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

1 MB·

by  Om S. Salafia

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.salafi[email protected] (OA Brera Merate), Inthefollowingyears,otherauthorsproposedincreasinglyrefined
o.salafi[email protected](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-intensitycorre-
lation(Ryde&Petrosian2002).Forthisreason,sincewefocuson ct
cT
theeffectoftheviewingangleratherthanondetailsofthepulse,
inwhatfollowswesetupasimplemodelofthepulsebasedonthe
shellcurvatureeffectonly. Figure3. Aspherestartsemittingradiationatt = t0 andstopsatt =
t0+T.Thelineofsightofadistantobserverisparalleltothezaxis.Left:
atimet<T afterthearrivalofthefirstphoton,theobserverhasreceived
radiationfromtheportionofthespherewithz > R−ct = Rcosθon;
3 PULSELIGHTCURVESANDTIMEDEPENDENT Right:laterwhent>T,theobserverhasstoppedreceivingradiationfrom
SPECTRA theportionofthespherewithz > R−c(t−T) = Rcosθoff.Thus
theeffectiveemittingsurfaceistheportionofthespherewithRcosθon<
3.1 Mainassumptions
z<Rcosθoff.
Basedontheargumentsoutlinedin§2.3,weassumethatthevari-
ationofthefluxseenbytheobserverduringasinglepulseisdue
Front view Top view
onlytotheangulartimedelaydescribedabove.TheluminosityL
oftheshellisassumedconstantduringanemissiontimeT andzero ΔR ΔR
beforeandafterthistimeinterval.Theemittingregionisassumed R
R
geometrically and optically thin. The emitted spectrum, as mea-
y y
suredbyalocallycomovingobserver,isassumedtobethesame
θ θ
foranyshellfluidelement. on off
Woods&Loeb(1999)andotherauthorsalreadyprovidedthe
EATS
necessaryformulasforthecomputationofthepulseshapeinthis x z
case.Fortheeaseofthereader,andforthepurposeofdeveloping
anintuitivephysicaldescriptionoftheresults,though,wewillgo Figure 4. The shaded regions represent the equal arrival time surface
throughsomedetailsofthederivationanyway,hereafterandinthe (EATS)oftheexpandingsphereatt > 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 <R/c,atsometimettheobserverwill“see”only
ifthenumberofemittingparticlesisconstantdespitetheincrease
theportionofspherecomprisedbetweencosθ =1−ct/Rand
on
of the surface area with the expansion. It would not be appropri-
cosθ = 1−c(t−T)/R(rightpartofFig.3).Thus,theEATS
off
atee.g.forexternalshocks,wherethenumberofemittingparticles
attimetisthisportionofthesphere.
insteadincreaseswithincreasingsurfacearea.
3.2.2 Anexpandingsphere
3.2 Equalarrivaltimesurfaces
Ifthesphereisexpanding,theaboveargumentisstillvalid,with
3.2.1 Asphere
somemodification.TheradiusnowisR(t) = R+βc(t−t )so
0
Consider a sphere of radius R. The surface of the sphere starts thatthelightinguptakesplaceatR(0)=Randtheturning-offat
emittingelectromagneticradiationatt=t andstopssuddenlyat R(T) = R+βcT ≡ R+∆R.Sincethelightinguphappensall
0
t=t +T (asmeasuredintheinertialframeatrestwithrespectto atthesameradius,theangleθ uptowhichtheobserverseesthe
0 on
thecentreofthesphere).Emittedphotonsreachadistantobserver surfaceonisstillgivenbycosθ =1−ct/R(thephotonsemitted
on
atdifferentarrivaltimes.Letthelineofsightbeparalleltothez att=t allcomefromthespherewithradiusR).Sincethesphere
0
MNRAS000,1–13(2015)
Singlepulsesfromoff-axisGRBs 5
thesameholdsiftheviewingangleθ issmallenoughsothatthe
v
angulardistanceθ −θ ofthelineofsightfromthejetborder
1.0 jet v
is still much larger than 1/Γ. Since the typical expected Lorentz
0.8 1.5 factor of GRB jets is Γ ∼ 100, this means that a viewing angle
afew0.01radianssmallerthanθ allowsonetoconsiderthejet
jet
Fmax0.6 0.6 pifrathceticLaollryenotnz-faaxcitso.rOisnltohweoetnhoeurghhan(id.,e.ififthθejejtetisiscovmerpyanraabrrleoww,iothr
F/ 1/Γ),thenthefinitehalfopeninganglemustcomeintoplay.Fig-
0.4 0.3
ure5showslightcurves(Eqs.2&3)ofpulsesfromanexpanding
sphere,orequivalentlyfromajetwithθ (cid:29)1/Γ;Figure6instead
jet
0.2 showslightcurves(Eqs.A14&A15)ofpulsesfromanon-axisjet
withθ =1/Γ.
jet
0.0 Suchlightcurvescanbecomputedanalyticallywithintheas-
0.0 0.5 1.0 1.5 2.0 2.5 3.0
t/τ sumptions stated in §3.1. Some natural scales emerge during the
derivation:
(i) theangulartimescale
Figure 5. Bolometric light curves of three pulses from an expanding
R
sphere.ThefluxisnormalizedtoFmax andtheobservertimeisinunits τ ≡ βc(1+β)Γ2
ofτ (seethetextforthedefinitionofthesequantities).Theratioof∆Rto
Risgivenneareachcurve.Theblackdashedlinerepresentsthesaturation (ii) thepulsepeaktime
flux,whichisreachedifT (cid:29)R/c,orequivalentlyif∆R(cid:29)R.
∆R
t ≡
peak βc(1+β)Γ2
1.0 (iii) thepulsesaturationflux
2πR2(1+β)3
F ≡ Γ2I(cid:48)
0.8 max 3d2 β 0
whereI(cid:48) isthecomovingbolometricintensityanddisthedistance
Fmax0.6 tjet 1.5 oInftthheejsep0thferorimcatlhceaoseb,sebrevfeorr.ethepeak(t (cid:54) t ),thebolometric
/ peak
F0.4 0.3 fluxrisesas
F(t) (cid:18) t(cid:19)−3
0.2 0.6 =1− 1+ (2)
F τ
max
then(t>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(<z)tot 00000.....45678 ΓΓΓ===511000,00,,θjθeθtje=t=55◦◦ p[EL10>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 red-
shift.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˙ (<z)ofobservableGRBswithin
tot enhance (in case the anisotropy favours backwards emission) the
redshiftzastheintegralofEq.13overredshiftfrom0tozandover
flux received by off-axis observers. By relaxing (ii), on the other
θ from0toπ/2;similarly,therateN˙ (< z)ofoff-axisGRBs
v off hand,onemayhavethatthepulseseparationdependsontheview-
withinredshiftz istheintegraloverredshiftfrom0toz andover
ingangleaswell.Onewouldthenneedtoexplain,though,whythe
theviewinganglefromθ +Γ−1 toπ/2.Sinceweareinterested
v observedpulsewidthdistributiondoesnotvaryintime(Ramirez-
intheratioofthesetwoquantities,wedonotneedtobotherabout
Ruiz & Fenimore 1999; Piran 2005), despite the change of the
thenormalization.
emissionradius.
InFig.16weshowthefractionofburstswithθ >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)