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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 luminous 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