FERMI GAMMA-RAY SPACE TELESCOPE OBSERVATIONS OF GAMMA-RAY OUTBURSTS FROM 3C 454.3 IN 2009 DECEMBER AND 2010 APRIL

The radio source 3C 454.3 is a well-known flat spectrum radio quasar (FSRQ) at redshift z = 0.859. It entered a bright phase starting in 2000 and has shown remarkable activity in the past decade. It underwent major outbursts in 2005, reaching an R-band magnitude of 12.0 and the largest apparent optical luminosity ever recorded from a blazar (Fuhrmann et al. 2006; Villata et al. 2006; Giommi et al. 2006). It also underwent major outbursts in 2007 (Ghisellini et al. 2007; Vercellone et al. 2009a) and 2008 (Vercellone et al. 2010; Jorstad et al. 2010).

First observations of 3C 454.3 with the Fermi-Large Area Telescope (LAT) began in 2008 July during Fermi's commissioning period, when the source was found at a high flux state with F_E>100 MeV ≅ 10 × 10⁻⁶  photon  cm⁻² s⁻¹ (Abdo et al. 2009). During this time, most observations were carried out in a pointed mode with 3C 454.3 being close to the edge of the field of view (≃55° off-axis), which did not allow for detailed spectral analysis during the brightest stage of the outburst. Observations in the decay stage, performed in the survey mode, revealed a timescale less than 2 d for the flux to decline by a factor of 2. The spectrum showed a spectral break around 2 GeV with a spectral steepening from Γ₁ = 2.3 to Γ₂ = 3.5. Such a break has now been found to be a common feature in bright FSRQs and in some low-synchrotron peaked BL Lacertae objects as well (Abdo et al. 2010d). Based on weekly light curves, a very moderate "harder when brighter" effect has also been observed, with the photon spectral index obtained with a single power-law (PL) fit varying by less than 0.3 for flux ratios varying by >7 (Abdo et al. 2010d). The source is listed as 1FGL J2253.9+1608 in the First-LAT active galactic nucleus (AGN) catalog (Abdo et al. 2010b).

Because of its brightness, 3C 454.3 is the first source for which daily resolved broadband spectral energy distributions (SEDs) with GeV data have been obtained (e.g., Abdo et al. 2009; Bonnoli et al. 2010). This offers a wealth of information on the source's spectral states. For example, the γ-ray emission has been observed to correlate with the optical and X-ray bands, pointing to the presence of a single emission zone (Bonning et al. 2009b; Bonnoli et al. 2010), with no significant lag between optical and γ-rays. Strong Lyα radiation has been observed from 3C 454.3 (Bonnoli et al. 2010), providing an external photon source for Compton scattering along with torus emission (Sikora et al. 2009). Modeling of such SEDs with single-zone models has been performed to derive jet powers, with Compton-scattered components from the accretion disk and the broad-line region (BLR; Finke & Dermer 2010). Bonnoli et al. (2010) studied the single-day broadband SEDs in the bright flare discussed here and in a lower state and found that a single-zone external Compton (EC) from the disk, X-ray corona, and torus plus synchrotron self-Compton (SSC) model adequately fitted the data. However, with a similar analysis using AGILE data, Pacciani et al. (2010) came to the conclusion that an additional component was required to fit the data.

The differential distribution of the fluxes of the daily light curves, measured over 18 months, increases as F^1.5 for low fluxes and peaks around F[E>100 MeV] ≃ 3 × 10⁻⁶ photon cm⁻² s⁻¹ and the fraction of the time where the source reaches a flux greater than 10 times the average flux is only 1% (Tavecchio et al. 2010). Foschini et al. (2010) claim γ-ray variability from 3C 454.3 on timescales as short as a few hours from the LAT data and implications on the size and distance to the black hole of the emitting region were discussed by Tavecchio et al. (2010). The results from the flaring state discussed above (as well as those presented in this paper) cannot be seen as typical for this source, as 3C 454.3's level of activity was much lower during the EGRET era, when its mean flux was F[E>100 MeV] ≃ 5.4 × 10⁻⁷ photon cm⁻² s⁻¹ (Hartman et al. 1999).

The continuous monitoring by the Fermi-LAT showed that the source activity faded continuously in early 2009 and then rose back up from June onward. It underwent an exceptional outburst in 2009 November–2010 January when it became the brightest γ-ray source in the sky for over a week, reaching a record daily flux level in the GeV band as seen by the LAT and AGILE (Striani et al. 2009a, 2009b; Escande & Tanaka 2009; Striani et al. 2010). At the same time, it also showed strong activity at optical frequencies (Villata et al. 2009; Bonning et al. 2009a; Sasada et al. 2009), in the Swift/XRT and Swift/BAT bands (Sakamoto et al. 2009; Krimm et al. 2009), and in the INTEGRAL/IBIS band (Vercellone et al. 2009b). Although its γ-ray flux reached record levels, the γ-ray luminosity did not, as larger luminosities have been observed for blazars located at larger distances such as PKS 1622-297 (Mattox et al. 1997). During the brightest flares, the power corresponding to the γ-ray luminosity was found to be similar to or greater than the accretion power of the disk (Bonnoli et al. 2010).

The source remained active afterward with a slowly decaying flux around 2 × 10⁻⁶ photon cm⁻² s⁻¹ until early 2010 April, when it brightened up again to a flux level of ≈16 × 10⁻⁶ photon cm⁻² s⁻¹, prompting the first Fermi-LAT target-of-opportunity (ToO) pointed observation beginning on MJD 55,291 (2010 April 5) lasting for 200 ks.

These two major events offer a unique opportunity to probe intra-day variability and the associated spectral changes in the γ-ray band, which is the focus of this paper. Particular attention is paid to the correlated spectral/flux variations on different time scales, both in terms of spectral hardness and position of the spectral break, which have not been investigated in detail before. In Section 2, observations and analysis of data from 3C 454.3 from 2009 August through 2010 April are presented. Section 3 gives results of the analysis and Section 4 provides interpretation. We summarize in Section 5. A flat ΛCDM cosmology with H₀ = 71 km s⁻¹ Mpc⁻¹, Ω_m = 0.27, Ω_Λ = 0.73 is used in this paper, implying a luminosity distance d_L = 1.69 × 10²⁸ cm to 3C 454.3.

The Fermi-LAT is a pair-conversion γ-ray telescope sensitive to photon energies greater than 20 MeV. In its nominal scanning mode, it surveys the whole sky every 3 hr with a field of view of about 2.4 sr (Atwood et al. 2009). The data presented in this paper (restricted to the 100 MeV–200 GeV range) were collected from MJD 55,070 (2009 August 27) to MJD 55,307 (2010 April 21) in the survey mode, except for a 200 ks period starting at MJD 55,291.82 (2010 April 5 19:38 UT) when the pointed mode was used, resulting in a gain of rate of accumulation of exposure by about a factor of 3.5 over the survey mode. During the pointed mode, the source direction was offset by 10° with respect to the LAT axis in order to limit adverse effects related to gaps in the detector that can affect on-axis photons. Over 3000 photons with E>100 MeV were collected in the pointed mode. To minimize systematics, only photons with energies greater than 100 MeV were considered in this analysis. In order to avoid contamination from Earth limb gamma-rays, a selection of events with zenith angle <105° was applied. This analysis was performed with the standard analysis tool gtlike, part of the Fermi-LAT ScienceTools software package (ver. v9r15p5). The P6_V3_DIFFUSE set of instrument response functions was used. This set includes a correction for the average reduction of the effective area due to pile-up effects as fewer photon events pass the rejection cuts. This correction is sufficient for integration times longer than a day. The residual energy and trigger-rate-dependent acceptance variations present for shorter times as established with Vela and Galactic diffuse emission data (typically amounting to less than 10% in the considered periods), have not been corrected for in this analysis.

Photons were selected in a circular region of interest (ROI) 10° in radius, centered at the position of 3C 454.3. The isotropic background, including the sum of residual instrumental background and extragalactic diffuse γ-ray background, was modeled by fitting this component at high galactic latitude (file provided with ScienceTools). The Galactic diffuse emission model version "gll_iem_v02.fit," was used, with both flux and photon spectral index left free in the fit (the Galactic longitude and latitude of 3C 454.3 are 861 and −382, respectively). All point sources lying within the ROI and a surrounding 5° wide annulus with a flux greater than about 1% of the quiescent level of 3C 454.3 were modeled in the fit with single PL distributions.

Although the actual spectral shape is better reproduced by a broken power law (BPL), the spectral variations were investigated using the photon indices resulting from single PL fits, as these indices are determined with a lower statistical uncertainty than those obtained from BPL fits. All light curves were produced using fluxes derived with PL fits.

Different analyses were performed by fitting the spectra with various models over the whole energy range covered by the LAT at E>100 MeV: a BPL $N(E)=N_0 (E/E_{\rm break})^{-\Gamma _{i}}$, with i = 1 if E < E_break and i = 2 if E>E_break, a log parabola function ($N(E)=N_0 (E/E_{p})^{-\alpha -\beta \log (E/E_p)}$ where E_p is fixed at 1 GeV) and a PL with exponential cutoff function (N(E) = N₀(E/E₀)^−Γexp(−E/E_cutoff)), and with a PL model over equally spaced logarithmic energy bins with the spectral index kept constant and equal to the value fitted over the whole range.

In case of fits with BPL models, the break energy (E_break), which separates the photon energy ranges where different photon indices Γ₁ (for E < E_break) and Γ₂ (for E>E_break) apply, could not be obtained directly from the fit because of convergence problems due to the non-smooth character of the BPL function at the break energy. It was instead computed from a log-likelihood profile fitting procedure, with a statistical uncertainty corresponding to a difference of −2ΔL = 1 in the log-likelihood function L with respect to its minimum.

In order to minimize spurious correlations between flux and spectral index, the fluxes $F_{ E>E_1}$ were also computed above the photon energy E₁ where the correlation between integrated flux and index is minimal. This energy was derived from a PL fit of the form $N(E)=F_{E>E_0}(\Gamma -1)\:E^{-\Gamma }/E_0^{-\Gamma +1}$ (with E₀=100 MeV). Since $F_{E>E_1}=F_{E>E_0} (E_1/E_0)^{-\Gamma +1}$, minimizing $\Delta F_{E>E_1}/F_{E>E_1}$ with respect to E₁ yields $\ln (E_1/E_0)=C_{F\Gamma }/(F_{E>E_0}\:C_{\Gamma \Gamma }$), where C_FΓ and C_ΓΓ are terms of the covariance matrix returned by the fit. In the 3C 454.3 case, E₁ has been found to be 163 MeV over the ToO time range. The same value has been used for the other time periods as well.

The estimated systematic uncertainty on the flux is 10% at 100 MeV, 5% at 500 MeV, and 20% at 10 GeV. The energy resolution is better than 10% over the range of measured E_break.

Figure 1 displays the daily light curves (red points) from MJD 55,070 to 55,307 (2009 August 27–2010 April 21) for fluxes above 100 MeV. The periods showing the fastest flux variations during the December flare, with fluxes changing by more than a factor of 2 in amplitude, are enlarged in the insets, with the E>100 MeV fluxes averaged over a daily, 6 hr, and 3 hr binning shown by the red, open blue, and green data points, respectively. The error bars are statistical only. Three flares displaying a flux variation greater than a factor of 2 over less than a day (MJD 55,167, 55,170, and 55,195) have been studied extensively during this period. These flares exhibit very different profiles and degrees of symmetry. The first two flares were fitted with the function (Abdo et al. 2010a)

$$\begin{equation} F=2F_0(e^{(t_0-t)/T_r}+e^{(t-t_0)/T_f})^{-1}+F_{\rm bgd}(t), \end{equation} \tag{ 1 }$$

where T_r and T_f are the rising and falling times, F₀ is the flare flux amplitude, and F_bgd(t) is a (slowly varying) background flux, while for the third one a constant plateau flux was allowed between the rising and falling phases. The characteristic flare duration can be estimated as T_r+T_f. We obtained T_r = 0.37 d and T_f = 0.06 d for the MJD 55,167 (December 2) flare, and T_r = 0.07 d and T_f = 0.26 d for the MJD 55,170 (December 5) flare. The MJD 55,195 flare, which is less prominent, shows a rise time of ≃0.16 d. In the two early flares, the flux variation F₀ was greater than 10⁻⁵ photon (>100 MeV) cm⁻² s⁻¹, and statistically significant factor of 2 variations take place on timescales as short as 3 hr.

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In Figure 1, the light curve of the 2008 July–August outburst (shifted by 511 d; see Abdo et al. 2009) is shown for comparison. The resemblance of the two light curves is notable, although the estimated fluences are different by 40%. The brightest periods of the outbursts last for about 10 d and are then followed by a long tail of fairly high activity upon which are superimposed minor flares lasting for a few days. Although the outburst in 2010 April is longer, lasting for > 30 d, it exhibits similar patterns as the 2009 December outburst, where its maximum level is preceded by an intermediate elevated flux lasting for about 5 d. This feature could serve as an alert for an imminent surge in flux.

The power density spectrum (PDS) of the MJD 55,140–552, 603 hr light curve has been calculated in a similar way as that reported in Abdo et al. (2010a) for the 11 month light curve (3 d bin), normalized to fractional variance per frequency unit (f) and white-noise subtracted. The time period is marked by lines in Figure 1. This PDS indicates a PL power density, 1/f^a with index a = 1.50 ±  0.06, i.e., intermediate between flickering (red noise) and shot noise (driven by Brownian processes). Such a result confirms over a wider frequency range the value found with the first 11 months of data. In Figure 2, the first-order structure function (SF), the PDS, and the Morlet continuous wavelet transform (CWT) of the continuous, unprecedented-resolution, γ-ray light curve of MJD 55,140–55,260 are reported. A break around 6.5 d is suggested by the SF analysis, the power-index slopes being a = 1.29 ± 0.10 between 3 hr and 6.5 d, and a = 1.64 ± 0.10 between 6.5 and about 26 d, while at longer lags spurious drops due to the finite range become apparent. The PDS analysis confirms these values (a = 1.40 ± 0.19 at high frequency and a = 1.56 ± 0.18 at low frequency). The temporal behavior of 3C 454.3 is therefore showing 1/f^1.5 universality from 3 hr to 11 month timescales. The 6.5 d break, consistently confirmed by the PSD and more clearly depicted by the SF, represents a steepening toward longer lags (flattening toward higher frequencies) and does not necessarily imply a local characteristic time scale, as it could simply be the point where two PDS components with different slopes are equally strong. Gamma-ray variability in 3C 454.3 can be seen as a short-term realization of a stochastic mechanism, where structures that are resolved in shorter observations are simply averaged out in long observations, and where big outbursts correspond to statistical tails of the same process.

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The CWT in Figure 2 provides a local and detailed time series analysis, through the two-dimensional energy density function (modulus of the transform, filled color contour) computed using a Morlet waveform, providing the best tradeoff between localization and period/frequency resolution. For timescales below 1 d no local peaks are found, although some marginal features in this time range are found during the outburst state. The big outburst of 2009 December is, in fact, localized and decomposed in a chain of well-defined power CWT peaks. The 6.5 d timescale is confirmed by the major peak out of the cone of influence (localized at about MJD 55,166, i.e., the onset of the outburst, MJD ∼ 55,166.2–55,172). A second energetic peak in this period is found at about 2.5 d manifesting another dominating timescale during the outburst, while in the second period of the outburst two minor power peaks at about 19 hr and 1.3 d are also visible. Based on this CWT local analysis, there is no evidence for structure on timescales shorter than about 12 hr but shorter timescales close to the sampling scale cannot be ruled out.

The 1 GeV daily light curve, shown in Figure 3 (top), closely resembles the 100 MeV light curve, hinting at little spectral variability. This behavior is confirmed by the very limited variation of the photon spectral index measured at E>E₁ = 163 MeV, displayed in Figure 3 (bottom) by the daily average photon index (open blue symbols) as well as the weekly averaged ones (solid black points). The near constancy of the spectrum is in accord with the results found from the 2008 July flare and the first 6 months of LAT data (Abdo et al. 2009, 2010d).

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The variation of the amplitude of the weekly photon indices is only ΔΓ = 0.35 (varying between 2.35 and 2.7) during the period under consideration, but the variation is statistically significant. Comparing the weekly photon indices to their weighted average returns a reduced χ² = 86.4/32, corresponding to a 10⁻⁷ probability for a nonvariable signal. Particularly notable is a slight softening of the spectrum during the periods of lowest fluxes, particularly during the periods of MJD 55,221–55,235 and 55,264–55,278. There is also a suggestion that a progressive hardening over several weeks precedes a major outburst, but more such events will be needed to establish whether this behavior is typical.

The correlation between photon index and flux was further investigated by computing a discrete cross-correlation function (DCCF) following Edelson & Krolik (1988). Error estimates were obtained by a Monte Carlo method (Peterson et al. 1998). The DCCF for the time of the December outburst and its decay (MJD 55,136–55,280) show evidence for a time lag such that index variations lead the flux by about 7 d. The photon flux above 163 MeV was used for this analysis. As stated above the light curve and photon index plots indicate a spectral softening at low flux levels. We therefore calculated a DCCF also using the logarithm of the flux, which is less dominated by the high flux values. The DCCF for this case (see Figure 4) gives a somewhat stronger correlation than for linear flux, supporting the impression that spectral softening at low flux levels has a clear contribution to the correlation. The time lag of the correlation peak, estimated by fitting a Gaussian function, was −6.8 ± 2.8 d when log(flux) was used and −7.9 ± 3.7 d for linear flux. As error values we used the standard deviation of Gaussian fits to the DCCF of Monte Carlo simulations, again along the lines of Peterson et al. (1998). As an additional test of the significance we divided the flare light curve into three segments (MJD 55,136–55,190, 55,190–55,240, and 55,240–55,280), each of which showed a time lag similar to that of the overall data. In contrast, the correlation during the 2010 April flare is weak and if anything shows a lag in the opposite sense. On the other hand, the data from the 2008 flare of this source do exhibit a time lag (−5.6 ± 3.4 d) similar to that of the 2009 December flare. The main uncertainty in the DCCF analysis is the possibility of spurious correlations. The fact that a similar time lag is seen in a number of independent data sets gives support for the, still tentative, detection of a lag between the flux and photon-index variations.

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Figure 5, top left (right) presents the weekly (daily) averaged PL photon spectral index versus flux above E₁ (the photon energy where the correlation between integrated flux and index is minimal). A weak "harder when brighter" effect shows up for weekly bins, but is almost washed out when considering daily bins. To make the trend clearer, photons were sorted in three daily-flux bins ($F_{E_1}<$2.5, $2.5<F_{E_1}<5$, and $5<F_{E_1}<10$, where $F_{E_1}$ is the photon flux above E₁ in units of 10⁻⁶ photon cm⁻² s⁻¹) and the analysis was repeated with the resulting photon files. The result, displayed as red points in the top right panel of Figure 5, still exhibits a slight harder when brighter effect. The data point at $F_{E_1}\cong 11$ corresponds to the MJD 55,167 flare and is consistent with the trend observed at lower flux. The three other panels in Figure 5 show the photon spectral index versus flux patterns for the three rapid flares, obtained either with a 6 hr binning (middle left and bottom left) or a 3 hr binning (middle right). The light curves at E>E₁ (not above 100 MeV, as in Figure 1) calculated at the time of the main flaring episodes are displayed in the insets of Figure 1. Despite the flux reaching the highest values for a non-gamma-ray burst (GRB) cosmic source, the statistical significance of these patterns is marginal, except for the MJD 55,167 flare. Instead of a well-defined, universal pattern, a variety of patterns is found. The MJD 55,167 flare is associated with a clockwise pattern, with a flux-doubling accompanied by an essentially constant (or weakly harder) photon spectral index. The reduced χ² for a constant fit of the photon spectral index is 28.6/9. The pattern is somewhat indicative of a "hard-lag" effect possibly reflecting particle acceleration (Kirk & Mastichiadis 1999). The short MJD 55,170 flare shows an indication for a counterclockwise loop (χ²_r = 6.0/7). The loop diagram for this flare on 3 hr timescales reveals the simplest ordered pattern, in this case a softening followed by a flux increase which then decays in flux and hardness. This could reflect the underlying timescale on which coherent variability takes place in 3C 454.3. Because this pattern does not recur in other flaring episodes, no strong conclusions can be made, however. The MJD 55,195 flare shows some softening during the plateau (points 3–6) following the flux rise as expected from a cooling behavior, but the significance is low (χ²_r = 8.6/9). While it is difficult to draw any firm conclusion on acceleration and cooling from these patterns, the lack of strong spectral variability still provides clues to the underlying physical processes.

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Figure 6 shows the flux and photon spectral index as a function of time in the period around (blue symbols) and during (red symbols) the time of the ToO when the Fermi-LAT was in the pointed mode (MJD 55,291.82–55,294.13). The binning is 6 hr and 3 hr for the survey and pointed modes, respectively. As expected by the 3.5-fold increase in exposure per unit time during the ToO, the statistical accuracy in the measurement of both parameters improves significantly. Although in a high state, the source was unfortunately fairly steady during this period. No indication for variability more rapid than that observed during the giant outburst is found during the ToO period, as already noted by Foschini et al. (2010). The photon spectral index versus flux above E₁ measured in 3 hr time bins is plotted in Figure 7. The time sequence is indicated by labels, the first and last points of the sequence being labeled as well in the light curve of Figure 6. The reduced χ² for a constant fit of the photon spectral index is 18.52/16, indicating that the data are consistent with a constant value. The correlation coefficient is 0.26.

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Figure 8 shows the integrated spectrum from Period 1, MJD 55,121–55,165; Period 2, MJD 55,166–55,173 (week of the giant outburst); Period 3, MJD 55,174–55,262; Period 4, MJD 55,280–55,300. The distributions have been fitted with a BPL, a log parabola function, and a PL with exponential cutoff function. The log parabola gives a worse fit than the BPL and the PL with exponential cutoff functions, which are difficult to discriminate for these periods. The fitted parameters are given in Tables 1–3 for the four periods.

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The variation of break energy (cutoff energy) with flux is displayed in the left (right) panel in Figure 9 at different observing periods. No strong evolution of either the break energy or the cutoff energy is found, but there is some indication of a slight hardening with flux. For a given flux, the position of the break energy is slightly different from that observed during the bright outburst in 2008, but all E_break (and E_cutoff) are constant within a factor of ≈2. For the same periods, Striani et al. (2010) found fairly large spectral variability with the AGILE data, with a photon spectral index as low as 1.66 ± 0.32 during big flares. This behavior is not confirmed by the present analysis. The discrepancy is at the 2σ level (considering statistical errors only). The reason for this discrepancy remains unclear. The systematic uncertainty affecting the AGILE results may possibly be a factor. (The advertized uncertainty is 10% on the flux in Pittori et al. (2009), it may be fairly large in the photon index as well.)

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The maximum photon energy found within the 95% containment radius from the location of 3C 454.3 during the period MJD 55,140–55,261 was a photon with E = 20.7 GeV at MJD 55,179.98, when F_E>100 MeV = 6 × 10⁻⁶ photon cm⁻² s⁻¹ and the variability time t_var ≈ 1 d. On MJD 55,167 (December 2), when the average of F_E>100 MeV over this day exceeded ≈20 × 10⁻⁶ photon cm⁻² s⁻¹, the highest energy was 8.5 GeV. As already noted, the flux variations during this flare were on timescales as short as a few hours or less. The energy of the most energetic photon during periods of rapid flux changes provides the strongest constraint on the γγ opacity, as discussed below.

Thanks to this series of outbursts observed with the Fermi-LAT, a much more accurate picture of the behavior of 3C 454.3 in flaring states has been obtained. The unprecedented correlated spectral and flux variability studies in the γ-ray band enabled by the high observed flux have revealed interesting features (universality of the PSDs for time scales from a few hours to several months, mild spectral variability, weak change in the spectral break with flux), which can reasonably be assumed to apply to some extent for many flaring FSRQs. However, the verification of this assertion might be difficult given the exceptional character of the fluxes considered here. Because of the Fermi-LAT characteristics, sources with a spectrum harder than that of 3C 454.3 can potentially be studied with a similar accuracy at a significantly lower flux, but only relatively few FSRQs have harder spectra than 3C 454.3 (Abdo et al. 2010b).

A photon flux of F_E>100 MeV =  22 ±  1 × 10⁻⁶ photon cm⁻² s⁻¹ from 3C 454.3 at z = 0.859 implies an apparent isotropic γ-ray luminosity above 100 MeV of L_γ ≅ 3.8 ± 0.2 × 10⁴⁹ erg s⁻¹. This is ∼3 times larger than the luminosity of the z = 1.839 blazar PKS 1502+106 during its 2008 August flare (Abdo et al. 2010c), but still lower than the luminosity of PKS 1622−297 (∼5.1 × 10⁴⁹ erg s^-1 with the current cosmological model) during the 1995 flare (Mattox et al. 1997). The time-averaged γ-ray luminosity (Tables) measured with the Fermi-LAT is ≈9 × 10⁴⁸ and ≈1.6 × 10⁴⁹ erg s⁻¹ for periods 1 and 2, respectively. When written in units of 10⁹M₉ Solar masses, estimates for the black-hole mass in 3C 454.3 range from M₉ ≈ 0.5 (Bonnoli et al. 2010) to M₉ ≈ 4 (Gu et al. 2001). To be radiating below the Eddington luminosity of 1.26 ×10⁴⁷(M₉ M_☉) erg s⁻¹ implies that the high-energy radiation is beamed into a jet with an opening angle θ_j ≲ 0.1.

The light curves from Figure 1 show evidence for a variability timescale of a few hours and the model fitted to a very brief subflare at MJD 55,170 implies a flux doubling timescale of (ln 4) × T_r ∼ 2.3 hr. However, if the bulk of the flare were characterized with such a short timescale, we would expect a very erratic light curve calculated with daily binning. Because the overall shape of the 2009 November/December flare is smooth, the dominating variability timescale is rather close to t_var ∼ 1 d. The short subflares may reflect existence within or outside the main emitting zone of geometrical substructures with comoving emission-region sizes smaller than

$$\begin{eqnarray*} R^\prime &\approx & {\delta ct_{\rm var}\over (1+z)} \approx 3.3\times 10^{15}\bigg ({\delta \over 25}\bigg)\bigg ({t_{\rm var}\over 2.3\,\rm hr}\bigg)\;{\rm cm}\nonumber\\ &\approx & {22\over M_9} \bigg ({\delta \over 25}\bigg)\bigg ({t_{\rm var}\over 2.3\,\rm hr}\bigg)R_{\rm g}, \end{eqnarray*}$$

where R_g = GM/c² is the gravitational radius, δ ≡ [Γ_b(1 − βcos θ)]⁻¹ is the Doppler factor, Γ_b is the bulk Lorentz factor and $\beta c= c\sqrt{\vphantom{A^A}\smash{\hbox{${1-(1/\Gamma _b^2)}$}}}$ is the speed of the jet plasma, and θ is the angle between the jet axis and observer's line of sight.

The minimum Doppler factor δ_min is defined by the condition that the optical depth τ_γγ(₁) of a photon with energy E₁ = ₁m_ec² to the γγ pair-production process is τ_γγ(₁) = 1, and can be estimated to ≈10% accuracy for target photon number indices <−1 compared to results of more detailed numerical calculations through the expression

$$\begin{equation} \delta _{\rm min}\cong \left[\frac{\sigma _T d_L^2(1 + z)^2 f_{\hat{\epsilon }} \epsilon _1}{4t_{\rm var}m_ec^4}\right]^{1/6} \end{equation} \tag{ 2 }$$

(Dondi & Ghisellini 1995; Ackermann et al. 2010). Here f is the νF_ν spectrum of 3C 454.3 measured at frequency ν = m_ec²/h. To estimate δ_min, the photon with maximum energy E₁ is used during the period in which f and variability time t_var = t_var,d d are measured. The νF_ν flux f in Equation (2) is evaluated at $\epsilon = \hat{\epsilon }={2\delta ^2}/{(1+z)^2\epsilon _1}$ from the pair-production threshold condition ' ≈ 2, where primes on quantities refer to values measured in the comoving frame, so '₍₁₎ = (1 + z)₍₁₎/δ. Writing f = 10⁻¹⁰ f₋₁₀ erg cm⁻² s⁻¹ in Equation (2) gives δ_min ≈ 13[f₋₁₀ E₁(10GeV)/t_var,d]^1/6 and $\hat{E}({\rm keV}) \cong 9.5 (\delta /25)^2/E_1$(10 GeV). Swift/XRT observations contemporaneous with the time that the 20 GeV photon was detected show a νF_ν ≈3 keV X-ray flux ≈5 × 10⁻¹¹ erg cm⁻² s⁻¹ (e.g., Bonnoli et al. 2010), so that δ_min ≈ 13.

The value of the Doppler factor can also be constrained from the argument that the energy flux from the SSC component should be below that observed at soft X-rays, in terms of isotropic luminosities L_SSC ⩽ L_X. SSC luminosity is related to the synchrotron (SYN) luminosity via L_SSC/L_SYN ∼ u'_SYN/u'_B in the Thomson limit, where u'_SYN ∼ L'_SYN/(4πR'²c) is the energy density of synchrotron radiation and u'_B = B'²/8π is the energy density of the magnetic field. Noting that Lorentz transformation of bolometric luminosity is L_SYN = δ⁴L'_SYN, we obtain

$$\begin{equation} \delta ^6 \ge \frac{2(1+z)^2L_{\rm SYN}^2}{L_{\rm X}c^3t_{\rm var}^2B^{\prime 2}}\,. \end{equation} \tag{ 3 }$$

This can be combined with a constraint from the EC and synchrotron luminosities ratio,

$$\begin{equation} \frac{L_{\rm EC}}{L_{\rm SYN}} \sim \delta ^2\frac{u_{\rm EXT}}{u^\prime _{\rm B}}\, \end{equation} \tag{ 4 }$$

(Dermer 1995), where u_EXT is the energy density of external radiation. Eliminating from these equations the magnetic field value, we obtain

$$\begin{eqnarray} \delta &\ge & 17.6 \bigg ({L_{\rm SYN}\over 10^{48}\, {\rm erg\, s}^{-1}}\bigg)^{1/8} \bigg ({L_{\rm EC}\over 10^{49}\, {\rm erg\, s}^{-1}}\bigg)^{1/8}\nonumber\\ &&\times \bigg ({L_{\rm X}\over 10^{47}\, {\rm erg\, s}^{-1}}\bigg)^{-1/8} t_{\rm var,d}^{-1/4} \left(\frac{u_{\rm EXT}}{0.015\,{\rm erg\, cm}^{-3}}\right)^{-1/8}\,.\quad \end{eqnarray} \tag{ 5 }$$

On the other hand, we can eliminate the Doppler factor and obtain a lower limit on the magnetic field

$$\begin{eqnarray} B^\prime ({\rm G})& \ge & 3.4\; \bigg ({L_{\rm SYN}\over 10^{48}\, {\rm erg\, s}^{-1}}\bigg)^{5/8} \bigg ({L_{\rm EC}\over 10^{49}\, {\rm erg\, s}^{-1}}\bigg)^{-3/8}\nonumber\\ &&\times\bigg ({L_{\rm X}\over 10^{47}\, {\rm erg\, s}^{-1}}\bigg)^{-1/8} t_{\rm var,d}^{-1/4} \left(\frac{u_{\rm EXT}}{0.015\,{\rm erg\, cm}^{-3}}\right)^{3/8}\,.\nonumber\\ \end{eqnarray} \tag{ 6 }$$

Order-of-magnitude reference values of isotropic luminosities L_SYN, L_X, and L_EC have been deduced from Figure 4 in Bonnoli et al. (2010). If the source region is located within the BLR, the energy density of external radiation is u_EXT ∼ L_BLR/(4πr²_BLRc), where L_BLR is the luminosity of the broad emission lines and r_BLR is the characteristic radius of the BLR. We approximate L_BLR with GALEX measurement of Lyα line, L_Lyα ≈ (2–4) × 10⁴⁵ erg s^-1 and adopt r_BLR ≈ 6 × 10¹⁷ cm based on reverberation scaling relations for C iv line (Bonnoli et al. 2010; Kaspi et al. 2007).

If the emission region is found deep within the broad-line region, the accretion-disk radiation can provide the dominant target photon source for γ-ray production. The target photon energy density is very sensitive to the BLR radius. Steady Lyα radiation as observed, e.g., in nearby radio-quiet AGN (Kaspi et al. 2007) could reflect a large radius for Lyα line photons, so other line radiations such as C iv λ1549 would instead dominate as the target photon source. In this case, u_EXT and the magnetic field given by Equation (6) would be smaller. Lower estimates of u_EXT imply smaller values of B'. Accretion-disk photons could make an important external radiation field, which would determine the magnetic field according to its contribution to u_ext and different estimates for the line fluorescence and scattered radiation fields. The Doppler factor is, however, still determined by the minimum value derived from gamma–gamma constraints, so it does not change much even if accretion disk photons make up the dominant target photon source, because internal SSC photons provide the dominant gamma–gamma opacity.

The analysis of radio observations made at a different epoch gives δ = 24.6 ± 4.5, bulk Lorentz factor Γ_b = 15.6 ± 2.2, and observing angle θ = 13 ± 12 obtained from superluminal observations (Jorstad et al. 2005). Both our constraints on the Doppler factor are consistent with this result and would become even tighter, if we adopt a shorter variability timescale. Hence, we take δ₂₅ = δ/25 ≈ 1 and Γ₁₅ = Γ_b/15 ≈ 1. The constraint on the value of the magnetic field B' is a few times the equipartition magnetic field B'_eq(G) ≈ 2(f_-10Λ₂)^2/7/(δ^13/7₂₅ν^1/7₁₃t^6/7_var,d), where f_SYN,peak = 10⁻¹⁰f₋₁₀ erg cm⁻² s⁻¹ is the observed synchrotron peak flux, ν_SYN,peak = 10¹³ν₁₃ Hz is the synchrotron peak frequency, and 100Λ₂ = (1 + ζ_pe)ln(₂/₁). Here ζ_pe is the ratio of proton to electron energy, and ₁ and ₂ bracket the νF_ν ∝ ν^1/2 portion of the SED from electron synchrotron radiation (e.g., Finke et al. 2008). The equipartition magnetic field is ≈3.7× smaller in a pure lepton jet.

For a conical geometry of the opening angle θ_j ≳ R/r ∼ 1/Γ_b, the location of the emitting region for the December flare is constrained to be at a distance r ≲ 2cΓ²_bt_var/(1 + z) ≈ 0.2Γ²₁₅t_var,d pc; i.e., toward the outer parts of the BLR. A narrow jet with an opening angle ≲0.2/Γ_b, places the emitting region further out, and could be consistent with a fragmentation of outflow (Marscher & Jorstad 2010). The most rapidly varying flaring episodes suggest locations at the sub-pc scale, contrary to conclusions from coherent optical polarization changes in 3C 279 (Abdo et al. 2010e) and PKS 1510-089 (Marscher et al. 2010) over timescales of tens of days, which suggest that the emitting regions are several pc from the black hole. Larger distances would be inferred if Γ_b ≫ 15 or θ_j ≪ 1/Γ_b. Higher bulk Lorentz factors would conflict with the expected source density of the parent population of FSRQs, as we discuss later. Narrow jets are suggested by radio observations (Pushkarev et al. 2009; Jorstad et al. 2005) and could result from recollimation shocks at the pc scale (e.g., Nalewajko & Sikora 2009; Bromberg & Levinson 2009) that reduce the characteristic size of the emission region. A multiwavelength radio, mm, optical, and X-ray campaign covering observations of 3C 454.3 between 2005 and 2008 show coherent optical and mm flux and polarization episodes consistent with the brightest events taking place at the mm core located ∼ pc beyond the acceleration and collimation zone (Jorstad et al. 2010). Without strong collimation or recollimation, the short observed variability timescale implies a very small value θ_j ∼ 2 × 10⁻³t_var,d/r_pc (Tavecchio et al. 2010) with an associated low radiative efficiency. Alternately, flaring episodes with short variability times might take place within the BLR, whereas the more slowly varying emissions could be radiated by jet plasma at larger distances. Spectral variations due to the different target photon sources might be concealed by mixing from components made at small and large radii. "Residual" collisions at large radii might also make a slowly varying underlying emission components, as similarly inferred for GRB observations (Li & Waxman 2008).

The large changes in flux exhibited by the γ-ray light curves in Figure 1 could be due to several different factors, including a changing mean magnetic field, electron number, Doppler factor, target photon density, and spectrum as the jet moves outward, or some combination of these factors. Any such explanation must also account for the moderate spectral changes, including the near constancy of E_break (Figure 9), while the flux changes by over an order of magnitude.

One possibility is related to scattering of a target photon field in the Klein–Nishina (KN) regime. Compton scattering takes place in the Thomson limit when the energy of the photon to be scattered is (in m_ec² units) '' ≲ 1/4 in the electron rest frame, denoted by the double primes on quantities. Since the radiation from ultrarelativistic electrons is seen only when they are moving very close to the line of sight, that energy for γ ≫ Γ_b is '' ≃ γδ_⋆, where _⋆ is the average energy of the target photons in the lab frame. Hence, the scattering is in the Thomson regime provided the upscattered photon energies are _C ≃ (4/3)δ²γ²_⋆/(1 + z) ≲ 1/[12_⋆(1 + z)], i.e., for E_C(GeV) ≲ 12/E_⋆(eV), independent of the Doppler factor (compare Georganopoulos et al. 2001).

If the break energy observed in 3C 454.3 at ≈2 GeV is due to the transition to scattering in the KN regime, then the underlying target photon energy E_⋆ ≈ 6 eV is close to the energy of the Lyα photon at 10.2 eV. We have tested this possibility by comparing the Compton-scattered spectrum from a PL electron distribution with a monochromatic Lyα photon source with the Fermi-LAT data. The spectrum is independent of δ, though it can depend on maximum and minimum electron Lorentz factors. As shown in Figure 10, the spectrum is too hard to fit the data and treatment of KN effects on cooling (Dermer & Atoyan 2002; Moderski et al. 2005; Sikora et al. 2009) or the addition of other soft photon sources will reduce the sharpness of the break. The difficulty of fitting the sharp spectral break with a single PL electron distribution is in accord with the conclusion of Abdo et al. (2009) that this break reflects a complex electron spectrum. To obtain a good spectral fit to the γ-ray spectrum of 3C 454.3, Finke & Dermer (2010) use a BPL electron distribution and a multi-component Compton-scattering model. The robustness of E_break in this model is due to similar, ≈r⁻³ dependence of accretion disk and external radiation energy density within the BLR.

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Although Abdo et al. (2009) excluded the possibility of explaining the spectral break with photon–photon pair absorption, Poutanen & Stern (2010) proposed that significant intrinsic absorption can be provided by He ii Lyα line at 40.8 eV and continuum at 54.4 eV. However, their model implies that the location of the gamma-ray production is in the inner edge of the BLR and inefficient scattering in the KN regime is predicted to lead to significant hardening of synchrotron spectra in IR/optical bands. Further studies are required to verify whether such effect is in contradiction with multiwavelength spectra created within a one-zone model.

If Γ_b ≅ 15 and the jet opening angle θ_j ≈ 1/Γ_b, then the beaming factor for a two-sided top-hat jet is f_b ≅ 1/2Γ²_b ≅ 1/(450Γ²₁₅). Within the framework of the unification hypothesis for radio galaxies (Urry & Padovani 1995), FRII radio galaxies are the parent population of FSRQs. The space density of FRII radio galaxies at 0.8 < z <1.5 is ρ ≅ 3 × 10⁻⁷ Mpc⁻³ (Gendre et al. 2010). At redshift unity, the volume of the universe ≅4π/3R³_H ≅ 3 × 10¹¹ Mpc³. Therefore, if all FRII radio galaxies were misaligned FSRQs similar to 3C 454.3, then the implied number N of FSRQs is N < f_b × 3 × 10⁻⁷ Mpc⁻³ × 3 × 10¹¹ Mpc³ ≈ 200/Γ²₁₅. This number is similar to the number, ≈100, of γ-ray blazars at redshift unity (Abdo et al. 2010b), so it is in accord with the unification scenario. This simple argument suggests that unless θ_j ≫ 1/Γ_b, in which case the total jet energy of an individual FSRQ blazar would be correspondingly larger, the typical bulk Lorentz factor Γ_b for FSRQs would not be ≫15.

The correlated spectral and temporal properties of 3C 454.3 during two very strong flaring episodes, during which the source was the brightest object in the γ-ray sky, have been studied. An important result of this work is that the significant spectral break between ≈2 and 3 GeV in the γ-ray spectrum of 3C 454.3 is very weakly dependent on the flux state, even when the flux changes by an order of magnitude. The light curves during periods of enhanced activity in 2008 July–August and 2010 December show strong resemblance, with a flux plateau of a few days preceding the major flare. The spectral index measured on a daily basis shows a very moderate "harder when brighter" effect for fluxes measured at E>E₁, where E₁ = 163 MeV is chosen to minimize spurious correlations resulting from the choice of the low-energy bound on the range. However, the weekly spectral index displays a more significant variation, but not exceeding ΔΓ = 0.35 when the flux varies by more than a factor of 10. Indication for a gradual hardening preceding a major flare and extending for several weeks is found. No recurring pattern in the photon spectral index/flux plane measured at E>E₁ during the course of flaring episodes has been identified as would be expected in acceleration and cooling scenarios. Flux variations of a factor of 2 have been observed over time scales as short as 3 hr, though only weak variability was observed during the time of the ToO pointing of the Fermi Telescope toward 3C 454.3.

From γγ opacity constraints, we derive a minimum Doppler factor δ_min ≈ 13 from the flux, variability time, and highest energy photon measurements. This value is in accord with independent measurements of δ from superluminal motion observations (Jorstad et al. 2005). The behavior of the break energy has also been investigated. Spectral softening due to the onset of KN effects in Compton scattering, which is independent of the Doppler factor for a PL electron distribution, was considered as the origin of the γ-ray spectrum. The magnitude of the softening for such a model was found to be inadequate to fit the spectrum without introducing intrinsic breaks in the underlying electron spectrum. An independent estimate of δ and the comoving magnetic field B' of the emission region also give values of δ ≳ 15; also B'≳ several G. The short flaring times suggest an origin of the gamma-ray emission region within the BLR, but possible recollimation shocks, small-scale reconnection events, or more complicated jet geometries are compatible with the production of gamma-ray emission from 3C 454.3 on the pc scale.

The Fermi-LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l'Energie Atomique and the Centre National de la Recherche Scientifique/Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council, and the Swedish National Space Board in Sweden. Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'Études Spatiales in France.