Table Of ContentMon.Not.R.Astron.Soc.000,1–13(2002) Printed29January2013 (MNLATEXstylefilev2.2)
SHOCK DYNAMICS IN RELATIVISTIC JETS
J. Canto´1, S. Lizano2⋆, M. Fern´andez-L´opez1,3, R. F. Gonz´alez2,
3 and A. Herna´ndez-G´omez1,4
1
0 1Instituto de Astronom´ıa, UNAM, Apdo. Postal 70-264, 04510 M´exico D. F., M´exico
2 2Centro de Radioastronom´ıa y Astrof´ısica, UNAM, Apdo. Postal 3-72, Morelia, Michoac´an 58089, M´exico
3Department of Astronomy, Universityof Illinois, 1002 West Green Street, Urbana, IL 61801, USA
n 4Instituto de CienciasF´ısicas, UNAM, Apdo. Postal 48-3, Cuernavaca, Morelos 62210, M´exico
a
J
6
January26,2013
2
]
E ABSTRACT
H We present a formalism of the dynamics of internal shocks in relativistic jets where
the source has a time-dependent injection velocity and mass-loss rate. The variation
.
h of the injection velocity produces a two-shock wave structure, the working surface,
p thatmovesalongthejet.Thisnewformalismtakesintoaccountthefactthatmomen-
- tum conservation is not valid for relativistic flows where the relativistic mass lost by
o
radiation must be taken into account, in contrast to the classic regime. We find ana-
r
t lytic solutions for the working surface velocity and radiated energy for the particular
s
a case of a step function variability of the injection parameters. We model two cases: a
[ pulse of fast material and a pulse of slow material (with respect to the mean flow).
Applying these models to gamma ray burst light curves, one can determine the ratio
1
v of the Lorentz factors γ2/γ1 and the ratio of the mass-loss rates m˙ 2/m˙ 1 of the up-
streamanddownstreamflows.Asanexample,weapplythismodeltothesourcesGRB
4
0 080413BandGRB070318andfindthevaluesoftheseratios.AssumingaLorentzfac-
2 torγ1 =100,wefurther estimate jetmass-lossratesbetweenm˙ 1 ∼10−5−1M⊙yr−1.
6 We alsocalculatethe fractionofthe injected mass lostby radiation.For GRB 070318
. this fractionis ∼ 7 %. In contrast,for GRB 080413Bthis fractionis largerthan 50%;
1
in this case radiation losses clearly affect the dynamics of the internal shocks.
0
3
Key words: hydrodynamics–shockwaves–relativity– galaxies:jets –gamma-ray:
1
bursts
:
v
i
X
r
a 1 INTRODUCTION Several authors have studied internal shocks in ultra-
relativistic outflows to explain the observed variability
Collimated outflows (with jet-like geometry) moving at rel-
of GRBs (e.g., Mochkovitch, Maitia & Marques 1995;
ativisticspeedsarecharacteristicofactivegalacticnuclei.It
Kobayashi,Piran&Sari1997;Daigne&Mochkovitch1998;
is commonly accepted that an extragalactic jet is produced
2000). In these models the flow is represented by a succes-
intheneighborhood ofamassiveblackholeinthecenterof
sionofshellswithdifferentvaluesoftheLorentzfactor.This
anactivegalaxy(e.g.Rees1984; Istomin2010). Theserela-
models reproduce the burst profiles and their short-time
tivistic jets are subject to the development of shock waves.
scalevariability.Kobayashietal.pointedoutthatvariations
Rees (1978) proposed that the observed knots in the ex-
of the relativistic flow velocity are strongly correlated with
tragalactic jet M87 correspond to the locations of internal
the temporal variations observed in the GRBs. Daigne &
shockswhichariseowingtovariationsintheoutflowvelocity
Mochkovitch(1998)studiedthedetailedradiationprocesses
ofabeamgeneratedinthenucleus.Later,Rees&M´esz´aros
to calculate the fraction of the kinetic energy dissipated in
(1994) pointed out that fluctuations of the Lorentz factor
the shocks that can be emitted in the form of gamma rays
arounditsmeanvalueinarelativisticoutflow,thatgiverise
andobtainedthatthetotalefficiencyisoftheorderofonly
tointernalshocks,candissipateasubstantialfractionofthe
afewpercent.Inaddition,Spadaetal.(2001)proposedthat
outflow energy into non-thermal radiation. They proposed
theinternalshockscenario canalsobeusedforblazars.For
thatthismechanismisoperatingintheso-calledgamma-ray
comprehensive reviews of the physical processes and obser-
bursts(GRBs).
vations of GRBs see, e.g., M´esz´aros (2002); Piran (2004) ;
and Gehrels et al. (2009).
Using mass and momentum conservation, Cant´o et al.
⋆ E-mail:[email protected] (2000) solved the dynamics of internal shocks of non rela-
2 Cant´o et al.
WS
tivisticjetswithtimedependentinjectionvelocityandmass-
loss rate. Mendoza et al. (2009) used this momentum con-
serving formalism in the case of relativistic jets and com-
pared with observed light curves of GRBs assuming a sinu-
soidal velocity variation. However, momentum is not con-
served in relativistic flows because radiative losses change
therelativisticmasssincetheLorentzfactordecreaseswhen
energyisradiatedaway.Inthispaperwepresentanewfor-
malism that describes the dynamics of internal shocks in a β
relativisticjettakingintoaccountthemomentumchangeby WS
radiation. In particular, we study the dynamics of internal
shocks for the case where the injection velocity and mass-
β β
loss rate are both step functions of time, when a fast wind 2 1
reachesaslowerflow.Forthistypeofvariability,wefindan-
alytic solutions for the dynamical evolution and luminosity Figure1.Schematicdiagramshowingaworkingsurfaceformed
of the shocks, which are implicit functions of time. by the interaction between two relativistic flows. The upstream
The organization of the paper is as follows. In §2 we and downstream flow velocities are β2 and β1 (with β2 > β1),
discuss the relevant relativistic equations. In §3 we present respectively.Theworkingsurfacemoveswithanintermediateve-
the model, and investigate the dynamical evolution of the locityβws.
internalWS.Theluminosities predictedbyourmodel from
relativistic jets are given in §4. A comparison between our
above, a WS is formed when fast material overtakes slow
analytic solutions and observations of extragalactic gamma
material. This WS travels downstream the jet flow with a
ray bursts are presented in §5. In §6 we evaluate the mass
velocity βws(t), where the time t is measured in the source
lostbyradiationintheWS.Finally,in§7wesummarizeour
reference frame. The slow and the fast material just enter-
conclusions.
ing the WS at time t were ejected at times τ1 and τ2, re-
spectively, with corresponding downstream and upstream
velocities β1 = β(τ1) and β2 = β(τ2), and mass-loss rates,
2 RELATIVISTIC EQUATIONS FOR m˙1 = m˙(τ1) and m˙2 =m˙(τ2). The time dependence of the
INTERNAL SHOCKS velocities and mass-loss rates is given through the time de-
Forafree-streaming flow,thenon-dimensionalvelocityata pendencesτ1(t), and τ2(t).
In Appendix A we discuss a simple example of the in-
distancexfromthesource,attimetmeasuredatthesource
elasticcollision of3relativisticparticlesthatradiateenergy
reference frame, is given by
after they collide. This example shows that the momentum
x
β(x,t)=β(0,τ)= , (1) of the final particle is not conserved because its relativistic
c(t−τ)
mass changes when energy is radiated away. Therefore, we
where the injection point is at x = 0, and τ is the time at introduce below the energy and momentum equations that
which the flow was ejected, and, as usual, β = v/c, where takeinto account theenergy lost by radiation.
v and c are the speeds of of the flow and of light, respec- ThetotalenergydissipatedbytheflowinteractionE(t)
tively . If the flow velocity at the injection point increases, is given by the difference between the total energy injected
the fast material will reach the slower material and form a intotheWSandtheenergycarriedbytheWSattheinstant
working surface (WS) bounded by two shock fronts (Raga t 2,
etal.1990).TheseWSstructuresaretheso-called“internal E(t) τ2
shocks”.Figure1showsaschematicdiagramofaWSformed c2 =Z m˙(τ)γ(τ)dτ−mws(t)γws(t), (3)
when a fast upstream material with velocity β2 reaches a τ1
slowerdownstreammaterialwithvelocityβ1.TheWSmoves where γws(t)=1/ 1−βws(t)2, and the rest mass injected
with velocity βws intermediate between β1 and β2. intothe WS is p
The internal WS forms at the distance x from the
i τ2
source given by
mws(t)= m˙(τ)dτ. (4)
Z
x β(τ)2 τ1
i = , (2)
c (cid:20)dβ(τ)/dτ(cid:21) We assume that dissipated energy E(t) is completely
min
radiatedaway(seeeq.[A9]);noneofthisenergyisstoredin
where we wrote for simplicity, β(τ)=β(0,τ),and the min- internal degrees of freedom. Therefore, the luminosity (the
imum is taken over the time interval where the velocity radiated energy per unit time in the WS) is given by the
ienjeccrteeadseast. tTimheeτWi,Sboisthfovramlueedsgaitvetnimbeyteiqfuraotmionth(e2)m1.aTtehreinal, tthimeeWdSeriisvadteisvceriLberd(t)by=thdeEe(nte)/rgdyt.eTquhaetni,ont,he dynamics of
onecanobtainthetime,t ,atwhichtheWSisformedfrom
equaCtioonns(id1e).r a relativistiic jet with time dependent injec- ddt(mws(t)γws(t))= ddtZ τ2m˙(τ)γ(τ)dτ− Lrc2(t), (5)
τ1
tion velocity, β(τ), and mass-loss rate, m˙(τ). As discussed
2 For simplicity, the internal energy of the particles that enter
1 First,τi isobtainedbyminimizingtheRHSofthisequation. theWSisignored.
SHOCK DYNAMICS IN RELATIVISTIC JETS 3
I) II) 1 t/tIcI
β(τ) β(τ)
β
2
Case II
β β
2 2 β
WS
β β
1 1 Case I
β
0 ∆τ τ −∆τ 0 τ 1
1 t/tI
c
Figure2.Injectionvelocityβasfunctionoftimeτ:I)Theinitial Figure3. QualitativebehavioroftheWSvelocityasafunction
velocity β1 suddenlyincreases to β2 atatimeτ =0, forafinite of normalized time t/tIc,II, for Case I and Case II, respectively.
timeinterval,∆τ,andtheninstantlyreturnsbacktoitsoriginal Seetextfordescriptionofthisfigure.
value. II) The initial velocity β2 instantly decreases to β1 at a
timeτ =−∆τ,andatτ =0,thefasterflowstartstobeinjected
both the slow and fast flows and it moves at a constant ve-
again.
locity, intermediate between the fast and slow speeds. The
second stage begins when one of the flows has been com-
obtained from the derivative of equation (3), and the mo- pletely incorporated into theWS; then,the WS accelerates
mentum equation is given by ordeceleratesdependingonwhichflow(fastorslow)contin-
d d τ2 uestofeedtheremainingshock.Asymptoticallyintime,the
(mws(t)γws(t)βws(t)) = m˙(τ)γ(τ)β(τ)dτ speedoftheWSinthesecondstagetendstothevelocityof
dt dtZ
τ1 theremaining flow.
−Lrc2(t)βws(t), (6) locityFiagsuraef3unschtoiownsothfenoqrumalailtiazteidvetibmeehafvorioCraosfethIe(pWulSsevoe-f
wherethelasttermintheRHSisthemomentumchangedue fast material) and Case II (pulse of slow material). In Case
to the relativistic mass lost by radiation. Combining equa- I,thetimeisnormalized tothecriticaltimetIc (eq.[14]).In
tions (5) and (6) one obtains the equation for the velocity Case II, the time is normalized to the critical time tIcI (eq.
of the WS, [26]). In both cases, the constant velocity phase ends when
t/tI,II =1.
mws(t)γws(t)dβws = d τ2m˙(τ)γ(τ)β(τ)dτ c Consider thematerialat timetthatenterstheWS,lo-
dt dtZτ1 catedattheposition xws(t),throughbothshocks.Theslow
d τ2 and fast material were ejected at times τ1 and τ2, respec-
−βws(t)dtZ m˙(τ)γ(τ)dτ tively,such that, according toequation (1)
τ1
=m˙(τ2)γ(τ2)[β(τ2)−βws(t)]ddτt2 τ˜1,2 =t˜− βx˜1w,2s and dτ˜d1t˜,2 =1− ββ1w,s2, (8)
−m˙(τ1)γ(τ1)[β(τ1)−βws(t)]dτ1, (7) where τ˜1,2 =τ1,2/∆τ,t˜=t/∆τ,and x˜ws =xws/(c∆τ).
dt Now,forastepfunctionvariability,therestmassofthe
that does not depend explicitly on the radiated energy,Lr. WS,given byequation (4), is
Givenatimevariabilityofthefunctionsβ(τ)andm˙(τ),
0 τ2
one can find the WS velocity, βws(t), from equation (7) to- mws(t) = m˙1(τ)dτ+ m˙2(τ)dτ
gether with equation (1), the latter giving the relation be- Zτ1 Z0
tween τ1,2 and t. We will show below how these equations = −m˙1τ1(t)+m˙2τ2(t). (9)
canbesolvedanalytically forthecase ofstepfunction vari-
ability of the injection parameters. Note that τ1 < 0 and τ2 > 0, and for now on, for sim-
plicity,we will drop thet dependenceof thefunctions.
The velocity of the WS is obtained from equation (7)
3 STEP FUNCTION VARIABILITY OF THE dβws (β2−βws)2 (β1−βws)2
INJECTION PARAMETERS mwsγws dt =m˙2γ2 β2 −m˙1γ1 β1 , (10)
Considerastepfunctionvariabilityfortheinjectionvelocity whereweusedequation(8).Thisequationhastheconstant
and mass-loss rate such that, for τ < 0, a slow flow has velocity,βws0, solution 3, such that dβws0/dt=0, then,
β =β1, and m˙ =m˙1; and for τ >0, a fast flow has β =β2 λ(β2−βws0)=±(βws0−β1), (11)
andm˙ =m˙2.Figure2showstwocases:I)whenfastmaterial
isinjected at τ >0for afinitetime, ∆τ;and II)when slow with
material is injected at τ <0, for a finite time interval, ∆τ.
m˙2γ2β1 br
In both cases, when the fast flow reaches the slow flow, the λ= = , (12)
rm˙1γ1β2 ra
WS isformed instantaneously at theinjection point:x=0,
t=0, and τ =0.
As we will show below, the dynamical evolution of the 3 Thissolutionexistsbecauseatt=0,theinitialconditionβws0
WSgoesthrough2stages.Inthefirststage,theWSisfedby makestheRHSoftheequationequalzero,sincemws(0)=0.
4 Cant´o et al.
where we defined the velocity ratio a = β2/β1, the mass- thatisanincreasingfunctionof|τ˜1|.Fromequation(18)one
loss rate ratio b = m˙2/m˙1, and gamma ratio r = γ2/γ1. can show that T(τ˜1) > 2. The constant CI is obtained by
The correct solution corresponds to the + sign 4, with the matching the solution βws(τ˜1,c) = βws0, where τ˜1,c is given
ordering β1 <βws <β2, where the constant WS velocity is byequation (15);thus, from equation (18) one gets
given by
CI = 2(γ1−βws0β1γ1+1/γws0). (21)
βws0 = λβλ2++1β1, and (b+aλ)(βws0−β1)
To find a relation between τ˜1 and t˜, we take the derivative
λ+1
Tγwhsi0s=velpoci(tλy/γco2r)r2e+sp2oλnd(1s−toβt1hβe2)fi+rst1/stγa12g.e of theevolu(t1io3n) oddfTt˜th=et−imCγe1Ifdudτ˜nt˜1ct=ionβ41CTγI13(e(Tq.2[2+0]4)Tβ,1T +4), (22)
of the WS,when it is fed (and bounded)by two shocks.
NowwewilldiscusstheevolutionoftheWSinthesec- whereweusedequations(8)and(19).Again,byseparation
ond stage of the two different cases I and II shown in Fig- of variables, this equation can be integrated as
ure2.Eventhoughtheformalism isthesame, theresulting T2 4CI
equations are different for each case. Thus, for clarity, we 2 +4β1T +4lnT = β1γ13t˜+DI, (23)
separate them in the nexttwo subsections.
wheretheconstantDI isobtainedevaluatingthisexpression
at t˜Ic and τ˜1,c, and is given by,
3.1 Case I: Decelerating WS
tThheefcaosntsmtaanttervieallocisitycopmhpalseeteelnydisnwcohrepnorτa2te=d∆inτt,oi.teh.,ewWheSn. DI = T2c2 +4β1Tc+4lnTc− β41CγI13a((aλ−+11)), (24)
Thishappensatacriticaltimeobtainedbysubstitutingthe where thecritical time function is
positiaon(λo+f t1h)eWS, x˜ws0 =βws0t˜, into equation (8), Tc=T(τ˜1,c)=CI(b+γ1aλ). (25)
t˜Ic = , (14)
a−1 Therefore, given τ˜1 one can evaluate the time function
which corresponds to ejection time T to obtain βws(τ˜1) from equations (19) and (20), and one
can obtain the WS velocity as a function of time t˜, βws(t˜),
τ˜1,c =t˜Ic(cid:18)1− ββw1s0(cid:19)=−aλ. (15) ionf tahteaWbuSla,rxfwosr(mt˜),,uissignigveenqubaytieoqnu(a2t3io)n.F(8in)a.lly, theposition
Inthissecondstage,t˜>t˜Ic,thefirsttermofequation(7)
is0becausedτ2/dt=0.Furthermore,substitutingτ2 =∆τ 3.2 Case II: Accelerating WS
in equation (9), therest mass of the WS is
In this case, the constant velocity phase ends when τ1 =
mws =m˙2∆τ −m˙1τ1=m˙1∆τ(b−τ˜1). (16) −∆τ,i.e.,whentheslowmaterialiscompletelyincorporated
CollectingtheseresultswewritetheWSvelocityinequation intothe WS.From equation (8), thecritical time is
(7) as function of τ˜1 as (λ+1)
t˜IcI = , (26)
dβws γ1(βws−β1) λ(a−1)
= . (17)
dτ˜1 γws(b−τ˜1) corresponding to theejection time
Foforvaβrwiasb6=leβs1a,sthisequation5canbeintegratedbyseparation τ˜2,c =t˜IcI(cid:18)1− ββw2s0(cid:19)= a1λ. (27)
2(γ1−(ββwwssβ−1γβ11+)1/γws) =CI(b−τ˜1), (18) For t˜> t˜IcI, one has dτ1/dt = 0. Also, from equation
(9), therest mass is
thatisaquadraticequationforβws(τ˜1),andhasthesolution
β1T2+4T +4β1 mws =m˙2τ2+m˙1∆τ =m˙1∆τ(bτ˜2+1). (28)
βws(τ˜1)= T2+4β1T +4 and Then, we write theWS velocity in equation (7) as function
of τ˜2 as
T2+4β1T +4
γws(τ˜1)=γ1(cid:18) T2−4 (cid:19), (19) dβws = γ2b(β2−βws). (29)
dτ˜2 γws(bτ˜2+1)
where we defined thetime function
Forβws 6=β2,thisequation6 can beintegrated bysep-
T(τ˜1)= CI(b−τ˜1), (20) aration of variables as
γ1
2(γ2−βwsβ2γ2+1/γws) =CII(bτ˜2+1), (30)
(β2−βws)
4 The−signinequation(11)isunphysicalbecause,inthissolu-
tion,βws0<β1,thatimpliesthatthereisnodownstreamshock.
5 Equation17hasaconstantvelocitysolution,βws=β1,thatis 6 Equation29hasaconstant velocity solution,βws=β2,which
trivialbecausethereisnoshock. istrivialbecause thereisnoshock.
SHOCK DYNAMICS IN RELATIVISTIC JETS 5
which isaquadraticequation for βws(τ˜2),andhasthesolu-
tion
β2T2−4T +4β2
βws(τ˜2)= T2−4β2T +4 , and
T2−4β2T +4
γws(τ˜2)=γ2(cid:18) T2−4 (cid:19), (31)
where thetime function is
CII(bτ˜2+1)
T(τ˜2)= , (32)
γ2
thatisanincreasingfunctionofτ˜2.Asinthepreviouscase,
onecanshowthat T(τ˜2)>2.TheconstantCII isobtained
by matching the solution βws(τ˜2,c) = βws0, where τ˜2,c is
givenbyequation(27).Substitutingtheseresultsintoequa-
tion (30) one gets
CII = 2aλ(γ2−βws0β2γ2+1/γws0). (33)
(b+aλ)(β2−βws0)
Wetake thederivativeof thetime function T (eq. [32]), Figure 4. Normalized luminosity L/L0, as function of the nor-
dT = CIIbdτ˜2 = 4CII T , (34) masasulimzeeddtγim1e=t/1t0c0faonrdCγa2se=I2a0n0d. CWaesealIsIo. aFsosrumtheedseλm=od1e,lst,huwse,
dt˜ γ2 dt˜ β2γ23 (T2−4β2T +4) thecriticaltimesarethesameforbothcases.
whereweusedequations(8)and(31).Byseparationofvari-
ables, thisequation can beintegrated as
T2 4CII ray radiation, L =ǫL (t), and express this GRB lumi-
2 −4β2T +4lnT = β2γ23t˜+DII, (35) nosity in non dimGRenBsionalrform as
where the constant DII is obtained evaluating this expres- L˜ = LGRB
sion at t˜IcI and τ˜2,c, and is given by, ǫc2γ1m˙1
DII = T2c2 −4β2Tc+4lnTc− 4βC2γI23I λ(λ(a+−11)), (36) = aλ2(γ2−γ2γws)ddτ˜t˜2 + (γwsγ−1 γ1)ddτ˜t˜1
mws dγws
− . (39)
where thecritical time function is m˙1γ1∆τ dt˜
(b+aλ)
Tc =T(τ˜2,c)=CII aλγ2 . (37) In the constant velocity phase, t˜< t˜Ic,II, the luminosity is
given by
Usingequations(8),(31),(32),and(35)onecanproceedas
in Case I to obtain βws(t˜) and xws(t˜) as functions of t in a L˜0 = λ2(γ2−γws0) a− βws0
tabular form. γ2 (cid:18) β1 (cid:19)
−(γws0−γ1) βws0 −1
γ1 (cid:18) β1 (cid:19)
4 LUMINOSITIES
= βws0 −1 λ(γ2−γws0) − (γws0−γ1) , (40)
Usingastepfunctionvariabilityoftheinjectionparameters (cid:18) β1 (cid:19)(cid:20) γ2 γ1 (cid:21)
in equation (5), theluminosity L of the WS is given by
r
where we substituted dτ˜1/dt˜ = 1−βws/β1 and dτ˜2/dt˜ =
Lrc2(t˜) = m˙2γ2ddτ˜t˜2 −m˙1γ1ddτ˜t˜1 1−βAwss/mβe2nftrioomnedeqiuna§ti3o.n1,(i8n).Case I,onceallthefast mate-
rialhasbeenbeencompletelyincorporatedintotheworking
+ m˙1dτ˜1 −m˙2dτ˜2 γws− mwsdγws surface, itisdecelerated bytheslowdownstream flow.This
(cid:16) dt˜ dt˜(cid:17) ∆τ dt˜ second stage starts at a timet˜I given byequation (14).For
c
= m˙2(γ2−γws)dτ˜2 +m˙1(γws−γ1)dτ˜1 this case, on has dτ˜2/dt˜=0 in equation (38); thus,one can
dt˜ dt˜ write theluminosity as,
mwsdγws
− . (38)
∆τ dt˜ L˜I = −(γws−γ1) βws0 −1 − mws dγws
where we haveused equation (9) for the rest mass mws. γ1 (cid:18) β1 (cid:19) m˙1γ1∆τ dt˜
Onlyafractionoftheenergyradiatedininternalshocks (βws−β1) 2 (γws−γ1)
willbeemittedasgammarays;thisfraction islow,ǫ∼0.01 = β1 (cid:20)βws(βws−β1)γws− γ1 (cid:21)(.41)
(e.g.,Daigne&Mochkovitch1998).Herewewillassumethat
a constant fraction ǫ of the luminosity will go into gamma where we used that dγws/dt˜ = βwsγw3sdβws/dt˜, and, from
6 Cant´o et al.
equation (7), theAppendixC,thevalueoftheslopeα,determineswhich
case (I or II) applies.
dβws m˙1γ1∆τ (β1−βws)2
=− . (42) Table 1 shows the model parameters that fit the light
dt˜ mwsγws β1 curvesofbothGRBs.ThenameoftheGRBisindicatedin
On the other hand, in Case II, once the slow material column 1; the applied model (Case I or Case II) is shown
has been completely incorporated at t˜II given by equation in column 2; column 3 shows the flux in the constant ve-
c
(26),dτ˜1/dt˜=0, and theluminosity is given by locity phase; column 4 and column 5 show the time of the
beginning of the pulse, t0, and the time of the end of the
L˜II = λ2(γ2−γws0) a− βws0 − mws dγws constant velocity phase, te; column 6 and column 7 show
γ2 (cid:18) β1 (cid:19) m˙1γ1∆τ dt˜ the coefficient, B and the exponent α of the power-law fit
2 (β2−βws) of the GRB light curve wing; column 8 gives the inferred
= λ a ×
β2 value of the critical time function, Tc; column 9 gives the
jump of the flux (or luminosity) at the end of the constant
2 (γ2−γws)
(cid:20)−βws(β2−βws)γws+ γ2 (cid:21), (43) velocity phase, J(te) defined by equations (B9) and (B10);
finally,column 10andcolumn 11givetheinferred valuesof
where, from equation (7), thegammaratio,r andthemass-lossrateratio,b,obtained
directly form thefit to thewind of thelight curve.
2
dβws = m˙2γ2∆τ (β2−βws) . (44) In our model, the WS approaches the observer at rel-
dt˜ mwsγws β2 ativistic speeds. As discussed in Appendix E, the observed
In the ultra-relativistic (UR) limit, γ ≫ 1, the expres- bolometric flux density of an approaching relativistic jet is
sions for the luminosities are simplified as shown in Ap- increased with respect toan emitterat rest byafactor δw4s,
pendixB.Inthefollowing section wewill applytheseequa- whereδws =1/[γws(1−βwscosθ)]andθistheanglebetween
tions for the GRB luminosities to describe the light curves thedirection oftherelativisticjet andtheobserver.Lind&
of two observed sources. Blandford (1985) obtained an amplification by a factor of
δw3s between theobserved fluxat frequency ν and theemit-
ted luminosity at frequency ν′ = ν/δws. An extra factor of
δws isobtainedwhenoneintegratesthefrequencytogetthe
5 PREDICTED FLUXES
observedbolometricflux(ortheobservedfluxinafrequency
Figure4showsthemodelluminosityforCaseIandCaseII, range)intermsoftheemittedbolometricluminosity.Thus,
normalizedtotheluminosityintheconstantvelocityphase, in our model the observed gamma ray flux in the constant
LI,II/L0,asafunctionofnormalizedtime,t/tIc,II.Themod- velocity phase is given by FG0RB =δw4s0L˜0URǫc2γ1m˙1/4πD2,
els have γ1 = 100, γ2 = 200, that correspond to the UR where L˜0UR is the normalized luminosity in equation (B5).
limit. Also, we choose λ ∼ (br)1/2 = 1, that implies a con- Inordertosolveforthemass-lossrate,weassumethat
stant energy injection rate m˙1γ1 = m˙2γ2. In this case, the the relativistic jet is seen almost along the jet axis, i.e.,
critical times (eqs. [14] and [26]) are equal. For this reason, cosθ∼1,thus,δws0 ∼2γws0.Also,assumingγ1 =100,both
wedropthesuperscriptsinthefollowing discussion.Asdis- L˜0UR and the Lorentz factor γws0 can be obtained from the
cussedabove,fort<t ,theWSisboundedby2shocksand modelparameters in Table 1. Inparticular, we obtain large
c
moves at a constant speed. Then, the relative velocity be- Lorentz factors for the WS, γws0 ∼ 802 for GRB 080413B,
tweentheincorporatedmaterialandtheWSand,therefore, and γws0 ∼ 120 for GRB 070318 from equation (B2). Fur-
the luminosity are constant. For t > t , in both cases one thermore, weestimate thedistance totheGRBs from their
c
shockdisappearsandtherelativevelocitiesbetweentheWS redshift. Thesource GRB070318 has an estimated redshift
and the new material that is incorporated decreases with z=0.836(Jaunsenetal.2007)andGRB080413Bhasared-
time; therefore, theluminosity, diminishes with time. shiftz=1.1(Vreeswijketal.2008).Assumingadarkenergy
Appendix C and D show that, in the UR approxima- density ΩΛ = 0.7, a matter density Ωm = 0.3, and a Hub-
tion, one can fit a power-law to the wing of the GRB light ble constant H =70kms−1Mpc−1, the luminosity distance
curve, and obtain the critical time function T , the gamma of GRB 070318 is D = 5300 Mpc and of GRB 080413B is
c
ratio, r = γ2/γ1, and the mass-loss rate ratio, b = m˙2/m˙1. D=7440Mpc.Withalltheseingredients,wecannowsolve
These quantities can be obtained without any further as- for themass-loss rate, and obtain m˙1 ∼9.7×10−6M⊙yr−1
sumptions.Asanexample,Figure5showsamodelfittothe for GRB 080413B, and m˙1∼1.4M⊙yr−1 for GRB 070318.
observedsourcesGRB08413BandGRB070318.Thefluxes BecausetheemittingWSmoveswith relativistic speed,the
were taken from Mendoza et al. (2009) that correspond to jet mass-loss rates m˙1 required to produce the gamma ray
observations between 15 and 150 keV with the Burst Alert flux observed at the Earth are much smaller than those
TelescopeonboardtheSWIFTsatellite.Wefirstfittheob- obtained, for example, in nucleosynthesis models of wind-
served flux density F directly and later discuss its relation driven supernovae and collapsar models (e.g., Fujimoto et
to the distance and luminosity of the GRB. We follow the al. 2008; Maeda & Toming 2009).
procedure described in Appendix D, and choose the value Note that in the model the time is measured in the
of the luminosity (or flux density) in the constant velocity frame of reference of the jet source, where the evolution
phase,thetimet0ofthebeginningofthevelocitypulse,and timescales for the WS are of the order of ∼ 105 − 106s.
time te =tc+t0 of the end of the constant velocity phase, Instead,thetimeoftheobservationsismeasuredattheob-
wheret isthecriticaltimeofthemodel.Wealsochoosethe server’s frame of reference and is only of the order of tens
c
fluxattheconstantvelocityphase,F0.Thewinglightcurve of seconds if the relativistic jet is seen almost along the
is then fit by a power-law F = B(t−t0)α. As discussed in jet axis. This happens because the arrival time is ∆ta =
SHOCK DYNAMICS IN RELATIVISTIC JETS 7
1)− 1x10−6 GCARSBE 0 I80413B 1)− GCARSBE 0 I7I0318
s s
· ·1x10 7
2− 2− −
m m
c c
g·5x10−7 g·
er er 5x10−8
( (
x x
u u
Fl 1x10−7 Fl 1x10−8
0 0
240 245 250 240 250 260 270 280
t(s) t(s)
Figure 5. Leftpanel:GRB080413B sourcewithadecelerating WSmodel(thicksolidline).Rightpanel:GRB070318 sourcewithan
acceleratingWSmodel(thicksolidline).Thethinlinesineachpanelshowthedurationoftheconstantvelocityphase.Theobservational
dataofbothsourceswastakenfromMendozaetal.(2009).
Table 1.ModelParameters
GRB Case ergFcG0mR−B2sa−1 (ts0) (tse) (10−7erBg±cm∆−B2s−1−α) α±∆α Tc J (γ2r) (m˙2b)
γ1 m˙1
080413B I 1.08×10−6 238.50 240.01 15.49±3.00 −2.03±0.30 2.57 0.38 22.49 233.73
070318 II 1.25×10−7 239.00 245.90 8.08±0.50 −1.08±0.05 5.35 0.20 2.64 0.12
a Fluxdensityintheconstant velocityphase,F0 =δ4L˜0 ǫc2γ1m˙1/4πD2.
GRB UR
∆tem/(γwsδws), where ∆tem is the emission time at the jet 6 FRACTION OF THE INJECTED MASS
source frame (e.g., Daigne & Mochkovitch 1998). Also, our LOST BY RADIATION IN THE WS
models are not meant to explain the shape variation of the
In this section, we evaluate the fraction of themass lost by
bursts with spectral band (e.g., Norris et al. 1996), which
radiationwithrespecttothemassinjectedintheWS,∆m.
would dependonfraction ofenegyradiated in gammarays,
From equation (5) thetotal mass lost byradiation is
ǫ.Infact,oneexpectsthatǫwilldependonenergyandtime,
as the WS decelerates and the energy is radiated at lower t τ2
spectral bands. Lr(t)c−2dt= m˙(τ)γ(τ)dτ−mws(t)γws(t). (45)
Z Z
0 τ1
IntheRHSofthisequation,thefirsttermisthetotalmass
ejected bytheflowthat hasbeenincorporated intotheWS
Finally, the model uses the simplest velocity variation and the second term is the actual mass of the WS. These
which allows an analytic solution: it assumes an instanta- two terms increase with time but their difference remains
neous jump in the velocity of the injected material (step finitebecause theshocks will weaken with time.
function),thus,itmakesthesimplificationthattheluminos- Onecandefinethefractionofmasslostbyradiation as
ity instantly achieves the maximum value. A more realistic
situation would be a gradual increase in the velocity of the ∆m= 0tLr(t)c−2dt =1− mwsγws . (46)
injected material which would producea gradual growth in Rτ2m˙(τ)γ(τ)dτ τ2m˙(τ)γ(τ)dτ
the luminosity. Here we show that the decay of two GRB τ1 τ1
R R
light curves can be fitted by the emission of an decelerated This fraction ∆m→0 when t→∞ because thetotal mass
or accelerated WS given by these very simple models. Al- lostbyradiationintheLHSofequation(45)isfinite.Thus,
though, this choice is intended to illustrate the formalism onehastoevaluate∆matafinitetimetodeterminetheim-
it is also true that the analytic solutions allow an explo- portance of radiation losses in the dynamicsof the working
ration of parameter space and give us an understanding of surface.
thedependanceoftheGRBemission onimportantphysical Thetotalmomentumlostbyradiation 0tLr(t)c−2βwsdt
parameters like the gamma ratio r and the mass-loss rate canbeobtainedfrom equation(6).InpartRicular, intheUR
ratio b. regime where βws ∼ 1, the mass and momentum losses are
8 Cant´o et al.
7 CONCLUSIONS
We have developed a new formalism that describes the dy-
namics of an internal WS in a relativistic jet produced by
variationsinthesourceinjectionvelocity.TheWSisformed
when a fast flow overtakes a previous slower flow. This for-
malism takes into account that the momentum is not con-
served becauserelativistic mass is lost by radiation, in con-
trast with non relativistic flows.
Assuming step function variations of the injection ve-
locity and mass-loss rate we find analytic solutions for the
WS velocity and luminosity. We consider two cases: when
a pulse of fast material reaches the slow downstream wind
(Case I); and when a pulse of slow material is pushed by
fast upstream wind (Case II). In the initial phase, the WS
is bounded by 2 shocks: one shock incorporates the mate-
rialfrom thefast (slow) pulse,theothershock incorporates
materialfrom theslow(fast)wind.Inthisphase,theveloc-
ity of the WS is constant. When thematerial from the fast
(slow) pulse is completely incorporated into the WS, only
oneshockremains:inCaseI,theWSisdeceleratedasmore
mass is added to the shock by the slow downstream wind;
inCaseII,theWSaccelerates,pushedbythefastupstream
Figure6.Fractionofmasslostbyradiationasafunctionofthe
wind. The WS luminosity in the constant velocity phase is
gamma ratio r = γ2/γ1. Each curve is labeled by the different
constant, and decreases with time when one of the shocks
valuesofthemass-lossrateratiob=m˙2/m˙1.
disappears.ToapplythesemodelstoobservedGRBsweas-
sume that a constant fraction ǫ of this energy is emitted in
gamma rays.
the same. In this regime, ∆m measures the importance of
both momentum and mass losses. In the UR limit, the ratio of the Lorentz factors r =
If ∆m ≪ 1, radiation losses will not change the rela- γ2/γ1 and the mass-loss rates b = m2/m1 of the relativis-
tivistic mass of the WS significantly. Note that in the con- tic flows that collide can be obtained directly by fitting
stant velocity phase, radiation losses, which change therel- the light curves of GRBs. As an example, we fit the light
ativistic mass of the WS according to equation (5), do not curves of the GRBs 080413B and 070318 with the Case
affect the velocity βws because the LHS of eq. (7) is zero I and Case II models, respectively. For GRB 080413B we
(since dβws/dt = 0). Nevertheless, these losses do change obtain the ratios γ2/γ1 = 22.5 and m˙2/m˙1 = 233.7; for
the momentum of the WS at the critical time and the dy- GRB 070318 the fit gives lower ratios, γ2/γ1 = 2.6 and
namics of the WS in the second decelerating/accelerating m˙2/m˙1 = 0.1. Since the WS is moving towards the ob-
phase of Case I and II,respectively. server at relativistic speeds (γws0 ∼ 100−800), one has to
We choose to evaluate ∆m at the critical times correct the observed gamma ray fluxes for Doppler boost-
(eqs. [14] and [26]), which correspond to the end of the ing. Assuming γ1 = 100, we estimate mass-loss rates of
coτ∆1ncτstma˙n(tτ)γv(eτlo)cdiτty=phm˙a1sγe.1∆Fτoarλ(C1a+seλI),; fτo2rc C=ase∆IτI,,τt1hcu=s, ijtsehtemkjuiencthestibscmetpawolelweerenrtmhi˙sa1nP∼ktihn1e0=−as5sγo−mc˙ic1a2tMed∼⊙yi1sro0−t41r4o−.p4Ni8ceortlgeusm−th1ina.otTstihthyies,
R−∆τ, thus, −τ2∆cτm˙(τ)γ(τ)dτ =m˙1γ1∆τ(1+λ). Then, us- Liso =4πD2FGRB/ǫ∼1053−54ergs−1, uncorrected for rela-
ing equationRs (16), (28), one can show that in both cases I tivisticeffects.Infact,theisotropicluminosityhasnophys-
and II, themass fraction is ical meaning for our relativistic jet models, which are able
γws (b+aλ) to produce the Doppler boosted gamma ray flux observed
∆m=1− . (47) at theEarth.
γ1 aλ(1+λ)
We also evaluate the fraction of the injected mass lost
IntheURregime,wherea∼1,λ∼(br)1/2,andγwsisgiven by radiation. For the model of thesource GRB 070318 this
by equation (B2), thisexpression reducesto fraction is ∼ 7%, while for the source GRB 080413B, one
finds that more than 50% of the injected mass is lost by
r(b+λ)
∆m=1− , (48) radiation. Therefore, in the latter source, radiation losses
λ(λ+r2)1/2(1+λ)1/2
change significantly the relativistic mass of the WS and af-
whichisafunctiononlyofthegammaratio,r=γ2/γ1,and fect its dynamics.
themass-loss rate ratio, b=m˙2/m˙1. Thestepfunction variabilityof thesourceinjection ve-
Figure6shows∆masafunctionofrforthebvaluesof locity and mass-loss rate is a simple approximation to the
themodelspresentedinTable1.Onecanseethatthemass realtimevariabilityoftheinjectionparameters.Otherfunc-
fraction is ∆m ∼ 7% for the model of GRB 070318, but it tional time variations of these parameters can beeasily im-
reaches values ∆m > 50% for the model of GRB 080413B. plemented in our formalism by integrating the equations in
In thelast case, radiation losses clearly affect thedynamics §3numerically.Nevertheless,intheURregimeouranalytic
of the WS:mass and momentum are not conserved. model is very useful to determine the ratios of important
SHOCK DYNAMICS IN RELATIVISTIC JETS 9
physicalparameters(thegammaratioandthemass-lossrate in a shock, momentum is not conserved because the rela-
ratio) without introducing any further assumptions. tivistic mass, that includes theLorentzfactor γ of internal
i
Inafutureworkwewillstudythehighenergyemission motions, is not conserved.
producedbytheseinternalshocks,inparticular,thefraction Consider two particles in the laboratory with masses
of energy emitted as gamma rays. m , velocity β , momentum P = m γ cβ , and energy,
k k k k k k
E = m γ c2, for particles k = 1,2. We consider the col-
k k k
lision between these two particles, where β2 > β1. In the
laboratoryreferenceframethemomentumconservation(di-
ACKNOWLEDGMENTS vided byc) gives,
J.C., S.L., M.F., R.F.G., and A.H. were supported by m1γ1β1+m2γ2β2 =(m1+m2)γi12γ12β12, (A1)
PAPIIT-UNAMIN100412andIN100511.J.C.wasalsosup-
portedbyCONACyT61547.WethankJoseIgnacioCabrera whereγi12 istheLorentzfactorthatcorrespondstointernal
for providing us with the observational data and Anabella motions,andγ12 correspondstothebulkmotionofthenew
Araudoforusefulsuggestions.Wealsothankananonymous particle1-2with velocityβ12.Energy conservation (divided
referee for useful comments that helped to improve the pa- byc2) gives
per.
m1γ1+m2γ2 =(m1+m2)γi12γ12. (A2)
Theratioofthesetwoequationsgivesthevelocityofparticle
1-2,
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m1γ1β1+m2γ2β2
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andtheLorentzfactorforinternalmotionsisobtainedfrom
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2009, MNRAS,395, 1403 and themomentum transformation gives
M´esz´aros, P. 2002, ARA&A,40, 137
Mochkovitch,R.,Maitia,V.,&Marques,R.1995,Ap&SS, Pk′ =mkγkγ12c(βk−β12). (A6)
231, 441 Using equation (A3), one can check that P1′+P2′ =0.
Norris, J. P., Nemiroff, R. J., Bonnell, J. T., et al. 1996,
In this system, after the collision, the new particle 1-2
ApJ,459, 393
will be at rest. Before any radiation is emitted, the initial
Piran, T. 2004, Reviews of Modern Physics, 76, 1143 energy before collision E′ is conserved; thus, after col-
Raga,A.C.,Binette,L.,Canto,J.,&Calvet,N.1990,ApJ, initial
lision it will become the internal energy of the particle 1-2,
364, 601 E′ ,
Rees, M. J. 1978, Nature,275, 516 internal
RReeeess,, MM.. JJ..,1&98M4,eAszRarAo&s,AP,.2129,9447,1ApJ, 430, L93 Ei′nitial =E1′ +E2′ = (m1γ1+γ1m2 2γ2)c2
Rybicki,G.B.,&Lightman,A.P.1979,NewYork,Wiley-
Interscience Publication =(m1+m2)γi′12c2 =Ei′nternal, (A7)
Spada, M., Ghisellini, G., Lazzati, D., & Celotti, A. 2001, where we used equation (A3) to calculate E′, and γi′12 is
MNRAS,325, 1559 associated with the internal motions of particle 1-2 in the
Vreeswijk, P.M.,Thoene, C. C.,Malesani, D.,et al. 2008, momentum center reference system. Solving this equation
GRB Coordinates Network, 7601, 1 for γi′12, one gets
γi′12 = (mm11γ+1+mm2)2γγ122 =γi12, (A8)
whereweusedequation(A4);i.e.,theLorentzfactorassoci-
APPENDIX A: COLLISION OF RELATIVISTIC
atedwithinternalmotionsisthesameinbothreferencesys-
PARTICLES
tems. Thus, after collision thenew particle 1-2 has a larger
Wediscussasimpleexampleofaninelasticcollisionof3rel- mass, (m1+m2)γi12.
ativistic particles that shows that when energy is radiated Now, let us assume that the particle 1-2 radiates
10 Cant´o et al.
isotropically(inthemomentumcentersystem)alltheavail-
able energy, thus, the momentum of particle 1-2 will not
change, P1′2 = 0. The maximum energy that can be radi-
ated byparticle 1-2 is
Er′ =Ei′nternal−E0′
c2
= [m1γ1+m2γ2−(m1+m2)γ12], (A9)
γ12
where E0′ = (m1+m2)c2 is the rest energy of particle 1-2,
and we used equation(A7).
Inthelaboratory referenceframe,thisparticle1-2that
hasradiatedalltheavailableenergynowhasanenergyand
momentum given bythe Lorentztransformations
E0=γ12 E0′ +cβ12P1′2 =(m1+m2)γ12c2; (A10)
(cid:0) (cid:1)
P0 =γ12 P1′2+ β12E0′ =(m1+m2)γ12cβ12. (A11)
c
(cid:16) (cid:17)
ComparingtheseequationswiththeRHSofequations(A1)
and (A2) one can see that after radiation, particle 1-2 has
lost energy and momentum.
On the other hand, the velocity of particle 1-2 after
radiation in thelaboratory reference frame is given by Figure A1. Luminosity jumps JI and JII as function of the
cP0 mass-loss rate ratio b for two γ ratios: r → 1 (solid lines) and
β≡ =β12. (A12) r=3(dashedlines).
E0
Comparing with equation (A3) one can see that, after ra-
diating all the available energy, particle 1-2 conserves its all the available energy before colliding with particle 3. In
velocity. this case, particle 1-2 has the momentum and the energy
Now,letusconsiderthecollisionofthreeparticleswith given by equations (A11) and (A10). Then, the equation of
restmassesm andvelocitiesβ ,respectively,fork=1,2,3. momentum conservation is
k k
We want to obtain the velocity of the particle that results
(m1+m2)γ2β12+m3γ3β3 =(m1+m2+m3)γi123γ123β123,(A16)
from the collision of thethree particles.
As discussed above, β2 > β1, such that particle 2 will and theconservation of energy is
collide with particle 1. The resulting velocity of the com-
(m1+m2)γ2+m3γ3 =(m1+m2+m3)γi123γ123. (A17)
bined particle 1-2 and internal Lorentz factor are given by
equations (A3) and (A4). Now we consider the collision of In this case, when energy is radiated by particle 1-2 before
particle 1-2 with particle 3 that has β3 >β12. collisionwithparticle3,thevelocityofthenewparticle1-2-3
If the new particle 1-2 collides with particle 3 before it is given by theratio of equations(A16) and (A17),
radiates away energy,theconservation of momentum gives β1r2a3d= (m1+m2)γ2β12+m3γ3β3. (A18)
(m1+m2)γi12γ2β12+m3γ3β3 = (m1+m2)γ2+m3γ3
This velocity is different from the velocity of the adiabatic
=(m1+m2+m3)γi123γ123β123, (A13) case, equation (A15). Therefore, the collision of particles
and the conservation of energy gives that radiate their internal energy does not conserve mo-
mentum!Thishappensbecausewhenenergyisradiatedbe-
(m1+m2)γi12γ2+m3γ3 =(m1+m2+m3)γi123. (A14) forecollision byparticle1-2above,itsmomentumdecreases
Inthiscase,wherenoenergyisradiatedbeforethecollision because its relativistic mass (m1+m2)γi12 decreases, i.e.,
of the 3 particles, the velocity of the new particle 1-2-3 is γi12 →1.
given by theratio of equations(A13) and (A14) Therefore, when energy is radiated in a shock in the
collision of 3 relativistic particles, one cannot obtain the
β1a2d3 = (m1+m2)γi12γ2β12+m3γ3β3 velocity of theshock byassuming momentumconservation.
(m1+m2)γi12γ2+m3γ3
m1γ1β1+m2γ2β2+m3γ3β3
= , (A15)
m1γ1+m2γ2+m3γ3 APPENDIX B: THE ULTRA-RELATIVISTIC
CASE
where we used equation (A1) to obtain the last equality.
Note that the RHS of the above equation is also obtained Herewewillconsidertheultra-relativistic(UR)limit,β≃1,
from the conservation of momentum and energy of the col- where the Lorentz factor γ ≫ 1. In this limit, a ∼ 1, and
lision of the 3 particles. If the new particle 1-2-3 radiates one can expand β ≃ 1−1/(2γ2). Thus, βws−β1 ≃ (γw2s−
all its internal energy, as shown in equation (A12), it will γ12)/2γ12γw2s, and a−1∼(γ22−γ12)/(2γ12γ22).
conserve this velocity. We define the normalized Lorentz factors r1 = γws/γ1
We now consider the case when particle 1-2 radiates and r2 =γws/γ2.
