The Cosmic Ray Precursor of Relativistic Collisionless Shocks: A Missing Link in Gamma-Ray Burst Afterglows

0.12 MB·English

by  Milos Milosavljevic

DRAFTVERSIONFEBRUARY5,2008
PreprinttypesetusingLATEXstyleemulateapjv.10/10/03
THECOSMICRAYPRECURSOROFRELATIVISTICCOLLISIONLESSSHOCKS:AMISSINGLINKINGAMMA-RAY
BURSTAFTERGLOWS
MILOŠMILOSAVLJEVIC´1,2ANDEHUDNAKAR1
DraftversionFebruary5,2008
ABSTRACT
Collisionlessshocksarecommonlyarguedtobethesitesofcosmicray(CR)acceleration. Westudytheinflu-
enceofCRsonweaklymagnetizedrelativisticcollisionlessshocksandapplyourresultstoexternalshocksin
gamma-rayburst(GRB)afterglows.ThecommonviewisthatthetransverseWeibelinstability(TWI)generates
asmall-scalemagneticfieldthatfacilitatescollisionalcouplingandthermalizationintheshocktransition.The
7 TWIfieldisexpectedtodecayrapidly,overafinitenumberofprotonplasmaskindepthsfromthetransition.
0 However,thesynchrotronemissionin GRBafterglowssuggeststhata strongandpersistentmagneticfieldis
0 presentintheplasmathatcrossestheshock;theoriginofthisfieldisakeyopenquestion.Herewesuggestthat
2
thecommonpictureinvolvingTWIdemandsrevision.Namely,theCRsdriveturbulenceintheshockupstream

n onscalesmuchlargerthantheskindepth.Thisturbulencegeneratesalarge-scalemagneticfieldthatquenches
a TWIandproducesamagnetizedshock. ThenewfieldefficientlyconfinesCRsandenhancestheacceleration
J efficiency.TheCRsmodifytheshocksinGRBafterglowsatleastwhiletheyremainrelativistic. Theoriginof

6 themagneticfieldthatgivesrisetothesynchrotronemissionisplausiblyintheCR-driventurbulence. Wedo
notexpectultrahighenergycosmicrayproductioninexternalGRBshocks.

2
Subjectheadings:accelerationofparticles—cosmicrays—gamma-rays:bursts—plasmas—shockwaves
v
8
4
1. INTRODUCTION Medvedevetal. 2005), survival of the field over larger dis-
5
tanceshasnotbeendemonstrated.
2 Collisionless shocks are observed on all astrophysical
According to the popular model of gamma-ray bursts
1 scales. The diffusive shock acceleration (DSA) mechanism
(GRBs; e.g., Piran 2005, and references therein), the after-
5 is believed to accelerate cosmic rays (CRs) in these shocks
gloworiginatesinarelativisticblastwavethatpropagatesinto
0 (Bell1978;Blandford&Ostriker1978;Blandford&Eichler
/ 1987). The CRs can carry a substantial fraction of the en- an ambient e- p plasma. The afterglow emission is ascribed
h to synchrotron radiation from nonthermal electrons that gy-
ergy of the shock and thus the CR pressure can influence
p
rate in the shock-generatedmagnetic field. Detailed studies
- the structure of the shock (Eichler 1979; Blandford 1980;
o Drury&Völk 1981; Ellison&Eichler 1984). This picture ofGRBspectraandlightcurves(Panaitescu&Kumar2002;
r hasrecentlyreceivedsupportfromX-rayobservationsofsu- Yostetal.2003)haveshownthatthemagneticenergydensity
t intheemittingregion(thedownstreamshockedplasma)isa
s pernova (SN) remnants (Warrenetal. 2005). Recently, Bell
a fractionǫ 10- 2to10- 3oftheinternalenergydensity.1 This
(2004, 2005) has shown that CRs in SNe can drive turbu- B
: field must p∼ersist overat least a few percentof the width of
v lenceandamplifymagneticfieldsintheshockupstream,and
theblastwave (Rossi&Rees2003). Itsoriginhasremained
i Dar&deRújula (2005) speculate that relativistic jets could
X akeyopenquestion.
dothesame.
r Thecurrentloreisthatweaklymagnetizedrelativisticcol- Compressionalamplificationoftheweakpre-existingmag-
a neticfield oftheinterstellar medium(ISM)yieldsǫ 10- 9
lisionlessshocksaremediatedbythetransverseWeibelinsta- B
∼
(Gruzinov 2001) and does not explain the field in the emit-
bility(TWI;Weibel1959;Fried1959). TWIproducesamag-
ting region. If TWI does develop in the transition layer, it
neticfieldnearequipartitionthatprovidescollisionalcoupling
generatesastrongfieldinthevicinityoftheshocktransition.
in the shock transition layer (Gruzinov&Waxman 1999;
However,asdiscussedabove,thereisnoevidencethatitcan
Medvedev&Loeb 1999). TWI is universally observed in
persistovertherequired 109λ fromtheshock(e.g.,Piran
two-andthree-dimensionalparticle-in-cell(PIC)simulations s
∼
2005andreferencestherein).
of unmagnetized colliding shells (e.g., Lee&Lampe 1973;
X-ray observations of GRB afterglows are modeled as
Gruzinov 2001; Silvaetal. 2003; Frederiksenetal. 2004;
an opticallythin synchrotronspectrum requiringnonthermal
Jaroscheketal.2004;Nishikawaetal.2005;Medvedevetal.
electronswith Lorentz factors as high as 106. The obser-
2005; Kato 2005). The resulting field is small-scale, of the
∼
vations indicate that electrons, and thus protonsas well, are
orderoftheprotonplasmaskindepthλ .
s efficientlyacceleratedintheshocktoproduceahardpower-
The role of TWI in particle acceleration and in the down-
law spectrum. DSA can achieve such Lorentz factors if the
streamthermodynamicsiscontroversial.Thefieldisexpected
circum-burstmedium is magnetized at 1 µG level, as ex-
to decay rapidly, within a few λ from the shock transition
s pected in the ISM. The weak magnetic∼field, however, does
(Gruzinov 2001; Milosavljevic´&Nakar 2006). This decay
not directly affect the structure of the shock (e.g., TWI can
is evidentin three dimensionalPIC simulations (Spitkovsky
stilldevelopintheshocktransition).
2005). Although there are claims that the decay saturates
It seems that a self-consistent picture of relativistic astro-
at distances of 100λ from the shock (Silvaetal. 2003;
∼ s physical shocks includes high energy particles (CRs) and at
1 Theoretical Astrophysics, Mail Code 130-33, California Institute of
Technology,1200EastCaliforniaBoulevard,Pasadena,CA91125. 1 Under some circumstances ǫB as low as ∼ 10- 5 can fit the data
2HubbleFellow. (Eichler&Waxman2005).
2 CosmicRayPrecursorinGRBShocks
leastaweaklymagnetizedupstream. Hereweexplorethein- B seed field
0
teraction between these components. In § 2, we derive the
conditions under which the CRs drive turbulence in the up-
stream;thisturbulencegeneratesalarge-scalemagneticfield
that increases the acceleration efficiency.2 In § 3, we argue Jret B
that this mechanism is a candidate solution of the origin of Jret
thefieldinferredfromtheafterglowemissioninexternalGRB
expanding shell
shocks. In § 4, we discuss implicationsforTWI andfor the
productionofultrahighenergycosmicrays(UHECRs).
J
cr
2. COSMICRAY–MAGNETICFIELDINTERACTION
shock
2.1. CosmicRayTrajectoriesandtheReturnCurrent
FIG.1.—Schematicrepresentationofthereactionoftheupstreamplasma
Consider a relativistic shock wave with Lorentz factor Γ totheCRstreamingaheadoftheshock. Thefigureisintheframeofthe
that accelerates particles to Lorentz factors γ Γ. The ac- shocksothattheupstreamplasmamovesverticallydownward. Aninitially
celerated particles are roughly isotropic in th≫e downstream weakmagneticfieldloopisstretchedbytheAmpèreforce(Jret×B)/cand
plasmaisacceleratedsidewaysasitflowstowardtheshockfront.
frame;a fractionofthemaremovingaheadofthe shock. In
theupstreamframe,thevelocitiesoftheacceleratedparticles
aredirectedwithinanangleθ √2/Γoftheshocknormal.3
∼ reachhigherenergiesandfartherfromtheshockintotheup-
IftheupstreamcontainsamagneticfieldB,thefielddeflects
stream. The upstream fluid sees a net positive CR current
chargedparticlesmovingaheadoftheshock.Whentheveloc-
J ev n in the region where it overlapswith the pre-
ityofaparticleisdeflectedtoanangle 1/Γfromtheshock | CR|∼ s CR
normal,theshockcatchesupandre-abs∼orbstheparticle. cursor. Here vs and nCR are the CR velocityand density. In
the upstream region, which is not accelerated to relativistic
The orbit of a particle in the upstream field can be quasi-
velocities by the CR pressure, the CR velocity is about the
circular or diffusive, depending on how the radial distance
speedoflight,v c;thus,wehave
X the particle traverses ahead of the shock compares to the s∼
relevantmagneticcorrelationlengthλ. Thetypicalangleby J ecn . (4)
CR CR
which the magnetic field deflects the particle is ∆θ X/r ∼
g
∼ To maintain overall charge neutrality, the upstream re-
when the orbit is quasi-circular and ∆θ (λX)1/2/r when
∼ g sponds to the CR current by supporting an opposite “return
theorbitisdiffusive,where
current” J that short-circuits the electric field associated
ret
γmc2 with the charge separation in the CR precursor.5 If nCR is
rg= (1) smallerthantheupstreamdensity,thereturncurrentapproxi-
eB(λ)
matelybalancestheCRcurrent,
isthegyroradiusoftheparticleexpressedintermsofthemag-
neticfieldB(λ)onscalesλ. Reabsorptionintheshockoccurs Jret JCR. (5)
∼
when∆θ 1/Γ,andthusthedistancetraversedbytheparti-
Iftheupstreamcontainsaweakmagneticfield,thereturncur-
∼
cleinthequasi-circularanddiffusiveregimeis
rentcoupleswiththisfield;nextweexploretheconsequences
r /Γ, (X .λ), ofthiscoupling.
X g (2)
∼(cid:26)(rg/Γ)2/λ, (X ≫λ). 2.2. TheDrivingofTurbulenceandFieldAmplification
Insphericalblastwaves,aparticlecanbeacceleratedbyre- Here we argue that the return current drives turbulence in
peatedshockcrossingifthetimeseparatingitsemissionand the upstream and suggest that this turbulence leads to mag-
re-absorptionin the shock X/c is smaller than the expan- netic field growth on scales much larger than the proton
∼
sion time of the shock R/c, where R is the shock radius.4 plasma skin depth. One type of interaction between the CR
∼
ThereforeX Risrequiredforacceleration. Themaximum precursorandthemagneticfieldoftheupstream,exploredby
≤
distancethattheparticlereachesfromthemovingshocktran- Bell(2004,2005) inthecontextofNewtonianshocks, isthe
sitionissubstantiallysmaller, Ampèreforce
J B
∆∼ ΓX2, (3) F= retc× (6)
whichacceleratestheupstreamperpendiculartotheshockve-
because the particle and the shock are both movingat about
locity. We assume that the initial weak magnetic field has
the speed of light. Another requirement for acceleration is
poweronallscales,asexpectedininterstellarturbulence,and
thatthecoolingtimebelongerthantheaccelerationtime(see
thatthepoweronrelevantscalesisroughlyscale-independent.
§3).
Consideraloopofradiusλofaweakinitialmagneticfield
Ingeneral, protonsand electronsare bothemittedinto the
B paralleltotheshocktransitionthatisdirectedclockwiseas
shockupstream.However,thesynchrotronandself-Compton 0
seenfromtheshock,asshowninFigure1,andassumethatthe
cooling inhibit the acceleration of electrons, and so the pro-
upstreamisinitiallystationary.TheAmpèreforceaccelerates
tons,whichwerefertoastheCRprecursoroftheshock,can
thefluidawayfromthecenteroftheloop. Bell(2004,2005)
carriedoutmagnetohydrodynamic(MHD)simulationsofthis
2Gruzinov(2001)andWaxman(2004)speculatethatCRscouldamplify
large-scalemagneticfieldsinexternalGRBshocks. process;webaseourestimatesonhisresults.
3Unlessotherwisenoted,quantitiesareevaluatedintheupstreamframe.
4 Itisalsoimplicitlyassumedthatsomefractionofaccelerated particles 5 Thereturncurrentisestablishedontheplasmatime.10mswhichis
candiffusebacktotheshockaftertheyfindthemselvesinthedownstream. instantaneouscomparedtothelightcrossingtimeoftheCRprecursor.
MILOSAVLJEVIC´ &NAKAR 3
The dynamics of the upstream fluid can be approximated Thereforewecanlimitthegeneratedmagneticfieldtoliebe-
using MHD, and thus flux freezing during the expansion of tweenB andB .
1 2
theloopimpliesthat Forsomeλ,theabovevaluesofBformallyimplysuperlu-
minal motion. We do not analyze the evolution of the mag-
B
θ = const, (7) netic field after its energy approaches equipartition with the
ρr rest energyof the upstream.6 Our treatment is applicable at
where B and ρ are the azimuthal magnetic field and the distances from the shock where the magnetic field does not
θ
density at radius r, respectively. If the expansion is non- reachequipartition,B<(4πρism)1/2c.
relativistic and if the thermal and magnetic pressure are ig-
2.3. TheCosmicRaySpectrumandtheQuasi-SteadyState
nored so that only the Ampère force influences the motion,
theradiusoftheloopacceleratesaccordingto The generated magnetic field can modify the spatial pro-
fileofJ byconfiningCRsclosertotheshockthanthepre-
d2r J B J B CR
ret θ CR 0r. (8) existing field. The maximum coherence length of the new
dt2 ∼ ρc ∼ λρ0c field cannot exceed the maximum distance CRs reach from
theshock R/Γ2. Thereforethepropagationofthemosten-
Therefore,theloopexpandsexponentiallyr(t) λeσt atarate ∼
∼ ergeticCRstraversingadistanceX Rbetweenemissionand
∼
J B 1/2 re-absorptionis always diffusivein the new field. If the CR
σ CR 0 , (9) motionisdominatedbythenewfield,7 theshocksettlesina
∼(cid:18) λρ0c (cid:19) quasi-steadystateinwhichtoeverydistance∆ R/Γ2from
≤
implyingavelocityofexpansionofdr/dt rσ. the shock corresponds a γ such that CRs with Lorentz fac-
The upstream accelerates until the pres∼sure overtakes the torγ areconfinedwithinadistance∆fromtheshockbythe
Ampèreforce(weassumethattheexpansionremainssublu- magneticfieldgeneratedbythestreamingofthesesameCRs.
minalatthispoint).Inarealisticenvironmentwherethemag- ThedensityoftheCRsatadistance∆is
neticfieldfluctuatesonallscales,weexpecttheexpansionon 1 dN
eachscaletosaturatewhenneighboringmagnetic“rings”on nCR(γ)∼ 4πR2∆(γ)dlnγ, (15)
similar scales collide, namely when r 2λ. This happens
when ∼ wheredN/dγ istheCRenergyspectrum. TheCRspectrum
σ∆ producedbytheDSAistypicallyapowerlaw
σt 1. (10)
∼ c ∼ dN
γ- p (16)
Compressionalamplificationofthemagneticfieldatthispoint dγ ∝
is of the order of unity. However it is commonly argued
in a range γ γ γ . We take the minimum Lorentz
(Kulsrud 2005, and references therein) that a turbulent dy- min max
≤ ≤
factorofCRstoequal(Achterbergetal.2001,andreferences
namo produces magnetic energy near equipartition with the
therein)
turbulentenergy.Therefore,theupstreamfluidthatisexposed
γ Γ2. (17)
toacurrentJ overatimetwilldevelopturbulentmotionon min
CR ∼
allscales Numerical simulations of acceleration in ultrarelativis-
λ.λ JCRB0t2. (11) tic shocks (Achterbergetal. 2001; Ellison&Double 2002;
max∼ ρ c Lemoine&Pelletier 2003) agree with the spectral index
0
p = 20 2.22 derived analytically by Keshet&Waxman
Eacheddy amplifiesthe magneticenergyonits ownscale 9 ≈
(2005) for isotropic diffusion. While the true spec-
on an e-folding time 1/σ(λ). The bulk of the kinetic en-
tral index will depend on the detailed interaction be-
ergy initially stored in expanding shells 1(σλ )2 turns
∼ 2 max tween CRs and magnetic turbulence (e.g., Ellison&Double
intoturbulentmotionandthemagneticfieldonscales λ .
∼ max 2004;Lemoine&Revenu2006;Niemiec&Ostrowski2006;
Thefieldinequipartitionwith theenergyinturbulenteddies
Lemoine,Pelletier,Revenu 2006) and may differ from this
equals
derivedvalue,weadopt
4πλJ B 1/2
B CR 0 . (12) 20
1∼(cid:18) cρ (cid:19) p= (18)
0 9
Thisistheminimumfieldstrengthgeneratedintheshockup- inwhatfollows.IfE isthetotalenergyinCRsintheshock
CR
streamonscalesλ λmax. TheinteractionbetweentheCRs wave,thentheCRspectrumisnormalizedsuchthat
≤
andthegeneratedfieldcouldcontinueto acceleratethe fluid
γmax dN
after the shells have collided once. This could result in ad- γm c2 dγ=E . (19)
p CR
ditionalturbulentdrivingandfieldgrowth. Thenasaturation Z dγ
γmin
occurswhentheAmpèreforcebalancesthemagnetictension
WerelatetheLorentzfactoroftheshocktothetotalenergy
B2 intheblastwave(Blandford&McKee1976)
F , (13)
| |∼ 4πλ E 1/2
Γ tot , (20)
whichimpliesalimitonthefinalfieldof ≈(cid:18)n m c2R3(cid:19)
0 p
4πλJ 4πλJ
B ret CR. (14) 6 Magnetic fieldexceeding equipartition withtherestenergy oftheup-
2
∼ c ∼ c streamwouldaffectthehydrodynamicprofileoftheshockandacceleratethe
Bell (2004), who considered Newtonian shocks, argues that ups7trTehaempirneetxhiestdinirgecmtiaognnoefticshfioeclkdpcoroupldaghaatvioenp,otwheerreobnytrheedluacrignegstvsscaanledsJ∼CRR.
such a field is generatedas a consequenceof CR streaming. andthusitcoulddominatetheCRmotionevenifthenewfieldisstronger.
4 CosmicRayPrecursorinGRBShocks
wheren ρ /m . Weassumefurtherthatafraction andtoexpressdistancefromtheshockintermsofthedimen-
0 0 p
≈
sionlessparameter
E
ǫCR≡ ECtoRt (21) ∆˜ ∆Γ2 1. (27)
oftheblastwaveisstoredintheCRs. ≡ R ≤
We proceed to outline the procedure by which the shock- Themaximumγ andλcorrespondtoCRsthatreachfarthest
acceleratedCRLorentzfactor,theshock-generatedmagnetic fromtheshock,i.e.,to∆˜ =1.
field coherence length, and the energy density in the shock
generatedmagneticfield arecalculatedself-consistentlyasa 3. GAMMA-RAYBURSTAFTERGLOWS
functionofdistancefromtheshock∆. In§3,weapplythis
Externalshocksin GRBs are the bestastrophysicalcandi-
proceduretoexternalGRBshocks.
dates of weakly magnetized relativistic collisionless shocks.
They have Lorentz factors Γ&100 when the blast wave is
2.4. TheSelf-ConsistentSolution
ataradiusR 1017cm(Piran2005,andreferencestherein).
Atanydistance∆fromtheshock,theCRcurrentisdom- The blast wa∼ve decelerates and becomes Newtonian at R
inated by the least energetic CRs that reach this distance. 1018cm. Theexternalshocksareinitiallybeamedinthefor∼m
CRs with smaller γ are confined to shorter distances from ofajetwithatypicalopeningangleθ 0.1rad. Theangular
j
the shock. The CR current JCR(∆) is larger at smaller ∆. sizeofthecausallyconnectedregioni∼ntheshockis1/Γ. At
Notethetimeoverwhichanupstreamelementisexposedto earlytimeswhenΓ>1/θ,theblastwavehasisotopicequiv-
j
JCR(∆) is proportionalto ∆≪R, i.e., since the cosmic rays alentenergyEtot=1052- 1054ergsandevolvesasaspherical
are confined so close to the shock front in an ultrarelativis- fragmentwith a Lorentz factor given by equation (20). The
tic shock, the time available for generation of the magnetic evolution at late times when Γ.1/θ is poorly understood:
j
field is short. The field is generated on scales λ for which the opening angle is expected to increase; Γ decays faster
σ(λ,∆)∆/c&1..8 In calculating σ, the distance CRs reach (perhaps exponentially)with R, and the isotropic equivalent
fromtheshockandtheCR numberandcurrentdensitiesare energyapproaches 1051ergs.
allcalculatedusingthegeneratedfieldB,ratherthanthepre- Here we explore∼the interaction between the CRs and the
existing field B0. A self-consistent solution for the CR and pre-existingmagneticfieldinthecircum-burstmediumwhile
magneticfieldprofileaheadoftheshockshowsthatCRswith theevolutionoftheblast-waveisquasi-spherical(Γ>1/θ).
j
smallerγ generateastrongermagneticfield(becauseoftheir We expect that our results are qualitatively applicable also
largerJCR)onsmallerscalesλ(becauseoftheshorterexpo- when Γ<1/θj by taking Etot 1051ergs and R 1018cm.
suretime)andareconfinedbytheirself-generatedfield. The WederivethemaximumLoren∼tzfactorγ ofCRst∼hatreacha
maximumCRLorentzfactorγmaxandthemaximumfieldcor- distance∆fromtheshock,themaximummagneticfieldcor-
relTaotiosonllveenfgothrγλ(m∆ax),aλre(∆ob),taainndedBf(o∆r)∆,w∼eRpr/oΓc2ee(dseaes§fo2l.l1o)w.s. erenleartgioyndleennsgittyhiλntahtethmisagdniesttaicncfiee,ldasnodnthsceaflerascλti.on ǫB of the
First,weeliminaterg andX fromequations(1),(2;thediffu- Followingthesolutionoutlinedin§2.4weobtain
sivecase),and(3)toobtain
γ2m2c4 γ∼5×107∆˜0.25 ǫ0- .162B0- .637E503.75R1- 71..54n-00.37,
λ∆∼ Γ4e2pB2 (22) λ∼1011∆˜0.70 ǫ0- .125B0- .655E5-30.7R317.0.45n0- 0.55cm
Next,weeliminatenCR,γmin,andECRfromequations(4)and ǫB∼10- 2∆˜- 1.21ǫ0- .199B0- .619E513.21R1- 73..584n-01.19, (28)
(15—21)toobtain where∆˜ =1 correspondsto the highestenergyCRs thatare
JCR= 18πΓγ4/191ǫ/C9mREctoRte2∆. (23) aticocneloefrattheedeinnerthgey Eshtootccka.rriHederbey, ǫCCRRs=ah0e.1aǫd- 1ofisthteheshforacck-,
p Etot = 1053E53ergs, B0 = B- 6 µG, nism = n0cm- 3, and R =
Next,weeliminateσfromequations(9)and(10)toobtain 1017.5R cm. We have used p= 20 but the dependenceon
17.5 9
pisweak. Evidently,intheshocksconsideredhere,thefield
J B 1/2∆
CR 0 1. (24) generated in the precursor has the capacity to confine CRs
(cid:18) λρ0c (cid:19) c ∼ withhigherenergiesthanapreexistingmicrogaussfield.9
Finally,wesetthestrengthofthesaturatedmagneticfieldto Note that λ λ 2 107n- 1/2cm. The fractional
max ≫ s ∼ × 0
B where the Ampère force balances magnetic tension (see magnetic energy density ǫ should be approximately pre-
2 B
§ 2.2; the results for saturation at B are qualitatively the served as the fluid passes the shock transition. The depen-
1
same) dence of ǫ on the radius and the field correlation length is
B
B 4πλJCR. (25) ǫB ∝R1.41λ- 1.73 (it is independentof Etot if ǫCR is constant).
∼ c Note that ǫB is larger on smaller scales since it is generated
We substitute Γ from equation (20) in equations (22) and atsmaller∆bythelargerJCR thatisdominatedbyCRswith
smallerγ.
(23). ThenwesubstituteJ fromequation(23)inequations
CR Sincetheupstreamisturbulentonscales.λ ,andsince
(24)and(25). We finallysolveforγ, λ, andB asafunction max
of∆andtheparametersoftheblastwave. Itisconvenientto this turbulence is accompanied by density inhomogeneities,
the shock transition and the fluid passing the transition are
expressthemagneticfieldstrengthintermsof
also expected to be turbulent on the same scales. The evo-
B2 lution of the magnetic field after it passes the transition is
ǫ (26)
B≡ 4πρc2
9 ThemaximumLorentzfactortowhichCRsareaccelerated inthepre-
8Inreality,fieldsaturationwillrequiremultiplee-foldings,σ∆/c∼ few. existingfieldoftheISMisγmax(B0)∼4.6×106E513/2B- 6n0- 1/2R-171./52.
MILOSAVLJEVIC´ &NAKAR 5
described by the laws governing MHD turbulence, and we Evenunderthemostoptimisticassumptions(ǫ 1onscales
B
do not attempt to describe it here. The value of ǫ in the λ R/Γ2),theresultingLorentzfactorsare ∼
B
∼
downstreamregionwillbethatreachedintheshockupstream,
modifiedbyanyevolutionthattheturbulenceexperiencesas γ<3 1010E1/3 n1/6, (29)
ittravelsdownstreampasttheshocktransition. × tot,51 0
The main premise of ourtreatmentis that the precursoris whereE =1051E ergsisthetotal,beamingcorrecteden-
tot tot,51
dominatedbyprotonsandthuscarriesanetpositivecurrent. ergyofthe blastwave. Thereforewe donotexpectUHECR
Tocompetewiththeprotonsinthedistancetheyreachahead accelerationinexternalGRBshocks.
oftheshock,theelectronswouldhavetohaveLorentzfactors
γe>γminmp/me Γ2mp/me.Electronswithγe 108- 1010in 5. CONCLUSIONS
∼ ∼
therestframeoftheupstreamarecooledbythesynchrotron
CRs accelerated in relativistic collisionless shocks excite
mechanism on timescales much shorter than the age of the
large-scales turbulence and magnetic field generation in the
shock, and thus their acceleration to even higher energies is
shock upstream. The field generated in the shock precursor
suppressed. Therefore,indeed,theshockprecursoratproton
haspoweronscalesmuchlargerthantheprotonplasmaskin
Lorentzfactorsγ &105 iscompletelydominatedbyprotons
depth.ThepropagationofCRsinthegeneratedfieldisdiffu-
andcarriesanetpositivecurrent.
sive. InexternalGRBshocks, CRs areacceleratedto higher
energies in the generated field than in the pre-existing field
4. DISCUSSION
oftheISM.Thegeneratedfieldreachesequipartitionwiththe
Thepicturepresentedherecastsdoubtonthe roleofTWI
energydensity in the fluid. The shocktransitionis turbulent
incollisionlessshocks. ItiscommonlyarguedthatTWIgen-
withahydrodynamicprofiledominatedbyCRpressure. The
erates the magnetic field that facilitates collisional coupling
commonlyinvokedTWI is probablyquenchedin relativistic
in the shock transition. However, if the upstream is weakly
collisionlessshocksbythemagneticfieldandtheturbulence
magnetized,the shockacceleratesCRs, and these CRs drive
generatedintheshockprecursor.ExternalGRBshocksdonot
upstreamturbulenceandmagneticfieldgeneration.Whenthe
accelerateUHECRs. PICsimulationsofcollisionlessshocks
upstreamfluidreachesthetransition—whereTWIisassumed
mustincludea weakupstreammagneticfield andsimulatea
totakeplace—thelarge-scalefieldgeneratedintheprecursor
spatialdomainaslargeastheCRgyroradiusintheupstream
willplausiblyquenchTWI.Hededal&Nishikawa(2005)find
toobservetheaccelerationofparticlesbeyondequipartition.
thatTWIisquenchedforratiosoftheelectronplasmatocy-
clotron frequencyω /ω <5. For nonrelativistic upstream
pe ce
electronsthis implies quenchingfor ǫ &2 10- 5, which is We thankJonathanAronsandAndreiGruzinovforinspir-
B
expectedfromouranalysis.10 × ingustoexploretheimportanceofupstreamturbulence. We
Recently, a question has been raised whether external are also indebted to Tony Bell, Don Ellison, and an anony-
GRB shocks are sites of UHECR acceleration (γ 1010- mousrefereeforsendingusdetailedcommentsandtoSteven
1011, Dermer 2002; Waxman 2004). The idea th∼at UHE- Cowley and Marc Kamionkowskifor illuminating conversa-
CRs are produced in GRBs (Waxman 1995; Vietri 1995; tions. M. M. thanks the Center for Cosmology and Particle
Milgrom&Usov 1995) is motivated by the observation that PhysicsatNewYorkUniversityforhospitalityandacknowl-
the cosmic energy densities in UHECRs and GRB fireballs edgessupportbyNASAthroughtheHubbleFellowshipgrant
are comparable. The presumed absence of strong upstream HST-HF-01188.01-A awarded by the Space Telescope Sci-
magneticfieldsinexternalGRBshockshasfocusedattention ence Institute, which is operated by the Association of Uni-
onUHECRproductioninshockspropagatingintorelativistic versities for Research in Astronomy, Inc., for NASA under
ejecta(Waxman2004;Gialis&Pelletier2005). Ouranalysis contract NAS5-26555. E. N. was supported at Caltech by a
of external shocks indicates that the maximum Lorentz fac- seniorresearchfellowshipfromtheShermanFairchildFoun-
tor of CRs producedthere is well below the UHECR range. dation.
10 Because of the finite transverse velocities of the CRs (∼c/Γ in the
frameoftheISM),theCRsthemselveswillnotexciteTWIintheupstream.
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