Meeting the Challenge from Bright and Fast Gamma-Ray Flares of 3C 279

TL;DR

This paper presents a model explaining rapid, bright gamma-ray flares in blazar 3C 279 through a clumpy jet structure with plasmoids, involving synchrotron emission, reflection, and inverse Compton scattering, matching observed flare characteristics.

Contribution

The model introduces a novel plasmoid-based jet structure with a moving mirror mechanism to explain fast gamma-ray flares without significant optical-UV counterparts.

Findings

Reproduces minute-scale gamma-ray flares at ~10^{18} cm from black hole

Achieves high Compton dominance factors up to 100

Predicts minimal optical-X-ray variability during gamma-ray flares

Abstract

Bright and fast gamma-ray flares with hard spectra have been recently detected from the blazar 3C 279, with apparent GeV luminosities up to $10^{49}$ erg/s. The source is observed to flicker on timescales of minutes with no comparable optical-UV counterparts. Such observations challenge current models of high-energy emissions from 3C 279 and similar blazar sources that are dominated by relativistic jets along our line of sight with bulk Lorentz factors up to $ \Gamma \sim 20$ launched by supermassive black holes. We compute and discuss a model based on a clumpy jet comprising strings of compact plasmoids as indicated by radio observations. We follow the path of the synchrotron radiations emitted in the optical - UV bands by relativistic electrons accelerated around the plasmoids to isotropic Lorentz factors $\gamma \sim 1000$. These primary emissions are partly reflected back by a leading member in the string that acts as a moving mirror for the approaching companions. Around the plasmoids, shrinking \emph{gap} transient overdensities of seed photons build up. These are upscattered into the GeV range by inverse Compton interactions with the relativistic electrons accelerated in situ. We show that such a combined process produces bright gamma-ray flares with minor optical to X-ray enhancements. Main features of our model include: bright gamma-ray flares with risetimes as short as a few minutes, occurring at distances of order $10^{18} $ cm from the central black hole; Compton dominance at GeV energies by factors up to some $10^2$; little reabsorption from local photon-photon interactions.

Equations

$$U^{\prime }=\frac{L^{\prime }}{4\pi c{d}^{\prime 2}},$$

$$U_{BLR}^{\prime }\simeq (7\times 10^{-3}\mathrm{erg}\mathrm{c}{\mathrm{m}}^{-3})\frac{\xi }{0.02}(\frac{R_{BLR}}{0.1\mathrm{pc}}{)}^{-2}{L}_{D,46}{\Gamma }^2,$$

$$d_g\equiv \tilde{r}-{\tilde{r}}_0=\frac{R_{BLR}}{(1+{\beta }_r{)}^2{\Gamma }_r^2},$$

$$U_m^{\prime }\simeq (510^{-5}\mathrm{erg}\mathrm{c}{\mathrm{m}}^{-3})\frac{f_{m,-1}{L}_{S,43}^{\prime }{\Gamma }_r^6}{(R_m^{\ast }-r{)}_{18}^2}.$$

$$U_m^{\prime }\lesssim (2\cdot 10^{-1}\mathrm{erg}\mathrm{c}{\mathrm{m}}^{-3})\frac{f_{m,-1}{L}_{S,43}^{\prime }{\Gamma }_r^4}{{\ell }_{m,16}^{\prime 2}}.$$

$$t_h^{\ast }=\frac{{\beta }_h{r}_{m0}-r_{h0}+r_h(1-{\beta }_h)}{c{\beta }_h(1-{\beta }_m)}$$

$${\tilde{t}}_{hk}=\frac{R_{hm}^{\ast }-r_{k0}+c{t}_h^{\ast }}{c(1+{\beta }_k)},$$

$$U_{hk}^{\prime }\simeq \frac{f(1-f{)}^{2N-h-k}{L}_{s,(h)}^{\prime }[(1+{\beta }_{r,h}){\Gamma }_{r,h}{]}^4{\Gamma }_{r,k}^2}{4\pi c(R_{hm}^{\ast }-r_h{)}^2},$$

$$t_v=(1+z)R_{BLR}/(8c{\Gamma }^2{\Gamma }_r^2).$$

Full text

gamma rays: general, sources – FSRQ Objects: individual: 3C 279.

1 Introduction

The Flat Spectrum Radio Quasar (FSRQ)3C 279 ($z=0.536$) is a blazar prominent in gamma rays. It was repeatedly detected above 100 MeV by EGRET (Hartman et al., 1992; Kniffen et al., 1993), AGILE (Giuliani et al., 2009), and Fermi-LAT (e.g., Hayashida et al. 2012, 2015), and also detected above 100 GeV by MAGIC (Albert et al., 2008). In 2015 June, 3C 279 was caught in a remarkably bright flaring state by Ackermann et al. 2016, hereafter A16.The apparent gamma-ray luminosity attained $10^{49}$ erg s*-1*, (under isotropy assumption), with flux variability timescales resolved down to 2-3 minutes and doubling times of $\sim 5$ minutes.

In fact, the overall gamma-ray activity observed in 3C 279 by Hayashida et al. 2015 and by A16 in 2015 June shows strongly enhanced radiation above 100 MeV lasting several days. Several features are to be noted: a very high ”Compton dominance” in the spectral energy distribution, marked by a ratio of gamma-ray to optical emission rising by factors 3 - 5 to attain values $q\sim 10^{2}$ in a few hours; very rapid flickering with resolved timescales down to a few minutes; a flat gamma-ray spectrum occasionally extending out to about 50 GeV (Paliya, 2015), with no evidence of local reabsorption. Such unprecedented features set a challenge hard to meet for most current radiative models of this source, and possibly of other blazars as well, such as PKS 1222+216 (Aleksić et al., 2011), PKS 1510-089 and 3C 454.3 (Coppi, 2016).

2 Modeling the Gamma-Ray Source

Blazars are dominated by a relativistic jet with a Lorentz boost $\Gamma\sim 10\,-\,20$, launched by a supermassive black hole (SMBH) along our line of sight. They feature widely extended non-thermal spectra, which are interpreted in terms of Synchrotron (S) emission and of Inverse Compton (IC) radiation (cf. Rybicki & Lightman 1979); these are produced by electrons accelerated in the jet to attain random Lorentz factors up to $\gamma_{b}\sim 10^{3}$. In particular, FSRQs are often marked by gamma-ray Compton dominance, though generally at less extreme degrees than 3C 279 itself.

The S emission is observed at energies $\epsilon_{s}\propto\gamma_{b}^{2}\,\Gamma\,B^{\prime}$ with a luminosity $L_{S}\sim c\,\sigma_{T}\,n^{\prime}\,\ell^{\prime 3}\,\gamma^{2}_{b}\,U_{B}^{\prime}\,\Gamma^{2}$. These are given in terms of the number density $n^{\prime}\sim 10^{3}$ cm*-3* of energetic electrons with $\gamma_{b}\sim 10^{3}$, within the source size $\ell^{\prime}\sim 10^{16}$ cm threaded by a magnetic field of strength $B^{\prime}\sim 0.1-1$ G with magnetic energy density $U_{B}^{\prime}=B^{\prime 2}\,/\,8\pi\simeq 10^{-2}\,\rm erg\,cm^{-3}$ (primed quantities refer to the comoving frame). Such emission is widely held to explain the continuum observed from the IR to *UV *bands. IC scattering is often taken into account for the gamma-ray yield from the same electron population. This operates on any density $U^{\prime}$ of soft“seed” photons present in the jet by conserving their number while upgrading their energy to yield observed luminosities $L_{IC}\sim c\,\sigma_{T}\,n^{\prime}\,\ell^{\prime 3}\,\gamma^{2}_{b}\,U^{\prime}\,\Gamma^{2}$.

On the other hand, all FSRQs also share with the other quasars two thermal features: strong, nearly isotropic broad emission lines shining in the optical band and produced in the Broad Line Region (BLR); a bright Big Blue Bump (BBB) comprising convolved continua produced by the hot inner rings of the accretion disk surrounding the central SMBH (e.g., Peterson 1997, 2006). To wit, 3C 279 features all radiative components observed in FSRQ blazars, but during its flares the gamma-ray band is enhanced and variable to an extreme degree.

In the present Letter we discuss the implications of these findings, and propose an interpretation of the gamma-ray flaring activity of 3C 279 in terms of a clumpy jet. The latter features strings of plasmoids as long recognized in the radio observations of this and other blazars (see Hovatta et al. 2009 and references therein). Indeed, radio observations of plasmoid kinematics in 3C 279 Hovatta et al. 2009 indicate a boost $\Gamma\simeq 20$ that we adopt in the present paper. In our model the jet has an opening angle $\sim\,\Gamma^{-1}$.

The short time scales $t_{v}\sim$ a few minutes observed in the June 2015 gamma-ray flares of 3C 279 point to emission from regions of small proper sizes $d^{\prime}\lesssim c\,t_{v}\;\Gamma^{2}\simeq\rm 10^{13}\,cm\,\,\Gamma^{2}$. High luminosity coupled with fast variability constitutes the challenge to be met by a viable source model.

A guide toward a satisfactory IC model is provided by considering the energy density of the seed photons after the pattern

[TABLE]

in terms of the effective luminosity $L^{\prime}$ of their source and of the size $d^{\prime}$ of the containing volume. It is seen that smaller size values not only enhance $U^{\prime}$, but also shorten the variability time scale, so as to achieve a substantial Compton dominance within short times. Meanwhile, smaller sizes decrease the local reabsorption of the IC radiation by photon-photon pair-producing interactions. In fact, our model focuses on physical conditions where small values of $d^{\prime}$ arise.

3 IC Radiation from BLR Seed Photons?

Meeting the above requirements while retaining the bound $\Gamma\lesssim 20$ for 3C 279 make the models limited to seed photons from the BLR problematic. Such models are based on combinations of S and IC radiated at radii $R\lesssim 0.1$ pc by a population of relativistic electrons with Lorentz factors $\gamma\lesssim 10^{3}$ as discussed by Hayashida et al. 2015 and A16. The electrons inhabit the jet and interact with the local magnetic field $B^{\prime}\lesssim 1$ G to produce S emission, while by the IC process upscatter any seed photons of density $U^{\prime}_{BLR}$ as seen in the jet frame.

In the Synchrotron-Self Compton approach (SSC, e.g., Maraschi et al. 1992), the very same S photons are upscattered into the GeV range, to yield a basal level of IC $\sim$ S luminosities resulting in $q\sim 1$. External radiation Compton (ERC) may yield larger IC emission by upscattering additional UV photons produced in the BLR at $R_{BLR}\sim 0.1$ pc by BBB light reprocessed/reflected by gas clouds that float around at some $10^{3}\,\rm km\,s^{-1}$ within the region. The resulting energy density is given by

[TABLE]

where we assumed a disk luminosity $L_{D}=10^{46}\,\rm erg\,s^{-1}$ appropriate for 3C 279, a BLR covering factor $a=0.2$ and a cloud reflectivity $f_{c}\simeq 0.1$ to produce $\xi=a\,f_{c}\simeq 0.02$. Such a value for $U^{\prime}_{BLR}$ may yield a ratio IC to S luminosities $q=U^{\prime}_{BLR}/U^{\prime}_{B}\simeq 50$, still insufficient for explaining the 2013 December and 2015 June bright and fast gamma-ray flares of 3C 279.

On route toward higher IC/S ratios, one may consider the more abundant IR photons that are radiated by the dusty torus around the accretion disk upon reprocessing the BBB light; however, such photons are less effective than the UV ones for upscattering into the GeV range (see Böttcher et al. 2013). Alternative scenarios just make do with lower values of $B$ and modest accelerations, but they have to assume $\Gamma\,>\,35$ to avoid reabsorption of the meager IC radiation by photon-photon interactions within the BLR (Hayashida et al. 2015, A16).

Straightforward consequences of these models include a very low magnetic field $B\,<\,0.1\,$G in the jet and the related bulk magnetization $\sigma_{j}\propto B^{2}/\Gamma^{2}\sim 10^{-4}$ (A16). This would imply (see e.g., Mignone et al. 2013; Yuan et al. 2016; Cavaliere et al. 2017) it has a jet that resists to forming MHD structures that are conducive to driving magnetic reconnections and micro-instabilities active for electron accelerations. In addition, these - if they somehow were driven - would be ineffective to attain high values of $\gamma\simeq 10^{3}$ (see next Sect.4). Alternative scenarios for FSRQ emissions consider special structures, such as spine-sheaths (Tavecchio & Ghisellini, 2008; Sikora et al., 2016) and ”rings of fire” (MacDonald et al., 2015).

We conclude that achieving substantial Compton dominance $q\sim 100$ coupled with very fast variability requires the source to have substantially higher values of $U^{\prime}$ compared with Eq. 1 within far smaller volumes than can be provided by the BLR, as anticipated in Sect. 2 and initially discussed in Vittorini et al. 2014. Moreover, the hard unabsorbed gamma ray spectra up to $10^{2}$ GeV often observed in such sources (see, e.g., Costamante 2017) point toward emission sites beyond the BLR.

4 Enhanced Seed Photons from Moving Mirrors

Thus, we are led to develop the scenario proposed in Tavani et al. 2015. In the present Letter, we base our model on a clumpy jet with a bulk magnetization $\sigma_{j}\gtrsim 1$; the jet comprises a sequence of plasmoids moving out with speeds $\beta$ and boosts $\Gamma\leq 20$. These plasmoids are related to tearing instability and reconnections of B-lines in the jet that are squeezed or even invert their polarity within a collisionless plasma layer. Such processes have been widely proposed, discussed, and numerically computed (cf. Kagan et al. 2015; Coppi 2016; Yuan et al. 2016; Lyutikov et al. 2016; Petropoulou et al. 2016; see also Burch et al. 2016 for observational evidence in an astrophysical context). They form strings of separate ”magnetic islands” that proceed to merge into larger and larger plasmoids. We refer to plasmoid sizes $\ell^{\prime}\sim 10^{16}$ cm at a distance $R<0.1$ pc from the central SMBH. Their origin is related to the changing topology of the B-lines that induces E fields localized between plasmoids in gaps of around $10^{16}$ cm, and shrinking. These **E **fields are known from a number of detailed kinetic simulations to accelerate electrons up to $\gamma\sim 10^{3}$ in conditions where the local electron magnetization is large, i.e., $\sigma_{e}>\sigma_{j}\,m_{p}/m_{e}\sim 10^{2}$.

The relative plasmoid distances in a string vary depending on their different speeds along the jet, with the leading one significantly slowed down primarily by snow-plow effects against jet material, in particular, BLR clouds. This constitutes an interesting condition as the leading member in a string can reprocess and partially reflect some of the primary S radiation emitted by the trailing members (photons ”mirrored” by slow clouds in the BLR were originally proposed by Ghisellini & Madau 1996).

Shrinking mirror-plasmoid gaps so produced provide compact sites with high seed photon densities for intense IC to take place. In a scenario with multiple emitting plasmoids (see Fig. 1), the primary S flux in the optical-UV bands is emitted by a plasmoid like $H$ starting within the BLR, that is, in the range $0<r_{h}<R_{BLR}$) The S flux is emitted forward and mirrored backward by the leading plasmoid $M$ toward an incoming plasmoid $K$. The whole process is subject to a ”causality condition” (Böttcher & Dermer 1998); the mirrored photons are received by an incoming plasmoid when the latter is at the receiving point that we denote with $\tilde{r}_{hk}$ (cf. Fig. 1).

The condition is easily evaluated when the same plasmoid moving with radial coordinate $r=r_{0}+\beta ct$ and boost $\Gamma$ first emits at $r$, and then receives the radiation reflected backward by the mirror moving with coordinate $r_{m}=r_{mo}+\beta_{m}ct$ and boost $\Gamma_{m}$. The receiving point reads $\tilde{r}=\tilde{r}_{0}+r[(1+\beta_{r})\Gamma_{r}]^{-2}$ in the AGN-frame, for primary photons emitted in the named range and reflected back at $R^{*}_{m}=R^{*}_{m0}+\beta_{m}(1-\beta)r\,/\,[\beta(1-\beta_{m})]$. Here $\beta_{r}=(\beta-\beta_{m})(1-\beta\beta_{m})^{-1}$ and correspondingly $\Gamma_{r}=\Gamma\Gamma_{m}(1-\beta\beta_{m})$ are the relative velocity and boost between the receiver and the mirror (note that $\Gamma_{r}\simeq\Gamma\,/\,[\Gamma_{m}(1+\beta_{m})]$). Moreover, $\tilde{r}_{0}=2(\beta r_{m0}-\beta_{m}r_{0})[(1+\beta)(1-\beta_{m})]^{-1}$ is the receiving point for primary photons emitted at $r\simeq 0$ and reflected at $R^{*}_{m0}=(\beta r_{m0}-\beta_{m}r_{0})/[(1-\beta_{m})\beta]$. Note that for small values of $r_{0}$ the approximation $R*_{m0}\simeq 2\Gamma_{m}^{2}\,r_{m0}$ holds; so the factor $\Gamma_{m}^{2}$ makes it possible to have relevant reflections beyond $R_{BLR}$. The IC radiative event is completed when the plasmoid $K$ crosses the effective gap defined by

[TABLE]

and lasts for an observer time $t_{v}=d_{g}/[c\beta(1+\beta)\Gamma^{2}]\simeq R_{BLR}/[8\,c\,\Gamma^{2}\,\Gamma_{r}^{2}]$: this turns out to be about 2 minutes when, e.g., $R_{BLR}\simeq 3\cdot 10^{17}\,\rm cm$, $\Gamma\simeq 20$, and $\Gamma_{r}\simeq 5$ hold.

In the initial stage, the primary emitter radiates from an inner position $r\ll R^{*}_{m}$ and the mirror surface $\pi\ell^{\prime 2}_{m}$ is small relative to the cross section of the emission beam $\pi R^{*}_{m}\,{}^{2}\,\Gamma^{-2}$ as to reduce the effective power received by the mirror to $4\,L^{\prime}_{S}\,[\ell^{\prime}_{m}\,/\,(R^{*}_{m}-r)]^{2}\,\Gamma_{r}^{4}$ in the head-on approximation and with $1+\beta_{r}\simeq 2$. A fraction $f_{m}\simeq n^{\prime}_{cold}\sigma_{T}\ell^{\prime}_{m}$ thereof is reflected back and is received by the incoming plasmoid when the latter is close to the reflection point, that is, in a cone of cross section of order $\pi\ell^{\prime 2}_{m}$. Then, in the gap we have

[TABLE]

The denominator represents the beam dilution of the primary photons, while the volume swept per unit time is $\pi\,c\,\ell^{\prime 2}_{m}$. Such values of $U^{\prime}_{m}$ are still insufficient to dominate over $U^{\prime}_{BLR}$ as given by Eq. 2; in addition, for 3C 279 flares very short risetimes are required by A16 observations.

As the emitting plasmoid travels toward and beyond $R_{BLR}$, the mirror surface fills up the primary emission cone, and the effective received power saturates to the full value $L^{\prime}_{S}\,\Gamma_{r}^{2}$. The fraction $f_{m}$ is reflected toward the incoming plasmoid until the receiving radius $\tilde{r}$ becomes very close to the reflection point. Thus, high-energy densities of seeds prevail in the comoving frame, and for primary photons emitted at $r\simeq R_{BLR}$ attain a maximum that reads

[TABLE]

We may compare the above Eqs. 4 and 5 with the photon energy density experienced by a plasmoid crossing the BLR, that is, with $U^{\prime}_{BLR}$ given by Eq. 2. Remarkably, if $\Gamma\simeq 20$ holds, we have $U^{\prime}_{m}>U^{\prime}_{BLR}$ for $\Gamma_{r}\simeq 5$ (corresponding to $\Gamma_{m}\simeq 2$); with such values an interesting configuration is obtained in the moving mirror scenario. We stress that high seed photon densities confined to the gap may well occur beyond the BLR, a welcome feature of our model that minimizes reabsorption by local photon-photon interactions.

5 Simulations

Our simulations treat in detail plasmoids ejected in a string along the jet with different velocities. Each member $H$ in the string has proper size $\ell^{\prime}_{h}$ and moves along the jet according to $r_{h}=r_{h0}+c\beta_{h}t$, $r_{h}$ being the distance from the SMBH at time $t$ in the AGN frame. Key elements are: (1) non-uniformity of the plasmoid boosts in the string, with a slower leading member; and (2) the leader acts as a moving mirror that partially reflects back S radiation toward an approaching companion $K$.

We numerically simulate the emissions resulting from a set of values $\Gamma_{h}$, for $h$ running from 1 to $6$, with the upper value denoting the mirror; we use the subscripts $h$ for the primary emitter, $k$ for the receiver, and $m$ for the mirror. For simplicity, we carry out the formalism in the AGN reference frame where $R^{*}_{hm}$ is the distance from the central BH when the photons emitted at $r_{h}$ by the plasmoid $H$ are reflected.

The primary S emission $L^{\prime}_{S,h}$ from $r_{h}$ occurs at the time $t_{h}=(r_{h}-r_{h0})/(\beta_{h}\,c)$, and is reflected back by the moving mirror at the time

[TABLE]

when the mirror is at the reflection point $R^{*}_{hm}\equiv r_{m0}+c\beta_{m}t_{h}^{*}$. We then take into account the causality condition for the receiving plasmoid $K$; accordingly, mirrored photons are received back at the time

[TABLE]

when the receiving plasmoid $K$ is at $\tilde{r}_{hk}=r_{k0}+c\beta_{k}\tilde{t}_{hk}$. Here, the mirrored photons are IC upscattered to gamma rays. The primary emission occurs in the range $0<r_{h}<R_{BLR}$; for $r_{h}$ spanning this range we obtain the related receiving points, subject to the obvious condition $\tilde{r}_{hk}<r^{*}_{k}$, in terms of the radius where plasmoid $K$ touches the mirror.

At the receiving point, the electron population of plasmoid $K$ upscatters after the IC processes the mirrored photons originally emitted by plasmoid $H$. The energy density of these seed photons in the IC scattering region is

[TABLE]

where $N$ is the number of emitters in our string, $\Gamma_{r,h}$ are the relative boosts of the plasmoids relative to the mirror; note that $U^{\prime}_{hk}$ can grow until $r_{h}$ becomes comparable with $R_{BLR}$, as anticipated in Sect. 4 and detailed by Eq. 5.

In Fig. 2 we present an example (with $N=5$ plus the mirror) of our simulated gamma-ray light curves that result from the sum of all mirroring events and include the contribution of the standard IC from the BLR. Note that such light curves feature several very bright and short spikes rising on timescales of a few minutes, as fit to explain the A16 observations of 3C 279.

6 Discussion and Conclusions

The super-bright, super-fast, hard spectrum gamma-ray flares from 3C 279 in 2015 June call for a strong, compact source beyond the BLR. These requirements challenge radiative processes and source structures proposed over the years in the context of blazar emissions. Features of our model that meet the challenge include:

• •

Power and Compton dominance. Very intense and fast gamma-ray flares are produced by the IC process beyond the BLR, arising from strongly enhanced density of reflected seed photons that are localized in mirror-plasmoid gaps along the jet. In fact, based on the photon energy density $U^{\prime}_{m}$ of Eq. 8, we derive the energetics of the Compton-dominant flare and its requirements. To account for the intrinsic gamma-ray power $L_{\gamma}\sim 2\cdot 10^{46}$erg s*-1*, we need an IC luminosity $L^{\prime}_{IC}\simeq 5\cdot 10^{43}\,\rm erg\,s^{-1}$ in the comoving frame. Since $L^{\prime}_{IC}\simeq n^{\prime}\,(4\pi/3)\,\ell^{\prime 3}\,\sigma_{T}\,c\,U^{\prime}\,\gamma_{b}^{2}$ holds, we require number densities of radiating electrons $n^{\prime}\sim\,10^{2}\,\div\,10^{3}\,\rm cm^{-3}$ with Lorentz factors up to $\gamma_{b}\,\sim 10^{3}$, and seed energy densities $U^{\prime}_{m}\sim 10\,\rm\,erg\,cm^{-3}$ within the gaps that considerably exceed those prevailing in the BLR after Eq. 2.

• •

Correlations. The moving mirror mechanism naturally decouples the IC gamma-ray radiation produced in the gaps from the optical-UV emissions prevailing inside the BLR. This decoupling occurs on timescales $\simeq 2(R^{*}_{m}\,-\,\tilde{r})\,/\,c\,\sim$ half a day, so very large gamma-ray flares with Compton dominance $q\sim$ some $10^{2}$ can be produced over and above a slowly varying background, as observed in 3C 279.

• •

Short timescales. We follow the photon paths as shown in Fig. 1: from S emission toward the mirror and backward from it. We obtain the risetime

[TABLE]

In our conditions given by $\Gamma\simeq 20$, $\Gamma_{m}\simeq 2$ and $R_{BLR}\simeq 3\cdot 10^{17}$cm, this comes to about 3 minutes, including the slowing down by the source redshift effect at $z=0.536$.

• •

Hard, unabsorbed gamma-ray spectra. We recall from the beginning of Sect. 4 that in the present context electron accelerations take place in the very same narrow gaps $d_{g}$ (cf. Eq. 3) where the highest seed density is built up and the induced $\mathbf{E}$-fields are largest. So the electrons can replenish their fast IC losses and retain a flat energy distribution. On the other hand, the IC upscattering of the mirrored photons occurs at radii beyond $R_{BLR}$; there we evaluate the optical depth to photon-photon pair-producing interactions in the AGN frame from $\tau_{abs}\simeq\sigma_{T}\,U_{m}\,d_{g}/(3\,\epsilon)\,\sim 0.2\,$ (where $\epsilon\sim 1$ eV is the typical energy of the soft target photons) and obtain values smaller than unity even with conservative values $\Gamma\sim 20$.

We assume the plasmoids move along the jet following its opening angle $\theta\,\sim\,\Gamma^{-1}$, so the transversal size is $l^{\prime}\,\simeq\,\theta\,R\,\simeq 10^{16}$cm at the BLR edge and becomes a few times larger at the receiving point $\tilde{r}\,\sim\,5\,10^{17}$cm. The angular alignment between plasmoids and mirror $l^{\prime}_{m}/\tilde{r}\sim 3\,10^{-2}$ implied by the model yields rare strong flickers over and above a much longer plateau, as indicated by the observations.

We note that the energetics required by the 2015 June event observed from 3C 279 that attained $E\simeq L^{\prime}_{\gamma}\,\Gamma^{2}\,t_{v}\,\simeq\,{\rm a\,few}\,10^{48}$ erg can be matched by a relativistic energy content of a plasmoid given by $E\simeq(4/3)\,\pi\,\ell^{\prime 3}\,n^{\prime}\,m_{e}\,c^{2}\,\gamma_{b}\,\sim$ a few $10^{48}$ erg, with $n^{\prime}\sim 10^{3}\,\rm cm^{-3}$. This implies a pressure balancing that of the magnetic field and can be maintained by continued accelerations from magnetic reconnections, as computed and discussed by Petropoulou et al. 2016.

On the other hand, a mirror Compton reflectivity given by $f\simeq n^{\prime}\,\sigma_{T}\,l^{\prime}_{m}\,\sim$ a few % requires electron densities in the mirror around a few $10^{6}$ cm*-3*; this leads to a proton-dominated kinetic power $\sim\,\pi\,l^{2}_{m}c\,n^{\prime}m_{p}c^{2}\Gamma^{2}_{m}\,\sim\,10^{47}$erg s*-1* of the mirror. Note that deceleration and snow-plough effects through the jet will affect the leading mirror and help it to slow down toward small values $\Gamma_{m}\,\sim$ 2 (see end of Sect. 4). Such low values ensure a mirror contribution to the jet power of the same order (or slightly below) that of the other plasmoids, in spite of its higher density. So the total power carried by the jet is $L_{j}\,\sim\,10^{47}$erg s*-1* close to the Eddington limit for a BH mass of $10^{9}M\odot$. Such a value is at the upper end of the FSRQ range discussed by Celotti & Ghisellini 2008.

Virtues of our model include: boost values within $\Gamma\simeq 20$, as provided by the radio observations of 3C 279; standard values of the magnetic field $B\sim 1$ G; electron magnetization $\sigma_{e}>10^{2}$, high enough as to allow efficient electron acceleration up to $\gamma\sim 10^{3}$ around or within plasmoids; and apparent gamma-ray luminosities up to $10^{49}$ erg s*-1* from compact regions.

We acknowledge partial support trough the ASI grant no. I / 028 / 12 / 2.