Mon.Not.R.Astron.Soc.000,1–12(1997) Printed5February2008 (MNplainTEXmacrosv1.6) Gamma-ray bursts from internal shocks in a relativistic wind: temporal and spectral properties F. Daigne and R. Mochkovitch Institut d’Astrophysique de Paris, CNRS, 98 bisBoulevard Arago, 75014 Paris, France 8 9 ABSTRACT 9 We construct models for gamma-ray bursts where the emission comes from internal 1 shocks in a relativistic wind with a highly non uniform distribution of the Lorentz n factor. We follow the evolution of the wind using a very simplified approach where a a large number of layers interact by direct collisions but where all pressure waves have J been suppressed. We suppose that the magnetic field and the electron Lorentz factor 6 reach large equipartition values in the shocks. Synchrotron photons emitted by the 2 relativistic electrons have a typical energy in the gamma-ray range in the observer frame. Synthetic bursts are constructed as the sum of the contributions from all the 1 internal elementary shocks and their temporal and spectral properties are compared v to the observations. We reproduce the diversity of burst profiles, the “FRED” shape 5 of individual pulses and the short time scale variability. Synthetic bursts also satisfy 4 the duration-hardnessrelationand individualpulses are found to be narrowerathigh 2 energy,inagreementwiththe observations.Theseresultssuggestthatinternalshocks 1 0 in a relativistic wind may indeed be at the origin of gamma-ray bursts. A potential 8 problem is however the relatively low efficiency of the dissipation process. If the rela- 9 tivisticwindispoweredbyaccretionfromadisc toastellarmassblackholeitimplies / that a substantial fraction of the available energy is injected into the wind. h p Key words: gamma-rays: bursts; radiation mechanisms: non thermal; shock waves; - accretion discs. o r t s a : v 1 INTRODUCTION ofcosmological effectsand indicatethatGRBshavetypical i redshiftsintherange0.3–1(Piran1992;Mao&Paczyn´ski X Since 1991 the BATSE experiment on board the Comp- 1992; Fenimore et al. 1993). r tonGROsatellite hasobservedmorethan1800 gamma-ray a bursts(hereafterGRBs).Theburstdistributionisisotropic ModellingGRBsisadifficulttaskduetotheextremedi- over the sky but non homogeneous in distance (Fishman versityofburstprofiles,thenonthermalspectraandthelack and Meegan, 1995 and references therein) which has been of clear signature for the emission processes involved. Most regarded as a strong indication that GRBs lie at cosmo- cosmological modelshoweversharesomecommoncharacte- logical distances (Paczyn´ski 1991). However the possibility ristics. The source, which must be able to release (between that GRBs belong to a large galactic halo (Hartmann et 10keVand10MeV)anenergyE > 1051 erg.sr−1onatime γ 4π al. 1994) could not be excluded a priori. This long stand- scale of seconds, is generally suppo∼sed to be a stellar mass ing controversy (see Nemiroff et al. 1995) about the burst black hole accreting material from a disc. Such a configu- distance scale may finally be solved by the recent observa- ration can result from the coalescence of two neutron stars tionsoftransientopticalcounterpartsfortwoGRBs.Inthe (Eichler et al. 1989; Paczyn´ski 1991; Narayan, Paczyn´ski & case of GRB 970228 (van Paradijs et al. 1997; Sahu et al. Piran1992),thedisruptionoftheneutronstarinaneutron 1997a) the point-like counterpart appears to be associated star – black hole binary (Narayan Paczyn´ski & Piran 1992; to an extended source which is probably a distant galaxy. Mochkovitch et al. 1993) or the collapse of a massive star ThecaseofGRB970508 isevenmorespectacular sincethe (Woosley 1993). The power emitted by cosmological GRBs spectrum of the counterpart shows Fe II and Mg II lines isordersofmagnitudelargerthantheEddingtonluminosity duetoabsorbers on theline of sight at aredshift z=0.835 andcannotcomedirectlyfromthediscsurface.Thereleased (Metzgeretal.1997).Iftheassociation ofthevisiblesource energyinstead drivesawindwhich hastobecomerelativis- to the burst is confirmed, GRB 970508 must be a very dis- ticbothtoproducegamma-raysandtoavoidphoton-photon tant object at z 0.835. If GRBs are placed at cosmologi- annihilation along the line of sight (Baring 1995; Sari & ≥ cal distances the LogN – LogP (peak flux) curve and the Piran 1997a). The Lorentz factor Γ must reach values of value of <V/V > can be naturally interpreted in terms 102 – 103 which limits the allowed amount of baryonic pol- max 2 F. Daigne and R.Mochkovitch lution in the flow to a very low level. A few mechanisms 2 A SIMPLE MODEL OF THE RELATIVISTIC which could possibly achieve such a severe constraint have WIND been proposed: (i) magnetically driven outflow originating 2.1 Description of the model from the disc or powered by the Blandford-Znajek (1977) effect (Thompson 1994; M´esz´aros & Rees 1997a), ii) recon- We do not discuss in this paper the nature of the source nection of magnetic field lines in the disc corona (Narayan, (coalescence of twoneutron stars, neutron star– black hole Paczyn´ski & Piran 1992) or (iii) neutrino-antineutrino an- binary,collapseofamassivestarorsomethingelse)whichis nihilation in a funnel along the rotation axis of the system initially responsible for theenergy release. Wealso suppose (M´esz´aros & Rees 1992; Mochkovitch et al. 1993, 1995). It that a relativistic wind carrying the energy has emerged is supposed in (i) and (ii) that the field has reached huge from the source, with an average Lorentz factor Γ¯ of a few values B > 1015 G. Concerning (iii) Ruffert et al. (1997) hundreds. have show∼n recently that νν¯ annihilation does not provide To study theevolution of the relativistic wind we have enoughenergytoaccountforcosmologicalGRBsexceptmay developed a very simple model where a succession of layers be for very massive discs, as those which could result from areemittedevery2mswithavaryingLorentzfactor,during thecollapse of a massive star. atotaltimet .Themassofthelayersisproportionalto1/Γ w so that the energy injection rate is constant. We follow the The energy initially stored in kinetic form within the layersasthewindexpandsandwhenarapidlayer(ofmass relativistic wind must then be converted into gamma-rays. m and Lorentz factor Γ ) catches up with a slower one 1 1 This can bedoneduringthedeceleration of thewind resul- (m , Γ <Γ ) they collide and merge to form a single shell 2 2 1 tingfrom itsinteraction withtheinterstellarmedium(Rees of resulting Lorentzfactor Γ . If thedissipated energy r & M´esz´aros 1992; M´esz´aros & Rees 1993) or a dense radia- e=m c2Γ +m c2Γ (m +m )c2Γ , (1) tionfield (Shemi1994). Inthefirstcase theemission comes 1 1 2 2− 1 2 r fromelectronsacceleratedintheforwardandreverseshocks can be radiated in the gamma-ray range on a time scale which then radiate synchrotron and inverse Compton pho- shorter than the shell expansion time (in the comoving tons in a magnetic field frozen in the wind or which has frame) cometoequipartitionwith theshockedmaterial(M´esz´aros, r Lamagbuiennat&radRieaetsio1n99fi3e)ld. I(ninththeesecceonntdralcarseeg,iopnhsotoofngsloobfutlhaer tex≃ cΓr , (2) clustersoractivegalacticnuclei)interactwiththeelectrons wherer istheshellradius,theburstprofilewillbemadeby (Shemi 1994; Shaviv & Dar 1995) or ions (Shaviv & Dar asuccession ofelementary contributionsof duration(in the 1996) of the wind and undergo a boost in energy by a fac- observerframe) tor Γ2. With Γ < 103, optical photons can be shifted to r the gamma-ray ra∼nge. In these two models the duration of ∆t≃ 2cΓ2 . (3) r the burst is t rdec where r is the deceleration ra- dius of the winbd∼. AncΓo2ther possibdielcity suggested by Rees & WeestimateΓrbyconsideringthatmostoftheenergyavail- ableinthecollision hasbeenalreadyreleased whentheless M´esz´aros (1994) consists to suppose that the Lorentz fac- massiveofthetwolayershassweptupamasscomparableto torinthewindisvariablesothatsuccessiveshellscanhave its own mass in the other layer (internal shocks being only large relative velocities leading to the formation of internal mildly relativistic). Then shocks. The energy which is dissipated in these shocks can then be radiated as gamma-rays via the production of pi- Γ √Γ Γ , (4) r 1 2 ≃ onsinproton-protoncollisions(Paczyn´ski&Xu1994)orby and synchrotron (or inverse Compton) emission of accelerated electrons. There are several important differences between e=mc2(Γ +Γ 2Γ ) , m=min(m ,m ). (5) 1 2 r 1 2 − the deceleration and internal shock models. In deceleration Afterthecompleteredistribution ofmomentumandenergy models, the duration and time profile of the burst depend theLorentz factor of themerged layer finally becomes on Γ, on the value of the deceleration radius and on the structure of the emitting shell. In internal shock models, m Γ +m Γ theduration of the burst is directly related to theduration Γf =rΓ1Γ2m1Γ1+m2Γ2 . (6) of energy injection at the source and the time profile is es- 1 2 2 1 sentially determined by thevariations of theLorentz factor Two more conditions haveto besatisfied for an elementary (Sari & Piran 1997a). shock to be produced and observed: i) the two layers must collideatarelativevelocitylargerthanthelocalsoundspeed Burst profiles and spectra have already been obtained andii)thewindmustbetransparenttotheemittedphotons. (sometimes in a rather detailed manner) in the case of de- The relative velocity of thetwo layers is given by celeration models (Fenimore, Madras & Nayakshin 1997; v Γ2 Γ2 Panaitescuetal.1997;Panaitescu&M´esz´aros1997)andthe rel 1− 2 , (7) c ≃ Γ2+Γ2 purposeofthisworkistopresentthesamekindofquantita- 1 2 tiveanalysisforinternalshockmodels.InSect.2wedescribe withΓ1 >Γ2 andΓ1,2 1.Wehaveadoptedasoundspeed ≫ our method to follow the evolution of the relativistic wind vs/c=0.1butwehavecheckedthatotherchoicesmakelittle andwediscusstheemission processes; Sect.3and4respec- differences in theresults since the main contribution to the tivelydealwiththetemporalandspectralpropertiesofour burst comes from shocks corresponding to large differences syntheticburst models and Sect. 5 is theconclusion. of theLorentz factor Γ1 >2. Γ2 ∼ Gamma-ray bursts from internal shocks in a relativistic wind 3 The transparency of the wind to the emitted photons Figure1.DistributionoftheLorentzfactorinthewindatdiffer- has been computed in the following way: in each collision enttimesinlagrangiancoordinatesm/mtot (mtot beingthetotal between layers a “photon shell” is generated which then massofthewind).Theinitialdistribution(t=0)consistsof5000 catchesupwithallthelayersaheadofit.Ittravelsthrough layerswithΓ=100 inthe1000firstemittedlayers andΓ=400 a total optical depth in the rest of the wind. Since the mass of the layers is propor- tionalto1/Γtheslowandrapidpartscontainanequalmass.At m τ =κ i , (8) t=3.4104 saforwardandareverseshock arepropagating into TiX>im 4πri2 trhiaenwsihnedll.oInfionucrresaisminpglemmaossd.eAl tthtey=a1r.e7l1o0c5atsedthoenfoarwsianrgdlesehuolcek- whereκ istheThomson opacity andr theradiusoflayer has alreadycrossedthewholeslowpartand at t=5.4105 sthe T i reverseshockreaches the endof therapidpart.The dashedline i when it is reached by the photon shell. The sum is over showstheinitialdistributionoftheLorentzfactor. all indices i larger than i , which corresponds to the two m merged layers. For theaverage Lorentz factor Γ¯ >100 used here it appears that the wind is transparent to t∼he emitted Figure 2.Physicalparameters ofthe elementaryshocks. Upper panel: (thick line) emission time te and (thin line) duration ∆t photonsexcept for a few early collisions. ofthesignal(intheobserverframe)bothasafunctionofarrival Thearrivaltimeofeachoftheelementarycontributions timeta.Lowerpanel:(thickline)energyedissipatedintheshock frominternalshocksiscalculatedrelativelytoasignalwhich in arbitrary unit and (thin line) Lorentz factor Γr at the shock would have travelled at the speed of light from the source location. to theobserver. It is given by r t =t , (9) 2.2 Emission processes a e− c The process by which the dissipated energy is finally ra- whereteistheemissiontimeandrthedistancetothesource diated depends on the energy distribution of protons and of the two merged layers. The evolution of the system is electrons in the shocked material and on the values of the followed until all the layers are ordered with Γ decreasing comoving density and magnetic field. The average energy fromthefronttothebackofthewind.Theefficiencyofthe which is dissipated per proton in a shock between two lay- dissipation process can then be obtained as ers of equal mass is given by e fd = Pmscs2Γ , (10) ǫ=(Γint−1)mpc2 i i i 1 Γ 1/2 Γ 1/2 wherePthees aretheenergiesreleasedineachoftheinternal with Γint = 2(cid:20)(cid:16)Γ21(cid:17) +(cid:16)Γ12(cid:17) (cid:21) , (11) elementaryshocksandthem ,Γ aretheinitialmassesand i i where Γ is the Lorentz factor for internal motions in the int Lorentz factors of the layers. shocked material. With Γ /Γ =4, which corresponds to a 1 2 This very simple approach is naturally very crude be- mildly relativistic shock (v /c=0.88), ǫ>200 MeV. rel causeit neglects allpressurewavespropagating throughout A mechanism which could directly ex∼tract the energy the layers. Nevertheless, we expect that it can still capture from the protons has been proposed by Paczyn´ski & Xu the basic features of the real process. As a first example (1994) sincethevalueofǫislargeenoughforpionsproduc- we have represented in Fig.1 the evolution of the Lorentz tioninppcollisions.Thepionsthendecaywiththeemission factor at different times when the initial distribution of Γ of gamma-rays. The global efficiency of this mechanism is consists of 5000 layers (i.e. tw = 10 s) with Γ(n) = 400 for low, of theorder of 10−3 for Γ /Γ =4. 1 2 n=1 to 4000 and Γ(n)=100 for n=4001 to 5000 (n=1 Another possibility, considered by Rees & M´esz´aros for the last emitted layer). Such a wind is then divided be- (1994),Papathanassiou&M´esz´aros(1996)andSari&Piran tween a “slow” part which is emitted first and a rapid part (1997a), consists to suppose that the electrons have come which will progressively collide against the slow part. The into (at least partial) equipartition with the protons. If a total masses injected in theslow and rapid parts are equal. fraction α of the dissipated energy goes to the electrons e After the first collision has occurred a separation layer of theircharacteristic Lorentzfactor will be increasingmassisformedbetweentheslowandrapidparts. ǫ We have plotted in Fig.2 the values of te, Γr, ∆t and e for Γe≃αemec2 , (12) the elementary shocks as a function of arrival time t . The a evolutionoftheshellsystemisessentiallycompletedaftera which,forαe =1/3(correspondingtoacompleteequiparti- time t t Γ¯2, where t is the characteristic time scale tionbetweenprotons,electronsandthemagneticfield)and e var var for the ∼variations of the Lorentz factor. As expected, the Γ1/Γ2 =4yieldsΓe 150.Theequipartitionmagneticfield ∼ plots of Γ , ∆t and e have two branches, corresponding to is given by r collisions takingplaceon bothsidesoftheseparation layer. B (8πα nǫ)1/2 , (13) eq B They mimic the forward and reverse shocks which, in our ≃ simple model, are located on a single surface. whereαB <1andnisthecomovingprotonnumberdensity ∼ The efficiency fd of the dissipation process is rather M˙ E˙ low, 10% in this specific case, typically less than 15%. This n≃ 4πr2Γ¯m c ≃ 4πr2Γ¯2m c3 . (14) is a severe problem, which can become even worse if only a p p fraction of the dissipated energy is radiated in the gamma- Assuming E˙ = 1052 erg.s−1, Γ¯ = 300, t = 1 s and α = var B ray range. This issue is examined in thenext section. 1/3 the equipartition magnetic field at a radius r ct Γ¯2 var ∼ 4 F. Daigne and R.Mochkovitch wheremostofthecollisionstakeplaceisB (102 103)G consider the scattering of electrons by turbulent magnetic eq ∼ − dependingon theratio Γ /Γ . field fluctuations. They get 1 2 Synchrotronemissionbytheacceleratedelectronsinthe 1/(3−µ) α ǫ magnetic field occurs at a typical energy (in the observer Γ M , (24) frame) e∼(cid:20)(cid:18) ζ (cid:19)(cid:16)mec2(cid:17)(cid:21) Γ B Γ 2 whereαM isthefractionofthedissipatedenergywhichgoes Esyn =50 30r0 1000G 10e0 eV. (15) into magnetic fluctuations, ζ the fraction of the electrons (cid:16) (cid:17)(cid:16) (cid:17)(cid:16) (cid:17) which are accelerated and µ the index of the fluctuation which corresponds to the UV range for Γe = 100. Gamma- spectrum. With 1.5 µ 2, α = 0.1 1, ζ 10−3 and M rayscanbeproducedbyinverseComptonscatteringonthe ǫ/m c2 500 (for Γ≤/Γ≤= 4) values o−f Γ in∼the range e 1 2 e synchrotron photons. Then 103 10∼4 can beobtained. − Γ B Γ 4 A fraction of the synchrotron photons will be shifted EIC≃EsynΓ2e =500 30r0 1000G 10e0 keV, (16) toeven higherenergy byinverseCompton scattering which (cid:16) (cid:17)(cid:16) (cid:17)(cid:16) (cid:17) now occurs in thelimit where andthefraction ofthetotalpowerwhich isradiatedbythe inverseCompton process is w= ΓeEs0yn 33 B Γe 3 , (25) mec2 ≃ (cid:16)1000G(cid:17)(cid:16)104(cid:17) τ Γ2 αIC = 1+⋆τeΓ2 , (17) is large, Es0yn = Esyn/Γr being the synchrotron energy in ⋆ e thecomovingframe.TheinverseComptonphotonshavean where energy τ⋆ = κ4TπMr⋆2⋆ , (18) EI0C≃Γemec2 =5(cid:16)1Γ0e4(cid:17) GeV, (26) is the optical depth of the shell of mass M⋆ and radius r⋆ in the comoving frame and carry a fraction whichcontainstherelativisticelectrons. AnestimateofM ⋆ τ Γ2/w is α = ⋆ e , (27) IC 1+τ Γ2/w t ⋆ e M = syn M˙ , (19) ⋆ 1+τ Γ2 shock of the dissipated energy. The optical depth τ⋆ has to be ⋆ e computedwiththeKlein-Nishinacross-sectionwhich,inthe where limit w 1, gives Γ −1 B −2 ≫ tsyn =6(cid:16)10e0(cid:17) (cid:16)1000G(cid:17) s, (20) τ⋆ ≃(cid:20)4πκr⋆T2(M1˙+shoτc⋆kΓts2ey/nw)(cid:21)(cid:16)83w(cid:17)[1+ln(2w)] , (28) is the synchrotron time of the relativistic electrons and M˙ the mass flow rate across the shock, both in the co- and therefore shock movingframeoftheshockedmaterial.Sincetheshockmoves τ Γ2 τ Γ2 E˙ with a Lorentz factor Γ Γ¯, M˙ can be approximated ⋆ e 1+ ⋆ e 810−4[1+ln(2w)] s ≃ shock w (cid:18) w (cid:19)≃ (cid:18)1052 erg.s−1(cid:19)× by t −2 Γ¯ −6 B −4 Γ −5 M˙ var e . (29) M˙shock ≃ Γ¯ . (21) (cid:16)1s(cid:17) (cid:18)300(cid:19) (cid:16)1000G(cid:17) (cid:16)104(cid:17) Theverylargeexponentswhichappearsin(29)showthata From (18), (19) and (21) we obtain an implicit expression for τ Γ2 small variation of the parameters can induce large changes ⋆ e in the relative importance of the synchrotron and inverse τ Γ2 = κTM˙shocktsynΓ2e . (22) Comptonprocesses.Inpractice,wefindthatαIC isgeneral- ⋆ e 4πr2(1+τ Γ2) ly small in the early part of a burst but increases at later ⋆ ⋆ e times essentially due to the reduction of the equipartition With r ct Γ¯2, relations (20) and (21) for t and M˙ ,⋆(1∼3) avnadr (14) for B and n, Eq. 22 givsyens (for magnetic field in shocks at large distances from the source shock eq (see Sect. 3.1 below). κ =0.2 cm2.g−1) T To be efficient theemission process must also occur on τ Γ2(1+τ Γ2) 0.3αe , (23) a time scale tem shorter than the shell expansion time tex ⋆ e ⋆ e ≃ αB (Eq.2). Using r ctvarΓ¯2, this condition becomes ∼ wlahrgicehr,frfoarctαioen∼ofαtBheydieisldsispaτt⋆eΓd2ee≃ne0rg.2y4caanndbαeIcCon≃ve0r.t1e9d.tAo 1+tsyQn <Γ¯tvar , (30) IC gamma-rays if the magnetic field does not reach equiparti- where Q =τ Γ2 (resp. τ Γ2/w) for w 1 (resp. w 1). tion. With for example αe = 100, τ Γ2 rises to 5 and α IC ⋆ e ⋆ e ≪ ≫ αB ⋆ e IC With expressions (11), (13), (14) and (20) above, (30) can to0.83.Smallervaluesofα couldstillincreaseα butthe B IC bewritten energy of theinverseCompton photonswould then become too small. E˙ −1 Conversely,iftheelectronLorentzfactorislargeenough 210−4α−B1(Γint−1)−1(cid:18)1052 erg.s−1(cid:19) × gamma-rays can bedirectly producedby synchrotronemis- sion. This will be the case if, instead of (12) one uses the tvar Γ¯ 5 Γe −1 <1+Q . (31) expression of Γe given by Bykov & M´esz´aros (1996) who (cid:16)1s(cid:17)(cid:18)300(cid:19) (cid:16)104(cid:17) IC Gamma-ray bursts from internal shocks in a relativistic wind 5 It is more easily satisfied when gamma-rays directly come Figure 3. Physical parameters governing the emission mecha- efrvoemn fsoyrncΓhrotr1o0n2etmhiesseimonisssiionncetitmheenreΓmea∼ins10s4m.aHlleorwetvhearn, nism:(a)fractionαsyn=1−αIC oftheenergywhichisradiated e ∼ bythesynchrotronprocess;(b)LorentzfactorΓeoftherelativis- the expansion time as long as the shocks are sufficiently ticelectrons;(c)equipartitionmagneticfieldBeq;(d)synchrotron strong (Γ1/Γ2 >2). energyEsyn.In(a), (b)and(d)thethicklinescorrespondtoΓe Some resu∼lts of a model where GRBs are produced by givenbyEq.24andthethinlinestoΓe=104. the inverse Compton process have been already presented elsewhere (Mochkovitch & Fuchs, 1996). In this study we limitourselvestosynchrotronemissionmodelsexceptwhen Figure4.BurstprofilesfortheinitialdistributionoftheLorentz we discuss the optical properties which strongly differ be- factor shown in Fig.1. The count rate C2+3 (in arbitrary unit) is given in the interval 50 – 300 keV, corresponding to BATSE tween the two cases. In synchrotron emission models the bands 2 and 3; (a) profile obtained with Eq. 24 for the electron total efficiency for the conversion of wind kinetic energy to Lorentz factor; (b) same as (a) with C2+3 in logarithmic scale gamma-rays below a few MeV is given by whichillustratestheexponentialdecayaftermaximum;(c)profile f =f α (1 α ). (32) obtainedwithaconstantΓe=104;(d)sameas(c)withC2+3 in tot d× e − IC logarithmicscale. where α is the fraction of the dissipated energy which is e transferredtotheelectrons.AccordingtoBykov&M´esz´aros 13 with α = 1/3), the electron Lorentz factor Γ (Eq. 24 B e e(1n9e9rg6y) αwehiicshcoisminpiatriaablllyeitnojetchteedfriancttoiomnaαgMne∼tic0fl.1u−ctu1aotfiotnhse. with αζM =1000 and µ=1.75) and the synchrotron energy E (Eq.15).ThesequantitiesarerepresentedinFig.3asa syn The total efficiency therefore does not exceed a few per- functionofarrivaltimet .AsinFig.2therearetwobranches a cent which imposes severe constraints on the energy source corresponding to the forward and reverse shocks. Typical or/and system geometry. values are 102 104 G for the magnetic field, 2 20 103 Theenergyavailablefromdiscaccretiontoablackhole − − × for theelectron Lorentz factor and 10 keV – 1 MeV for the synchrotron energy. 1 M E ≃ 6MDc2 =3 1053(cid:18)M⊙D(cid:19) erg, (33) Thephotonsproducedintheelementaryshocksaredis- tributed according to a synchrotron spectrum where M is the disc mass, can be less than 1053 erg for D −x dn(E) e E the coalescence of two neutron stars since numerical simu- , (34) lations indicate that MD 0.2 0.3 M⊙ (Rasio & Shapiro dE ∝ Esyn (cid:18)Esyn(cid:19) ∼ − 1994; Davies et al. 1994; Ruffert, Janka & Schaefer 1996). with x = 2/3 for E < E , 2 < x < 3 for E > E and Theconversionofdiscgravitationalenergyintowindkinetic syn syn where e is the energy which is dissipated in the shock (Eq. energy should then be very efficient in order to account for 5).TheresultingburstprofileisshowninFig.4aforahighe- cosmological GRBs. Naturally, if the wind is beamed in a nergyindexx=2.5.Cosmological effects(timedilationand solidangleδΩalongthesystemaxistheenergyrequirement redshift) have been taken into account, assuming that the issmallerbyafactor δΩ butonehasnowtofaceastatistical problem since even op2tπimistic estimates of theneutron star burst is located at z = 0.5. The duration t90 which, accor- ding to the definition used for the BATSE data is the time mergingratearenotconsiderablylargerthantheburstrate duringwhich90% ofthetotalfluenceisreceived(excluding (Phinney 1991; Narayan, Piran & Shemi 1991; Tutukov & thefirstandlast5%)is10.04s,verysimilartotheduration Yungelson1993; Lipounovet al.1995).Thesituation isless ofwindemissiont .Thedecayaftermaximumisclosetoan critical if the disc results from the disruption of a neutron w exponentialascanbeseenintheplotofthelogarithmofthe starbyablackholeorthecollapseofamassivestarbecause countrateversustimeinFig.4bwherethereisaquasilinear themassofthedisccanbeMD >1M⊙ (andevenMD >10 declineaftermaximumbetweent 12sandt 20s.How- M⊙). ∼ ∼ evertherisetimeinourprofileisn≃otmuchsho≃rterthanthe decaytimeandtheburstisthereforenotverydissymmetric if we except the low intensity exponential tail after t = 15 3 TEMPORAL PROPERTIES s. We found that a profile much closer to the characteris- 3.1 Burst profiles ticFRED(FastRiseExponentialDecay)shapeobservedin manybursts(orinindividualpulsesinsideacomplexburst) We first study the temporal properties of our burst models canbeobtainedbyassumingthattheelectronLorentzfactor when the Lorentz factor in the wind has the simple shape variesmoreslowlywithǫ(thedissipatedenergyperproton) shown in Fig.1. We inject an energy E˙ = 1052 erg.s−1.sr−1 thanapower-lawofindex1/(3 µ)(Eq.24).Thiswouldbe 4π − which, for an efficiency of conversion into gamma-rays of the case if the fraction ζ of accelerated electrons increased a few percent, yields a total energy E 1051 erg.sr−1 for with ǫ. Such a behavior is observed in simulations of col- ≃ 4π a burst lasting a few seconds. We obtain the burst profiles lisionless non relativistic shocks (Bykov, private communi- (in numberofphotonspersecond between 50and 300 keV, cation) and we have supposed that it remains valid in the whichcorrespondstoBATSEbands2and3)byaddingthe relativisticlimit.Moreover,wehavemadethesimplechoice contributions of all the internal elementary shocks which ζ ǫ which leads to a characteristic Lorentz factor for the ∝ occurduringtheexpansionofthewindasexplainedinSect. electronswhichisindependentofǫ.Thesynchrotronenergy 2. For each shock we compute α = 1 α (Eq. 27) intheelementary shockshasbeenrepresentedin Fig.3d for syn IC − whereα isthefractionoftheenergywhichgoestoinverse a constant Γ = 104 and the corresponding burst profile is IC e Comptonphotons,theequipartitionmagneticfieldB (Eq. shown in Fig.4c. It has now a typical FRED shape with a eq 6 F. Daigne and R.Mochkovitch the initial distribution of the Lorentz factor is made of a Figure 5. Full line: ratio of the decay time to the rise time of rapid component (with an average value of Γ of a few hun- burstprofilesobtainedwithaninitialdistributionoftheLorentz dreds) and of some slower layers (with Γ 100). The total factor homothetic to that shown in Fig.1 but with different du- ≃ rations. For t90 <1.65s the profiles decay faster than they rise. mass in the slow layers has to be comparable to the mass Dashed line: the discontinuity in the initial distribution of the in the rapid component in order to keep the efficiency at a Lorentz factor has been replaced by a smoother transition (Eq. reasonablelevel.Afewexamplesofsyntheticprofilesarepre- 36). Now, only bursts with t90 < 0.65 s decay faster than they sentedinFig.6.Itcanbeseenthatagreatdiversityofburst rise. shapes can be obtained if the distribution of the Lorentz factor in thewind varies from one eventto theother. Figure 6. Burst profiles for three initial distributions of the Lorentz factor in the wind. In all three cases a rapid compo- 3.2 Short timescale variability nentwithΓ=400isdeceleratedbyaseriesofslowerlayers.The masses in the rapid component and slower layers are compara- The profiles shown in Fig.4 and 6 have a satisfactory gene- ble. (a) Relatively simple profile with four layers which produce ralshape butdonotexhibit anyvariability on a short time four intensity pulses two of which partially overlap; (b) and (c) scale. Rapid temporal variations cannot result from small morecomplexprofileswith15slowlayers.Noticethatthedistri- irregularities of the emitting surface since photons coming butions of the Lorentz factor in (b) and (c) are homothetic, (c) beingsimplytentimeslongerthan(b). from many different regions in space and time are received atasametimebytheobserverwhichleadstoalossofcohe- rence of the temporal variations (Woods & Loeb 1995; Sari Figure7.SameasFig.6witharandomfluctuation(ofmaximum amplitude ± 20%) added to the average value Γ = 400 of the &Piran1997b,c).Instead,onehastoconsideragainthatthe rapidcomponent of the wind. Theresultingprofiles now exhibit Lorentz factor itself can fluctuate at the millisecond level. variabilityonashorttimescale. This is not unrealistic if the flow at the origin of the wind is veryirregular andturbulentsince one millisecond corres- ratio of the decay time to the rise time ττdr = 3.4 where τr pondstothetypicaldynamicaltimescale of adisc orbiting and τ are definedrespectively by a stellar mass black hole. Nothing being known about the d temporalspectrumofthefluctuationswehavesimplyadded τ =t t , τ =t t , (35) r max− 5% d 95%− max arandomfluctuationtotheLorentzfactorofeachofthelay- wheret isthetimeofmaximumcountrateandt (resp. ersinitially injected (every2 ms) in thewind. Theadopted max 5% t ) the time when 5% (resp. 95%) of the total fluence amplitude for these fluctuations is 10 – 20% of the average 95% has been received. We have tested with our model the ten- value of Γ. The resulting profiles represented in Fig.7 now dency for short bursts (or short pulses within a complex show the rapid temporal variations which are seen in most burst) to become more symmetric (Norris et al. 1996). We observed bursts. have computed the profiles obtained when the duration of wind emission t is varied while the initial distribution of w the Lorentz factor remains homothetic to a given shape for 4 SPECTRAL PROPERTIES whichwechoosetheonerepresentedinFig.1wherethewind 4.1 Burst spectrum consists of a slow part (Γ = 100) followed by a rapid part (Γ = 400) both containing the same mass. The results (for The overall burst spectrum is the sum of all the elemen- z=0.5) areshowninFig.5 wherewehaveplottedtheratio tary contributions (Eq. 34) from the internal shocks. The τd as a function of t . It appears that τd decreases from spectrum corresponding to the profile shown in Fig.7a is τr 90 τr about 3 when t 10 s to about 0.3 when t < 0.5 s represented in Fig.8 (again a cosmological redshift z = 0.5 90 90 with τd = 1 for t ≃ 1.65 s. We therefore reprod∼uce the has been assumed). Its shape can be easily understood: let observτerd behavior9b0u≃t the effect is even exagerated for the Esmyinn and Esmyanx be theminimum and maximum of thesyn- shortestpulseswhichdecayfasterthantheyrise.Webelieve chrotron energy for the whole set of elementary shocks. As that this might be a consequence of the crude assumptions long as E is smaller (resp. larger) than Esmyinn (resp. Esmyanx) madeinoursimplemodelandweexpectthatmoredetailed the number of photons n(E) per unit energy interval is a hydrodynamical simulations (Daigne and Mochkovitch, in power-law of index 2/3 (resp. 2.5) and in the interme- − − preparation)couldhelptoimprovetheprofiles.Already,we diate region the current index evolves from 2/3 to 2.5 − − found that better results can be obtained (with τd = 1 for as E goes beyond the value of Esyn for a growing number τr t 0.65 s) if the discontinuity between the two extreme of elementary shocks. In the four BATSE bands, between 90 ≃ valuesofΓ isreplaced byasmoother transition oftheform 20keVanda fewMeV, thespectrum canbewell described with Band’s formula (Band et al. 1993) Γ(t/t )=250+150cos[2.5π(t/t 0.6)], (36) w w − E α E tfohre0e.ffi6c≤ientc/ytwf≤o1f tahneddΓiss=ipa4t0i0onfoprrot/ctewss<is0t.h6e.nHroewdeuvceerd, n(E)=A(cid:16)100keV(cid:17) exp(cid:16)−E0(cid:17) for (α−β)E0≥E , d by nearly a factor of 2. (α β)E α−β E β n(E)=A − 0 exp(β α) Norrisetal.(1996)haveshownthatinmostcasescom- (cid:20) 100keV (cid:21) − (cid:16)100keV(cid:17) plex bursts can be analyzed in terms of a series of (possi- for (α β)E E . (37) bly overlapping) pulses. In the same way we build complex − 0 ≤ bursts with our model by the addition of intensity pulses The parameters α, β and E have been adjusted to ob- 0 formed in the deceleration of rapid parts of the wind by tain the best possible fit of the spectrum in Fig.8. We get sloweroneswhichwereemittedpreviously.Wesupposethat α = 1.33, β = 2.31 and E = 544 keV, in agreement 0 − − Gamma-ray bursts from internal shocks in a relativistic wind 7 Figure 8.Spectrum of theburstcorrespondingtotheprofileof Figure9.Duration-hardnessratio(HR32)relationsforasimple Fig.7a.Thenumberofphotonsperenergyintervaln(E)andthe (onepulse)burstandacomplexburstwithfivepulses.Therela- productE2n(E)areshowninarbitraryunits.Thedashedlineis tions are shown for three values of the high energy index x=2, afitofthespectrum withBand’sformulaintheinterval10keV 2.5 and 3 of the elementary synchrotron spectrum and two red- –10MeV.TheproductE2n(E)ismaximumatthepeakenergy shiftsz=0.3and1.Itcanbeseenthattheeffectoftheredshift Ep=365keV. isnegligible. Figure 10. Evolution of hardness with time: the instantaneous with typical values found in observed bursts. The product value of the peak energy Ep (thick line) is represented together E2n(E) is also shown in Fig.8. It is maximum at the peak withtheburstprofile(thinline).Upperpanel:forthesimpleone energy Ep = 365 keV where the bulk of the emission takes pulseburstofFig.4c.Lowerpanel:forthemorecomplexburstof place. Fig.6a. laterpulsestendtobesofterthanearlierpulses,evenifthey 4.2 Duration-hardness ratio relation havea greater intensity. Shorterburstsareexpectedtobeharderinourmodel.Inter- We have compared the spectral evolution of our syn- nalshocksareformed atan approximateradiusr ctvarΓ¯2 theticburstmodelstotheseobservationalresults.Thehard- ∼ andarethereforeclosertothesourceiftheburstevolveson ness can be obtained as a function of time through the es- a short time scale. Assuming that the injected power E˙ is timation of the instantaneous value of E , the energy of p independentofthedurationofwindinjectiontwtheequipar- the peak of E2n(E). We first considered the simple burst titionmagneticfieldisstrongerandthesynchrotronenergy of Fig.4c which has a characteristic FRED shape and a du- is larger in the dissipation region for shorter bursts. To ob- ration t = 10.23 s. The hardness and count rate evolve 90 taintheduration-hardnessrelationwecomputethehardness similarly (see Fig.10) but their maxima are separated by ratioHR32(definedastheratioofthenumberofphotonsre- a time lag of 0.89 s, the maximum of Ep occurring before ceived in BATSE band 3 to that in band 2) for burstswith that of the count rate. In complex bursts the hardness in- homothetic initial distributions of the Lorentz factor but creases during intensity pulses and also precedes the count different valuesof tw.Fora given duration t90 thehardness rate (Fig.10). ratio is a function of the high energy index of the elemen- The correlation between spectral hardness and count tary spectrum (Eq. 34) (theburstsbecoming softer when x rateisthereforecorrectlyreproducedinourmodels.Ween- increasesfrom2to3),ofthedetailedhistoryoftheLorentz counteredmoredifficultieswith theglobalhard-to-soft evo- factor and of the cosmological redshift z. Complex bursts lution which was observed in 70% of the sample of bright tend to be harder because the distribution of the Lorentz longburstsstudiedbyFordetal.(1995).Insyntheticbursts factor varies on a shorter time scale at given tw and dissi- (Fig.10)itispresentaslongastheprofilesremainrelatively pation therefore begins earlier than in more regular bursts simple (i.e. dominated by one main pulse or made of just a ofsameduration.Finallycosmological effectsshiftburstsin few pulses) while in complex bursts with many pulses only theHR32–t90diagramdownward(redshift)andtotheright the correlation between hardness and count rate is clearly (time dilation). However the shift is nearly colinear to the visible. Also, the hardness of successive pulses remains es- duration-hardness relation which therefore remains practi- sentiallycorrelatedtotheintensityinsteadofdecreasinglike cally unchanged at different z. This is illustrated in Fig.9 in 50% of theFord et al. (1995) sample. where the duration-hardness relation has been represented for several values of x and z and for a simple (single pulse) and a complex burst. In agreement with the observations 4.3.2 Pulse shape as a function of energy (Kouveliotou et al. 1993; Dezalay et al. 1996) a transition occurs at t 2 s. The shortest burstsreach a limit Whenobservedinspectralbandsofincreasingenergy,pulses 90 ∼ becomenarrowerasshown byNorrisetal. (1996) whoana- 3001/3 1001/3 HR32 ≃ 1001/3− 501/3 ≃2.1, (38) lTyhzeedy faoulanrdgethnautmtbheeriro(fhpaulflsmesaixnimtuhme )fowuridBthATcaSnEbbeanwdesll. − whenE islargeenoughforbothBATSEbands2and3tolie represented by a power-law p in the region where n(E) E−2/3. The longer bursts tend ∝ W(E) E−0.4 . (39) to various limiting values of the hardness ratio depending ∝ on thechoice made for the high energy index x. The same relation was obtained by Fenimore et al. (1995) who used the autocorrelation of averaged burst profiles in- 4.3 Spectral evolution steadofindividualpulses.Inoursyntheticburstmodelsthe pulse width also decreases at high energy. Figure 11 repre- 4.3.1 Instantaneous hardness sents a single pulse burst in the four BATSE bands. The The spectral evolution of GRBs shows a few trends which width can be fitted by power-laws such (39) but not with arefollowed byamajorityofburstsbutalsosuffersomeex- a unique exponent p = 0.4 for all the pulses. We indeed − ceptions and are therefore not universal (Bhat et al. 1994; get p 0.4 for pulses of 2 – 10 s but for pulses of 0.1 – ≃ − Fordetal.1995).First,spectralhardnessandcountrateap- 1 s, p> 0.2. Norris et al. (1996) obtained p= 0.4 as an peartobecorrelated. Withinintensitypulsesbothincrease averag∼eo−nacollectionofpulseswithdurationran−gingfrom and decrease together, the hardness usually preceding the 0.1 to10 sbutdo not providethevalueof pfor theshorter count rate. Anothertrend is a global hard-to-soft evolution andlongerpulsesseparatelywhichdoesnotallowadetailed overthecourseoftheburstoutsideintensitypulses.Finally, comparison with ourtheoretical results. 8 F. Daigne and R.Mochkovitch Figure 11. Upper panel: normalized profiles for the burst of Fig.4c now represented in all four BATSE bands. Lower panel: Figure12.Ep-fluencerelation(thickline)fortheburstofFig.6a. The peak energy is represented in logarithmic scale to show the halfmaximumwidthsoftheprofilesasafunctionofenergy.The fitofthemodelresults(dashedline)hasaslopep=−0.39. sectionoflineardeclinefollowingthemomentofmaximumcount rateinthefourintensitypulses.Thephotonfluencecorresponds totheintegratedphotonfluxinBATSEbands2and3.Thethin lineshowsthecountrateinthesamebands. represented in Fig.8 which corresponds to a burst with 4.4 The E - fluence relation p t = 10.55 s we get a V magnitude of 18.4 for F 90 γ Liang&Kargatis(1996)discoveredinasampleof37BATSE 210−6 erg.cm−2. A larger fluence would naturally produc≃e burstsanexponentialdependenceofthepeakenergyE on a brighter counterpart, possibly up to V 15 16 for photon fluence Fph during the declining part of intenpsity Fγ > 2 10−5 erg.cm−2. An optical emission∼in th−is lumi- pulses i.e. nosi∼tyrange should bedetectablebythenextgeneration of counterpartsearchinstruments,suchastheTAROTproject E exp( aF ). (40) p ∝ − ph (Boer 1997). The photon fluence is defined as the integral of the photon InX-raysthecalculatedfluxesfortheBeppoSAXWide fluxfromthebeginningoftheburstandaistheslopeofthe Field Camera are in reasonable agreement with the obser- LogE –F relation.Incomplexburststheslopestaysap- vations. In the case of GRB 970228 the X and gamma-ray p ph proximately constant from one pulse to another. Synthetic fluencesduringthefirst100sareFX(2 10keV) 1.210−6 bursts also follow a relation such (40) but the slope in suc- erg.cm−2 andFγ(40 700keV) 1.11−0−5 erg.cm≃−2 (Costa − ≃ cessive pulses can somewhat vary, especially if they have etal.1997a).Forthesamegamma-rayfluencethemodelpre- very different intensity or duration (see Fig.12). dicts FXmodel ≃0.7−1.110−6 erg.cm−2 depending on burst hardness(the upperand lower limits of Fmodel respectively X correspond to burstswith E =100 and 350 keV). 4.5 X-ray and optical counterparts p In GRB 970508 the peak X-ray flux and the gamma- The X-ray and optical counterparts recently discovered in ray fluence are (2 10 keV) 1.2 10−8 erg.cm−2.s−1 X two GRBs have been interpreted in the context of cosmo- and F (50 30F0 keV−) 1.1 1≃0−6 erg.cm−2 (Costa et γ − ≃ logical models as the emission coming from a relativistic al. 1997b; Kouveliotou et al. 1997). The calculated X-ray shell expanding in the interstellar medium (Wijers, Rees & flux for a burst of comparable hardness, model 8 10−9 M´esz´aros 1997; Vietri 1997; Waxman, 1997a,b). If however erg.cm−2.s−1,isclosetotheobservedvalueFXwhilet≃heinitial GRBs are produced by internal shocks in a wind, X-ray to X-ray emission predicted by afterglow models appears to optical photons should also be emitted together with the be an order of magnitude weaker (Sahu et al. 1997b). This gamma-raysbeforetheafterglowresultingfromtheinterac- can be considered as an indication that both the gamma- tion with theinterstellar medium. raysandtheinitialX-rayemissionareproducedbyinternal To compute these early counterparts we consider a ty- shocks. pical burst where an energy E = 21052 erg.sr−1 has been The evolution of the afterglow which follows the emis- 4π injected into the wind with a 5% efficiency for the conver- sion from internal shocks can be obtained from a solution sion to gamma-rays between 50 keV and 300 keV. For a of therelativistic Sedov problem (M´esz´aros & Rees 1997b). GRB located at 2 Gpc (z 0.5) the observed fluence in SuchasolutionapplieswhentheexpandingshellofmassM BATSE bands 2 and 3 will b∼e Fγ ≃210−6 erg.cm−2. This has swept up a mass MISM∼ MΓ in the interstellar medium value of F is then used to normalize a synthetic spectrum which occurs after a deceleration time γ from which the expected flux in the X-rays and visible can t 180E1/3n−1/3Γ−8/3 s, (41) be finally obtained. In practice these fluxes are highly vari- dec≃ 52 1 2 able (as the gamma-rays) and have been averaged over the where E is the shell energy in units of 1052 erg.sr−1, n 52 4π 1 duration t90 of the burst. the density of the interstellar medium in atom.cm−3 and We did not consider in this paper inverse Compton Γ = Γ .Most oftheenergyisradiated atthesynchrotron 2 100 emission models but we now briefly discuss their opti- frequency of therelativistic electrons cal properties since they greatly differ from synchrotron ν 31016α2α E1/2t−3/2 Hz, (42) emission models. When the gamma-rays come from in- s≃ e B 52 day verse Compton scattering, a fraction αsyn ∼ 1 − αIC of where αe and αB are the fractions of the dissipated energy the total power is emitted at a typical synchrotron energy which go to the electrons and magnetic field respectively. Ep/Γ2e ∼ 10−100 eV where Ep is the peak energy of the Thefluxatthesynchrotronfrequencyisgivenby(Waxman, gamma-ray spectrum. The fraction αsyn is fixed by the ra- 1997a,b). tio α /α through Eq. (17) and (23). Preliminary results e B 6.510−26√n α E D−2 erg.cm−2.s−1.Hz−1 , (43) indicate that for αe/αB = 1 the emission in the visible of Fs ≃ 1 B 52 Gpc a burst of fluence Fγ and duration t90 = 10 s could be as and for ν >νs by bright as V =5 6 which is already excluded by the limit − ν −β set by the ETC and GROCSE instruments (Krimm, Van- = , (44) derspek&Ricker1996; Leeetal.1997). Withα /α =100 Fν Fνs(cid:16)νs(cid:17) e B thepredictedmagnitudebecomesV =8 9still withinthe where β = q−1, q being the exponent of the power-law dis- reach of ETC and GROCSE. − tribution of t2he accelerated electrons (N(Γ ) Γ−q). The e ∝ e In synchrotron emission models the optical counter- fluxinthevisibledecreasesfromamaximum max fol- part is much weaker. Taking for example the spectrum lowingapower-lawofindex −3β thevalueofFβV( 0≃.6Fs0.8) 2 ∼ − Gamma-ray bursts from internal shocks in a relativistic wind 9 beingobtainedfromtheobservations.ForGRB970508afit 4) Syntheticspectra can be fitted with Band’s formula ofthedataalsoprovides max 610−28erg.cm−2.s−1.Hz−1 with parameters α, β and E comparable to those of ob- FV ≃ 0 (Sahu et al. 1997b). With thegamma-ray fluencemeasured served bursts. for this burst and assuming a 5% efficiency for the conver- 5) The duration-hardness relation is a natural conse- sion of wind kinetic energy togamma-rays we obtain quenceofthemodelsinceinshort burstsdissipation occurs closer to the source where the magnetic field and the syn- √n α 310−2 . (45) 1 B ≃ chrotron energy are larger. If we adopt this value as typical we find that the afterglow 6) Spectral hardness and count rate are correlated du- of the burst (given in example above) with E = 2 and ring burst evolution, the hardness generally preceding the 52 D = 2 has a V magnitude of 18.9 before the phase of count rate. Gpc power-law decline.IninverseCompton emission models the 7) The pulse width decreases with increasing energy initial optical counterpart produced by internal shocks is following a power law W(E) E−p with p 0.4 for pulses ∝ ∼ considerablybrighterthantheafterglow butinsynchrotron of duration 2 – 10 s. emission models the two contributions have comparable 8) In the declining part of intensity pulses the peak brightness. However, the optical signal will present a short energy Ep decreases exponentially with photon fluence. interruption if the duration of theburst is smaller than the Some other properties of observed GRBs are however deceleration time.Conversely,ift >t thetwocontribu- not so well reproduced byour model: w dec tionsoverlapandformacontinuous∼signal.Inanycase,ade- 1) The shortest pulses tend to decay faster than they tailed photometric follow-up in the optical range beginning rise instead of being symmetric. beforetheendofthegamma-rayburstwould certainlypro- 2) Synthetic bursts do not show a global hard to soft videcrucial informations about theemission mechanisms. evolution as frequently as real bursts. 3) The slope of the E – fluence relation can differ p among pulses inside thesame burst. It is not yet clear whether these difficulties represent 5 CONCLUSIONS real problems for our model or are simply a consequence Wehavedevelopedasimplemodeltocomputethetemporal of some of the crude assumptions we have made. In parti- andspectralpropertiesofGRBsundertheassumptionthat cular, the evolution of the wind should be followed with a they originate from internal shocks in a relativistic wind. detailedhydrodynamicalcode(DaigneandMochkovitch,in Wehavenotdiscussedthecriticalpointofhowsuchawind preparation)ratherthanwiththesimplemethodusedhere. could form but the recent observations of optical counter- A more fundamental issue may be the rather low effi- parts for GRB 970228 and GRB 970508 seem to indicate ciency for the conversion of wind kinetic energy to gamma- that a relativistic shell was indeed present in theseobjects. rays. If GRBs result from the coalescence of neutron stars The distribution of the Lorentz factor in the wind has thewindcannotbestronglybeamed sincethemergingrate noreasontobeuniformandvariationsonseveraltimescales isnotverymuchgreaterthantheburstrate.Alargefraction (downtoonemillisecond whichcorrespondstothedynami- oftheenergyreleasedinthecoalescenceshouldthereforebe cal timescale of a relativistic disc orbiting a stellar mass injected intothe wind. black hole) can be expected. Layers of different velocities We believe that the present work has shown that if a will collide and form internal shocks within the relativistic relativistic wind carrying enough energy can be produced, windandtheenergydissipatedintheseshockscanbeemit- theinternalshock modelappearsasaconvincingcandidate ted in the form of gamma-rays. We have followed the evo- to explain GRBs. 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