Ultrahigh energy cosmic ray nuclei from remnants of dead quasars

1.1 MB·

by  Roberto J. Moncada

Ultrahighenergycosmicraynucleifromremnantsofdeadquasars
RobertoJ.Moncada,1,2 RafaelA.Colon,1,3 JuanJ.Guerra,1,3 MatthewJ.O’Dowd,1,3,4 andLuisA.Anchordoqui1,3,4
1DepartmentofAstrophysics,AmericanMuseumofNaturalHistory,CentralParkWest79St.,NY10024,USA
2Department of Physics, City College, City University of New York, NY 10031, USA
3DepartmentofPhysicsandAstronomy,LehmanCollege,CityUniversityofNewYork,NY10468,USA
4DepartmentofPhysics,GraduateCenter,CityUniversityofNewYork,365FifthAvenue,NY10016,USA
(Dated:January2017)
Were-examinethepossibilityofultrahighenergycosmicraysbeingacceleratedinnearbydormantquasars.We
particularizeourstudytoheavynucleitoaccommodatethespectrumandnuclearcompositionrecentlyreported
bythePierreAugerCollaboration. ParticleaccelerationisdrivenbytheBlandford-Znajekmechanism,which
wiresthedormantspinningblackholesasFaradayunipolardynamos. Wedemonstratethatenergylossesare
dominatedbyphotonuclearinteractionsontheambientphotonfields. Wearguethatthelocaldarkfossilsof
7
thepastquasaractivitycanbeclassifiedonthebasisofhowsourceparameters(massofthecentralengineand
1
0 photonbackgroundsurroundingtheaccelerator)impactthephotonuclearinteraction. Inthisclassificationitis
2 possibletodistinguishtwounequivocaltypeofsources:thoseinwhichnucleiarecompletelyphotodisintegrated
beforeescapingtheaccelerationregionandthoseinwhichphotopionproductionisthemajorenergydamping
r
mechanism. We further argue that the secondary nucleons from the photodisintegrated nuclei (which have a
p
A steepspectralindexatinjection)canpopulatetheenergyregionbelow“theankle”featureinthecosmicray
spectrum, whereas heavy and medium mass nuclei (with a harder spectral index) populate the energy region

3 beyond“theankle”,allthewaytothehighenergyendofthespectrum.Inaddition,weshowthatfivepotential
quasarremnantsfromourcosmicbackyardcorrelatewiththehot-spotobservedbytheTelescopeArray.

]
E
H I. INTRODUCTION scopesornon-imagingCherenkovdetectorsconsistentlyfind
. a predominantly light composition at around 109 GeV [13]
h
andthecontributionofprotonstotheoverallcosmicrayflux
p The origin(s) of ultrahigh energy cosmic rays (UHECRs)
is(cid:38)50%inthisenergyrange[14–17]. Duetheabsenceofa
- remainsacentralquestioninhighenergyastrophysics[1,2].
o largeanisotropyinthearrivaldirectionofcosmicraysbelow
Three principle observables drive the search for and charac-
r the ankle [18, 19], we can conclude that these protons must
t terization of cosmic ray sources: the energy spectrum, the
s be of extra-galactic origin. At energies above 1010 GeV, the
a nuclear composition, and the distribution of arrival direc-
high-statistics data from the Pierre Auger Observatory sug-
[ tions. Thecosmicrayenergyspectrumencompassesaplum-
meting flux that drops from 104 m−2s−1 at E ∼ 1 GeV to gests a gradual increase of the fraction of heavy nuclei in
2 10−2 km−2yr−1 at E ∼ 1011 GeV. Its shape is remarkably the cosmic ray flux [14–17]. Within uncertainties, the data
v
from the Telescope Array (TA) is consistent with these find-
3 featurelessandcanbedescribedbyabrokenpowerlaw: the
5 spectrumsteepensgraduallyfromE−2.7 toE−3.3,thenflattens ings [20, 21]. Moreover, the Telescope Array has observed
0 to E−2.6 near 109.6 GeV forminga feature known as“the an- a statistically significant excess in cosmic rays with energies
0 kle”,anddropssharplyataround1010.5GeV[3–5].Thesmall above 1010.7 GeV in a region of approximately 1150 square
0 degrees centered on equatorial coordinates (R.A. = 146.7◦,
variations of the spectral index can be interpreted either as a
. Dec. =43.2◦)–theTAhotspot[22]. Theabsenceofacon-
2 transitionbetweencosmicraypopulations,orasanimprintof
0 cosmicraypropagationeffects. centrationofnearbysourcesinthisregionoftheskyfavorsa
7 heavynucleushypothesis,wherebyafewlocalsourceswithin
The simplest interpretation of the ankle is that above
1 the GZK sphere can produce the hot spot through deflection
109.6 GeV a new population emerges which dominates the
: in the Galactic B-field. All in all, the apparent dominance
v moresteeplyfallingGalacticpopulationofheavynuclei. The
i of heavy nuclei in the vicinity of 1010 GeV strongly argues
X extragalactic component can be dominated either by pro-
againsttheinterpretationsoftheanklegivenabove.
tons [6] or heavies [7, 8]. Proton-dominance beyond the an-
r
a kle is ultimately limited by the onset of photopion produc- One can (of course) accommodate the data with the addi-
tion on the cosmic microwave background, whereas domi- tion of an ad hoc light extragalactic component below the
nance of a heavy composition is restricted by nucleus pho- ankle, with a steep injection spectrum [23]. However, a
todisintegration through the giant dipole resonance – the more natural explanation of the entire spectrum and compo-
so called Greisen-Zatsepin-Kuz’min (GZK) suppression at sitionemergeswhileaccountingforthe“post-processing”of
around1010.5GeV[9,10]. Ithasalsobeenadvocatedthatthe UHECRsthroughphotodisintegrationintheenvironmentsur-
anklefeaturecouldbewellreproducedbyaproton-dominated rounding the source [24–27]. In such models relativistic nu-
power-lawspectrum,wheretheankleisformedasadipinthe cleiacceleratedbyacentralenginetoextremelyhighenergies
spectrum from the energy loss of protons via Bethe-Heitler remain trapped in the turbulent magnetic field of the source
pair production [11, 12]. In this case extra-galactic protons environment. Their escape time decreases faster than the in-
would already have started to dominate the spectrum some- teraction time with increasing energy, so that only the high-
whatbeyond108.7GeV. est energy nuclei can escape the source unscathed. In effect,
Opticalobservationsofairshowerswithfluorescencetele- thesourceenvironmentactsasahigh-passfilteronthespec-
2
trumofcosmicrays. Allnucleibelowtheenergyfilterinter- black hole and determine plausible source conditions for ac-
act, scattering off the far-infrared photons in the source en- celerationofUHECRnuclei. InSec.IVweexaminethevari-
vironment. These photonuclear interactions produce a steep ousenergy-lossmechanismsofone-shotacceleration,includ-
spectrum of secondary nucleons, which is overtaken by the ingradiativeprocessesandphotonuclearinteractionswiththe
harder spectrum of the surviving nucleus fragments above ambientphotons. Weshowthat,ingeneral,photonuclearen-
about109.6 GeV. Theseoverlappingspectracouldthencarve ergylossesaredominant. Afterthatweproposeanovelphe-
anankle-likefeatureintothesourceemissionspectrum. The nomenologicalmodeltoexplaintheevolutionoftheUHECR
spectrum above the ankle exhibits a progressive transition to spectrum and composition with energy, without fine-tuning.
heavynuclei,astheescapeofnon-interactingnucleibecomes Finally,inSec.Vwepresentourconclusions.
efficient. In such models, the source evolution with redshift
playsaparamountroleonthesourceparameters. Formodels
with positive redshift evolution (a greater number of sources II. DORMANTBLACKHOLESAFTERASHININGPAST
percomovingvolumeathighredshift),boththeimpositionof
adominantlightextragalacticcomponentbelowtheankleand The first extensive radio surveys of the 1950s revealed a
thesource-anklemodelfavorahardinjectionspectrum∝ E−1. new family of radio sources that, by the early 60s, had been
However,ithasbeenrecentlynotedthatfittingthehighendof associated with point-like optical counterparts [38]. These
thecosmicray(CR)spectrumandcompositionwithnegative quasi-stellarradiosources,orquasars,revealedverylargesys-
sourceevolutionallowsforsofterinjectionspectra[28]. temic redshifts indicating cosmological distances [39]. This
Onaseparatetrack,itwasrecentlypointedoutthatFermi- impliedluminositiesuptohundredsoftimesthatoftheMilky
LATobservations[29,30]severelyconstrainthedistribution Way,apparentlyemittedfromverysmallsize-scales[40],sug-
ofUHECRsources[31]. ThisisbecauseonthewaytoEarth gesting an extremely energetic emission mechanism. It has
UHECRsinteractwiththeradiationfieldspermeatingtheuni- nowbeenestablishedthatquasarsareultimatelypoweredby
verse and give rise to energetic electron/positron pairs and the accretion of matter onto the 107 −1010M(cid:12) supermassive
photons, which in turn feed electromagnetic cascades result- blackholes(SMBHs)atcoresofgalaxies[41–45].
inginadiffusegamma-raybackgroundradiation. Therecent Quasars and their relativistically-beamed counterparts,
studyin[31]suggeststhatalocal“fog”ofUHECRsacceler- blazars, are the most luminous sub-class of active galactic
atedinnearbysourcesmustexistsoastonotoverproducethe nuclei (AGNs), which is the general name for an actively-
cascadegamma-rayfluxbeyondFermi-LATobservations. accreting SMBH. In fact, these objects are the most lumi-
nous continuously-emitting objects in the Universe. Quasar
In consideration of these new results, it seems interesting
luminosity is, to first order, constrained by the Eddington
to explore whether remnants of dead quasars, which are dis-
limit [46]. This is the maximum luminosity of a spherically
tributed preferentially in the low-redshift universe, could ac-
accretingobject,determinedbythepointatwhichoutwardra-
celerateheavynucleiuptothehighestobservedenergies. Ac-
diationpressurehaltsgravitationalinfall.TheEddingtonlimit
tually,nearbyquasarremnantshavelongbeensuspectedtobe
isproportionaltothemassoftheaccretingobject. Scaledfor
sources of UHECRs [32–35]. Particle acceleration proceeds
aquasarpoweredbyaSMBHof109 M ,thisluminosityis:
via the Blandford-Znajek mechanism, which wires dormant (cid:12)
spinning black holes as Faraday unipolar dynamos [36, 37].
M
Detailedmodelingwithreasonableassumptionsonsourcepa- L =1.3×1047 erg/s∼102L , (1)
rameters suggests that protons can be efficiently launched at Edd 109M(cid:12) galaxy
E (cid:38) 1012 GeV. Photopionproductionincollisionswitham- where M is the solar mass and L is the bolometric lu-
(cid:12) galaxy
bientphotonsbecomesarelativelyimportanteffectduringthe minosity of large galaxies like the Milky Way. Note that ac-
final phase of the acceleration process, and could damp the cretionbythestandardα-disk,whichistypicallyassumedfor
maximumattainableenergytovalueswellbelowthatimposed quasars,isalsoEddingtonlimited[47].
bythevoltagedrop. ThelargechargeZeofheavynucleifa- Itisgenerallyacceptedthatthenumberdensityofquasars
cilitates their acceleration to highest energy (scaling linearly peaked at redshift z ∼ 2, when the universe was one-fifth
with Z). However, heavy nuclei could also interact with the of its present age [48]. Locally, essentially all galaxies with
ambientphotonsandmaysuffersignificantspallation. Inthis spheroidal components contain a SMBH [49–51]. The close
paper we study in detail the dominant processes for nucleus relationships between the masses of these SMBHs and their
energy losses, while searching for plausible source parame- hostgalaxy’sproperties,includingstellarvelocitydispersion,
ters which may allow acceleration of heavy nuclei up to the spheroidalluminosity,anddynamicalmass,indicatesthatthe
highestobservedenergies. growth of SMBHs and their hosts are closely tied [52–56].
Thelayoutofthepaperisasfollows. InSec.IIweprovide These relationships also allow us to make a relatively direct
anassessmentofwhatisknownobservationallyabouttheevo- determinationofthelocalSMBHmassfunction[57–61].
lution of quasar activity and then discuss the general frame- Another approach to deriving the local SMBH mass func-
work use to relate the quasar population with massive dark tionistointegrateSMBHaccretionovercosmictimebasedon
objectsatthecenteroflocalinactivegalaxies. Wealsoshow theevolvinghigh-zquasarluminosityfunction[60–67]. Such
thatfivenearbyrelicsofsuchpastactivitycorrelatewiththe models are able to reproduce the “observered” low-z mass
TAhot-spot. InSec.IIIwereviewthegeneralitiesofparticle functionforquiescentSMBHsverysuccessfullybyassuming
accelerationneartheeventhorizonofaspinningsupermassive thetypicalradiativeefficienciesforanα-disk(0.1(cid:46)η(cid:46)0.2),
3
andreasonableEddingtonfractions—quasarluminositiesas This magnetic field may be taken as a fiducial strength for
a fraction of the Eddington limit (Eddington fraction) — of quasars remnants at the upper end of the local SMBH mass
0.2 (cid:46) L/L (cid:46) 1 [59, 60, 65, 66, 68]. This general agree- andspinfunctions.
Edd
mentsupportsfortheideathataccretioninAGNmodeisan This represents the magnetic field strength developed by
important,andperhapstheprinciplemechanismbywhichlo- a maximally rotating black hole at the peak of its accretion
cal SMBHs grow [48, 60, 64, 65]. Thus, it is reasonable to phaseandattheupperendofthemassfunction.Inthefollow-
assume that many of the more massive quiescent SMBHs in ingcalculationsweworkwiththisfiducialstrength,yielding.
the local universe are ”remnant” quasars [48, 63], and that Itwillbeseenthatarelativelyhighfieldstrength,B ∼104G,
0
thesegreatlyoutnumberactivequasarsinthemodernepoch. isnecessarytoachieveasignificantfluxof1011 GeVsingle-
In their active phases, accreting SMBHs are expected to protonCRsarounda109M SMBH.However,whenwecon-
(cid:12)
support strong magnetic fields. Assuming a rapidly-rotating sider black holes up to 1010M , and heavy nuclei CRs, the
(cid:12)
black hole, which may be common in luminous quasars [65, required field strength drops by up to two orders of magni-
69],thismagneticfieldisexpectedtosurvivetheexpirationof tude. Wealsoassumethatasignificantfractionofthemaxi-
the quasar phase. We calculate a typical relic magnetic field mumfieldstrengthissustainedasaccretionratedropsintothe
strengthforarapidlyrotatingSMBHbyassumingthatitcor- remnantquasarphase. Furtherdataonthespatialassociation
respondstomaximumfieldstrengthacquiredduringitsquasar ofUHECRswithSMBHswilltestthisassumption.
phase.Thisinturncorrespondstoitsmaximumaccretionrate.
Significantmagneticfieldsdoappeartosurviveinthecase
To determine this accretion rate we assume the lower end of
of least one class of remnant quasar. BL Lacertae objects
α-diskradiativeefficiency,η=0.1(correspondingtoahigher
are AGNs of the blazar class – broadly, AGNs with jets that
accretion rate per unit luminosity), and that the quasar radi-
are closely aligned with our line of sight [45, 71]. BL Lacs
atesattheEddingtonlimit. Themassaccretionrateneededto
show little or no signs of broad emission lines or thermal
sustaintheEddingtonluminosityis
emission [45, 72, 73]. This implies that at least a subset
of BL Lacs are starved of gas, and perhaps in a post-quasar
L
M˙Edd = ηEcd2d (cid:39)22M9 M(cid:12)yr−1. (2) phase. Under this interpretation, flat-spectrum radio quasars
(FSRQs) become the progenitor population of low-accretion
whereM = M/(109M ). Theaccretingplasmaisassumedto BL Lacs [74–76]. This evolutionary argument is supported
9 (cid:12)
support an axisymmetric magnetic field configuration due to by BL Lacs’ negative cosmological evolution; their number
thegenerationofcurrents. Following[32],wegetanestimate densitiesriseataroundthesameepochasthedeclineofFS-
of the characteristic magnetic field strength B by assuming RQs [77]. BL Lacs exhibit BZ-powered jets [78], indicating
0
pressure equilibrium between the magnetic field and the in- thatsignificantmagneticfieldsmaybesustainedlongafterthe
fallingmatter[70] cessationofsignificantaccretion.
Hanny’s Voorwerp contains perhaps the first direct probe
(cid:32) M˙ (cid:33)1/2 of quasar evolution [79]. Observations in the optical, ultra-
B ∼104 M−1/2 G. (3)
0 9 M˙ violet, and X-ray indicate that this unusual object near the
Edd
spiral galaxy IC 2497 contains highly ionized gas [80]. The
emission-lineproperties,andlackofX-rayemissionfromIC
2497,seemtoindicatethisspiralgalaxyisahighlyobscured
AGN with a novel geometry arranged to allow photoioniza-
tion of Hanny’s Voorwerp but not the galaxy’s own circum-
nucleargas[81],oralternativelytheluminosityofthecentral
sourcehasdecreaseddramaticallywithinthelast105 yr[82].
Shouldthisbethecase,Hanny’sVoorwerpmayexemplifythe
firstdetectionofaquasarlightecho. Interestingly,IC2497is
insidetheTAhotspot;seeFig.1.However,themeasuredred-
shiftz=0.05ofIC2497,setsthisdormantquasaroutsideour
local GZK–horizon (∼ 100 Mpc). To sample sources inside
theGZK–sphereweadoptadistance-limited(z (cid:46) 0.02)com-
pilationofquasarremnantcandidates,capableofaccelerating
FIG.1.ComparisonofUHECReventlocationswithquasarremnants
UHECRs [34, 87]. It is compelling that five of the objects
inequatorialcoordinates,withR.A.increasingfromrighttoleft.The
contained in this sample are also inside the TA hot spot; see
circlesindicatethearrivaldirectionsof231eventswithE>52EeV
Fig.1.
andzenithangleθ < 80◦ detectedbythePierreAugerObservatory
from 2004 January 1 up to 2014 March 31 [83–86]. The squares CentaurusA(CenA)isacomplexradio-loudsourceiden-
indicate the arrival directions of 72 events with E > 57 EeV and tifiedatopticalfrequencieswiththegalaxyNGC5128. Cen
θ < 55◦ recordedfrom2008May11to2013May4withTA[22]. A is the closest radiogalaxy to Earth (distance ∼ 3.4 Mpc)
Thestarsindicatethelocationofnearbyquasarremnants[34,87]. andhaslongbeensuspectedtobeapotentialUHECRaccel-
The shaded region delimits the TA hot-spot. The locations of the erator[88,89]. ThePierreAugerCollaborationhassearched
sources inside the TA hot-spot and the closest object in the survey for anisotropies in the direction of Cen A scanning the en-
(NGC5128)areexplicitelyidentified. ergythresholdbetween1010.6GeVand1010.9GeVandcount-
4
ing events in angular radii ranging from 1◦ to 30◦ [83]. wherer =r /2isthegravitationalradius. Theergosphereis
g S
The strongest departure from isotropy (post-trial probability theregiondescribedbytherelation
∼ 1.4%) has been observed for E > 58 EeV in a window of
15◦: 14 events (out of a total of 155) have been observed in r <r<r =r (cid:16)1+ √1−a2cos2θ(cid:17) , (7)
h max g
such an angular window while 4.5 are expected on average
fromisotropicdistributions.
where θ is the angle to the polar axis; see e.g. [95, 96]. In-
Astatisticalanalysistoascertainthesignificanceofapossi-
sidetheergosphere(whichisellipsoidalinshape),spacetime
blecorrelationbetweenUHECRsandSMBHswouldrequire
ispulledheavilytowardsthedirectionoftheblackholerota-
using data from Auger and TA. However, if the high energy
tion(framedragging),andconsequentlynostaticobservercan
end of the cosmic ray spectrum is the same in the north and
exist because the particles must co-rotate with the hole. The
in the south, there appears to be a difference in the energy
blackhole’sangularvelocityΩcoincideswiththeangularve-
scalebetweenthetwodetectors[90], makingsuchstudyun-
locityofthedraggingofinertialframesatthehorizon[96]
reliableatpresent. Futureexperimentsthatwillobserveboth
hemispheres using the same apparatus, such as the Probe Of (cid:32) (cid:33)
c
Extreme Multi-Messenger Astrophysics (POEMMA), would |Ω|=a . (8)
makearobuststatisticalanalysismorestraightforward.1 2rh
Forconvenience,wesubdividetheSMBHsampleintotwo
subcomponents: one with masses (cid:38) 1010M and one with Ithaslongbeenknownthatarotatingblackholecanradi-
(cid:12)
masses (cid:46) 1010M . The rationale for using 1010M as the ate away its available reducible energy [97, 98]. In this con-
(cid:12) (cid:12)
demarcation mass will become evident in Sec. IV. Roughly text Blandford and Znajek (BZ) proposed a model of elec-
speaking,16%ofSMBHshavemassabove1010M [87]. In- tromagneticextractionofblackhole’srotationalenergybased
(cid:12)
sidetheTAhotspotoneofsix(fivenearby)candidatesources ontheanalogywiththeclassicalFaraday(unipolarinduction)
isconsistentwithamass(cid:38)1010M . Thus,wecanassumethat dynamo phenomenon [36, 37]. In the BZ mechanism, the
(cid:12)
theratiosinsidethehotspotarerepresentativeoftheratiosof magnetic field B near the horizon taps the rotational energy
theuniverseatlarge. oftheblackholeandgeneratespowerfuloutflowsofelectro-
magnetic (Poynting) energy. BZ argue that spacetime frame
dragging induces an electric field E that is strong enough to
III. UHECRACCELERATIONINQUASARREMNANTS breakthevacuumandestablishanelectron-positronforce-free
magnetosphere.
In realistic astrophysical situations involving quasars, the The particulars of the black hole magnetosphere could be
blackholeitselfisuncharged,andthegravityoftheaccretion very complicated, distinctively if the poloidal magnetic field
diskispracticallynegligible. Thismeansthatthegeometryof near the horizon is misaligned with the black hole rotation
spacetimeisdescribedbytheKerrmetricandthereforedeter- axis [35]. For simplicity, herein we assume that the B-field
minedbytwofreeparameters: theblackholemass M andits isalignedwiththeaxisofrotation. Thismanageablesystem
angularmomentum allows a transparent analytic calculation which captures the
GM2 essenceofmostaspectsoftheBZmechanismandleadstoa
|J|=aJ =a , (4) correctorderofmagnitude.
max c
Conduction electrons within the magnetosphere undergo
where0≤a≤1isthedimensionlessspinparameter[92].Be-
collisionswiththeorbitingatoms,andthiscausesthemtotake
foreproceedingwenotethatforaSchwarzschildblackhole,
up the rotational motion. At each point the electrons have a
a=0[93],whereasforamaximallyspinning(a.k.a. extreme netdriftvelocity,v=Ω∧r,where risthepositionvectorof
Kerr) black hole, a = 1. As a matter of fact, the increase of
thepointinquestionrelativetoanoriginwhichischosentolie
thespinparameterwouldstopata ≈ 0.998becausephotons
onthemagneticaxis. Theconductionelectronsexperiencea
emitted from the disk on retrograde paths are more likely to
magneticforce−ev∧B,whichismainlydirectedtowardsthe
becapturedbytheholethanprogradeones,hencede-spinning
centralaxis;asaresultanegativechargeappearsinthecoreof
thehole[94].
themagnet,andapositivechargeonitscurvedoutersurface.
For static black holes, the scale characterizing the event
Inequilibriumanelectrostaticfieldissetup,suchthattheto-
horizonr istheSchwarzschildradius
h talLorentzforceontheconductionelectronsiszero. Namely,
(cid:32) (cid:33)
2GM M F=−eE−ev∧B=0,andso
r = (cid:39)3×1014 cm. (5)
S c2 109M
(cid:12)
E=−v∧B. (9)
Forspinningblackholes,the(spherical)eventhorizonsurface
isdefinedby
(cid:16) √ (cid:17) WenowcalculatethepotentialdifferenceV correspondingto
rh =rg 1+ 1−a2 , (6) thiselectricfield. Itisstraightforwardtoseebyinspectionof
(9)thatalongtheaxisofrotationE=0,andthereforeV =0at
allpointsalongthisaxis. Foramaximallyspinninghole,the
1POEMMAhasbeenrecentlyselectedbyNASAforanin-depthprobemis- potential drop between the central (r = 0) and the marginal
sionconceptstudyinpreparationforthenextdeceadalsurvey[91]. (r = r ) magnetic surfaces passing through the horizon is
max
5
foundtobe Z
(cid:90) (cid:90) rmax 1 G
V =− E·ds= ΩB rdr∼ MB
0 4 c 0 3
0
16.7×10−111.9×1030
∼ MB m2s−1, (10)
4 3×108 M 0 2.5
(cid:12)
where B is the magnetic field strength. Now, recalling that
0
104G=1T=1Vsm−2,wecanrewrite(10)as 2
V ∼1020M B V, (11) p
9 4
1.5
whereB = B /104G.
4 0
Farawayfromthehorizon,theelectromagneticfieldcanbe
expressedas[99] 1 e
rl→im∞Brˆ = B0cosθ, rl→im∞Bθˆ =−B0sinθ, (12) 0.5 e
and p
B aM(3cos2θ−1) 0 0.5 1 1.5 2 2.5 3 X
lim E =− 0 , lim E =O(r−4). (13)
r→∞ rˆ r2 r→∞ θˆ
FIG.2. Magnetic(bluesolid)andelectric(redsolid)fieldlinesas-
The electric field has a quadrupole topology, which is dis- sociatedtotheasymptoticallyhomogeneousBfieldalignedwiththe
tinctlyseeninFig.2.Thevoltagedifferencebetweenthehori- rotationaxisofamaximallyspinningblackhole.Thewideblueand
zonandr = ∞isoforderV. Withsuchahugevoltagedrop redarrowsshowthedirectionsofaccelerationofelectronsandpro-
along field lines, one expects that electrons roaming in the tonsindifferentregions. Theaccelerationcanbeavoidedonlyifa
particleresidesatasurfaceatwhichtheelectricfieldisorthogonalto
vicinityofthehorizonwouldbeacceleratedtohugeenergies
and,uponcollidingwithstrayphotonsand/orpositrons,pro- themagneticfield,theso-calledforce-freesurfaces. Thicksolidand
dashedlinesindicatetheconicalandequatorialforce-freesurfaces.
duceacascadeofe+e− pairs. Veryquicklytherefore,thevac- Theforce-freesurfacesseparatedifferentaccelerationregions,with
uumsurroundingtheblackholewouldbefilledwithahighly
electricfielddirectedalongoroppositelytothemagneticfield.Taken
conducting plasma. The plasma-loaded field lines can serve from[35].
as wires to complete a circuit between the pole and equator
ofthehorizon. Theelectromotiveforcearoundthiscircuitis
numericallycomparableto(11). Thus,ifacosmicraybaryon In a realistic situation the energy losses of the accelerated
canfullytapthispotentialaccelerationuptoextremeenergies, particleslimitthemaximalenergiestothevaluesbelowthees-
timateof(14). Itisthisthatwenowturntostudy. Beforepro-
E =ZeV ∼1011ZM B GeV∼1011ZM1/2GeV, (14)
max 9 4 9 ceedingthough,itisinterestingtonotethattheaxisymmetric
stationaryflowinthevicinityoftherotation-poweredcompact
would become possible. However, the charge density in the
objectcouldprovideasourceofultrarelativisticnucleiwithan
vicinity of accreting black holes could be so high that a sig-
extremelyflatinjectionspectrum(typically∝ E−1[100,101]),
nificantfractionofthispotentialwouldbescreenedandsono
in good agreement with the requirements to accommodate
longeravailableforparticleacceleration. Therefore,itseems
Augerobservations[17,24]. Moreover,asteepenedinjection
more appropriate to define an effective potential where the
spectrumwouldarisequitenaturallyifthemediumsurround-
availablelengthscale,thegapheightζ,isexplicitlytakeninto
ingthecompactobjectwereleakyattheselateages.
account. Following[96],wetake
(cid:32)ζ (cid:33)2
Veff ∼V r . (15) IV. MECHANISMSOFENERGYDISSIPATION
h
Accordingly,thecharacteristicrateofenergygainisfoundto
IntheabsenceofenergylossesthemaximumLorentzfactor
be
isgivenby
(cid:12)
dE(cid:12) c
dt (cid:12)(cid:12)(cid:12) =ZeVeff ζ , (16) Z (cid:32)ζ (cid:33)2
acc γacc ∼1011 M B , (18)
max A 9 4 r
h
andsotheaccelerationtimescaleisgivenby
(cid:34)dE(cid:12)(cid:12) (cid:35)−1 E ζ whereE =γAmp,withmp theprotonmassandAthenuclear
τacc = E dt (cid:12)(cid:12)(cid:12)acc = ZeVeff c . (17) bthaerycounrvneudmmbaegrn.eWticitfiheinldtlhienepsoatnendtsiaoledmroitpc,utrhveatnuurcel-eraidfioaltlioown
photons. Theenergylossrateortotalpowerradiatedawayby
6
TABLEI.Relevantsourceparametersandnormalizationfactornβ,forβ=−4,−5.
0
Source L /L M n−4/(MeVcm3) n−5/(MeVcm3) z References
FIR (cid:12) 9 0 0
NGC2768 4.8×108 0.16 7.8×1023 1.0×1024 0.0051 [34,114,115]
NGC2832 <1.7×108 11.4 3.0×1019 3.9×1019 0.0194 [87,115,116]
NGC3610 4.7×107 0.05 6.9×1023 8.9×1023 0.0066 [34,115,117]
NGC3613 <3.6×108 0.16 4.8×1023 6.2×1023 0.0066 [34,116,117]
NGC4125 3.0×108 0.28 1.5×1023 1.9×1023 0.0063 [34,115,117]
12 12
NGC 2768 56Fe NGC 2768 28Si
photonuclear interactions photonuclear interactions
10 curvature radiation 10 curvature radiation
ζ=rh ζ=rh
ζ=0.1rh ζ=0.1rh
8 ζ=0.01rh 8 ζ=0.01rh
) )
s s
/ 6 / 6
nt nt
acc,i 4 acc,i 4
( (
g g
o o
l 2 l 2
0 0
2 2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
log(E/EeV) log(E/EeV)
12 12
NGC 2768 56Fe NGC 2768 28Si
photonuclear interactions photonuclear interactions
10 curvature radiation 10 curvature radiation
ζ=rh ζ=rh
ζ=0.1rh ζ=0.1rh
8 ζ=0.01rh 8 ζ=0.01rh
) )
s s
/ 6 / 6
nt nt
acc,i 4 acc,i 4
( (
g g
o o
l 2 l 2
0 0
2 2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
log(E/EeV) log(E/EeV)
FIG. 3. Comparison of acceleration and interaction time scales for NGC 2768. The bumpy curve indicates the interaction time scale for
photonuclear interactions. The straight line with negative slope indicate the time scale due to curvature radiation. The straight lines with
positiveslopecorrespondtoaccelerationtimescaleswithdifferentζ.Theupperpanelscorrespondtoβ=−4andthelowerpanelstoβ=−5.
asinglecosmicrayis[102] Acceleration gains are balanced by radiative losses. In the
absenceofotherdampingmechanisms,theradiationreaction
dE(cid:12)(cid:12)rad 2Z2e2c
(cid:12)(cid:12) = γ4, (19) limitis[103,104]
dt (cid:12) 3 r2
loss c (cid:18)B (cid:19)1/4 (cid:32)r (cid:33)1/2(cid:32)ζ (cid:33)1/4
wherer isthecurvatureradiusofthemagneticfieldlines,and γrad ∼1011 4 M1/2 c . (21)
c max Z 9 r r
γistheLorentzfactoroftheradiatingparticles. Thecharac- h h
teristiccoolingtimescaleisthengivenby All in all, direct electric field acceleration on the black hole
magnetospherewouldallowLorentzfactorsupto
(cid:34)dE(cid:12)(cid:12)rad(cid:35)−1
τrinadt =γAmp dt (cid:12)(cid:12)(cid:12)loss . (20) γmax =min (cid:110)γmacacx, γmraadx(cid:111) . (22)
7
12 12
NGC 2832 56Fe NGC 2832 28Si
photonuclear interactions photonuclear interactions
10 curvature radiation 10 curvature radiation
ζ=rh ζ=rh
ζ=0.1rh ζ=0.1rh
8 ζ=0.01rh 8 ζ=0.01rh
) )
s s
/ 6 / 6
nt nt
acc,i 4 acc,i 4
( (
g g
o o
l 2 l 2
0 0
2 2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
log(E/EeV) log(E/EeV)
12 12
NGC 2832 56Fe NGC 2832 28Si
photonuclear interactions photonuclear interactions
10 curvature radiation 10 curvature radiation
ζ=rh ζ=rh
ζ=0.1rh ζ=0.1rh
8 ζ=0.01rh 8 ζ=0.01rh
) )
s s
/ 6 / 6
nt nt
acc,i 4 acc,i 4
( (
g g
o o
l 2 l 2
0 0
2 2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
log(E/EeV) log(E/EeV)
FIG.4.ComparisonofaccelerationandinteractiontimescalesforNGC2832.ConventionsareasinFig.3.
NotethatforourfiducialvaluesB ∼1, M ∼1,andr ∼r , Namely, the interaction times are comparable if the photon
4 9 c h
we have γacc ∼ γrad if ζ ∼ r . However, by taking ζ < r density is assumed to follow a (modified) black body spec-
max max h h
one can always suppress the emission of curvature photons. trum[24]. Formoredistancesources,itisnottrivialtodisen-
Moreover, asweshowinwhatfollowsphotonuclearinterac- tangle which photons are near the SMBH. As a first approx-
tionsontheambientphotonbackgroundbecomethedominant imation, we use the galaxy-wide average far infra-red (FIR)
mechanismforenergylosses. fluxreportedintheNASAextragalacticdatabasetonormalize
The estimation of the photon density in the vicinity of the thespectrumandestimatethespectralindices[107]. Ifacor-
SMBH is not straightforward. Studies for Cen A are quite relationbecomesevidentinthefutureamoredetailedanalysis
extensive [105, 106], and the photon spectrum around the wouldberequired. Byaveragingoverthethermalspectraof
SMBHisconsistentwithathermalorigin: mostlydustemis- thesourcesstudiedin[108]wetakeα=2and−5≤β≤−4.
sion heated by the nucleus [105]. For simplicity, in our cal- We normalize the spectrum to the FIR luminosity in the
culations we approximate the photon spectrum as a broken vicinityoftheSMBH
power-law,
(cid:90)
 LBH =4πr2c n(ε)εdε
n(ε)=n0((εε//εε00))αβ εot<heεrw0ise, (23) FIR =4πrg2cn (cid:40)(cid:90) ε(0ε/ε )αεdε+(cid:90) ∞(ε/ε )βεdε(cid:41),(24)
g 0 0 0
εmin ε0
which is also consistent with the data. Here ε is the photon
energyandthemaximumofthedensityisatanenergyofε . whereε = 10−4 eVandε = 10−2 eV. Thereisageneral
0 min 0
The broken power-law spectrum allows a complete analytic consensusthattheextendedFIRemissionalongthedustlane
treatment of photonuclear interactions, and the global result ofCenA,L ∼2×1010L ,cannotonlybeattributedtodust
FIR (cid:12)
does not depend on the exact shape of the photon spectrum. heatedbytheactivenucleus[109–112]. Opticaldetectionsof
8
12 12
NGC 3610 56Fe NGC 3610 28Si
photonuclear interactions photonuclear interactions
10 curvature radiation 10 curvature radiation
ζ=rh ζ=rh
ζ=0.1rh ζ=0.1rh
8 ζ=0.01rh 8 ζ=0.01rh
) )
s s
/ 6 / 6
nt nt
acc,i 4 acc,i 4
( (
g g
o o
l 2 l 2
0 0
2 2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
log(E/EeV) log(E/EeV)
12 12
NGC 3610 56Fe NGC 3610 28Si
photonuclear interactions photonuclear interactions
10 curvature radiation 10 curvature radiation
ζ=rh ζ=rh
ζ=0.1rh ζ=0.1rh
8 ζ=0.01rh 8 ζ=0.01rh
) )
s s
/ 6 / 6
nt nt
acc,i 4 acc,i 4
( (
g g
o o
l 2 l 2
0 0
2 2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
log(E/EeV) log(E/EeV)
FIG.5.ComparisonofaccelerationandinteractiontimescalesforNGC3610.ConventionsareasinFig.3.
luminous HII regions in relatively unoscured regions of the we also study the case of NGC 5128. This case is special,
dark lane [113] suggest that recently formed stars embedded as the photon spectrum peaks in the mid-infrared. We take
in the dust lane must be responsible for most of the broadly (cid:15) = 0.13 eV and L /L = 1.3×108. The mass of the
0 mid−IR (cid:12)
extended FIR emission. Only about 1% of the FIR emission SMBHissomewhatuncertain[119–121]. Inourcalculations
comes from the active nucleus. With this in mind, we take we adopt M = 0.1, yielding n−4 = 3.0×1023 MeV−1cm−3
9 0
LFBIHR/LFIR ∼10−2. Thenormalizationfactoristhen andn−05 =3.9×1023MeV−1cm−3.
Theinteractiontimeforahighlyrelativisticnucleusprop-
n = LFBIHR  ε−0α (cid:104)εα+2−εα+2(cid:105)− ε20 −1 . (25) agating through an isotropic photon background with energy
0 4πrg2c(α+2) 0 min (β+2) ε and spectrum n(ε(cid:82)), normalized so that the total number of
photonsinaboxis n(ε)dε,isgivenby[122]
In Table I we summarize the relevant properties of a sub-
group of 5 candidate sources shown in Fig. 1. In the case 1 c (cid:90) ∞ n(ε) (cid:90) 2γε
= dε dε(cid:48)ε(cid:48)σ(ε(cid:48)), (26)
of no-detection in the FIR we normalize the photon flux to τAγ 2 0 γ2ε2 0
the reported upper-limit. To determine the distance to these int
sources we adopt the usual concordance cosmology of a flat
whereσ(ε(cid:48))isthephotonuclearinteractioncrosssectionofa
universe dominated by a cosmological constant, with ΩΛ = nucleusbyaphotonenergyε(cid:48)intherestframeofthenucleus.
0.692 ± 0.012 and a cold dark matter plus baryon compo-
We have found that for the considerations in the present
nent Ω = 0.308±0.012; the Hubble parameter as a func-
m work,thecrosssectioncanbesafelyapproximatedbythesin-
tion of redshift is given by H2(z) = H02[Ωm(1 + z)3 + ΩΛ], glepoleofthenarrow-widthapproximation,
normalizedtoitsvaluetoday,H =100hkms−1Mpc−1,with
0
h=0.678[118]. Γ
To characterize the population of nearby quasar remnants σ(ε(cid:48))=π σres 2res δ(ε(cid:48)−ε(cid:48)res), (27)
9
12 12
NGC 3613 56Fe NGC 3613 28Si
photonuclear interactions photonuclear interactions
10 curvature radiation 10 curvature radiation
ζ=rh ζ=rh
ζ=0.1rh ζ=0.1rh
8 ζ=0.01rh 8 ζ=0.01rh
) )
s s
/ 6 / 6
nt nt
acc,i 4 acc,i 4
( (
g g
o o
l 2 l 2
0 0
2 2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
log(E/EeV) log(E/EeV)
12 12
NGC 3613 56Fe NGC 3613 28Si
photonuclear interactions photonuclear interactions
10 curvature radiation 10 curvature radiation
ζ=rh ζ=rh
ζ=0.1rh ζ=0.1rh
8 ζ=0.01rh 8 ζ=0.01rh
) )
s s
/ 6 / 6
nt nt
acc,i 4 acc,i 4
( (
g g
o o
l 2 l 2
0 0
2 2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
log(E/EeV) log(E/EeV)
FIG.6.ComparisonofaccelerationandinteractiontimescalesforNGC3613.ConventionsareasinFig.3.
where σ is the resonance peak, Γ its width, and ε(cid:48) the Substituting(23)into(28)wefinallyfind[24]
res res res
pole in the rest frame of the nucleus. The factor of 1/2 is
introduced to match the integral (i.e. total cross section) of
theBreit-Wignerandthedeltafunction[123].
Themeaninteractiontimecannowbereadilyobtainedsub-
stitutingEq.(27)intoEq.(26),
1 cπσ ε(cid:48) Γ (cid:90) ∞ dε
≈ res res res n(ε) Θ(2γε−ε(cid:48) )
τAγ(E) 4γ2 0 ε2 res
int
cπσ ε(cid:48) Γ (cid:90) ∞ dε
= res res res n(ε). (28)
4γ2 (cid:15)r(cid:48)es/2γ ε2

1 = 1  (Eb/E)β+1 (cid:104) (cid:105) E ≤ Eb , (29)
τAγ(E) τb (1−β)/(1−α) (Eb/E)α+1−(Eb/E)2 +(Eb/E)2 E > Eb
int
where The parameters characterizing the photodisintegration cross
2E (1−β) ε(cid:48) Am section are: σres ≈ 1.45×10−27 cm2A, Γres = 8 MeV, and
τ = b and E = res p. (30) (cid:15)(cid:48) =42.65A−0.21(0.925A2.433)MeV,forA>4(A≤4)[124].
b cπσ Am Γ n b 2ε res
res p res 0 0
10
12 12
NGC 4125 56Fe NGC 4125 28Si
photonuclear interactions photonuclear interactions
10 curvature radiation 10 curvature radiation
ζ=rh ζ=rh
ζ=0.1rh ζ=0.1rh
8 ζ=0.01rh 8 ζ=0.01rh
) )
s s
/ 6 / 6
nt nt
acc,i 4 acc,i 4
( (
g g
o o
l 2 l 2
0 0
2 2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
log(E/EeV) log(E/EeV)
12 12
NGC 4125 56Fe NGC 4125 28Si
photonuclear interactions photonuclear interactions
10 curvature radiation 10 curvature radiation
ζ=rh ζ=rh
ζ=0.1rh ζ=0.1rh
8 ζ=0.01rh 8 ζ=0.01rh
) )
s s
/ 6 / 6
nt nt
acc,i 4 acc,i 4
( (
g g
o o
l 2 l 2
0 0
2 2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
log(E/EeV) log(E/EeV)
FIG.7.ComparisonofaccelerationandinteractiontimescalesforNGC4125.ConventionsareasinFig.3.
The parameters for the photopion production cross section one-shotaccelerationmechanisminquasarremnantsaccord-
are: σ (cid:39) 5.0×10−28 cm2A, Γ = 150 MeV, and ε(cid:48) = ingtohowsourceparameters(massofthecentralengineand
res res res
(m2 −m2)/(2m )(cid:39)340MeV[118]. theambientphotonbackgrounds)impactthephotonuclearin-
∆ p p
In Figs. 3, 4, 5, 6, 7, and 8 we compare the characteris- teraction. In the first type nuclei are completely photodis-
tic time scales of acceleration and energy losses for the var- integrated before escaping the acceleration region producing
ious representative sources given in Table I and Cen A. We a flux of secondary nucleons, e.g. NGC 2768, 3610, 3613,
consider 56Fe and 28Si as fiducial nuclear species. In gen- 4125, and 5128. In the second type photopion production
eral,photonuclearinteractionsdominatetheenergylosses.By is the major energy damping mechanism and nuclei are able
comparingtheresultsforβ=−5andβ=−4,wecanseethat to escape without suffering significant spallation, e.g. NGC
there is almost no dependence on the photon spectral index. 2832(whichisnearthecenteroftheTAhot-spot). Combin-
ForE (cid:38)1011GeV,theacceleratednucleireachthephotopion ing the ideas put forward in [23, 24] we can now formulate
productionthreshold,andthesignificanceofthisprocessbe- a new hybrid model to explain the spectral shape of extra-
comesmoreimportantwithincreasingenergy. Thephotopion galacticcosmicrays,includingthecriticalregionoftheankle.
productionthresholdcorrespondstoanenergy-per-nucleonof Namely,thesecondarynucleonsproducedinthephotodisinte-
E/A ∼ 1010 GeV. Note that the onset of photopion produc- grationprocesswouldhaveasoftspectralindexatthesource,
tion is above the required maximum energy=109.5GeV for andconsequentlycanexplaintheenergyregionbelowthean-
thefiducialmodelin[24]. Therefore,weconcludethatsome kle. Note that after the photodisintegration process the sec-
quasarremnantsarecapableoflaunchingheavy(andmedium ondaryprotonscanstillbeacceleratedtoreachenergiesnear
mass)nucleiuptotheobservedmaximumenergies. theankle. Ontheotherhand,heavyandmediummassnuclei
areemittedwithaharderspectralindex, andtherefore(upon
ThoughFigs.3,4,5,6,7,and8provideonlyana¨ıveillus-
propagationtoEarth)canpopulatethespectrumabovethean-
tration of particle acceleration with energy dissipation in the
kle,allthewaytotheGZKcutoff.Weestimatethatabout16%
blackholedynamoitisevidentthatitpossibletoclassifythe