CONSTRAINING THE LOCATION OF GAMMA-RAY FLARES IN LUMINOUS BLAZARS

We assume that the emitting region has a characteristic size R, which is related to the co-moving variability timescale via $R \simeq ct_{\rm var}^{\prime }$. The variability timescale scales like $t^{\prime }_{\rm var} = \mathcal {D}t_{\rm var,obs}/(1+z)$, where z is the blazar redshift. The most reliable estimate of the observed variability timescale t_{var, obs} is the flux-doubling timescale measured 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 parameter Γθ. We can now write the source Lorentz factor as a function of Γθ:

$$\begin{equation} \Gamma (r,\Gamma \theta) \simeq \left(\frac{\mathcal {D}}{\Gamma }\right)^{-1/2}\left[\frac{(1+z)(\Gamma \theta)r}{ct_{\rm var,obs}}\right]^{1/2} \,. \end{equation} \tag{ 1 }$$

There are strong observational and theoretical indications that Γ_jθ_j < 1 for blazar jets. A jet opening angle on a scale of tens of parsecs was measured in a substantial sample of blazars using VLBI imaging, with the typical result of Γ_jθ_j ∼ 0.1–0.2 (Pushkarev et al. 2009; Clausen-Brown et al. 2013). Numerical simulations of acceleration and collimation of external-pressure-supported relativistic jets also find that after the acceleration is complete, Γ_jθ_j ≲ 1 (Komissarov et al. 2009; Tchekhovskoy et al. 2010). However, the relation between the collimation parameter of the jet Γ_jθ_j and the collimation parameter of the emitting region Γθ is unclear. On one hand, we expect that θ ⩽ θ_j, but on the other hand, it is possible that Γ > Γ_j. Therefore, here we adopt a relatively conservative collimation constraint, defined as Γθ ≲ 1.

We assume that the gamma-ray emission is produced by Comptonization of external radiation (ERC) by a population of ultrarelativistic electrons, and that the apparent gamma-ray luminosity L_γ (hereafter understood as the peak of νL_{γ, ν} spectral energy distribution (SED), as opposed to the bolometric luminosity L_{γ, bol} = ∫L_{γ, ν}dν) measured by Fermi/LAT represents L_ERC, the peak luminosity of the ERC component. The same electrons produce the synchrotron and the SSC components, of which at least the former should contribute to the observed SEDs as indicated by fast optical/IR flares often correlated with the gamma rays. The three luminosities—L_ERC, L_syn, and L_SSC—can be related to the co-moving energy densities of external radiation $u_{\rm ext}^{\prime }$, magnetic fields $u_{\rm B}^{\prime } = B^{\prime 2}/(8\pi)$, and synchrotron radiation $u_{\rm syn}^{\prime } \simeq L_{\rm syn}/(4\pi c\mathcal {D}^4R^2)$, respectively. On one hand, we have $L_{\rm SSC}/L_{\rm syn} \simeq g_{\rm SSC}(u_{\rm syn}^{\prime }/u_{\rm B}^{\prime })$, where g_SSC = (L_SSC/L_syn)/(L_{SSC, bol}/L_{syn, bol}) ≃ 3/4 is a bolometric correction factor (mainly due to spectral shape and source geometry). On the other hand, we can define a Compton dominance parameter

$$\begin{equation} q = \frac{L_\gamma }{L_{\rm syn}} \simeq g_{\rm ERC}\left(\frac{\mathcal {D}}{\Gamma }\right)^2\left(\frac{u_{\rm ext}^{\prime }}{u_{\rm B}^{\prime }}\right)\,, \end{equation} \tag{ 2 }$$

where g_ERC = (L_ERC/L_syn)/(L_{ERC, bol}/L_{syn, bol}) ≃ 1/2 is a bolometric correction factor (mainly due to Klein–Nishina effects), and the $(\mathcal {D}/\Gamma)^2$ factor reflects the beaming 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 luminosity L_d via

$$\begin{equation} u_{\rm ext}^{\prime } \simeq \frac{\zeta (r)\Gamma ^2L_{\rm d}}{3\pi cr^2}\,. \end{equation} \tag{ 3 }$$

Here, ζ(r) is a function that describes the composition of external radiation fields, including contributions from the broad-line region (BLR), the dusty torus producing infrared emission (IR), and the direct accretion disk radiation:

$$\begin{equation} \zeta (r) \simeq \frac{0.4\xi _{\rm BLR}(r/r_{\rm BLR})^2}{1+(r/r_{\rm BLR})^4} +\frac{0.4\xi _{\rm IR}(r/r_{\rm IR})^2}{1+(r/r_{\rm IR})^4} +\frac{0.21R_{\rm g}}{r}\,, \end{equation} \tag{ 4 }$$

where ξ_BLR is the covering factor of the BLR of characteristic radius r_BLR, ξ_IR and r_IR are the analogous parameters of the dusty torus, and R_g is the gravitational radius of the SMBH (we explain the origin of this function in Appendix A). In this work, we adopt the following scaling laws: $r_{\rm BLR} \simeq 0.1\,L_{\rm d,46}^{1/2}\;{\rm pc}$, and $r_{\rm IR} \simeq 2.5\,L_{\rm d,46}^{1/2}\;{\rm pc}$, where L_{d, 46} = L_d/(10⁴⁶ erg s⁻¹) (Sikora et al. 2009). Putting the above relations together, we obtain a constraint on Γ:

$$\begin{eqnarray} \Gamma (r,L_{\rm SSC}) &\simeq & \left[3\left(\frac{g_{\rm SSC}}{g_{\rm ERC}}\right)\left(\frac{L_{\rm syn}}{L_{\rm SSC}}\right)\left(\frac{L_\gamma }{\zeta (r)L_{\rm d}}\right)\right]^{1/8} \nonumber \\ \ms && \times \left(\frac{\mathcal {D}}{\Gamma }\right)^{-1}\left[\frac{(1+z)r}{2ct_{\rm var,obs}}\right]^{1/4}\,. \end{eqnarray} \tag{ 5 }$$

The SSC component in the SEDs of luminous blazars peaks at the observed photon energy of $E_{\rm SSC,obs} \simeq 20\;{\rm neV}\times \mathcal {D}B_0^{\prime } \gamma _{\rm peak}^4 /(1+z)$, where $B_0^{\prime } = B^{\prime }/(1\;{\rm G})$ and γ_peak is the characteristic random Lorentz factor of electrons contributing to the SED peaks. We can estimate γ_peak from the observed photon energy of the SED peak of the ERC component $E_{\rm ERC,obs} \simeq \mathcal {D}\Gamma \gamma _{\rm peak}^2E_{\rm ext}(r)/(1+z)$, where E_ext(r) is the energy of external radiation photons. In order to account for the transition between the BLR and IR external radiation fields, we use the following approximation (see Appendix A):

$$\begin{equation} E_{\rm ext}(r) \simeq \frac{E_{\rm BLR}}{1+(r/r_{\rm BLR})^3} + \frac{E_{\rm IR}}{1+(r/r_{\rm IR})^3}\,, \end{equation} \tag{ 6 }$$

where E_BLR ≃ 10 eV and E_IR ≃ 0.3 eV. The magnetic field strength can be found from Equations (2) and (3):

$$\begin{equation} B^{\prime } \simeq \frac{\mathcal {D}}{r}\left[\frac{8g_{\rm ERC}\zeta (r)L_{\rm d}}{3qc}\right]^{1/2}\,. \end{equation} \tag{ 7 }$$

Combining the above formulas, we find:

$$\begin{eqnarray} E_{\rm SSC,obs} &\simeq & \frac{20\;{\rm neV}}{r}\frac{(1+z)}{\Gamma ^2}\left[\frac{E_{\rm ERC,obs}}{E_{\rm ext}(r)}\right]^2 \nonumber \\ &&\times \left[\frac{8g_{\rm ERC}\zeta (r)L_{\rm d}}{3qc}\right]^{1/2}\,. \end{eqnarray} \tag{ 8 }$$

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 = 0.1, L_d = 3 × 10⁴⁵ erg s⁻¹, and q = 10, we find E_{SSC, obs} ≃ 6(1 + z) keV. Because SSC spectral components are very broad, in most cases they should peak around, or contribute significantly to, the soft/hard X-ray band. Some blazars show spectral softening in the soft X-ray part of their SEDs, which was interpreted as a signature of the SSC component (Bonnoli et al. 2011). However, in many sources the observed X-ray emission is harder than it would be if it were dominated by the SSC component (Sikora et al. 2009). Also, the observed X-ray variability is usually not well correlated with variability in the gamma-ray and optical bands (Hayashida et al. 2012). In the case that the SSC component dominates the X-ray emission, we would expect that X-ray variability should be stronger than the optical/IR variability. For example, in a simple scenario of varying number of energetic electrons at constant magnetic field we have $L_{\rm SSC} \propto L_{\rm syn}^2$. As this is not the case for luminous blazars, we can only use the observed X-ray luminosity as an upper limit for the SSC luminosity (Ackermann et al. 2010). Therefore, our SSC constraint is defined as L_SSC ≲ L_X.

Rapid gamma-ray variability of blazars, with roughly time-symmetric light curve peaks, and tight energetic requirements for the brightest observed gamma-ray flares, indicate very efficient cooling of the underlying ultrarelativistic electrons. The radiative cooling of electrons in luminous blazars is dominated by the ERC process with a cooling timescale $t_{\rm cool}^{\prime }(\gamma) \simeq 3m_{\rm e}c/(4\sigma _{\rm T}\gamma u_{\rm ext}^{\prime })$, where γ is the electron random Lorentz factor. In general, $t_{\rm cool}^{\prime }(\gamma)$ should be compared with the variability timescale $t_{\rm var}^{\prime }$ (which is associated with the observed flux doubling timescale; see Section 2.1), and adiabatic cooling timescale $t_{\rm ad}^{\prime }$.⁶ Observations of roughly time-symmetric flares indicate that the cooling timescales do not exceed the observed flux decaying timescales, i.e., that $t_{\rm cool}^{\prime }(\gamma) \lesssim t_{\rm var}^{\prime }$.⁷ We calculate a characteristic electron Lorentz factor γ_cool such that $t_{\rm cool}^{\prime }(\gamma _{\rm cool}) \simeq t_{\rm var}^{\prime }$,⁸ and a corresponding observed ERC photon energy $E_{\rm cool,obs} \simeq \mathcal {D}\Gamma \gamma _{\rm cool}^2E_{\rm ext}(r)/(1+z)$. Taking the above together, we obtain the following constraint on Γ:

$$\begin{eqnarray} \Gamma (r,E_{\rm cool,obs}) &\simeq & \left(\frac{\mathcal {D}}{\Gamma }\right)^{-1/4}\left[\frac{9\pi m_{\rm e}c^2r^2}{4\sigma _{\rm T}\zeta (r)L_{\rm d}t_{\rm var,obs}}\right]^{1/2} \nonumber \\ &&\times \left[\frac{(1+z)E_{\rm ext}(r)}{E_{\rm cool,obs}}\right]^{1/4}\,. \end{eqnarray} \tag{ 9 }$$

Since the gamma-ray light curves based on the Fermi/LAT data are typically calculated for photon energies E > 100 MeV, our cooling constraint is defined as E_{cool, obs} ≲ 100 MeV.

Alternatively, the cooling timescale as a function of photon energy potentially can be estimated directly from gamma-ray observations, but this is only feasible for the very brightest events (Dotson et al. 2012).

The maximum observed gamma-ray photon energy E_{max, obs} is constrained at least by the pair-production absorption process due to soft radiation produced in the same emitting region (e.g., Dondi & Ghisellini 1995). The peak cross section for the pair-production process is σ_γγ ≃ σ_T/5 for soft photons of co-moving energy $E_{\rm soft}^{\prime } \simeq 3.6(m_{\rm e}c^2)^2/E_{\rm max}^{\prime }$. In the observer frame, the soft photon energy is

$$\begin{eqnarray} E_{\rm soft,obs} &\simeq & \frac{3.6(m_{\rm e}c^2)^2\mathcal {D}^2}{(1+z)^2E_{\rm max,obs}} \nonumber \\ &\simeq & \frac{38\;{\rm keV}}{(1+z)^2}\left(\frac{E_{\rm max,obs}}{10\;{\rm GeV}}\right)^{-1}\left(\frac{\mathcal {D}}{20}\right)^2\,. \end{eqnarray} \tag{ 10 }$$

The optical depth for gamma-ray photons is:

$$\begin{equation} \tau _{\rm \gamma \gamma } = \sigma _{\rm \gamma \gamma }n_{\rm soft}^{\prime }R \simeq \frac{(1+z)^2\sigma _{\rm T}L_{\rm soft}E_{\rm max,obs}}{72\pi (m_{\rm e}c^2)^2c^2\mathcal {D}^6t_{\rm var,obs}}\,. \end{equation} \tag{ 11 }$$

As the observed soft photon energy E_{soft, obs} may fall outside any observed energy range, we relate the target soft radiation luminosity to the observed X-ray luminosity via a spectral index α such that

$$\begin{equation} L_{\rm soft} = L_{\rm X}\left(\frac{E_{\rm soft,obs}}{E_{\rm X}}\right)^{1-\alpha } \simeq \frac{[3.6(m_{\rm e}c^2)^2]^{1-\alpha }\mathcal {D}^{2-2\alpha }L_{\rm X}}{(1+z)^{2-2\alpha }E_{\rm X}^{1-\alpha }E_{\rm max,obs}^{1-\alpha }}\,. \end{equation} \tag{ 12 }$$

Substituting this into Equation (11), we obtain:

$$\begin{equation} \tau _{\rm \gamma \gamma } \simeq \frac{(1+z)^{2\alpha }\sigma _{\rm T}L_{\rm X}E_{\rm max,obs}^{\alpha }}{20\pi [3.6(m_{\rm e}c^2)^2]^{\alpha }c^2\mathcal {D}^{4+2\alpha }E_{\rm X}^{1-\alpha }t_{\rm var,obs}}\,. \end{equation} \tag{ 13 }$$

For gamma-ray observations of blazars, it is typical to associate E_{max, obs} with τ_γγ ≃ 1. This leads to the following constraint on Γ:

$$\begin{eqnarray} \Gamma (r,E_{\rm max,obs}) &\simeq & \left\lbrace \frac{(1+z)^{2\alpha }\sigma _{\rm T}L_{\rm X}E_{\rm max,obs}^{\alpha }}{20\pi [3.6(m_{\rm e}c^2)^2]^{\alpha }c^2E_{\rm X}^{1-\alpha }t_{\rm var,obs}}\right\rbrace ^{\frac{1}{4+2\alpha }} \nonumber \\ && \times \left(\frac{\mathcal {D}}{\Gamma }\right)^{-1}\,. \end{eqnarray} \tag{ 14 }$$

In Section 4, we will demonstrate that the internal gamma-ray opacity constraint is relatively weak compared to the SSC constraint.

An additional potential source of gamma-ray opacity is from the BEL. To the first order of approximation, this would affect photons of observed energy:

$$\begin{equation} E_{\rm max,BLR,obs} \simeq \frac{3.6(m_{\rm e}c^2)^2}{(1+z)E_{\rm BLR}} \simeq \frac{94\;{\rm GeV}}{(1+z)}\left(\frac{E_{\rm BLR}}{10\;{\rm eV}}\right)^{-1}\,, \end{equation} \tag{ 15 }$$

with the peak optical depth of

$$\begin{eqnarray} &&\tau _{\rm \gamma \gamma,BLR}(r) \simeq \frac{\xi _{\rm BLR}\sigma _{\rm T}L_{\rm d}(r_{\rm BLR}-r)}{20\pi c r_{\rm BLR}^2E_{\rm BLR}} \simeq 71\left(\frac{\xi _{\rm BLR}}{0.1}\right) \nonumber \\ && \times \left(\frac{L_{\rm d}}{10^{46}\;{\rm erg\,s^{-1}}}\right)^{1/2}\left(\frac{r_{\rm BLR}-r}{r_{\rm BLR}}\right)\left(\frac{E_{\rm BLR}}{10\;{\rm eV}}\right)^{-1} \end{eqnarray} \tag{ 16 }$$

(again using the scaling $r_{\rm BLR} \simeq 0.1\,L_{\rm d,46}^{1/2}\;{\rm pc}$). The high value of the peak optical depth indicates that absorption should already become noticeable at the threshold observed energy of (m_ec²)²/[(1 + z)E_BLR] ≃ 26 GeV/(1 + z)/(E_BLR/10 eV).⁹ The actual strength of the BLR absorption features depends significantly on the BLR geometry, and determining it requires detailed calculations (e.g., Donea & Protheroe 2003; Reimer 2007; Tavecchio & Ghisellini 2012). We will briefly comment on the expected significance of the BLR absorption in those cases from Section 4, which allow for the emitting region to be located within r_BLR.

Synchrotron radiation is subject to SSA process, which can produce a sharp spectral break. This is a powerful probe of the intrinsic 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:

$$\begin{equation} \nu _{\rm SSA}^{\prime } \simeq \frac{1}{3}\left(\frac{eB^{\prime }}{m_{\rm e}^3c}\right)^{1/7}\frac{L_{\rm syn}^{\prime 2/7}}{R^{4/7}}\,, \end{equation} \tag{ 17 }$$

where we approximated the synchrotron luminosity at $\nu _{\rm SSA}^{\prime }$ with the synchrotron energy distribution peak luminosity $L_{\rm syn}^{\prime }$ (i.e., we assumed a flat synchrotron SED in the mid-IR/millimeter band; in any case, $\nu _{\rm SSA}^{\prime }$ depends only weakly on the spectral index of unabsorbed synchrotron emission). Substituting relevant relations from previous sections, we find a constraint on Γ:

$$\begin{eqnarray} \Gamma (r,\nu _{\rm SSA,obs}) &\simeq & \left[\frac{8g_{\rm ERC}e^2\zeta (r)L_{\rm d}L_\gamma ^4}{3^{15}q^5m_{\rm e}^6c^{11}(1+z)^6\nu _{\rm SSA,obs}^{14}t_{\rm var,obs}^8r^2}\right]^{1/8} \nonumber \\ &&\times \left(\frac{\mathcal {D}}{\Gamma }\right)^{-1}\,. \end{eqnarray} \tag{ 18 }$$

In luminous blazars, the SSA spectral break is typically observed in the sub-millimeter/radio band. As the synchrotron radiation observed in this band probes lower electron energies than the ∼GeV gamma-ray radiation, a connection between these bands should be verified by studying variability correlations. These are very challenging observations, and for most cases studied in Section 4 such data are not available. Therefore, in this work, the SSA constraint is limited to provide a prediction of what ν_{SSA, obs} should be for each studied case.

We can constrain the energy content of blazar jets underlying the observed gamma-ray flares by estimating two of its essential ingredients: the radiation energy density dominated by the gamma rays $u_\gamma ^{\prime }$, and the magnetic energy density $u_{\rm B}^{\prime }$. Because the production of gamma-ray radiation through the ERC process is very efficient, $u_\gamma ^{\prime }$ closely probes the high-energy end of the electron energy distribution. Additional jet energy may be carried by cold/warm electrons and protons, the contribution of which is very uncertain. For example, the number of cold electrons can be constrained by modeling the broadband SEDs, but the low-energy electron distribution index is usually one of the most uncertain parameters. On the other hand, the energy content of protons in blazar jets can be constrained only indirectly, by combining arguments such as interpretation of (hard) X-ray spectra of luminous blazars, and energetic coupling between the protons and electrons (Sikora 2011). Rather than introducing extra parameters with highly uncertain values, we choose to discuss a firm lower limit L_{j, min} on the jet power required to produce the observed gamma-ray flares of blazars together with their synchrotron and SSC counterparts.

The radiation energy density can be written as:

$$\begin{equation} u^{\prime }_\gamma \simeq \frac{L_\gamma }{4\pi c\mathcal {D}^4R^2} \simeq \left(\frac{\mathcal {D}}{\Gamma }\right)^{-6}\frac{(1+z)^2L_\gamma }{4\pi c^3\Gamma ^6t_{\rm var,obs}^2}\,. \end{equation} \tag{ 19 }$$

The magnetic energy density $u_{\rm B}^{\prime }$ can be derived from the synchrotron luminosity L_syn, which is related to the gamma-ray luminosity L_γ through the Compton dominance parameter q = L_γ/L_syn:

$$\begin{equation} u_{\rm B}^{\prime } \simeq \left(\frac{\mathcal {D}}{\Gamma }\right)^2\frac{g_{\rm ERC}u_{\rm ext}^{\prime }}{q} \simeq \left(\frac{\mathcal {D}}{\Gamma }\right)^2\frac{g_{\rm ERC}\zeta (r)\Gamma ^2L_{\rm d}}{3\pi cqr^2}\,. \end{equation} \tag{ 20 }$$

Instead of using these two energy densities separately, we will analyze their more useful combinations: their ratio and their sum. The ratio of the two energy densities is a measure of energy equipartition between the magnetic fields and the ultra-relativistic electrons. One can show that (cf. Sikora et al. 2009):

$$\begin{equation} \frac{u_\gamma ^{\prime }}{u_{\rm B}^{\prime }} \simeq \frac{L_\gamma L_{\rm SSC}}{g_{\rm SSC}L_{\rm syn}^2}\,, \end{equation} \tag{ 21 }$$

therefore, this energy density ratio is proportional to L_SSC, and it follows the same dependence on r and Γ. The sum of the two energy densities constitutes a lower limit on the jet energy density $u_{\rm j,min}^{\prime } = u_\gamma ^{\prime } \,{+}\, u_{\rm B}^{\prime }$. The corresponding minimum jet power is given by $L_{\rm j,min} \simeq \pi c\Gamma ^2R^2u^{\prime }_{\rm j,min}$. Therefore, we can write L_{j, min} = L_{j, γ, min} + L_{j, B, min}, where

$$\begin{equation} L_{\rm j,\gamma,min} = \left(\frac{\mathcal {D}}{\Gamma }\right)^{-4}\frac{L_\gamma }{4\Gamma ^2}\,, \end{equation} \tag{ 22 }$$

$$\begin{equation} L_{\rm j,B,min} = \left(\frac{\mathcal {D}}{\Gamma }\right)^4\frac{g_{\rm ERC}\Gamma ^6\zeta (r)L_{\rm d}}{3q}\left[\frac{ct_{\rm var,obs}}{r(1+z)}\right]^2\,. \end{equation} \tag{ 23 }$$

The dependence of the magnetic jet power on Γ is much steeper than for the radiative jet power. Thus, we can derive approximate constraints on Γ in two limits. For $u_\gamma ^{\prime } \gg u_{\rm B}^{\prime }$, we find

$$\begin{equation} \Gamma (L_{\rm j,\gamma,min}) = \left(\frac{\mathcal {D}}{\Gamma }\right)^{-2}\left(\frac{L_\gamma }{4L_{\rm j,\gamma,min}}\right)^{1/2}\,; \end{equation} \tag{ 24 }$$

and for $u_\gamma ^{\prime } \ll u_{\rm B}^{\prime }$, we find

$$\begin{eqnarray} \Gamma (r,L_{\rm j,B,min}) &=& \left(\frac{\mathcal {D}}{\Gamma }\right)^{-2/3}\left(\frac{3q L_{\rm j,B,min}}{g_{\rm ERC}\zeta (r)L_{\rm d}}\right)^{1/6} \nonumber \\ &&\times \left[\frac{r(1+z)}{ct_{\rm var,obs}}\right]^{1/3} \,. \end{eqnarray} \tag{ 25 }$$

In Section 4, we will investigate the values of $u_\gamma ^{\prime }/u_{\rm B}^{\prime }$ and L_{j, min} for individual blazar flares. Again, we stress that contributions from cold/warm electrons and protons should be included to obtain total jet energies.

3C 454.3 (z = 0.859, d_L ≃ 5.49 Gpc) provided us with 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} ≃ 2.1 × 10⁵⁰ erg s⁻¹ (Abdo et al. 2011). We convert the bolometric peak luminosity L_{γ, bol} into the peak νL_ν luminosity L_γ, using a bolometric correction factor g_{γ, bol} = L_{γ, bol}/L_γ ∼ 4.5 calculated from the best-fit spectral model (power-law with exponential cutoff), resulting in L_γ ≃ 4.7 × 10⁴⁹ erg s⁻¹. The flare temporal template fitted by Abdo et al. (2011) has a flux doubling timescale of t_{var, obs} ≃ 8.7 h ≃ 3.13 × 10⁴ s. Vercellone et al. (2011) showed that this gamma-ray flare was accompanied by simultaneous outbursts, smaller in amplitude by a factor ∼3, in soft X-ray, optical, and millimeter bands. They compiled an SED from which we can estimate the simultaneous luminosity ratios q = L_γ/L_syn ≃ 30, L_syn/L_X ≃ 10. These ratios are used to derive the simultaneous soft X-ray luminosity L_X ≃ 1.6 × 10⁴⁷ erg s⁻¹. We can also estimate the spectral index of the X-ray part of the spectrum as α ≃ 0.65. The bolometric accretion disk luminosity is taken as L_d ≃ 6.75 × 10⁴⁶ erg s⁻¹ (Bonnoli et al. 2011), from which we find the characteristic radii of external radiation components r_BLR ≃ 0.26 pc and r_IR ≃ 6.5 pc. The black hole mass of 3C 454.3 is uncertain; here we adopt the value of M_BH ∼ 5 × 10⁸ M_☉ after Bonnoli et al. (2011).

In Figure 1, we plot the constraints on r and Γ corresponding to fixed values of Γθ, L_SSC, E_{cool, obs}, λ_{SSA, obs} and E_{max, obs}, as well as the energetics parameters $u_\gamma ^{\prime }/u_{\rm B}^{\prime }$ and L_{j, min}. We assumed here that ξ_BLR ≃ ξ_IR ≃ 0.1. The yellow area is defined by the following three conditions: Γθ < 1, L_SSC < L_X, and E_{cool, obs} < 100 MeV. The intersection of the first two of these constraints gives the marginal solution—the minimum Lorentz factor Γ_min ≃ 30 and the minimum distance scale r_min ≃ 0.16 pc. For (r_min, Γ_min), other constraints yield the following predictions: λ_{SSA, obs} ≃ 125 μm, E_{max, obs} ≳ 10 TeV, $u_\gamma ^{\prime }/u_{\rm B}^{\prime } \simeq 3.3$, and L_{j, min} ≃ 1.7 × 10⁴⁶ erg s⁻¹ ≃ 0.25 L_d. On the other hand, in the IR region (r ∼ r_IR), the SSC constraint is much stronger and hence there are no solutions with Γ < 50. Therefore, in this case, the dissipation region is clearly constrained to be located not far from r_BLR. The minimum required jet power is one order of magnitude higher than the kinetic jet power estimated by Meyer et al. (2011).

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VLBI measurements of the jet of 3C 454.3 yield Γ_j ≃ 20, $\mathcal {D} \simeq 33$ (Hovatta et al. 2009), and Γ_jθ_j ≃ 0.3 (Pushkarev et al. 2009). Adopting $\mathcal {D}/\Gamma _{\rm j} \simeq 1.67$ would shift the marginal solution to r_min ≃ 0.09 pc and Γ_min ≃ 18. The VLBI-derived solution of r ≃ 0.34 pc and Γ ≃ 20 would be consistent with our E_{cool, obs} constraint, and marginally consistent with our L_SSC constraint. On the other hand, for $\mathcal {D}/\Gamma = 1$, the SSC constraint also implies that jet collimation parameter is Γθ > 0.5.

Abdo et al. (2011) estimated the minimum Doppler factor of the emitting region responsible for this flare as $\mathcal {D}_{\rm min} \simeq 16$, using the gamma-ray opacity constraint for the maximum observed photon energy of E_{max, obs} = 31 GeV. Our opacity constraint for the same E_{max, obs} yields $\Gamma _{\rm min} = \mathcal {D}_{\rm min} \simeq 13$. The main reason for this discrepancy is that we use the 3.6 factor in Equation (10), which is neglected in numerous studies. We point out that the SSC constraint is stronger than the opacity constraint (see Ackermann et al. 2010). We also note that our minimum distance scale is compatible with the estimate of r_min ≃ 0.14 pc obtained by calculating gamma-ray opacity due to the broad-line photons (Abdo et al. 2011).

The SSA break is predicted to fall in the far-IR range, both at the BLR and IR distance scales. 3C 454.3 was observed by Herschel PACS and SPIRE instruments during and after the peak of this gamma-ray flare (Wehrle et al. 2012). While the period of the highest gamma-ray state was sparsely covered in the far-IR band, a very good correlation between the 160 μm data and the Fermi/LAT gamma rays was found. Such a correlation implies that the gamma-ray producing region is transparent to SSA, i.e., that λ_{SSA, obs} ≳ 160 μm. Such a condition can be easily satisfied, together with our collimation and SSC constraints, even at BLR distance scales. However, Wehrle et al. (2012) also showed that 1.3 mm data from SMA, of much better sampling rate, correlate well with the gamma rays. This is very difficult to explain in a one-zone model—the 1.3 mm SSA line satisfies all three constraints only at r ≃ 27 pc and Γ ≃ 400. The most reasonable way to accommodate this observation is to consider a different variability timescale of the 1.3 mm emission. Indeed, data presented in Figure 10 of Wehrle et al. (2012) indicate that the flux-doubling timescale corresponding to the fastest observed increase of the 1.3 mm flux is t_{var, mm} ≃ 7.5 d. Adopting this variability timescale, the 1.3 mm photosphere can be located already at r_mm ≃ 5 pc and Γ = 38, which are much more reasonable parameters. Therefore, we need to consider an extended, possibly structured emitting region for the gamma rays observed during this event, with the rapidly flaring component produced at sub-parsec scales, and a more slowly varying component correlated with the 1.3 mm emission at supra-parsec scales. This scenario is similar to the one proposed for PKS 1510-089 by Nalewajko et al. (2012b).

A previous flare of 3C 454.3, peaking at MJD 55168 (2009 December 3), also attracted considerable interest (e.g., Pacciani et al. 2010; Ackermann et al. 2010; Bonnoli et al. 2011). The apparent peak bolometric gamma-ray luminosity was L_{γ, bol} ≃ 3.8 × 10⁴⁹ erg s⁻¹ (Ackermann et al. 2010), which corresponds to the νL_ν luminosity L_γ = L_{γ, bol}/g_{γ, bol} ≃ 8.4 × 10⁴⁸ erg s⁻¹. The variability timescale was estimated at t_{var, obs} ≃ 1 d, although episodes were observed with a flux-doubling timescale as short as ≃ 2.3 h. From the SED compiled by Bonnoli et al. (2011), we deduce q = L_γ/L_syn ≃ 14, L_syn/L_X ≃ 10, and α ≃ 0.55. We use the same values of ξ_BLR, ξ_IR, L_d, and M_BH as for the MJD 55520 flare.

Our constraints for this event are shown in Figure 2. We find the marginal solution at r_min ≃ 0.17 pc and Γ_min ≃ 19. This solution corresponds to λ_{SSA, obs} ≃ 215 μm, $u_\gamma ^{\prime }/u_{\rm B}^{\prime } \simeq 1.6$, and L_{j, min} ≃ 10⁴⁶ erg s⁻¹ ∼ 0.14 L_d. While solutions within r_BLR are allowed, the maximum observed photon energy is E_{max, obs} ≃ 21 GeV (Ackermann et al. 2010), so the effect of BLR absorption is expected to be lower than in the case of the MJD 55520 flare (Section 4.1).

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Bonnoli et al. (2011) modeled the SEDs of 3C 454.3 for several epochs close to MJD 55168, probing different luminosity levels. They noted that the gamma-ray luminosity scales with the X-ray and UV luminosities roughly as $L_\gamma \propto L_X^2 \propto L_{\rm UV}^2$. Therefore, they proposed that the location of the gamma-ray emitting region shifts outward with increasing gamma-ray luminosity. For the highest state at MJD 55168, they suggested a distance scale of r ≃ 0.06 pc at Γ ≃ 20 (see Figure 2). It is critical to note at this point that they adopted a variability timescale of t_{var, obs} ≃ 6 hr and a Doppler-to-Lorentz factor ratio of $\mathcal {D}/\Gamma \simeq 1.45$. We have checked that for such parameters our constraints are marginally consistent with their result; our model predicts $u_\gamma ^{\prime }/u_{\rm B}^{\prime } \simeq 0.84$, λ_SSA ≃ 118 μm, and L_{j, min} ≃ 2.7 × 10⁴⁵ erg s⁻¹.

AO 0235+164 (z = 0.94, d_L ≃ 6.14 Gpc) is an LBL-type blazar, which was active in 2008–2009. The highest gamma-ray state, achieved between MJD 54700 and MJD 54780, was analyzed in detail by Ackermann et al. (2012). They estimated the observed gamma-ray luminosity as L_γ ≃ 6.7 × 10⁴⁷ erg s⁻¹; the observed variability timescale t_{var, obs} ≃ 3 d = 2.6 × 10⁵ s; the Compton dominance q = L_γ/L_syn ≃ 4; the synchrotron to X-ray luminosity ratio L_syn/L_X ≃ 6; the accretion disk luminosity L_d = 4 × 10⁴⁵ erg s⁻¹; and the characteristic radii of external radiation components r_BLR ≃ 0.06 pc and r_IR ≃ 1.6 pc. For the black hole mass, they adopted M_BH ∼ 4 × 10⁸ M_☉. The X-ray spectral index is very uncertain, as very soft X-ray spectra were observed by Swift/X-ray Telescope during the gamma-ray activity. Here we adopt α ≃ 1.

In Figure 3, we plot the constraints on the location of the gamma-ray flare, adopting ξ_BLR = ξ_IR = 0.1. The marginal solution is located at r_min ≃ 0.65 pc and Γ_min ≃ 22. The predictions for this solution are λ_{SSA, obs} ≃ 920 μm, $u_\gamma ^{\prime }/u_{\rm B}^{\prime } \simeq 0.7$, and L_{j, min} ≃ 8.5 × 10⁴⁴ erg s⁻¹ ≃ 0.2L_d. The gamma-ray emitting region is certainly located outside the BLR, in the region where external radiation is dominated by the dusty torus emission. The jet is predicted to be at least moderately magnetized at r ∼ 3 × 10⁴ R_g. The required minimum jet power is higher by factor ≃ 4 than the estimate of Meyer et al. (2011).

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VLBI measurements of the jet of AO 0235+164 imply that $\mathcal {D}/\Gamma _{\rm j} \simeq 1.98$ and Γ_jθ_j ≃ 0.04 (Hovatta et al. 2009; Pushkarev et al. 2009). This rather extreme solution of a very narrow and perfectly aligned jet is inconsistent with both the L_SSC and E_{cool, obs} constraints. For $\mathcal {D}/\Gamma = 1$, the combination of L_SSC and E_{cool, obs} constraints implies that Γθ > 0.4.

Agudo et al. (2011b) presented a detailed discussion of the same event, and they argued that this flare was produced at the distance scale of ∼12 pc, based on the VLBI imaging and cross-correlation between the gamma rays and the millimeter data. Ackermann et al. (2012) used a simple variability timescale argument to show that locating the emitting region at 12 pc would require a very high jet Lorentz factor Γ ≃ 50. Here, we find that the SSC constraint leads to a similar limit on Γ already at r ≃ 9 pc. Moreover, the cooling constraint is even stronger at distances larger than ≃ r_IR, implying that energetic electrons injected at the distance of 12 pc have no 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 will be transparent to wavelengths shorter than ≃ 1 mm. Agudo et al. (2011b) calculated the discrete correlation function (DCF) between the gamma rays and the 1 mm light curve, showing multiple peaks in the range of delays between 0 and −50 days (the latter meaning that the gamma rays lead the millimeter signals). Our result is thus not in conflict with the gamma—1 mm DCF. However, our model does not allow for the possibility that the emitting region producing three-day long gamma-ray flares is transparent at 7 mm, which is the wavelength of Very Long Baseline Array observations reported by Agudo et al. (2011b). In our model, even for Γ = 100 the 7 mm photosphere would fall at a very large distance of ≃ 90 pc. Just like in the case of 3C 454.3 (see Section 4.1), the solution to this apparent paradox is that the variability timescale of the 7 mm radiation has to be much longer than three days. Indeed, the 7 mm light curves presented in Agudo et al. (2011b) indicate variability timescale of the order of ≃ 80 days. When we used this timescale to calculate the collimation (Γθ) and the SSA (λ_{SSA, obs}) constraints, we obtained the following solution: the Γθ = 1 line crosses the 7 mm photosphere at r_7mm ≃ 6.7 pc and Γ_{j, 7mm} ≃ 14. This is consistent with the detection around this epoch of a superluminal radio element of apparent velocity β_app ∼ 13 (Agudo et al. 2011b).

The close observed correspondence between the gamma-ray flares and the activity at the 7 mm wavelength does not necessarily indicate that the gamma rays should be produced co-spatially with the 7 mm core. In Appendix B, we present a simple light travel time argument according to which the gamma rays could still be produced at the distance of ∼1 pc.

Our results indicate that the 12 pc scenario cannot be constrained by energetic requirements, as the required minimum jet power is only L_{j, min} ∼ 3 × 10⁴⁴ erg s⁻¹ in this case. However, even a moderate jet magnetization implied by the SSC constraint puts into question the efficiency of the reconfinement/conical shock that is proposed by Agudo et al. (2011b) as the physical mechanism behind the 7 mm core.

3C 279 (z = 0.536, d_L ≃ 3.07 Gpc) produced a gamma-ray flare peaking at MJD 54880 that was extensively studied in Abdo et al. (2010b) and Hayashida et al. (2012). The gamma-ray flux doubling timescale can be estimated as t_{var, obs} ≃ 1.5 d, and the half-peak gamma-ray luminosity is L_γ ≃ 2.6 × 10⁴⁷ erg s⁻¹. Following Hayashida et al. (2012), we adopt L_d ≃ 2 × 10⁴⁵ erg s⁻¹, q ≃ 7.5, L_syn/L_X ≃ 9.2, M_BH ≃ 5 × 10⁸ M_☉, and α ≃ 0.7. This implies that r_BLR ≃ 0.045 pc and r_IR ≃ 1.1 pc.

In Figure 4, we plot the constraints on r and Γ for this flare. The marginal solution is r_min ≃ 0.62 pc and Γ_min ≃ 27, which locates the gamma-ray emission firmly outside the BLR, and close to r_IR. The predictions for this solution are λ_{SSA, obs} ∼ 1.03 mm, $u_\gamma ^{\prime }/u_{\rm B}^{\prime } \sim 0.3$, L_{j, min} ∼ 4 × 10⁴⁴ erg s⁻¹ ∼ 0.2L_d. The required jet power is roughly half of the estimate of Meyer et al. (2011).

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The MOJAVE jet kinematics solution yields $\mathcal {D} \simeq 24$, Γ_j ≃ 21 (Hovatta et al. 2009), and Γ_jθ_j ≃ 0.22 (Pushkarev et al. 2009). The implied Doppler-to-Lorentz factor ratio of $\mathcal {D}/\Gamma \simeq 1.15$ is fairly close to unity. This solution is inconsistent with both the L_SSC and E_{cool, obs} constraints. For $\mathcal {D}/\Gamma = 1$, the combination of the L_SSC and E_{cool, obs} constraints implies that Γθ > 0.7.

Hayashida et al. (2012) proposed two scenarios for the gamma-ray emission. One of them emphasized the connection to a 20 days scale polarization event, which implicated the location at 1–4 pc. The other was based on mid-IR spectral structure detected by Spitzer, which was interpreted as a SSA turnover. The latter implicated sub-parsec scales (r_BLR) for the main synchrotron/gamma-ray component, with an additional emitting region located at ∼4 pc. Our results show very clearly that location of the gamma-ray flare at r_BLR is not consistent with the variability timescale of days, rather it would require a variability timescale of several hours. With the relatively moderate peak gamma-ray flux of 3C 279, such short timescales could not be probed with Fermi/LAT. Such timescales are essential in order to interpret the Spitzer spectral feature in terms of SSA. On the other hand, the distance of 1 pc is fully consistent with all constraints, however, shifting the emitting region to the distance of 4 pc would violate the E_{cool, obs} constraint.

Dermer et al. (2014) presented a detailed model of the radiation of blazars which was applied to the 3C 279 data from Hayashida et al. (2012). They concluded that this gamma-ray flare was produced at r ∼ 0.1–0.5 pc for Γ ∼ 20–30. This is still outside the BLR, but according to Figure 4, their parameter region extends well into the Γθ > 1 regime. However, they assumed a very short variability timescale of t_{var, obs} ∼ 10⁴ s = 2.8 h. We have checked the consequences of adopting t_var = 10⁴ s in our model. For 20 ⩽ Γ ⩽ 30, we found a range of possible locations r ∼ 0.025–0.11 pc, which are closer to the black hole than the solutions of Dermer et al. (2014). In that work, the location of the gamma-ray emitting region was constrained by calculating u_BLR from SED modeling, and comparing it with the level u_{BLR, 0} expected for r < r_BLR. By noting that u_BLR < u_{BLR, 0}, they concluded that r > r_BLR. However, it is difficult to provide a precise estimate of r in this way, because it depends on the uncertain shape of the $u_{\rm BLR}^{\prime }(r)$ function for r > r_BLR. Because these authors allowed for higher values of the accretion disk luminosity, up to L_d = 10⁴⁶ erg s⁻¹, they also have higher values of $r_{\rm BLR} \propto L_{\rm d}^{1/2} \lesssim 0.1\;{\rm pc}$. Taking these differences into account, the discrepancy between their and our results does not appear to be significant.

PKS 1510-089 (z = 0.36, d_L ≃ 1.92 Gpc), the second most active blazar of the Fermi era (Nalewajko 2013), has been monitored extensively in the X-ray, optical/NIR, and radio/millimeter bands. In early 2009, it produced a series of gamma-ray flares, peaking at MJD 54917 (2009 March 27), MJD 54948 (2009 April 27), and MJD 54962 (2009 May 11) (Abdo et al. 2010a; D'Ammando et al. 2011). The first and the last flares were accompanied by sharp optical/UV flares, but none of them had a clear X-ray counterpart. A cross-correlation analysis indicates that the optical signal could be delayed with respect to the gamma-ray signal by ≃ 13 days, in which case the major optical flare peaking at MJD 54961 would be associated with the second gamma-ray event at MJD 54948. However, in our work we are primarily concerned with the gamma-ray emitting regions as they are when they produce a gamma-ray flare, and thus we use strictly simultaneous multiwavelength data. Therefore, we will focus on the case of MJD 54948, ignoring the optical flare that follows it. As usual, there is some ambiguity about establishing the flare parameters, and for this purpose we carefully examine the results of Abdo et al. (2010a), and compare them with our own analysis. We adopt the νL_ν gamma-ray luminosity of L_γ ≃ 5.4 × 10⁴⁷ erg s⁻¹, the gamma-ray variability timescale of t_{var, obs} ≃ 0.9 d (Nalewajko 2013), the accretion disk luminosity of L_d ≃ 5 × 10⁴⁵ erg s⁻¹ (Nalewajko et al. 2012b), the Compton dominance parameter of L_γ/L_syn ≃ 100, the X-ray luminosity of L_X ≃ 5 × 10⁴⁴ erg s⁻¹, the X-ray spectral index of α ≃ 0.3, the black hole mass of M_BH ≃ 4 × 10⁸ M_☉, the covering factors of ξ_BLR = ξ_IR ≃ 0.1, and the external radiation fields radii r_BLR ≃ 0.07 pc and r_IR ≃ 1.8 pc.

Our constraints for the MJD 54948 flare of PKS 1510-089 are presented in Figure 5. The SSC constraint is particularly strong in this case, since L_γ/L_X ≃ 1000. The marginal solution is r_min ≃ 0.37 pc at Γ_min ≃ 26, which is well outside the BLR. The predictions for this solution are λ_{SSA, obs} ≃ 1.4 mm, $u_\gamma ^{\prime }/u_{\rm B}^{\prime } \simeq 12$, and L_{j, min} ≃ 2.2 × 10⁴⁴ erg s⁻¹ ∼ 0.045 L_d, which is slightly lower 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.

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Abdo et al. (2010a) argued that this gamma-ray flare was produced within the BLR, as they found that the gamma-ray and optical luminosities are related roughly as $L_\gamma \propto L_{\rm opt}^{1/2}$, which favors the ERC(BLR) mechanism of gamma-ray production over ERC(IR). Their SED models were calculated for Γ ≃ 15, and their SSC components peak significantly below L_X. This would be in strong disagreement with our results, if not for two crucial assumptions: they adopted $\mathcal {D}/\Gamma \simeq 1.4$ and t_var ≃ 0.25 days. When these parameters are used in our model, we obtain r_min ≃ 0.035 pc at Γ_min ≃ 12, which is consistent with their result. We note that VLBI observations indicate that $\mathcal {D}/\Gamma \simeq 0.8$ (Hovatta et al. 2009), so our choice of $\mathcal {D}/\Gamma = 1$ seems to be more conservative. Abdo et al. (2010a) used the intrinsic gamma-ray opacity constraint to derive a limit on the Doppler factor $\mathcal {D} \gtrsim 8$, which we find very conservative, and certainly weaker than the SSC constraint. They also estimated the jet power, and for this particular flare they obtained L_j ≃ 4.8 × 10⁴⁵ erg s⁻¹, about 60% of which is in the magnetic form, and only ∼8% in the radiative form. This indicates that in their model $u_\gamma ^{\prime }/u_{\rm B}^{\prime } \simeq 0.13$, which is consistent with their low L_SSC, but this solution is likely to require Γθ > 1. The energetic requirements discussed by Abdo et al. (2010a) can be significantly relaxed by bringing their model closer to equipartition.

Marscher et al. (2010) presented an independent analysis of the activity of PKS 1510-089 in early 2009, including more detailed VLBI analysis and optical polarization data. The VLBI observations at 43 GHz revealed a superluminal knot of apparent velocity 22c, which was projected to pass the stationary core at MJD ∼54959, simultaneous with the major optical flare. This optical flare was accompanied by a sharp increase of the optical polarization degree, up to ∼37%, and apparently preceded by a gradual (∼50 days timescale) rotation of the optical polarization angle by ∼720°. They interpreted the gamma-ray activity of PKS 1510-089 as directly related to the emergence of the superluminal radio/millimeter feature, with optical polarization rotation indicating either stochastic or helical structure of the jet. This interpretation implies a ∼10–20 pc distance scale for the gamma-ray flares, at which the ERC mechanism based on IR photons is inefficient. Instead, it was proposed that the gamma rays are produced by Comptonization of synchrotron radiation produced in slower outer jet layers (spine-sheath models, Ghisellini et al. 2005). In Appendix C.1, we show that in fact the spine-sheath model offers no advantage over the ERC model in explaining strongly beamed gamma-ray emission.

Chen et al. (2012) performed time-dependent SED modeling of the 2009 March flare of PKS 1510-089, investigating three scenarios for the gamma-ray emission: ERC(BLR), ERC(IR), and SSC. The ERC(BLR) scenario was demonstrated to require very low values of the covering factor, ξ_BLR ∼ 0.01. The other two scenarios produce reasonable fits to the observed SEDs, and each scenario has its own moderate problems. The problem of localization of the gamma-ray emitting region was not directly addressed. We note that since the ERC(BLR) model should be located at r ≲ r_BLR, it requires Γθ ≫ 1, especially for the adopted variability timescale of 4 d. The SSC models are difficult to localize, because their parameters are independent of the external radiation fields. However, in order to suppress the ERC component, they require a significantly lower Lorentz factor, Γ ≲ 10, than the ERC models. We briefly discuss the constraints on SSC models in Section 6.4.

During the active state in 2009, PKS 1510-089 was detected in the very high energy (VHE) gamma-ray band, up to 300 GeV, by the H.E.S.S. observatory (H.E.S.S. Collaboration 2013). Opacity constraints due to BEL imply that the VHE emission must be produced outside the BLR (Barnacka et al. 2013), which is fully consistent with our results for the GeV emission.

PKS 1222+216 (z = 0.432, d_L ≃ 2.4 Gpc) was in a very active gamma-ray state in 2010, producing major GeV flares peaking at MJD 55317 (2010 May 1) and MJD 55366 (2010 June 19) (Tanaka et al. 2011). Shortly before the latter event, the MAGIC observatory detected VHE emission (up to 400 GeV) of extremely short variability timescale, ∼9 min (Aleksić et al. 2011), which proved to be very challenging to explain (Tavecchio et al. 2011, 2012; Dermer et al. 2012; Nalewajko et al. 2012a; Giannios 2013). Arguably, the only certain result concerning this VHE event is that it should be produced at the distance scale beyond r_{min, VHE} ∼ 0.5 pc in order to avoid the absorption of the VHE photons by the BLR radiation. Here, we focus on the GeV flare peaking at MJD 55366, for which the variability timescale was estimated as t_{var, obs} ≃ 1 d, and the gamma-ray luminosity as L_γ ≃ 10⁴⁸ erg s⁻¹ (Tanaka et al. 2011). Following Tavecchio et al. (2011), we adopt L_d ≃ 5 × 10⁴⁶ erg s⁻¹, ξ_BLR ≃ 0.02, ξ_IR ≃ 0.2, q = L_γ/L_syn ≳ 100, L_X ≃ 10⁴⁵ erg s⁻¹, α ≃ 0.6, r_BLR ≃ 0.22 pc, and r_IR ≃ 5.6 pc. There is significant uncertainty in the value of q, as the simultaneous Swift/UVOT spectra are dominated by the thermal component. The black hole mass was recently estimated as M_BH ≃ 6 × 10⁸ M_☉ (Farina et al. 2012).

Our constraints for the GeV flare of PKS 1222+216 are presented in Figure 6. The marginal solution is found at r_min ≃ 0.18 pc and Γ_min ≃ 17. This location is within the BLR, and significantly closer to the black hole than the minimum location of the VHE emission. The predictions for this solution are: λ_{SSA, obs} ≃ 0.76 mm, $u_\gamma ^{\prime }/u_{\rm B}^{\prime } \simeq 11$, and L_{j, min} ≃ 9.5 × 10⁴⁴ erg s⁻¹ ≃ 0.019 L_d, which is slightly above the estimate by Meyer et al. (2011). When we increase the Compton dominance parameter to 300, we obtain r_min ≃ 0.13 pc and Γ_min ≃ 14. When we use a shorter variability timescale of ≃ 6 h (Foschini et al. 2011), we obtain r_min ≃ 0.08 pc and Γ_min ≃ 23.

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The VLBI kinematic solution is rather peculiar, with $\mathcal {D}/\Gamma _{\rm j} \simeq 0.11$ (Hovatta et al. 2009), which would indicate that PKS 1222+216 is not a blazar. When we decrease our Doppler-to-Lorentz factor ratio merely to $\mathcal {D}/\Gamma = 0.5$, a minimum Lorentz factor of Γ_min ≃ 52 is required. Therefore, adopting $\mathcal {D}/\Gamma \simeq 1$ seems to be the most reasonable option in this case.

Tavecchio et al. (2011) modeled the broadband SED of PKS 1222+216 for this particular event, considering three scenarios: a single compact emitting region for both VHE and GeV emission, separate emitting regions located outside the BLR, and separate emitting regions with the GeV radiation produced within the BLR. For the GeV emitting regions, they adopted a Lorentz factor Γ = 10 and a Doppler factor $\mathcal {D} \simeq 20$. However, because they fixed the jet opening angle, at different distances they adopted different radii for the emitting regions, corresponding to different variability timescales. For the GeV emitting region located within the BLR, their model predicts a variability timescale of ≃ 10 h, and for the region located outside the BLR, it predicts a variability timescale of ≃ 3 d. When using these timescales, and a Doppler-to-Lorentz factor ratio of $\mathcal {D}/\Gamma = 2$, our constraints are entirely consistent with the model parameters adopted by Tavecchio et al. (2011) in either scenario.

A characteristic feature of all the models of Tavecchio et al. (2011) is that the magnetic component of the jet power is strongly dominated by the particle component, which in turn is dominated by protons. However, considering only the electrons, they predict that $u_{\rm e}^{\prime }/u_{\rm B}^{\prime } \simeq 6$. Even if only a moderate fraction of the energy of electrons can power the gamma-ray emission, their model is consistent with our result that $u_\gamma ^{\prime }/u_{\rm B}^{\prime } \lesssim 10$. We find that at moderate values of the Lorentz factor Γ the jet can only be weakly magnetized. If the extremely rapid VHE variability is due to processes powered by relativistic magnetic reconnection (Nalewajko et al. 2012a; Giannios 2013), this requires a high jet magnetization, which is possible at the ∼pc scale, but only for very high Lorentz factors (Γ ≳ 40). Alternatively, the required regions of very high magnetization may only occupy a small fraction of the jet cross-section.

PKS 0208-512 (z = 1.003, d_L ≃ 6.7 Gpc) showed several gamma-ray flares of moderate luminosity, which were studied in detail by Chatterjee et al. (2013a). What is interesting about these flares is that they show significantly variable Compton dominance parameter. Here we discuss the constraints on the parameters of one of the brightest gamma-ray flares produced by this source, peaking around MJD 55750. Preliminary results for this event were presented in Chatterjee et al. (2013b). Following that work, we adopt the following parameter values: L_γ ≃ 1.7 × 10⁴⁷ erg s⁻¹, t_{var, obs} ≃ 2 d, q = L_γ/L_syn ≃ 3.3, L_X ≃ 3.5 × 10⁴⁵ erg s⁻¹, α ≃ 0.7, L_d ≃ 8 × 10⁴⁵ erg s⁻¹, ξ ≃ 0.1, r_BLR ≃ 0.09 pc, and r_IR ≃ 2.2 pc. While Chatterjee et al. (2013b) adopted $\mathcal {D}/\Gamma \simeq 1.4$, here we will use $\mathcal {D}/\Gamma = 1$ as we do for all other sources. We also adopt a black hole mass of M_BH ≃ 1.6 × 10⁹ M_☉ (Fan & Cao 2004).

Our constraints for the gamma-ray flare in PKS 0208-512 are shown in Figure 7. The marginal solution is uncertain in this case, because the L_SSC constraint is almost tangent to the collimation constraint, nevertheless, we adopt r_min ≃ 0.2 pc and Γ_min ≃ 15. With a relatively massive black hole, we have r_min ≃ 2500 R_g. The predictions of this solution are: λ_{SSA, obs} ≃ 0.65 mm, $u_\gamma ^{\prime }/u_{\rm B}^{\prime } \simeq 0.26$, and L_{j, min} ≃ 0.92 × 10⁴⁵ erg s⁻¹ ≃ 0.12L_d. The cooling constraint, which was not considered by Chatterjee et al. (2013b), is rather strong, indicating that the jet cannot be strongly collimated, with Γθ ≳ 0.3. This means that the emitting region must be located beyond r_BLR, and possibly close to r_IR.

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While we have imposed an upper limit on the collimation parameter Γθ ≲ 1, the SSC and cooling constraints provide a firm lower limit. In some analyzed cases (Figure 4), this limit is as strong as Γθ ≳ 0.7. In other cases (Figure 6), values of Γθ ≃ 0.1 can be obtained only for Lorentz factors Γ ≳ 25. Such tight lower limits may be in conflict with VLBI radio observations that imply significantly tighter upper limits, with Γ_jθ_j ≲ 0.3 (Pushkarev et al. 2009; Clausen-Brown et al. 2013). However, these radio observations probe the jet geometry at many parsec scales, and it is not clear whether these results are relevant for parsec-scale jets. Also, the Lorentz factor Γ of the emitting region may be larger than the jet Lorentz factor Γ_j. In any case, we can securely conclude that very narrow opening angles of the gamma-ray emitting regions are excluded by the SSC and cooling constraints. This makes any model of energy dissipation in jets which operates on a small fraction of the jet cross-section, in particular reconfinement shocks leading to very narrow nozzles (e.g., Bromberg & Levinson 2009), inconsistent with the ERC scenario. This also challenges models of strongly structured jets, e.g., the spine-sheath models (Ghisellini et al. 2005), or models involving strongly localized dissipation sites, e.g., minijets (Giannios et al. 2009), unless they can be distributed uniformly across a large fraction of the jet cross-section. While these models can still explain the most extreme modes of blazar variability, in particular the sub-hour VHE gamma-ray flares (Aleksić et al. 2011), they may not be responsible for the bulk of the gamma-ray emission of blazars.

The intersection between the collimation constraint and the SSC constraint defines the marginal solution (r_min, Γ_min), which sets firm lower limits on both r and Γ. One can derive the marginal solution from Equations (1) and (5):

$$\begin{eqnarray} r_{\rm min} &\simeq & \frac{ct_{\rm var,obs}}{(1+z)} \left[\frac{3}{4}\left(\frac{g_{\rm SSC}}{g_{\rm ERC}}\right)\left(\frac{L_{\rm syn}}{L_{\rm X}}\right)\left(\frac{L_\gamma }{\zeta (r_{\rm min})L_{\rm d}}\right)\right]^{1/2} \nonumber \\ && \times\left(\frac{\mathcal {D}}{\Gamma }\right)^{-2} \,, \end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray} \Gamma _{\rm min} &\simeq & \left[\frac{3}{4}\left(\frac{g_{\rm SSC}}{g_{\rm ERC}}\right)\left(\frac{L_{\rm syn}}{L_{\rm X}}\right)\left(\frac{L_\gamma }{\zeta (r_{\rm min})L_{\rm d}}\right)\right]^{1/4} \nonumber \\ &&\times \left(\frac{\mathcal {D}}{\Gamma }\right)^{-3/2} \,. \end{eqnarray} \tag{ 27 }$$

Because of the dependence of ζ on r, Equation (26) is not explicit, but the solutions discussed below are calculated self-consistently. One can see that the minimum distance scale r_min is proportional to the observed variability timescale t_{var, obs}. Both r_min and Γ_min depend strongly on the Doppler-to-Lorentz ratio, and they are weak functions of the broadband SED shape. The marginal solutions for the cases analyzed in Section 4 are listed in Table 1. Even with this very small sample, we can point to some general trends and differences. The minimum distance ranges between 0.16 ≲ r_min [pc] ≲ 0.65. In terms of gravitational radii, the range is 2500 ≲ r_min/R_g ≲ 33,000, which is much wider than the spread of black hole mass estimates for the six analyzed blazars—4 × 10⁸ ≲ M_BH/M_☉ ≲ 1.6 × 10⁹. In terms of the BLR radii, the range is 0.62 ≲ r_min/r_BLR ≲ 14. Interestingly, the range of absolute values of r_min is much narrower than the ranges of relative values of r_min/R_g and r_min/r_BLR. Flares with relatively large r_min happen to be both long and faint. The minimum Lorentz factor ranges between 15 ≲ Γ_min ≲ 30. It does not show an obvious trend with the gamma-ray luminosity L_γ or with the timescale t_{var, obs}.

Table 1. Parameters of Our Constraints, the Marginal Solutions (Minimum Distances), and the Maximum Distances for All Blazar Flares Studied in Section 4

Object	3C	3C	AO	3C	PKS	PKS	PKS
	454.3	454.3	0235+164	279	1510-089	1222+216	0208-512
MJD	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)
Lγ (1048 erg s−1)	47	8.4	0.67	0.26	0.54	1	0.17
tvar (d)	0.36	1	3	1.5	0.9	1	2
q = Lγ/Lsyn	30	14	4	7.5	100	100	3.3
Lγ/LX	300	140	24	69	1000	1000	49
Ld (1046 erg s−1)	6.75	6.75	0.4	0.2	0.5	5	0.8
Mbh (108 M☉)	5	5	4	5	4	6	16
rmin (pc)	0.16	0.17	0.65	0.62	0.37	0.18	0.2
rmin/Rg [103]	6.6	7	33	26	19	6.2	2.5
rmin/rBLR	0.62	0.65	10	14	5.3	0.8	2.2
Γmin	30	19	22	27	26	17	15
λSSA, obs (mm)	0.125	0.215	0.92	1.03	1.4	0.76	0.65
$u_\gamma ^{\prime }/u_{\rm B}^{\prime }$	3.3	1.6	0.7	0.3	12	11	0.26
Lj, min (1045 erg s−1)	17	10	0.85	0.4	0.22	0.95	0.92
Lj, M11 (1045 erg s−1) (s)	2	2	0.2	0.9	0.3	0.8	⋅⋅⋅
rmax (pc) (*)	0.8	8.5	3.4	1.7	2.4	10.7	4
rmax/rmin	5	50	5.2	2.8	6.4	59	20
rmax/rIR	0.12	1.3	2.1	1.5	1.4	1.9	1.8

Notes. (*) Calculated for Γ_max = 50, and in the case of 3C 279 for Γ_max ≃ 46. (a) Abdo et al. 2011; (b) Vercellone et al. 2011; (c) Wehrle et al. 2012; (d) Ackermann et al. 2010; (e) Bonnoli et al. 2011; (f) Pacciani et al. 2010; (g) Ackermann et al. 2012; (h) Agudo et al. 2011b; (i) Abdo et al. 2010b; (j) Hayashida et al. 2012; (k) Dermer et al. 2014; (l) Abdo et al. 2010a; (m) D'Ammando et al. 2011; (n) Marscher et al. 2010; (o) Tanaka et al. 2011; (p) Tavecchio et al. 2011; (q) Chatterjee et al. 2013a; (r) Chatterjee et al. 2013b; (s) Meyer et al. 2011.

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The energy density ratio of the gamma-ray radiation to the magnetic fields for the marginal solution is given by (cf. Equation (21)):

$$\begin{equation} \frac{u_\gamma ^{\prime }}{u_{\rm B}^{\prime }} \simeq \frac{L_\gamma L_{\rm X}}{g_{\rm SSC}L_{\rm syn}^2}\,. \end{equation} \tag{ 28 }$$

One can see that it depends only on the broadband SED shape. From Table 1, we find that it ranges between $0.26 \lesssim u_\gamma ^{\prime }/u_{\rm B}^{\prime } \lesssim 12$. Values lower by about order of magnitude are possible for other solutions, which also have lower values of L_SSC. The energy density ratio generally increases with the Compton dominance parameter q. The gamma-ray radiation density $u_\gamma ^{\prime }$ closely probes the high-energy end of the electron population, and provides a lower limit on the total electron energy density $u_{\rm e}^{\prime }$. Assuming very roughly that $3 \lesssim u_{\rm e}^{\prime }/u_\gamma ^{\prime } \lesssim 10$, we can expect that $u_{\rm e}^{\prime }/u_{\rm B}^{\prime } \sim 0.08\hbox{--}120$. In this sense, our constraints are not in conflict with the equipartition condition $u_{\rm e}^{\prime }/u_{\rm B}^{\prime } \simeq 1$, which is sometimes imposed on blazar models (e.g., Böttcher et al. 2009; Dermer et al. 2014). This also indicates that (sub-)parsec scale jets are at most moderately magnetized. Very high magnetization values would require violating the jet collimation constraint, i.e., Γθ > 1.

The minimum required jet power for the marginal solution is given by (cf. Equations (22) and (23)):

$$\begin{eqnarray} L_{\rm j,min} &=& \frac{L_\gamma }{4} \left[\frac{3}{4}\left(\frac{g_{\rm SSC}}{g_{\rm ERC}}\right)\left(\frac{L_{\rm syn}}{L_{\rm X}}\right)\left(\frac{L_\gamma }{\zeta (r_{\rm min})L_{\rm d}}\right)\right]^{-1/2} \nonumber \\ &&\times \left(\frac{\mathcal {D}}{\Gamma }\right)^{-1} \left(1+\frac{u_{\rm B}^{\prime }}{u_\gamma ^{\prime }}\right) \,. \end{eqnarray} \tag{ 29 }$$

One can see that it depends primarily on the gamma-ray luminosity, relatively weakly on the Doppler-to-Lorentz ratio, and to some degree also on the broadband SED shape. Our estimates of the minimum jet power for the analyzed cases (Table 1) range between 2.2 × 10⁴⁴ ≲ L_{j, min} [erg s⁻¹] ≲ 1.7 × 10⁴⁶, which is significantly narrower than the range of apparent gamma-ray luminosities 1.7 × 10⁴⁷ ≲ L_γ [erg s⁻¹] ≲ 4.7 × 10⁴⁹. In terms of the accretion disk luminosity, we find 0.019 ≲ L_{j, min}/L_d ≲ 0.25. There is a trend for this ratio to be higher for lower Compton dominance q (and higher jet magnetization). For five blazars (excluding PKS 0208-512), we compare L_{j, min} with the estimates L_{j, M11} of total jet power by Meyer et al. (2011). We find that in many cases our lower limits significantly exceed L_{j, M11}, with 0.44 ≲ L_{j, min}/L_{j, M11} ≲ 8.5. Since our estimates do not take into account the contributions from cold/warm electrons and protons, the total jet powers required to power the observed gamma-ray flares may be comparable to, or even exceed, the accretion disk luminosity (in agreement with Ghisellini et al. 2009), and they are certain to be significantly higher than the estimates of Meyer et al. (2011). This indicates that the total jet powers in blazars are strongly variable, and that the values estimated from energetics of the brightest gamma-ray flares (this work) can exceed by more than order of magnitude higher the average values inferred from the low-frequency (300 MHz) radio luminosity (Meyer et al. 2011).

The SSA wavelength for the marginal solutions ranges between 0.125 ≲ λ_{SSA, obs} [mm] ≲ 1.4. For other allowed solutions λ_SSA will be somewhat larger. The SSA threshold appears to be better correlated with the gamma-ray luminosity L_γ than with the observed variability timescale t_{var, obs}. For five events with marginal λ_{SSA, obs} > 0.5 mm, a fairly close correlation between the gamma rays and the mm data can be expected (Sikora et al. 2008). However, for the bright flares of 3C 454.3, where marginal λ_{SSA, obs} ≲ 0.2 mm, we expect that the millimeter signal should be significantly delayed with respect to the gamma-ray signal. The gamma-ray emitting regions for the analyzed events cannot be transparent at the 7 mm wavelength. Because of the weak dependence of λ_{SSA, obs} on either r or Γ, SSA can potentially provide very strong additional constraints on the parameters of gamma-ray emitting regions in blazars.

The intrinsic gamma-ray opacity does not provide a significant constraint in the analyzed cases, with Γ(E_{max, obs} = 100 GeV) ≲ 10. In every analyzed case, the SSC constraint gives a stronger lower limit on Γ, as first noted by Ackermann et al. (2010).

For a given value of the Lorentz factor Γ, the maximum distance r_max(Γ) is determined either by the SSC constraint, or by the cooling constraint. Eventually, at some Γ_max there is a solution where the cooling constraint crosses the collimation constraint, which gives an absolute upper limit r_max(Γ_max). However, the values of Γ_max can be extremely high (Γ_max ≫ 50), especially for sources with high accretion disk luminosity L_d (3C 454.3 and PKS 1222+216), for which the cooling constraint is relatively weak. Therefore, the effective maximum distance scale depends on how high values of Γ one would accept.¹⁰ For a rather high Γ_max = 50 (Γ_max ≃ 46 in the case of 3C 279), we obtain 0.8 ≲ r_max [pc] ≲ 10.7 (see Table 1). In terms of the IR radii, the range is 0.12 ≲ r_max/r_IR ≲ 2.1. The ratio of maximum to minimum distances is in the range 2.8 ≲ r_max/r_min ≲ 59. The distance 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 the swinging motion of the emitting region, we can still place significant limits on the far-dissipation scenario by using solely the SSC constraint. In most analyzed cases, locating the gamma-ray emitting regions at r ≃ 10 pc would require Γ > 50.

In Section 4.3, we discussed the tension between the constraints imposed by the ERC model and the far dissipation (∼10 pc) scenarios motivated by the observed gamma-ray/millimeter-radio connection. We showed that the SSC constraint requires very high Lorentz factors, Γ ≳ 50, in order for gamma-ray flares with a variability timescale of ∼1 day to be produced at the distance scale of ∼10 pc. These solutions are also characterized by inefficient electron cooling (E_{cool, obs} ≫ 100 MeV), which would result in strongly asymmetric gamma-ray light curves with long flux-decay timescales, unless there are fast variations in the local Doppler factor. Alternative sources of external radiation at large distance scales were proposed as a way around these problems. In Appendix C, we discuss two such ideas—spine-sheath models (e.g., Marscher et al. 2010), and extended BLRs (León-Tavares et al. 2011).

While far less popular than the ERC model, the SSC model is still being considered when modeling FSRQ blazars (e.g., Finke & Dermer 2010; Zacharias & Schlickeiser 2012; Chen et al. 2012). It was suggested that the SSC model is most relevant for FSRQs with 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 in this case L_ERC may be suppressed, being strongly in the Klein–Nishina regime due to higher electron energies). With minor modifications, we can use our constraints to identify the parameter space region where these conditions can be satisfied. Our SSC constraint (Equation (5)) is more generally a constraint on the luminosity ratio L_SSC/L_ERC, which increases systematically with decreasing 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 > 1 corresponding to moderate and low Lorentz factors, Γ ≲ 10. The parameter space of the SSC model is clearly separated from the parameter space of the ERC model. At distances of ∼10 pc, SSC model may be favored over the ERC model, the latter requiring extreme values of Γ.

The jet collimation constraint is the same for the ERC and SSC models, as it does not depend on any kind of luminosity. Because of the lower Lorentz factor characterizing the SSC model, it corresponds to very strong jet collimation, with Γθ ≲ 0.1, especially at larger distances. An SSC model operating at the distance scale of 10 pc requires significant jet recollimation or sharp jet substructure. Other constraints are distance-independent, as they no longer depend on the distribution of external radiation fields. According to Equation (21), the "equipartition" parameter is $u_{\rm B}^{\prime }/u_\gamma ^{\prime } \simeq g_{\rm SSC}/q^2 \ll 1$, therefore the SSC model implies a strongly particle-dominated emitting region (Sikora et al. 2009). The minimum required jet power, dominated by the radiative component, is comparable to or slightly larger than that in the ERC model.

There are additional very strong constraints on the SSC model from the observed broad-band SEDs of luminous blazars. While being very successful in explaining the emission of low-luminosity HBL blazars, SSC models can have serious difficulties in matching the observed SEDs of FSRQs (e.g., Joshi et al. 2012). In order to match the characteristic frequencies of the two main spectral components, SSC models typically require very low magnetic field strength and high average electron random Lorentz factor, which independently suggests a particle-dominated emitting region. A more detailed analysis of the spectral constraints on the SSC model is beyond the scope of this work.

In the spine-sheath model, the jet consists of a highly relativistic spine surrounded by a mildly relativistic sheath (Ghisellini et al. 2005). Let us denote the spine co-moving frame with $\mathcal {O}^{\prime }$, and the sheath co-moving frame with $\mathcal {O}^{\prime \prime }$. Consider that the gamma-ray flares are produced in the spine by Comptonization of synchrotron radiation originating from the sheath, and that the synchrotron radiation from the spine region contributes significantly to the observed optical/IR emission. The required energy density of the sheath radiation in $\mathcal {O}^{\prime }$ is $u_{\rm sh}^{\prime } \simeq 0.1\;{\rm erg\,cm^{-3}}$. If the sheath propagates with Lorentz factor Γ_sh in the external frame, the radiation energy density in $\mathcal {O}^{\prime \prime }$ is $u_{\rm sh}^{\prime \prime } \simeq u_{\rm sh}^{\prime } / (4\Gamma _{\rm rel}^2/3)$, where Γ_rel = Γ_shΓ(1 − β_shβ) ≃ Γ/(2Γ_sh) is the relative Lorentz factor of $\mathcal {O}^{\prime \prime }$ in $\mathcal {O}^{\prime }$ (the approximation is done in the limit where 1 ≪ Γ_sh ≪ Γ). We can calculate the apparent luminosity of the sheath radiation for an external observer aligned with the jet spine as $L_{\rm sh,obs} \simeq 4\pi c\Gamma _{\rm sh}^4R_{\rm sh}^2u_{\rm sh}^{\prime \prime }$, where R_sh ≃ θ_shr is the sheath radius parameterized by the sheath opening angle θ_sh. Putting this all together, we find:

$$\begin{eqnarray} L_{\rm sh,obs} &\simeq & \frac{12\pi c\Gamma _{\rm sh}^4(\Gamma _{\rm sh}\theta _{\rm sh})^2r^2u_{\rm sh}^{\prime }}{\Gamma ^2} \simeq 2.7\times 10^{47}\;{\rm erg\,s^{-1}} \nonumber \\ && \times\Gamma _{\rm sh}^4(\Gamma _{\rm sh}\theta _{\rm sh})^2\left(\frac{r}{10\;{\rm pc}}\right)^2\left(\frac{\Gamma }{20}\right)^{-2}\,. \end{eqnarray} \tag{ C3 }$$

Assuming that Γ_shθ_sh ∼ 1, even for very moderate values of Γ_sh, we would have L_{sh, obs} > L_γ, and even for higher values of Γ it is very likely that L_{sh, obs} > L_syn. The fact that L_{sh, obs} is a strongly increasing function of Γ_sh means that the spine-sheath model actually offers no advantage in providing soft photons for Comptonization to the observed gamma-ray emission over static sources of external radiation.

Here we estimate a possible contribution to the external radiation energy density from a BLR extended along the jet to supra-parsec distance scales (León-Tavares et al. 2011). For the purpose of first-order estimates, we will approximate the extended BLR as a sphere of radius R_BLR* centered on the jet at the distance scale r_BLR* ∼ 10 pc ≫ R_BLR*. Let L_BLR* be the luminosity of emission lines produced in this region, not taking into account any lines produced elsewhere. These emission lines are expected to be significantly narrower from conventional BEL, and there is little observational evidence for their existence in the line profiles of radio-loud quasars. Therefore, we will adopt an upper limit of L_BLR* ≲ 10⁴⁴ erg s⁻¹, so that it constitutes only a small fraction of the total luminosity of BEL. This luminosity will contribute the external radiation density $u^{\prime }_{\rm BLR*} \simeq \Gamma ^2L_{\rm BLR*}/(3\pi cR_{\rm BLR*}^2)$ at the center of the sphere in the co-moving frame of the gamma-ray emitting region. We consider two sources of radiation illuminating the extended BLR—(1) the direct accretion disk radiation of luminosity L_d; and (2) the jet synchrotron radiation produced at an arbitrary distance scale r_syn < r_BLR* and of apparent luminosity L_syn. We consider two types of covering factors—the geometric factor ξ_geom, and the intrinsic factor ξ_int—such that L_BLR* = ξ_intξ_geomL_d(syn).

In case (1), assuming that the accretion disk radiation is roughly isotropic, the geometric factor is given by ξ_geom ≃ (R_BLR*/2r_BLR*)², hence:

$$\begin{eqnarray} u_{\rm BLR*}^{\prime } &\simeq & \frac{\xi _{\rm int}\Gamma ^2L_{\rm d}}{12\pi cr_{\rm BLR*}^2} \simeq 3.7\times 10^{-4}\;{\rm erg\,cm^{-3}} \nonumber \\ &&\times L_{\rm d,46}\left(\frac{\xi _{\rm int}}{0.1}\right)\left(\frac{\Gamma }{20}\right)^2\left(\frac{r_{\rm BLR*}}{10\;{\rm pc}}\right)^{-2}\,. \end{eqnarray} \tag{ C4 }$$

This value is more than two orders of magnitude too small to satisfy the L_SSC and E_{cool, obs} constraints for typical parameter values.

In case (2), the jet synchrotron radiation is strongly beamed, and effectively it can illuminate a region of radius R_BLR* ≃ (r_BLR* − r_syn)/Γ. Assuming that all of the illuminated region is filled with the gas, we adopt ξ_geom ≃ 1.¹² Since we normalize the synchrotron luminosity to L_syn ≃ 10⁴⁷ erg s⁻¹, for consistency we adopt ξ_int ∼ 10⁻³ in order to have L_BLR* ≃ ξ_intL_syn ∼ 10⁴⁴ erg s⁻¹:

$$\begin{eqnarray} u_{\rm BLR*}^{\prime } &\simeq & \frac{\xi _{\rm int}\Gamma ^4L_{\rm syn}}{3\pi c(r_{\rm BLR*}-r_{\rm syn})^2} \simeq 0.24\;{\rm erg\,cm^{-3}} \nonumber\\ &&\times L_{\rm syn,47}\left(\frac{\xi _{\rm int}}{10^{-3}}\right)\!\left(\frac{\Gamma }{20}\right)^4\!\left(\frac{r_{\rm BLR*}-r_{\rm syn}}{5\;{\rm pc}}\right)^{-2}. \qquad \end{eqnarray} \tag{ C5 }$$

In principle, this mechanism can provide enough of external radiation density to satisfy the L_SSC and E_{cool, obs}. However, we suggest that more observational support for the existence of such emission lines is necessary.