Spectral and temporal signatures of ultrarelativistic protons in compact sources

0.2 MB·English

by  A. Mastichiadis

Astronomy&Astrophysicsmanuscriptno.2161paper February2,2008
(DOI:willbeinsertedbyhandlater)
Spectral and temporal signatures of ultrarelativistic protons in
compact sources
5
0 I. Effects of Bethe-Heitler pair production
0
2
n A.Mastichiadis1,R.J.Protheroe2,andJ.G.Kirk3
a
J
1 DepartmentofPhysics,UniversityofAthens,Panepistimiopolis,GR15783,Zografos,Greece
0
e-mail:[email protected]
1
2 DepartmentofPhysicsandMathematicalPhysics,UniversityofAdelaide,NorthTerrace,Adelaide,SA5005,Australia

1 3 Max-Planck-Institutfu¨rKernphysik,Postfach103980,69029Heidelberg,Germany
v
6 Received?;accepted?
5
1
Abstract. Wepresentcalculationsofthespectralandtemporalradiativesignaturesexpectedfromultrarelativisticprotonsin
1
compactsources.ThecouplingbetweentheprotonsandtheleptoniccomponentisassumedtooccurviaBethe-Heitlerpairpro-
0
duction.ThisprocessistreatedbymodelingtheresultsofMonte-Carlosimulationsandincorporatingtheminatime-dependent
5
kineticequation,thatwesubsequentlysolvenumerically.Thus,thepresentworkis,inmanyrespects,anextensionofthelep-
0
/ tonic‘one-zone’modelstoincludehadrons.Severalexamplesofastrophysicalimportancearepresented,suchasthesignature
h resultingfromthecooling ofrelativisticprotons onanexternal black-body fieldand thatof theircooling inthepresenceof
p radiationfrominjectedelectrons.Wealsoinvestigateandrefinethethresholdconditionsforthe’PairProduction/Synchrotron’
-
o feedbackloopwhichoperateswhenrelativisticprotonscoolefficientlyonthesynchrotronradiationoftheinternallyproduced
r Bethe-Heitlerpairs.Wedemonstratethatanadditionalcomponentofinjectedelectronslowersthethresholdforthisinstability.
t
s
a Keywords.Radiationmechanisms:nonthermal;Radiativetransfer;Galaxies:active
:
v
i
X
1. Introduction part of the SED (synchrotron self-Compton model) possibly
r supplementedby inverseComptonscattering of externalpho-
a
Thespectralenergydistribution(SED)ofpowerfulAGNsuch
tons(externalComptonmodels)forexamplefromthedisk,ei-
asflat-spectrumradioquasarsandblazarshasadoublehumped
ther directly or scattered by clouds. Despite these successes,
appearancewiththelowenergypartextendingfromtheradio
thequestionoftheroleofapossiblerelativistichadroniccom-
toUV (orinextremecasestoX-rays),andahighenergypart
ponentremainsanopenone.
extendingfromX-raystoγ-rays.InAGNwithrelativisticjets
In principle, sites of electron acceleration may accelerate
closelyalignedtothelineofsighttheemissionisdominatedby
protonsaswell.Consequently,modelsinwhichthehighenergy
non-thermalradiation,withthelowenergyhumpbeingmainly
part,andsomefractionofthelowenergypart,oftheSEDisdue
synchrotronradiation.Ifthealignmentisnotsoclose,athermal
to acceleration and interaction of protons in the jet have also
componentofUV radiationfroman accretiondiskmaydom-
beenproposed.Someofthesemodelsinvokeinteractionswith
inate. The non-thermal components can be strongly variable,
ambientmatter(Beall & Bednarek1999;Pohl& Schlickeiser
probablyoriginatinginthejet.
2000;Schusteretal.2002)buttheyrequirehighmassdensities
These observations indicate that the jets of blazars act as
in thejettobe viable.Herewe concentrateonhadronicmod-
efficientparticleaccelerators.Furthermorethegamma-rayob-
els in which the protonsinteractwith low energyphotonsvia
servations in the GeV (Hartman et al. 1999) and TeV regime
Bethe-Heitlerpairproduction.
(Horan & Weekes 2004) imply that the accelerated particles
As with leptonic models, the target photons may be pro-
can reach veryhigh energies.Modelsinvolvingelectronradi-
duced inside the emission region in the jet or may originate
ation can adequately explain both this high energy emission
fromoutsideofthejet,e.g.,fromanaccretiondisk(Protheroe
andthecoordinatedmultiwavelengthcampaigns(Mastichiadis
1997; Bednarek & Protheroe 1999; Atoyan & Dermer 2001;
&Kirk1997;Tavecchioetal.2001;Krawczynskietal.2002).
Neronov & Semikoz 2002). For internally produced target
The usual assumption is then that the high energypart of the
photons, synchrotron emission by a co-accelerated popula-
SED is due to inverse Compton scattering of the low energy
tion of electrons is assumed (Mannheim & Biermann 1992;
Sendoffprintrequeststo:A.Mastichiadis Mannheim 1993, 1995). The high energy hump of the SED
2 Mastichiadisetal.:Hadrons
then results from electromagnetic cascading of gamma-rays Heitler process and show how these can be incorporated in
from π0 decay and electrons from Bethe-Heitler pair produc- thekineticequations.InSection4wepresentsomeresultsfor
tion and π± → µ± → e± decay in the radiation and mag- the case in whichrelativistic protonsinteractwith an external
netic field of the blob. Neutrinos and cosmic rays would also black-bodyradiation field. In Section 5, we present a numer-
be emitted as a result of the neutrinos from the π± and µ± ical analysis of the ’Pair-Production/Synchrotron’ instability.
decays, and neutrons produced in pγ → nπ+ interactions if The case of simultaneous injection of relativistic protons and
the threshold for pion photoproduction is exceeded (Eichler electrons is examined in Section 6 and the main conclusions
& Wiita 1978;Sikora et al. 1987,1989; Kirk & Mastichiadis aresummarizedinSection7.
1989; Begelman et al. 1990; Giovanoni & Kazanas 1990;
Protheroe & Szabo 1992; Szabo & Protheroe 1994; Waxman
2. Thekineticequationsforelectrons,protonsand
& Bahcall 1999; Mannheim et al. 2001; Atoyan & Dermer
photons
2003; Protheroe 2004). For protons to be accelerated to en-
ergies sufficient to exceed the Bethe-Heitler and photo-pion- Thekineticequationsdescribingahomogeneoussourceregion
production thresholds, relatively high magnetic fields are re- containingprotons,electronsandphotonswereformulatedand
quired.Inprotonsynchrotronblazarmodelsthemagneticfield solvednumericallybyMastichiadis&Kirk(1995,henceforth
is sufficiently high such that the the high energy part of the MK95). We followthesamemethod,usinganimprovedde-
SEDhasamajorcontributionalsoduetosynchrotronradiation scription of the microscopic processes. The equations to be
byprotons(Mu¨cke&Protheroe2000,2001;Aharonian2000; solvedcanbewritteninthegenericform
Reimeretal.2004).Alloftheworkdescribedaboveassumes
∂n
theemissionhasreachedasteadystate,andthatthetargetpho- i +L +Q =0 (1)
∂t i i
tonfieldsaresteady.Inreality,thestrongvariabilitydisplayed
by these sourcesmandatesa time-dependentcalculation,that, where the index i can be any one of the subscripts ‘p’, ‘e’ or
ideally, should be done self-consistently, with internally pro- ‘γ’referringtoprotons,electronsorphotonsrespectively.The
duced radiation fields contributing alongside external ones to operatorsLidenotelossesandescapefromthesystemwhileQi
thetargetradiationfield. denoteinjectionandsourceterms.Thesearedefinedbelow.
Time-dependentcodesthatsolvethekineticequationsde- Theunknownfunctionsni arethedifferentialnumberden-
scribingelectronsandphotonsandtheirinteractionshavebeen sitiesofthethreespecies,normalisedasfollows:
developed and successfully applied to AGN (Mastichiadis &
Protons:
Kirk 1997; Krawczynski et al. 2002). However, codes of this E
typethatalso accountforhadronicinteractionshave beenne- n∗p(γp,t)dγp = σTRnp(Ep,t)dEp withγp = m pc2 (2)
p
glected. One reason for this is that whereas the modeling of
Electrons:
leptonicprocessesisrelativelystraightforward(e.g.,Lightman E
n∗(γ ,t)dγ =σ Rn (E ,t)dE withγ = e (3)
& Zdziarski1987;Coppi& Blandford1990),photo-hadronic e e e T e e e e m c2
e
and hadron-hadron interactions are much more complex. To
Photons:
ǫ
date,allattemptshaveusedapproximationsofuncertainaccu- n∗(x,t)dx = σ Rn (ǫ ,t)dǫ withx = γ (4)
racy(e.g.,Stern&Svensson1991)buttheuseofMonte-Carlo γ T γ γ γ mec2
eventgeneratorswhichmodelindetailelectromagnetic(Szabo
andthetimethasbeennormalisedinallequationstothelight-
& Protheroe 1994; Protheroe & Johnson 1996) and hadronic crossingtimeofthesourcet =R/c.
cr
interactions(Mu¨ckeetal.2000)opensupthepossibilityofex-
Thephysicalprocessesto beincludedin thekinetic equa-
tracting accurate descriptions of the fundamental interactions
tionsare:
suitableforincorporationintoakineticcode.
Motivated by these developments we investigate the con- 1. Proton-photon (Bethe-Heitler) pair production which acts
sequences of the presence of relativistic hadrons in compact as a loss term for the protons(LBH) and an injection term
p
sourcesbyincorporatingnewresultsfromMonte-Carlosimu- fortheelectrons(QBH)
e
lationsintoatime-dependentcodewhichfollowstheevolution 2. Synchrotronradiationwhichactsasanenergylosstermfor
of relativistic hadrons, electrons and photons by solving the electrons(Lsyn)andasasourcetermforphotons(Qsyn)
e γ
appropriate kinetic equations. In the present paper we inves- 3. Synchrotron self absorption which acts as an absorption
tigateasafirststep,thecaseinwhichtheonlychannelofcou- termforphotons(Lssa)
γ
pling between hadrons and leptons is the Bethe-Heitler pair- 4. Inverse Compton scattering (in both the Thomson and
production process, leaving the investigation of photo-meson Klein-Nishinaregimes) which acts as an energyloss term
productionforafuturepaper.Althoughthisisnota complete for electrons (Lics) and as a source term for high energy
e
description of hadronic models it nevertheless enables one to photons and a loss term for low energy photons, both ef-
drawusefulandinterestingnewresults. fectsincludedinQics
γ
Thepresentpaperisstructuredasfollows:InSection2we 5. Photon-photonpair production which acts as an injection
presentthenumericalcodethatsolvessimultaneouslyinaself- termforelectrons(Qγγ)andasanabsorptiontermforpho-
e
consistent manner the coupled, time-dependent kinetic equa- tons(Lγγ)
γ
tions for each species, i.e. protons, electrons and photons. In 6. Electron-positronannihilationwhichactsasasinktermfor
Section 3 we present the Monte-Carlo results for the Bethe- electrons(Lann)andasasourcetermforphotons(Qann)
e γ
Mastichiadisetal.:Hadrons 3
7. Compton scattering of radiation on the cool pairs, which Inanalogousfashionwecanalsodefinethecompactnessofthe
impede the free escape of photons from the system. This externallyinjectedelectrons
effect is treated approximately by multiplying the photon
1
escapetermbythefactor(1+H(1−x)τ /3)−1,whereτ the ℓext = dγ(γ−1)Qext(γ). (10)
T T e 3Z e
Thomsonopticaldepth,whileH(1−x)isthestep-function.
(Lightman&Zdziarski1987) Finallyin ordertocalculatetheThomsonopticaldepthofthe
coolpairsweuse
Withtheinclusionoftheabovetermsthekineticequations
become, for each species (from now on we refer only to nor- 1.26
τ = dγn (γ), (11)
malisedquantitiesand,forconvenience,droptheasterisks) T e
Z
1
– Protons Closing this section we note that the treatment of syn-
chrotron and inverse Compton scattering has been improved
∂n n
p +LBH+ p = Q (5) over that described by MK95 in that the full emissivities
∂t p t p
p,esc are incorporated, rather than delta-function approximations.
– Electrons However, the main improvement is in the treatment of the
Bethe-Heitlerpair-productionprocess,usingMonte-Carlosim-
∂n n
e +Lsyn+Lics+Lann+ e = Qext+QBH+Qγγ (6) ulations,asdescribedinthefollowingsection.
∂t e e e t e e e
e,esc
– Photons
3. Bethe-Heitlerpairproduction
∂n n
∂tγ + 1+τγ /3 +Lγγγ+Lsγsa = Qsγyn+Qiγcs+Qaγnn (7) 3.1.Monte-Carlosimulations
T
For an isotropic target comprising monoenergetic photons of
Wenotethefollowingregardingtheaboveequations
energyε = xm c2 theeffectivecross-sectionforinteractionof
e
1. Whenthevarioustermsabovearewrittenexplicitly,equa- aprotonofenergyE =γpmpc2isgivenby
tions (5),(6) and (7) forma non-linearsystem of coupled
1 cosθmin(γp,x)
integro-differentialequations. hσ (γ ,x)i= (1−β cosθ)σ(s)dcosθ (12)
BH p p
2. The various rates conserve the energy exchange between 2Z−1
the species – for example, the amount of energy lost per whereθistheanglebetweentheprotonandphotondirections,
secondbyelectronsateachinstantduetosynchrotronradi- θ (γ ,x) is the minimumvalue of this angle consistentwith
min p
ationisequaltothepowerradiatedinsynchrotronphotons, thethreshold,
(fordetailsseeMK95)
3. In the absence of the Bethe-Heitler pair-production term s=m2pc4+2εE(1−βpcosθ) (13)
QBH, equations (6) and (7) decouple from the protons
e is the centre of momentum (CM) frame energy squared, β c
(Eq.5)andthesystembecomesidenticaltothe’one-zone’ p
is the proton’s velocity, and σ the total cross-section for
time-dependentleptonicmodels BH
whichweusetheRacahformulaasparameterizedbyMaximon
4. ProtonsareinjectedviathetermQ withaprescribeddis-
p (1968)(see Formula3D-0000in Motz et al. 1969).Changing
tribution in energy.Thus, in contrastto MK95, we do not
variables,oneobtainstheangle-averagedcross-section
investigatetheeffectsofparticleacceleration.
5. Electronsmayalsobeinjectedexternallythroughthepre- 1 smax(γp,x)
sthceribeeledctterormn eQqeeuxta.tHioonw(eQvBeHrtahnedtwQoγγo)thaerreindjeetcetrimoninteedrmseslifn- hσBH(γp,x)i= 8βpE2ε2 Zsmin σ(s)(s−m2pc4)ds, (14)
e e
consistentlyfromtheprotonandphotondistributions. where
6. Both protons and electrons (we make no distinction be-
s =(m c2+2m c2)2 ≈0.882GeV2, (15)
tweenelectronsandpositronsinthepresenttreatment)es- min p e
capefromthesourceregiononthetimescalestp,escandte,esc and
respectively(giveninunitsoft )
cr s (γ ,x)=m2c4+2γ m c2xm c2(1+β ) (16)
max p p p p e p
We can also define various compactnesses related to the
corresponding to a head-on collision. The angle-averaged
mostimportantoftheabovequantities.Sowedefinethephoton
cross-sectionisplottedas a functionofthe productofphoton
compactnessas(seeMK95)
energyand protonenergyin Fig. 1. For x ≪ 1, the threshold
1 nγ(x,t) conditionimpliesγ ≫ 1andβ ≈ 1,sothattoafirstapprox-
ℓ = dxx (8) p p
γ 3Z tcr(1+H(1−x)τT/3) imation,thecross-sectionisafunctionof xγp,ratherthanof x
andγ separately.
while the scaled to electron rest-mass compactness of exter- p
ExaminationoftheintegrandinEquation14showsthatthe
nallyinjectedprotonsis
squareofthetotalCMframeenergyisdistributedas
m
p
ℓp = dγ(γ−1)Qp(γ). (9) p(s)∝σ(s)(s−m2c4), (17)
3me Z p
4 Mastichiadisetal.:Hadrons
Fig.1. The angle-averaged Bethe-Heitler pair-production cross Fig.2.Thedistributionofelectronenergiesisshownforx=10−6and
section hσ (γ ,x)i (solid curve), the angle-averaged Bethe- γ = 106.3 (dashedhistogram),107 (solidhistogram)and109 (dotted
BH p p
Heitler pair-production inelasticity ξ (γ ,x) (dashed curve), and histogram).
BH p
ξ (γ ,x)hσ (γ ,x)i(dottedcurve) areplottedasafunctionofγ x
BH p BH p p
forx≪1.
3.2.NumericalModeling
3.2.1. Electron/positronPairProduction
in the range s ≤ s ≤ s . The Monte Carlo rejection
min max
The pair-production spectra were calculated using the results
technique is used to sample s, and Equation 13 is used to
oftheMonteCarlocodedescribedinSection2.1.Protonsofa
find θ. We then Lorentz transform the interacting particles to
specificenergywereallowedtointeractwithisotropicmonoen-
the proton rest frame and sample the positron’s energy from
the single-differential cross-section, dσ/dE , for which we ergetic target photons and the energies of the products were
+
tabulated. The photon target energies used were x = 10−6,
usetheBethe-Heitlerformulaforanunscreenedpointnucleus 0
10−4, 10−2 and1 (inunitsofelectronrestmass). Protonener-
(Formula3D-1000inMotzetal.1969).Finally,thepositron’s
directionissampledfromthedouble-differentialcross-section, giesrangedfromγp = x−01 (sothethresholdrequirementcould
dσ/dE+dΩ+ forwhichweusetheSauter-Gluckstern-Hullfor- bemet)uptoγp =104x−01inlogarithmicstepsof0.1.
mula for an unscreened point nucleus (Formula 3D-2000 in Thepair-creationrateisthengiven
Motzetal. 1969),anditslaboratoryframeenergyis obtained
by a Lorentz transformation. For a range of proton energies, QBeH(γe,t) = dγpnp(γp,t) dxnγ(x,t)×
Z Z
thesimulationisrepeatedalargenumberoftimestobuildup
f(γ ,γ ,x)hσ (γ ,x)i (20)
distributions in energy of positrons producedin BH pair pro- p e BH p
duction. where f(γ ,γ ,x) is the distribution found from the Monte-
p e
The distribution in energy γemec2 of electrons (of either Carlomodelling(seeeq.18),normalisedsuchthat
charge),
dγ f(γ ,γ ,x)=2. (21)
e p e
dN Z
f(γ ;γ ,x)≡ e, (18)
e p dγe Assimpleparameterisationsofthesecurvesdonotgiveac-
ceptable fits, we tabulated the spectra and used interpolations
istakentobetwicethatforpositrons,asdiscussedinProtheroe
toderivetheproducedpair-injectionspectrumfromaparticular
& Johnson (1996), and is plotted in Fig. 2 for x = 10−6 and
proton-photoncollision.
threeγ values.Inasimilarcalculationusingblack-bodytarget
p We notethatMK95usedthe approximation f(γ ,γ ,x) =
p e
photons(Protheroe&Johnson1996),themeaninelasticitywas
2δ(γ −γ )correspondingtoξ =m /m .Foraprotondistribu-
e p e p
foundto be in excellentagreementwith those calculatedana-
tionn (γ )whichdecreasesasγ increases,thisoverestimates
p p p
lytically (Blumenthal1970;Chodorowskietal. 1992;Rachen
theproductionrateofelectronsofagivenenergy.
&Biermann1993).Themeaninelasticity,givenby
m γ 3.2.2. Protonlosses
ξ(γ ,x)= e dγ e f(γ ;γ ,x) (19)
p mp Z eγp e p Since the proton loses a small amount of energy (typically
givenby∆γ ∼m /m ineachpair-producingcollisionwecan
is plotted as a function of the product of photon energy and p e p
treatthelossesasacontinuousprocessandwrite
protonenergyinFig.1(dashedline)togetherwithitsproduct
(dottedcurve)with the cross-section,to show how the proton ∂
LBH = γ n (γ ,t)Y(γ ,t) (22)
energylossratedependsonenergy. p ∂γ p p p p
p h i
Mastichiadisetal.:Hadrons 5
where
Y(γ ,t)= dxn (x,t)hσ (γ ,x)iξ(γ ,x) (23)
p γ BH p p
Z
000
istheconvolutionofthenormalisedcollisionratewiththein-
elasticityξgivenbyeqn.(19).
4. Protoninjectionandblack-bodyphotonfield
As a first example, we examinethe case of relativistic proton
---555
injectionandsubsequentcoolingonablack-bodyphotonfield.
Weassume,asusualinastrophysicalcases,apower-lawproton
injectionoftheform
Q (γ)= Q γ−βH(γ−γ )H(γ −γ) (24)
p p,0 p,min p,max
whereγ andγ arerespectivelytheloweranduppercut-
p,min p,max ---111000
offoftheprotondistribution.Wealsoassumethattheprotons
canescapeataratet−1 fromthesourceregion,takentobea
p,esc
sphereofradiusR.Thenthestationarysolution,intheabsence
---555 000 555
ofprotonlossesissimply
n (γ)= Q t γ−βH(γ−γ )H(γ −γ). (25)
p p,0 p,esc p,min p,max Fig.3.Steady-statephotonspectrumresultingfromapower-lawpro-
ton injection and subsequent proton-photon pair production on a
We assumenextthatasignificantnumberofexternalpho-
black-body photon field in the case where photon-photon pair pro-
tons is present and that these have a black-body distribution
duction(i)hasbeenignored(dashedline)and(ii)hasbeentakeninto
oftemperatureT .TheenergydensityU ofthesephotons
BB BB account(fullline).Thedottedlinecurvedepictstheelectron/positron
inthesourcecanbeparameterisedintermsoftheblack-body
distributionfunctionatproduction.Forthisparticularruntheprotons
compactnessℓBB,definedbytherelation wereassumedtobeinjectedwithapower-lawofslopeβ=2,between
ℓBB = UBmBσc2TR (26) athnedlitmp,eistcs=γp,mtcirnh=av1e00b.1eeanndaγsspu,mmaxe=d.1T0h6e.Abllasockt-hbeovdaylupehsoℓtpon=fi2e.5ld10p−a3-
e rameterswereT =105K(Θ=1.710−5)andℓ =1.Synchrotron
BB BB
Thisemissioncanarise,forexample,fromthesurfaceofanac- losseswereneglected.
cretiondisklocatedclosetothesource.Ingeneral,ifthesource
isirradiatedbyablackbodyattemperatureT whosesurface
BB 4.1.Case1:ℓ ≪ ℓ andnegligiblesynchrotron
occupiesasolidangleof∆Ωasseenfromthesource,then p BB
losses
∆Ω
Ubb = 4π!aradTB4B (27) In this case, the photonsproducedby electronscreated in the
BHprocessarenotimportantastargetsandtheresultingpho-
where arad is the radiation density constant. In terms of com- tonspectrumisquitesimple.Itisasinglepower-lawof(num-
pactness: ber)spectralindex−3/2,extendinguptoanenergyx ,thatis
max
approximatelytheinverseofthetemperatureoftheblack-body
∆Ω T 4 R
ℓ = 615× BB (28) field, i.e. x ≃ Θ−1, where Θ = (kT /m c2). The explana-
BB 4π!(cid:18)105K(cid:19) (cid:18)1015cm(cid:19) tionofthismsapxectrumisstraight-forwarBdB:proetonspair-produce
We further assume that initially there are no other pho- on the black-body field and, since they are produced with
tons or electrons present (except maybe in very small num- high energies, the pairs cool on the black-body photons ini-
bers) and that there is no external electron injection. Since tiallybyComptonscatteringintheKlein-Nishinaregime.This
the Bethe-Heitler pair production has a threshold condition produces γ-rays that are above the threshold for pair produc-
γx ≥ 1+m /m (correspondingto head-oncollisions) it fol- tionontheblack-bodyphotonsand,therefore,arere-absorbed.
e p
lowsthat,inthecasewherey = γ (kT /m c2) ≪ 1,there Lowerenergypairs, which coolby Comptonscatteringin the
p,max BB e
will be negligible pair production.Therefore,for all practical Thomsonregime,producephotonswhicharebelowthethresh-
purposes,thesolutionoftheprotonequationwillstillbegiven oldforphoton-photonpairproduction.Thisnaturallyproduces
byEq.(25).However,fory∼1,theproton-photonreactionrate an electron distribution function n ∝ γ−2 and thus a photon
e
issubstantialandcannotbeneglectedasaprotonlossmecha- spectrum n ∝ x−3/2. The value of x , therefore, is set by
γ max
nism. Moreover,the produced electron/positronpairs provide theconditionτ (x ) ≃ 1,whereτ istheopticaldepthfor
γγ max γγ
an injection term for the electron equation and lose energy photon-photonpair productionon thebackgroundblack-body
mainly by inverse Compton scattering and/or synchrotron ra- field.
diation.Asaresultaphotonspectrumisformed.Toinvestigate Since for the present case there is neither escape of elec-
thepropertiesofthisemission,suchasitsluminosityandspec- tronsfromthesystemnoranysinkofenergy,(e.g.synchrotron
tralshapeetc.,wemustdistinguishbetweentwocases: self-absorption) other than photon escape, the radiated lumi-
6 Mastichiadisetal.:Hadrons
nosity (or, equivalently, ℓ ) equals the luminosity injected in
γ
pairsonceastatesteadyisachieved,i.e.ℓ ≃ℓBH,where -6
γ e
1
ℓBH = dγ(γ−1)QBH (29)
e 3Z e
isthecompactnessofthecreatedpairsfromtheBethe-Heitler
pair production. Therefore, the overall photon luminosity de- -6.5
pendsontheparameterydefinedaboveand,asweshowbelow,
alsoonℓ andℓ .
BB p
As an illustrativecase we take ℓ = 1 ≫ ℓ = 2.5 10−3.
BB p
Here also T = 105 K, while γ = 100.1, γ = 106.
BB p,min p,max
The dotted line in Fig. 3 depicts the electron injection func-
-7
tion which shows a broad maximum. The same figure shows
the photon spectrum which is obtained from the cooling of
these electrons and the ambientblack-bodyfield. The dashed
lineshowsthespectruminthecasewheretheγγpairproduc-
tionhasbeenartificiallyswitchedoff.Thespectrumisflatand
peaksathighenergies.Wenotethatboththeelectronproduc-
-7.5
tion function and the unabsorbed photon spectrum extend to 2 4 6
abouttwoordersofmagnitudeaboveγ ,aneffectthatisdue
p,mx
tothekinematicsoftheBethe-Heitlerpairproduction–seeFig.
Fig.4. Proton steady-state spectra for various external black-body
3.Atlowerenergiesitproducesthecharacteristic−3/2power-
photoncompactnesses.Allcasesaretakenforthesameinjectionpro-
law. The full line shows the final photon spectrum which in- ton parameters(ℓ = 2.5 10−3, γ = 106, t = 1) and for the
p p,max p,esc
cludesγγabsorption.Ithasstillthesamepower-law,however sametemperatureoftheexternalblack-bodyfield(T =105 K)The
BB
all the details of the electron injection spectrum which were plottedprotonspectraareshownwhenℓ =0(no-losscase,fullline),
BB
evident in the unabsorbed spectrum have disappeared due to ℓ = 1(dotted line),ℓ = 102 (short-dashed line) andℓ = 104
BB BB BB
the intense attenuation.Neverthelessthe overallluminosity is (long-dashedline).
conserved(asitshouldbe)andastheelectromagneticcascade
redistributesthepowertolowerenergies,thefluxisincreased
there. Figure 4 shows the effects of losses on the steady-state
proton distribution for these parameters (ℓ = 1) and two thesynchrotronmechanism.Therefore,asℓB progressivelyin-
BB
highervaluesofℓ ,comparedtotheloss-freecase. creases,thepeakisshiftedtowardsthelowerenergies,showing
BB
TheefficiencyoftheBHprocess,i.e.,theratiooftheproton theincreasingimportanceofsynchrotronradiationasanelec-
tronenergyloss/photonemissionmechanism.AsinFig.3,the
powerturnedinto pairsto thetotalpowerinjectedasprotons,
spectrumcontinuestobestronglyabsorbedforphotonenergies
isshowninFig.5asafunctionofthemaximumLorentzfactor
ofinjectionγ .Theinelasticityofthisprocessissmall(ξ ∼ x> xmax ≃Θ−1.
p,max
10−3),so thathighefficiencycan onlybeachievedifa proton
interactsmanytimesbeforeescape.Thisisindeedthecasefor
highblack-bodycompactnesses—theefficiencyexceeds40% 5. ThePair-Production/SynchrotronInstability
forℓ >104andγ >107.
BB p,max
5.1.Marginalstability
4.2.Case2:ℓp ≪ℓBB andnon-negligiblesynchrotron Kirk & Mastichiadis (1992, henceforth KM92) have shown
losses thatultra-relativisticprotonscan,undercertainconditions,be-
come unstable to various types of radiative instability. They
In the more realistic case, where synchrotron losses cannot
showed explicitly the necessary conditions for one of them
be neglected, the spectrum of the electrons becomes more
to occur, namely the Pair-Production/Synchrotron instability
complicated, as now the electrons cool by a combination of
(henceforth PPS). To understand the basic idea, assume that
synchrotron radiation and inverse Compton scattering. Fig. 6
protons are confined in a region of characteristic radius R
shows the obtained spectra in the cases where the magnetic
where a magneticfield of strength B is also present. Assume,
compactnessℓ ,definedaccordingto
B moreover,thattheprotonsarerelativisticandhaveLorentzfac-
B2 torssuchthatiftheyphoto-pairproduce,thesynchrotronpho-
ℓB = σTR 8πm c2! (30) tons radiated from the created pairs are sufficiently energetic
e
fortheprotonstoproducemorepairsonthem.Makingthesim-
is comparable to ℓ . The redistribution of the radiated lu- plifying assumptions that (i) the created pairs have the same
BB
minosity to increasingly lower energies is evident, the rea- Lorentzfactorsastheprotonsand(ii)thesynchrotronphotons
son being that, for this particular choice of T and B, in- are allemittedatthecriticalfrequency,KM92showedthatin
BB
verseComptonscatteringproducesmuchharderphotonsthan orderforprotonstobeabletoinitiatethislooptheyshouldhave
Mastichiadisetal.:Hadrons 7
0 222
000
-2
---222
---444
-4
---666
---888
-6
---111000
4 5 6 7 ---555 000 555
Fig.5. Proton efficiency, i.e. fraction of power lost to pairs to total Fig.6. Steady state photon spectra for the same parameters used in
powerinjectedinprotonsasafunctionoftheuppercutoffoftheproton Fig. 3 but including synchrotron radiation. The magnetic field used
distributionγ .Therestoftheprotoninjectionparametersarekept wasB=100Gandthemagneticcompactnesseswereℓ =0.01(full
p,max B
constant(ℓ = 2.510−3,γ = 100.1,t = 1),whiletheexternal line),ℓ =0.1(dashedline)andℓ =1.Theblack-bodycompactness
p p,min p,esc B B
black-body field has temperature T = 105 K. and compactnesses wasℓ =1.
BB BB
(bottomtotop)ℓ =.01, 1, 100, 104.
BB
cm,a magneticfield B = 103 Gaussanda protondistribution
Lorentzfactorsaboveacriticalvaluegivenby functionoftheformn (γ )=n γ−β withβ=2fromthelow-
p p p,0 p
2 1/3 estallowedprotonenergyγp,min = 100.1toamaximumenergy
γ = (31) γ .We notethatinthiscasetheprotondistributionisheld
crit b! p,max
constant throughout each run, i.e. protons do not evolve. For
where b = B/B with B the critical value of the magnetic variousvaluesofγ ,werunthecodefordifferentvaluesof
cr cr p,max
field(B = m2c3/e~ = 4.4131013 Gauss).Forthislooptobe theonlyremainingfreeparameter(n ).AccordingtoKM92,
cr e p,0
self-sustaineditisnecessarythatatleastoneofthesynchrotron thetimeevolutionofthephotonandelectrondistributionfunc-
photonsshouldproducea pairbeforeescapingthesourceand tionsisoftheformn (t)∝n ∝estwiths>0whentheprotons
γ e
thisconditionnaturallyleadsto a criticalprotonnumberden- areintheunstableregime.Thusinordertoverifynumerically
sity(seeKM92,eq.6). theexistenceofthePPSinstabilityweseekavaluencrit above
p,0
MK95 presenteda numericalsimulation of the PPS insta- whichtheinternallyproducedelectron/positronpairsandpho-
bilityin thecase whereprotonsare acceleratedfromlow mo- tonsstartincreasingwithtime.
mentabyaFermi-typeaccelerationscheme.Oncetheprotons The onset of instability for γ = 106 can be seen in
p,max
(assumedtohaveadensityexceedingthecriticalnumberden- Fig.7whichdepictsthephotoncompactnessl asafunctionof
γ
sity) reachedenergiesaboveγ , theconditionsfortheinsta- timet (expressed,asalways,inunitsoft )forvariousvalues
crit cr
bilityloopwerecompleteandtheinternallyproducedphotons of n aroundncrit. When theprotonsarein the stable regime
p,0 p,0
increased,saturatingtheacceleration,anddrivingthesystemto thereissomepairproductionbetweentheprotonsandthesyn-
equilibrium. However,the code used by Mastichiadis & Kirk chrotronphotonsproducedfromtheinitialelectrondistribution
(1995) is limited by the simplifying assumptions mentioned but the system, for times larger than the synchrotron cooling
in the previous paragraph. Here we re-examine this problem time,settles toa steady-state.Thusfort > t we get s = 0.
cool
with the improvedversion of the code that uses, as described However,ascanbeseenfromthefigure,asn increases(from
p,0
insection3,theBethe-Heitlerpair-productionspectraasgiven bottomto top)the photonsstart to growexponentially,with s
byMonte-Carlocodeandthefullsynchrotronemissivity.The increasingwithincreasingn .Itisworthmentioningthateach
p,0
objective is to find an accurate estimate of the critical proton curve correspondsto a value of n thatis largerby only 2%
p,0
numberdensityabovewhichthePPSinstabilityoccurs,i.e.,a than its previousvalue;therefore,this figuredepictsthe rapid
numericalversionoftheanalytical(butapproximate)Fig.1of onsetoftheinstability.
KM92. Fig.8showsthebehaviourof sasafunctionoftheproton
To make ourresultsdirectlycomparablewith those ofthe normalisation n for three values of γ . It is evident that
p,0 p,max
aforementionedfigure, we have set the parametersto the val- once the instability sets in, s is a very sharp function of n .
p,0
ues prescribedthere. For this we took a source size R = 1015 Therefore,wefindthatincreasingn approximatelybyafac-
p,0
8 Mastichiadisetal.:Hadrons
1
-46
-48
0
-50
-1
-52
-54
-2
0 200 400 600 800 1000 4 5 6 7
Fig.7. Plot of the internally produced photon compactness ℓ as a Fig.9.Plotofthenumericallyobtainedmarginalstabilityasafunction
γ
function of time for values of n around the critical value ncrit. In ofγ (fullline).Thedashedlineshowsthes= 1locus,whilethe
p,0 p,0 p,max
eachcurvethevalueofn isincreased2%fromitspreviousvalue. dottedlinedepictsthemarginal stabilityresultsofKM92. Notethat
p,0
Fortheserunsγ =106. theKM92approximationisreasonablygoodatlowγ .
p,max p,max
n whichcorrespondto s= .05.Duetothesteepdependence
p,0
of s on n this value can be considered as very close to the
1.5 p,0
marginal stability one). Note that this curve is in very good
agreementwiththecurveestimatedbyKM92(plottedhereas
adottedline)forvaluesofγclosetothethreshold,butexceeds
itbyafactorofabout2forathighenergy.Thisdifferencecan
be attributed to the overestimation of the electron production
1
rate in the BH process as discussed in Section 2. For maxi-
mum valuesof the protonLorentz factor close to γ , all the
crit
BH interactions occur close to threshold, so that the assump-
tionusedbyMKisaccurate.(Thesmalldifferenceintheshape
of the KM curve and our present result in this energy range
0.5
canbeattributedpartlytothekinematicsofBethe-Heitlerpair
production and partly to the fact that the full expression for
thesynchrotronemissivitywasused).However,oncetheupper
cut-offoftheprotondistributionsubstantiallyexceedsγ ,the
crit
effectofpairsinjectedwithγ > γ becomesimportant.These
0 p
-1 -0.8 -0.6 -0.4 -0.2 arenottakenintoaccountintheapproximationusedbyMK92,
butaretreatedaccuratelyinthesimulation-basedmethodused
here. It is evidentthat the basic conceptof the feedbackloop
Fig.8.Behaviouroftheinstabilitygrowthindexsasafunctionofn
p,0 remainsunalteredbyourmoreaccuratetreatment.Thequanti-
forvaluesofγ =105, 106, 107(righttoleft).
p,max
tativeimplications,however,arediscussedbelow.
tor of 2 above its critical value, s becomes greater than one,
5.2.Protoninjection
i.e.thedensityofphotonsstartsgrowingonatimescaleshorter
thanthecrossingtimeofthesource.This,asweshallseeinthe In order to see the effects of the PPS instability when proton
nextsection,hascatastrophicconsequencesforthehighenergy lossesaretakenintoaccountweassumeonceagain(seesection
protonsasthespontaneouslygrowingphotonsmakethemlose 4)thatprotonsareinjectedinaregionofradiusRwithapower-
theirenergies. law (eq. 24). However, to make the picture less complicated,
Fig.9showsthemarginalstabilitycurveasisobtainedfrom weassumethatthereisnoexternalphotonfieldpresent.Thus
the presentcode(in practice we have calculatedthe valuesof oneexpectsthattheprotonspectrumwillreachanequilibrium
Mastichiadisetal.:Hadrons 9
stategivenbytheno-losssolution(eq.25).Thisisindeedtrue
aslongasthissteady-statesolutionisbelowthecriticaldensity 4
forthePPSinstability,i.e.whenn = Q t <ncrit(γ ),
p,0 p,0 p,esc p,0 p,max
withn thenormalisationoftheprotonsandncrit(γ )given
p,0 p,0 p,max 2
from Fig. 9. In this case also there is a very small number of
internally produced photons. However, as soon as the condi-
tion n = Q t > ncrit(γ ) holds, the criteria for the
p,0 p,0 p,esc p,0 p,max
0
PPS instability are satisfied. As a result, the photonsincrease
exponentially,protonslose energydue to pair productionand
thesystembehaviourdependsonthechoiceoftheparameters
Q andt . -2
p,0 p,esc
As a first example, we show in Fig. 10 the case where
β = 2, γ = 106, t = 1 with a proton injection rate
p,max p,esc -4
of Q = 0.22.Theprotoncompactnessparameter,according
p,0
toeq.9,isℓ =1710.TheabovecombinationofQ andt ,
p p,0 p,esc
accordingtoFig.8,correspondstoafeedbackloopthatcauses
-6
the photon density to increase with s = 0.7. The dotted and
short-dashedlinesshow,respectively,theevolutionofthelow-
50 100
est and highest differential density bins of the proton energy.
InagreementwiththeanalyticalsolutionofEq.(5)intheno-
loss case, the numberof particlesin these binsincreasesvery Fig.10.Plotoftheevolutionofthesysteminthecasewhereprotons
quickly(inaboutonet )toasteadystate,which,however,cor- areinjectedwithβ = 2, γ = 106, t = 1,whilethecombina-
cr p,max p,esc
respondstoanunstableprotonconfiguration.Thelong-dashed tionoftheprotoninjectionrate(Q =0.22)andt issuchthatthe
p,0 p,esc
lineshowsthephotoncompactnesswhenonlysynchrotronra- steady-stateprotondistributionintheno-losscasecorrespondstoan
diation is taken into account. This increases as ℓ ∝ est with unstableprotondistribution.Thesolidlineshowsthephotoncompact-
γ
s = 0.7 until it reaches a steady state. The dot-dashed line nesswhenallprocessesaretakenintoaccount,whilethelongdashed
anddot-dashedlinesshowthephotoncompactnesswhencertainpro-
showsthe effectsonthe photoncompactnesscaused fromthe
cessesareomitted(fordetailsseetext).Thedottedandshortdashed
additionofinverseComptonscatteringandphoton-photonpair
linesshowtheevolutionofthefirstandlastprotonoccupationnumber
production.Astheseprocessesarequadraticwithrespecttothe
bininthecasewherealltherelevantprocessesaretakenintoaccount.
photonandelectron/positronnumberdensitiestheydonotaf-
fect the slope of the compactness at early stages, i.e. as long
as ℓ ≪ 1. However, at the later stages of evolution these
γ source.Thebehaviourofthelowestandhighestprotonenergy
processes become important and, because they are quadratic,
binsisalsoshown(dottedanddashedlinesrespectively).Itis
theycausethephotoncompactnesstoincreaseevenfasterand
clear that photonsand high energyprotonsare anticorrelated,
reachsaturationsooner.Finallythesolidlineshowstheevolu-
inthesensethatwhenonepopulationishigh,theotherislow.
tionofthephotoncompactnesswhen,inadditiontotheabove
These cycles are similar to those found by Stern & Svensson
processes, photon trapping due to the high density of created
(1991)usingMonte-Carlotechniques.
pairs is taken into account. This leads to higher photon com-
pactnessesasrelativelymoreenergyisextractedfromthehigh
6. Simultaneouselectronandprotoninjection
energyprotons.Thiscurvemustbeconsideredthe‘correct’one
asitcontainsalltherelevantprocesses.Ontheotherhandthe
As a last case we examine the situation where electrons and
photonincrease causes the high energyprotons(short-dashed
protonsareinjectedsimultaneously,i.e.,bothprotonandelec-
line) to lose energy and settle in a new steady state. These
tron kinetic equations (5) and (6) have an external injection
losses do not affect naturally the low energy protons (dotted
term. We assume that both of these terms are in power-law
line)whichmaintaintheiroriginalsteadystate.
form, so that, in addition to Eqn. (24) that describes proton
At the other extreme, one can envisage a case where pro- injection, we prescribe electron injection using a similar ex-
tons are injected slowly, but have a very long escape time. In pression:
thiscasewetakeQ = .2210−3 andt = 103whichcorre-
p,0 p,esc Qext(γ )= Q γ−βH(γ −γ )H(γ −γ ). . (32)
spondstothesameno-losssteadystateasbefore.Theresultis e e e,0 e e,min e,max e
showninFig.11,wherequasi-periodicbehaviourtypicalofa To lower the number of free parameters we assume that
relaxationoscillatorisapparent.Theprotonsaccumulateinthe bothelectronsandprotonsareinjectedwiththesamespectral
source and, once their density rises abovethe criticaldensity, indexβandwiththesamemaximumLorentzfactor,i.e.weset
the photon density grows rapidly (on the timescale of a few arbitrarilyγ =γ .Moreover,whensolvingtherelevant
e,max p,max
timest ) anddepletethe highenergypartofthe protonspec- kinetic equations, we assume that the two species have equal
cr
trum.Oncetheprotonshavelosttheirenergy,thereisnothing escapetimes,i.e.,t =t .
p,esc e,esc
lefttosustaintheloopandthephotonsescapefromthesystem. Dependingontheparticularchoiceofparameters,highen-
This cycle is repeated as the protons accumulate again in the ergy electron injection can result in a synchrotron and/or an
10 Mastichiadisetal.:Hadrons
particularchoiceofthe parametersthe primaryelectronscool
fastandthe systemquicklyreachesa steady state.In thenext
twocases,theinjectedprotoncompactnessisℓ = 200(long-
2 p
dashed line) and ℓ = 400 (short-dashed line) respectively.
p
Again a steady state is quickly reached. However the photon
compactness ℓ increases as ℓ increases because of the BH
γ p
0 pairs created and subsequently cooled. In both of these cases
thefeedbackloopdoesnotoperateinthesensethattheproton
losses remain low — or, equivalently, ℓ ≃ ℓext +ℓBH ≪ ℓ .
γ e e p
-2 However, for the last two cases which are for ℓp = 800 (full
line)andℓ = 1600(dot-dashline)theeffectsofthefeedback
p
areevident.Inthecaseofthesolidcurve,andforthefirst300
or so light crossing times, there is a gradualincrease of pairs
-4
and photons in the system until their numbers are built to a
levelthatallowsacatastrophicreleaseofthe energystoredin
protons.Afterthat,asteadystateisreached,butatalevelmuch
-6 higherthanthatofthepreviouscases:ℓ ≃ℓext+ℓBH ≃ℓ .Itis
γ e e p
worthmentioningthatthisloopoperatesataprotoncompact-
1000 2000 3000 4000 5000 nesswhichisbelowthecriticalthresholdobtainedinSection5.
Therefore, the presence of external electrons helps to initiate
thecatastrophicprotonenergylossesatlowerprotondensities.
Fig.11. Plot of the evolution of the system in the case where alow
Finally,theuppermostcurvecorrespondstoaprotoninjection
protoninjectionrate(Q = 2.210−4)combineswithaslow proton
p,0 whichisabovethecriticalthresholdforthePPSinstabilityand
escape time(t = 103) insuch away that the steady-state proton
p,esc thephotonsgrowveryquicklyasdiscussedinSection5—see
distribution in the no-loss case to correspond to an unstable proton
alsoFig.10.
distribution.Therestoftheparametersareasinthepreviousfigure.
Solidlineshowsℓ ,whiledottedandshortdashedlinesshowtheevo-
γ
lutionoftheprotonspectrumatγ =10.0.1andγ =106respectively.
p p
The photonspectra correspondingto the steady states ob-
tained in each of these runs are shown in Fig. 13. The bot-
tom curve (dotted line) corresponds to the injection of elec-
inverse Compton component (see, for example, Mastichiadis trons only whereas the four others are for electron and pro-
& Kirk 1997). When protons are injected as well, these will ton injection, correspondingto the curvesof Fig. 12. The ex-
interactwiththeaforementionedphotonscausingasecondary tra component due to BH pair production is apparent on the
injectionofBethe-Heitlerpairs.Thisleadstoanon-linearsit- twolowerspectrathatincludeprotons(longandshort-dashed
uation. To see this, one should compare the compactness of lines).Thetwouppermostcurvescorrespondtothesteady-state
the externally injected electrons ℓext (eq. 10) with the corre- spectrawhentheloopwasabletoextractmostoftheprotonen-
e
spondingcompactnessoftheinternallyproducedpairsviathe ergy.Duetotheresultinghighphotoncompactnessthespectra
Bethe-HeitlerprocessℓBH (Eq.29).Asthelatteris,ingeneral, arestronglyabsorbedatenergiesabove1MeVduetophoton-
e
afunctionofbothℓp andℓeext,whenℓeext < ℓeBH thesystemop- photonpairproduction.
eratesinthenon-linearregime,inthesensethatthecoolingof
theprotonsoccursmainlyontheinternallyproducedphotons.
Toinvestigatetheeffectdescribedaboveweproceedasfol- TheabovefindingsaresummarisedinFig14whichshows
lows: we keep the electron injection parameters constant and thephotoncompactnessofthesystemversustheprotoncom-
changeonlythenormalisationoftheprotoninjectionrateQ pactnessforvariousinjectedelectroncompactnesses.Forlow
p,0
ineq(24)—or,equivalently,wechangetheprotoncompact- valuesofℓ (i.e.lessthan30),thephotonsofthelow-frequency
p
ness parameterℓ (Eq. 9). When ℓ ≪ ℓext the resulting pho- part of the SED produced come almost exclusively from the
p p e
ton spectrum is simply that producedfrom the cooling of the presence of the electrons, i.e. the protonscannotsignificantly
externally injected electrons. However as we increase ℓ , the influencethelow-frequencybehaviourofthesystem.Wenote
p
quantity ℓBH increases as well and, at some stage, it becomes herethatforsynchrotronprotonblazarmodelfitstotheSEDof
e
comparabletoℓext.Abovethispointthesystementersthenon- BLLacObjectsobservedatgamma-rayenergies(e.g.(Mu¨cke
e
linear regime, as the cooling of the protons occurs primarily et al. 2003)) the proton compactness is ℓ ∼ 10−3–10−2 im-
p
onitsownradiation.Finally,abovesomecriticalprotoncom- plying that Bethe-Heitler pair production, and any associated
pactness, a loop analogous to the one described in Section 5 instability, is unimportant in these models. For intermediate
operates,andtheprotonsconvertasubstantialfractionoftheir valuesofℓ (i.e.between30and300)theinternallyproduced
p
energy content to electron/positron pairs and radiation. This Bethe-Heitler pairs make their presence visible in that their
is depicted in Fig. 12 which shows the evolution of the pho- cooling increasingly dominates the photon spectrum. Finally
tonswhenℓext iskeptconstant,whileℓ varies.Thefirstcurve atevenhighercompactnessestheprotonsbecomeunstableand
e p
(dotted line) assumes that no protonsare injected. Due to the coolefficientlyontheir“own”radiation.