Bow Ties in the Sky I: The Angular Structure of Inverse Compton Gamma-ray Halos in the Fermi Sky

31 Pages·2016·3.86 MB·English

DRAFTVERSIONSEPTEMBER5,2016
PreprinttypesetusingLATEXstyleemulateapjv.01/23/15
BOWTIESINTHESKYI:
THEANGULARSTRUCTUREOFINVERSECOMPTONGAMMA-RAYHALOSINTHEFermiSKY
AVERYE.BRODERICK1,2,PAULTIEDE2,MOHAMADSHALABY1,2,3,CHRISTOPHPFROMMER4,
EWALDPUCHWEIN5,PHILIPCHANG6,ANDASTRIDLAMBERTS7
1DepartmentofPhysicsandAstronomy,UniversityofWaterloo,200UniversityAvenueWest,Waterloo,ON,N2L3G1,Canada
2PerimeterInstituteforTheoreticalPhysics,31CarolineStreetNorth,Waterloo,ON,N2L2Y5,Canada
3DepartmentofPhysics,FacultyofScience,CairoUniversity,Giza12613,Egypt
4HeidelbergInstituteforTheoreticalStudies,Schloss-Wolfsbrunnenweg35,D-69118Heidelberg,Germany
6 5InstituteofAstronomyandKavliInstituteforCosmology,UniversityofCambridge,MadingleyRoad,Cambridge,CB30HA,UK
1 6DepartmentofPhysics,UniversityofWisconsin-Milwaukee,1900E.KenwoodBoulevard,Milwaukee,WI53211,USA
0 7TAPIR,Mailcode350-17,CaliforniaInstituteofTechnology,Pasadena,CA91125,USA
2 DraftversionSeptember5,2016

p
ABSTRACT
e
S ExtendedinverseComptonhalosare generallyanticipatedaroundextragalacticsourcesofgammarayswith en-
ergiesabove100GeV. These result frominverseComptonscattered cosmic microwavebackgroundphotonsby
1
apopulationofhigh-energyelectron/positronpairsproducedbytheannihilationofthehigh-energygammarays

ontheinfraredbackground. Despitetheobservedattenuationofthehigh-energygammarays,thehaloemission
]
E has yet to be directly detected. Here, we demonstrate that in most cases these halos are expected to be highly
anisotropic,distributingtheup-scatteredgammaraysalongaxesdefinedeitherbytheradiojetsofthesourcesor
H
orientedperpendiculartoaglobalmagneticfield. Wepresentapedagogicalderivationoftheangularstructurein
h. theinverseComptonhaloandprovideananalyticformalismthatfacilitatesthegenerationofmockimages. We
p discussexploitingthisfactforthepurposeofdetectinggamma-rayhalosinasetofcompanionpapers.
- Keywords:BLLacertaeobjects: general–gammarays: general–radiationmechanisms: non-thermal–gamma
o
rays:diffusebackground–infrared:diffusebackground–plasmas
r
t
s
a
1. INTRODUCTION moving units near z 1 due to the peak in the cosmological
[ ≈
starformationratearoundthattime. Assuch,observationsof
The extragalactic gamma-ray sky above 100 GeV is domi-
1 theabsorbedVHEGRspectraofnearbysourceshaveresulted
nated by the unresolvedemission of a subset of active galac-
v indirectmeasurementsoftheEBL(Biteau&Williams2015).
tic nuclei (AGNs) (Ackermannetal. 2016b). Of these, the
7 Fromtheseitisclearthateventodaytheuniverseiseffectively
8 vastmajorityareblazars–objectswithrelativisticjetspointed opticallythicktoVHEGRs,with
3 in our direction (see, e.g., Table 5 of Ackermannetal.
0 2011). While the mechanisms by which these very-high E 1+z - ζ E - 1
0 energy gamma rays (VHEGRs) are produced remain un- Dpp(Eγ,z)=Dpp,0(z)E0 =35 2 1TγeV Mpc,
. clear(Mannheim1993;Ghisellinietal.1998;Böttcher2007), γ (cid:18) (cid:19) (cid:18) (cid:19)
9 (1)
their propagation through the cosmos has provided an in-
0 valuable means by which to probe the intervening universe whereE0isafiducialenergyandζ=4.5forz<1andζ=0for
6 z 1(Kneiskeetal.2004;Neronov&Semikoz2009).
1 (Gould&Schréder 1967; Steckeretal. 1992; deJageretal. ≥More recently, the evolution of the electron-positron pairs
1994; Salamon&Stecker 1998; Domínguezetal. 2011;
: hasprovideda meanstoprobethemagnetizationofthe inter-
v Gilmoreetal.2012;Vovketal.2012).
veningcosmos.ThehomogeneityoftheEBLcoupledwiththe
i Extragalactic VHEGR sources are observed to be strongly
X largeD placesmanyofthesepairswithinintergalacticvoids,
biased towards low redshifts, with the number of known pp
where their propagation can be affected by the intergalactic
r sources peaking at a redshift of 0.1-0.2 (see, e.g., the
a magneticfield(IGMF).Thesepairsthusostensiblypermitthe
redshift distribution of high-syncrotron-peak sources in
onlymeasurementofthelargescaleIGMFlocatedinthemean
Ackermannetal. 2011, 2015). This is a natural consequence
densityregionsfarremovedfromgalacticactivity.
of the annihilation of VHEGR on the nearly-homogeneous
The high energy of VHEGRs imply similarly high-energy
infrared-ultravioletextragalactic background light (EBL) that
pairs,whichcorrespondtoLorentzfactorsof
permeates the universe, generated by previous generationsof
stars and quasars (Gould&Schréder 1967), which can thus γ Eγ 106 Eγ . (2)
beprobedbypropagatingVHEGRs(Ackermannetal.2012d; ≈ 2m c2 ≈ 1TeV
e
Dwek&Krennrich 2013). The center of momentum energy
If nothing else happens, these pairs will cool on the cosmic
of the VHEGRs and EBL photon exceeds the pair-creation
microwavebackground(CMB),producinganinverseCompton
threshold, i.e., E E & 4m2c4; and thus VHEGRs can an-
γ IR e cascade(ICC)ofphotonswithtypicalenergiesof
nihilate as they propagate through the EBL. In practice, the
mboetahnefnreeregpyaathn,dDrpepd,sohfifVt dHeEpGenRdsentot,adbespoernpdtiionngoonntthheeEeBvoLlvis- EIC 2γ2ECMB=2 Eγ 2 ECMB GeV. (3)
≈ 1TeV 1meV
ingdensityandspectrumoftheEBL.HigherEBLdensitiesat (cid:18) (cid:19)
largerredshiftscorrespondto shorterD , whichpeaksin co- Thatis, theICCeffectivelyreprocessesaninitialTeVgamma
pp
ray into many GeV gammarays. It is the non-observationof
2 BRODERICK ET AL.
thisICCcomponentinknownVHEGRsourcesthathavepro- regarding the duration of VHEGR outbursts and thereby re-
videdthestrongestlowerlimitsontheIGMFtodate(see,e.g., duce the uncertainty on the IGMF lower limits substantially
Neronov&Semikoz2009). (Dermeretal. 2011). Even in the presence of an IGMF the
In a number of extragalactic VHEGR sources the intrin- ICCsaredeflectedawayfromtheaxisalongwhichtheoriginal
sic gamma-ray spectrum can now be constructed after mak- VHEGRemissionisbeamed.ThustheICCcomponentshould
ing weak assumptions either about the intrinsic spectrum or be visible for observers who either do not see the VHEGR
the absorption on the EBL, and thus the resulting ICC emis- emission,orseeonlyweakVHEGRemission.
sion estimated. This is then limited directly by observations EffortstodirectlydetecttheICCcomponentarefundamen-
by the Fermi gamma-ray space telescope, which has ruled tally complicated by the large mean free paths of the VHE-
out the presence of the ICC component with extraordinary GRs, typically resulting in halos that extend over many de-
confidence (e.g., Neronov&Semikoz 2009). This is natu- greesandthereforehavinga lowsurfacebrightness(see, e.g.,
ral if an IGMF is present – within the IGMF the electron- Figure 1). Therefore, all efforts to detect this emission to
positron pairs deflect away from the line of sight and there- date have stacked multiple Fermi gamma-ray images to in-
fore the up-scattered gamma rays are beamed away from crease the significance with which the halo emission can be
us. Based upon this scenario typical estimates for the IGMF separatedfromthatduetothecentralsourceandbackground.
rangefrom10- 17–10- 15 G,dependingonassumptionsonduty Theseeffortsarefurthercomplicatedoperationallybytheun-
cycles (see e.g., Neronov&Semikoz 2009; Neronov&Vovk certaintiesinthepointspreadfunction(PSF)ofFermiandthe
2010; Tavecchioetal. 2010, 2011; Dermeretal. 2011; spatially varyinggamma-raybackground(e.g., Neronovetal.
Tayloretal. 2011; Takahashietal. 2012; Dolagetal. 2011; 2011;Ackermannetal.2013). Asaresult, thisprocedurehas
H.E.S.S.Collaborationetal. 2014; Prokhorov&Moraghan ledtonowdisprovendetectionsofanexcess(Ando&Kusenko
2016). 2010). A more recent attempt that utilizes the most recent
This argument for non-zero IGMFs is predicated on three PSF,reportedinChenetal.(2015),neverthelessexhibitssimi-
keyassumptions: larsensitiviestotheuncertaininstrumentresponse.
All such efforts have ignored the possibility of structure
1. TheintrinsicTeVemissionisnarrowlybeamed. withinthegamma-rayhalo. However,such structureis a nat-
ural consequenceof either the original beaming of the VHE-
2. TheintrinsicTeVemissionspectrumcanbereasonably
GRs responsible for the generation of the pairs or the ori-
approximated,usuallybyanexponentiallycut-offpower
entation of the IGMF where the pairs are created (see, e.g.,
law.
Long&Vachaspati 2015). Thus for a wide range of parame-
3. Nootherprocessescontroltheevolutionoftheelectron- tersforanIGMFweexpecthighlyanisotropicICChalo. Here
positronpairs. wepresentsemi-analyticalcomputationsofthehalostructure,
explicitlydemonstratingthepresenceofthestructure,identify-
Thefirstiswellsupportedbytheprevalenceofblazarsamong ingitsorigin,andcreatingthefacilitytogeneratemockimages
VHEGR-brightAGNspecifically,andgamma-raybrightAGN ofVHEGRsourceswithrealisticICChalostructures. Inprin-
generally (Ackermannetal. 2011, 2015; Wakely&Horan ciple, knowledgeofthe halostructurecanaidsubstantiallyin
2008), which immediately implies that VHEGR emission is effortstodirectlydetecttheICCcomponent. Wereportonan
localized near the axis of the radio jet. The second is rea- explicitimplementationof a methodto do so in a companion
sonably well supported by the gamma-ray spectra of nearby paper(Tiedeetal.2016a).Inacompanionletter,weapplythis
VHEGR-bright AGN, the systematic softening of observed formalismtoFermi-LATdataanddiscusstheconsequencesof
gamma-rayspectrawith increasingredshift(Ackermannetal. thismeasurementfortheIGMF(Tiedeetal.2016b).
2012d),andtheunderlyingassumptionthattheintrinsicspec- In Section 2 we describe qualitatively the origin of the
traonlyweaklyevolve. anisotropyandhowitrelatestothestructureoftheIGMF.Gen-
The third remains unclear. Should any alternate cooling eralexpressionsdescribingthegenerationandevolutionofthe
mechanism dominate inverse Compton cooling it would pre- energeticpairsarepresentedinSection3.Applicationstocases
empt the generation of ICCs directly. A recently suggested of highlytangled andorderedfieldsare describedin Sections
example would be cooling mediated by large-scale beam 4 and 5 with typicalapplicationsshown. The constructionof
plasma instabilities driven by the bulk motion of the rela- mockFermiimages,includingvariouscomponentsandinstru-
tivistic electron-positronpairs through the ionized intergalac- mentaleffectsisdicussedinSection6. Thedependenceofthe
tic medium (Brodericketal. 2012; Schlickeiseretal. 2012, mockICChalosforatypicalbrightFermisourceareexplored
2013; Changetal. 2014). While the nonlinear development inSection7. Finally,concludingremarksarecollectedinSec-
of these instabilities is uncertain, there are a variety of lines tion8. Inordertostreamlinethepaper,mostofthedemonstra-
of astronomical evidence that suggest a cooling mechanism tionsarelefttotheAppendix.
with very similar properties is at work (Changetal. 2012;
Pfrommeretal. 2012; Puchweinetal. 2012; Brodericketal.
2. QUALITATIVEORIGINOFSTRUCTURE
2014a; Lambertsetal. 2015). Regardless of the origin of the
additionalcooling,however,shouldtheICCsbepreemptedthe BeforedescribingthecreationofphysicallyrealisticICCha-
resulting gamma-ray spectra would necessarily be consistent loswe beginwith a summaryof the key ideas underlyingthe
withthecurrentlackofadetectionofanICCinknownextra- structureswewillfind. Thisispredicatedonthestandardpic-
galacticVHEGRsources,independentofanIGMF. ture of the ICC halo formationdescribedin Section 1: VHE-
The situation would be immediately clarified by the direct GRsareemittedfromAGNsandtravelcosmologicaldistances
detectionoftheICCcomponent,layingsquarelywithintheen- priortogeneratingenergeticelectron-positronpairsontheEBL
ergyrangeprobedthebylarge-areatelescope(LAT)onFermi. viaphotonannihilation.ThosepairstheninverseComptonup-
Not only would doing so obviate the aboveassumptions (im- scattertheCMBtoGeVenergiesoveracomparablyshortdis-
portantly including the third), it would also settle questions tance. However,fortwoindependentreasons,theseICChalos
ANGULARSTRUCTURE OF GAMMA-RAY HALOS 3
102 3. FORMATIONANDEVOLUTIONOFINTERGALACTICPAIRS
The generation of the halos involves two critical steps that
imprintanisotropyupontheresultinggamma-raysky:thegen-
) eration of the initial pairs by the VHEGR emission and their
s
ree 101 100GeV subsequentevolutionduringtheinverseComptonup-scattering
eg of the CMB. To simplify the computationwe will exploitthe
d
( disparityinscalesbetweentheinverseComptoncoolinglength
s
u (0.01–0.1Mpc)andtheVHEGRmeanfreepath,andthusas-
di
Ra 1TeV sume that the ICC emission is generated in situ follwing pair
r 100 production.Therefore,thenecessaryelementsare:
a
ul
g 1. ThespatialandenergydistributionofVHEGRphotons,
n
A andthecorrespondingdistributionfortheinjectedpairs.
10TeV
10−1 2. Modelsfortheconsequencesofpairenergylossandde-
flectionintheIGMF.
0.0 0.5 1.0 1.5 2.0
z 3. TheinverseComptoncascade(ICC)spectrumduetoup-
scatteringtheCMB.
Figure1. AngularsizeofVHEGRmeanfreepathasafunctionofredshift We treat each generally first here, and then specialize to spe-
andenergy.
cificlimitsofinterestinSections4and5.
arenotisotropic.
3.1. SpatialandEnergyDistributionofInjectedPairs
First, we consider the case of a small-scale tangled IGMF
that would isotropize the generated electron-positron pairs Theoriginoftheultra-relativisticpairsistheVHEGRemis-
in combination with a non-aligned non-thermally dominated sionfromAGN.ThatnearlyallofthebrightextragalacticTeV
AGN, i.e., that the line of sight is not intersecting the jet sources are blazars implies that the VHEGR emission from
opening angle. The VHEGRs are originally beamed along TeV-luminous AGN is strongly beamed along the jet. Thus
the jet axis. This is evidenced by the overwhelming domi- we begin with a description of the anisotropic VHEGR flux
nance of blazars in the extragalactic gamma-ray AGN sam- fromtheAGNitself,beforeannihilationontheEBLmodifies
ple(Ackermannetal.2011,2015). BecausetheVHEGRmean it. Specifically,we startwith the numberof VHEGRphotons
free path is long in comparisonto the inverse Compton cool- passingthroughasolidangledΩ′ inthedirectionxˆ′, withen-
ingtimeoftheresultingpairsthisimpliesthattheemissionis ergiesbetweenE′andE′+dE′,overatimeintervaldt′:
essentially local, and therefore arises from a biconical region
dN
indicated by the radio jet of the source AGN. If the inverse AGN (E′,xˆ′). (4)
Compton gamma rays are isotropically emitted, arising, e.g., dt′dE′dΩ′
fromahighlytangledIGMF,thespatialstructureinthegamma Foroursystemofcoordinates,primedquantitiescorrespondto
raysgeneratesa resultantstructureinthe GeVimage. Thisis theframeoftheAGN(includingthesitesofpairproduction),
shownexplicitlyintheleft-handpanelofFigure2,alongwith assumedtobeatafixedredshift,andcenteredupontheAGN
theassociatedgamma-rayimageoftheICChalo. itself. Unprimedquantitiescorrespondto the observerframe,
Alternatively,the processof gyrationin the IGMFalso can centeredupontheobserver. Notethatx′ andΩ′ correspondto
impartstructureontheimageifweconsiderablazarwherethe thepositioninthree-dimensionalspaceandsolidanglerelative
VHEGRsarebeamedtowardsus. InthepresenceofanIGMF totheAGN,respectively. Theformerisrelatedtoangularpo-
thatishomogeneousonscalescomparabletoD electronsand sitiononthesky,α,anddistancealongthelineofsight,inthe
pp
positronswillgyrateonfixedtrajectoriesthatemittowardsan directionℓˆ,by
observeronlyforasubsetofinitialinjectionpositions. Thisis x′=D (z)α+ℓ′ℓˆ. (5)
stillsuperimposedonthejetstructure,resultinginanasymmet- A
ricimagestructureifthelineofsightdoesnotcoincidewiththe whereDAistheangulardiameterdistance.Hence,forexample
boresightofthejet,shownintheright-handpanelofFigure2. radial distance from the AGN is given by r′2 =D (z)α2+ℓ′2
A
Gamma rays on opposite sides of the original AGN are pro- andtheanglerelativetothejetaxis(thatisorientedalongzˆ′),
ducedpredominantlybydifferentleptonspecies,i.e.,positrons locatedatanangleΘ′relativetothelineofsightandΦ′relative
ononesideandelectronsontheother. toafiducialdirectiononthesky(e.g.,north)definingthex-axis
Hence we see that different AGN populations can be used intheimageplane,isgivenby
toprobetheexistenceofsmall-orlarge-scaletangledIGMFs:
whilenon-alignednon-thermallydominatedAGNaresuitable ′ ′ ′ ℓ′cosΘ′+DAsinΘ′ αxcosΦ′+αysinΦ′
cosθ =zˆ xˆ = .
for probingsmall-scale fields, blazar geometriesare ideal for · r′
exploringlarge-scaleIGMFsbecauseinbothcasestheassoci- (cid:0) (cid:1)(6)
atedstronganisotropyofthepairhalosenablesefficientstack- Notethatalongthelineofsight,i.e.,atα=0,r′=ℓ′ andθ′=
ing of the angular power spectra. In the following sections Θ′.
we make these instances explicit, computing physically real- As VHEGRs propagate,they annihilateuponthe EBL, and
istic halofluxdistributionsthatconnecttheenergy-dependent thusboththefluxandspectrumoftheVHEGRsdeviatesfrom
flux distributions to the underlying physical properties of the thatemittedbytheAGN.DuetothehomogeneityoftheEBL,
VHEGRemissionandIGMFgeometry. theopticaldepthtoannihilationisgivenby
r′
′ ′
τ(E ,z,r )= , (7)
D (E′,z)
pp
4 BRODERICK ET AL.
B
Θ
Figure2. CartoonsofthemechanismsbywhichanisotropyintheICChalosisgenerated,distinguishedbythestructureoftheunderlyingIGMF.Left:ForanIGMF
tangledonsmallscales(correlationlengthsofλB.3Mpc)theanisotropyisduetothestructureofthegamma-rayjet. Right: ForanIGMFthatisuniformacross
thegamma-rayjet(correlationlengthλB&30Mpc)theanisotropyisduetothegeometryofthegyrating,relativisticpairs. Inthelatter,inverseComptongamma
raysfromelectronsandpositronsareshowninredandgreen,respectively. Alldistancesarenottoscaleandopeninganglesareexaggeratedforvisualpurposes.
101 are set by the typical values for the high synchrotron peak
sources (HSPs) in the Fermi sample: Γ 1.8 and Γ 2.5
l h
≈ ≈
(Ackermannetal. 2011, 2015, 2016a). The 1σ range about
these values for HSPs is given by the dotted lines in Figure
3. Fortheangularfluxdistribution,weassumeaGaussianjet
100
profilewithopeningangleθ . Thatis,weset
dN j
E′2
dE′
dNAGN (E′,xˆ′)= f G(θ′) (E′/Ep)- Γl E′<Ep (9)
10−1 dt′dE′dΩ′ 0 (cid:26)(E′/Ep)- Γh E′≥Ep,
where
G(θ′)=e(cosθ′- 1)/θ2j +e- (cosθ′+1)/θ2j. (10)
10−120−4 10−3 10−2 10−1 100 101 102
The two terms correspond to the two oppositely oriented jets
Energy(TeV)
with angular structuresset by G(θ), which for θ 1 rad, as
j
≪
istypicallythecasefortheAGNconsideredhere(seeSection
Figure3. TypicalassumedintrinsicspectrumforabrightFermiblazar,witha
spectralbreakat1TeV.Dottedlinesshowthe1σvariationsinthelow-energy 7), correspond to Gaussian jet profiles. Note that despite the
andhigh-energyphotonspectralindexesΓl andΓh,respectively, amongthe factthatweareconsideringaversionoftheGaussianjets,the
hardgamma-rayblazars. Thedark-greyshadedregionindicates the1GeV–
approximateformsimplifiessignificantlyasaconsequenceof
100GeVregionforwhichwegenerateICChalorealizations; thelight-grey
shadedregionindictesthe1TeV–10TeVVHEGRbandprimarilyresponsible writingthisintermsofcosθ′.
fortheICChalos. This emission is naturally normalized by the 1 GeV–
100GeVfluxobservedbyFermi. Thisnecessarilydependson
wherethemeanfreepathisgiveninEquation(1)anddepends Θ′,thoughforFermiobjects,inpracticedoessoonlyweakly;
uponbothVHEGRenergyandredshift(z)1. Theresultingflux
the fact that the hard Fermi AGN are blazars implies that the
distributionofVHEGRsfarfromtheAGNisthen
viewingangleissmallerthantherelativisticbeamingangleof
dNVHEGR (E′,x′)= dNAGN (E′,xˆ′)e- τ(E′,z,r′). (8) the jet (i.e., Θ′ .Γ-je1t). Thus, in the absence of a dominant
dt′dE′dΩ′ dt′dE′dΩ′ inverseComptonhalo,assumingaroughlyfixedeffectivearea
from1GeVto100GeV,thegamma-rayfluxobservedbyFermi
For concreteness we will assume that the spectral and spa-
duetotheintrinsicemissionofthesourcebetweentheseener-
tialdistributionsofthe VHEGRsareseparable. Thismaynot
giesis
beagoodapproximationatlargeanglesiftheemissionvaries
throughoutthejet.However,fortheapplicationsofprimaryin-
terest,wherethelargesteffectsoccuratviewingangleswithin F = 100GeVdE dΩ′dt′dEV′HEGR dNAGN
therelativisticbeamingpatternofthejet,i.e.,Γ-je1t,thisiswell 35 Z1GeV ZΩ′Fermi dt dE dt′dE′dΩ′
motivated.
1 100GeV dN
Forthespectrum,weassumeabrokenpowerlaw,character- = dE′ dΩ′ AGN (11)
(iFzΓiegld)usrbeoym3d.eifpfTeihvreeontatespnshueomrgtoeydn(Esvpapel)uc;tearsnaloeifxnatdhmeexpeplsehaosbptooenvctesrpu(Γemcht)irsaanslhdinobwdeenlxoeiwns =(11++zzZ)-13GADeeV2ff f0EpN(ZΓΩl′F,erzm)iG(Θ′d)t,′dE′dΩ′
A
1 Inprinciple, theopticaldepthshouldbeintegratedoverr′(andthusz′).
However, in practice, in the cases of interest Dpp is much shorter than the where the source-frame solid angle subtended by Fermi is
Hubblelength,justifyingoursimplerexpression. A /D2(1+z)2andN(Γ,Γ ,E ,z)isanormalizationfactorde-
eff A l h p
ANGULARSTRUCTURE OF GAMMA-RAY HALOS 5
pendingonlyonthespectralshape: (seeAppendixB).Thatis,inthislimit
N(Γl,Γh,Ep,z)= 100(1+z)GeV/Epdx x- ΓlΘ(1- x)+x- ΓhΘ(x- 1) , f˙scat≈- ∇p·(Wfe±), (16)
Z(1+z)GeV/Ep where
(cid:2) (12)(cid:3) W=- 4σTuspp - pp , (17)
in which Θ(x) is the Heaviside function. For our pur- 3m2c2 ≡ t m c
e IC e
poses here we will presume that E > 100 GeV gener-
ally, resulting in the simplified exprpession N(Γl,z) = (1+ in which tIC = 3mec/4σTus ≈ 2.4×1012(1+z)- 4 yr is the
z)1- Γl (1GeV/Ep)1- Γl- (100GeV/Ep)1- Γl /(Γl- 1). There- asymptotic inverse Compton cooling time for a nonrelativis-
ticlepton. Asaresult,theBoltzmannequationsimplifiestoa
fore,thenormalizationisrelatedtotheobservedfluxvia
(cid:2) (cid:3) strictlypartialdifferentialequation.
f = F35 D2A (1+z)3 . (13) Strong variability has been observed in many extragalactic
0 E A N(Γ,z)G(Θ′) VHEGRsources,indicatingthatvariableemissionischaracter-
p eff l
istic. However,whenthisoccursovertimescalesmuchshorter
TherateatwhichpairsarecreatedbytheVHEGRissetby thanthetypicalcoolingtime,theimpactonthepairdistribution
the rate at which VHEGRs are annihilated. Each generated ˙
issmall. Thatis,eveninthepresenceofarapidlyvarying f ,
pair has an energy ′ =E′/2.2 Thus, the production rate of inj
thepairdistributioncanbewellapproximatedbyastationary
electrons/positrons wEith energies between ′ and ′+d ′ at ˙
positionxˆ′is E E E solution.Thusweset fe± =0.
The large disparity between the inverse Compton cooling
dN lengthofthepairs(700kpcat0.5TeV)andthepairproduction
dt′d e′±d3x′(E′,x) mean free path of the VHEGR photon (currently 800 Mpc at
EdΩ′dr′dE′ dN dτ 1 TeV) implies that this is a fundamentally local and nearly
= VHEGR (2 ′,x′) (2 ′,z) (14) homogeneous process. The latter immediately implies that
d3x′ d ′ dt′dE′dΩ′ E dr′ E v ∇f issmall,andmaybeneglectedhenceforth.Theformer
= 2 e- τ(2EE′,z) dNAGN (2 ′,xˆ′), im·pliees±thattheenergydistributionof f˙inj isgivenbyEquation
r′2D (2 ′,z)dt′dE′dΩ′ E (14); we defera discussion of the angulardescription to Sec-
pp
E tions4and5.
wheretheadditionalfactorofdτ/dr′ istherateatwhichpairs Therefore,inpracticeweseektosolve
are being producedlocally. It is straightforwardto show that
power and number flux are conserved in the above relation, p˙ ∇pfe±+∇p (Wfe±)= f˙inj. (18)
· ·
andisshownexplicitlyinAppendixA.1.1. Fornon-pathologicalp˙(p)andW(p),thisisimmediatelysolv-
Atthispointwehaveonlytheinjectionspectrumofpairsasa
able via the method of characteristics (see Appendix C). For
functionofposition. Toprogressweneedamodelforhowthe
deflectionsinalocallyfixedmagneticfielddescribedbyalo-
pairsevolveandtheirsubsequentinverseComptonemission. cal spherical coordinate system (q′,ϑ′,ϕ′), it is convenientto
expressthis in terms of a set of polar momentumcoordinates
3.2. EvolutionofthePairDistributionFunctions alignedwiththefield(p′,ϑ′,ϕ′),yielding
p p
We have greatly simplified the computation of the gamma
ray flux at the expense of pushing the difficulty onto the de- m ct ∞
′ ′ e IC ′ ′2
tteiornmiisnaotciocnurorifngthuenedveorltvheedaeclteioctnroonftdhirsetreibpuroticoens.seTs:hiisnjeevcotiloun- fe±(x,p)= p′4 Zp′ dqq
by the VHEGRs, possible gyration in an IGMF, and inverse f˙ x′,q′,ϑ′,ϕ′ ωBtICsinϑ′p m2ec2 - m2ec2 , (19)
Comptonscattering.Differentassumptionsregardingthescale × inj p p± 2 q′2 p′2
andstrengthoftheIGMFwillimpacttheevolutionandthere- (cid:20) (cid:18) (cid:19)(cid:21)
fore the image. Throughout we will make use of the ultra- whereω =eB/m cisthecyclotronfrequency.
B e
relativisticapproximationeverywherepermissible,andthuspc
andE areusedinterchangeably. 3.3. InverseComptonEmission
Generally the evolution of the distribution function of the Whilewewillconsideravarietyofpotentialmodelsforthe
pairs, fe±,isdescribedbytheBoltzmannequation, evolutionof the pairsafterproduction,their inverseCompton
f˙e±+v·∇fe±+p˙ ·∇pfe± = f˙scat+f˙inj, (15) eevmoilsustiioonnmeanytebrsedsoelseclryibaesdabnyaunsiknngolewfnraemleecwtroornkainndwhpiocshittrhoins
where f˙ describestheimpactofinverseComptonscattering distributionfunction, fe±(x′,p′). Weconstructthisframework
˙ scat here, paying particular attention to identifying a number of
and f is the injected distribution of pairs. p and v are the
inj simplifyingassumptions.
momentumandvelocityofthepairs,respectively.
Thefirstoftheseisstationarity,whichinturncorrespondsto
In general the scattering term must be written in terms of
a presumption regarding the duty cycle of VHEGR sources.
an integraloverinitial and final particle states, describingthe
Short time variability will necessarily impart similar vari-
multitudeof waysin which particlesmayscatter into and out
ability on the energy dependence and spatial distribution of
ofthestateofinterest. However,inthesoftseedphotonlimit
the pair population. Both of these will be smoothed, how-
the scatteringterm can be substantially simplifiedas a conse-
ever, if the source is active for a sufficient period. In the
quenceofthesmallmomentumchangesduringeachscattering
case of the energy dependence this is set by the inverse
Compton cooling timescale, roughly 106 yr. For the spatial
2Toavoidconfusionwewillrefertotheenergiesoftheemittedgammarays,
includingtheVHEGRs,asE′,leptonenergiesasE′,andobservedgamma-ray distribution this is determined by the propagation-dependent
energiesasE. time delay, which for 2◦ halos ranges from 102 yr to 4
×
6 BRODERICK ET AL.
103 yr, depending on redshift and gamma-ray energy. Thus, sincethepairdistributionfunctionswillevolvethisneednotbe
if VHEGRactivitypersistsforlongerthan106 yrwemayas- inthedirectionoftheoriginalVHEGR.
sume the underlyingpair population has reached steady state Giventheseassumptions,therateatwhichICCgammarays
in both terms. Note that this is comfortably short in com- withmomentaq′areproducedis
parison to typical radio duty cycles of a few times 107 yr to
2ian0gf0e5twh;aMttimtchNeissami1s0aw8raeyelrltja(uAl.stlie2fix0ea0dn5.d;eASrh&sasbLuaeclahah,eywtae1l9.w827i0l;l0Nn8o)u,tlscseounngesgitedsaetlr-. dtd′dN3IxC′,de±3q′ =σ2Tqc′sucs(cid:18)2mq2e′scq2′(cid:19)3/2fe± x′, m2ecqq′sq′′!(cid:12)(cid:12)(cid:12)q′s=EC′M(B2/2c),
non-stationary transients (though see Menzler&Schlickeiser p (cid:12)
thederivationofwhichcanbefoundinAppendixD.(cid:12)Fromthis
2015).
wemayconstructthefluxasseenatEarthfromasourcealong
The inverse Compton process is simplified by three addi-
′
tional assumptions, relaxing any of which will make at most a line of sight in the direction ℓˆ, and thus corresponding to
′
smallchangestotheresults. Thefirstisthatasinglegeneration momentaq′=E′ℓˆ/c,
ofpairsiscreated,i.e.,thatinverseComptonscatteredgamma
′
rsauybssedqouennottgtehneemrasteiolvnessoafnandihdiiltaiotenaolnpathires.ETBhLisacnodrrgeesnpeornadtes dt′dd2Nx′γd3q′ = dℓ′dtd′dN3IxC′,de±3q′ x′,Ec′ℓˆ , (23)
toajointconstraintontheenergyoftheVHEGRsconsidered e+,e- Z !
X
and the seed photon populationthat is inverse Compton scat-
obtainedbydirectlyintegratingtheVlasovequation(see,e.g.,
tered,effectivelylimitingittotheCMB.Theformerconstitutes
Broderick 2006). In the above we have neglected any subse-
a conservativeassumption, limitingourattentionto VHEGRs
quentscattering or absorptionof the ICC gammarays, which
withenergieslessthan10TeV.Thelatterissuppressedbythe
iswelljustified bytheirtypicalenergy,givenbyEquation(3)
ratioofthe numberdensitiesofCMB andEBL photons–the
andgenerally.100GeV.
EBL providesthe only substantial alternative population – of
The above may immediately be converted into a surface
at least 104, and thus is rare in practice (Dwek&Krennrich
brightnessasmeasuredbyFermi:
2013). Including additional generations of pairs will modify
theGeVsignalbyacomparableamount,justifyingitsneglect. dN dt′dE′ q′2dΩ′ d2x′ dN
theTCheMinBv,ewrsitehCtyopmicpatolennceorogliiensgǫ′oft1h0e-p3a(1ir+szis)epVri.mFaorrilVyHdEueGtRo dtdEdγ2α = dt dE ZΩ′Fermi c q d2αdt′d2x′γd3q′
leonwergthieastEth′e≪enmer2egcy4/oǫf′∼the3×up1s0c2a/t≈t(e1r+edz)pTheoVtotnhiissiwsseullffiacpipernotxly- = (1A+efzf)4163tmE4ec25 2EE (24)
imated by Equation (3). Typically we will assume that only eX+,e- IC CMBr CMB
VthHisEisGwRsellwjiuthstiefineedr.giWeselewsisllthfuarnth1e0raTsesVumaerethrealtetvhaentC,MheBncies × dℓ′fe± x′,mec 2EE ℓˆ′ ,
characterizedbya monoenergeticphotondistribution. Thatis Z (cid:18) r CMB (cid:19)
theseedphotondistributionfunctionis whereE′ =(1+z)E andE′/E′ =E/E wereused.
CMB CMB CMB CMB
At this point it is necessary to explicitly define the evolution
u E′
f (x′,q′)= s δ q′- CMB , (20) model for the electrons and positrons and the corresponding
s s 4πq′s3c (cid:18) s c (cid:19) fe±(x,p). Thus, we now turnto explicitlyconsideringthe in-
jectionmodeldescribedbyEquation(14)inthecontextoftwo
whereE′ =0.7(1+z)meVandu arethetypicalenergyand
CMB s limiting evolutionary models. We defer until Sections 6 and
energy density of CMB photons, respectively. Relaxing this
7 the construction of mock images, which include additional
makes little difference to the resulting emission spectrum at
contributions(centralsourceandgamma-raybackground),in-
theexpenseofcomplicatingtheanalysissubstantially.
strumentaleffects(i.e.,thePSF),anddrawexplicitsubsamples
Thehighenergiesofthepairsincomparisontothesoftseed
withtheappropriatestatistics.
photonsresponsiblefortheircoolingimpliesthattheresulting
ThephysicalpictureofICCsreprocessingtheVHEGRemis-
inverseComptonemissionishighlybeamedinthedirectionof
sionfromthecentralsourcetolower-energygammaraysper-
theleptonpropagation.Wewillassumethatthisisexclusively
mitsavarietyofintegralrelationshipsbetweenthetwo. These
the case, i.e., the inverse Compton photons propagate along
are used to check both the results of this section as well as
the directionofthescatteringelectronsandpositrons.3 Inthe
thefollowingtwowheretheenergy-dependenthalostructures,
isotropic approximationthis implies that the differentialscat-
differingintheassumedstructureoftheIGMF,inAppendixA.
tering cross section for up-scattering a seed photon with mo-
Atthesametimeacleardiscussionoftheanticipatedredshift
mentumq′ toagamma-raywithmomentumq′ byanelectron
withinitialsandfinalmomentump′ andp′ is dependencemaybefoundinAppendixA.3.
f
dσ 2q p′ 4. GAMMA-RAYHALOSFROMTANGLEDFIELDS
=σ δ3 q′- s p′ δ3(p′ - p′+q′),
d3q′d3q′d3p′d3p′ T m2c2 f The first limit we consider assumes rapid isotropization
s f (cid:18) e (cid:19) of the pairs, i.e., the pair momenta become isotropic on a
(21)
timescale short in comparison to the inverse Compton cool-
where the first δ-function encodes both the forward-
ing time. This naturally occurs if the IGMF is sufficiently
propagation approximation and the soft-seed limit, while the
strong and tangled. This occurs in two steps, first isotropiz-
secondδ-functionenforcesmomentumconservation.Notethat
ing in the azimuthaldirectionaboutthe magneticfield due to
3Thisisanexcellentapproximationforasinglescatteringduetothehigh gyration,andsecondisotropizinginthepoloidaldirectiondue
energiesoftheleptons.Thatthisremainstrueaftermanyscatteringsisshown tovariationsinthelocalmagneticfieldorientation.Inpractice,
inAppendixE. thismayoccuronlystatistically shouldthegyrationradiusbe
ANGULARSTRUCTURE OF GAMMA-RAY HALOS 7
smallincomparisontothecorrelationlength,i.e.,aftercoarse- geometry(i.e.,θ ,Θ,Φ,andΓ )itissufficienttonumerically
j h
grainingoverthecoolingscale,orviadiffusionifthegyration computeΨ (R). ThisfunctionisshowninFigure4forvarious
k
radiusislargeincomparisontothecorrelationlength,λ . viewingangles.
B
Quantitatively,theIGMFstrengthandvalueofλ atwhich In terms of Ψ , the full energy-dependent,gamma-raysur-
B k
thepairseffectivelyisotropizedependsonthegamma-rayen- facebrightnessdistributionisgivenby
ergyand sourceredshift. Pairswith highenergieswill gyrate
the slowest, placeingthe strongestlimitonIGMFstrength; at dN m c2 3(1+z)- 1 m c2 1- Γh
10 TeV a lepton will make a full gyration within the inverse dtdEdγ2α =F35δ2Ee2 4πN(Γ,z)G(Θ′) Ee
ComptoncoolingtimewhenB 10- 15G. Ifthecurrentlower- A CMB l (cid:18) p (cid:19)
limitsonthestrengthoftheIGM≥Faretakenatfacevalue,they 2E - Γh/2 2E α
i2m01p1ly).thTisheissltiakteislytictaolbiseotthroepcizaasetio(nhorweqevueirre,scft.anDgelirnmgeroefttahle. ×(cid:18)ECMB(cid:19) Ψ2- Γh rECMBδA!, (29)
IGMF on scales comparable to the jet opening angle at dis- where we have summed over the two species and δ
A
tancesofDpp.Inturn,thisrequiresλB θjDpp,whichissmall- (E /m c2)D (E ,z)/D (z)isameasureoftheangularsizeo≡f
≪ p e pp p A
estat10TeVandz=0:λ 3Mpc. Therefore,nominallythe
B meanfreepathofVHEGRs,modulothelargeenergyratioap-
≪
applicabilityoftherapidisotropizationlimitdependssolelyon
pearing in the prefactor. Note that the normalization is fully
thestructureoftheIGMF.
determinedbytheobservedF andtheassumedintrinsicspec-
35
tralshape.Thatthisreproducestheintegratedfluxintheparent
4.1. PairDistributionFunction
VHEGRsisverifiedinAppendixA.2.1.
Within this context we approximate the effect of rapid ThespatialstructureoftheICChalofollowsthatoftheun-
isotropization as an instantaneous redefinition of the pair in- derlyinggamma-rayjet.Thatis,sincethepairsareisotropized,
jectionmodel: and therefore emit via inverse Compton isotropically, their
gamma-ray emission maps out the pair injection sites. Since
f˙inj,e±(x′,p′)= 4πcp′2dt′ddNe′±d3x′ p′c,x′ . (25) tchoensVeHquEeGntRlyesmoisissitohneiirsecmonisfisnioend.toAjseats,ressoualtr,eththeeopriaeinrst,atainodn
E
(cid:0) (cid:1) oftheICChalosinthislimitfollowsthatofthegamma-rayjet
Theisotropyrenderstheazimuthalevolutionofthedistribution
andpresumablythemuchsmaller-scaleradiojets.
functionduringinverseComptoncoolingmoot,andthus
GenerallytheICChalospectrumwillvaryacrosstheimage
t m c2 ∞ dN due to the different mean free paths of the underlying VHE-
fe±(x′,p′)= I4Cπpe′4 Zp′ dp˜′dt′dEe′±d3x′ p˜′c,x′ . (26) GICRCs.haHlooswpeevcetrru,mit.iΨs p(oRs)sipbelaektsonecahraraafictxeerdizRe,tfhaellilnarggrea-pscidallye
(cid:0) (cid:1) k
FortheexpressionforthepairinjectionspectruminEquation thereafter,correspondingtotheexponentialsuppressionofthe
(14)thismaybeintegratedexplicitlyateachr′,producing injection of pairs by the VHEGR after a mean free path. As
aresult,forsufficientlyextendedimagestheinternalstructure
fe±(x′,p′)= tI4CπmEecp45dt′ddNEe′±d3x′(cid:18)E2p,x′(cid:19)eτ(Ep,z) toifonΨakl(Rto)Eis- 1u-nΓihm/2p.orTtahnetraensdoltvheedICspCechtaraloasrpeegcetrnuemralilsyphroaprdoer-r
E′ - 4 ∞d ′ 2E′ 1- Γhe- τ(2E′,z) duFeotrotthheensemarallyleflraDtpspp.ectrumsourcesunderconsiderationhere
×(cid:18)Ep(cid:19) ZE′ E (cid:18)Ep (cid:19) typicalvaluesforthelow-andhigh-energyphotonspectralin-
= tICmec5 dNe± Ep,x′ eτ(Ep,z) cdaelxIeCsCarheaΓlolp≈ho1t.o8nasnpdecΓtrhal≈in2d.e5x.isG2iv.2e5n,tihneterlmatteedriathteetotytphie-
8πE3 dt′d ′d3x′ 2
p E (cid:18) (cid:19) two, andimportantly,intermediatebetweenthatofthe source
′ - (Γh+2) ′ Γh- 2 2 ′ and the soft background(which has a typical photon spectral
E E Γ 2- Γh, E . index of 2.4). This has two relevant consequences. First, it
× E E E
(cid:18) p(cid:19) (cid:18) D(cid:19) (cid:18) D(cid:19) is difficult to spectrally separate the ICC halo emission from
(27)
thesourceorthebackground.Second,sincetheICChalosare
whereE E /τ(E ,z)andΓ(k,x)istheincompleteGamma
D≡ p p marginally harder than the backgroundthe marginal value of
functionoforderk(Abramowitz&Stegun1972).
low-energygammarays(.1 GeV) is small, typically adding
morenoisethanICChalosignal. Fortechniquesthatleverage
4.2. Gamma-rayHaloStructure
the anisotropic structure of the halos this is compounded by
Withthedistributionfunctioninhandwemaynowcompute thegrowthofthePSFatlowenergies. Hence,ourdecisionto
the gamma-ray halo flux distribution from Equation (24) ex- restrictourattentiontogammarayswithenergiesabove1GeV.
plicitly.Thisissimplifiedinthiscasebythefactthat fe±(x′,p′)
is separable in x′ and p′. Up to an energy-dependent coef- 4.3. NumericalImplementation
ficient, the flux distribution can be expressed as an energy-
The creation of mock Fermi images will ultimately re-
dependent rescaling of a single spatial function, i.e., the im-
quirethe rapidgenerationofsamplesdrawnfromthe energy-
ages at differentenergiesform a homologousclass of images
dependentflux distributionin Equation(29). Despite its sim-
describedby
plicityincomparisontoMonteCarlopropagationschemes,this
∞ remains nontrivial, requiring a handful of practical optimiza-
Ψk(R′) dℓ′r′- k- 2Γ(k,r′)G(θ′), (28) tions.
where R′ =D (z)α,≡r′Z- ∞√R′2+ℓ′2, and θ′ is givenby Equa- latWineg btheeginresbuyltcolongsatrruitchtminigcaΨll2y- Γinh(RR), nraunmgeinrigcaflrloyman1d0t-a8buto-
A
tion (6). Hence, in pra≡ctice, for a given source structure and 104. ThisisthenintegratedoverRandE between1GeVand
8 BRODERICK ET AL.
Fofigthuereje4t..AΨtka(cRu)tefovrievwarinioguasnvgileewsi(nΘga.ng1l0e◦s,)Θge,oamndetrkic=f-o0re.6sh(oi.rete.nΓinhg=c2u.r6ta).ilsAtthoisblsitqruuectvuireew,ainngdabnygΘles≈(Θ2◦&it1i0s◦n)eathrleyainsiostorotrpoipci.cstructureoffollowscloselythat
Figure5. ExamplerealizationofICChalointhepresenceofasmall-scaletangledmagneticfieldviewedatajetinclinationof60◦ (left),30◦ (middle),and2◦
(right)before(top)andafter(bottom)convolvingwiththeFermiPass8R2_V6-frontPSF.Inallcases,theon-axisnumberofgammarayswasassumedtobe5000
andthesourceplacedatz=0.3,θj=3◦,andΓh=2.6. Theenergyofthegammaraysareindicatedbycolor: with1GeV-3GeVindarkred,3GeV-10GeVin
orange,10GeV-30GeVingreen.
ANGULARSTRUCTURE OF GAMMA-RAY HALOS 9
100 GeV to obtain the ratio between the number of gamma sufficientlynearbyobjectsλ islimitedbytheproperdistance,
B
raysinthehalotothatcontainedwithinthedirectsourcecom- D – for Mkn 421 this is 130 Mpc. For more distant objects
P
ponent, i.e., the halo-core flux ratio. We then construct the therelevantlimitisset byD . Fora typicalICC gammaray
pp
followingadditionalmarginalizedprobabilitydistributionsfor (10GeV)fromatypicalmoderate-redshiftsource(z=0.2)this
ahalophotonontheimage: is also roughly 130 Mpc. Thus, typically, the IGMF may be
treatedasuniformifλ 102Mpc.
P(E) d2α dNγ ThispresentsanaturBa≫lcounter-pointtotheisotropizedcase.
∝ dtdEd2α Italsopermitsasimplifiedgeometricpicturesinceonlyasmall
Z
dN subsetofthegyratingpairswillgyrateintoourlineofsightand
P(αE) dθ γ (30)
| ∝ αdtdEd2α thereforegenerategammaraysinourdirection.
Z The characteristic radial extent of halos in this case is de-
dN
P(θ E,α) γ , terminedby both the deflection angle of typicalpairs and the
α| ∝ dtdEd2α openingangle of the jet. For observed gamma-rayenergyE,
whereα= α, θ is thepolarangleonthe skyrelativetothe theformertypicallylimitstheangularextentofthehalosto
α
| |
pTrhoejeficrtsetdijsetalarxeiasd,yankdnothwennaonrmalayltiizcaatliloyn,sPa(Ere)setEap- p1-rΓohp/r2i.atTehlye. ∆α ωBtIC 1◦.5 B E - 1(1+z)- 4,
lattertwoareconstructednumericallyinterms∝ofintegralsover def≈ γ2 ≈ 10- 16G 10GeV
(cid:18) (cid:19)(cid:18) (cid:19)
Ψ2- Γh(R) and tabulated. To generate a gamma-ray event the which for z .1 is generally substantial for fields consis(t3e1n)t
procedureisthen
with the arguments of Neronov&Vovk (2010). This is not
1. Draw a randomnumber to determine if the gamma ray surprising given that those require similar size deflections to
isinthehaloorthesourcebaseduponthehalo-coreflux explain the absence of an ICC component in the spectra of
ratio. Ifitisinthehalo: VHEGR sources. The latter sets a typical angular scale for
blazarjetsindependentofIGMFstrengthof
2. DrawanenergyfromP(E).
θ D D
∆α j pp 2◦ pp , (32)
3. DrawavalueforαfromP(αE). jet≈ D - D ≈ D - D
A pp A pp
|
whichiscomparabletothelimitfromthemagneticdeflections
4. Drawavalueforθ fromP(θ α,E).
α α
| fornearbysources(z.0.15)anddominatestheICChalosizes
fordistancesources.Asaresult,onceagainthestructureofthe
4.4. ExampleHaloRealizations
IGMFdominatesthegrossfeaturesoftheICChaloforalarge-
Examplerealizationsof ICC halos associated with a small-
scale, homogeneousIGMF that is consistent with the current
scale,tangledIGMFareshowninFigure5forvariousviewing
lowerlimitsonitsstrength.
angles. Asanticipatedimplicitlybytheenergydependenceof
D andexplicitlybyEquation(29),theenergyofthegamma 5.1. PairDistributionFunction
pp
raysare stronglycorrelatedwiththe distancefromthecentral In this case the pairs retain the memory of their injection
source. High-energy halo photons are found at the smallest momentum,andthus,
angular offsets, while low-energy halo photons compose the
buAlktolafrtgheevdiieswtainntghaanlgol.es,theexpectedangularstructureofthe f˙inj,e±(x′,p′)= p′2scinϑ′ dt′ddNe′±d3x′ p′c,x′
halo is easily visible, both before and after convolution with p E
the Fermi Pass P8R2_V6 PSF for front-convertedevents (see δ(ϑ′(cid:0)- ϑ′)δ (cid:1)ϕ′ - ϕ′ . (33)
× p p
Section 6.1). Small viewing angles, comensurate with those
impliedbyobservationsofgamma-rayblazars,resultinanex- Perhapscounter-intuitivelytheϕp(E′)depend(cid:2)onE′a(cid:3)saresult
of the definition of p adopted in Appendix C, which effec-
tremeforeshorteningthateliminatesmuchoftheanisotropy. 0
tively describes the orientation of the injected leptons not at
Inallcasesthetotalnumberofintrinsicsourcegammarays
injectionbutbythatofafictitiousleptoninjectedatinfiniteen-
seenbyanobserverlookingdownthegamma-rayjetaxis,i.e.,
ergyand inverse Comptoncooled until E′ (which necessarily
theon-axissourceflux,washeldfixed,correspondingtoaflu-
occursoverafinitetime). Thecorrespondingstationarydistri-
enceof5000ph,chosentomatchatypical,bright,hardFermi
butionfunctionduringinverseComptoncoolingis
source. Thecomparativelysmallhalofluenceisaresultofthe
isotropyoftheICCcomponent,whichfollowsfromtherapid
t m c2
iinsoittiraolplyizhaitgiohnlyo-bfethaempeadiVrsH,aEnGdRdijleuttebsytahefaecntoerrgoyffl2uθx-j2f.rFoomrtthhee fe±(x′,p′)= p′I4Csineϑ′pδ(ϑ′p- ϑ′)
parametersemployedhere, thiscorrespondsto a reductionby ∞ dN
afactorofnearly7×102. ×Zp′ dq′dt′dEe′±d3x′ q′c,x′ δ ϕ′p(q′)- ϕ′ , (34)
(cid:0) (cid:1) (cid:2) (cid:3)
5. GAMMA-RAYHALOSFROMORDEREDFIELDS where we have subsumed all of the azimuthal argument in
Equation19intoϕ′(q′),i.e.,
The second limit we consider assumes that the distribution p
fcuonoclitniogntiimnemuinnidmeraltlhyeisaocttrioopnizoefd,aeuvnoilfvoirnmg IoGvMerFt.hWe eenftuirre- ϕ′p(q′)=ϕ′p+ωBtIC2sinϑ′p mq2e′c22- mp2e′c22 . (35)
ther assume that this occurs coherently across the jet, requir- (cid:18) (cid:19)
ing large IGMF coherence lengths. Both the magnitude and Again the integrationmay be performedexplicitly for the in-
natureofthelimitonλ dependsonredshiftandenergy. For jection spectrum in Equation(14), now with the aid of the δ-
B
10 BRODERICK ET AL.
Figure6. 1GeVgamma-raybrightnessdistributionassociatedwithauniformIGMFbeforeanyinstrumentalresponsefortheICChalogeneratedbybothelectrons
and positrons (left), onlyelectrons (center), andonlypositrons (right). Thecolor indicates the logarithmic brightness, normalized relative tothehighest value
(concentratedatthesource);inwhitearetheiso-brightnesscontoursenclosing90%,99%,and99.9%ofthetotalflux.Extremevaluesofthesourceparameterswere
chosentoaccentuatekeyqualitativefeatures(B=2.5×10- 14Glocatedat45◦tothelineofsight,θj=10◦,Θ=6◦.7).
Figure7. ExamplerealizationofICChalointhepresenceofalarge-scaleorderedmagneticfieldwithstrengths10- 15G,10- 16G,10- 17G,before(top)andafter
(bottom)convolvingwiththeFermiPass8R2_V6ULTRACLEANVETO-frontPSF.Inallcases,theon-axisnumberof1GeV–100GeVsourcegammarayswas
assumedtobe5000andthesourceplacedatz=0.3,θj=3◦,Θ=2◦,andΓh=2.6.Theenergyofthegammaraysareindicatedbycolor:with1GeV-3GeVindark
red,3GeV-10GeVinorange,10GeV-30GeVingreen,and30GeV-100GeVinblue.