Constraining the Location of Gamma-Ray Flares in Luminous Blazars
aa r X i v : . [ a s t r o - ph . H E ] M a y Draft version September 29, 2018
Preprint typeset using L A TEX style emulateapj v. 04/17/13
CONSTRAINING THE LOCATION OF GAMMA-RAY FLARES IN LUMINOUS BLAZARS
Krzysztof Nalewajko , Mitchell C. Begelman , and Marek Sikora Draft version September 29, 2018
ABSTRACTLocating the gamma-ray emission sites in blazar jets is a long-standing and highly controversial issue.We investigate jointly several constraints on the distance scale r and Lorentz factor Γ of the gamma-ray emitting regions in luminous blazars (primarily flat spectrum radio quasars, FSRQs). Working inthe framework of one-zone external radiation Comptonization (ERC) models, we perform a parame-ter space study for several representative cases of actual gamma-ray flares in their multiwavelengthcontext. We find a particularly useful combination of three constraints: from an upper limit on thecollimation parameter Γ θ .
1, from an upper limit on the synchrotron self-Compton (SSC) luminos-ity L SSC . L X , and from an upper limit on the efficient cooling photon energy E cool , obs .
100 MeV.These three constraints are particularly strong for sources with low accretion disk luminosity L d . Thecommonly used intrinsic pair-production opacity constraint on Γ is usually much weaker than theSSC constraint. The SSC and cooling constraints provide a robust lower limit on the collimationparameter Γ θ & . − .
7. Typical values of r corresponding to moderate values of Γ ∼
20 are inthe range 0 . − t var , obs .Alternative scenarios motivated by the observed gamma-ray/mm connection, in which gamma-rayflares of t var , obs ∼ a few days are located at r ∼
10 pc, are in conflict with both the SSC and coolingconstraints. Moreover, we use a simple light travel time argument to point out that the gamma-ray/mm connection does not provide a significant constraint on the location of gamma-ray flares. Weargue that spine-sheath models of the jet structure do not offer a plausible alternative to externalradiation fields at large distances, however, an extended broad-line region is an idea worth exploring.We propose that the most definite additional constraint could be provided by determination of thesynchrotron self-absorption (SSA) frequency for correlated synchrotron and gamma-ray flares.
Keywords: galaxies: active — galaxies: jets — gamma rays: galaxies — quasars: general — radiationmechanisms: non-thermal INTRODUCTIONBlazars are a class of active galaxies, whose broad-band emission is dominated by non-thermal compo-nents produced in a relativistic jet pointing towardus (Urry & Padovani 1995). Due to the relativis-tic luminosity boost, many of these sources outshinetheir host galaxies by orders of magnitude, mak-ing them detectable at cosmological distances. Thebrightest blazars, belonging to the subclasses knownas flat-spectrum radio quasars (FSRQs) and low-synchrotron-peaked BL Lacertae objects (LBLs), ra-diate most of their energy in MeV/GeV gamma-rays(Fossati et al. 1998). The origin of this gamma-rayemission has been debated for a long time, with pro-posed mechanisms including external-radiation Comp-tonization (ERC; Dermer et al. 1992; Sikora et al. 1994),synchrotron self-Comptonization (SSC; Maraschi et al.1992; Bloom & Marscher 1996), and hadronic pro-cesses ( e.g. , Mannheim & Biermann 1992; Aharonian2000; M¨ucke & Protheroe 2001). The emerging con-sensus favors the ERC process (Ghisellini et al. 1998;Mukherjee et al. 1999; Hartman et al. 2001; Sikora et al. JILA, University of Colorado and National Institute of Stan-dards and Technology, 440 UCB, Boulder, CO 80309, USA; [email protected] NASA Einstein Postdoctoral Fellow Department of Astrophysical and Planetary Sciences, Uni-versity of Colorado, UCB 389, Boulder, CO 80309, USA Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland r & .
01 pc(Ghisellini & Madau 1996). At these smallest alloweddistances, the dominant external radiation componentin the jet co-moving frame is the direct emission of theaccretion disk ( e.g. , Dermer & Schlickeiser 2002). Atdistances of r ∼ . e.g. ,Sikora et al. 1994). For an emitting region propagatingwith a typical Lorentz factor of Γ ≃
20, the observedvariability time scale ∼ r/ (Γ c ) expected from radiationproduced at such distances is several hours, which is con-sistent with the shortest variability time scales probed bythe Fermi
Large Area Telescope (LAT) (Tavecchio et al.2010; Saito et al. 2013; Rani et al. 2013). The likelydissipation mechanism at these distances depends onthe efficiency of energy flux conversion from magnetic(Poynting flux) to inertial (kinetic energy flux) forms(Sikora et al. 2005). In particle-dominated jets, internalshocks can operate with reasonable efficiency, provided
Nalewajko, Begelman & Sikora that the jet acceleration mechanism is strongly modu-lated (Spada et al. 2001). In magnetically dominatedjets, shocks are generally expected to be weak (but seeKomissarov 2012), however, in right circumstances thejet magnetic fields could be dissipated directly in the pro-cess of magnetic reconnection ( e.g. , Giannios & Spruit2006; Giannios et al. 2009).At distances of r & λ syn , obs . e.g. ,Sikora et al. 2008), or between the gamma-ray and mil-limeter signals ( e.g. , Wehrle et al. 2012). At these dis-tances, reconfinement shocks arising from the interactionof the jet with the external medium provide an alterna-tive dissipation mechanism ( e.g. , Nalewajko 2012).The structure of blazar jets can be at least par-tially resolved with interferometric radio/mm observa-tions. Typically, it includes a stationary core and asuccession of knots propagating superluminally down-stream from the core. The core could be a photo-sphere due to the synchrotron self-absorption process(certainly at wavelengths longer than 7 mm), or an opti-cally thin physical structure presumably resulting fromreconfinement shocks (Marscher 2009). There is sub-stantial evidence that many major gamma-ray flares inblazars are accompanied by radio/mm outbursts, and/orejection (estimated moment of passing through the ap-parent position of the core) of superluminal radio/mmknots ( e.g. , Marscher et al. 2012). While radio/mmoutbursts are typically much longer ( ∼ weeks/months)than gamma-ray flares ( ∼ hours/days), the gamma-rayflares are often found between the onset and the peakof the mm outbursts (L¨ahteenm¨aki & Valtaoja 2003;Le´on-Tavares et al. 2011). This gamma-ray/mm connec-tion is used to argue for gamma-ray flares being producedat distance scales of r ∼ −
20 pc ( e.g. , Agudo et al.2011a,b; see also Sikora et al. 2008). At these distances,the external radiation field is still likely dominated bythermal dust emission, although its energy density is ex-pected to fall off rapidly with r . In order to explainshort variability time scales of gamma-ray flares at suchdistances, very strong jet collimation is required.In this work, we study the parameter space of location r and Lorentz factor Γ of the emitting regions respon-sible for major gamma-ray flares in luminous blazars. We use 5 direct observables — gamma-ray luminosity L γ , gamma-ray variability time scale t var , obs , synchrotronluminosity L syn (or the Compton dominance parame-ter q = L γ /L syn ), X-ray luminosity L X , and accretiondisk luminosity L d — and a minimal number of assump-tions — in particular the Doppler-to-Lorentz factor ratio D / Γ, and the external radiation sources covering factors ξ BLR , ξ IR — to derive 4 constraints in the ( r, Γ) planerelated to the following parameters — collimation pa- In some blazar studies, multiple emitting regions were deemednecessary ( e.g. , Nalewajko et al. 2012b). However, in any casewhere a coherent gamma-ray flare is observed, one can consideronly the emitting region dominating the gamma-ray emission. rameter Γ θ , synchrotron self-Compton luminosity L SSC ,observed ERC photon energy corresponding to efficientelectron cooling threshold E cool , obs , and observed ERCphoton energy corresponding to intrinsic pair-productionabsorption threshold E max , obs — and 2 predictions forthe following parameters — synchrotron self-absorptioncharacteristic observed wavelength λ SSA , obs , and mini-mum required jet power L j , min . These constraints arethen applied in several case studies of actual gamma-rayflares of prominent blazars for which detailed multiwave-length data are available, and for which all 5 observablescan be securely estimated. Most of these cases have al-ready been discussed in the literature, but here they aresystematically and critically compared for the first time.We begin by deriving our constraints in Section 2, fol-lowed by additional predictions in Section 3. Then wepresent the case studies in Section 4. We consider thesensitivity of our constraints to the most uncertain pa-rameters in Section 5. Our results are discussed in Sec-tion 6 and summarized in Section 7. CONSTRAINTS ON r AND ΓWe consider an emitting region located at distance r from the central supermassive black hole (SMBH),propagating with velocity β = v/c and Lorentz factorΓ = (1 − β ) − / . Parameters measured in the co-moving frame of the emitting region will be denoted witha prime. We should stress here that the Lorentz factorof the emitting region Γ does not need to coincide withthe jet Lorentz factor Γ j . While simple models explic-itly assume that Γ ≃ Γ j , in some scenarios a significantdifference between these values is inferred, e.g., in thespine-sheath model (Ghisellini et al. 2005), and in theminijet model (Giannios et al. 2009).For an observer located at viewing angle θ obs withrespect to the emitting region velocity vector, theDoppler factor of the observed radiation is D = [Γ(1 − β cos θ obs )] − . In blazars, the value of D is of the sameorder as Γ, but the actual ratio D / Γ is a major sourceof uncertainty in constraining r and Γ. In the case ofa very compact emitting region, for θ obs ≃ / Γ we have D / Γ ≃
1, and for θ obs ≃ D / Γ ≃
2. However,in a conical jet, elements of the emitting region may spana significant range of θ obs , and thus a significant range of D / Γ. The effective value of D / Γ depends not only on themean θ obs of the emitting region, but also on its openingangle θ . In particular, for emitting regions with Γ θ ∼ D / Γ . j and D for individual sources can beevaluated independently by analyzing the radio structureof jets observed with VLBI techniques (Jorstad et al.2005), and many such results are available for the MO-JAVE sample (Hovatta et al. 2009). Therefore, it is nowpossible to make an informed choice of D / Γ j for manystudied sources. However, as we will discuss later, thisdoes not work equally well for all sources. In this work,we decided to adopt D / Γ = 1 for all analyzed sources,and we evaluate the effect of varying the value of D / Γ inSection 5. 2.1.
Collimation constraint
We assume that the emitting region has characteristicsize R , which is related to the co-moving variability time ocating gamma-ray flares in blazars R ≃ ct ′ var . The variability time scale scales like t ′ var = D t var , obs / (1 + z ), where z is the blazar redshift.The most reliable estimate of the observed variabilitytime scale t var , obs is the flux-doubling time scale mea-sured with respect to the flare peak. We can also relate R to the location of the emitting region via R ≃ θr .Again, we distinguish θ from the jet opening angle θ j ,demanding only that θ ≤ θ j . It is convenient to combine θ with the Lorentz factor Γ to define the collimation pa-rameter Γ θ . We can now write the source Lorentz factoras a function of Γ θ :Γ( r, Γ θ ) ≃ (cid:18) D Γ (cid:19) − / (cid:20) (1 + z )(Γ θ ) rct var , obs (cid:21) / . (1)There are strong observational and theoretical indica-tions that Γ j θ j < j θ j ∼ . − . j θ j . j θ j and thecollimation parameter of the emitting region Γ θ is un-clear. On one hand, we expect that θ ≤ θ j , on the otherhand, it is possible that Γ > Γ j . Therefore, here we adopta relatively conservative collimation constraint , definedas Γ θ .
1. 2.2.
SSC constraint
We assume that the gamma-ray emission is producedby Comptonization of external radiation (ERC) by apopulation of ultrarelativistic electrons, and that the ap-parent gamma-ray luminosity L γ (hereafter understoodas the peak of νL γ,ν SED, as opposed to the bolometricluminosity L γ, bol = R L γ,ν d ν ) measured by Fermi /LATrepresents L ERC , the peak luminosity of the ERC com-ponent. The same electrons produce the synchrotronand the SSC components, of which at least the formershould contribute to the observed spectral energy dis-tributions (SEDs) as indicated by fast optical/IR flaresoften correlated with the gamma rays. The three lu-minosities — L ERC , L syn and L SSC — can be relatedto the co-moving energy densities of external radiation u ′ ext , magnetic fields u ′ B = B ′ / (8 π ), and synchrotronradiation u ′ syn ≃ L syn / (4 πc D R ), respectively. Onone hand, we have L SSC /L syn ≃ g SSC ( u ′ syn /u ′ B ), where g SSC = ( L SSC /L syn ) / ( L SSC , bol /L syn , bol ) ≃ / q = L γ L syn ≃ g ERC (cid:18) D Γ (cid:19) (cid:18) u ′ ext u ′ B (cid:19) , (2)where g ERC = ( L ERC /L syn ) / ( L ERC , bol /L syn , bol ) ≃ / D / Γ) factor reflects the beam-ing profile of the ERC component in the case of flat νL ν SED (Dermer 1995). The co-moving energy density of external radiation is related to the accretion disk lumi-nosity L d via u ′ ext ≃ ζ ( r )Γ L d πcr . (3)Here, ζ ( r ) is a function that describes the compositionof external radiation fields, including contributions fromthe broad-line region (BLR), the dusty torus producinginfrared emission (IR), and the direct accretion disk ra-diation: ζ ( r ) ≃ . ξ BLR ( r/r BLR ) r/r BLR ) + 0 . ξ IR ( r/r IR ) r/r IR ) + 0 . R g r , (4)where ξ BLR is the covering factor of the BLR of character-istic radius r BLR , ξ IR and r IR are the analogous parame-ters of the dusty torus, and R g is the gravitational radiusof the SMBH (we explain the origin of this function inAppendix A). In this work, we adopt the following scalinglaws: r BLR ≃ . L / , pc, and r IR ≃ . L / , pc, where L d , = L d / (10 erg s − ) (Sikora et al. 2009). Puttingthe above relations together, we obtain a constraint onΓ: Γ( r, L SSC ) ≃ (cid:20) (cid:18) g SSC g ERC (cid:19) (cid:18) L syn L SSC (cid:19) (cid:18) L γ ζ ( r ) L d (cid:19)(cid:21) / × (cid:18) D Γ (cid:19) − (cid:20) (1 + z ) r ct var , obs (cid:21) / . (5)The SSC component in the SEDs of luminous blazarspeaks at the observed photon energy of E SSC , obs ≃
20 neV × D B ′ γ / (1 + z ), where B ′ = B ′ / (1 G) and γ peak is the characteristic random Lorentz factor of elec-trons contributing to the SED peaks. We can estimate γ peak from the observed photon energy of the SED peakof the ERC component E ERC , obs ≃ D Γ γ E ext ( r ) / (1 + z ), where E ext ( r ) is the energy of external radiation pho-tons. In order to account for the transition between theBLR and IR external radiation fields, we use the follow-ing approximation (see Appendix A): E ext ( r ) ≃ E BLR r/r
BLR ) + E IR r/r IR ) , (6)where E BLR ≃
10 eV and E IR ≃ . B ′ ≃ D r (cid:20) g ERC ζ ( r ) L d qc (cid:21) / . (7)Combining the above formulas, we find: E SSC , obs ≃
20 neV r (1 + z )Γ (cid:20) E ERC , obs E ext ( r ) (cid:21) × (cid:20) g ERC ζ ( r ) L d qc (cid:21) / . (8)One can see that E SSC , obs is a sensitive function of E ERC , obs and Γ. However, for Γ = 20, E ERC , obs =100 MeV, r = 1 pc, E ext = 1 eV, ζ = 0 . L d = 3 × erg s − , and q = 10, we find E SSC , obs ≃ z ) keV.Because SSC spectral components are very broad, in Nalewajko, Begelman & Sikora most cases they should peak around, or contribute signif-icantly to, the soft/hard X-ray band. Some blazars showspectral softening in the soft X-ray part of their SEDs,which was interpreted as a signature of the SSC com-ponent (Bonnoli et al. 2011). However, in many sourcesthe observed X-ray emission is harder than it would be ifit were dominated by the SSC component (Sikora et al.2009). Also, the observed X-ray variability is usually notwell correlated with variability in the gamma-ray andoptical bands (Hayashida et al. 2012). In the case thatthe SSC component dominates the X-ray emission, wewould expect that X-ray variability should be strongerthan the optical/IR variability. For example, in a sim-ple scenario of varying number of energetic electrons atconstant magnetic field we have L SSC ∝ L . As this isnot the case for luminous blazars, we can only use theobserved X-ray luminosity as an upper limit for the SSCluminosity (Ackermann et al. 2010). Therefore, our SSCconstraint is defined as L SSC . L X .2.3. Cooling constraint
Rapid gamma-ray variability of blazars, with roughlytime-symmetric light curve peaks, and tight energetic re-quirements for the brightest observed gamma-ray flares,indicate very efficient cooling of the underlying ultrarel-ativistic electrons. The radiative cooling of electrons inluminous blazars is dominated by the ERC process withcooling time scale t ′ cool ( γ ) ≃ m e c/ (4 σ T γu ′ ext ), where γ is the electron random Lorentz factor. In general, t ′ cool ( γ ) should be compared with the variability timescale t ′ var (which is associated with the observed flux dou-bling timescale, see Section 2.1), and adiabatic coolingtime scale t ′ ad . Observations of roughly time-symmetricflares indicate that the cooling time scales do not ex-ceed the observed flux decaying time scales, i.e., that t ′ cool ( γ ) . t ′ var . We calculate a characteristic electronLorentz factor γ cool such that t ′ cool ( γ cool ) ≃ t ′ var , and acorresponding observed ERC photon energy E cool , obs ≃D Γ γ E ext ( r ) / (1 + z ). Taking the above together, weobtain the following constraint on Γ:Γ( r, E cool , obs ) ≃ (cid:18) D Γ (cid:19) − / (cid:20) πm e c r σ T ζ ( r ) L d t var , obs (cid:21) / × (cid:20) (1 + z ) E ext ( r ) E cool , obs (cid:21) / . (9)Since the gamma-ray light curves based on the Fermi /LAT data are typically calculated for photon en-ergies
E >
100 MeV, our cooling constraint is defined as E cool , obs .
100 MeV.Alternatively, the cooling time scale as a function ofphoton energy potentially can be estimated directly from The adiabatic loss time scale is t ′ ad ≃ r/ ( A Γ c ) ≃ t ′ var / ( A Γ θ ),where A ≤
1. Therefore, as long as the collimation constraintΓ θ . t ′ ad & t ′ var . Alternatively, the time-symmetric gamma-ray flares may in-dicate that the velocity vector of the emitting region is rapidlyswinging relative to the line-of-sight. In such case, both the fluxrise and decay time scales would be determined primarily by vari-ations in the Doppler factor. This is different from a cooling break which is obtained byequating the radiative and adiabatic energy loss rates. gamma-ray observations, but this is only feasible for thevery brightest events (Dotson et al. 2012).2.4.
Internal gamma-ray opacity constraint
The maximum observed gamma-ray photon energy E max , obs is constrained at least by the pair-productionabsorption process due to soft radiation produced inthe same emitting region ( e.g. , Dondi & Ghisellini 1995).The peak cross section for the pair-production processis σ γγ ≃ σ T / E ′ soft ≃ . m e c ) /E ′ max . In the observer frame, thesoft photon energy is E soft , obs ≃ . m e c ) D (1 + z ) E max , obs ≃
38 keV(1 + z ) (cid:18) E max , obs
10 GeV (cid:19) − (cid:18) D (cid:19) . (10)The optical depth for gamma-ray photons is: τ γγ = σ γγ n ′ soft R ≃ (1 + z ) σ T L soft E max , obs π ( m e c ) c D t var , obs . (11)As the observed soft photon energy E soft , obs may fall out-side any observed energy range, we relate the target softradiation luminosity to the observed X-ray luminosityvia a spectral index α such that L soft = L X (cid:18) E soft , obs E X (cid:19) − α ≃ [3 . m e c ) ] − α D − α L X (1 + z ) − α E − α X E − α max , obs . (12)Substituting this into Equation (11), we obtain: τ γγ ≃ (1 + z ) α σ T L X E α max , obs π [3 . m e c ) ] α c D α E − α X t var , obs . (13)For gamma-ray observations of blazars, it is typical toassociate E max , obs with τ γγ ≃
1. This leads to the fol-lowing constraint on Γ:Γ( r, E max , obs ) ≃ ( (1 + z ) α σ T L X E α max , obs π [3 . m e c ) ] α c E − α X t var , obs ) α × (cid:18) D Γ (cid:19) − . (14)In Section 4, we will demonstrate that the internalgamma-ray opacity constraint is relatively weak com-pared to the SSC constraint.Additional potential source of gamma-ray opacity isfrom the broad emission lines. To the first order of ap-proximation, this would affect photons of observed en-ergy: E max , BLR , obs ≃ . m e c ) (1 + z ) E BLR ≃
94 GeV(1 + z ) (cid:18) E BLR
10 eV (cid:19) − , (15)with the peak optical depth of τ γγ, BLR ( r ) ≃ ξ BLR σ T L d ( r BLR − r )20 πcr E BLR ≃ (cid:18) ξ BLR . (cid:19) × (cid:18) L d erg s − (cid:19) / (cid:18) r BLR − rr BLR (cid:19) (cid:18) E BLR
10 eV (cid:19) − (16) ocating gamma-ray flares in blazars r BLR ≃ . L / , pc). The highvalue of the peak optical depth indicates that absorptionshould become noticeable already at the threshold ob-served energy of ( m e c ) / [(1 + z ) E BLR ] ≃
26 GeV / (1 + z ) / ( E BLR /
10 eV). The actual strength of the BLRabsorption features depends significantly on the BLRgeometry, and determining it requires detailed calcu-lations ( e.g. , Donea & Protheroe 2003; Reimer 2007;Tavecchio & Ghisellini 2012). We will briefly commenton the expected significance of the BLR absorption inthose cases from Section 4, which allow for the emittingregion to be located within r BLR . PREDICTIONS FOR GIVEN r AND Γ3.1.
Synchrotron self-absorption
Synchrotron radiation is subject to synchrotron self-absorption (SSA) process, which can produce a sharpspectral break. This is a powerful probe of the intrin-sic radius of the source of synchrotron emission ( e.g. ,Sikora et al. 2008; Barniol Duran et al. 2013). In the co-moving frame, the SSA break is expected at: ν ′ SSA ≃ (cid:18) eB ′ m c (cid:19) / L ′ / R / , (17)where we approximated the synchrotron luminosity at ν ′ SSA with the synchrotron energy distribution peak lu-minosity L ′ syn (i.e., we assumed a flat synchrotron SEDin the mid-IR/mm band; in any case ν ′ SSA depends onlyweakly on the spectral index of unabsorbed synchrotronemission). Substituting relevant relations from previoussections, we find a constraint on Γ:Γ( r, ν
SSA , obs ) ≃ " g ERC e ζ ( r ) L d L γ q m c (1 + z ) ν , obs t , obs r / × (cid:18) D Γ (cid:19) − . (18)In luminous blazars, the SSA spectral break is typicallyobserved in the sub-mm/radio band. As the synchrotronradiation observed in this band probes lower electron en-ergies than the ∼ GeV gamma-ray radiation, a connec-tion between these bands should be verified by studyingvariability correlations. These are very challenging obser-vations, and for most cases studied in Section 4 such dataare not available. Therefore, in this work the
SSA con-straint is limited to provide a prediction of what ν SSA , obs should be for each studied case.3.2. Jet energetics
We can constrain the energy content of blazar jets un-derlying the observed gamma-ray flares by estimatingtwo of its essential ingredients: the radiation energy den-sity dominated by the gamma rays u ′ γ , and the magneticenergy density u ′ B . Because the production of gamma-ray radiation through the ERC process is very efficient, Considering the ionized Helium lines with E BLR ≃
54 eV,the threshold observed energy would shift to ≃ . / (1 + z )(Poutanen & Stern 2010), however, this is only relevant for dis-tance scales r ≪ r BLR that are not of interest here. u ′ γ closely probes the high-energy end of the electron en-ergy distribution. Additional jet energy may be carriedby cold/warm electrons and protons, the contributionof which is very uncertain. For example, the number ofcold electrons can be constrained by modeling the broad-band SEDs, but the low-energy electron distribution in-dex is usually one of the most uncertain parameters. Onthe other hand, the energy content of protons in blazarjets can be constrained only indirectly, by combining ar-guments such as interpretation of (hard) X-ray spectraof luminous blazars, and energetic coupling between theprotons and electrons (Sikora 2011). Rather than intro-ducing extra parameters with highly uncertain values,we choose to discuss a firm lower limit L j , min on the jetpower required to produce the observed gamma-ray flaresof blazars together with their synchrotron and SSC coun-terparts.The radiation energy density can be written as: u ′ γ ≃ L γ πc D R ≃ (cid:18) D Γ (cid:19) − (1 + z ) L γ πc Γ t , obs . (19)The magnetic energy density u ′ B can be derived fromthe synchrotron luminosity L syn , which is related to thegamma-ray luminosity L γ through the Compton domi-nance parameter q = L γ /L syn : u ′ B ≃ (cid:18) D Γ (cid:19) g ERC u ′ ext q ≃ (cid:18) D Γ (cid:19) g ERC ζ ( r )Γ L d πcqr . (20)Instead of using these two energy densities separately, wewill analyze their more useful combinations: their ratioand their sum. The ratio of the two energy densities isa measure of energy equipartition between the magneticfields and the ultra-relativistic electrons. One can showthat ( cf. Sikora et al. 2009): u ′ γ u ′ B ≃ L γ L SSC g SSC L , (21)therefore, this energy density ratio is proportional to L SSC , and it follows the same dependence on r andΓ. The sum of the two energy densities constitutes alower limit on the jet energy density u ′ j , min = u ′ γ + u ′ B . The corresponding minimum jet power is givenby L j , min ≃ πc Γ R u ′ j , min . Therefore, we can write L j , min = L j ,γ, min + L j , B , min , where L j ,γ, min = (cid:18) D Γ (cid:19) − L γ , (22) L j , B , min = (cid:18) D Γ (cid:19) g ERC Γ ζ ( r ) L d q (cid:20) ct var , obs r (1 + z ) (cid:21) . (23)The dependence of the magnetic jet power on Γ is muchsteeper than for the radiative jet power. Thus, we canderive approximate constraints on Γ in two limits. For u ′ γ ≫ u ′ B we findΓ( L j ,γ, min ) = (cid:18) D Γ (cid:19) − (cid:18) L γ L j ,γ, min (cid:19) / ; (24) Nalewajko, Begelman & Sikora and for u ′ γ ≪ u ′ B we findΓ( r, L j , B , min ) = (cid:18) D Γ (cid:19) − / (cid:18) qL j , B , min g ERC ζ ( r ) L d (cid:19) / × (cid:20) r (1 + z ) ct var , obs (cid:21) / . (25)In Section 4, we will investigate the values of u ′ γ /u ′ B and L j , min for individual blazar flares. Again, we stressthat contributions from cold/warm electrons and protonsshould be included to obtain total jet energies. CASE STUDIESIn this section, we apply the constraints derived in Sec-tion 2 to several well-studied cases of powerful gamma-ray flares in blazars with excellent multiwavelength cov-erage. We would like to emphasize the value of havingextensive simultaneous spectral coverage of these sources,however, each case is different and the data quality is notuniform enough to warrant a broader study.4.1.
3C 454.3 at MJD 55520
3C 454.3 ( z = 0 . d L ≃ .
49 Gpc) provided uswith the most spectacular gamma-ray flares in the
Fermi era (Nalewajko 2013). On MJD 55520 (2010 Nov 20)it produced a flare of apparent peak bolometric (
E >
100 MeV) luminosity of L γ, bol ≃ . × erg s − (Abdo et al. 2011). We convert the bolometric peak lu-minosity L γ, bol into the peak νL ν luminosity L γ , using abolometric correction factor g γ, bol = L γ, bol /L γ ∼ . L γ ≃ . × erg s − .The flare temporal template fitted by Abdo et al. (2011)has a flux doubling time scale of t var , obs ≃ . ≃ . × s. Vercellone et al. (2011) showed that thisgamma-ray flare was accompanied by simultaneous out-bursts, of amplitude smaller by factor ∼
3, in soft X-ray,optical and millimeter bands. They compiled an SEDfrom which we can estimate the simultaneous luminos-ity ratios q = L γ /L syn ≃ L syn /L X ≃
10. Theseratios are used to derive the simultaneous soft X-ray lu-minosity L X ≃ . × erg s − . We can also estimatethe spectral index of the X-ray part of the spectrum as α ≃ .
65. The bolometric accretion disk luminosity istaken as L d ≃ . × erg s − (Bonnoli et al. 2011),from which we find the characteristic radii of externalradiation components r BLR ≃ .
26 pc and r IR ≃ . M BH ∼ × M ⊙ after Bonnoli et al.(2011).In Figure 1, we plot the constraints on r and Γ corre-sponding to fixed values of Γ θ , L SSC , E cool , obs , λ SSA , obs and E max , obs , as well as the energetics parameters u ′ γ /u ′ B and L j , min . We assumed here that ξ BLR ≃ ξ IR ≃ .
1. Theyellow-shaded area is defined by the following 3 condi-tions: Γ θ < L SSC < L X , and E cool , obs <
100 MeV.The intersection of the first two of these constraintsgives the marginal solution — the minimum Lorentz fac-tor Γ min ≃
30 and the minimum distance scale r min ≃ .
16 pc. For ( r min , Γ min ), other constraints yield thefollowing predictions: λ SSA , obs ≃ µ m, E max , obs &
10 TeV, u ′ γ /u ′ B ≃ .
3, and L j , min ≃ . × erg s − ≃ . L d . On the other hand, in the IR region ( r ∼ r IR ),the SSC constraint is much stronger and hence there areno solutions with Γ <
50. Therefore, in this case the dis-sipation region is clearly constrained to be located not farfrom r BLR . The minimum required jet power is one orderof magnitude higher than the kinetic jet power estimatedby Meyer et al. (2011).VLBI measurements of the jet of 3C 454.3 yield Γ j ≃ D ≃
33 (Hovatta et al. 2009), and Γ j θ j ≃ . D / Γ j ≃ .
67 wouldshift the marginal solution to r min ≃ .
09 pc and Γ min ≃
18. The VLBI-derived solution of r ≃ .
34 pc and Γ ≃ E cool , obs constraint, andmarginally consistent with our L SSC constraint. On theother hand, for D / Γ = 1, the SSC constraint also impliesthat jet collimation parameter is Γ θ > . D min ≃
16, using the gamma-ray opacity constraintfor the maximum observed photon energy of E max , obs =31 GeV. Our opacity constraint for the same E max , obs yields Γ min = D min ≃
13. The main reason for thisdiscrepancy is that we use the 3 . r min ≃ .
14 pc obtained by calcu-lating gamma-ray opacity due to the broad-line photons(Abdo et al. 2011).The synchrotron self-absorption break is predicted tofall in the far-IR range, both at the BLR and IR dis-tance scales. 3C 454.3 was observed by Herschel PACSand SPIRE instruments during and after the peak of thisgamma-ray flare (Wehrle et al. 2012). While the periodof the highest gamma-ray state was sparsely covered inthe far-IR band, a very good correlation between the160 µ m data and the Fermi /LAT gamma rays was found.Such a correlation implies that the gamma-ray produc-ing region is transparent to synchrotron self-absorption,i.e., that λ SSA , obs & µ m. Such a condition canbe easily satisfied, together with our collimation andSSC constraints, even at BLR distance scales. However,Wehrle et al. (2012) also showed that 1 . . r ≃
27 pc andΓ ≃ . . t var , mm ≃ . . r mm ≃ . ocating gamma-ray flares in blazars Γ E m a x , ob s [ T e V ] rr BLR r IR R g R g R g
3C 454.3
MJD 55520L γ = 4.7e+49 erg s -1 t var,obs = 8.7 h Γ θ = Γ θ = . Γ θ = . L S S C = e r g s - ( u γ ’ / u B ’ = . ) L S S C = e r g s - ( u γ ’ / u B ’ = . ) E c oo l = M e V E c oo l = M e V λ SSA = µ m L j,min = 10 erg s -1 L j,min = 3x10 erg s -1 Figure 1.
Parameter space of distance scale r and Lorentz factor Γ of the emitting region responsible for the major gamma-ray flareof 3C 454.3 that peaked at MJD 55520. Five classes of constraints are indicated: the collimation constraint ( solid red lines ; Eq. 1), theSSC constraint ( dashed blue lines ; Eq. 5), the cooling constraint ( dotted magenta lines ; Eq. 9), the synchrotron self-absorption constraint( dot-dashed orange lines ; Eq. 18), and the intrinsic gamma-ray opacity constraint (denoted by the maximum escaping photon energylabeled along the right-hand vertical axis; Eq. 14). We also show predictions for the jet energetics: the equipartition parameter ( u ′ γ /u ′ B ,shown together with the SSC constraint; Eq. 21), and the minimum required jet power ( double-dot-dashed green lines ; Eqs. 22, 23). On theupper horizontal axis, we show the distance scale in terms of the gravitational radius of the supermassive black hole, and the characteristicradii for main external radiation components (BLR and IR). Yellow-shaded area marks the parameter space allowed by the conditionsΓ θ < L SSC < L X , and E cool , obs <
100 MeV. (2012b). 4.2.
3C 454.3 at MJD 55168
A previous flare of 3C 454.3, peaking at MJD 55168(2009 Dec 3), also attracted considerable inter-est ( e.g. , Pacciani et al. 2010; Ackermann et al. 2010;Bonnoli et al. 2011). The apparent peak bolometricgamma-ray luminosity was L γ, bol ≃ . × erg s − (Ackermann et al. 2010), which corresponds to the νL ν luminosity L γ = L γ, bol /g γ, bol ≃ . × erg s − . Thevariability time scale was estimated at t var , obs ≃ ≃ . q = L γ /L syn ≃ L syn /L X ≃
10, and α ≃ .
55. We use the same val-ues of ξ BLR , ξ IR , L d , and M BH as for the MJD 55520flare.Our constraints for this event are shown in Figure 2.We find the marginal solution at r min ≃ .
17 pc andΓ min ≃
19. This solution corresponds to λ SSA , obs ≃ µ m, u ′ γ /u ′ B ≃ .
6, and L j , min ≃ erg s − ∼ . L d . While solutions within r BLR are allowed, themaximum observed photon energy is E max , obs ≃
21 GeV(Ackermann et al. 2010), so the effect of BLR absorptionis expected to be lower than in the case of the MJD 55520flare (Section 4.1).Bonnoli et al. (2011) modeled the SEDs of 3C 454.3 for several epochs close to MJD 55168, probing differ-ent luminosity levels. They noted that the gamma-rayluminosity scales with the X-ray and UV luminositiesroughly like L γ ∝ L X ∝ L . Therefore, they proposedthat the location of the gamma-ray emitting region shiftsoutwards with increasing gamma-ray luminosity. For thehighest state at MJD 55168, they suggested a distancescale of r ≃ .
06 pc at Γ ≃
20 (see Figure 2). It is criti-cal to note at this point that they adopted a variabilitytime scale of t var , obs ≃ D / Γ ≃ .
45. We have checked that forsuch parameters our constraints are marginally consis-tent with their result; our model predicts u ′ γ /u ′ B ≃ . λ SSA ≃ µ m, and L j , min ≃ . × erg s − .4.3. AO 0235+164 at MJD 54760
AO 0235+164 ( z = 0 . d L ≃ .
14 Gpc) is an LBL-type blazar, which was active in 2008–2009. The high-est gamma-ray state, achieved between MJD 54700 andMJD 54780, was analyzed in detail by Ackermann et al.(2012). They estimated the observed gamma-ray lumi-nosity as L γ ≃ . × erg s − ; the observed variabil-ity time scale t var , obs ≃ . × s; the Comp-ton dominance q = L γ /L syn ≃
4; the synchrotron toX-ray luminosity ratio L syn /L X ≃
6; the accretion diskluminosity L d = 4 × erg s − ; and the characteristicradii of external radiation components r BLR ≃ .
06 pc
Nalewajko, Begelman & Sikora Γ E m a x , ob s [ T e V ] rr BLR r IR R g R g R g
3C 454.3
MJD 55168 L γ = 8.4e+48 erg s -1 t var,obs = 1.0 d Γ θ = Γ θ = . Γ θ = . L SS C = e r g s - ( u γ ’ / u B ’ = . ) L SS C = e r g s - ( u γ ’ / u B ’ = . ) E c oo l = M e V E c oo l = M e V λ SSA = µ m λ SSA = µ m L j,min =3x10 erg s -1 L j,min = 10 erg s -1 L j,min = 3x10 erg s -1 Figure 2.
Parameter space of r and Γ for the major flare of 3C 454.3 that peaked at MJD 55168. See Fig. 1 for detailed description.The diamond indicates the solution obtained by Bonnoli et al. (2011). and r IR ≃ . M BH ∼ × M ⊙ . The X-ray spectral indexis very uncertain, very soft X-ray spectra were observedby Swift /XRT during the gamma-ray activity. Here weadopt α ≃ ξ BLR = ξ IR = 0 . r min ≃ .
65 pcand Γ min ≃
22. The predictions for this solution are λ SSA , obs ≃ µ m, u ′ γ /u ′ B ≃ .
7, and L j , min ≃ . × erg s − ≃ . L d . The gamma-ray emitting region iscertainly located outside the BLR, in the region whereexternal radiation is dominated by the dusty torus emis-sion. The jet is predicted to be at least moderatelymagnetized at r ∼ × R g . The required minimumjet power is higher by factor ≃ D / Γ j ≃ .
98 and Γ j θ j ≃ .
04 (Hovatta et al. 2009;Pushkarev et al. 2009). This rather extreme solution of avery narrow and perfectly aligned jet is inconsistent withboth the L SSC and E cool , obs constraints. For D / Γ = 1,the combination of L SSC and E cool , obs constraints impliesthat Γ θ > . ∼
12 pc, based on theVLBI imaging and cross-correlation between the gammarays and the mm data. Ackermann et al. (2012) useda simple variability time scale argument to show thatlocating the emitting region at 12 pc would require avery high jet Lorentz factor Γ ≃
50. Here, we find thatthe SSC constraint leads to a similar limit on Γ already at r ≃ ≃ r IR , implying thatenergetic electrons injected at the distance of 12 pc haveno chance to cool down efficiently. On the other hand,we show that if the emitting region is located at r IR and has a moderate Lorentz factor of Γ ≃
24, it willbe transparent to wavelengths shorter than ≃ −
50 days (the latter meaning that thegamma rays lead the mm signals). Our result is thus notin conflict with the gamma – 1 mm DCF. However, ourmodel does not allow for the possibility that the emittingregion producing 3-day long gamma-ray flares is trans-parent at 7 mm, which is the wavelength of VLBA obser-vations reported by Agudo et al. (2011b). In our model,even for Γ = 100 the 7 mm photosphere would fall at avery large distance of ≃
90 pc. Just like in the case of3C 454.3 (see Section 4.1), the solution to this apparentparadox is that the variability time scale of the 7 mmradiation has to be much longer than 3 days. Indeed,the 7 mm light curves presented in Agudo et al. (2011b)indicate variability time scale of the order of ≃
80 days.When we used this time scale to calculate the collimation(Γ θ ) and the synchrotron self-absorption ( λ SSA , obs ) con-straints, we obtained the following solution: the Γ θ = 1line crosses the 7 mm photosphere at r ≃ . j , ≃
14. This is consistent with the detectionaround this epoch of a superluminal radio element of ap-parent velocity β app ∼
13 (Agudo et al. 2011b).The close observed correspondence between thegamma-ray flares and the activity at the 7 mm wave- ocating gamma-ray flares in blazars Γ E m a x , ob s [ T e V ] rr BLR r IR R g R g R g AO 0235+164
MJD 54760L γ = 6.7e+47 erg s -1 t var,obs = 3.0 d Γ θ = Γ θ = . Γ θ = . L S S C = e r g s - ( u γ ’ / u B ’ = . ) L S S C = e r g s - ( u γ ’ / u B ’ = . ) E c oo l = M e V E c oo l = M e V λ SSA = µ m λ SSA = mm L j,min = 10 erg s -1 L j,min = 3x10 erg s -1 Figure 3.
Parameter space of r and Γ for the major flare of AO 0235+164 that peaked at MJD 54760. See Fig. 1 for detailed description. length does not necessarily indicate that the gamma raysshould be produced co-spatially with the 7 mm core. InAppendix B, we present a simple light travel time argu-ment according to which the gamma rays could still beproduced at the distance of ∼ L j , min ∼ × erg s − in this case. However, even a moderate jet magnetiza-tion implied by the SSC constraint puts into question theefficiency of the reconfinement/conical shock that is pro-posed by Agudo et al. (2011b) as the physical mechanismbehind the 7 mm core.4.4.
3C 279 at MJD 54880
3C 279 ( z = 0 . d L ≃ .
07 Gpc) produced a gamma-ray flare peaking at MJD 54880 that was extensivelystudied in Abdo et al. (2010a) and Hayashida et al.(2012). The gamma-ray flux doubling time scale can beestimated as t var , obs ≃ . L γ ≃ . × erg s − . FollowingHayashida et al. (2012), we adopt L d ≃ × erg s − , q ≃ . L syn /L X ≃ . M BH ≃ × M ⊙ , and α ≃ . r BLR ≃ .
045 pc and r IR ≃ . r and Γ forthis flare. The marginal solution is r min ≃ .
62 pc andΓ min ≃
27, which locates the gamma-ray emission firmlyoutside the BLR, and close to r IR . The predictions forthis solution are λ SSA , obs ∼ .
03 mm, u ′ γ /u ′ B ∼ . L j , min ∼ × erg s − ∼ . L d . The required jet poweris roughly half of the estimate of Meyer et al. (2011).The MOJAVE jet kinematics solution yields D ≃
24, Γ j ≃
21 (Hovatta et al. 2009), and Γ j θ j ≃ . D / Γ ≃ .
15 is fairly close to unity. Thissolution is inconsistent with both the L SSC and E cool , obs constraints. For D / Γ = 1, the combination of the L SSC and E cool , obs constraints implies that Γ θ > . − Spitzer , which wasinterpreted as a synchrotron self-absorption turnover.The latter implicated sub-pc scales ( r BLR ) for the mainsynchrotron/gamma-ray component, with an additionalemitting region located at ∼ r BLR isnot consistent with the variability time scale of days,rather it would require a variability time scale of sev-eral hours. With the relatively moderate peak gamma-ray flux of 3C 279, such short time scales could not beprobed with
Fermi /LAT. Such time scales are essentialin order to interpret the
Spitzer spectral feature in termsof synchrotron self-absorption. On the other hand, thedistance of 1 pc is fully consistent with all constraints,however, shifting the emitting region to the distance of4 pc would violate the E cool , obs constraint.Dermer et al. (2014) presented a detailed model of theradiation of blazars which was applied to the 3C 279 datafrom Hayashida et al. (2012). They concluded that thisgamma-ray flare was produced at r ∼ . − . ∼ −
30. This is still outside the BLR, but accord-ing to Figure 4 their parameter region extends well intothe Γ θ > t var , obs ∼ s = 2 . t var = 10 s in ourmodel. For 20 ≤ Γ ≤
30, we found a range of possible0
Nalewajko, Begelman & Sikora Γ E m a x , ob s [ T e V ] rr BLR r IR R g R g R g
3C 279
MJD 54880L γ = 2.6e+47 erg s -1 t var,obs = 1.5 d Γ θ = Γ θ = . Γ θ = . L S S C = e r g s - ( u γ ’ / u B ’ = . ) L S S C = e r g s - ( u γ ’ / u B ’ = . ) E c oo l = M e V E c oo l = M e V λ SSA = µ m λ SSA = mm L j,min = 3x10 erg s -1 L j,min = 10 erg s -1 Figure 4.
Parameter space of r and Γ for the major flare of 3C 279 that peaked at MJD 54880. See Fig. 1 for detailed description. The black box indicates roughly the parameter space region constrained by Dermer et al. (2014). locations r ∼ . − .
11 pc, which are closer to theblack hole than the solutions of Dermer et al. (2014). Inthat work, the location of the gamma-ray emitting regionwas constrained by calculating u BLR from SED model-ing, and comparing it with the level u BLR , expected for r < r BLR . By noting that u BLR < u
BLR , , they con-cluded that r > r BLR . However, it is difficult to providea precise estimate of r in this way, because it depends onthe uncertain shape of the u ′ BLR ( r ) function for r > r BLR .Because these authors allowed for higher values of theaccretion disk luminosity, up to L d = 10 erg s − , theyalso have higher values of r BLR ∝ L / . . PKS 1510-089 at MJD 54948
PKS 1510-089 ( z = 0 . d L ≃ .
92 Gpc), the sec-ond most active blazar of the
Fermi era (Nalewajko2013), has been monitored extensively in the X-ray,optical/NIR, and radio/mm bands. In early 2009,it produced a series of gamma-ray flares, peaking atMJD 54917 (2009 Mar 27), MJD 54948 (2009 Apr27), and MJD 54962 (2009 May 11) (Abdo et al. 2010c;D’Ammando et al. 2011). The first and the last ofthem were accompanied by sharp optical/UV flares, butnone of them had a clear X-ray counterpart. A cross-correlation analysis indicates that the optical signal couldbe delayed with respect to the gamma-ray signal by ≃
13 d, in which case the major optical flare peak-ing at MJD 54961 would be associated with the secondgamma-ray event at MJD 54948. However, in our workwe are primarily concerned with the gamma-ray emit-ting regions as they are when they produce a gamma- ray flare, and thus we use strictly simultaneous multi-wavelength data. Therefore, we will focus on the caseof MJD 54948, ignoring the optical flare that follows it.As usual, there is some ambiguity about establishing theflare parameters, and for this purpose we carefully exam-ine the results of Abdo et al. (2010c), and compare themwith our own analysis. We adopt the νL ν gamma-ray lu-minosity of L γ ≃ . × erg s − , the gamma-ray vari-ability time scale of t var , obs ≃ . L d ≃ × erg s − (Nalewajko et al. 2012b), the Compton dominance pa-rameter of L γ /L syn ≃ L X ≃ × erg s − , the X-ray spectral index of α ≃ .
3, the black hole mass of M BH ≃ × M ⊙ , thecovering factors of ξ BLR = ξ IR ≃ .
1, and the externalradiation fields radii r BLR ≃ .
07 pc and r IR ≃ . L γ /L X ≃ r min ≃ .
37 pc at Γ min ≃ λ SSA , obs ≃ . u ′ γ /u ′ B ≃
12, and L j , min ≃ . × erg s − ∼ . L d , which is slightlylower than the total jet power estimate by Meyer et al.(2011). Therefore, we suggest that the jet of PKS 1510-089 is only weakly magnetized.Abdo et al. (2010c) argued that this gamma-ray flarewas produced within the BLR, as they found that thegamma-ray and optical luminosities are related roughlylike L γ ∝ L / , which favors the ERC(BLR) mech-anism of gamma-ray production over ERC(IR). TheirSED models were calculated for Γ ≃
15, and their SSC ocating gamma-ray flares in blazars Γ E m a x , ob s [ T e V ] rr BLR r IR R g R g R g PKS 1510-089
MJD 54948L γ = 5.4e+47 erg s -1 t var,obs = 0.9 d Γ θ = Γ θ = . Γ θ = . L SS C = e r g s - ( u γ ’ / u B ’ = . ) L SS C = e r g s - ( u γ ’ / u B ’ = . ) E c oo l = M e V E c oo l = M e V λ SSA = mm L j,min = 3x10 erg s -1 L j,min = 10 erg s -1 Figure 5.
Parameter space of r and Γ for the major flare of PKS 1510-089 that peaked at MJD 54948. See Fig. 1 for detailed description. components peak significantly below L X . This wouldbe in strong disagreement with our results, if not fortwo crucial assumptions: they adopted D / Γ ≃ . t var ≃ .
25 d. When these parameters are used in ourmodel, we obtain r min ≃ .
035 pc at Γ min ≃
12, which isconsistent with their result. We note that VLBI obser-vations indicate that D / Γ ≃ . D / Γ = 1 seems to be more conservative.Abdo et al. (2010c) used the intrinsic gamma-ray opac-ity constraint to derive a limit on the Doppler factor D &
8, which we find very conservative, and certainlyweaker than the SSC constraint. They also estimatedthe jet power, and for this particular flare they obtained L j ≃ . × erg s − , about 60% of which is in the mag-netic form, and only ∼
8% in the radiative form. Thisindicates that in their model u ′ γ /u ′ B ≃ .
13, which is con-sistent with their low L SSC , but this solution is likely torequire Γ θ >
1. The energetic requirements discussedby Abdo et al. (2010c) can be significantly relaxed bybringing their model closer to equipartition.Marscher et al. (2010) presented an independent anal-ysis of the activity of PKS 1510-089 in early 2009, in-cluding more detailed VLBI analysis and optical polar-ization data. The VLBI observations at 43 GHz revealeda superluminal knot of apparent velocity 22 c , which wasprojected to pass the stationary core at MJD ∼ ∼ ∼
50 d time scale) rotation of theoptical polarization angle by ∼ ◦ . They interpretedthe gamma-ray activity of PKS 1510-089 as directly re-lated to the emergence of the superluminal radio/mmfeature, with optical polarization rotation indicating ei- ther stochastic or helical structure of the jet. This inter-pretation implies a ∼ −
20 pc distance scale for thegamma-ray flares, at which the ERC mechanism basedon IR photons is inefficient. Instead, it was proposedthat the gamma rays are produced by Comptonization ofsynchrotron radiation produced in slower outer jet lay-ers (spine-sheath models, Ghisellini et al. 2005). In Ap-pendix C.1, we show that in fact the spine-sheath modeloffers no advantage over the ERC model in explainingstrongly beamed gamma-ray emission.Chen et al. (2012) performed time-dependent SEDmodeling of the March 2009 flare of PKS 1510-089, in-vestigating three scenarios for the gamma-ray emission:ERC(BLR), ERC(IR), and SSC. The ERC(BLR) sce-nario was demonstrated to require very low values of thecovering factor, ξ BLR ∼ .
01. The other two scenariosproduce reasonable fits to the observed SEDs, each sce-nario having its own moderate problems. The problemof localization of the gamma-ray emitting region was notdirectly addressed. We note that since the ERC(BLR)model should be located at r . r BLR , it requires Γ θ ≫ .
10, thanthe ERC models. We discuss briefly the constraints onSSC models in Section 6.4.During the active state in 2009, PKS 1510-089 wasdetected in the Very High Energy (VHE) gamma-rayband, up to 300 GeV, by the H.E.S.S. observatory(H.E.S.S. Collaboration 2013). Opacity constraints dueto broad emission lines imply that the VHE emissionmust be produced outside the BLR (Barnacka et al.2
Nalewajko, Begelman & Sikora Γ E m a x , ob s [ T e V ] r PKS 1222+216
MJD 55366 r
BLR r IR R g R g R g L γ = 1.0e+48 erg s -1 t var,obs = 1.0 d Γ θ = Γ θ = . Γ θ = . L S S C = e r g s - ( u γ ’ / u B ’ = ) L S S C = e r g s - ( u γ ’ / u B ’ = . ) E c oo l = M e V E c oo l = M e V λ SSA = mm L j,min = 10 erg s -1 L j,min = 3x10 erg s -1 r m i n , V H E Figure 6.
Parameter space of r and Γ for the major flare of PKS 1222+216 that peaked at MJD 55366. See Fig. 1 for detailed description.The vertical solid gray line indicates the minimum distance for the production of VHE radiation observed by MAGIC. PKS 1222+216 at MJD 55366
PKS 1222+216 ( z = 0 . d L ≃ . ∼ r min , VHE ∼ . t var , obs ≃ L γ ≃ erg s − (Tanaka et al. 2011). FollowingTavecchio et al. (2011), we adopt L d ≃ × erg s − , ξ BLR ≃ . ξ IR ≃ . q = L γ /L syn & L X ≃ erg s − , α ≃ . r BLR ≃ .
22 pc, and r IR ≃ . q , as thesimultaneous Swift /UVOT spectra are dominated by thethermal component. The black hole mass was recentlyestimated as M BH ≃ × M ⊙ (Farina et al. 2012).Our constraints for the GeV flare of PKS 1222+216 arepresented in Figure 6. The marginal solution is found at r min ≃ .
18 pc and Γ min ≃
17. This location is within theBLR, and significantly closer to the black hole than the minimum location of the VHE emission. The predictionsfor this solution are: λ SSA , obs ≃ .
76 mm, u ′ γ /u ′ B ≃ L j , min ≃ . × erg s − ≃ . L d , whichis slightly above the estimate by Meyer et al. (2011).When we increase the Compton dominance parameterto 300, we obtain r min ≃ .
13 pc and Γ min ≃
14. Andwhen we use a shorter variability time scale of ≃ r min ≃ .
08 pc andΓ min ≃ D / Γ j ≃ .
11 (Hovatta et al. 2009), which would indi-cate that PKS 1222+216 is not a blazar. When wedecrease our Doppler-to-Lorentz factor ratio merely to D / Γ = 0 .
5, a minimum Lorentz factor of Γ min ≃
52 isrequired. Therefore, adopting D / Γ ≃ D ≃
20. However,because they fixed the jet opening angle, at different dis-tances they adopted different radii for the emitting re-gions, corresponding to different variability time scales.For the GeV emitting region located within the BLR,their model predicts a variability time scale of ≃
10 h,and for the region located outside the BLR, it predictsa variability time scale of ≃ D / Γ = 2,our constraints are entirely consistent with the model ocating gamma-ray flares in blazars Γ E m a x , ob s [ T e V ] rr BLR r IR R g R g R g PKS 0208-512
MJD 55750L γ = 1.7e+47 erg s -1 t var,obs = 2.0 d Γ θ = Γ θ = . Γ θ = . L SS C = e r g s - ( u γ ’ / u B ’ = . ) L S S C = e r g s - ( u γ ’ / u B ’ = . ) E c oo l = M e V E c oo l = M e V λ SSA = mm λ SSA = µ m L j,min = 3x10 erg s -1 L j,min = 10 erg s -1 Figure 7.
Parameter space of r and Γ for the major flare of PKS 0208-512 that peaked at MJD 55750. See Fig. 1 for detailed description. parameters adopted by Tavecchio et al. (2011) in eitherscenario.A characteristic feature of all the models ofTavecchio et al. (2011) is that the magnetic componentof the jet power is strongly dominated by the particlecomponent, which in turn is dominated by protons. How-ever, considering only the electrons, they predict that u ′ e /u ′ B ≃
6. Even if only a moderate fraction of theenergy of electrons can power the gamma-ray emission,their model is consistent with our result that u ′ γ /u ′ B . ∼ pc scale, but only for very highLorentz factors (Γ & PKS 0208-512 at MJD 55750
PKS 0208-512 ( z = 1 . d L ≃ . L γ ≃ . × erg s − , t var , obs ≃ q = L γ /L syn ≃ . L X ≃ . × erg s − , α ≃ . L d ≃ × erg s − , ξ ≃ . r BLR ≃ .
09 pc, and r IR ≃ . D / Γ ≃ .
4, here we will use D / Γ = 1 as we do forall other sources. We also adopt a black hole mass of M BH ≃ . × M ⊙ (Fan & Cao 2004).Our constraints for the gamma-ray flare in PKS 0208-512 are shown in Figure 7. The marginal solution isuncertain in this case, because the L SSC constraint is al-most tangent to the collimation constraint, nevertheless,we adopt r min ≃ . min ≃
15. With a rela-tively massive black hole, we have r min ≃ R g . Thepredictions of this solution are: λ SSA , obs ≃ .
65 mm, u ′ γ /u ′ B ≃ .
26, and L j , min ≃ . × erg s − ≃ . L d .The cooling constraint, which was not considered byChatterjee et al. (2013b), is rather strong, indicatingthat the jet cannot be strongly collimated, with Γ θ & . r BLR , and possibly close to r IR . SENSITIVITY TO ASSUMPTIONSThere are parameters in our constraints, as in everymodel of blazar emission, that may not be well deter-mined from observations. In practice, even an informedchoice of the values of these parameters is to some degreean arbitrary assumption. In this section, we will discussthe sensitivity of our constraints to three such parame-ters: the Doppler-to-Lorentz factor ratio D / Γ, the cover-ing factor of external radiation sources ξ ext (where ‘ext’stands for either BLR or IR), and the observed variabilitytime scale t var , obs .In fact, each of these three parameters can be es-timated observationally to some degree. As we men-tioned at the beginning of Section 2, D and Γ j can bededuced independently from the pc-scale jet kinemat-ics probed by VLBI radio observations (Jorstad et al.4 Nalewajko, Begelman & Sikora Γ E m a x , ob s [ T e V ] rr BLR r IR R g R g R g reference Γ E m a x , ob s [ T e V ] rr BLR r IR R g R g R g Doppler factor x2 Γ E m a x , ob s [ T e V ] rr BLR r IR R g R g R g covering factor x2 Γ E m a x , ob s [ T e V ] rr BLR r IR R g R g R g variability time scale /2 Figure 8.
Illustration of the sensitivity of our constraints to the assumptions on the Doppler-to-Lorentz factor ratio D / Γ, the externalradiation source covering factor ξ , and the observed variability time scale t var , obs . See Fig. 1 for detailed description. D / Γ j ≪
1, which is inconsistent with a blazar(PKS 1222+216, see Section 4.6 and references therein).The covering factors ξ ext can be estimated in thosesources where both the accretion disk continuum andbroad emission lines or the infrared thermal componentcan be observed directly (PKS 1222+216, see Section4.6), however, the geometry of external radiation sources(spherical — planar) is uncertain, and it has a strong ef-fect on the local energy densities (Tavecchio & Ghisellini2012; Sikora et al. 2013). Thus, for most sources weadopted a fiducial value of ξ ext ≃ .
1. The variabilitytime scale t var , obs is a direct observable, however, it isa common situation that different values are adopted inindependent studies of the same events (see Sections 4.2,4.4, 4.5).In Figure 8, we show our constraints for 4 closely re-lated fiducial models. The reference model is calculatedfor L γ = 10 erg s − , t var , obs = 1 d, q = L γ /L syn = 10, L syn /L X = 10, L d = 10 erg s − , D / Γ = 1, ξ BLR = ξ IR = 0 .
1, and M BH = 10 M ⊙ . The second model dif-fers from the reference model by having D / Γ ≃
2. Thethird model differs from the reference model by having ξ BLR = ξ IR = 0 .
2. Finally, the fourth model differs fromthe reference model by having t var , obs = 12 h.The effect of increasing the Doppler-to-Lorentz factor ratio D / Γ is to significantly relax the SSC constraint,allowing for much lower values of Γ. The collimationconstraint is somewhat stronger, but the net effect ofthese two constraints is to decrease r min . This can beunderstood from the fact that Γ( r, Γ θ ) ∝ ( D / Γ) − / andΓ( r, L SSC ) ∝ ( D / Γ) − (see Eqs. 1 and 5). The coolingconstraint is affected only slightly, since Γ( r, E cool , obs ) ∝ ( D / Γ) − / (see Eq. 9). The relation between the‘equipartition’ parameter u ′ γ /u ′ B and the SSC constraintis independent of D / Γ (see Eq. 21), therefore lines ofconstant L SSC correspond to the same values of u ′ γ /u ′ B as in the reference model. The dependence of the in-trinsic opacity constraint Γ( E max , obs ) on D / Γ (see Eq.14) is the same as that of the SSC constraint. Althoughthe gradients of E max , obs in the ( r, Γ) space are large,the value of E max , obs for the marginal solution decreasesonly slightly. The minimum jet power is also signif-icantly relaxed, especially in the region dominated bythe radiation energy density. However, in the more rel-evant region dominated by the magnetic energy density,Γ( r, L j , B , min ) ∝ ( D / Γ) − / (see Eq. 25), and the lines ofconstant L j , min are aligned roughly parallel to the linesof constant L SSC . Because of steep gradients of L j , B , min in the ( r, Γ) space, its value is very sensitive to the exactlocation within the allowed region. Finally, the depen-dence of the SSA constraint Γ( r, ν
SSA ) on D / Γ (see Eq. ocating gamma-ray flares in blazars ν SSA , obs in the ( r, Γ) space are very small.Therefore, the predicted SSA characteristic frequency forthe marginal solution will be only weakly affected. Weconclude that while the allowed parameter space regionfor higher D / Γ is significantly extended towards lowervalues of r and Γ, most parameter values correspondingto the marginal solution ( r min , Γ min ) are not very sensi-tive to the choice of D / Γ.The effect of increasing the covering factor ξ ≡ ξ BLR = ξ IR is relatively minor. Our constraints scale with ξ like:Γ( r, Γ θ ) ∝ ξ , Γ( r, L SSC ) ∝ ξ − / , Γ( r, E cool ) ∝ ξ − / ,Γ( r, ν SSA , obs ) ∝ ξ / , Γ( E max , obs ) ∝ ξ , Γ( L j ,γ, min ) ∝ ξ , and Γ( r, L j , B , min ) ∝ ξ − / . The cooling constraintis moderately relaxed, extending the allowed parameterspace region towards higher values of r . Other scalingsare very weak, and therefore we conclude that the choiceof ξ is not critical in our analysis.The effect of decreasing the observed variability timescale t var , obs is quite significant. Our constraints scalewith t var , obs like: Γ( r, Γ θ ) ∝ t − / , obs , Γ( r, L SSC ) ∝ t − / , obs , Γ( r, E cool ) ∝ t − / , obs , Γ( r, ν SSA , obs ) ∝ t − , obs ,Γ( E max , obs ) ∝ t − / (4+2 α )var , obs , Γ( L j ,γ, min ) ∝ t , andΓ( r, L j , B , min ) ∝ t − / , obs . The allowed parameter space re-gion is shifted towards smaller values of r due to relaxedcollimation constraint and tighter cooling constraint, andthe SSA is noticeably stronger, but other parameters arenot strongly affected.In summary, the uncertainty in the Doppler-to-Lorentzfactor ratio is the most significant unknown in our model,but the general conclusions that we draw for each caseanalyzed in Section 4 are securely robust. DISCUSSIONWe have demonstrated that it is possible to signifi-cantly constrain the parameter space of distance fromthe central SMBH r and Lorentz factor Γ of emitting re-gions responsible for bright gamma-ray flares of luminousblazars in the framework of the ERC mechanism, using5 direct observables: gamma-ray luminosity L γ , gamma-ray variability time scale t var , obs , synchrotron luminos-ity L syn , X-ray luminosity L X , and accretion disk lumi-nosity L d . A combination of the collimation constraint(Γ θ . L SSC . L X ), and the cool-ing constraint ( E cool , obs .
100 MeV) defines a parame-ter space region such that for each value of Γ > Γ min ,the range of r is limited to factor ∼ −
10. This isa significant improvement over previous studies, whichare typically limited to deciding between the BLR andIR regions, with r IR /r BLR ∼
30 ( e.g. , Sikora et al. 2009;Dotson et al. 2012; Brown 2013). Moreover, we evaluatethe effect on our results of the most uncertain parame-ters like Doppler-to-Lorentz factor ratio D / Γ, or coveringfactor ξ of external radiation sources. Further progressis possible with improved multiwavelength observationsof blazars, if they can be used to securely pinpoint thesynchrotron self-absorption frequency ν SSA , obs .6.1. Collimation parameter
While we have imposed an upper limit on the colli-mation parameter Γ θ .
1, the SSC and cooling con- straints provide a firm lower limit. In some analyzedcases (Figure 4), this limit is as strong as Γ θ & . θ ≃ . &
25. Such tightlower limits may be in conflict with VLBI radio observa-tions that imply significantly tighter upper limits, withΓ j θ j . . j . In any case, we can se-curely conclude that very narrow opening angles of thegamma-ray emitting regions are excluded by the SSC andcooling constraints . This makes any model of energy dis-sipation in jets which operates on a small fraction of thejet cross-section, in particular reconfinement shocks lead-ing to very narrow nozzles ( e.g. , Bromberg & Levinson2009), inconsistent with the ERC scenario. This alsochallenges models of strongly structured jets, e.g. thespine-sheath models (Ghisellini et al. 2005), or modelsinvolving strongly localized dissipation sites, e.g. mini-jets (Giannios et al. 2009), unless they can be distributeduniformly across a large fraction of the jet cross-section.While these models can still explain the most extrememodes of blazar variability, in particular the sub-hourvery high energy gamma-ray flares (Aleksi´c et al. 2011),they may not be responsible for the bulk of the gamma-ray emission of blazars.6.2. Marginal solutions
The intersection between the collimation constraintand the SSC constraint defines the marginal solution( r min , Γ min ), which sets firm lower limits on both r andΓ. One can derive the marginal solution from Equations(1) and (5): r min ≃ ct var , obs (1 + z ) (cid:20) (cid:18) g SSC g ERC (cid:19) (cid:18) L syn L X (cid:19) (cid:18) L γ ζ ( r min ) L d (cid:19)(cid:21) / × (cid:18) D Γ (cid:19) − , (26)Γ min ≃ (cid:20) (cid:18) g SSC g ERC (cid:19) (cid:18) L syn L X (cid:19) (cid:18) L γ ζ ( r min ) L d (cid:19)(cid:21) / × (cid:18) D Γ (cid:19) − / . (27)Because of the dependence of ζ on r , Eq. (26) is notexplicit, but the solutions discussed below are calculatedself-consistently. One can see that the minimum distancescale r min is proportional to the observed variability timescale t var , obs . Both r min and Γ min depend strongly onthe Doppler-to-Lorentz ratio, and they are weak func-tions of the broad-band SED shape. The marginal so-lutions for the cases analyzed in Section 4 are listedin Table 1. Even with this very small sample, we canpoint to some general trends and differences. The mini-mum distance ranges between 0 . . r min [pc] . . . r min /R g . × . M BH /M ⊙ . . × . In terms of the6 Nalewajko, Begelman & Sikora
Table 1
Parameters of our constraints, the marginal solutions (minimum distances), and the maximum distances for all blazar flares studied inSection 4. object 3C 3C AO 3C PKS PKS PKS454.3 454.3 0235+164 279 1510-089 1222+216 0208-512MJD 55520 55168 54760 54880 54948 55366 55750(a,b,c) (d,e,f) (g,h) (i,j,k) (l,m,n) (o,p) (q,r) L γ [10 erg s − ] 47 8.4 0.67 0.26 0.54 1 0.17 t var [d] 0.36 1 3 1.5 0.9 1 2 q = L γ /L syn
30 14 4 7.5 100 100 3.3 L γ /L X
300 140 24 69 1000 1000 49 L d [10 erg s − ] 6.75 6.75 0.4 0.2 0.5 5 0.8 M bh [10 M ⊙ ] 5 5 4 5 4 6 16 r min [pc] 0.16 0.17 0.65 0.62 0.37 0.18 0.2 r min /R g [10 ] 6.6 7 33 26 19 6.2 2.5 r min /r BLR min
30 19 22 27 26 17 15 λ SSA , obs [mm] 0.125 0.215 0.92 1.03 1.4 0.76 0.65 u ′ γ /u ′ B L j , min [10 erg s − ] 17 10 0.85 0.4 0.22 0.95 0.92 L j , M11 [10 erg s − ] (s) 2 2 0.2 0.9 0.3 0.8 — r max [pc] (*) 0.8 8.5 3.4 1.7 2.4 10.7 4 r max /r min r max /r IR max = 50, and in the case of 3C 279 for Γ max ≃ BLR radii, the range is 0 . . r min /r BLR .
14. In-terestingly, the range of absolute values of r min is muchnarrower than the ranges of relative values of r min /R g and r min /r BLR . Flares with relatively large r min happento be both long and faint. The minimum Lorentz factorranges between 15 . Γ min .
30. It does not show an ob-vious trend with the gamma-ray luminosity L γ or withthe time scale t var , obs .The energy density ratio of the gamma-ray radiationto the magnetic fields for the marginal solution is givenby (cf. Equation 21): u ′ γ u ′ B ≃ L γ L X g SSC L . (28)One can see that it depends only on the broad-band SEDshape. From Table 1, we find that it ranges between0 . . u ′ γ /u ′ B .
12. Values lower by about order ofmagnitude are possible for other solutions, which alsohave lower values of L SSC . The energy density ratio gen-erally increases with the Compton dominance parameter q . The gamma-ray radiation density u ′ γ closely probesthe high-energy end of the electron population, and pro-vides a lower limit on the total electron energy density u ′ e . Assuming very roughly that 3 . u ′ e /u ′ γ .
10, wecan expect that u ′ e /u ′ B ∼ . − ourconstraints are not in conflict with the equipartition con-dition u ′ e /u ′ B ≃
1, which is sometimes imposed on blazarmodels ( e.g. , B¨ottcher et al. 2009; Dermer et al. 2014).This also indicates that (sub-)pc scale jets are at most moderately magnetized.
Very high magnetization valueswould require violating the jet collimation constraint, i.e.Γ θ > L j , min = L γ (cid:20) (cid:18) g SSC g ERC (cid:19) (cid:18) L syn L X (cid:19) (cid:18) L γ ζ ( r min ) L d (cid:19)(cid:21) − / × (cid:18) D Γ (cid:19) − (cid:18) u ′ B u ′ γ (cid:19) . (29)One can see that it depends primarily on the gamma-rayluminosity, relatively weakly on the Doppler-to-Lorentzratio, and to some degree also on the broad-band SEDshape. Our estimates of the minimum jet power for theanalyzed cases (Table 1) range between 2 . × . L j , min [erg s − ] . . × , which is significantly nar-rower than the range of apparent gamma-ray luminosi-ties 1 . × . L γ [erg s − ] . . × . In terms of theaccretion disk luminosity, we find 0 . . L j , min /L d . .
25. There is a trend for this ratio to be higher forlower Compton dominance q (and higher jet magneti-zation). For 5 blazars (excluding PKS 0208-512), wecompare L j , min with the estimates L j , M11 of total jetpower by Meyer et al. (2011). We find that in manycases our lower limits significantly exceed L j , M11 , with0 . . L j , min /L j , M11 . .
5. Since our estimates do nottake into account the contributions from cold/warm elec-trons and protons, the total jet powers required to powerthe observed gamma-ray flares may be comparable to, or ocating gamma-ray flares in blazars even exceed, the accretion disk luminosity (in agreementwith Ghisellini et al. 2009), and they are certain to besignificantly higher than the estimates of Meyer et al.(2011). This indicates that the total jet powers in blazarsare strongly variable, and that the values estimated fromenergetics of the brightest gamma-ray flares (this work)can exceed by more than order of magnitude higherthe average values inferred from the low-frequency (300MHz) radio luminosity (Meyer et al. 2011).The synchrotron self-absorption (SSA) wavelengthfor the marginal solutions ranges between 0 . . λ SSA , obs [mm] . .
4. For other allowed solutions λ SSA will be somewhat larger. The SSA threshold appearsto be better correlated with the gamma-ray luminos-ity L γ than with the observed variability time scale t var , obs . For 5 events with marginal λ SSA , obs > . λ SSA , obs . . λ SSA , obs on either r or Γ, SSA can potentially provide very strong additionalconstraints on the parameters of gamma-ray emitting re-gions in blazars.
The intrinsic gamma-ray opacity does not providea significant constraint in the analyzed cases, withΓ( E max , obs = 100 GeV) .
10. In every analyzed case,the SSC constraint gives a stronger lower limit on Γ, asfirst noted by Ackermann et al. (2010).6.3.
Maximum distance scale
For a given value of the Lorentz factor Γ, the maxi-mum distance r max (Γ) is determined either by the SSCconstraint, or by the cooling constraint. Eventually, atsome Γ max there is a solution where the cooling con-straint crosses the collimation constraint, which gives anabsolute upper limit r max (Γ max ). However, the values ofΓ max can be extremely high (Γ max ≫ L d (3C 454.3and PKS 1222+216), for which the cooling constraint isrelatively weak. Therefore, the effective maximum dis-tance scale depends on how high values of Γ one wouldaccept. For a rather high Γ max = 50 (Γ max ≃
46 inthe case of 3C 279), we obtain 0 . . r max [pc] . . . . r max /r IR . .
1. The ratio of maximum to mini-mum distances is in the range 2 . . r max /r min .
59. Thedistance scale is best constrained for the flare in 3C 279,which is characterized by the lowest value of L d .If the cooling constraint can be relaxed due to theswinging motion of the emitting region, we can still placesignificant limits on the far-dissipation scenario by usingsolely the SSC constraint. In most analyzed cases, locat-ing the gamma-ray emitting regions at r ≃
10 pc wouldrequire Γ > VLBI observations indicate that jet Lorentz factors for lu-minous blazars are Γ j .
35 (Hovatta et al. 2009). Moreover,Γ j ≫
15 would contradict the blazar beaming statistics ( e.g. ,Ackermann et al. 2010). However, in this work we explicitly al-low for Γ > Γ j . Limits to the ERC model
In Section 4.3, we discussed the tension between theconstraints imposed by the ERC model and the fardissipation ( ∼
10 pc) scenarios motivated by the ob-served gamma-ray/mm-radio connection. We showedthat the SSC constraint requires very high Lorentz fac-tors, Γ &
50, in order for gamma-ray flares with variabil-ity time scale of ∼ ∼
10 pc. These solutions are also characterizedby inefficient electron cooling ( E cool , obs ≫
100 MeV),which would result in strongly asymmetric gamma-raylight curves with long flux-decay time scales, unless thereare fast variations in the local Doppler factor. Alterna-tive sources of external radiation at large distance scaleswere proposed as a way around these problems. In Ap-pendix C, we discuss two such ideas — spine-sheath mod-els ( e.g. , Marscher et al. 2010), and extended broad-lineregions (Le´on-Tavares et al. 2011).While far less popular than the ERC model, theSSC model is still being considered when mod-eling FSRQ blazars ( e.g. , Finke & Dermer 2010;Zacharias & Schlickeiser 2012; Chen et al. 2012). It wassuggested that the SSC model is most relevant for FSRQswith relatively low kinetic jet power (Meyer et al. 2012).Such models can be characterized by two conditions: L SSC = L γ and L ERC < L
SSC (one should note that inthis case L ERC may be suppressed, being strongly in theKlein-Nishina regime due to higher electron energies).With minor modifications, we can use our constraintsto identify the parameter space region where these con-ditions can be satisfied. Our SSC constraint (Eq. 5)is more generally a constraint on the luminosity ratio L SSC /L ERC , which increases systematically with decreas-ing Γ. One can extrapolate from the lines of constant L SSC shown on Figures 1 - 7 ( L SSC /L ERC = L SSC /L γ ≪
1) to the case of L SSC /L ERC = L γ /L ERC > . ∼
10 pc, SSC model may be favored over theERC model, the latter requiring extreme values of Γ.The jet collimation constraint is the same for the ERCand SSC models, as it does not depend on any kind ofluminosity. Because of the lower Lorentz factor charac-terizing the SSC model, it corresponds to very strong jetcollimation, with Γ θ . .
1, especially at larger distances.
An SSC model operating at the distance scale of
10 pc re-quires significant jet recollimation or sharp jet substruc-ture.
Other constraints are distance independent, as theyno longer depend on the distribution of external radia-tion fields. According to Eq. (21), the ‘equipartition’parameter is u ′ B /u ′ γ ≃ g SSC /q ≪
1, therefore the SSCmodel implies a strongly particle-dominated emitting re-gion (Sikora et al. 2009). The minimum required jetpower, dominated by the radiative component, is com-parable to or slightly larger than that in the ERC model.There are additional very strong constraints on theSSC model from the observed broad-band SEDs of lu-minous blazars. While being very successful in explain-ing the emission of low-luminosity HBL blazars, SSCmodels can have serious difficulties in matching the ob-served SEDs of FSRQs ( e.g. , Joshi et al. 2012). In or-der to match the characteristic frequencies of the two8
Nalewajko, Begelman & Sikora main spectral components, SSC models typically requirevery low magnetic field strength and high average elec-tron random Lorentz factor, which independently sug-gests a particle-dominated emitting region. A more de-tailed analysis of the spectral constraints on the SSCmodel is beyond the scope of this work. CONCLUSIONSWe investigated several constraints on the location r and the Lorentz factor Γ of gamma-ray emitting re-gions in the jets of luminous blazars, assuming that thegamma-ray emission is produced by the external radia-tion Comptonization (ERC) mechanism. In Section 2,we defined 4 such constraints, based on: collimation pa-rameter Γ θ , synchrotron self-Compton (SSC) luminosity L SSC , observed photon energy corresponding to efficientcooling threshold E cool , obs , and maximum photon energy E max , obs due to intrinsic gamma-ray opacity. In Section3, we also considered specific predictions for given ( r, Γ)— synchrotron self-absorption (SSA) frequency ν SSA , obs ,and minimum jet power L j , min including only contri-butions from high-energy electrons and magnetic field.In practical application, these constraints require 5 di-rect observables — gamma-ray luminosity L γ , gamma-ray variability time scale t var , obs , synchrotron luminosity L syn (or Compton dominance parameter q = L γ /L syn ),X-ray luminosity L X , and accretion disk luminosity L d —and a small number of assumptions: Doppler-to-Lorentzfactor ratio D / Γ, and covering factors of external radia-tion sources ξ BLR , ξ IR . The sensitivity of the constraintsto the assumptions was evaluated in Section 5. In Sec-tion 4, we applied these constraints to several well-knowngamma-ray flares for which extensive multiwavelengthdata are available. For each studied case, we plot theparameter space ( r, Γ) to illustrate our results (Figures1 – 7).We find that the most useful constraints on r and Γcan be derived from the combination of three conditions:Γ θ . L SSC . L X , and E cool , obs .
100 MeV. Theydefine a characteristic region in the parameter space an-chored at the marginal solution ( r min , Γ min ). In the an-alyzed cases, we found that 0 . . r min [pc] . .
65 and15 . Γ min .
30. Larger distances are possible only forhigher Lorentz factors, but eventually they are limited bythe cooling constraint. The size of the allowed parame-ter space region is particularly small for sources with lowaccretion disk luminosity L d .Our constraints challenge the far-dissipation scenariosinspired by the observed gamma-ray/mm connection. Aswe show in Appendix B, light travel time effects can eas-ily explain the temporal coincidence between gamma-rayflares and the radio/mm activity, even when the gamma-ray emitting region is located far upstream from the ra-dio/mm core. As we show in Appendix C.1, externalradiation fields cannot be substituted at large distancesby synchrotron radiation from a slower jet sheath. How-ever, as we discuss in Appendix C.2, a scenario involvingan extended broad-line region (Le´on-Tavares et al. 2011)may provide an alternative source of external radiation.The upper limit on L SSC can be translated into a lowerlimit on the collimation parameter, Γ θ & . − .
7, whichmeans that dissipation cannot be limited to very com-pact jet substructures like reconfinement nozzles, spines,minijets, etc. Our results support the idea that pc-scale blazar jets should be close to energy equipartition be-tween the particle and magnetic components.The intrinsic opacity constraint on the Lorentz fac-tor is always weaker than the SSC constraint. The syn-chrotron self-absorption constraint can significantly im-prove the determination of the parameters of gamma-rayemitting regions, if sufficient multiwavelength data canbe collected, possibly resolving the degeneracy in the val-ues of the Doppler and covering factors.We thank the anonymous referee for valuable com-ments on the manuscript, and Alan Marscher for dis-cussions. K.N. thanks the staff of the Nicolaus Coperni-cus Astronomical Center for their hospitality during thepreparation of this manuscript. This project was partlysupported by NASA through the Fermi Guest Investi-gator program, and by the Polish NCN through grantDEC-2011/01/B/ST9/04845. K.N. was supported byNASA through Einstein Postdoctoral Fellowship grantnumber PF3-140112 awarded by the Chandra X-ray Cen-ter, which is operated by the Smithsonian AstrophysicalObservatory for NASA under contract NAS8-03060.REFERENCES
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APPENDIX A. EXTERNAL RADIATION DISTRIBUTION
In this work we adopt a specific geometry of the broad-line region and the dusty torus where external radiationfields are produced (see Figure 9). Both regions are as-sumed to be symmetric with respect to the jet axis, andthey span a distance range (measured from the SMBH) of r ext , min ≤ r ext ≤ r ext , max and an equatorial angle range(measured from the accretion disk plane, perpendicularto the jet axis) of − α ext , max ≤ α ext ≤ α ext , max . Thefraction of accretion disk radiation reprocessed over unitradius d r ext is assumed to scale like ξ ( r ext ) ∝ r − β ext ext , andit is normalized so that the effective covering factor is ξ ext = R r ext , max r ext , min ξ ( r ) d r ext .This simple model has only 4 significant parameters: r ext , min , α ext , max , ξ ext , and β ext . Two of them canbe robustly constrained from standard observational ar-guments — r ext , min = r BLR(IR) ∝ L / , and ξ ext = ξ BLR(IR) ∼ . α ext , max and β ext determine the scale height and the radial stratifica-tion of the external radiation emitting region, respec-tively. These parameters are poorly understood, but For β ext ≫
1, the value of r ext , max is of minor importance.Here we adopt r ext , max = 30 r ext , min . the results are sensitive mainly to the former. In thiswork we assume that α ext , max = 45 ◦ and β ext = 4.In the case of planar geometry, with α ext , max . ◦ (Tavecchio & Ghisellini 2012), we would need to intro-duce an additional geometrical correction factor of order ∼ . − . r from the SMBH (point A).Consider an infinitesimal volume element d V = d A d r ext located within the adopted geometry at some ( r ext , α ext )(point B). The energy density of direct accretion diskradiation at point B is u d ( r ext ) ≃ L d / (4 πcr ). Theluminosity of the radiation reprocessed by this volumeelement is d L ext = ξ ( r ext ) u d ( r ext ) c d A . Its contribu-tion to the co-moving energy density of external radi-ation at point A is d u ′ ext = D d L ext / (4 πcd ), where D ext = Γ(1 + β cos θ ext ) is the Doppler factor of point Bwith respect to point A in the emitting region co-movingframe, tan θ ext = r ext cos α ext / ( r ext sin α ext − r ) gives thezenithal angle of point B with respect to point A, and d = ( r ext cos α ext ) + ( r − r ext sin α ext ) gives the dis-tance between points A and B. We also calculate thecharacteristic co-moving photon energy E ′ ext = D ext E ext ,where E ext is independent of ( r ext , α ext ). We integratefunction u ′ ext ( E ′ ext ) over the entire volume of the adoptedgeometry, and we identify its peak value u ′ ext , peak (inthe Eu ′ E sense), and the corresponding photon energy0 Nalewajko, Begelman & Sikora d ext α ext r ext α ext , max ABr θ ext r ext , min r ext , max Figure 9.
Geometry of the external radiation emitting regionadopted in this work for both the broad-line region and the dustytorus. See Appendix A for details. E ′ ext , peak .Finally, we identify simple analytical forms that canreasonably well approximate the numerically calculatedfunctions u ′ ext ( r ) and E ′ ext , peak ( r ). These forms are pre-sented in Equations (4) and (6) in Section 2.2. B. GAMMA-RAY/MM CONNECTION
As we discussed in Section 4.3, the observational con-nection between many major gamma-ray flares and ra-dio/mm activity of blazar jets has been used to arguethat gamma-ray flares should be produced close to thelocation of radio/mm cores, at the distance scale r core ≃
10 pc. Here, we use a very simple light travel time argu-ment to demonstrate that this inference is not valid. Theradio/mm activity typically consists of a t mm ∼
100 dlong radio/mm outburst and a superluminal radio/mmknot propagating downstream from the core, whose esti-mated moment of crossing the radio core coincides withthe radio/mm outburst. We approximate the superlu-minal knot by a shell of fixed thickness l mm propagatingwith the Lorentz factor Γ mm = (1 − β ) − / ≃
20. Werelate the shell thickness to the radio/mm outburst dura-tion by l mm ≃ β mm ct mm ≃ .
084 pc. We choose the timecoordinate such that at t = 0 the front of the shell crossesthe location of the radio/mm core, and thus the tail of theshell crosses the radio/mm core at t = t mm (see Figure10). A gamma-ray flare is ‘observed’ (gamma-ray pho-tons cross the radio/mm core) at t γ, obs = kt mm , where0 < k .
1. However, we assume that the gamma-rayflare was produced at r γ ≃ t γ, em = t γ, obs − ( r core − r γ ) /c .At that time, the front of the shell was located at r ≃ r γ + ( r core − r γ ) / (2Γ ) + kl mm , and its tail at r = r − l mm . We can see that r > r γ , while thecriterion for r < r γ , which means that the gamma-ray r [ p c ] t [pc] (c = 1) r core r γ t = t = k t mm t = t mm Γ mm = 3t mm = 1.5 pck = 0.5radio/mm knot γ -ray photonmm photons Figure 10.
Spacetime diagram illustrating the ambiguity of usingthe observational gamma-ray/mm connection to infer the locationof the gamma-ray flares in blazars. In this example, we adopt exag-gerated values of Γ mm and t mm to clearly distinguish the photonsfrom the radio/mm knot. Red diamonds indicate 2 events (out ofmany) consistent with the production of a gamma-ray flare withinthe radio/mm knot, but at widely different distances along the jet.See Appendix B for detailed description. emission site was within the shell, is: k < − r core − r γ β mm ct mm . (B1)As long as ( r core − r γ ) ≪ β mm ct mm , it is easy tohave the gamma-ray flare produced within the shell. Forour fiducial parameters, this criterion is k < .
87. Onecan see that temporal coincidence, and even causality,between the gamma-ray flares and the radio/mm out-burst does not imply that they are produced co-spatially. C. FAR-DISSIPATION SOLUTIONS
We showed that two of our constraints, the L SSC con-straint and the E cool , obs constraint, are likely violatedat large distance scales. Here, we consider formal re-quirements to satisfy these constraints for an arbitrary( r , Γ ). From the L SSC constraint (Equation 5), we findthe following condition: u ′ ext > .
09 erg cm − × (1 + z ) (cid:18) D Γ (cid:19) − (cid:18) Γ (cid:19) − × (cid:18) t var , obs (cid:19) − (cid:18) L γ, L syn , L X , (cid:19) . (C1)And from the E cool , obs constraint (Equation 9), we find: u ′ ext > .
11 erg cm − × (1 + z ) / (cid:18) D Γ (cid:19) − / × (cid:18) t var , obs (cid:19) − (cid:18) E ext
10 eV (cid:19) / . (C2)Both the L SSC and E cool , obs constraints can be sat-isfied for a sufficiently high external radiation den-sity. For comparison, typical co-moving energy densi-ties of BLR and IR components are of order u ′ BLR ∼
15 erg cm − ( ξ BLR / . / for r . . u ′ IR ∼ .
024 erg cm − ( ξ IR / . / for r . . ocating gamma-ray flares in blazars e.g. , Marscher et al. 2010), and an extended broad-lineregion (Le´on-Tavares et al. 2011).C.1. Spine-sheath models
In the spine-sheath model, the jet consists of a highly-relativistic spine surrounded by a mildly-relativisticsheath (Ghisellini et al. 2005). Let us denote the spineco-moving frame with O ′ , and the sheath co-movingframe with O ′′ . Consider that the gamma-ray flaresare produced in the spine by Comptonization of syn-chrotron radiation originating from the sheath, and thatthe synchrotron radiation from the spine region con-tributes significantly to the observed optical/IR emis-sion. The required energy density of the sheath radiationin O ′ is u ′ sh ≃ . − . If the sheath propagateswith Lorentz factor Γ sh in the external frame, the radia-tion energy density in O ′′ is u ′′ sh ≃ u ′ sh / (4Γ / rel = Γ sh Γ(1 − β sh β ) ≃ Γ / (2Γ sh ) is the relative Lorentzfactor of O ′′ in O ′ (the approximation is done in the limitwhere 1 ≪ Γ sh ≪ Γ). We can calculate the apparent lu-minosity of the sheath radiation for an external observeraligned with the jet spine as L sh , obs ≃ πc Γ R u ′′ sh ,where R sh ≃ θ sh r is the sheath radius parametrized bythe sheath opening angle θ sh . Putting this all together,we find: L sh , obs ≃ πc Γ (Γ sh θ sh ) r u ′ sh Γ ≃ . × erg s − × Γ (Γ sh θ sh ) (cid:18) r
10 pc (cid:19) (cid:18) Γ20 (cid:19) − . (C3)Assuming that Γ sh θ sh ∼
1, even for very moderate val-ues of Γ sh , we would have L sh , obs > L γ , and even forhigher values of Γ it is very likely that L sh , obs > L syn .The fact that L sh , obs is a strongly increasing function ofΓ sh means that the spine-sheath model actually offers noadvantage in providing soft photons for Comptonizationto the observed gamma-ray emission over static sourcesof external radiation.C.2. Extended broad-line region
Here we estimate a possible contribution to the ex-ternal radiation energy density from a broad-line re-gion extended along the jet to supra-pc distance scales(Le´on-Tavares et al. 2011). For the purpose of first-orderestimates, we will approximate the extended BLR as asphere of radius R BLR ∗ centered on the jet at the dis-tance scale r BLR ∗ ∼
10 pc ≫ R BLR ∗ . Let L BLR ∗ be the luminosity of emission lines produced in this region, nottaking into account any lines produced elsewhere. Theseemission lines are expected to be significantly narrowerfrom conventional broad emission lines, and there is lit-tle observational evidence for their existence in the lineprofiles of radio-loud quasars. Therefore, we will adoptan upper limit of L BLR ∗ . erg s − , so that it consti-tutes only a small fraction of the total luminosity of broademission lines. This luminosity will contribute the exter-nal radiation density u ′ BLR ∗ ≃ Γ L BLR ∗ / (3 πcR ∗ ) atthe center of the sphere in the co-moving frame of thegamma-ray emitting region. We consider two sources ofradiation illuminating the extended BLR — 1) the directaccretion disk radiation of luminosity L d ; and 2) the jetsynchrotron radiation produced at an arbitrary distancescale r syn < r BLR ∗ and of apparent luminosity L syn . Weconsider two types of covering factors — the geometricfactor ξ geom , and the intrinsic factor ξ int — such that L BLR ∗ = ξ int ξ geom L d(syn) .In case 1), assuming that the accretion disk radia-tion is roughly isotropic, the geometric factor is givenby ξ geom ≃ ( R BLR ∗ / r BLR ∗ ) , hence: u ′ BLR ∗ ≃ ξ int Γ L d πcr ∗ ≃ . × − erg cm − × L d , (cid:18) ξ int . (cid:19) (cid:18) Γ20 (cid:19) (cid:18) r BLR ∗
10 pc (cid:19) − . (C4)This value is more than 2 orders of magnitude too smallto satisfy the L SSC and E cool , obs constraints for typicalparameter values.In case 2), the jet synchrotron radiation is stronglybeamed, and effectively it can illuminate a region of ra-dius R BLR ∗ ≃ ( r BLR ∗ − r syn ) / Γ. Assuming that all of theilluminated region is filled with the gas, we adopt ξ geom ≃ Since we normalize the synchrotron luminosity to L syn ≃ erg s − , for consistency we adopt ξ int ∼ − in order to have L BLR ∗ ≃ ξ int L syn ∼ erg s − : u ′ BLR ∗ ≃ ξ int Γ L syn πc ( r BLR ∗ − r syn ) ≃ .
24 erg cm − × (C5) L syn , (cid:18) ξ int − (cid:19) (cid:18) Γ20 (cid:19) (cid:18) r BLR ∗ − r syn (cid:19) − . In principle, this mechanism can provide enough of ex-ternal radiation density to satisfy the L SSC and E cool , obs .However, we suggest that more observational support forthe existence of such emission lines is necessary. The synchrotron radiation beam should extend significantlybeyond the jet boundaries (otherwise ξ geom < j θ j ≪
1, or that the jet accelerates significantly be-tween r syn and r BLR ∗∗