Scenarios for ultrafast gamma-ray variability in AGN
aa r X i v : . [ a s t r o - ph . H E ] M a y Draft version March 13, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
SCENARIOS FOR ULTRAFAST GAMMA-RAY VARIABILITY IN AGN
F.A. Aharonian , , , M.V. Barkov , , , D. Khangulyan Dublin Institute for Advanced Studies, 31 Fitzwilliam Place, Dublin 2, Ireland Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany National Research Nuclear University MEPhI, Kashirskoje Shosse, 31, 115409 Moscow, Russia Deutsches Elektronen-Synchrotron (DESY), Platanenallee 6, D-15738 Zeuthen, Germany Astrophysical Big Bang Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Department of Physics and Astronomy, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907-2036, USA Department of Physics, Rikkyo University, Nishi-Ikebukuro 3-34-1, Toshima-ku, Tokyo 171-8501, Japan
Draft version March 13, 2018
AbstractWe analyze three scenarios to address the challenge of ultrafast gamma-ray variability reported fromactive galactic nuclei. We focus on the energy requirements imposed by these scenarios: (i) externalcloud in the jet, (ii) relativistic blob propagating through the jet material, and (iii) production ofhigh-energy gamma rays in the magnetosphere gaps. We show that while the first two scenariosare not constrained by the flare luminosity, there is a robust upper limit on the luminosity of flaresgenerated in the black hole magnetosphere. This limit depends weakly on the mass of the central blackhole and is determined by the accretion disk magnetization, viewing angle, and the pair multiplicity.For the most favorable values of these parameters, the luminosity for 5-minute flares is limited by2 × erg s − , which excludes a black hole magnetosphere origin of the flare detected from IC 310.In the scopes of scenarios (i) and (ii), the jet power, which is required to explain the IC 310 flare,exceeds the jet power estimated based on the radio data. To resolve this discrepancy in the frameworkof the scenario (ii), it is sufficient to assume that the relativistic blobs are not distributed isotropicallyin the jet reference frame. A realization of scenario (i) demands that the jet power during the flareexceeds by a factor 10 the power of the radio jet relevant to a timescale of 10 years. Subject headings:
Gamma rays: galaxies - Galaxies: jets - Radiation mechanisms: non-thermal INTRODUCTIONThe hypothesis of supermassive black holes (SMBHs)as powerhouses of active galactic nuclei (AGN) has beenproposed (Salpeter 1964; Zel’dovich & Novikov 1966;Lynden-Bell 1969) to explain the immense luminositiesof AGN and quasars by the release of the gravitationalenergy through the process of gas accretion. The radia-tion power of the accreting plasma is limited by the Ed-dington luminosity, L Edd = 1 . × M erg s − , where M = M bh / M ⊙ is the mass of the black hole in theunities of 10 solar masses.The apparent luminosities of radiation of many AGN, L app = 4 πD L f ( f is the detected energy flux and D L is the source luminosity distance) may exceed the Ed-dington luminosity of SMBHs by orders of magnitude.However, the “energy crisis” can be overcome if one as-sumes that the observed radiation is highly anisotropic,namely, that it is produced in a collimated outflow (jet)close to the line of sight (see, e.g., Begelman et al. 1984).The concept of relativistically beamed emission offersnot only an elegant scheme for the unification of variousclasses of AGN (see, e.g., Urry & Padovani 1995), butalso provides a natural interpretation of enormous fluxesof their nonthermal emission. Indeed, the assumption ofproduction of radiation in a source relativistically mov-ing toward the observer with a Doppler factor δ ≫ intrinsic luminosities of AGNdubbed blazers , reducing them by orders of magnitudecompared to the apparent luminosity, L app = δ L int .This assumption also allows a larger (more “comfort- able”) size of the production region that is demanded bythe observed variability of radiation: l ≤ c ∆ t var δ . Theserelations apply to all electromagnetic wavelengths, butthey are crucial, first of all, for gamma-ray loud AGN,the apparent luminosities of which during strong flares,e.g., in 3C454.3 (Striani et al. 2010; Abdo et al. 2011)and 3C 279 (Hayashida et al. 2015), can achieve the levelof L γ ∼ − erg s − . Strong Doppler boosting is alsoneeded for prevent severe internal gamma-ray absorp-tion, especially at VHE energies (see, e.g., Celotti et al.1998).The gamma-ray emission of blazers is strongly vari-able, with fluxes that match the sensitivity of the Fermi
Large Area Telescope (LAT) in the MeV/GeV bandwell, and the current arrays of imaging atmosphericCherenkov telescopes (IACT), H.E.S.S., MAGIC,
Ver-itas , in the VHE band. During the strongest flares ofBL Lac objects like Mkn 421, Mkn 501 and PKS 2155-304, the energy fluxes of VHE gamma-rays often ex-ceed f VHE = 10 − erg cm − s − . Such fluxes can bestudied with IACT arrays with huge detection areasthat are as large as 10 m in almost background freeregime, allowing variability studies on timescales of min-utes . Although the fluxes of flaring powerful quasars atMeV/GeV energies can be significantly larger, f VHE =10 − erg cm − s − , because of the small detection area ofspace-borne instruments ( ≃ ), the capability of thelatter of probing the brevity of such strong AGN flareshas until recently been limited by timescales of hoursand days. However, after the release of the latest soft-ware tools by the Fermi
LAT collaboration, whivh allowa significant increase in the gamma-ray photon statistics,the variability studies at GeV energies for exceptionallybright flares can be extended down to minute timescales.This potential recently has recently been demonstratedby the
Fermi
LAT collaboration for the giant 2015 Juneoutburst of 3C 273 (Ackermann et al. 2016).It is straightforward to compare these timescales withthe minimum time that characterizes a black hole sys-tem as an emitter, namely, the light-crossing time of thegravitational radius of the black hole: τ = r g /c ≈ × M s . (1)Note that r g = GM bh /c = 1 . × M cm is the grav-itational radius corresponding to the extreme Kerr blackhole, i.e. twice smaller than the Schwarzschild radius.Thus, for the mass range of black holes M ≥ M ⊙ ,the current gamma-ray detectors have a potential toexplore the physics of AGN that is close to the eventhorizon on timescales shorter than τ . Such ultra-fast gamma-ray flares have previously been detectedfrom four AGN: PKS 2155 −
304 (Aharonian et al. 2007),Mkn 501 (Albert et al. 2007), and IC 310 (Aleksi´c et al.2014) at TeV energies, and 3C 279 at GeV energies(Ackermann et al. 2016). In addition, a flare with a dura-tion comparable to the BH horizon light-crossing time, ∼ τ , was observed from a missaligned radio galaxy M87,in which the jet Doppler factor is expected to be small(Gebhardt & Thomas 2009). For comparison, it is in-teresting to note that the characteristic timescales of theeven the shortest GRBs ( ∼ τ by severalorders of magnitude.The detection of variable VHE gamma-ray emissionfrom AGN on timescales significantly shorter than τ is an extraordinary result and requires a careful treat-ment and interpretation. The masses of SMBHs indistant AGN are typically derived from the empiricalFaber-Jackson law (also known as the M − σ relation,see Ferrarese & Merritt 2000; Gebhardt et al. 2000). Al-though this statistical method is characterized by a smalldispersion, scatter for individual objects may be signif-icant, which consequently leads to uncertainties of τ .On the other hand, it follows from Eq. (2) that for theminute-scale flares reported from PKS 2155 −
304 andIC 310, the variability time can exceed τ only for massesof the BHs than are lower than 3 × M ⊙ . For bothobjects, different methods of estimating M bh give signif-icantly higher values, and therefore τ < τ .If the emission is produced in a relativistically mov-ing source with a velocity β em , the variability time-scale for the observer is shortened by the Doppler fac-tor δ em = 1 / Γ em (1 − β em cos θ em ); Γ em = 1 / p (1 − β )is the Lorentz factor and θ em is the angle between thesource velocity and the line of sight. Thus if we wish toincrease the proper size of the emitter R ′ (the source sizein the comoving reference frame) to a physically reason-able value of R ′ ≥ r g , the Doppler factor should be large, δ em >
10. For example, in the case of PKS 2155 − For a recent summary of ultrafast gamma-ray flares of AGNsee Vovk & Babi´c (2015) where the mass of SMBH is expected to be high, M ∼
10, the VHE variability sets a lower limit on the value ofthe Doppler factor: δ em ≥
25. However, there is anotherissue of conceptual importance that cannot be ignored.The problem is that if the perturbations originate in thecentral engine and then propagate in the jet, e.g. in theform of sequences of blobs ejected with different Lorentzfactors (leading to internal shocks), the size of the emit-ter in the laboratory frame, R = R ′ / Γ j , would not dependon the Doppler factor and it should exceed the gravita-tional radius: R ≥ r g . Let us present the proper sizeof the production region as R ′ = λ Γ j r g , where Γ j is thejet bulk Lorentz factor, and λ is a dimensionless param-eter, which corresponds to the ratio of the productionregion size in the laboratory frame to the gravitationalradius. The causality condition provides a limitation onthe variability timescale t var ≥ τ λ Γ j Γ em . (2)The variability of t var = 0 . τ inferred from the VHEflares of PKS 2155 −
304 (Aharonian et al. 2007) requiresΓ em ≃ λ Γ j , i.e., the emitter should move relativisti-cally in the frame of the jet, which in turn moves rel-ativistically toward the observer. The jet-in-jet modelsuggested by Giannios et al. (2009) can be considered asa possible realization of this general scenario. Alterna-tively, if the source of the flare does not move relativisti-cally relative to the jet (Γ em ≃ Γ j ), the size of the sourcein the laboratory frame should be much smaller than theblack hole gravitational radius: λ ≃ . λ >
1. Thus, the condition of λ < external origin, i.e., are not directlylinked to the central black hole. This scenario can berealized when a star or a gas cloud of radius R ∗ ≪ r g enters the jet from outside and initiates perturbations onscales smaller than the black hole gravitation radius r g (Barkov et al. 2012a).Finally, it has been suggested that the flarescan be produced in the BH magnetosphere(Neronov & Aharonian 2007; Levinson & Rieger 2011;Rieger 2011). In this case, the production site does notmove relativistically with respect to the observer, andEq. (2) is reduced to t var > τ λ , where λ = R/r g . Thus,the flare originates in a compact region that occupiesa small fraction of the black hole magnetosphere. Ananalogy for this possibility could be the emission ofradio-loud pulsars. It is believed that in these objectsthe radio pulses are produced in the polar cap region,which constitutes only a small part of the pulsar surface.Note that for the typical pulsar radius R psr of 10 km, τ = R psr /c ∼ µ s is too small to be probed throughthe variability of the radio emission. We note here thatalthough the production site of relativistic motion doesnot allow to reducing the minimum variability time (seeEq. (2)), the relativistic beaming effect allows significantrelaxation of the energetics required to produce the To overtake this constraint, some models involve a BH binarysystem as the central engine in PKS 2155 −
304 (Rieger & Volpe2010) flare. Thus, magnetospheric scenarios should havehigher energy requirements then the jet scenarios.In this paper we discuss in rather general terms threepossible scenarios for the production of ultrafast (”sub-horizon” scale) variability in AGNs:(i) The source of the flare is a magnetosphericgap occupying a small volume in the proxim-ity of the black hole close to the event horizon(Neronov & Aharonian 2007; Levinson & Rieger2011).(ii) The emitter moves relativistically in the jet ref-erence frame. The most feasible energy sourcefor this motion is magnetic field reconnectionin a highly magnetized jet (Lyubarsky 2005;Giannios et al. 2009; Petropoulou et al. 2016).(iii) Flares are initiated by penetration of external ob-jects (stars or clouds) into the jet (Araudo et al.2010; Barkov et al. 2012a).Apparently, any model designed to explain the ultra-fast variability on timescales t var < τ should addresssome other key issues. In particular, the required overallenergy budget should be feasible, the source should beoptically thin for gamma-rays, and of course, the pro-posed radiation mechanism(s) should be able to explainthe reported spectral features of gamma-ray emission. ADDRESSING THE “SUBHORIZON” SCALEVARIABILITY2.1.
The Magnetospheric Model
Magnetospheres of the central SMBHs in AGN can besites of production of gamma-rays with spectra extendingto VHE energies (see, e.g., Beskin et al. 1992; Levinson2000; Neronov & Aharonian 2007; Rieger & Aharonian2008; Levinson & Rieger 2011). At low accretion rates,the injection of charges into the BH magnetosphere isnot sufficient for a full screening of the electric field in-duced by the rotation of the compact object. The re-gions with unscreened electric field, referred to as gaps,are capable of effective acceleration of charged particles.Such a scenario may result in a variability of the sourceon “subhorizon” timescales since the size of the gap ismuch smaller than the gravitational radius. The attrac-tiveness of this scenario is its applicability to the non-blazar-type AGN. On the other hand, because of boththe low accretion rate and the lack of Doppler boosting,the gamma-ray luminosities of such objects are expectedto be quite modest when compared to blazers. Thereforethe detectability of the black hole magnetospheric radi-ation is most likely limited by a few nearby objects. Inparticular, the radio galaxy M87, as well as the compactradio source Sgr A* in the center of our Galaxy, can beconsidered as suitable candidates for the realization ofsuch a scenario (see, e.g., Levinson & Rieger 2011).The energy release in the entire magnetosphere is lim-ited by the BZ luminosity. Below we follow a simpli-fied treatment that allows us to estimate the energyrelease in a thin vacuum gap formed in the SMBHmagnetosphere. The rotation of a magnetized neu-tron star or BH in vacuum induces an electric field, E , in the surrounding space (Goldreich & Julian 1969; Blandford & Znajek 1977). If a charge enters this re-gion, the electric field should accelerate it. In an astro-physical context the unscreened electric field is usuallystrong enough to boost the particle energy to the do-main where the particle starts to interact with the back-ground field and thus initiates an electron-positron paircascade. The secondary particles move in the magne-tosphere in a way that tends to screen the electric field(Sturrock 1971; Ruderman & Sutherland 1975). Even-tually, an electric-field-free configuration of the magne-tosphere can be formed. However, one should note thatthere are differences between the structures of the pulsarand BH magnetospheres, and consequently, the theoret-ical results obtained for pulsar magnetospheres cannotbe directly applied to the BH magnetosphere. In partic-ular, while in the case of the pulsar magnetosphere thesource of the magnetic field is well defined, in the BHmagnetosphere the magnetic field is generated by cur-rents in the disk and magnetosphere. The configurationof the field is determined by the structure of the accre-tion flow. Thus, a change of the accretion flow can resultin the formation of charge-starved regions (gaps) in theBH magnetosphere.The charge density required for the screening isknown as the Goldreich-Julian density (GJ), ρ GJ (Goldreich & Julian 1969). However, the process of thepair creation is expected to be highly non-stationary (see,e.g., Levinson et al. 2005; Timokhin 2010, for theoreticand numerical considerations of the pair creation in vac-uum gaps, respectively), thus even if am electric-field-freestate of the magnetosphere is possible, it cannot be sta-ble (Sturrock 1971). The gaps, i.e., regions in which thecharge density is not sufficient for the electric field screen-ing, may appear sporadically in the magnetosphere, forexample, in the vicinity of the stagnation surface, i.e., atthe boundary that separates acretion and ejection trendsin the flow.The vacuum electric field strength E determines themaximum electric field in the gap, thus the maximumacceleration rate of a particle with charge e in the gap is mc ˙ γ < ecE . The total power of particle accelerationcan be expressed as˙ E < Z gap dV e ( n e + n e + ) cE , (3)where n e and n e + are densities of electrons and positrons.If all the energy gained by the particles in the gap is emit-ted in gamma rays, Eq. (3) also corresponds to the upperlimit of the gamma-ray luminosity. Note that electronsand positrons move in opposite directions in the gap, andonly one of these species generates emission detectable bya distant observer.For a thin spherical gap, R < r < R + h , the luminosityupper limit is L γ < πR hen e cE , (4)where the electrons are assumed to emit outward. Theparticle density can be expressed as a fraction of theGoldreich-Julian density: en e = κρ GJ , where κ is themultiplicity. The condition for the electric field screen-ing, e | n e − n e + | = ρ GJ , allows charge configurations withhigh multiplicity and still non-screened electric field. Toobtain a more detailed estimate of the generated pairs’influence on the electric field in the gap, it is necessaryto consider the electromagnetic cascade in the gap.The numerical simulations of Timokhin (2010) andTimokhin & Arons (2013) demonstrate an importanttendency. When the multiplicity becomes significant, κ ∼
1, the charges in the gap start to generate an electricfield that is comparable to E , and the accelerating fieldvanishes. Thus, for effective charge acceleration, the fol-lowing condition should be fulfilled: κ ≪
1. Thus, thetotal energy release in the gap of thickness, h , can beestimated as L γ < πR hκρ GJ cE . (5)The electrical field in the gap is estimated as E ≈ B g R Ω F sin θc , (6)where B g is the magnetic field in the vacuum gap, Ω F isthe angular velocity of the frame, R is the radius, and θ is the polar angle. In fact, the actual electric fieldin the drop is smaller by a factor h/R than the valuegiven by Eq. (6) (see, e.g. Blandford & Znajek 1977;Levinson & Rieger 2011). This factor accounts for theinfluence of the magnetospheric charges located outsidethe gap. Eq. (6) does not account for this contribu-tion. Since these charges, even if they remain outsidethe gap, tend to decrease the electrical field in the gap,Eq. (6) provides a strict upper limit on the gap electricfield strength.The Goldreich-Julian density is also determined by thesame parameters: ρ GJ = Ω F B g sin θ/ (2 πc ) . (7)For a Kerr BH with the maximum angular momentum,the angular velocity Ω F is estimated asΩ F c ≃ r g . (8)Substituting Eq. (6) - (8) to Eq. (5), one obtains L γ < B R r κhc sin θ . (9)The upper limit on the luminosity from a vacuum gapdepends on the factor R B that is expected to decreasewith R . For sake of simplicity below it is adopted that R B ≃ r g B , where B bh is the magnetic field at theBH horizon. Thus, one obtains L γ < B r g κhc sin θ . (10)We should note that for h → r g , the luminosity esti-mate provided by Eq. (10) (after averaging over the polarangle θ ) exceeds the Blandford-Znajek (BZ) luminosity(Blandford & Znajek 1977; Beskin 2010) by a factor of2. This is imposed by several simplifications in our treat-ment. The most important contribution is caused by theusage of the electric field upper limit, Eq. (6), as theaccelerating field.Thus, Eq. (10) can be considered as a safe upperlimit for the luminosity of magnetospheric flares. Asimilar estimate has been obtained by Rieger (2011) and Levinson & Rieger (2011). However, the numer-ical expression in Levinson & Rieger (2011) containssome uncertain geometrical factor ( η in the notations ofLevinson & Rieger 2011). Eq. (10) allows us to estimateits value: this geometrical factor should be small, ∼ − (see also Eq. 52 in Rieger 2011).Finally, Broderick & Tchekhovskoy (2015) argued thatfor the full screening of the electric field in a thin gap, thecharge density should exceed the Goldreich-Julian valueby a factor R/h , which should lead to an enhancement ofthe gap radiation. To illustrate the physical reason forthe existence of this factor, Broderick & Tchekhovskoy(2015) computed the divergence of the electric field inthe gap. However, as the gap electric field they adopteda field determined by an expression similar to Eq. (6), i.e.,a value that overestimates the true field by the factor
R/h (see, e.g., Blandford & Znajek 1977; Levinson & Rieger2011; Rieger 2011, for a more accurate introduction of theelectric field in the gap). Thus, the factor suggested byBroderick & Tchekhovskoy (2015) seems to be stronglyoverestimated.Finally, the thickness of the gap, h , in Eq. (10) is con-strained by the variability time scale, h ∼ t var c . To pro-duction the emission variable on a 5-minute time-scale, t var = 5 t var , min, the gap thickness, h = 10 t var , cm,should be smaller than the gravitational radius of theSMBH with a mass M >
1. Thus, the estimatedgamma-ray luminosity cannot exceed the following value: L γ < × κB M t var , sin θ erg s − , (11)where B bh = 10 B G.Eq. (11) contains two parameters that are determinedby properties of the advection flow in the close vicinityof the BH: pair multiplicity, κ , and the magnetic fieldstrength, B . Importantly, these parameters are essen-tially defined by the same property of the flow, morespecifically, by the accretion rate. The magnetic fieldat the BH horizon needs to be supported by the accre-tion flow. Therefore the field strength is directly deter-mined by the accretion rate. The accretion rate also de-fines the intensity of photon fields in the magnetosphere,and consequently, the density of electron-positron pairsproduced through gamma-gamma interaction (see, e.g.,Levinson & Rieger 2011). If the multiplicity parameter, κ , approaches unity, the gap electric field vanishes (see,e.g., Timokhin & Arons 2013). This sets an upper limiton the accretion rate, and consequently on the magneticfield strength.In previous studies (Levinson & Rieger 2011;Aleksi´c et al. 2014) the maximum accretion ratecompatible with the existence of a vacuum gap in themagnetosphere was estimated as˙ m < × − M − / , (12)where ˙ m is the accretion rate in the Eddington units:˙ M edd = 4 πm p GM bh ηcσ t . (13)Here m p , σ t , and η are the proton mass, the Thompson Note that Broderick & Tchekhovskoy (2015) used a differentnotation for the gap thickness, ∆. cross-section, and the accretion efficiency factor, respec-tively.To derive the estimate provided by Eq. (12),Levinson & Rieger (2011) adopted a value of η = 0 . B bh = 1 . × ( ˙ m/M ) / G , (14)where we rescaled the numerical coefficient to the nor-malization used throughout our paper. For this magneticfield strength, Eq. (11) yields L γ < × κM − / t var , sin θ erg s − . (15)In some cases, e.g., for IC 310, the energy requirementsare rather close to the obtained upper limit, thereforewe consider a somewhat more accurate treatment of thecase of a magnetosphere around a Kerr BH below.The strength of the magnetic field at the BH horizoncan be obtained by extrapolating the field at the lastmarginally stable orbit. Let us define the magnetic fieldin the disk as B d = p πβ m p g , (16)where β m and p g are the disk magnetization and gaspressure in the accretion disk that confines the magneticfield at the horizon. The gas pressure can be estimatedusing the solution for a radiatively inefficient accretionflow (see Narayan & Yi 1994) as p g = √
10 ˙ M √ GM bh πα ss R / , (17)where α ss is the nondimensional viscosity of the disk(Shakura & Sunyaev 1973). Eq. (16) for R → r g pro-vides an estimate for the magnetic field at the BH hori-zon: B bh = 1 . β / ( ˙ M c ) / ( α ss ) / r g . (18)The magnetic field strength provided by Eq. (18) to-gether with Eq. (10) yields in L γ < √ β m κ ( h/r g ) sin θ ˙ M c α ss . (19)The multiplicity parameter, κ , at the Kerr radius isdetermined as ( see Appendix A for details) κ ≡ n ± n GJ ≈ × ˙ m / M / ( ηα ss ) / β / . (20)The condition κ < m < − ηα ss β / M / . (21)Eq. (21) and (13) substituted into Eqs. (18) and (19) givean upper limit for the magnetic field that is consistentwith the existence of vacuum gaps: B bh < × (cid:18) β m M (cid:19) / G , (22) A more accurate treatment of the accretion flow reveals a cor-rection by less than 30% (see Narayan & Yi 1995a) as comparedto the height-averaged treatment in Narayan & Yi (1994) and consequently, the maximum luminosity of particlesaccelerated in the gap does not depend on α ss and η : L γ < × β / κt var , M − / sin θ erg s − . (23)This estimate is obtained for the thick-disk accretion(in the ADAF-like regime). The limit on the accre-tion rate given by Eq. (12) is consistent with the re-alization of this accretion regime. For higher accretionrates, ˙ m ≥ .
1, the accretion flow is expected to convergeto the thin-disk solution (Bisnovatyi-Kogan & Blinnikov1977; Abramowicz et al. 1988). In this regime, the tem-perature of the disk is expected to be significantly below1 MeV, thus the pair creation by photons supplied by theaccretion disk should be cease. This effectively mitigatesthe constraints imposed by the accretion rate. How-ever, the change of the accretion regime also significantlyweakens the strength of the magnetic field at the BHhorizon (Bisnovatyi-Kogan & Lovelace 2007), and con-sequently decreases the available power for accelerationin the gap.To derive Eq.(23), we assumed that the gap thicknessis determined by the variability time-scale; this corre-sponds to the energetically most feasible configuration.In a more realistic treatment, one should also take intoaccount the interaction of the particles that are accel-erated in the gap with the background radiation field.For high and ultrahigh energies of electrons,
E > λ γγ ≤ λ IC . Thus, computation of the TeVemission requires a detailed modeling of the electromag-netic cascade (see, e.g., Broderick & Tchekhovskoy 2015;Hirotani et al. 2016). Furthermore, the production andevacuation of the cascade-generated pairs may follow acyclic pattern and the inductive electric field may becomecomparable to the vacuum field (Levinson et al. 2005).A detailed consideration of this complex dynamics is be-yond the scope of this paper, but we note that the char-acteristic length of such a cascade-moderated gap shouldbe small, ∼ p λ IC λ γγ , resulting in a reduction of theavailable power (see also Beskin et al. 1992).Eq. (23) determines the maximum luminosity of vac-uum gaps that can collapse quicker than t var . It hasbeen assumed for its derivation that the magnetic fieldis determined by an accretion regime that its in turndetermines the intensity of the photon field in the mag-netosphere. In the case of a steady accretion, this seemsto be a very feasible approximation. However, this maylook less certain in the case of a rapidly changing ac-cretion rate, since the processes that govern the varia-tion of the accretion rate and escape of the magneticfield from the BH horizon may have different character-istic timescales. Therefore we provide some estimates forthese two timescales below.The dominant contribution to the photon field comesfrom plasma located at distances r ∼ r g , and the char-acteristic viscous accretion time (density decay time inthe flow) is t ρ, decay ≃ r g cα ss ≃ α − , − M s . (24)When the accretion fades, the decay of the magnetic fieldis determined by the magnetic field reconnection rate(Komissarov 2004): t B , decay ≃ πr g . c ∼ M s . (25)Since these two time-scales are essentially identical, itis natural to expect that the field strength and the diskdensity will decay simultaneously. Thus, Eq. (23) shouldalso be valid for the time-dependent accretion regime.2.2. Relativistically Moving Blobs
The properties of radiation generated in jets may besignificantly affected if some jet material moves relativis-tically with respect to the jet local comoving frame. Forexample, the magnetic field reconnection may be ac-companied by the formation of slow shocks (see, e.g.,Lyubarsky 2005) that in the magnetically dominatedplasma produce relativistic flows (Komissarov 2003). Ifsuch a process is realized in AGN jets, it can lead togamma-ray flares in blazar-type AGN with variabilitytimescale significantly shorter than r g /c (Giannios et al.2009). Another implication of this scenario is relatedto short gamma-ray flares detected from missaligned ra-dio galaxies (Giannios et al. 2010). Indeed, the conser-vation of momentum requires that for each plasmoid di-rected within the jet-opening cone, there should exista counterpart that is directed outside the jet-beamingcone. While the radiation of the plasmoid directedalong the jet appears as a short flare, the emission asso-ciated with its counterpart outflow can be detected as abright flare by an off-axis observer. The latter processmay have a direct implication on the interpretation offlares from nearby missaligned radio galaxies, e.g., M87(Giannios et al. 2010).If a process, operating in a region of the jet with co-moving volume V ′ , results in the ejection of plasmoids,some fraction, ξ , of the energy contained in the volumeis transferred to the outflow. The conservation of energycan be written as ξV ′ ǫ ′ j = S co Γ v co ∆ t ′ (4 / ǫ e ) . (26)Here ǫ ′ j and ∆ t ′ are the energy density of the jet plasmaand duration of the ejection, as seen in the jet comovingreference frame. Γ co = 1 / p − ( v co /c ) , S co , and ˜ ǫ arethe plasmoid Lorentz factor, the outflow cross-section,and the internal energy, respectively. The outflow cross-section can be estimated as S co ≃ S/ (2Γ ), where S ≃ V ′ / (∆ t ′ v co ) is the surface of the volume V ′ . Thus, oneobtains an estimate for the energy density of the ǫ ′ j ≃ ξ ˜ ǫ e . (27)For simplicity, in what follows we take ξǫ ′ j ≃ ˜ ǫ e . Theefficiency of the energy transfer, ξ , depends on a specificrealization of the scenario. For example, it seems that thefor the reconnection of the magnetic field, the efficiencymight be high ξ ∼
1, as follows from an analytic treat-ment by Lyubarsky (2005) and the results of numericalsimulations by Sironi et al. (2016) . We note, however, From Figure 2 of Sironi et al. (2016) it follows that n lab ≃ σn that in the presence of a guiding field, the magnetiza-tion of the ejected plasmoids should be high (Lyubarsky2005).On the other hand, the energy density in the plas-moid can be estimated through the emission variabilitytime and the luminosity level (see Eq. (9) Giannios et al.2009): ˜ ǫ e = E em Γ em ˜ l , (28)where the variability time-scale determines the size ofthe production region: ˜ l em = c ∆ t Γ em , and the fluxlevel defines the energy content in the plasmoid: E em = L γ ∆ t/ (4 f Γ ) (here f < ξ ). Thus, oneobtains ˜ ǫ e = L γ c ∆ t . (29)On the other hand, the energy density in the jet is ǫ ′ j = L j ∆Ω r c Γ , (30)where ∆Ω ≃ π/ Γ is the jet propagation solid angle. Thecomparison of these equations allows us to estimate therequired true luminosity of the jet as L j = L γ Γ πr ξc ∆ t . (31)The above equation is consistent with Eq. (10) fromGiannios et al. (2009). Note, however, a difference in thenotations: throughout this paper, L j is the true jet lu-minosity, while in Giannios et al. (2009) L j correspondsto the isotropic luminosity.If the viewing angle is small, the mini-jet Lorentz factorcan be expressed as Γ em = 2Γ j Γ co / (1+ α ) where α = θ Γ j is the viewing angle expressed through the jet-openingangle (see Appendix B). Thus, the above equation canbe simplified as L j = L γ Γ Γ (cid:0) α (cid:1) πr ξc ∆ t (32)or L j = 1 . × − L γ (cid:18) α (cid:19) Γ − , Γ − , ξ − − r M t − , . (33)Here it was assumed that the flare originates at a dis-tance r = 100 r g from the central BH with mass M bh =10 M ⊙ M .The above estimate describes the jet luminosity re-quirement to generate a single short flare of duration t var . Observations in HE and VHE regimes show thatAGNs often demonstrate a rather long period of activity(as compared to the duration of a single peak): T ≫ t var .If the mini-jets are isotropically distributed in the jet co-moving frame, the probability for an observer to be in the and < γ > lab n lab ≃ σ n , thus the internal energy in the plasmoidis ˜ ǫ ≃ ˜ n < ˜ γ > ≃ σn mc . mini-jet beaming cone depends weakly on the observerviewing angle , and this probability can be estimated as P ≃ (2Γ co ) − (Giannios et al. 2010). If the mini-jet for-mation is triggered by some spontaneous process, thenthe comoving size of the region responsible for the flareis l ′ = δ j T c , and the energy contained in this region is E ′ = Sl ′ e ′ j (here S is the jet cross-section). The energyof a single mini-jet in the comoving frame is E ′ mj = L γ t var Γ co ξ Γ . (34)The total number of mini-jets during a flaring episodecan be estimated as N ≈ Φ T /P t var , where Φ is the so-called filling factor.The total dissipated energy for the flare should besmaller than the energy that is contained in the dissi-pation region: E ′ mj Φ TP t var < L j T δ j Γ . (35)This implies a requirement for the jet luminosity L j > . ζ δ − , L γ ξ − − , (36)here ζ = Γ j /δ j , or L j > . (cid:0) α (cid:1) Γ − , L γ ξ − − , (37)where the small viewing angle limit was used for the ratioof Lorentz and beaming factors: ζ = Γ j /δ j ≃ (1 + α ) / ξ , which accounts for the conversion effi-ciency from jet material to the outflow, and from outflowto the radiation. While the letter can be high, ∼
1, ifa good target for nonthermal particles exists, the valueof the former efficiency depends on the process behindthe outflow formation. For example, it was argued thatif the outflow is formed by the Petschek-type relativis-tic reconnection (Lyubarsky 2005), the energy transferis expected to be high, ∼
1. However, the efficiency ofthe transfer can be significantly suppressed if the guidingfield is present in the reconnection domain (Lyubarsky2005; Barkov & Komissarov 2016).On the other hand, this requirement can be somewhatrelaxed if the velocity direction of the plasmoids is notrandom, e.g., is controlled by the large-scale magneticfield (Giannios et al. 2009), or is triggered by some per-turbation propagating from the base of the jet. In theformer case the mini-jet detection probability, P , may behigher, and in the latter case, the comoving distance be-tween the mini-jets may be larger than l ′ . Let us assumethat the flare trigger propagates with Lorentz factor Γ ′ tr in the jet comoving frame, then the comoving region sizeis larger by a factor of Γ ′ tr . If Γ co > Γ j , this statement is correct for observers located in θ view < π/
2, otherwise for tan θ view < v co / q / Γ − / Γ . This follows from Eq.(6) in (Giannios et al. 2009), i.e., ˜ ǫ e = σρ ′ j c ≃ ǫ ′ j for σ ≫ Cloud-in-Jet Model
In the framework of the cloud-in-jet scenario, wedeal with the nonthermal emission generated at theinteraction of a jet with some external obstacle, e.g.,a BLR cloud or a star (see, e.g., Blandford & K¨onigl1979; Bednarek & Protheroe 1997; Araudo et al. 2010;Barkov et al. 2010, 2012b). Debris of the obstacle mat-ter, produced at such an interaction, can be caught bythe jet flow. This debris should form dense blobs orclouds in the jet, and the emission generated during theiracceleration may be detected as a flare (Barkov et al.2012a; Khangulyan et al. 2013). If this interpretation iscorrect, each peak of the light curve can be associatedwith emission produced at the acceleration of some indi-vidual blob. The peak profile and its duration are deter-mined by the condition of how quickly this blob can beinvolved into the the jet motion, i.e., by the dimensionand mass of the blob. Light blobs with a mass satisfyingthe condition M c c < P j πR r (38)(here P j and R c are the jet ram pressure and the cloudradius, respectively) are accelerated on length scalessmaller than the distance to the SMBH, r (for a con-sideration of heavy blobs that can be accelerated over adistance comparable to r , see Khangulyan et al. 2013).If this condition is fulfilled, the variability timescale can be estimated as (Barkov et al. 2012a;Khangulyan et al. 2013) t var ≃ cM c Γ P j πR δ j . (39)This estimate ignores perturbations in the jet that aregenerated by the obstacle-jet interaction, which is prob-ably very complex, and their influence can be exploredonly by numerical simulations, which are beyond thescope of this paper (see, e.g., de la Cita et al. 2016).In the case of a cloud with a mass that satisfies toEq. (38), the variability can be very short: t var
3, which is related to the correction function F e defined in Barkov et al. 2012a;Khangulyan et al. 2013). Thus, if the cloud dynamicsdetermines the variability, then the luminosity of theemission appears to be independent of the mass of thecloud: L γ ≃ cP πR ξδ . (41)Since L j > cP πR , the above equation allows us to ob-tain a lower limit on the jet luminosity required for theoperation of the star-jet interaction scenario: L j > . ζ δ − , L γ ξ − − , (42)or L j > . (cid:0) α (cid:1) Γ − , L γ ξ − − , (43)which is a factor of 4 / Φ larger than the estimate for thejet-in-jet scenario (see Eq. (36)).Eqs. (39) and (40) contain four parameters P , Γ j (wetreat δ j as a related parameter), M c , and R c , and there-fore formally allow a solution even if two of these param-eters are fixed. For example, for given properties of thejet (i.e., for specific values of the parameters Γ j and P )and the parameters characterizing the flare (the total en-ergy and the variability), the characteristics of the cloudcan be determined as M c c = E γ ξδ , πR = 4 ζ δ E γ ξt var cP . (44)However, the determined parameters of the cloud maynot necessary be physical, and their feasibility should beexamined by dynamical estimates.The first dynamical limitation is related to the abil-ity of a cloud to penetrate the jet and become involvedin the jet motion. According to the estimates given byBarkov et al. (2012a) and Khangulyan et al. (2013), forthe typical jet parameters these constraints do not im-pose any strong limitations. The heaviest blobs that canbe accelerated by a jet with luminosity 10 erg s − canresult in flares with a total energy release of 10 erg.If the cloud is light enough to be caught by the jet,then one should consider two main processes: the cloudexpansion, and its acceleration. At the initial stage, thecloud cross-section is not sufficiently large to provide itsacceleration to relativistic velocities. On the other hand,the intense jet-cloud interaction at this stage leads toa rapid heating and expansion of the cloud. The cloudsize-doubling time can be estimated as t exp ≈ A (cid:18) M c γ g R c P j (cid:19) / . (45)where γ g = 4 / A is a constantof about a few (Gregori et al. 2000; Nakamura et al.2006; Pittard et al. 2010; Bosch-Ramon et al. 2012).When the size of the cloud becomes large enough foracceleration to relativistic energies, the intensity of thejet-cloud interaction fades away, and the cloud expan-sion proceeds in the linear regime. Since the time scalefor acceleration to relativistic velocity is t ac ≃ M c c πR cP j , (46)the size of the cloud relevant for the flare generation canbe obtained by balancing Eqs. (45) and (46): R c = A exp (cid:18) M c c P j γ g π A (cid:19) / . (47)Here the constant A exp accounts for the cloud expansionin the linear regime. The dynamical limitation given by Eq. (47) togetherwith Eq. (44) allows determination of the jet ram pres-sure: P j = πA ξγ A (2 ζ ) E γ t c . (48)The actual value of the coefficient in the above equa-tion, in particular the value of A exp , can be revealedonly through the numerical simulations given the com-plexity of the jet-cloud interaction. However, if one as-sumes that the expansion proceeds very efficiently, i.e.,the cloud size achieves a value close to the light-crossinglimit, R c ≃ δ j t var c , then the expression for the jet rampressure becomes P j = ζδ ! E γ πξt c . (49)Since each flaring episode should correspond to specificjet parameters , the above equation implies that the en-ergy emitted in an individual peak of a flare should beproportional to the cube of its duration: E γ ∝ t (or L γ ∝ t ). Obviously, the study of individual peaks ina statistically meaningful way requires a detailed lightcurve that can be obtained with future observations, inparticular with CTA (see, e.g., Romoli et al. 2017).2.4. Energetic Constraints for Detected ExceptionalFlares
So far, several super-fast gamma-ray flares have beendetected in the VHE or HE regimes from different typesof AGNs. The peculiarity of the signal is related bothto the duration of the flare and to the released energy.Below we consider several cases that are summarized inTable 1. 2.4.1.
PKS 2155 − The July 2006 flare of PKS 2155 −
304 is characterizedby a very short variability of 180 s and the a intrinsicVHE gamma-ray luminosity at the level of 10 erg s − (Aharonian et al. 2007).The mass of the central BH is estimated to be M BH , ≃
10 (Aharonian et al. 2007, and reference therein). To-gether with the short variability time, this constrains theluminosity of (potential) gamma-ray flares produced bymagnetosphere gaps at the level of L γ, ms < erg s − and thus excludes any magnetospheric origin of theseflares.PKS 2155 −
304 is a typical representative of high-energy peaked BL Lacs. It is expected that the jet isaligned along the line of sight: α ≈
0. For a typicalvalue of the jet Lorentz factor, Γ j = 20, the jet powerrequired for the realization of the jet-in-jet scenario is L j , jj > Φ − . Γ − , . ξ − − . (50)It follows from the comparison of Eqs. (37) and (43)that the lower limit of the jet power in the cloud-in-jetscenario is higher by a factor ∼ / Φ, i.e., L j , cj > × Γ − , . ξ − − erg s − . (51) We note, however, that across a magnetically driven jet onemay expect strong gradients of the jet ram pressure (see, e.g.,Beskin & Nokhrina 2006; Komissarov et al. 2007, 2009)).
TABLE 1Comparison of models for different sources
Source IC 310 a M87 b
3C 454.3 c
3C 279 d PKS 2155 − e M BH , t τ L γ , erg s − × × × Φ 0.1 0.3 0.7 0.3 0.7Γ j
10 10 20 20 20Γ co
10 10 10 10 10 α L γ /L γ,ms
10 5 × − × × L j , jj × × L j , cj × × × × Note . — M BH , = M BH / M ⊙ is the SMBH mass, t = t/
300 s is the variability time, τ = tc/r g is the nondimensional variabilitytime in units of gravitation radius light-crossing time, L γ is the maximum luminosity in gamma-rays, Γ j is the jet Lorentz factor, Γ co isthe Lorentz factor of the mini-jet, α = θ/ Γ j is the normalized viewing angle, L γ,ms is the upper limit of the gamma-ray luminosity for amagnetospheric model, L j , jj is the minimal jet power for the jet-in-jet model, L j , cj is the minimum jet power for cloud-in-jet model. a Aleksi´c et al. (2014) b Gebhardt & Thomas (2009), Wang & Zhou (2009), Abramowski et al. (2012). c Striani et al. (2010), Abdo et al. (2011) d Hayashida et al. (2015) e Aharonian et al. (2007)
We note here that constraints imposed by the radiationmechanism enhance the required jet power by a factor of ∼
10 for the external inverse Compton scenario and by ∼ for proton synchrotron emission (see Barkov et al.2012a), which exceeds the Eddington luminosity.2.4.2. IC 310
In 2012 November, the MAGIC collaboration detecteda bright flare from IC 310 (Aleksi´c et al. 2014). Theflare consisted of two sharp peaks with a typical durationof ∼ .
2, extending up to ∼ × erg s − .The mass of the BH powering activity of IC 310has been estimated to be M IC = (cid:0) +4 − (cid:1) × M ⊙ (Aleksi´c et al. 2014), i.e., the measured variability timescale is as short as 20% of τ .According to the estimate provided by Eq. (23), theluminosity of flares generated in the BH magnetospheredepends weakly on the mass of the BH and is determinedby the disk magnetization, the viewing angle, and thepair multiplicity . Since all these parameters are smallerthan unity, from Eq. (23) we have L γ, ms < × erg s − (52)This upper limit is an order of magnitude below the re-quired value (Aleksi´c et al. 2014). Thus, we concludethat the ultrafast flare detected from this source cannothave a magnetospheric origin.Assuming that mini-jets are distributed isotropicallyin the jet frame and that the detection of two pulses isnot a statistical fluctuation, one can estimate the truejet luminosity using Eq. (37). For the relevant flare pa- Eq. (23) does not account for relativistic effects that shouldbe small unless the gap is formed close to the horizon. However,if the vacuum gap is close to the horizon, then the gravitationalredshift should make more robust the constraints imposed by thevariability time. rameters (i.e., t var = 4 . L γ = 2 × erg s − ) and M = 3 L j , jj > Φ − (cid:18) α (cid:19) Γ − , ξ − − . (53)If the mini-jets are not distributed isotropically, the re-quirement on the jet power can be a few orders of mag-nitude weaker; see Eq. (33).The cloud-in-jet scenario requires a higher jet luminos-ity; from Eq. (43) it follows that L j , cj > × Γ − , (cid:18) α (cid:19) ξ − − erg s − . (54)2.4.3. M87
In 2010, a bright flare has been recorded duringa multiintrument campaign in the VHE energy band(Abramowski et al. 2012). The variability time duringthe VHE transient was about 0 . erg s − . This source is characterized by alarge jet-viewing angle of θ j ≈ o and a Lorentz factorof about Γ j ≈ ∼ × M ⊙ (Gebhardt & Thomas 2009).Given the heavy central BH and the relatively longduration of the VHE flare, which allows high values ofthe gap size, the energy constraint in the magnetospherescenario is quite modest: L γ, ms < × erg s − . (55)M87 might be an interesting candidate for a detection ofmagnetosphere flares.For the flare parameters (i.e., t var = 0 . L γ =10 erg s − ) and M = 60, Eq. (37) constrains the re-quired jet true luminosity at the level L j , jj > Φ − . (cid:18) α (cid:19) Γ − , ξ − − . (56)On the other hand, the mulitwavelength properties of0the gamma-ray flares detected from M87 seem to be quitediverse, with no detected robust counterparts at otherwavelengths. Thus, if the VHE emission is produced bya single mini-jet, then a much weaker constraint, pro-vided by Eq. (33), is applied. In this case, the variabilitydetected with Cherenkov telescopes should correspond tothe mini-jet variability, thus the mini-jet comoving sizeshould be˜ l em = ∆ tc Γ em = 2∆ tc Γ j Γ co α ∼ cm , (57)which is about the jet cross-section at a parsec distancefrom the central BH. We should also note that the typi-cal spectra emitted by plasmoids are dominated by syn-chrotron radiation, which seems to be inconsistent withthe multiwavelength observations of M87. Moreover, thepeculiar light curve that has been detected with H.E.S.S.has not yet been explained in the framework of the jet-in-jet scenario.Formally, for the parameters of the flare detected fromM87, the minimum jet luminosity required by the cloud-in-jet scenario is L j , cj > × Γ − , (cid:18) α (cid:19) ξ − − erg s − . (58)However, it has been argued that the light curve and theVHE spectrum is best explained if the TeV is producedthrough p-p interactions induced by the jet collision witha dense cloud. In this case, the required jet power isabout L j ≈ × erg s − (Barkov et al. 2012b).2.4.4.
3C 454.3 and 3C 279
In 2010 November, an exceptionally bright flare wasdetected from 3C 454.3 by
AGILE and
Fermi
LAT(Striani et al. 2010; Abdo et al. 2011). The minumumdetected variability time and gamma-ray luminosity were4 . × erg s − , respectively.Several similarly bright flares were detected form3C 279 in the period from 2013 December to 2014 April(Hayashida et al. 2015). The GeV gamma-ray luminos-ity reached a level of 6 × erg s − , and the flux variedon a time scale of 0.7 hr.The magnetosphere gap luminosity is limited by10 erg s − and 2 × erg s − for 3C 454.3 and 3C 279,respectively.Therefore the magnetospheric origin of theseflares is excluded for both sources.3C 279 and 3C 454.3 are distant quasars, thus it is safeto fix α = 0 for both cases. By adopting a standard jetLorentz factor, Γ j = 20, one can obtain the jet luminosityrequired for the realization of the jet-in-jet scenario: L j , jj > Φ − . Γ − , . ξ − − (59)for 3C 454.3, and L j , jj > × Φ − . Γ − , . ξ − − (60)for 3C 279. Both these estimates appears to be belowthe corresponding Eddington luminosity limits of 1 . × erg s − and 6 × erg s − for M BH , = 10 and M BH , = 5 in 3C 454.3 and 3C 279, respectively.The star-in-jet scenario requires higher jet luminosi-ties, which seem to exceed the Eddington limit for 3C 454.3. Namely, one obtains L j , cj > Γ − , . ξ − − erg s − . (61)This value agrees with the estimate L j , cj ≈ erg s − that is obtained within a more accurate model that pro-vides also provides an interpretation for the so-calledplateau phase and the spectrum (Khangulyan et al.2013). Such a high luminosity of the jet, L j ≈ . L γ ,has also been obtained in the framework of the one-zoneexternal Compton model of Bonnoli et al. (2011).For 3C 279, the lower limit on the jet luminosity is L j , cj > × Γ − , . ξ − − erg s − , (62)which is close to the Eddington limit. A similar estimatewas obtained by Hayashida et al. (2015). DISCUSSION AND CONCLUSIONS3.1.
Gamma-ray Flare Detected from IC 310
IC 310 is a radio galaxy with redshift z ≃ . θ j < ◦ (see, e.g.,Kadler et al. 2012). Arguments based on the absenceof the a contra-jet and assuming that the true lengthof the jet is smaller than 1Mpc allowed to further con-strain the viewing angle 10 ◦ < θ j < ◦ (for detailssee Aleksi´c et al. 2014). Finally, observations of super-luminal motion allowed us to constraint the Dopplerfactor to δ j ∼
5. The radio luminosity has been esti-mated to be at the level of L IC ,radio ≃ erg s − (Sijbring & de Bruyn 1998). This implies a minimumenergy in relativistic electrons of 5 . × erg, thusthe power required for the supply of emitting electronsyields L IC ,e ≃ × erg s − , and the total jet lu-minosity can be estimated as L IC ≃ erg s − (seeAleksi´c et al. 2014). We note that these estimates repre-sent values for the minimum required energetics averaged over 10 years.In 2012 November, MAGIC detected a bright flarefrom IC 310 (Aleksi´c et al. 2014). The flare consistedof two sharp peaks with a typical duration of ∼ .
2, extending up to ∼
10 TeV. The energy releasedduring that event has been estimated to be at a level of2 × erg s − .The mass of the BH powering activity of IC 310has been determined to be M IC = (cid:0) +4 − (cid:1) × M ⊙ (Aleksi´c et al. 2014), i.e., the measured variability timescale is as short as 20% of τ ; therefore one can expect arealization of some unconventional mechanism for VHEemission production. Aleksi´c et al. (2014) have consid-ered possible scenarios (see Sect. 2) for the flare produc-tion and found that jet-in-jet and star-in-jet interactionmodels face certain difficulties. Based on this, one con-cluded that the magnetosphere origin remains the onlypossible option, and no further verification of that sce-nario has been provided.Hirotani & Pu (2016) have performed simulations ofthe gamma-ray spectrum produced in a stationary gapfor different accretion rates and concluded that the emis-sion generated in the vacuum gap could closely reproducethe spectral properties of the TeV emission detected dur-ing the flare. The strength of the magnetic field has1been fixed at a level of 10 G. Hirotani & Pu (2016)have emphasized that the generation of such a strongmagnetic field requires an accretion rate exceeding thevalues compatible with the existence of vacuum gaps bya factor of 100. In addition, we note that if the thick-ness of the gap is determined by the variability time, h ≃ t var c , this scenario requires an even higher efficiency(by a factor of ∼
20, since the best-fit parameters forthe magnetospheric scenario require ˙ m ∼ × − andconsequently h gap ∼ r g ( ∼ t var c ), see Figures 8 and 17of Hirotani & Pu 2016).Since one does not expect any significant focusingor enhancement of the emission produced in the mag-netosphere, the measured energy should correspondto the real energetics of the processes responsible forthe emission generation. Thus, the feasibility of gen-erating such a powerful flare in the vacuum gap isclosely related to the general efficiency of processestaking place in BH magnetosphere. Currently, theBZ mechanism (Ruffini & Wilson 1975; Lovelace 1976;Blandford & Znajek 1977) represents the most promi-nent energy extraction mechanism that can operate inthe BH magnetosphere. The efficiency of this mecha-nism is determined by the strength of the magnetic fieldthat is accumulated at the BH horizon, which in turn isdetermined by the accretion rate. Hirotani & Pu (2016)argued that the efficiency of the BZ mechanism can bevery high, up to a level of 900%, as compared to the ac-cretion rate ˙ M c . This assumption is based on 2D sim-ulations presented by McKinney et al. (2012). However,3D simulations for a similar setup presented in the samepaper reveal a significantly lower efficiency, ∼ α ss = 0 . . Since all these parameters are smallerthan one, then the numerical coefficient in Eq. (23) canbe taken as a strict upper limit for the flare luminos-ity for the given variability time. This upper limit ap-pears to be approximately an order of magnitude belowthe value measured with MAGIC (Aleksi´c et al. 2014).Thus, we conclude that it seems very unfeasible thatthese processes are indeed behind the bright flaring ac-tivity recorded from IC 310 (which is also consistent withmodeling presented by Hirotani et al. 2016).Our simplified analysis does not allow us to robustlyrule out two other scenarios for the flare production inIC 310. If one adopts the minimum averaged jet lumi-nosity as a reasonable constraint for the present jet lu-minosity (as done in Aleksi´c et al. 2014), then both sce-narios formally do not allow reproducing the observedproperties. However, if one assumes that there is someanisotropy in the mini-jet distribution, then jet-in-jetmodel provides an energetically feasible scenario. Wenote that the spectral energy distributions currently ob- Eq. (23) does not account for relativistic effects, which shouldbe small unless the gap is formed close to the horizon. If the vac-uum gap is close to the horizon, then gravitational redshift shouldeven strengthen the constraints imposed by the variability time. tained for the jet-in-jet models feature the dominant ex-cess in the UV band (Petropoulou et al. 2016), which isnot consistent with the observations from IC 310. Thismight be either a fundamental constraint or just a sys-tematic underestimation of the inverse Compton contri-bution that is due to the small scale of the simulations.Thus, it seems that more detailed large-scale simulationsare required to verify the applicability of the jet-in-jetscenario for IC 310. On the other hand, if the present-day jet luminosity is significantly higher than the aver-aged value, the star-in-jet scenario may also meet theenergetic requirements.3.2.
Comparison of Scenarios for Ultrafast Variability
In this paper we considered three scenarios for the pro-duction of ultrafast AGN flares with variability timesshorter than the Kerr radius light-crossing time: gamma-ray emission of gaps in the SMBH magnetosphere(Neronov & Aharonian 2007; Levinson & Rieger 2011),the jet-in-jet realization (Giannios et al. 2009), and theemission caused by penetration of external dense clouds(Barkov et al. 2012a).The production of gamma rays in the BH magneto-sphere has several unique properties. In particular, thisscenario can be invoked to explain emission from off-axisAGNs and orphan gamma-ray flares. On the other hand,the luminosity of the magnetospheric gap has a robustupper limit that depends weakly on the SMBH mass.Moreover, the magnetospheric emission is not enhancedby the Doppler-boosting effect, and this seems to be cru-cial for explaining short flares from distant AGN. Onthe other hand, some nearby SMBHs(Levinson & Rieger2011), e.g., the Sagittarius A star or M87, might bevery promising candidates to produce gamma-ray flares(see, however, Li et al. 2009; Levinson & Rieger 2011;Cui et al. 2012, for the discussion of gamma-gamma at-tenuation in magnetosphere).In general terms, there can be little doubt that thenonthermal radiation of powerful AGN is related, in oneway or another, to relativistic jets. The ultrafast gamma-ray flares might be linked to the formation of relativis-tically moving features (plasmoids or mini-jets) insidethe major outflow, the jet originating from the centralblack hole. Depending on the orientation of the mini-jets to the jet axis, the radiation of the mini-jet can befocused within the jet cone or outside. This scenario hasbeen suggested to interpret the variable emission fromAGN (Giannios et al. 2009, 2010). It has been shownthat under certain conditions, magnetic field reconnec-tion can result in the formation of relativistic outflows(Lyubarsky 2005; Sironi et al. 2016). We note, however,that formation of a relativistic outflow is not an indis-pensable feature of reconnection. Thus, ejection of rela-tivisitcally moving plasmoids may require a specific con-figuration of the magnetic field. Independently, to formoutflows with large Lorentz factors, Γ co ≥
10, an initialconfiguration with high magnetization, σ ≃ Γ ≥ σ init ≫ . Jets with such a highmagnetization should have an extremely low mass load,which seems to be inconsistent with the properties ofAGN jets at large distances (see, however, Komissarov1994; Stern & Poutanen 2006; Araudo et al. 2013, and2references therein).Finally, the SED of the emission produced by plas-moids formed at reconnection contains a dominating syn-chrotron component that peaks in the UV energy band(Petropoulou et al. 2016). This feature is not consistentwith the SEDs obtained from AGNs during the ultra-fast flares. The presence of a guiding magnetic field cansignificantly enhance the magnetization of plasmoids, re-sulting in a further enhancement of the synchrotron com-ponent and perhaps in the extension of the synchrotroncomponent to the gamma-ray band. The examination ofthis scenario requires detailed modeling, since the guid-ing filed also impacts the Lorentz factor of plasmoids.The jet-in-jet scenario quantitatively implies a mod-est requirement for the jet intrinsic luminosity, however;it can be even further relaxed if one assumes that themini-jets are not distributed isotropically in the majorjet comoving frame. Such an anisotropy can be realized,for example, by focusing the outflow along the directionof the reconnecting magnetic field.An important issue to realize the jet-in-jet scenariowhat is the triggering mechanism for the reconnection.If the jet is launched by the BZ mechanism, it is ex-pected to be magnetically dominated at the initial stage,thus reconnection is a thermodynamicaly favored pro-cess. However, the reasoning based on equipartition ar-guments, without any particular energy transfer mecha-nism, can hardly be valid. The time-scale of such ther-modynamic processes may be enormous; thus they mightbe irrelevant for astrophysical jets. In recent years, sig-nificant progress has been achieved in PIC simulationsfor the reconnection in magnetic field configurations withalternating polarities. Such configurations should natu-rally appear in the pulsar outflows close to the currentsheet. However, it is less obvious how such regions wouldform in AGN jets. Several scenarios can be considered.The first is a change in the magnetic field polarity inthe jet caused by a change in magnetic field polarity in the accretion disk and, consequently, in the BH magneto-sphere (Barkov & Baushev 2011; McKinney & Uzdensky2012). However, such a change takes a long time, ofabout ∼ r g /c , and it is hard to expect ultrafast vari-ability caused by such a configuration of the magneticfield. An arrangement with alternating magnetic fieldpolarities can also be a result of a growth of MHD insta-bilities in the jet (see, e.g., Barniol Duran et al. 2016).However, an intense instability growth leads to the aflow disruption on a scale of several dynamical lengths(Porth & Komissarov 2015). So it is hard to obtain anintensive reconnection event close to the base of a jetthat extends a significant distance beyond the reconnec-tion region. Finally, the reconnection can be caused bya sudden compression and mixing of a small part of thejet, which, for example, can be due to an external ob-stacles in the jet. In such a case, a short but intensivelocal reconnection episode may occur without disruptingthe entire flow. This specific case represents an interest-ing synergy of two models: the formation of a relativisticmini-jet by reconnection of the magnetic field triggeredby a star in the jet. The feasibility of this scenario needsto be tested with detailed numerical simulations.The star-in-jet scenario, the third possibility consid-ered in the paper, requires significantly higher jet lumi-nosity than the jet-in-jet scenario. In many cases, the jetluminosity, needed to realize the star-in-jet scenario, ex-ceeds the Eddington limit. It was also shown that somedetails of the GeV light curve obtained from 3C 454.3with Fermi
LAT, e.g., the plateau phase, can be readilyinterpreted in the framework of the star-in-jet scenario(Khangulyan et al. 2013). It is also important to notethat the emission produced by the interaction of a cloudwith the AGN jet should be characterized by a universalrelation between the luminosity and the duration of indi-vidual peaks of the flare: L / ∝ ∆ t . To verify this rela-tion observationally, a high photon statistics is required,which may possibly be achieved with future observationswith CTA.APPENDIX A. CALCULATION OF THE PAIR MULTIPLICITY FROM A RADIATIVELY INEFFICIENTACCRETION FLOW
Levinson & Rieger (2011) have estimated the density of electron-positron pairs produced by photon-photon annihi-lation in a radiatively inefficient accretion flow (RIAF, Narayan & Yi 1995b). For the sake of consistency, a similarconsideration is present below, but for a Kerr BH and explicitly accounting for the nondimensional viscosity parameter α ss and radiation efficiency off an accretion flow η . Identically to Levinson & Rieger (2011), we relie on the solutionobtained by Narayan & Yi (1995b), assuming that the advection parameter is small: 1 − f ≪ v r = 3 α ss (cid:18) GM bh r (cid:19) / . (A1)The ion density n i can be estimated as n i ( r ) = ˙ M πrHm p v r = 5 √
106 ˙ mηα ss ( GM bh ) / cσ T r / , (A2)given the revealed height of the accretion disk, H ≃ c s / Ω k (for details and notations see Narayan & Yi 1994).The total cooling rate of the ion-electron plasma can be estimated (Narayan & Yi 1995c; Levinson & Rieger 2011)as q ff = q ee + q ei ≈ − n e θ e erg s − cm − (A3)3for relativistic electron temperature θ e = kT e /m e c &
1. Since the density of the ions exceeds the GJ density, thepair production does not provide any sensible contribution to the disk electron density in configurations allowing theexistence of vacuum gaps, thus in what follows we assume the number densities of electrons and ions to be equal, n i = n e .For such a hot electron plasma the emission appears in the MeV energy band, and the luminosity of the inner partof the accretion flow is (Levinson & Rieger 2011) L ff ≈ Z r g r g πr q ff dr . (A4)A lower limit on the number density of these MeV photons is n γ ≈ L ff πc (2 r g ) e γ ≈ . q ff r g cr g e γ ∼ ˙ m α ss η M cm − , (A5)where e γ = 3 θ e m e c . The production rate of e ± pairs inside the magnetosphere due to γγ -annihilation is approximately σ γγ n γ c (4 π/ r g ) , where σ γγ ≈ σ T / ∼ πc (2 r g ) n ± , allowung us to estimate the n ± & ˙ m η α M cm − . (A6)The GJ density (ignoring its polar angle dependence) is determined by the accretion rate, ˙ m , via Eqs. (18) and (8): n GJ = Ω B πec = 0 . β m ˙ mηα ss M , ! / cm − . (A7)Thus, the multiplicity parameter is κ & n ± n GJ ≈ × ˙ m / M / , η / α / β / m . (A8)The condition κ < m . − ηα ss β / m M / . (A9)Eq. (A4) allows an estimation of the characteristic inverse Compton cooling time of electrons in the photon fieldprovided by the disk. For an electron with energy E = 10 m e c γ λ IC = 34 σ t mc n γ e γ γ ∼ α η M θ e × cm γ ˙ m & . r g γ M / β / m θ e . (A10) B. SMALL ANGLE LIMIT
It is believed that the emission from AGNs is mostly detected by observers located within the jet-beaming cone θ ≤ Γ − , but there are also some examples when the flares are detected from off-axis radio galaxies, e.g., M87 andIC 310. Sometimes it is convenient to measure the viewing angle in the jet-opening units: θ = α Γ − . (B1)In particular, this parameter gives a simple relation between the jet Doppler factor and the Lorentz factor: δ j = 1Γ j (1 − β j cos θ ) = 2Γ j α , (B2)which is valid for α ≪ Γ j .If the production region moves relativistically in the jet, then its Lorentz factor with respect to the observer is (see,e.g., Giannios et al. 2009) Γ em = Γ j Γ co (1 + β j β co cos θ ′ ) , (B3)where θ ′ is the angle between the jet velocity and the outflow velocity. This angle is related to the angle in the observerframe via the aberration formula, tan θ = β co sin θ ′ Γ j ( β co cos θ ′ + β j ) . (B4)4The latter equation allows us to express the angle in the comoving reference framecos θ ′ = − α β j β co ± r α β β + (cid:16) − α β β (cid:17) (1 + α )1 + α . (B5)One should account for the kinematic constraints that naturally appear in the above equation: | cos θ ′ | ≤
1. Taking thesolution with the + sign (which corresponds to a stronger enhancement in the region where two solutions are allowed)in the limit Γ j , co ≫
1, one obtains β co cos θ ′ ≃ − α α − , (B6)and consequently, the emitter Lorentz factor is Γ em ≃ j Γ co α . (B7) ACKNOWLEDGMENTS
The authors are grateful to the anonymous referee for the insightful comments and helpful suggestions. We wouldlike to thank Andrey Timokhin and Frank Rieger for productive discussions. The authors appreciate the supportby the Russian Science Foundation under grant 16-12-10443. D.K. acknowledges financial support by a grant-in-aidfor Scientific Research (KAKENHI, No. 24105007-1) from the Ministry of Education, Culture, Sports, Science andTechnology of Japan (MEXT). M.B. acknowledges partial support by the JSPS (Japan Society for the Promotion ofScience): No.2503786, 25610056, 26287056, 26800159. M.B. also acknowledges MEXT: No.26105521 and for partialsupport by NSF grant AST-1306672 and DoE grant de-sc0016369.
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