MNRAS000,1–13(2015) PreprintJune24,2016 CompiledusingMNRASLATEXstylefilev3.0 Light curves and spectra from off-axis gamma-ray bursts O. S. Salafia1,3(cid:63), G. Ghisellini3, A. Pescalli2,3, G. Ghirlanda3, F. Nappo2,3 1Universita`degliStudidiMilano-Bicocca,PiazzadellaScienza3,I-20126Milano,Italy 2Universita`degliStudidell’Insubria,ViaValleggio,11,I-22100Como,Italy 3INAF-OsservatorioAstronomicodiBreraMerate,viaE.Bianchi46,I–23807Merate,Italy 6 Draftversion,June24,2016 1 0 2 ABSTRACT If gamma-ray burst prompt emission originates at a typical radius, and if material pro- n ducing the emission moves at relativistic speed, then the variability of the resulting light u curve depends on the viewing angle. This is due to the fact that the pulse evolution time J scale is Doppler contracted, while the pulse separation is not. For off-axis viewing angles 3 θ (cid:38) θ +Γ−1,thepulsebroadeningsignificantlysmearsoutthelightcurvevariability. 2 view jet This is largely independent of geometry and emission processes. To explore a specific case, ] wesetupasimplemodelofasinglepulseundertheassumptionthatthepulseriseanddecay E aredominatedbytheshellcurvatureeffect.Weshowthatsuchapulseobservedoff-axisis(i) H broader,(ii)softerand(iii)displaysadifferenthardness-intensitycorrelationwithrespectto . thesamepulseseenon-axis.Foreachoftheseeffects,weprovideanintuitivephysicalexpla- h nation.Wethenshowhowasyntheticlightcurvemadebyasuperpositionofpulseschanges p withincreasingviewingangle.Wefindthatahighlyvariablelightcurve,(asseenon-axis)be- - o comessmoothandapparentlysingle-pulsed(whenseenoff-axis)becauseofpulseoverlap.To r testtherelevanceofthisfact,weestimatethefractionofoff-axisgamma-rayburstsdetectable t s by Swift as afunction of redshift, finding that a sizablefraction (between 10%and 80%) of a nearby (z < 0.1) bursts are observed with θ (cid:38) θ +Γ−1. Based on these results, we view jet [ arguethatlowluminosityGamma-RayBurstsareconsistentwithbeingordinaryburstsseen 3 off-axis. v Keywords: relativisticprocesses-gamma-rayburst:general-gamma-rayburst:individual 5 (GRB980425,GRB031203,GRB060218,GRB100316D)-methods:analytical 3 7 3 0 . 1 1 INTRODUCTION 2002;Hakkila&Preece2011;Luetal.2012;Basak&Rao2014) 0 performedcarefulanalysesoflightcurvesandtimeresolvedspec- Despite more than 40 years of observation and modelling, many 6 tralookingforpatternsandforhintsaboutsuchfundamentalbuild- features of Gamma-Ray Bursts (GRBs hereafter) still lack firm 1 ingblocks.Asearlyas1983,Golenetskiietal.foundevidenceof : andunanimousexplanations.ThediversityandcomplexityofGRB a correlation between spectral peak energy and photon flux dur- v prompt emission light curves is often used to illustrate the diffi- i ing the decay of pulses. Such correlation was later confirmed by X cultyintheclassificationofthesesourcesandintheunificationof Kargatisetal.(1994),Kargatis&Liang(1995)andBorgonovo& theirproperties.Anaturalapproachtogetinsightintosuchcom- r Ryde(2001)andbecameknownasthehardness-intensitycorrela- a plexityistolookforglobalandaverageproperties,likefluxtime tion(Ryde&Svensson1998).Norrisetal.(1986)waspresumably integral(i.e.fluence),totalduration,averagespectrum,peakflux. the first to systematically decompose the light curves into pulses Alternatively,onecantrytobreakdownthelightcurveintosim- andtolookforpatternsinthepropertiesoftheseputativebuilding plerpartsfollowingsomepattern.Ifafundamentalbuildingblock blocks.Someyearslater,Woods&Loeb(1999)developedtoolsto was identified, the analysis of single blocks could be the key to calculatetheemissionfromarelativisticallyexpandingjet,includ- theunificationanddisentanglementofpropertiesoftheunderlying ingthecaseofanoff-axisviewingangle.Ioka&Nakamura(2001) processes.Manyauthors(e.g.Imhofetal.1974;Golenetskiietal. tookadvantageofthisformulationtomodelthesinglepulse,find- 1983;Norrisetal.1986;Linketal.1993;Ford etal.1995;Kar- ingthatthespectrallag-luminosityandvariability-luminositycor- gatis & Liang 1995; Liang & Kargatis 1996; Preece et al. 1998; relations found by Norris et al. (2000) and Reichart et al. (2001) Ramirez-Ruiz&Fenimore1999;Leeetal.2000;Ghirlandaetal. canbeexplainedasviewingangleeffects.Thepulsemodelatthat stageassumedemissionfromauniqueradiusandfromaninfinites- imallyshorttimeinterval(i.e.adeltafunctioninradiusandtime). (cid:63) E-mail: omsharan.salafia@brera.inaf.it (OA Brera Merate), Inthefollowingyears,otherauthorsproposedincreasinglyrefined o.salafia@campus.unimib.it(Univ.Milano-Bicocca) (cid:13)c 2015TheAuthors 2 O.S.Salafia,G.Ghisellini,A.Pescalli,G.Ghirlanda,F.Nappo modelsofthepulse(e.g.Dermer2004;Genet&Granot2009),but R R neglectedthepossibilityforthejettobeobservedoff-axis. on off Theviewingangle,i.e.theanglebetweenthejetaxisandthe line of sight, is usually assumed to be smaller than the jet semi- aperture, in which case the jet is said to be on-axis. For larger viewingangles,i.e.foroff-axisjets,thefluxisseverelysuppressed becauseofrelativisticbeaming.Nevertheless,itcanbestillabove detection threshold if the viewing angle is not much larger than the jet semi-aperture, especially if the burst is at low redshift. In T Pescallietal.(2015)wehaveshownthatoff-axisjetsmightindeed Δ c dominatethelowluminosityendoftheobservedpopulation. The idea that nearby low luminosity GRBs could be off- axis events has been a subject of debate since the observation of GRB980425. Soderberg et al. (2004) rejected such possibil- on axis off axis ity,basedonradioobservationsofGRB980425andGRB031203, F but soon later Ramirez-Ruiz et al. (2005) presented an off-axis ΔT modelfortheafterglowofGRB031203whichseemstofitbetter theobservations(includingradio)withrespecttotheusualon-axis modelling.Usingthesameoff-axisafterglowmodel,Granotetal. overlap (2005)extendedtheargumenttotwoX-RayFlashes,thusincluding F ΔT 0 theminthecategoryofoff-axisGRBs.Basedonpromptemission properties,anoff-axisjetinterpretationofX-RayFlasheshadbeen F/b4 0 already proposed by Yamazaki et al. (2002) and Yamazaki et al. t (2003),followingtheworkbyIoka&Nakamura(2001).Ghisellini Δt bΔt 0 0 etal.(2006)arguedthattheoff-axisinterpretationofGRB031203 andGRB980425isnotpracticable,becausetheirtrueenergywould then be on the very high end of the distribution, implying a very Figure1.Upperleftpanel:twopointsources(blueandreddots)moveat lowlikelihoodwhencombinedwiththelowredshiftofthesetwo equalconstantspeedalongthezaxis,separatedbyadistanceβc∆T.Each events.Morerecently,theideathatsucheventsaremembersofa startsemittingatz=Ronandstopsemittingatz=Roff.Theblueandred separateclass(e.g.Liangetal.2007;Zhang2008;Heetal.2009; circlesrepresentwavefrontsoftheemittedlight.Thefirstbluewavefront andthefirstredwavefrontreachanyobserverwithatimedifference∆T. Bromberg et al. 2011; Nakar 2015) has gained popularity. Our results about the GRB luminosity function (Pescalli et al. 2015), Upperrightpanel:closeup.Dependingontheviewingangleθv,adistant observerseestheblueandredsignalseparated(θv < θov)oroverlapped though,stillpointtowardstheunificationoftheseeventswithor- (θv >θov).Theangleθovistheanglebetweenthezaxisandthenormal dinaryGRBsbasedontheoff-axisviewingangleargument.With toaplanetangenttoboththefirstredwavefrontandthelastbluewavefront. thepresentworkweaddresstheissuefromanotherpointofview, Lowerpanel:sketchofthebolometriclightcurveasseenbyon-axis(θv= byfocusingontheapparentlysinglepulsed,smoothbehaviourof 0)andoff-axis(θv>θov)observers.Lettingb=(1−βcosθv)/(1−β), promptemissionlightcurvesofthesebursts,tryingtofigureoutif thesinglepulsefluxasmeasuredbytheoff-axisobserverisdecreasedby suchbehaviourisexpectedinthecaseofanoff-axisviewingangle. afactorb4withrespecttotheon-axisone,whilethedurationisincreased Thestructureofthepaperisasfollows:insection§2weex- byafactorb.Thepulseseparation∆T,though,doesnotdependonthe plain why an off-axis GRB is always less variable than the same viewingangle,beingtheemissiontimedifferenceatafixedradius.This causesthepulsestooverlapasseenbytheoff-axisobserver. GRBseenon-axis;in§3wediscussthemainassumptionsofour simplepulsemodelandwepresentthepredictionsforanon-axis (§3.3)andanoff-axisobserver(§3.4).In§4webuildasuperposi- tionofpulses(torepresentasyntheticpromptemissionlightcurve) candependontheviewingangle.Toseethis,considertwopoint andshowhowitspropertieschangewithincreasingoff-axisview- sourcesmovingatequalconstantspeedβcalongthezaxis,sepa- ingangle,comparingthemwiththosefoundinlightcurvetimere- ratedbyadistanceβc∆T,asinFig.1.Eachsourcestartsemitting solvedspectralanalysis.Asexpected,themodelpredictsthatvari- atradiusR andstopsemittingatR .Anobserveralongthez on off abilityissuppressedinoff-axisGRBsbecauseofpulsebroadening axis (viewing angle θ = 0) sees two separated pulses of equal v andoverlap.In§5off-axisGRBsareshowntobeasignificantfrac- duration ∆t and peak flux F , the second starting a time ∆T 0 0 tionofnearbyobservedGRBs.Basedontheobtainedresults,we after the start of the first. Because of relativistic Doppler effect, conclude(§5.1)thatlightcurvesoflowluminosityGRBsarecon- an observer with another θ (cid:54)= 0 measures a lower (bolometric) v sistentwiththeoff-axishypothesis.Wethensummarizeanddraw peak flux F = F /b4 and a longer pulse duration ∆t = b∆t , 0 0 ourconclusionsin§6. where b = (1 − βcosθ )/(1 − β) is the ratio of the on-axis v relativistic Doppler factor δ(0) = Γ−1(1−β)−1 to the off-axis one δ(θ ) = Γ−1(1−βcosθ )−1 (Rybicki & Lightman 1979; v v 2 PULSES:BUILDINGBLOCKSOFGRBLIGHT Ghisellini 2013). The difference in pulse start times ∆T, on the CURVES otherhand,isnotaffectedbytheviewingangle,becausetheemis- sion of both pulses begins at the same radius: it can be thought 2.1 Pulseoverlapandlightcurvevariability of as emission from a source at rest (for what concerns arrival Inahighlyvariablelightcurve,pulsesmustbeshortandnotover- times). The pulses overlap if ∆t > ∆T, which corresponds to laptoomuch.Ifpulsesareproducedatatypicalradiusbymate- θ > θ ≈ Γ−1(cid:112)∆T/∆t −1.Considerthecaseinwhichthe v ov 0 rialmovingclosetothespeedoflight,thentheamountofoverlap pulse separation is equal to the pulse duration, i.e. ∆T = 2∆t 0 MNRAS000,1–13(2015) Singlepulsesfromoff-axisGRBs 3 7 scenario,discontinuousactivityinthecentralengineproducesase- 6 quenceofshellswithdifferentLorentzfactors.Whenfastershells 5 catchupwithslowerones,shocksdevelopandparticlesareheated. 4 on-axis Iftheplasmaisopticallythinandsomemagneticfieldispresent, 3 theenergygainedbytheelectronsispromptlyandefficientlyradi- 2 atedawaybysynchrotron(andinverseCompton)emission.Each 1 pulseisthustheresultofthemergeroftwoshellsbeyondthepho- 0 tospheric radius R . The strength of the shock, and thus the ef- F0 0.07 ficiency of the elecpthron heating, depends strongly on the relative / 0.06 F 0.05 Lorentzfactorofthemergingshells(aradiativeefficiencyofafew 00..0043 θv= 3Γ−1 ppearircsenwtiitshascmhiaelvlerdeloantilvyefLororΓernetlz(cid:38)fac3t,orLsazmzeartigeetlaatle.r19(t9h9e)y.Snheeeldl 0.02 p moretimetocatchupwitheachother),thusthehighestefficiency 0.01 is achieved for shells merging just after the photospheric radius. 0.00 0 20 40 60 80 100 Thisexplains,withinthisframework,whythetypicalpulsewidth t [s] isnotseentogrowwithtime:thebulkoftheemissionhappensat afixedradius,regardlessoftheexpansionofthejethead. Figure2.Examplelightcurvesconstructedbysuperpositionofpulses.All pulsesareequal.Thepulseshapeisadouble-sidedGaussian(Norrisetal. 2.3 Timescales 1996), which is a common phenomenological description of GRB pulse Threemaintimescalesariseintheinternalshockscenario: shapes.ThepeakfluxisF0,andtherisetodecaytimeratiois1:3.The start times of the pulses are the same for the two light curves and have • theelectroncoolingtimeτ ,i.e.thetimeneededbyelec- cool been sampled from a log-normal distribution with mean 20s and sigma tronstoradiateawaymostoftheenergygainedfromtheshock; 0.35dex.Pulsesinthelowerlightcurvearebroadenedbyafactorof4and • theangulartimescaleτ ,i.e.thedifferenceinarrivaltime theirfluxisloweredbyafactorof256withrespecttotheupperlightcurve, ang √ betweenphotonsemittedatdifferentlatitudes; whichcorrespondstotheeffectofanoff-axisviewingangleθv= 3Γ−1 • the shell crossing time τ , i.e. the time needed for the two asdiscussedinthetext. sc shellstomerge. andθ ≈Γ−1.Increasingtheviewingangle,theamountofpulse Theelectroncoolingtimescale,asmeasuredinthelabframe, ov is τ ∼ Γ−1γ/γ˙, where Γ is the bulk Lorentz factor, γ is the overlapincreases,reachinghalfofthepulsewidthassoonasb=4, cool √ whichcorrespondstoθ ≈ 3Γ−1.Withthisviewingangle,the typicalelectronLorentzfactorasmeasuredinthecomovingframe, v fluxofthesinglepulseisreducedbyb4 =256,butthefluxinthe andγ˙ isthecoolingrate.Forsynchrotronemission,itisoftheorder ofτ ∼10−7sfortypicalparameters1(Ghisellinietal.2000). overlappedregionishigherbyafactoroftwo,sothatthepeakflux cool Theangulartimescaleariseswhenonetakesintoaccountthe effectivelydecreasesby128. arrivaltimedifferenceofphotonsemittedatthesametimebyparts Thepurposeofthissimpleargumentistoshowthatifpulses of the shell at different latitudes. It is defined as the arrival time areproducedbymaterialmovingatrelativisticspeed,andifatypi- differencebetweenapairofphotons,oneemittedatzerolatitude calemissionradiusexists,thentheapparentvariabilityofthelight andtheotheratΓ−1 latitude.Givenatypicalphotosphericradius curvecanbesignificantlysmearedoutbypulseoverlapasseenby (Daigne & Mochkovitch 2002) R ∼ 1012 cm, this difference anoff-axisobserver(seealsoFig.2).Theviewingangleneededfor ph isτ ∼ R/Γ2c ≈ 3×10−3 sR /Γ2 (weadoptthenotation thistohappenisstillsmallenoughforthefluxnottobeheavilysup- ang 12 2 Q =Q/10xincgsunits). pressedbyrelativistic(de-)beaming.Onemayarguethattheprob- x The shell crossing time is τ ∼ w/c, where w is the typ- abilitytohaveaviewingangleintherightrangeforthistohappen sc ical shell width. Being linked to the central engine activity, one withoutfallingbelowthelimitingfluxoftheinstrumentisvanish- mayassumewtobeoftheorderofafewSchwarzschildradii.The inglysmall.Toaddressthispoint,in§5wegiveanestimateofthe Schwarzschildradiusofa5M blackholeisR ≈1.5×106cm, rateofsuchevents,showingthatasignificantfraction(∼40%)of (cid:12) s nearbybursts(z<0.1)arelikelyobservedwithθ >θ +Γ−1. thusanestimatemightbeτsc ∼ 5×10−5 sw6.Inthiscase,we v jet haveτ > τ ,i.e.theeffectofshellcurvaturedominatesover Beingbasedsolelyongeometryandrelativity,theabovear- ang sc (i.e. smears out) intrinsic luminosity variations due to shock dy- gumentdoesnotrelyonaspecificscenario,e.g.internalshocks. namics,whichtakeplaceovertheτ timescaleorless. Anymodelinwhichphotonsareproducedatatypicalradius,be- sc Temporal analysis of GRB light curves, though, along with ingthephotosphericradius(e.g.subphotosphericdissipationmod- simplemodellingofinternalshocks(Nakar&Piran2002b,a),seem elslikethosedescribedinRees&Me´sza´ros2005;Giannios2006; to indicate that the shell width must be comparable to the initial Beloborodov2010)orbeyond(e.g.magneticreconnectionmodels, shellseparation.Takingthetwoasequal,thetimeneededfortwo Lazarianetal.2003;Zhang&Yan2010)eventuallymusttakeinto shells to collide is the same as the shell crossing time, and thus accountthepulseoverlapasseenbyoff-axisobservers. the shell merger is completed within a doubling of the radius. In this case, the shell crossing time and the angular time scale are 2.2 Pulsesintheinternalshockscenario thesame(Piran2005).Thismeansthatdetailsofthepulseshape ThepulsewidthinGRBlightcurvesisroughlyconstantthrough- out the burst duration (Ramirez-Ruiz & Fenimore 1999). The in- 1 bytypicalparameterswemeanΓ=100,Γrel =afew,Urad =UB,a ternalshockscenario(Rees&Meszaros1994)providesanatural typicalsynchrotronfrequencyof1MeVandweassumeequipartition.See frameworkfortheunderstandingofthiskindofbehaviour.Inthis Ghisellinietal.(2000)andreferencesthereinforacompletetreatment. MNRAS000,1–13(2015) 4 O.S.Salafia,G.Ghisellini,A.Pescalli,G.Ghirlanda,F.Nappo andspectralevolutioncannotbeexplainedasjustbeingduetothe x x shell curvature effect. Indeed, discrepancies between predictions basedonshellcurvatureonlyandobservationshavebeenpointed R out(e.g.Dermer2004). Nevertheless, the description of the pulse in terms of shell θon to observer θon to observer curvaturequalitativelyreproducesthemainfeaturesofmanylong z R θoff z GRBpulses,namelythefastriseandslowerdecay,thehard-to-soft spectralevolution,andthepresenceofahardness-intensitycorrelation(Ryde&Petrosian2002).Forthisreason,sincewefocuson ct cT theeffectoftheviewingangleratherthanondetailsofthepulse, inwhatfollowswesetupasimplemodelofthepulsebasedonthe shellcurvatureeffectonly. Figure3. Aspherestartsemittingradiationatt = t0 andstopsatt = t0+T.Thelineofsightofadistantobserverisparalleltothezaxis.Left: atimet R−ct = Rcosθon; 3 PULSELIGHTCURVESANDTIMEDEPENDENT Right:laterwhent>T,theobserverhasstoppedreceivingradiationfrom SPECTRA theportionofthespherewithz > R−c(t−T) = Rcosθoff.Thus theeffectiveemittingsurfaceistheportionofthespherewithRcosθon< 3.1 Mainassumptions z toff.Thelineofsightisparal- appendix. leltothez-axis.ThespherestartedemittingwhenitsradiuswasR,and stoppedwhenitwasR+∆R. LettheradiusoftheshellbeRatthebeginningoftheemis- sionandR+∆Rattheendofit.ThebolometricfluxF(t)(specific fluxF (t))iscomputedbyintegrationoftheintensityI (specific ν axis(asinFig.3).Thefirstphotontoreachtheobserveristheone intensityI )overtheappropriateequalarrivaltimesurfaceS(t), ν emittedatt=t fromthetipofthesphereatz=R.Lett=0be 0 namely itsarrivaltimeasmeasuredbytheobserver.Aphotonemittedatthe (cid:90) sametimet=t byapointofthesurfaceatz=Rcosθ reaches F(t)= I(s)ds/r2 (1) 0 on theobserveratalatertimet = R(1−cosθ )/c.Thus,despite S(t) on thesurfaceturnedonallatthesametimet = t ,atagiventime 0 where r is the distance between the element ds of S(t) and the t the observer has received radiation only from the portion with observer. z/R > cosθ = 1−ct/R(leftpanelofFig.3).Thiscanbevi- on Assuming isotropic emission in the comoving frame, in the sualizedaseachpointonthespherebeingturnedonbythepassage approximation of infinitesimal shell thickness, the intensity is re- ofaplanetravelinginthe−zdirectionwithspeedc,startingfrom latedtothecomovingone(primedquantitiesthroughoutthepaper z = Ratt = 0.Thesamereasoningappliestotheturning-offof will always refer to the comoving frame) through I = δ4I(cid:48) (or thesphere:eachpointisturnedoffbythepassageofaplanetravel- I (ν) = δ3I(cid:48)(ν/δ)forthespecificintensity).Notethatthecon- ν ν inginthe−zdirectionwithspeedc,startingfromz=Ratt=T. stantluminosityassumptionimpliesI(cid:48) ∝ R−2:thisisconsistent Asaresult,ifT t )itdecreasesas 0.0 peak 0.0 0.5 1.0 1.5 2.0 2.5 3.0 t/τ F(t) =(cid:18)1+ t−tpeak(cid:19)−3−(cid:18)1+ t(cid:19)−3 (3) F τ +t τ max peak When the finite jet opening angle θ is taken into account, jet Figure6. Bolometriclightcurvesofthreepulsesfromanon-axisjetwith thelightcurveisgiveninsteadbyEqs.A14&A15. θjet =1/Γ.Theratioof∆RtoRisreportedneareachcurve.Theblack dashedlinerepresentsthetimetjetatwhichthejetborderfirstcomesinto sight.Inthiscasetjetequalsτ. 3.3.1 Spectraandhardness-intensitycorrelation Since we are mainly interested in how the peak of the observed isexpanding,itssurface“runsafter”theemittedphotons,causing spectrum evolves with time, we assume a simple form of the co- thearrivaltimedifferencebetweenthefirstandthelastphotonto movingspectralshape,namely Rcwoh(ntet)rr,eatchΓte.I=anrrpi(va1ratl−ictiumβlae2r),d−fiof1fr/e2trheienscfiterhsietsaLtnoodffrel=ansttzTpf(ha1oc−ttoonrβo)emf=tihtTtee/de(xf1rpo+amnβszi)oΓ=n2. Iν(cid:48)(ν(cid:48))=n(a,b)νI00(cid:48)(cid:48) (cid:34)(cid:18)νν0(cid:48)(cid:48)(cid:19)−a+(cid:18)νν0(cid:48)(cid:48)(cid:19)−b(cid:35)−1 (4) Forthisreason,theangleuptowhichtheobserverseesthesurface wheren(a,b)isanormalizationconstantwhichdependsuponthe turnedoffisgivenbycosθ =1−c(t−t )/(R+∆R). off off highandlowspectralindicesaandb;clearlyI(cid:48) ∝ν(cid:48)aforν(cid:48) (cid:28)ν(cid:48) The resulting geometry is not spherical (see Fig. 4), but the ν 0 andI(cid:48) ∝ν(cid:48)bforν(cid:48) (cid:29)ν(cid:48).Ifa>0andb<−1,thenormalization assumptionofconstantluminositygreatlysimplifiesthemathemat- ν 0 n(a,b)canbedefinedsothat icaltreatmentinthatitallowsonetoperformallintegrationsover angularcoordinatesonly. I(cid:48) =(cid:90) ∞I(cid:48)(ν(cid:48))dν(cid:48) (5) 0 ν 0 3.3 On-axisjet Thebreakfrequencyν0(cid:48) isrelatedtothecomovingν(cid:48)Fν(cid:48) peaken- ergyE(cid:48) through peak A radially expanding (homogeneous) jet seen on-axis is indistin- guishablefromanexpandingsphereaslongasitshalfopeningan- E(cid:48) =(cid:18)−a+1(cid:19)a−1b hν(cid:48) (6) gleθjetismuchlargerthan1/Γ(e.g.Rhoads1997).Asacorollary, peak b+1 0 MNRAS000,1–13(2015) 6 O.S.Salafia,G.Ghisellini,A.Pescalli,G.Ghirlanda,F.Nappo 1 x jet x Fmax R 10-1 F/ θon to observer θon to observer max 0 2 4 6 8 θv z θoff z ν)/F10-2 t/τ θ A A νF( actual emitting ct surface 10-3 ctoff Figure9. Theoff-axisjetcanbethoughtofasbeingpartofanexpanding 10-4 sphere.Forsimplicity,theEATSoftheexpandingsphere(hatchedarea)is 1 10 100 1000 representedasinFigure3,butitisactuallythesameasinFigure4.The E/E' actualEATSofthejetistheinterceptionbetweenthejetsurfaceandthe peak sphereEATS. Figure7. Spectraatdifferenttimesofapulsefromajetseenon-axis,with Γ=100and∆R=R.ThecomovingspectralshapeisgiveninEq.4.The (ii) Pulsedecay:afterthepulsepeak,thetipofthejetturnsoff, colouredcirclesintheinsetshowatwhichpointinthepulseeachspectrum causingE todrop.Atthistimethevisiblepartofthejetisan peak (identifiedbythecolour)wascalculated. annulus(seeFig.4):thespectralpeakisdeterminedmainlybythe maximumDopplerfactorδ (t) = Γ−1[1−βcosθ (t)]−1 of max off thevisiblearea,whichcorrespondstotheinnermostcircleofthe 2.4 annulus,sothatE ∼δ E(cid:48) .ThefluxF inturndecreases peak max peak approximatelyasδ4 timestheangularsizeoftheannulus.The max 2.2 max latterisproportionaltocosθoff −cosθon,whichcanbeshownto )k F/F be /E'peakpea 2 0 2 t /4τ 6 8 cosθoff −cosθon ≈ ∆RRβΓδ1max (7) E 1.8 Asaresult,wehavethatF ∝δ3 ,whichexplainswhyE ∝ g( max peak o F1/3. L 1.6 3.4 Off-axisjet 1.4 -3 -2.5 -2 -1.5 -1 -0.5 0 Alsotheoff-axisjet(θv >θjet)canbetreatedusingtheformalism introduced by Woods & Loeb (1999). In the appendix (§A3) we Log(F/F ) max giveanalternativederivationbasedontheideathattheoff-axisjet can be thought of as being part of an expanding sphere, and that Figure8. Peakoftheobservedspectrumversusthebolometricflux,fora wecanworkouttheproperEATSastheintersectionbetweenthe pulsewithΓ=100and∆R=R.Aclearhardness-intensitycorrelation expandingsphereEATSandthejetsurface. ispresent.Theslopeoftheblackdashedlineis1/3.Theinsetisthesame asinFig.7. 3.4.1 Alongerpulseduration If the jet is off-axis, relativistic beaming of the emitted radiation where h is Planck’s constant. All the examples in this paper will causesboththefluxandE tobemuchlowerthantheon-axis assume the above comoving spectral shape, with a = 0.2 and peak counterparts. For the same reason, the duration of the pulse be- b = −1.3,whichrepresentaveragehighandlowspectralindices comeslonger.Thiscanbeunderstoodintuitivelyasfollows:asin ofFermiGRBspectra(Navaetal.2011). the on-axis case, the jet surface is not seen to turn on all at the Figure7showsspectrafromanon-axisjetatsixrepresentative sametime,butprogressivelyfromthenearest-to-the-observerpoint times,computedusingEq.A32.Theevolutionisclearlyhard-to- (pointAinFig.9)downtothefarthest.Thesameholdsfortheturn- soft(i.e.thepeakenergydecreasesmonotonicallywithtime),and ingoff.ThuspointAisthefirsttoturnon,andalsothefirsttoturn thelowandhighenergyspectralindicesarethesameasthoseofthe off.Asaconsequence,theeffectiveemittingareaincreasesaslong comovingspectrum.Figure8showsthatafterthepeakofthelight aspointAisseenemitting,thenitdecreases.Inotherwords,the curvethepeakenergyE oftheobservedνF spectrumvaries peak ν withthebolometricfluxF followingroughlyE ∝ F1/3,i.e. peaktimeequalstheemissiontimeofpointA,whichisgivenby peak themodelpredictsahardnessintensitycorrelationwithindex1/3 t (θ ,θ )=T[1−βcos(θ −θ )] (8) peak v jet v jet duringthedecayofthepulse.Letusinterprettheseresults: thusitsratiototheon-axispeaktimeis (i) Pulse rise: the maximum of E (t) is at the very begin- peak t (θ ,θ ) 1−βcos(θ −θ ) ningofthepulse,whenonlyasmallareapointingdirectlytowards peak v jet = v jet (9) t 1−β theobserver(the“tip”ofthejetatzerolatitude)isvisible.Asthe peak visibleareaincreases,lessbeamedcontributionsfrompartsatin- Figure10showsaplotofthisratioasafunctionofθ −θ for v jet creasinglatitudecomeintosight,reducingE slightly. differentvaluesofΓ.Theoff-axispulseisthusintrinsicallybroader peak MNRAS000,1–13(2015) Singlepulsesfromoff-axisGRBs 7 (iv) ifθ isonlyslightlylargerthanθ ,thefluxisdominated 105 v jet bythecontributionofthejetborder,whoseDopplerfactorisδ = B Γ−1[1−βcos(θ −θ )]−1,thusF ∝δ4 F∗; v jet p B 104 (v) asθ increasestowardsθ (cid:29)θ ,therelativedifferencein v v jet peak Dopplerfactorbetweendifferentpartsofthejetisreduced,andthe θ) / tjet 103 Γ = 300 flinugxlyciomnptroibrtuatniot.nTshoifscpoamrtspeonthsaetresthiannpathrtethbeorddee-rbebaemcoimngeoifntchreeajse-t (θ,peakv 102 Γ = 1Γ 0=0 50 bthoerdeeffre,cthtieveeeffmecitttibneginsgurmfaocreeapreraonisoulanrcgeedr.for larger jets, because t 10 Basedontheseconsiderations,anempiricalanalyticalformulacan beconstructedtodescribehowthepeakfluxdependsontheview- ingangleθ andonthejethalfopeningangleθ .Anexampleof v jet 1 suchanempiricalformulais 0.1 1 10 90 θv - θjet [deg] Fp/F∗ ≈ 11−Γ(θv−θj∗et)/2 θj∗et <θθvv (cid:54)(cid:54)θθjj∗eett oEFnaig-cauhxrciesur1pv0ue.lsreeRfpeaertsiaoktootafimdtheifefteporefefan-kat.xviTaslhupeeujolesftethhpaeelafLkoorpteiemnntieznfgtapcaetnaogkr,l(efθrvios,mθθjjΓeett=)=to505th◦toe.  21(cid:18)(1+δBβ)Γ(cid:19)(4−√2θj1e/t3) θv >θjet Γ=300withastepof50. (10) whereθ∗ = θ −1/Γ.Thedefinitionforθ∗ < θ (cid:54) θ is jet jet jet v jet justalineardecreasefromF∗toF∗/2;theexponentofδ inthe 100 B definitionforθ >θ is4reducedbyanamount2whichdepends 10-1 on θ , in ordevr to tjaetke into account the flux loss compensation jet 10-2 explainedinpoint(v)above. x a m 10-3 IntheAppendix(§A3)weshowthatthefluxattimetofthe F )/ 10-4 pulsefromanoff-axisjetisgivenbytheintegralinEq.A25,which θjet 10-5 howeverhasnoanalyticalsolutionforθv >0.Thecolouredsolid ,v linesinFigure11representF ascomputedbynumericalintegra- F(θpeak 1100--76 thiaolnfoopfeEnqin.gAa2n5glaets.tT=hetopreaankg(peθvd,aθsjheet)d,lfinoersfiavreejpeltostswoitfhEdqi.f1fe0refonrt 10-8 the corresponding parameter values, showing that the best agree- 10-9 mentisforhalfopeningangles5◦ (cid:46)θjet (cid:46)10◦. 10-10 3 5 10 15 40 θ [deg] v 3.4.3 Spectralpeakenergy,hardness-intensitycorrelation Withthesameassumptionsasintheon-axiscase,wecomputedthe Figure11. Peakfluxesofpulsesfromjetswithfourdifferenthalfopening spectrafromtheoff-axispulseatdifferenttimes.Thespectrumat angles,namelyθjet =3◦,5◦,10◦,and15◦(indicatedbythethinvertical eachtimeisdominatedbythepartoftheEATSwiththestrongest dottedlines),assumingΓ=100and∆R=R.Theorangedashedcurves beaming. At time t , such part is the border of the jet nearest peak representthecorrespondingempiricalparametrizationgiveninEq.10. totheobserver,thusoneexpectsE (t )todecreasewithθ peak peak v as the Doppler factor of the jet border, i.e. E (t ) ∝ δ . peak peak B Figure12isaplotofE (t )forthreevaluesofΓ,obtained peak peak thanitson-axiscounterpart.Theeffectivedurationasseenbythe byusingEq.A32tocomputethespectra,anditshowsthatindeed observer,though,dependsonthelimitingfluxandontheamount E isapproximatelyproportionaltoδ .Ingeneral,E isa peak B peak ofoverlapwithotherpulses. little lower than δ E(cid:48) because of the “blending in” of softer B peak spectrafromlessbeamedpartsofthejet. Figure13showstheevolutionofE asafunctionofthe 3.4.2 Alowerpeakflux peak fluxF duringthepulse,forfourdifferentoff-axisviewingangles. The decrease of the pulse peak flux F with increasing viewing A “hardness-intensity” correlation during the pulse decay is still p anglecanbeunderstoodasfollows: apparent,withaslightlysteeperslope(∼ 0.5)justafterthepulse peak,gettingshallowerasthefluxdecreasesandeventuallyreach- (i) whenthejetisobservedonaxis,thebulkofthefluxcomes ing∼1/3asintheon-axiscase. fromaringofangularradius1/Γcentredonthelineofsight.Let usindicatethispeakfluxwithF∗; (ii) aslongasθ < θ −1/Γ,wehavethatF isessentially v jet p equaltoF∗; 2 ThecoefficientandexponentofθjetinEq.10havebeenchosentoget (iii) ifθv =θjet,abouthalfoftheringisstillvisible,thusFpis agoodagreementwiththeresultsfromthesemi-analyticalformulationde- reducedtoaboutF∗/2; velopedintheAppendix. MNRAS000,1–13(2015) 8 O.S.Salafia,G.Ghisellini,A.Pescalli,G.Ghirlanda,F.Nappo 0 0.45 1 0.4 ]peak -0.5 Γ = 30 0.35 10-1 E' )/2Γpeak -1 max 0 .02.53 10-2 (tpeak -1.5 Γ = 100 F/F 0.2 10-3 E 0.15 g[ -2 10-4 Lo Γ = 300 0.1 0.1 1 10 -2.5 0.05 5 10 15 0 0 1 2 3 4 5 6 θ [deg] v t/τ Figure12. EpeakatthepulsepeaktimeforthreejetswithR=1013cm, Figure14. Lightcurvesofapulsefromajetwithθjet=5oandΓ=100. θjet =5◦andthreevaluesofΓ,namely(fromredtogreen)Γ=30,100 Eachcurvereferstoadifferentviewingangleinthesequence(fromgreen and300.TheblackdashedlinesareplotsofδB/2Γforthecorresponding tored)θv=5o,5.2o,5.4o,5.6o,5.8oand6o.Fmaxreferstotheon-axis valuesofΓ. jet.Theinsetshowsthesamecurvesplottedwithlogarithmicaxes. 2.5 suchsuperposition.Figure15showsfourlightcurvesofthesame series of N = 100 pulses seen at four different viewing angles. Allpulsesareequalindurationandpeakflux.Theirstartingtimes 2 havebeensampledfromauniformdistributionwithina2seconds E')peak 1.5 tcimmeanspd√a∆n.RTh=ejeRt.paTrhaemvetieewrsinargeaΓng=les10a0re,θθjvet==05,◦θ,jRet=+11/0Γ13, g(E/peak 1 0.5 slopes θsaatjem3t2e+amssb3ree/fsoΓorleaun(tEidoqnθ.j4feot).r+Tah2be/ertΓtee.sruTclhtoienmgcpolaimgrihostvocinnugwrvisetphsehacactrvtauelabslheGeaRnpeBbiinslingthehdet o L 0.5 F1/3 0.45 curves.Foreachlightcurve,theEpeakofthespectrumineachtime 0.4 binisalsogiven(thinorangehistograms).Thefollowingfeatures 0.35 shouldbeapparent: 0 (i) astheviewingangleincreases,variabilityissmearedoutby -7 -6 -5 -4 -3 -2 -1 0 thepulsebroadening; Log(F/F ) max (ii) theshapeoftheoveralllightcurvetendstoresemblea(long) singlepulsewhentheviewingangleislargeenough; (iii) thesuperpositionofpulsesmasksthehard-to-softspectral Figure13. LogarithmicplotofEpeakversusFluxofthesamepulseseen evolutionofthesinglepulses,turningitintoanintensitytracking atdifferentoff-axisviewingangles.Thejethasθjet = 5◦,Γ = 100and behaviour:thisisduetothesuperpositionofspectrawithdifferent R = 1013 cm.Thefourseriesofpoints(fromgreentored)correspond peakenergies; toθv = 5.1◦,5.5◦,6◦and7◦.Theinsetshowstheslopeoftherelation (iv) the variation of E leads slightly the variation in flux, duringthedecayofthepulseforeachviewingangle. peak becauseofthehard-to-softnatureofthesinglepulses; (v) thereisageneralsofteningofE intimeovertheentire peak 3.4.4 Lightcurves lightcurve. Figure14showsplotsofbolometriclightcurvesofthesamepulse Thesefeaturesarestrikinglysimilartothosefoundintimeresolved seen at different viewing angles, computed by numerical integra- spectralanalysisofrealGamma-RayBursts(e.g.Fordetal.1995; tion of Eq. A25 using a Runge-Kutta IV order scheme. Both the Ghirlandaetal.2002).Wedonotadvocatethisasaproofofthecor- peak flux decrease and the duration increase discussed in §3.4.1 rectnessofourmodel,whichiscertainlyoversimplified,butrather and§3.4.2areapparent.Theoverallshapeisqualitativelyinsensi- asafurtherindicationthatsomefeaturesofGRBlightcurvesmight tiveoftheviewingangle,apartfromthepeakbeingsharperinthe beexplainedadmittingthatthejetisseenatleastslightlyoff-axis. on-axiscase. Theoff-axisviewinganglefavoursthebroadeningandsuperposi- tionofpulses,whichisthenecessaryingredienttosomeofthefea- turesenumeratedabove.Itcanalsocontributeinasimplewayto explainwhytheslopeofthehardness-intensitycorrelationchanges 4 MULTI-PULSELIGHTCURVES frombursttoburst,beinginfluencedbytheviewingangle(§3.4.3). Nowthatwehaveadetailed(thoughsimple)modelofthesingle Figure15showsthatthesimpleargumentsoutlinedin§2.1are pulse,wecanproceedtoconstructa“synthetic”GRBlightcurve validnotonlyifpulsesareproducedbypointsources,butalsoin bysuperpositionofpulses.Somenon-trivialfeaturesemergefrom presenceofanextendedgeometry. MNRAS000,1–13(2015) Singlepulsesfromoff-axisGRBs 9 1.0 0.06 200 100 0.05 0.8 80 150 0.04 0.6 V] Lmax 0.03 60 [ke L/ 0.4 100 40 Epeak 0.02 0.2 0.01 20 50 0.0 0.00 00..00 00..55 11..00 11..55 22..00 22..55 00..00 00..55 11..00 11..55 22..00 22..55 0.010 0.006 50 40 0.005 35 0.008 40 30 0.004 L/Lmax00..000046 2300 00..000023 122505E[keV]peak 10 0.002 10 0.001 5 0.000 0.000 00..00 00..55 11..00 11..55 22..00 22..55 33..00 00..00 00..55 11..00 11..55 22..00 22..55 33..00 33..55 t [s] t [s] Figure15. Lightcurves(thickhistograms)andspectralpeakevolution(thinorangehistograms)ofasequenceof100pulsesfromajetwithθjet = 5o, Γ=100,R=1013cmand∆R=R.ThepeakoftheνFν comovingspectrumisEp(cid:48) =1keV.Thepulsestarttime√sarerandomlydistributedwithinthe first2softheobservertime.Eachpanelreferstoadifferentviewingangleinthesequenceθv=0,θjet+1/Γ,θjet+ 3/Γ,θjet+2/Γ(fromlefttoright, toptobottom).Lmaxreferstothepeakluminosityoftheon-axislightcurve. 5 THENUMBEROFOFF-AXISGRBS functionisindeedwelldescribedbyabrokenpowerlaw,withthe breakaroundtheaverageon-axisluminosity(Pescallietal.2015); We can obtain an estimate of the number of off-axis GRBs over (v) since the result is sensitive to the assumed typical Lorentz theobservedpopulationbythesimplifyingassumptionthatalljets factorΓandhalf-openingangleθ ,weexplorethecasesΓ=50, jet sharethesameintrinsicproperties,andthattheirfluxinanobserver 100and300,andθ =5◦and10◦. jet bandisuniquelydeterminedbytheviewingangleandtheredshift. WeassumethatthemajorityofGRBsareobservedon-axis,andwe We then define the effective luminosity L(θ ) following v choosethefollowingparametersinanattempttomatchtheaverage Eq.10,namely propertiesoftheon-axispopulation:  1 θ (cid:54)θ∗ (i) Epeak,0 = 560keV as the typical (on-axis, rest frame)  1−Γ(θv−θj∗et)/2 θj∗et (cid:54)θvv<θjjeett pFeearmkis/GpeBcMtralsaemnperlgey(cid:10),Ecpohbeoasske(cid:11)n∼to1m86aktcehVth(Neaavvaeeratgael.v2a0l1u1e)omfutlhtie- L(θv)=L0× 21(cid:18)(1+δBβ)Γ(cid:19)(cid:16)4−√2θj1e/t3(cid:17) θv >θjet pliedbyatypicalredshift(cid:104)1+z(cid:105)∼3; (ii) α=−0.86andβ =−2.3astypicallow-andhigh-energy (11) spectralindices(Navaetal.2011); (iii) Ψ(z)=(0.0157+0.118z)/(1+(z/3.23)4.66)asthered- withθj∗et =θjet−Γ−1,andtheeffectivepeakenergy shiftdistribution,i.e.theGRBformationrateasgiveninGhirlanda etal.(2015); E (θ )= Epeak,0 ×(cid:26) 1 θv (cid:54)θjet (12) (iv) L0 =2.5×1052ergs−1asthetypical(on-axis)luminosity, peak v 1+z (1+δBβ)Γ θv >θjet whichcorrespondstothebreakofthebrokenpowerlawluminosity function(modelwithnoredshiftevolution)oftheBAT6complete asin§3.4.3.Withtheseassumptionsandprescriptions,wecancom- sample (Salvaterra et al. 2012). This choice is motivated by the putetheobservedrateofGRBswithaviewingangleintherange fact that if GRBs can be observed off-axis, then their luminosity (θ ,θ +dθ ),intheredshiftrange(z,z+dz),assumingalimiting v v v MNRAS000,1–13(2015) 10 O.S.Salafia,G.Ghisellini,A.Pescalli,G.Ghirlanda,F.Nappo ulation of low luminosity Gamma-Ray Bursts exists, based on 0.9 somecommonfeaturesofGRB060218,GRB980425,GRB031203 ˙/N(p5e=a−0k,.2104.5=5p0h×5k6se1−0V0k15bce2maVenr−gd2s]−1 afsSohthemnfiirgendrolhceimoGked(teahhRaijsileoBgollvrih1tigesrte0unohyr0ctpts3ohtihcc1fpaub6eetnruhDqrvreute.suhsitTnvees(ailhteorwleaevwcsetneoeertnmevftlosuaeofaramvtbi“taiueinanbncrogieoallrossiumvtiwsyitoaney)ll,c,ru”oelamaufdnGndsesdaRehep,iilBnfaepatcsapt)lt(rhaoi,zeorewnbtnutU(cid:46)fltreyanovfwfism0eveier.nec1atgrtdhg)ssl,oee.eetTnlphiEoshouewpetlvsereseaairenredktyee--., ˙Noff 0.3 Γ=300,θjet=10◦ sults discussed in this paper suggest that the apparently peculiar 0.2 Γ=300,θjjeett==510◦ fineadtiucraetisoonftthhaetstehGeyRwBesrceaonbbseerivnetedrporfef-teadxiisn.sMteoadreaosvebre,iinngPjuesstcathllei 0.1 ◦ etal.(2015)wehaveshownthattheobservedrateoflowluminos- ity GRBs is consistent with what one would expect if they were 0.0 0.01 0.1 1 2 3 just ordinary bursts seen off-axis. Based on these considerations, z wearguethatthereisnoneedtoinvokeanewseparatepopulation oflowluminosityGRBs. Figure 16. Fraction of off-axis GRBs over the total within a given redshift.ThecurvesrepresentanestimateofthefractionofGRBswithred- shift lower than z observable by Swift/BAT (i.e. with photon flux p > 0.4phs−1cm−2inthe15–150keVband)whoseviewingangleislarger 6 DISCUSSIONANDCONCLUSIONS thanθjet+Γ−1. In this work we set up a simple physical model of a single GRB pulsebasedonshellcurvatureonly,asatooltoexploretheeffect photonfluxpliminagivenband,as oftheviewingangleonGRBlightcurves.Comparedtoothersim- dN˙ Ψ(z) dV ilar(andmorerefined)models(e.g.Dermer2004;Genet&Granot dθ dzdθvdz= 1+z dzP(θv,z,plim)dθvdz (13) 2009),ourmodelincludestheeffectofanoff-axisviewingangle. v Weshowthattheinclusionofsucheffectisimportantbecauseasig- whereP(θ ,z,p )istheviewingangleprobability,dV/dzisthe v lim nificantfraction(from10%upto80%)ofnearbybursts(z <0.1) differential comoving volume, and the factor 1+z accounts for are likely observed off-axis. Admittedly, the assumptions behind cosmologicaltimedilation.Theviewingangleprobabilityis the pulse model are at best a very rough approximation of real- P(θ ,z,p )=(cid:26) sinθv θv (cid:54)θv,lim(z,plim) (14) ity.Thegeneraltrendoftheeffectoftheviewingangle,though,is v lim 0 θv >θv,lim(z,plim) largelyinsensitiveofthesimplificationsadopted:aslightlyoff-axis viewingangleisenoughtoproduceasignificantpulsebroadening, Thelimitingviewingangleθ corresponds(throughEq.11)to v,lim withoutaffectingthepulseseparation.Thisinturnleadstopulse thelimitingluminosityL computedas lim overlap,whichsmearsoutvariabilityatallfrequencies,resultingin (cid:82)∞ dNEdE L =4πd2 p 0 dE (15) asmootherlightcurveandspectralevolution. lim L lim(cid:82)(1+z)Ehigh dN dE Thisismainlyaconsequenceoftwoassumptions:(i)thatthe (1+z)Elow dE emissionisisotropicinthecomovingframeand(ii)thatallpulses whereE (E )isthelower(upper)limitoftheobserverband, low high are produced around a typical radius. By relaxing (i), i.e. allow- d istheluminositydistance,anddN/dE(E ,α,β)istherest L peak ingforastronglyanisotropicemissioninthecomovingframe,one framespectrum. couldreduce(incasetheanisotropyfavoursforwardemission)or WedefinethetotalrateN˙ (θ +Γ−1 v jet Giventheaboveconsiderations,weconcludethat: atredshiftlowerthanz forvariouschoicesofΓandθ ,assum- jet ing a limiting flux plim = 0.4phs−1cm−2 in the 15-150keV (i) if the GRB jet is seen off-axis, single pulses appear longer band, to reproduce the Swift/BAT band and limiting flux. Stan- andtheirspectrumappearssofterthanintheon-axiscase; dardflatΛCDMcosmologywasassumed,withPlanckparameters (ii) ifaburstismadeupofasuperpositionofpulses,itsvari- H0 = 67.3kms−1Mpc−1 and Ωm,0 = 0.315 (Planck Collab- ability is smeared out by pulse broadening if the jet is observed oration2013).Theseresultsclearlyindicatethatatlowredshifta off-axis,withrespecttotheon-axiscase; significantfractionofGRBsislikelyseenoff-axis. (iii) ifsinglepulsesfeatureanintrinsichard-to-softspectralevo- lution,pulseoverlapcanturnitintoanintensity-trackingbehaviour. 5.1 LowluminosityGRBs Asdiscussedin§5.1,theresultssupporttheideathatprompt Recently, some authors (e.g. Liang et al. 2007; Zhang 2008; He emissionpropertiesofso-calledlowluminosityGRBscanbeinter- et al. 2009; Bromberg et al. 2011) argued that a unique pop- pretedasindicationsthattheyarejustordinaryburstsseenoff-axis. MNRAS000,1–13(2015)

Light curves and spectra from off-axis gamma-ray bursts

Similar Light curves and spectra from off-axis gamma-ray bursts