations areneeded todetermined thedependence ofthe efficiencyoftheenergytransferonmagnetization. Wealsowanttoinvestigatethelong-term evolution of the blast wave and determine when the shell profile relaxes to the Blandford- McKeesolution [7]. 4. Lightcurves: Finally,wewishtocomputesyntheticmulti-wavelengthafterglowlightcurves. Lightcurvesareverysensitivetothemagneticfieldcontent oftheshell,shockstrength and, especially, detailed radial profile g (R) of the Lorentz factor as the shell propagates into the external medium. They also depend on the distribution of shock accelerated particles. To see why g (R) is crucial for the light curve, consider the difference in the arrival time to the observer ofsignals emittedsimultaneously fromtwopointswithradiiRandR+dR,respec- tively. Itturnsoutthatthedifferenceisdt≈dRg (R)−2forarelativisticshell. Ascanbeseen, asuddendropinaLorentzfactorinonemodelascomparedtotheotherwillproducefeatures which have longer observed duration. Theresulting light curves oftwo models can be very different. High-resolution simulations are needed toresolve sufficiently short timeintervals in order tobe able to study theinfluence ofmagnetization on g (R)and the subsequent light curve. Wehavedevelopedarelativisticmagnetohydrodynamic codeMRGENESIS[29,27,28]which 6 AnRMHDstudyoftransitionbetweenpromptandafterglowGRBphases PetarMimica Figure 3: Same as Fig. 2, but for the magnetized (s =1) shell. Note that the reverse shock crosses the 0 ejectafasterwithrespecttothenon-magnetizedcaseinagreementwithanalyticalexpectations. consists of a finite-volume, high-resolution, shock-capturing scheme GENESIS [1, 2, 21] which solves for the conservation laws of relativistic magnetohydrodynamics, and a module which fol- lowsthetransport,evolutionandradiationfromnon-thermalparticles. Forthepurposeofanalyzing dynamicsandemissionfromafterglowshells, averyhighresolution isneeded. Numericalconver- gencetestshaveshownthat,inordertosufficiently resolvetheshell, weneedtoresolvethescales of theorder ofD g −2. Thismeansthat weneed touse atleast g 2 zones withinthe shell. Weuse a 0 0 0 gridre-mappingproceduredescribedin[29],whichcanbethoughtofasaguidedmeshrefinement centered on the shell. Even with this procedure the actual number of zones for a typical model is expected tobeseveralmillions. Weareperforming simulations onsupercomputers oftheSpanish Supercomputing Network. 4. Conclusions Weareperforminghigh-resolution numericalstudiesofthetransitionfromprompttotheearly afterglowphaseofgamma-raybursts. Theafterglowisbeingmodeledastheradiationfromtherel- ativisticshellexpanding intothehomogeneous externalmedium. Westudythedifferencebetween thefireball(unmagnetized shell)andthePoynting-flux dominated (magnetized shell)models. In the context of the early optical afterglow we have shown analytically that even a moder- ate magnetization of the flow can suppress the existence of a reverse shock, and thus explain the apparent paucity of the optical flashes for a large number of early afterglows. We are currently performing simulations to study the formation and suppression of relativistic shocks in detail. To study the later phases of the afterglow we aim to determine the influence of the initial shell mag- 7 AnRMHDstudyoftransitionbetweenpromptandafterglowGRBphases PetarMimica netization on the energy content, transfer of energy from the shell to the forward shock, and the long-term flowstructure. We have developed a novel scheme for treating non-thermal processes in relativistic magne- tohydrodynamic simulations. Thisschemeisused tocompute multi-wavelength lightcurves from numerical simulations. Theaim isto study the influence of the initial magnetization on the short- andlong-term lightcurves. Acknowledgements PMisattheUniversityofValenciawithaEuropeanUnionMarieCurieIncomingInternational Fellowship(MEIF-CT-2005-021603). MAAisaRamónyCajalFellowoftheSpanishMinistryof Education and Science. He also acknowledges partial support from the Spanish Ministry of Edu- cationandScience(AYA2004-08067-C03-C01, AYA2007-67626-C03-01). Theauthorsthankfully acknowledgethecomputerresources, technicalexpertiseandassistanceprovidedbytheBarcelona Supercomputing Center-CentroNacionaldeSupercomputación. References [1] M..A.Aloy,J.M.Ibáñez,J.M.Martí,E.Müller,GENESIS:AHigh-ResolutionCodefor Three-dimensionalRelativisticHydrodynamics,APJs(1999)122,151 [2] M.A.Aloy,J.A.Pons,J.M.Ibañez,Anefficientimplementationoffluxformulaeinmultidimensional relativistichydrodynamicalcodes,Comp.Phys.Comm.(1999)120,115 [3] M.A.Aloy,E.Müller,J.M.Ibañez,J.M.Martí,A.MacFayden,Relativisticjetsfromcollapsars,ApJ (2000)531,L119 [4] M..A.Aloy,H.-Th.Janka,E.Müller,Relativisticoutflowsfromremnantsofcompactobjectmergers andtheirviabilityforshortgamma-raybursts,A&A(2005)436,273 [5] M.A.Aloy,M.Obergaulinger,RelativisticOutflowsinGamma-RayBursts,arXiv:astro-ph/0701187 [6] R.Birkel,M.A.Aloy,H.-Th.Janka,E.Müller,Neutrinopairannihilationnearaccreting, stellar-massblackholes,A&A(2007)463,51 [7] R.D.Blandford,C.F.McKee,Fluiddynamicsofrelativisticblastwaves,PhFl(1976)19,1130 [8] R.D.Blandford,D.G.Payne,Hydromagneticflowsfromaccretiondiscsandtheproductionofradio jets,MNRAS(1982)199,983 [9] R.D.Blandford,R.L.Znajek,ElectromagneticextractionofenergyfromKerrblackholes,MNRAS (1977)179,433 [10] J.K.Cannizzo,N.Gehrels,E.T.Vishniac,ANumericalGamma-RayBurstSimulationUsing Three-DimensionalRelativisticHydrodynamics:TheTransitionfromSphericaltoJetlikeExpansion, ApJ(2004)601,380 [11] F.Daigne,R.Mochkovitch,Gamma-rayburstsfrominternalshocksinarelativisticwind: temporal andspectralproperties,MNRAS(1998)296,275 [12] G.Drenkhahn,H.C.Spruit,EfficientaccelerationandradiationinPoyntingfluxpoweredGRB outflows,A&A(2002)391,1141 8 AnRMHDstudyoftransitionbetweenpromptandafterglowGRBphases PetarMimica [13] D.Giannios,PromptemissionspectrafromthephotosphereofaGRB,A&A(2006)457,763 [14] D.Giannios,H.C.Spruit,TheroleofkinkinstabilityinPoynting-fluxdominatedjets,A&A(2006) 450,887 [15] D.Giannios,H.C.Spruit,Spectralandtimingpropertiesofadissipativegamma-rayburst photosphere,A&A(2007)469,1 [16] D.Giannios,P.Mimica,M.A.Aloy,OntheexistenceofareverseshockinmagnetizedGRBejecta, A&A(2007)(inpress),arXiv:0711.1980[astro-ph] [17] J.Goodman,Aregamma-ray-burstsopticallythick?,ApJ(1986)308,L47 [18] J.Granot,M.Miller,T.Piran,W.M.Suen,P.A.Hughes,LightCurvesfromanExpanding RelativisticJet,inproceedingsofGamma-rayBurstsintheAfterglowEra(2001),312 [19] S.Kobayashi,T.Piran,R.Sari,HydrodynamicsofaRelativisticFireball: TheCompleteEvolution, ApJ(1999)513,669 [20] P.Kumar,A.Panaitescu,Aunifiedtreatmentofthegamma-rayburst021211anditsafterglow, MNRAS(2003)346,905 [21] T.Leismann,L.Antón,M..A.Aloy,E.Müller,J.M.Martí,J.A.Miralles,J.M.Ibáñez,Relativistic MHDsimulationsofextragalacticjets,A&A(2005)436,503 [22] A.Levinson,D.Eichler,BaryonPurityinCosmologicalGamma-RayBurstsasaManifestationof EventHorizons,ApJ(1993)418,386 [23] M.Lyutikov,R.D.Blandford,GammaRayBurstsasElectromagneticOutflows,arXiv arXiv:astro-ph/0312347 [24] M.V.Medvedev,A.Loeb,GenerationofMagneticFieldsintheRelativisticShockofGamma-Ray BurstSources,ApJ(1999)526,697 [25] Z.Meliani,R.Keppens,F.Casse,D.Giannios,AMRVACandrelativistichydrodynamicsimulations forgamma-rayburstafterglowphases,MNRAS(2007)376,1189 [26] P.Mészáros,M.J.Rees,OpticalandLong-WavelengthAfterglowfromGamma-RayBursts,ApJ (1997)476,232 [27] P.Mimica,M.A.Aloy,E.Müller,W.Brinkmann,Whichphysicalparameterscanbeinferredfrom theemissionvariabilityofrelativisticjets?,A&A(2005)451,103 [28] P.Mimica,M.A.Aloy,E.Müller,Internalshocksinrelativisticoutflows:collisionsofrelativistic shells,A&A(2007)466,93 [29] P.Mimica,M.A.Aloy,E.Müller,W.Brinkmann,SyntheticX-raylightcurvesofBLLacsfrom relativistichydrodynamicsimulations,A&A(2004)418,947 [30] B.Paczyn´ski,Gamma-rayburstersatcosmologicaldistances,ApJ(1986)308,L43 [31] M.J.Rees,P.Mészáros,Unsteadyoutflowmodelsforcosmologicalgamma-raybursts,ApJ(1994) 430,L93 [32] R.Sari,T.Piran,HydrodynamicTimescalesandTemporalStructureofGamma-RayBursts,ApJ (1995)455,L143 [33] C.Thompson,DecelerationofaRelativistic,Photon-richShell:EndofPreacceleration,Dampingof MagnetohydrodynamicTurbulence,andtheEmissionMechanismofGamma-RayBursts,A&A(2006) 651,333 9 AnRMHDstudyoftransitionbetweenpromptandafterglowGRBphases PetarMimica [34] V.V.Usov,Millisecondpulsarswithextremelystrongmagneticfieldsasacosmologicalsourceof gamma-raybursts,Nature(1992)357,472 [35] S.W.Woosley,Gamma-rayburstsfromstellarmassaccretiondisksaroundblackholes,ApJ(1993) 405,273 [36] B.Zhang,S.Kobayashi,P.Mészáros,Gamma-RayBurstEarlyOpticalAfterglows: Implicationsfor theInitialLorentzFactorandtheCentralEngine,ApJ(2003)595,950 [37] B.Zhang,S.Kobayashi,Gamma-RayBurstEarlyAfterglows: ReverseShockEmissionfroman ArbitrarilyMagnetizedEjecta,ApJ(2005)628,315 10