A TIGHT CONNECTION BETWEEN GAMMA-RAY OUTBURSTS AND PARSEC-SCALE JET ACTIVITY IN THE QUASAR 3C 454.3

The γ-ray data are collected with the Large Area Telescope (LAT) of the Fermi Gamma-ray Space Telescope. We construct a daily γ-ray light curve of the quasar using Pass 7 photon and spacecraft data, version V9r23p1 of the Fermi Science Tools, and the instrument responses for the gal_2yearp7v6_v0 and iso_p7v6clean.txt diffuse source models. We model the γ-ray emission from 3C 454.3 and other point sources within 15 deg radius of the quasar with spectral models, as found in the 2FGL catalog of sources detected by the LAT. We have fixed the catalog's spectral parameters of sources within the area and searched for values of flux normalization parameters with 1 day integration intervals of photons between 0.1 and 200 GeV using the standard unbinned likelihood analysis. This produces a γ-ray light curve with 823 measurements, with 59 values representing only upper limits to the γ-ray emission. The flux is considered detected if the test statistic, TS, provided by the analysis exceeds 10, which corresponds to approximately a 3σ detection level (Nolan et al. 2012).

We have acquired X-Ray Telescope (XRT) and Ultraviolet and Optical Telescope (UVOT) data of 3C 454.3 from 2009 April 25 to 2011 August 1 from the Swift archive and processed them with the HEAsoft version 6.11 software package. We have obtained 201 measurements of the flux at 0.3–10 keV. The XRT observations were carried out in a mixture of Photon Counting (PC) and Windowed Timing (WT) modes (Hill et al. 2004). We counted photons with apertures as proposed by Raiteri et al. (2011), a 30 pixel circular region (∼71 arcsec) and an annular region with inner and outer radii of 110 and 160 pixels for the source and background measurements, respectively. All of the observations collected in PC mode were near or exceeded 0.5 counts s⁻¹, indicating possible photon pile-up that was corrected by eliminating 3–5 central pixels. A new exposure map was generated using the Swift XRT task xrtexpomap. For each observation collected in WT mode, we created a box-shaped extraction region individually sized to exclude the point at which the pixels dropped to less than 2 counts. The Swift XRT task xrtmkarf was applied on all extracted spectra. The spectra were rebinned with the FTOOLS task grppha to include a minimum of 20 photons in each channel. We used XSPEC version 12.7.0 to fit the data with a single power-law model while the hydrogen column density was fixed at 1.34 × 10²¹ cm⁻² (Villata et al. 2006).

For the UVOT data, we extracted the magnitude and its error using the tool UVOTSOURCE, specifying a region with a circular aperture of 7 arcsec, and a background annulus region centered on the object with inner and outer radius of 22 and 25 arcsec, respectively. The magnitudes were corrected for Galactic extinction following the procedures outlined in Cardelli et al. (1989), with A(V) = 0.355 mag and E(B − V) = 0.107 mag (Schleger et al. 1998). We converted the magnitudes to fluxes using the central wavelengths for each filter as calibrated by Poole et al. (2008).

The optical photometric data in BVRI bands were collected at various telescopes: (1) the 1.83 m Perkins telescope of Lowell Observatory (Flagstaff, AZ); (2) the 1.54 m Kuiper and 2.3 m Bok telescopes of Steward Observatory (Mt. Bigelow and Kitt Peak, AZ); (3) the 70 cm AZT-8 telescope of the Crimean Astrophysical Observatory (Nauchnij, Ukraine); (4) the 40 cm LX-200 telescope of St. Petersburg State University (St. Petersburg, Russia); (5) the 2.2 m telescope of Calar Alto Observatory (Almería, Spain); (6) the 2 m Liverpool telescope of the Observatorio del Roque de Los Muchachos (Canary Island, Spain); (7) the 1.25 m telescope of Abastumani Astrophysical Observatory (Mt. Kanobili, Georgia); (8) the 60 cm telescope of the Maria Mitchell Association (Nantucket, MA); (9) the 1.3 m telescope at the Cerro Tololo Inter-American Observatory (CTIO), and (10) the UVOT of Swift.

Near-infrared, JHK, photometric data (near-IR) were collected at the 1.1 m telescope of the Main (Pulkovo) Astronomical Observatory of the Russian Academy of Sciences located at Campo Imperatore, Italy (Larionov et al. 2008). The optical and near-IR data were supplemented by measurements by the SMARTS consortium, posted at their website (Bonning et al. 2012). The data have been corrected for Galactic extinction. The conversion factors calculated by Mead et al. (1990) were used to convert magnitudes into flux densities.

Mid-infrared (mid-IR) observations were carried out on 2010 November 3 at the NASA Infrared Telescope Facility (IRTF) with the MIRSI camera (Kassis et al. 2008). The observations were performed in three bands centered at 4.9, 10.6, and 20.7 μm with total on-source integration times of 480, 720, and 960 s, and frame times of 200, 24, and 4 ms, respectively. The comparison star 63 Peg from the MIRAC3 Users's Manual was observed before and after each observation of the quasar to provide the flux density calibration. The data were reduced with an IDL script supplied by the IRTF staff. This resulted in flux measurements of 293 ± 21, 699 ± 37, and 1293 ± 197 mJy at 4.9, 10.6, and 20.7 μm, respectively.

Far-infrared (far-IR) photometric data were collected from 2010 December 25 to 2011 January 10 at 250, 350, and 500 μm with the SPIRE photometer (13 measurements at each wavelength) and with the PACS photometer at 70 and 160 μm (15 measurements at each wavelength) on board the Herschel satellite. The details of the observations and data reduction along with a table of flux densities can be found in Wehrle et al. (2012).

The 0.85 mm (350 GHz) and 1.3 mm (230 GHz) measurements were obtained at the Submillimeter Array (SMA), Mauna Kea, Hawaii, within a monitoring program of compact extragalactic radio sources that can be used as calibrators at mm and submm wavelengths (Gurwell et al. 2007). The data at 0.85 mm (47 data points) and 1.3 mm (215 data points) are supplemented by measurements at the IRAM telescope at 1.3 mm (22 data points) and 3.5 mm (28 data points). The data reduction procedure of the IRAM data can be found in Agudo et al. (2010). We will refer to the combined light curve at 1.3 mm as "the 1 mm light curve."

The 8 mm (37 GHz) observations were performed with the 13.7 m telescope at Metsähovi Radio Observatory of Aalto University, Finland (307 measurements). The flux density calibration is based on observations of DR 21, with 3C 84 and 3C 274 used as secondary calibrators. A detailed description of the data reduction and analysis is given in Teräsranta et al. (1998).

Figure 1 shows the γ-ray, X-ray, UV, optical R-band, and radio light curves from 2009 April 25 to 2011 August 1 (RJD: 4947–5775). Simple visual inspection of the light curves reveals a prolonged, ∼600 day, state of high activity at γ-rays, from 2009 August (RJD ∼ 5050) to 2011 March (RJD ∼ 5650), that coincides with a high state seen in the 1 mm light curve. Within this active state three major γ-ray outbursts occurred, in 2009 December, 2010 April, and 2010 November.

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The optical polarization measurements were performed at telescopes 1–5 as listed above in R band, except for LX-200 of St. Petersburg State University, where the observations were carried out without a filter with the central wavelength λ_eff ∼ 670 nm, and the spectropolarimetric observations at Steward Observatory (see below). The observations at the Calar Alto Observatory were carried out within the Monitoring AGN with Polarimetry at the Calar Alto Telescopes²⁰ program. The details of optical polarization observations and data reduction can be found in Jorstad et al. (2010).

The spectropolarimetric observations of 3C 454.3 at Steward Observatory were part of a currently operating program to monitor bright γ-ray blazars from the Fermi LAT-monitored blazar list.²¹ The observations were performed using the CCD Imaging/Spectro-polarimeter (SPOL; Schmidt et al. 1992), yielding spectra that span the range of 4000–7550 Å with a dispersion of 4 Å pixel⁻¹. Depending on the width of the slit used for the observation, the resolution was typically between 16 and 24 Å. The flux density averaged over 5400–5600 Å was scaled to agree with that determined from the synthetic V-band photometry performed on the same night. As a result, 181 calibrated spectra of the quasar were obtained during the period from 2009 April 25 to 2011 August 1. In polarization mode, the full-resolution Stokes spectra were obtained to calculate the linear polarization parameters within 5000–7000 Å (244 spectra). Details of the spectropolarimetric data reduction can be found in Smith et al. (2009). The combined optical polarization data obtained from the telescopes used for this study consist of 523 measurements of the degree, P, and position angle, χ_opt, of polarization. The data are displayed in Figure 2.

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Polarization observations of the quasar at 1.3 and 3 mm were obtained at the IRAM 30 m telescope within the MAPI²² and POLAMI²³ polarimetric programs. Each program performs monthly monitoring of a sample of γ-ray blazars, with both samples including 3C 454.3. The data were reduced in the same manner as described in Agudo et al. (2010). The values of P at all wavelengths were corrected for statistical bias (Wardle & Kronberg 1974).

We observed 3C 454.3 with the VLBA in the course of a program of monthly monitoring of bright γ-ray blazars at 43 GHz (7 mm)²⁴ and more dense monitoring during campaigns in 2009 October 12–25, 2010 April 7–20, and 2010 October 31–November 13. Within these campaigns, the quasar was observed three times. During the period from 2009 April to 2011 August, we obtained 35 total and polarized intensity images at a resolution of ∼0.3 × 0.1 mas. We performed the data reduction in the manner of Jorstad et al. (2005) using the Astronomical Image Processing System (AIPS) and Difmap (Shepherd 1997). The electric vector position angle (EVPA) was calibrated by different methods. Over the period 2009 April–2009 December we used the NRAO polarization database²⁵ that provides EVPAs at 43 GHz for several sources in our sample (0420−014, 0528+134, OJ287, 1156+295, 3C 279, BL Lacertae, and 3C 454.3) obtained with the Very Large Array (VLA), which we compared with VLBA integrated EVPAs at simultaneous or nearly simultaneous epochs. We obtained polarization measurements during the campaigns with the VLA on 2009 October 14 (sources: 0235+164, 0528+134, BL Lacertae, and 3C 454.3) and with the EVLA on 2010 April 10 (sources: 1156+295, 3C 279, 1308+326, and OT+081) and on 2010 November 2 (sources: 0235+164, 0528+134, 0716+710, and OJ287). At epochs where VLA/EVLA data were not available, we used the D-terms method (Gómez et al. 2002). The calibration was checked for consistency between epochs by comparing EVPAs of polarized jet features located ⩾1 mas from the core in 0528+134, 3C 273, 3C 345, CTA102, and BL Lacertae that had stable EVPAs based on VLBA observations with simultaneous VLA/EVLA observations. The accuracy of the EVPA calibration is within 5–10 deg. We have corrected the EVPA values of the core using the most recent estimate of the Faraday rotation measure in the core region of 3C 454.3, RM = 1320 ± 170 rad m⁻² (Algaba et al. 2011). The accuracy of the flux density calibration as revealed by comparison between the VLBA integrated flux and VLA/EVLA flux obtained at simultaneous epochs is within 5%.

We have calculated γ-ray light curves with a 3 hr integration interval during outbursts I, II, and III using the same approach as described in Section 2.1. This results in 646 (38), 526 (24), and 908 (39) measurements for outbursts I, II, and III, respectively, with the numbers in parentheses being non-detections. We have ignored the non-detections in our analysis since they represent a small fraction of the data. The light curves, presented in Figure 3, are normalized to the maximum flux density of each outburst, with time t = 0 set to the date of the maximum. Figure 3 shows that the structure of the outbursts is similar, although values of the maximum flux differ. We identify three flares, a, b, and c, within each outburst, separated by troughs with durations comparable to those of the flares. Peaks b and c have similar delays (within 0.5–3 days) with respect to the main peak of flare a. The primary difference in the profiles of the three outbursts is connected with the shape of flare a. We define the duration of the main flare, a, as the FWHM of a Gaussian that fits the flare profile near maximum flux, $\Delta T_\gamma ^{\rm a}\sim 11$, 20, and 5 days for outbursts I, II, and III, respectively. All three outbursts have pre-flare and post-flare "plateaus" of enhanced γ-ray emission, as discussed by Abdo et al. (2011). We determine the duration of a plateau as the time interval within which the standard deviation of the average γ-ray flux does not exceed 2σS_ave, where σS_ave is the average uncertainty of individual measurements. The duration of the plateaus differs from outburst to outburst, although the duration of the pre-flare plateau, $\Delta T_\gamma ^{\rm pre}$, is almost equal to the duration of the post-flare plateau, $\Delta T_\gamma ^{\rm post}$, for each outburst. In addition, the flux levels of the pre- and post-flare plateaus are comparable, except for outburst III. The entire duration of flare a, which includes $\Delta T_\gamma ^{\rm pre}$, $\Delta T_\gamma ^{\rm a}$, and $\Delta T_\gamma ^{\rm post}$, is comparable for all outbursts (28, 33, and 31 days for outbursts I, II, and III, respectively), as is the period between the peaks of flares a and c, equal to 46, 46, and 48 days for outbursts I, II, and III, respectively. The similarity in structure of the γ-ray outbursts argues in favor of the same mechanism(s) and location of γ-ray production for all three events. Jorstad et al. (2010) have previously reported a triple flare structure of optical outbursts in 3C 454.3 that coincide with the time of passage of superluminal knots through the mm-wave core of the jet. The measured time interval between the first and third peaks of these earlier events was ∼50 days, which is only slightly longer than the interval between the peaks of flares a and c observed for γ-ray outbursts I, II, and III. Parameters of the γ-ray outbursts studied here are given in Table 1.

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Table 1. Parameters of Gamma-Ray Outbursts

Parameter	Outburst I	Outburst II	Outburst III
ΔTγ (days)	82	66	113
〈Sγ〉 (10−6 photons cm−2 s−1)	5.84 ± 3.79	5.94 ± 3.34	12.36 ± 11.66
〈σγ〉 (10−6 photons cm−2 s−1)	1.04	1.05	1.87
$\Delta T_\gamma ^{\rm a}$ (days)	11	20	5
$T_\gamma ^{\rm max}$	2009 Dec 2	2010 Apr 8	2010 Nov 20
$T_\gamma ^{\rm max}$ (RJD)	5168.343	5294.593	5520.593
$S_\gamma ^{\rm max}$ (10−5 photons cm−2 s−1)	2.41 ± 0.23	1.60 ± 0.15	8.38 ± 0.40
$\alpha _\gamma ^{\rm max}$	1.32 ± 0.04	1.34 ± 0.05	1.20 ± 0.02
$\tau _\gamma ^{\rm a}$ (hr)	5.2	4.3	5.4
$f_\gamma ^{\rm a}$	1.79	2.02	1.74
$\Delta T_\gamma ^{\rm pre}$ (days)	8	7	13
$S_\gamma ^{\rm pre}$ (10−6 photons cm−2 s−1)	6.98 ± 1.05	5.54 ± 0.83	11.03 ± 1.02
$\alpha _\gamma ^{\rm pre}$	1.34 ± 0.10	1.36 ± 0.11	1.30 ± 0.06
$\tau _\gamma ^{\rm pre}$ (hr)	3.9	3.1	4.4
$f_\gamma ^{\rm pre}$	2.18	2.64	1.97
$\Delta T_\gamma ^{\rm post}$ (days)	9	6	13
$S_\gamma ^{\rm post}$ (10−6 photons cm−2 s−1)	6.51 ± 0.92	5.34 ± 0.89	20.91 ± 4.20
$\alpha _\gamma ^{\rm post}$	1.31 ± 0.08	1.41 ± 0.07	1.35 ± 0.06
$\tau _\gamma ^{\rm post}$ (hr)	5.6	3.5	7.3
$f_\gamma ^{\rm post}$	1.71	2.35	1.51
$T_\gamma ^{\rm b}$	2009 Dec 29	2010 May 5	2010 Dec 20
$T_\gamma ^{\rm b}$ (RJD)	5195.47	5322.22	5550.84
$S_\gamma ^{\rm b}$ (10−5 photons cm−2 s−1)	1.07 ± 0.16	1.14 ± 0.16	2.53 ± 0.22
$\alpha _\gamma ^{\rm b}$	1.36 ± 0.04	1.36 ± 0.03	1.29 ± 0.02
$T_\gamma ^{\rm c}$	2010 Jan 18	2010 May 18	2011 Jan 6
$T_\gamma ^{\rm c}$ (RJD)	5214.59	5340.59	5568.22
$S_\gamma ^{\rm c}$ (10−6 photons cm−2 s−1)	8.1 ± 1.4	6.5 ± 1.1	22.3 ± 4.4
$\alpha _\gamma ^{\rm c}$	1.38 ± 0.04	1.32 ± 0.04	1.26 ± 0.02
$\tau _\gamma ^{\rm min}$ (hr)	3.9	3.1	4.4
$T_\gamma ^{\tau _{\rm min}}$ (RJD)	5161.343	5281.593	5506.093
〈τγ, 2〉 (hr)	19	21	22

Notes. ΔT_γ: duration of γ-ray outburst (see text Section 3.1); 〈S_γ〉: the average flux density during the outburst and its standard deviation; 〈σ_γ〉: the average 1σ uncertainty of an individual measurement during the outburst; $\Delta T_\gamma ^{\rm a}$: duration of the main sub-flare in flare a (FWHM); $S_\gamma ^{\rm max}$: γ-ray flux at the peak of flare a at 0.1–200 GeV calculated with a 3 hr integration interval; $\alpha _\gamma ^{\rm max}$: spectral energy index at 0.1–200 GeV calculated for a simple power-law model for a 1 day integration interval centered at $T_\gamma ^{\rm max}$; $\tau _\gamma ^{\rm a}$: minimum timescale of variability of γ-ray flux during the main flare; $f_\gamma ^{\rm a}$: factor of the γ-ray flux change over $\tau _\gamma ^{\rm a}$; $\Delta T_\gamma ^{\rm pre}$: duration of the pre-flare plateau during an a flare; $S_\gamma ^{\rm pre}$: the average γ-ray flux and its standard deviation over period of $\Delta T_\gamma ^{\rm pre}$; $\alpha _\gamma ^{\rm pre}$: spectral index at 0.1–200 GeV averaged over $\Delta T_\gamma ^{\rm pre}$; $\tau _\gamma ^{\rm pre}$: minimum timescale of variability of γ-ray flux during $\Delta T_\gamma ^{\rm pre}$; $f_\gamma ^{\rm pre}$: factor of the γ-ray flux change over $\tau _\gamma ^{\rm pre}$; $\Delta T_\gamma ^{\rm post}$, $S_\gamma ^{\rm post}$, $\alpha _\gamma ^{\rm post}$, $\tau _\gamma ^{\rm post}$, and $f_\gamma ^{\rm post}$: parameters for the post-flare plateau obtained in the same manner as for the pre-flare plateau; $T_\gamma ^{\rm b},S_\gamma ^{\rm b}$, and $\alpha _\gamma ^{\rm b}$: epoch, maximum flux, and spectral index, respectively, for flare b, calculated in the same manner as for flare a; $T_\gamma ^{\rm c}, S_\gamma ^{\rm c}$, and $\alpha _\gamma ^{\rm c}$ are epoch, maximum flux, and spectral index, respectively, for flare c; $\tau _\gamma ^{\rm min}$: minimum timescale of variability of γ-ray flux during an outburst; $T_\gamma ^{\rm \tau }$: epoch of the start of an event with minimum timescale of variability; 〈τ_{γ, 2}〉: typical timescale of flux doubling (see text).

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We have determined timescales of γ-ray flux variability, τ_γ, using the formalism suggested by Burbidge et al. (1974): τ ≡ Δt/ln  (S₂/S₁), where S_i is the flux density at epoch t_i, with S₂ > S₁, and Δt = |t₂ − t₁|. We have calculated the timescale of variability for all possible pairs of flux measurements within 3 days of each other if, for a given pair, S₂ − S₁ > 3(σS₁ + σS₂)/2, where σS_i is the uncertainty of an individual measurement, and if the test statistic of the γ-ray measurement TS > 25 for both measurements. Using the derived values of τ_γ, we have searched for a minimum timescale of variability among pairs. Table 1 gives minimum timescales of ∼3–4 hr with the γ-ray flux changing by a factor of ⩾2. Table 1 shows that a short timescale of variability can occur at different stages of an outburst, although the occurrence of $\tau _\gamma ^{\rm min}$ takes place during the pre-flare plateau for all three outbursts. The range is consistent with the minimum timescales of variability reported by Ackermann et al. (2010) and Abdo et al. (2011), as well as Foschini et al. (2011), who apply different methods for estimation of τ_γ. Note that Foschini et al. (2011) have calculated γ-ray light curves using a time bin equal to the good time interval (GTI). Such a method should produce the most accurate flux estimates at short timescales for observations performed in scanning mode, since it allows one to find shortest intervals for binning with a sufficient number of photons for a good statistic. Agreement between our results suggests that $\tau _\gamma ^{\rm min}\sim 3$–5 hr might be an upper limit for the minimum timescale of variability defined by the GTI. In addition, Foschini et al. (2011) report the time of the global peak of outburst III to be RJD: 5520.573–5520.627, which matches $T_\gamma ^{\rm max}$ very well. We have also analyzed the distribution of τ_γ values that fall within the range 0–72 hr to determine a typical timescale, τ_{γ, 2}, for the γ-ray emission to change by a factor of ⩾2. Table 1 shows that this timescale of variability is similar for all three outbursts, τ_{γ, 2} = 20 ± 1 hr.

Figure 4 displays the X-ray light curves for outbursts I and III normalized to the maximum flux density of each X-ray outburst and centered with respect to the date of the maximum of corresponding γ-ray outburst. Table 2 lists the parameters of the X-ray outbursts. Although the X-ray data are much more sparsely sampled than the γ-ray light curves, the global X-ray and γ-ray peaks of outbursts I and III coincide within ∼1 day, with the γ-ray peak of outburst I leading by ∼1.2 day while the γ-ray peak of outburst III is delayed by ∼1.0 day. In addition, the duration of the main X-ray event in flare a is similar to $\Delta T_\gamma ^{\rm a}$ for both outbursts, and flare a of outburst III has pre-flare and post-flare plateaus contemporaneous with their γ-ray counterparts. There are also indications of the presence of a post-flare plateau and flare c in outburst I (the durations of the plateaus were determined in the same manner as for the γ-ray events). The main difference between the X-ray and γ-ray outbursts is the timescale of variability τ_X, calculated in the same manner as τ_γ, except for the condition for the test statistic. The fastest events were observed when the X-ray flux changed by a factor of 1.8 in 27 hr, which corresponds to $\tau _{\rm X}^{\rm min}\simeq 6\tau _{\gamma }^{\rm min}$, and the typical flux doubling timescale is ∼2 days, which gives τ_{X, 2} ≃ 2τ_{γ, 2}

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Table 2. Parameters of X-ray Outbursts

Parameter	Outburst I	Outburst III
M	33	96
〈SX〉 (10−11 erg cm−2 s−1)	8.46 ± 3.54	7.50 ± 3.26
〈σX〉 (10−11 erg cm−2 s−1)	0.45	0.40
$\Delta T_{\rm X}^{\rm a}$ (days)	13.5	5
$T_{\rm X}^{\rm max}$	2009 Dec 4	2010 Nov 19
$T_{\rm X}^{\rm max}$ (RJD)	5169.531	5519.594
$S_{\rm X}^{\rm max}$ (10−11 erg cm−2 s−1)	16.73 ± 0.40	17.92 ± 0.64
$\Delta T_{\rm X}^{\rm pre}$ (days)	...	24
$S_{\rm X}^{\rm pre}$ (10−11 erg cm−2 s−1)	...	6.48 ± 0.64
$\Delta T_{\rm X}^{\rm post}$ (days)	7	12
$S_{\rm X}^{\rm post}$ (10−11 erg cm−2 s−1)	6.61 ± 0.68	7.85 ± 0.71
$\tau _{\rm X}^{\rm min}$ (hr)	33.1	26.2
fX (hr)	1.71	1.80
$T_{\rm X}^{\tau _{\rm min}}$ (RJD)	5176.89	5518.28
〈τX, 2〉 (hr)	48	57

Notes. M: number of X-ray measurements at 0.3–10 keV obtained during the outburst; 〈S_X〉: the average flux density during the outburst and its standard deviation; 〈σ_X〉: the average 1σ uncertainty of an individual measurement during the outburst; $\Delta T_{\rm X}^{\rm a}$: duration of the main sub-flare in flare a (FWHM); $T_{\rm X}^{\rm max}$: epoch of the global maximum; $S_{\rm X}^{\rm max}$: the flux density at the peak of flare a; $\Delta T_{\rm X}^{\rm pre}$: duration of the pre-flare plateau during an a flare, the values in parentheses indicating the average uncertainty of an individual measurement of the flux density in 10⁻¹¹ erg cm⁻² s⁻¹; $S_{\rm X}^{\rm pre}$: the average flux density and its standard deviation over period of $\Delta T_{\rm X}^{\rm pre}$; $\Delta T_{\rm X}^{\rm post}$ and $S_{\rm X}^{\rm post}$: similar parameters for the post-flare plateau; $\tau _{\rm X}^{\rm min}$: minimum timescale of variability of X-ray flux during an outburst; f_X: factor of the flux change over $\tau _{\rm X}^{\rm min}$; $T_{\rm X}^{\tau _{{\rm min}}}$: epoch of the start of an event exhibiting the minimum timescale of variability; 〈τ_{X, 2}〉: typical timescale of flux doubling.

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We perform the same analysis of the structure of the R-band optical light curves (Figure 5 and Table 3) during outbursts I, II, and III as for the γ-ray and X-ray light curves. Figure 5 shows that the two well-sampled optical outbursts, I and III, have a complex structure of the main flare, a: flare a of outburst I has 2 peaks, a1 and a2 (Figure 5, the top insert), and flare a of outburst III has at least three peaks, a1, a2, and a3 (the bottom insert in Figure 5). Remarkably, flare a at γ-ray energies possesses similar structure for both outbursts, with the global γ-ray peak (a1 for outburst I and a3 for outburst III) coinciding with a prominent optical counterpart within the 3 hr γ-ray sampling, although the ratio of the fluxes of the γ-ray peaks can be different from those at optical wavelengths. The similarity in structure of flare a at optical and γ-ray frequencies implies that the flaring emission at the two wavelengths originates in the same region. The difference between the relative amplitudes of γ-ray versus optical peaks can be explained as the result of differences in relativistic boosting of γ-ray and optical emission, as proposed by Raiteri et al. (2011), or by variations in the density of seed photons available for scattering to γ-ray energies, as suggested by Vercellone et al. (2011). The latter is additionally supported by existence of orphan optical outbursts, for example, a very sharp spike at ∼10 days before the maximum when the fastest optical variability during outburst III was observed (Table 3) without an obvious counterpart in the γ-ray light curve (Figure 5, the bottom insert).

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Table 3. Parameters of Optical Outbursts

Parameter	Outburst I	Outburst II	Outburst III
M	306	103	2767
〈Sopt〉 (mJy)	6.49 ± 3.08	3.68 ± 1.43	10.93 ± 5.72
〈σopt〉 (mJy)	0.14	0.07	0.19
$\Delta T_{\rm opt}^{\rm a}$ (days)	10	...	8
$T_{\rm opt}^{\rm max}$	2009 Dec 6	2010 Apr 10	2010 Nov 20
$T_{\rm opt}^{\rm max}$ (RJD)	5172.273	5297.010	5520.673
$S_{\rm opt}^{\rm max}$ (mJy)	12.71 ± 0.19	6.71 ± 0.14	24.40 ± 1.10
$\Delta T_{\rm opt}^{\rm pre}$ (days)	6	...	7
$S_{\rm opt}^{\rm pre}$ (mJy)	6.29 ± 0.59	...	8.30 ± 0.72
$\Delta T_{\rm opt}^{\rm post}$ (days)	10	...	6
$S_{\rm opt}^{\rm post}$ (mJy)	7.57 ± 0.99	...	7.42 ± 0.34
$T_{\rm opt}^{\rm b}$ (RJD)	5193.153	5324.521	5541.180
$S_{\rm opt}^{\rm b}$ (mJy)	6.62 ± 0.15	6.60 ± 0.10	7.12 ± 0.15
$T_{\rm opt}^{\rm c}$ (RJD)	5215.144	5338.522	...
$S_{\rm opt}^{\rm }$ (mJy)	4.97 ± 0.05	3.46 ± 0.06	...
$\tau _{\rm opt}^{\rm min}$ (hr)	18.3	...	23.7
fopt (hr)	1.44	...	2.57
$T_{\rm opt}^{\tau _{\rm min}}$ (RJD)	5168.26	...	5509.55
〈τX, 2〉 (hr)	58	...	40

Notes. M: number of optical measurements in R band obtained during the outburst; 〈S_opt〉: the average flux density during the outburst and its standard deviation; 〈σ_opt〉: the average 1σ uncertainty of an individual measurement during the outburst; $\Delta T_{\rm opt}^{\rm a}$: duration of the main sub-flare in flare a (FWHM); $S_{\rm opt}^{\rm max}$: the flux density in R band at the peak of flare a; $\Delta T_{\rm opt}^{\rm pre}$: duration of the pre-flare plateau during an a flare; $S_{\rm opt}^{\rm pre}$: the average flux in R band and its standard deviation over period of $\Delta T_{\rm opt}^{\rm pre}$; $\Delta T_{\rm opt}^{\rm post}$ and $S_{\rm opt}^{\rm post}$: parameters for the post-flare plateau obtained in the same manner as for the pre-flare plateau; $T_{\rm opt}^{\rm b}$ and $S_{\rm opt}^{\rm b}$: epoch and maximum flux, respectively, for flare b; $T_{\rm opt}^{\rm c}$ and $S_{\rm opt}^{\rm c}$: epoch and maximum flux, respectively, for flare c; $\tau _{\rm opt}^{\rm min}$: minimum timescale of variability of optical flux during an outburst; f_opt: factor of the flux change over $\tau _{\rm opt}^{\rm min}$; $T_{\rm opt}^{\tau _{{\rm min}}}$: epoch of the start of an event with minimum timescale of variability; 〈τ_{opt, 2}〉: typical timescale of flux doubling (see text).

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Figure 5 and Table 3 show that optical outbursts I and III have pre-flare and post-flare plateaus that are contemporaneous with the corresponding γ-ray plateaus, although the relative flux level of the pre-flare plateau is higher with respect to the global maximum at optical wavelengths than at γ-ray energies. The durations of the optical plateaus are similar to the γ-ray values. The durations of the plateaus in R band were determined by the criterion that the flux variations within a plateau should not exceed 30% of the average flux value. This criterion is similar to that used for the γ-ray data analysis, since the average 1σ uncertainty of a γ-ray measurement is ∼17% (Table 1) while the 1σ uncertainty of an optical flux is ∼2% (Table 3). Note that the duration of the pre-flare plateau of outburst III is a factor of 2 shorter than $\Delta T_{\gamma }^{\rm pre}$. However, $\Delta T_{\rm opt}^{\rm pre}$ would match $\Delta T_{\gamma }^{\rm pre}$ if the pre-flare plateau were not interrupted by the orphan optical flare mentioned above. For outbursts I and III the entire optical flare a ($\Delta T_{\rm opt}^{\rm a}$+$\Delta T_{\rm opt}^{\rm pre}$+$\Delta T_{\rm opt}^{\rm post}$) has a similar duration as its γ-ray counterpart. The post-flare optical variability does not correspond as closely to the γ-ray variations as during the pre-flare and main flare stages. Nevertheless, optical outbursts I and II have counterparts to γ-ray flares b and c (see Table 3 and Figure 5) that peak within 0.5–5 days of the corresponding γ-ray flares. Flare b is distinct during outburst III as well, although it precedes the γ-ray flare b by ∼9 days.

Table 3 shows that the minimum timescale of variability is ∼18–24 hr with the optical flux changing by a factor of 1.5–2.5 (see also Raiteri et al. 2011; Vercellone et al. 2011). Comparison of Tables 1–3 reveals that the timescale of optical variability is different from τ_γ and similar to τ_X, which is longer by a factor of five than the minimum timescale of the γ-ray flux. Note that during flare a of outburst III (from RJD: 5500 to 5540) we obtained ∼2400 measurements in R band, which is suitable for revealing a timescale of variability as short as <1 hr. The typical doubling timescale is also different for γ-ray and optical variations, with τ_{opt, 2} ≈ (2–3)τ_{γ, 2}.

Parameters of outbursts I and III at 1 mm are presented in Table 4. Figure 6 shows the structure of outbursts I and III with respect to $T_\gamma ^{\rm max}$ of the corresponding γ-ray outburst. (Unfortunately, as in the case of the X-ray and optical light curves, observations at 1 mm miss outburst II.) Strikingly, the global peaks of the mm-wave and γ-ray outbursts coincide within hours for both events, while during the dramatic outburst in 2005 the global peak at 1 mm was delayed with respect to that at optical wavelengths by ∼2 months (Raiteri et al. 2008; Jorstad et al. 2010). Moreover, the duration of flare a is similar at mm-waves and γ-rays for both outbursts, and pre-flare and post-flare plateaus are apparent for outburst III. In addition, both outbursts at 1 mm contain flare b, which coincides with the corresponding flare b at γ-rays within 2 days, and flare c is seen in the 1 mm light curve of outburst III only 3.5 days later than γ-ray flare c. Taking into account the dramatic difference in the opacity at γ and mm wavelengths, such similarity requires the γ-ray events to take place in a region that is optically thin at 1 mm. The main differences between the γ-ray and mm-wave events are connected with the amplitude and timescale of variability. According to Tables 1 and 4, the size of the emission region at 1 mm is ∼100–300 times larger than that at γ-rays.

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Table 4. Parameters of Outbursts at 1 mm

Parameter	Outburst I	Outburst III
M	25	44
〈Smm〉 (Jy)	22.32 ± 4.27	36.27 ± 7.04
〈σmm〉 (Jy)	1.17	1.63
$\Delta T_{\rm mm}^{\rm a}$ (days)	14	6
$T_{\rm mm}^{\rm max}$	2009 Dec 3	2010 Nov 20
$T_{\rm mm}^{\rm max}$ (RJD)	5168.726	5520.657
$S_{\rm mm}^{\rm max}$ (Jy)	27.70 ± 1.39	51.77 ± 4.31
$T_{\rm mm}^{\rm b}$ (RJD)	5199.668	5548.587
$S_{\rm mm}^{\rm b}$ (mJy)	27.49 ± 1.38	37.73 ± 3.46
$T_{\rm mm}^{\rm c}$ (RJD)	...	5571.764
$S_{\rm mm}^{\rm c}$ (mJy)	...	39.91 ± 3.88
$\tau _{\rm mm}^{\rm min}$ (days)	41	10
fmm (days)	1.34	1.35
$T_{\rm mm}^{\tau _{\rm min}}$ (RJD)	5168.726	5534.64

Notes. M: number of measurements at 1 mm obtained during the outburst; 〈S_opt〉: the average flux density during the outburst and its standard deviation; 〈σ_opt〉: the average 1σ uncertainty of an individual measurement during the outburst; $\Delta T_{\rm mm}^{\rm a}$: duration of flare a (FWHM); $T_{\rm mm}^{\rm max}$: epoch of the global maximum; $S_{\rm mm}^{\rm max}$: the flux density at the peak of flare a; $T_{\rm mm}^{\rm b}$ and $S_{\rm mm}^{\rm b}$: epoch and maximum flux, respectively, for flare b; $T_{\rm mm}^{\rm c}$ and $S_{\rm mm}^{\rm c}$: epoch and maximum flux, respectively, for flare c; $\tau _{\rm mm}^{\rm min}$: minimum timescale of flux variability during an outburst; f_mm: factor of the flux change during $\tau _{\rm mm}^{\rm min}$; $T_{\rm mm}^{\tau _{{\rm min}}}$: epoch of the start of an event exhibiting the minimum timescale of variability.

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We perform a discrete cross-correlation analysis between the γ-ray and optical light curves and between the X-ray and optical light curves. For the purpose of this analysis, we construct a γ-ray light curve with an integration time of 12 hr in the same manner as described in Section 2.1. We use the original sampling of the X-ray light curve, which corresponds to a minimum time interval between two measurements of ∼12 hr, and bin the optical light curve with a 12 hr minimum interval, although the light curves have gaps ranging from days to months. We calculate the discrete cross-correlation function (DCF) using the algorithm developed by Edelson & Krolik (1988), and determine the significance of the correlation with the approach suggested by Chatterjee et al. (2008) and Max-Moerbeck et al. (2010). We follow Timmer & Koenig (1995) by simulating 5000 light curves with the same mean and standard deviation as the observed light curves. The statistics of the flux variations are described by a power spectral density, PSD, with a power-law shape, PSD∝f^−b, where a different value of b from 1 to 2.5, in steps of 0.2, is adopted for each set of simulations (Chatterjee et al. 2012). Figure 7 shows the DCF between the γ-ray and optical light curves (left) and between the X-ray and optical light curves (right). The DCF between the γ-ray and optical light curves is symmetric within ±2 days of the peak, with no delay between variations at two wavelengths >12 hr, since such a delay would produce, at least, an asymmetry in the DCF peak. The peak is significant at a level >99.7%. The DCF between the X-ray and optical light curves has a peak near zero delay as well; however, the position of the maximum of the centroid calculated for points exceeding 99.7% significance gives a delay of the X-ray with respect to optical variations of 0.5 ± 1 days. This result is consistent with the finding of Raiteri et al. (2011) that the X-rays lag the optical flux variations by 1.0 ± 1.0 days during the period 2008–2009.

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A correlation analysis between the γ-ray light curve from 2008 August 5 to 2011 October 21 and 1 mm light curve from 2008 January 12 to 2011 October 27 was performed by Wehrle et al. (2012). These authors found a significant correlation between variations at the two wavelengths for delays from −1.5 to +3.5 days, which suggests that mm-wave variations are either simultaneous with γ-ray variations or slightly precede the latter.

Our analysis of the multi-frequency light curves therefore reveals a strong similarity in the general structure of contemporaneous γ-ray, X-ray, optical, and mm-wave outbursts and a statistically significant correlation between variations at different wavelengths. This suggests that:

• 1.  
  The same ensemble of relativistic electrons that produces variable synchrotron optical emission participates in the production of variable γ-ray emission. This follows from the lack of significant time delays between the variations at these two wavebands. In fact, for values of the magnetic field typically inferred in the flaring regions of jets, ∼0.1–1 G, essentially the same energies of electrons are involved in both optical and γ-ray emission.
• 2.  
  Given the above similarity in electron energies, the existence of orphan optical flares implies that either the density of seed photons changes with time, or changes in the strength and/or direction of the magnetic field cause such flares.
• 3.  
  The emission regions at all four wavelengths are at least partially co-spatial, with the γ-ray emission exhibiting the fastest variability, $\tau _\gamma ^{\rm min}\approx 1/5\tau _{\rm opt}^{\rm min}$, while $\tau _{\rm opt}^{\rm min}\approx \tau _{\rm X}^{\rm min}$ and $\tau _{\rm mm}^{\rm min}\approx 30\tau _{\rm opt}^{\rm min}$. Either the γ-ray flux is more sensitive to changes in the physical parameters than is the optical/X-ray flux, or the γ-ray emitting plasma fills ∼1/5 of the optical/X-ray emission region. The similar X-ray and optical timescales of variability suggest that the corresponding emission regions are fully co-spatial. Although the mm-wave emission region is ∼100 times larger than that at γ-rays, a strict correspondence between the global maxima of the γ-ray and mm-wave outbursts implies that the mm-wave region has sub-structures of different sizes, e.g., ∼0.01 pc (the size of a turbulent cell within which the magnetic field is considered to be uniform; Wehrle et al. 2012), ⩽0.4 pc (the size of the mm-wave core; see Section 4), and ∼1 pc (the size of a superluminal knot), with the most compact and variable features co-spatial with the γ-ray emission region.
• 4.  
  The delay of X-ray with respect to optical flux variations, plus the similarity of the timescales of variability, imply a delay in the arrival of seed photons before they are scattered to X-ray energies. This favors a synchrotron origin of the seed photons from a location near to, but not coincident with, the scattering electrons. Such a situation can occur in the synchrotron self-Compton (SSC) mechanism for X-ray production, since there is a light-travel delay of synchrotron seed photons from the flare as they cross the flaring region if the angle to the line of sight is close to zero (Sokolov et al. 2004), as in 3C 454.3 (Θ_○ ∼ 03–13; see Section 4).
• 5.  
  The dramatic difference in the amplitudes of the γ-ray and optical outbursts, the presence of orphan flares, and the tight correlation between γ-ray and optical variations without a significant delay favor the process of scattering of external photons by relativistic electrons that produce synchrotron optical emission up to γ-rays (the external Compton mechanism, EC) as the main mechanism for production of the γ-ray emission during the outbursts. However, a contribution from SSC cannot be avoided, since synchrotron seed photons are also produced by the relativistic electrons involved in the γ-ray production.

Figure 11 shows the results of modeling of the total intensity images within 1 mas of the core. We have identified features that can be associated with moving knots K1, K2, and K3, as well as quasi-stationary feature C identified in Jorstad et al. (2010). In comparison with the results reported in Jorstad et al. (2010), knot K1 appears to have accelerated by a factor of ∼2, although at a number of epochs the knot is confused with either K2 or C. Knot K2 moves with the same slow apparent speed, ∼3 c, as seen during the later epochs analyzed in Jorstad et al. (2010). Although knot K3 has faded dramatically (S_7 mm ∼ 0.3 Jy), it has the same speed ∼4 c as reported previously. After the appearance of the new, very bright feature K09 at the end of 2009, knots K2 and K3 became too weak to be detected with a dynamic range of ∼1500:1. Knot K09 was as bright as the core in the beginning of 2010 and even dominated the flux of the parsec scale jet in 2010 Summer. According to the modeling, knot K10 appeared to be ejected at the end of 2010.

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The apparent motions of knots K09 and K10 are complex, as seen in Figure 11. Knot K09 appears to decelerate at a distance of ∼0.15 mas from the core and then accelerate at a distance of ∼0.2 mas. However, this is almost surely an artifact of the blending of K10 with the core, which shifts the apparent centroid of the core downstream for some time, after which the separation of K09 from the core rejoins the line representing a ballistic trajectory. Knot K10 appears in the jet at Θ ≈ −136°, south of the usual jet direction (Figure 9), then its trajectory curves into the direction of the average projected jet axis, Θ_jet ≈ −92° (Figure 12). Using the technique developed in Jorstad et al. (2005), we have calculated for K09 and K10 the apparent speed, β_app, acceleration along and perpendicular to the jet, $\dot{\mu }_\parallel$ and $\dot{\mu }_\perp$, time of ejection T_○,²⁶ timescale of flux variability, τ_var, Doppler factor, δ, Lorentz factor, Γ, and viewing angle, Θ_○. The values of these parameters are given in Table 5. According to Table 5, components K09 and K10 have similar apparent speeds, ∼9 c, a value that falls within the range of β_app observed previously in 3C 454.3 (Jorstad et al. 2001, 2010; Kellermann et al. 2004; Lister et al. 2009). Both K09 and K10 execute an acceleration perpendicular to the jet, which can be related to the apparent change of position angle near the core. Figure 12 suggests that the knots were ejected along different position angles, K09 to the north and K10 to the south with respect to the average jet axis. K09 also increases its proper motion along the jet, which can be attributed to an intrinsic acceleration (Homan et al. 2009). The values of the parameters δ, Γ_b, and Θ_○ of K09 are very close to those derived by Jorstad et al. (2005) from the kinematics of the jet in 1998–2001, while the Doppler factor of K10 is extreme, δ ∼ 50. The latter yields a much smaller viewing angle for K10 with respect to K09, in agreement with the different projected trajectories of the knots, which differ by ∼38° (Figure 12). According to Table 5, the main difference in the derived values of δ results from the timescale of variability. K10 fades faster than K09 by a factor of 2.5.

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Table 5. Parameters of Knots K09 and K10

Parameter	K09	K10
μ (mas yr−1)	0.21 ± 0.02	0.19 ± 0.03
$\dot{\mu }_\parallel$ (mas yr−2)	0.10 ± 0.01	...
$\dot{\mu }_\perp$ (mas yr−2)	0.13 ± 0.02	1.10 ± 0.22
βapp (c)	9.6 ± 0.6	8.9 ± 1.7
T○, yr	2009.86 ± 0.05	2010.95 ± 0.07
T○ (RJD)	5146 ± 18	5543 ± 25
Smax (Jy)	17.00 ± 0.45	7.10 ± 0.16
τvar (yr)	0.67 ± 0.06	0.24 ± 0.02
a (mas)	0.12 ± 0.02	0.08 ± 0.01
δ	27 ± 3	51 ± 4
Γb	15 ± 2	26 ± 3
Θ○ (deg)	1.35 ± 0.2	0.4 ± 0.1
N	24	7

Notes. μ: proper motion; $\dot{\mu }_\parallel$: angular acceleration along the jet; $\dot{\mu }_\perp$: angular acceleration perpendicular to the jet; β_app: apparent speed; T_○: time of ejection; S_max: maximum flux; τ_var: timescale of flux variability; a: angular size of component at epoch of maximum flux; δ: Doppler factor, Γ_b: Lorentz factor; Θ_○: angle between velocity of component and line of sight; N: number of epochs at which component was detected.

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Figure 13 displays the overall 1 mm and 7 mm light curves of individual components in the inner jet. The total 7 mm flux is calculated as the sum of A0, K09, K10, and C, depending on which feature is present at a given epoch according to the modeling of the images. The light curve of the core follows a smooth version of the variations at 1 mm, although the contribution of other jet components to the 1 mm flux is significant, since in general the flux at 1 mm exceeds the core flux at 7 mm throughout the majority of epochs. Comparison of the 1 mm and inner jet light curves shows that during RJD: 5100–5200 and RJD: 5500–5550 the flux at 1 mm is higher than that at 7 mm from the inner jet, and the opposite is observed within RJD: 5300–5500 and after RJD: 5600. The bright 1 mm states, relative to 7 mm, are modeled to be contemporaneous with the times when knots K09 and K10 were passing through the core. The lower 7 mm flux can be explained by a temporary suppression of the 7 mm flux outburst resulting from opacity increases in the 7 mm core as a superluminal knot moves through it. After a significant increase of the 1 mm flux in 2009 Autumn, the 1 mm flux remains at a high level for more than a year, a circumstance that we associate with the appearance in the jet of the very bright knot K09. Although the flux of the core decreased significantly after the ejection of K09, the core was still brighter than during quiescent states in 2009 Spring and 2011 Summer when there is good agreement between the 1 mm and 7 mm core light curves. This implies that the contribution of the jet outside 1 mas to the emission at 1 mm is negligible.

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Figure 13 displays the degree of linear polarization P versus time at mm wavelengths. The polarization from the whole source at 1 and 3 mm changes from 0% to ∼10% and agrees very well with that of the inner jet (dashed line). This presents another argument in favor of the emission at 1–3 mm arising mostly from the inner jet that includes the core and components within 1 mas of the core. The polarization of the core ranges from 1% to ∼5%, with a modest increase of P during the outbursts. The polarization of moving knots K09 and K10 lies within 2%–6%; however, as K09 approaches stationary feature C, P increases significantly for both knots and reaches 12% and 30% for K09 and C, respectively (see Figure 9). This supports the idea that the knots experience an interaction with the ambient jet, most likely in the form of a shock, since the position angle of polarization of both knots aligns with the jet direction. This is a primary signature of a transversely oriented shock, with the magnetic field compressed along the shock front (Hughes et al. 1985, 1989). At this time, knots K09 and C appear to contribute significantly to the polarized emission at 1 and 3 mm and P_1 mm rises up to 10%. Although the flux of C increases slightly, the total flux density continues to fade at mm wavelengths along with the very long baseline interferometry (VLBI) core.

During outbursts I and III the polarization position angle χ_core of the core at 7 mm rotates from +16° to −44° and from +91° to −11°, respectively (Figure 13). In addition, χ_3 mm rotates in a similar manner, although the range of rotation is greater than for χ_core, especially during outburst I. Figures 8, 9, and 13 show that the polarization vectors of both K09 and K10 undergo rotations, as well. Figure 14 compares the evolution of χ_K09 and χ_K10 with distance from the core. Both knots appear in the jet with polarization vectors oriented perpendicular to the jet axis, but between separations of 0.05 and 0.18 mas the EVPAs of both knots swing, although the rotations are in the opposite direction at the same distance from the core. These rotations might be a signature of a large intrinsic Faraday rotation measure near the core caused by a toroidal structure of the magnetic field. In this case the different directions of rotation of χ_K09 and χ_K10 can be readily explained by a change in the sign of the magnetic field with respect to the line of sight, assuming that the knots are propagating along different sides of the jet, as can be inferred from the knot's trajectories (Figure 12). At ∼0.2 mas from the core, χ_K09 ≈ χ_K10 ≈ 50°, oblique to the jet direction, and at distances >0.2 mas K09 has a stable EVPA, aligned with the jet axis, as well as with the EVPA of stationary knot C. Such a behavior is consistent with the knots being transversely oriented shocks propagating down the jet, which has a turbulent magnetic field (Hughes et al. 1985, 1989). Figure 15 shows the composite structure of the parsec-scale jet emission from 2009 April to 2011 August. The image is obtained by summing Stokes I, Q, U parameter maps over all 35 epochs obtained during this period (each map was convolved with the same restoring beam). The image represents an active state of the inner jet, since the I_peak is a factor of ∼10 brighter than during a quiescent state. A spiral-type structure is apparent in polarized intensity up to ∼0.4 mas from the core, with a different direction of the EVPAs on the southwest and northwest edges of the polarized emission region. Farther down the jet, the position angle of polarization aligns with the jet direction, as expected if a turbulent magnetic field becomes partially ordered along the front of a transverse shock. This region of the jet is dominated by the contribution from knots K09 and C. This picture is consistent with the scenario, proposed by Jorstad et al. (2007), that the mm-wave core is located at the end of the acceleration zone, where the jet energy density is dominated by the Poynting flux of a toroidal magnetic field. Near the core the flow becomes kinetic energy dominated, with the magnetic field becoming turbulent.

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We analyze the relation between the γ-ray flux, S_γ, and the 43 GHz flux density of the VLBI core, S_core. Figure 16 shows S_γ versus S_core for all VLBA epochs (35), with the γ-ray photons integrated over the 24 hr centered on the VLBA observation. There is a strong correlation between variations at γ-rays and in the core. The linear Pearson correlation coefficient, r = 0.77, is statistically significant at the 99.9% confidence level. The flux density of the core increases by a factor of 16 while the γ-ray flux rises by a factor of ∼65. This implies that the γ-ray flux is relativistically beamed in the same manner or more strongly than the radio flux, as suggested previously by Jorstad et al. (2001) and Kovalev et al. (2009). The relation between S_γ and S_core does not fit a simple linear dependence, rather, two relationships can be inferred: (1) for S_core ≲ 14 Jy the dependence is almost linear, and (2) for S_core ≳ 14 Jy the dependence is roughly quadratic. In addition, Wehrle et al. (2012) have found a statistically significant correlation between the γ-ray and 1 mm light curves, with no lag.

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Enhanced flux density of the core region in a VLBI image of a blazar usually corresponds to the emergence of a new disturbance into the flow in the radio-emitting zone of the jet (e.g., Savolainen et al. 2002). Comparison of Tables 1 and 5 indicates that knots K09 and K10 passed through the mm-wave VLBI core close to the time of the γ-ray peaks of outbursts I and III (within the 1σ uncertainty of the ejection time of 18 and 25 days, respectively). The duration of these γ-ray outbursts is comparable to the time needed for a knot to go through the core, 90 ± 15 days, for the average core size of 0.05 mas and average proper motion of knots of 0.20 mas yr⁻¹. In addition, during the γ-ray outbursts we observe an increase in the core opacity, as revealed by comparison of the 1 mm and 7 mm light curves, and rotation of the polarization vector in the core and at 3 mm. These trends argue in favor of the γ-ray outbursts coinciding with the passage of superluminal knots through the core. Although we did not detect a superluminal knot associated with γ-ray outburst II, Figure 13 shows that both the core and K09 underwent flares during outburst II (RJD: 5275–5349), with the flux reaching S ∼ 18 Jy. Moreover, we observed an increase of the degree of polarization in the core and at 1 mm and 3 mm, along with a rotation of χ at 1 and 3 mm. Based on the history of 3C 454.3, it is likely that this increase in the flux and polarization of the core during the γ-ray event was caused by the passage of a new superluminal knot through the core, with a flux of ⩽7 Jy. Summer–Autumn 2010 would have been the best period to detect this hypothetical new component in the jet. However, during this interval K09 was still close to the core, between 0.12 mas and 0.20 mas (Figure 11). Although K09 is distinct from the core at the high resolution of our observations (see Figure 8), the resolution is insufficient to identify a weaker knot situated between the core and an extremely bright knot within 0.20 mas. The increase in the flux of K09 along with the core during γ-ray outburst II might be an artifact of modeling with circular Gaussians of such complex structure, but K09 continued to be brighter than the core as the core faded in mid-2010 (Figure 13). This suggests that K09 might have been blended with the putative superluminal knot. The subsequent dramatic increase of the flux in the core in 2010 November and ejection of K10 further reduced the possibility of detecting any propagating disturbance associated with γ-ray outburst II. Overall, comparison of the γ-ray and mm-wave behavior provides strong evidence of a tight connection between enhanced γ-ray emission and the passage of superluminal knots through the mm-wave core located at a distance ∼15–20 pc from the central engine (Jorstad et al. 2010; Pushkarev et al. 2012).

We have derived weekly γ-ray spectral indices α_γ between 0.1–200 GeV using a simple power-law model with variable photon index and normalization ("prefactor") to represent the quasar emission. Frozen spectral parameters (values from the 2FGL catalog; Nolan et al. 2012) were used for other sources within 15° radius of 3C 454.3 to guide the maximum likelihood routine. Figure 17 shows the derived values of α_γ, which varies from 1.2 to 1.8 with an average 〈α_γ〉 = 1.44 and standard deviation σ_γ = 0.08, which is less than the average uncertainty of an individual measurement $\sigma _\gamma ^1=0.13$. However, the spectral indices averaged over a month-long interval centered on the peak of outburst are 1.33 ± 0.03 (flare I), 1.37 ± 0.03 (flare II), and 1.28 ± 0.03 (flare III). This implies a harder γ-ray spectrum during the highest states of γ-ray emission, a trend studied in detail by Ackermann et al. (2010) and Abdo et al. (2011).

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We have derived 201 spectral indices, α_X, between 0.3 and 10 keV from 2009 April 25 to 2011 August 1. The X-ray spectral index varies from 0.3 to 0.8 with an average value of 〈α_X〉 = 0.65 ± 0.12, while the mean uncertainty of an individual measurement $\sigma _{\rm X}^1=0.10$ (Figure 17). This behavior is consistent with the conclusion of Raiteri et al. (2011) that the spectral variations are dominated by the noise of the measurement uncertainties. Indeed, during both outbursts for which X-ray data are available (I and III), α_X exhibits random fluctuations around 〈α_X〉. However, ∼3 months before the peak of outburst I, the spectral index flattens over 19 consecutive measurements from RJD: 5065 to RJD: 5090, with an average value of 0.44 ± 0.08.

We use the data collected at the SMA, IRAM, and Metsähovi Radio Observatory facilities to derive the mm-wave spectral index (α_mm) between 1.3 and 8 mm. We have obtained 92 spectral indices from measurements simultaneous within 1 day. Figure 17 shows α_mm versus epoch. The spectral index changes from +0.3 to −0.25, very similar to the range observed in 2004–2008 (Jorstad et al. 2010). The figure also shows times of ejection of knots K09 and K10 from the core. Although α_mm is slightly steeper during quiescent states than $\alpha _{\rm mm}^{\rm q}=0.18\pm 0.04$ measured in 2004–2008, the main feature of the α_mm behavior—that the mm-wave spectral index reaches a local minimum just before the ejection of a superluminal knot—holds for both knots K09 and K10. This pattern was also observed for knots K1, K2, and K3 (Jorstad et al. 2010). Unfortunately, there is a gap in the radio data during γ-ray outburst II that prevents us from using changes in α_mm to search for signs of the ejection of the hypothetical knot discussed in Section 4.3. During the events associated with knots K09 and K10, the spectrum becomes inverted with spectral indices of −0.25 ± 0.05 ($\alpha _{\rm mm}^{\rm K09}$) and −0.08 ± 0.01 ($\alpha _{\rm mm}^{\rm K10}$), implying an increase in opacity at mm wavelengths during the events. The fact that $\alpha _{\rm mm}^{\rm K09}<\alpha _{\rm mm}^{\rm K10}$ suggests a significant contribution of optically thin emission from K09 to the total flux at mm-wavelengths during the K10 event.

Figure 17 reveals similarity in variations of α_γ and α_mm. Both spectral indices flatten at/near the time of ejection of superluminal knots and steepen during quiescent periods (2009 Spring, 2010 Summer, and 2011 Spring). However, the similarity is not sufficient to claim a statistically significant correlation between the indices. Although the measurements of α_X are sparse and noisy, a flattening of α_X at RJD: 5065–5090, which is significant at the 2σ level, coincides with the period of the flattest α_mm (RJD: 5075–5110) that precedes the ejection of K09.

Figure 18 shows four spectra of the quasar at the epochs of the brightest (or next to brightest) γ-ray states during outbursts I, II, and III, and at a quiescent state in 2011 Summer. The most prominent emission features in the Steward Observatory spectra (2150–4060 Å in the rest frame) are broad Mg ii (λ = 2800 Å) and blended Fe ii multiplets. A time series analysis between variations in the line and continuum emission is important for determining the size, geometry, and location of the broad emission line region (BLR). A number of studies have found that the size of BLR is proportional to the AGN luminosity, $R_{\rm BLR}\propto L_{\rm 5100}^{\kappa }$, where κ varies from 0.5 to 0.7 (e.g., Kaspi et al. 2000), with R_BLR of low-luminosity AGNs falling in the range of 10–100 lt-day. This implies that the size of BLR in luminous quasars ⩾1 lt-yr. In general, analyses of emission line variations in blazars, including 3C 454.3, do not reveal a connection between line and continuum flux variations (3C 454.3: Raiteri et al. 2008, 1222+216: Smith et al. 2011, and 1633+384: Raiteri et al. 2012). However, a recent analysis by León-Tavares et al. (2013) of the optical spectra of 3C 454.3 obtained at Steward Observatory during 2008–2011 suggests the existence of significant variations in the Mg ii line, corresponding to a flare-like event, during outburst III that challenges the standard model of the BLR in blazars. We possess the same data as analyzed by León-Tavares et al. (2013). Unfortunately, the data miss the 2-week period from 2010 November 16 to 30 during the brightest γ-ray and optical state in 3C 454.3. Since the Yale University blazar monitoring group obtained optical spectra of the quasar during this period (C. M. Urry 2013, private communication), we defer to an analysis of time variations of emission lines to the work by the Yale University blazar group.

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We have corrected the spectra of 3C 454.3 for Galactic extinction, performed Gaussian fits to the emission features, subtracted the fits from each of 181 spectra, and approximated the residual continuum by a power law, $S_\nu \propto \nu ^{-\alpha _{\rm cont}}$, within the log₁₀ν range from 14.65 to 14.85 (λ from 4235 to 6712 Å) to avoid the atmospheric O₂ absorption feature and noisy edges of the spectra (the wavelength range is from 2278 to 3611 Å in the quasar rest frame). Figure 19 plots α_cont versus flux in V band from simultaneous measurements (within ∼1 hr). Table 6 lists average values of α_cont with their standard deviations for different flux intervals. The spectral index of the continuum steepens by a factor of two, from ∼0.6 to ∼1.2, as the flux increases from ∼1.5 mJy to 3.5 mJy, while the scatter of α_cont values does not exceed 0.15 within each interval of averaging. A further increase of the flux up to ∼10 mJy does not change the average spectral index significantly, although a slight steepening, up to 1.35, toward higher flux is observed. Such behavior of α_cont is a direct indication that the continuum consists of at least two components, blue and red. The blue component dominates at low brightness states and can be attributed to big blue bump (BBB) emission produced by the accretion disk (Smith et al. 1988; Raiteri et al. 2007). The red component is non-thermal optical emission that originates in the relativistic jet. Figure 19 can be interpreted within the assumption that the blue component is constant during our observations both in flux and in spectral index, while the red component varies significantly in flux and slightly (within ± 0.15) in spectral index. The latter can account for the scatter observed within different intervals of averaging. We derive a linear approximation of α_cont = (0.18 ± 0.04) + (0.30 ± 0.02)S_V for S_V from 0 to 4.0 mJy, which is shown in Figure 19 by a solid line. According to the Pearson's χ² test the linear fit corresponds to the data sufficiently well with χ² = 1.83 for 103 measurements. Vanden Berk et al. (2001) have created a composite continuum spectrum of a quasar using a homogeneous data set consisting of over 2200 quasar spectra from the Sloan Digital Sky Survey (SDSS), which covers a rest-wavelength range from 800 to 8555 Å. The quasars were selected from both optical and radio criteria. The authors have found that at wavelengths from ∼1300 to 5000 Å the continuum is well fit by a power law with α_ν = 0.44, which we denote as α_disk, assuming that this continuum should represent the emission from the accretion disk. Although it is not clear what fraction of quasars in the sample are radio loud, the index is in good agreement with values found in optically selected quasar samples (e.g., Natali et al. 1998). Under the assumption that the accretion disk in 3C 454.3 has similar properties as the disk of a generic quasar and, therefore, when α_cont = α_disk the contribution of the nonthermal component to the continuum is negligible, we can use the linear fit for α_cont to estimate the flux of the accretion disk, $S^{\rm V}_{\rm disk}=0.85\pm 0.15$ mJy, which corresponds to $S^{\rm R}_{\rm disk}=0.91\pm 0.16$ mJy. As expected, these values are slightly less than the minimum flux of the quasar observed in V and R bands, respectively.

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A method proposed by Hagen-Thorn (1997) allows one to subtract from the total near-IR to UV emission the contribution of components that either are constant or vary on long timescales. These are expected to include the accretion disk, BLR, and dusty torus. The result is a relative SED, RSED, of the component responsible for variable emission on timescales of hours, days, or weeks. Since it is synchrotron emission that varies on such short timescales in a blazar, we refer to this as the "synchrotron component." The method is based on simultaneous multicolor observations and assumes a linear dependence between variations at two different bands, S_N = A_N + C_NS_r, where N is the band at which the flux is measured, and S_r is the flux at a reference band (see below). Figures 20 and 21 reveal approximately linear flux–flux relations at different bands that are statistically significant at a level ⩾99.9% according to the χ² criterion. We consider that two measurements at different wavelengths are simultaneous if they are performed within 2 hr of each other. Table 7 shows the number of simultaneous observations at different wavelengths with respect to R band for outbursts I and III, as well as the values of χ² for a linear fit to the flux–flux relations. Figures 20 and 21 show that the simultaneous measurements cover a wide range of variability during both outbursts. Unfortunately, in the case of outburst II there are no observations in the UV region and only 5–6 observations at near-IR wavelengths over a narrow range of flux levels. R band is chosen as the reference band for the construction of the RSED, since the largest number of observations was obtained in this band. In the case of UV observations, B band serves as a primary reference band (see Figures 20 (left) and 21 (left)) and the linear dependence between the B and R fluxes is used to derive the coefficients C_U, C_UVW1, C_UVM2, and C_UVW2. Similarly, J band is the primary reference band for measurements in H and K bands. The C coefficients are given in Table 7. The dependence of the coefficients on frequency represents an RSED. Figure 22 shows the RSED for outbursts I and III. Both RSEDs can be approximated by a power law, $S\propto \nu ^{-\alpha _{\rm opt}^{\rm syn}}$, with similar spectral indices, $\alpha _{\rm opt}^{\rm synI}=1.77\pm 0.05$ and $\alpha _{\rm opt}^{\rm synIII}=1.71\pm 0.04$. This synchrotron emission represents a red spectral component that dominates the optical continuum when the source is brighter than S_V ∼ 3.5 mJy, as discussed in Section 5.2. The spectral index of the red component is significantly steeper than the spectral index of the optical continuum, α_cont ∼ 1.30, during high optical states, hence the disk emission is always significant for the quasar SED.

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We have applied the same method as described in Section 5.3 for simultaneous far-IR data obtained during outburst III with the Herschel PACS and SPIRE photometers at 70, 160, 250, 350, and 500 μm, along with 0.85 and 1 mm measurements obtained with the SMA and IRAM. In this case we treat two observations as simultaneous if they are performed within 24 hr of each other. We use 250 μm as a reference wavelength relative to which the RSED is constructed, except for measurements at 70 μm, for which 160 μm serves as a preliminary reference wavelength, with the linear dependence between fluxes at 160 μm and 250 μm employed to derive C₇₀. Figure 23 (left) shows that the

flux–flux relations follow linear dependences quite well. Table 8 lists the number of simultaneous observations at different wavelengths, values of χ² for linear fits, and coefficients C that represent the slopes of the linear fits. The dependence of the coefficients on frequency is plotted in Figure 23 (right), which shows a good correspondence (χ² = 1.19) of far-IR points (from 70 to 850 μm) with a power law of spectral index $\alpha _{\rm IR}^{\rm syn}=0.79\pm 0.06$. The 1 mm measurement deviates slightly from the fit, falling a factor of 0.84 below an extrapolation of the fit to the 70–850 μm RSED. This deviation most likely indicates that the variable component responsible for the synchrotron emission at 1 mm is partially optically thick. The good correlation between far-IR and 1 mm flux variations (Wehrle et al. 2012 and Figure 23, left) and between 1 mm and 7mm core flux and polarization behavior (Section 4.2) suggests that the mm-wave VLBI core is the region where the synchrotron far-IR emission originates.

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We construct SEDs of the quasar from UV to mm wavelengths for three epochs during outburst III: on 2010 November 3/4 (RJD: 5503), at the beginning of flare a; on November 19 (RJD: 5519), at the maximum of the outburst; and on December 7 (RJD: 5537), during the fading branch of the outburst. We model each SED of the quasar assuming that (1) the optical continuum from UV to near-IR wavelengths consists of a blue component with a constant spectral index α_disk = 0.44 and constant flux, $S^{\rm V}_{\rm disk}=0.85$ mJy plus a variable red (synchrotron) component with a constant spectral index $\alpha _{\rm opt}^{\rm syn}=1.71$, and (2) the mid- and far-IR continuum is dominated by a variable (synchrotron) component with a constant spectral index $\alpha _{\rm IR}^{\rm syn}=0.79$ with a contribution from jet knot K09. We use Herschel observations to determine the minimum wavelength where the contribution from K09 is still significant, ∼80 μm. We determine the flux density of the red variable component in V band as $S_{\rm V}^{\rm syn}=S_{\rm V}^{\rm obs}-S^{\rm V}_{\rm disk}$, where $S_{\rm V}^{\rm obs}$ is the observed flux density at a given epoch. The flux density of the variable synchrotron component at 1.3 mm is calculated as $S_{\rm 1\,mm}^{\rm syn}={\rm const}\times [S_{\rm 1\,mm}^{\rm obs}-S_{K09}(\nu _{\rm 1\,mm}/\nu _{\rm 7\,mm})^{-\alpha _{K09}}]$, where S_K09 is the flux of knot K09 at 7 mm derived from the VLBA images. We adopt α_K09 = 0.7, and set const = 1.0 for November 3/4 and 1.19 after November 4 to correct for the higher opacity at mm wavelengths with respect to the far-IR measurements (see Section 5.5). Figure 24 shows the observed and modeled SEDs, as well as SEDs for different components: the variable synchrotron component, the accretion disk, and K09. Table 9 gives the parameters of the SED's peak according to modeling. Using the values of $S^{\rm V}_{\rm disk}$ and α_disk, we estimate the luminosity of the accretion, L_disk ≈ 2.6 × 10⁴⁶ erg s⁻¹, with integrating from 6500 to 2000 Å.

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Table 9. Parameters of SEDs during Outburst III

Parameter	Nov 3/4	Nov 19	Dec 7
$S_{\rm V}^{\rm obs}\ ({\rm mJy}$)	7.59 ± 0.28	16.52 ± 0.15	6.00 ± 0.12
$S_{\rm 1\,mm}^{\rm obs}\ ({\rm Jy}$)	34.9 ± 1.8	48.8 ± 2.40	29.6 ± 1.5
SK09 (Jy)	15.5 ± 0.3	11.1 ± 0.2	11.0 ± 0.2
νpeak (Hz)	4.66E+13	4.78E+13	2.39E+13
λpeak (μm)	6.4	6.3	12.5
Speak (mJy)	493	1079	895

Notes. Parameters of SEDs in the observer's frame—$S_{\rm V}^{\rm obs}$: flux density observed in V band; $S_{\rm 1\,mm}^{\rm obs}$: flux density observed at 1.3 mm, S_K09: flux density of knot K09 at 7 mm; ν_peak: frequency of peak of SED, λ_peak: wavelength of peak of SED, S_peak: flux density at peak of SED. L_disk: luminosity of accretion disk integrated from 6500 to 2000 Å.

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Figure 24 shows good agreement between the observations and models for all three epochs. Especially promising is the close correspondence between the mid-IR measurements obtained with the IRTF on 2010 November at 4.9, 10.6, and 20.7 μm and the modeled SED, since these data-points were not involved in modeling. This supports the assumption that one synchrotron component with a constant spectral index and variable flux is responsible for the outburst at optical/IR wavelengths. The IRTF observations cover wavelengths affected by an IR excess if 3C 454.3 has a dust torus of similar properties as found in the γ-ray quasar 1222+216 (Malmrose et al. 2011). Although neither the IRTF nor Herschel observations suggest the presence of an additional component between 5 and 160 μm, the measurement at 20.7 μm—where the peak of a dust component (with temperature ∼1200 K) is expected in the observer's frame for 3C 454.3—is slightly higher than the model value. However, this measurement has substantial uncertainties. The maximum deviation from the model flux defines an upper limit to the luminosity of the dust torus of L_dust < 5 × 10⁴⁶ erg s⁻¹. This is quite high, allowing the dust torus to possess a higher luminosity than that of the accretion disk, while in the case of 1222+216 IR emission from hot dust has a luminosity of only 0.22 times that of the accretion disk (Malmrose et al. 2011).

The peak of the synchrotron SED corresponds to a break in the spectrum from a spectral index α_low < 1 to α_high > 1. The magnitude of the break Δα ≡ α_high − α_low. During outburst III, $\alpha _{\rm IR}^{\rm syn}=0.79$ and $\alpha _{\rm opt}^{\rm syn}=1.71$, hence Δα = 0.92. This is considerably greater than the value of 0.5 expected from radiative energy losses while relativistic electrons are constantly injected into the emission region. Interestingly, according to the data listed in Table 9, the wavelength of peak flux, λ_peak, remains essentially constant as the outburst proceeds to its global maximum on 2010 November 19. If the value of λ_peak were determined by a balance between the timescale of radiative losses and the time for the energized electrons to cross the emission region, λ_peak would increase as the intensity of seed photons for IC scattering rises. On the other hand, between 2010 November 19 and December 7, λ_peak increases by a factor of two while the optical flux drops by a factor of 2.75, consistent with the general trend expected from radiative losses as the rate of injection of relativistic electrons subsides.

Figure 25 presents SEDs of the quasar at the epochs of the γ-ray maximum, $T_\gamma ^{\rm max}$, of each outburst as well as during a more quiescent state. The γ-ray fluxes during the outbursts are calculated using spectral parameters given in Table 2 of Ackermann et al. (2010) and Table 1 of Abdo et al. (2011) for a LogParabola spectrum. To analyze the SEDs, we apply the Doppler factors derived for knots K09 and K10 (Table 5) for outbursts I and III, respectively. For outburst II we assume δ_II to be slightly less than δ_I (Table 10) because, on one hand, the outbursts have very similar SEDs, while on the other hand both the optical and γ-ray fluxes of outburst II are slightly less than those of outburst I. For the quiescent state we use a minimum Doppler factor obtained by Jorstad et al. (2005) in 1998–2001 when the flux of the VLBI core at 43 GHz did not exceed 5 Jy. All SEDs have a double peak shape typical of blazars with a low-frequency peak $S_{\rm peak}^{\rm LE}$ at frequency $\nu _{\rm peak}^{\rm LE}$, representing enhanced synchrotron emission, and a high-frequency peak, $S_{\rm peak}^{\rm HE}$ at frequency $\nu _{\rm peak}^{\rm HE}$, which we attribute to inverse Compton scattering. The high-energy peak dominates the SEDs during the active states.

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Table 10. Parameters of SEDs during Outbursts and Quiescent State

Parameter	Outburst I	Outburst II	Outburst III	Quiescent
δ	27	25	51	13
$\nu _{\rm peak}^{\rm LE}$ (Hz)	2.53E+13	2.34E+13	4.78E+13	1.22E+13
$\lambda _{\rm peak}^{\rm LE}$ (μm)	11.8	12.8	6.3	24.6
$S_{\rm peak}^{\rm LE}$ (mJy)	760	750	1100	160
LLE (erg s−1)	6.9E+47	6.8E+47	1.9E+48	7.0E+46
Lsyn (erg s−1)	8.2E+47	6.0E+47	1.0E+49	4.4E+46
$\nu _{\rm peak}^{\rm HE}$ (Hz)	3.16E+22	2.63E+22	6.02E+22	1.51E+22
$E_{\rm peak}^{\rm HE}$ (GeV)	0.13	0.11	0.25	0.06
$S_{\rm peak}^{\rm HE}$ (mJy)	9.0E-6	7.6E-6	1.2E-5	2.3E-7
LHE (erg s−1)	1.0E+49	7.2E+48	2.6E+49	1.2E+47
LIC (erg s−1)	2.2E+48	1.6E+48	2.8E+49	1.2E+47

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Since neither low- nor high-energy peaks of the SEDs are restricted by the measurements, we use the values of S_peak and ν_peak for outburst III derived from the modeling (Table 9) and adopt the values of $S_{\rm peak}^{\rm HE}$ and $\nu _{\rm peak}^{\rm HE}$ for outburst I from the SED presented by Bonnoli et al. (2011) for 2009 December 2. Taking into account that ν = δν', where ν' is the frequency of the emission emitted by the source and ν is the frequency of the emission received by the observer, we have estimated $\nu _{\rm peak}^{\rm LE}$ and $\nu _{\rm peak}^{\rm HE}$ for each SED assuming that a change in the Doppler factor is the main factor responsible for the outburst's increase in energy output. Using such peak frequencies, along with spline approximations of the SED's data points, we estimate values of $S_{\rm peak}^{\rm LE}$ and $S_{\rm peak}^{\rm HE}$ for each SED. The values are given in Table 10 and marked in Figure 25 by the symbol "⊥." Note that in 2011 June the mm-wave emission was still in some moderately active state with respect to the higher energy emission. We calculate the apparent luminosity of 3C 454.3 at the low and high energies for each activity state as $L\approx 4{\pi }D_{\rm L}^2S_{\rm peak}\nu _{\rm peak}$, where D_L is the luminosity distance, 5.489 Gpc. The values of L^HE for the outbursts listed in Table 10 are by a factor of 2–5 lower than those presented by Ackermann et al. (2010) and Abdo et al. (2011) because we use flux densities at the peak frequency only to estimate the luminosity. Table 10 shows that during the quiescent state the luminosity at low energies is comparable to the luminosity of the accretion disk. We estimate the luminosity of a synchrotron component of the emission at the quiescent state as $L_{\rm q}^{\rm syn}\approx L_{\rm q}^{\rm LE}-L_{\rm disk}$. Attributing an enhanced emission at low energies during the outbursts to a change of the Doppler factor, we derive the luminosity of the synchrotron component for each outburst as $L^{\rm syn}\approx L_{\rm q}^{\rm syn}(\delta /\delta _{\rm q})^4$. Making the same assumption that enhanced luminosity at high energies is caused by a change of the Doppler factor with respect to the quiescent state and adopting $L_{\rm q}^{\rm IC}=L_{\rm q}^{\rm HE}$, we calculate values of the luminosity at high energies expected within such assumptions, $L^{\rm IC}\approx L_{\rm q}^{\rm IC}(\delta /\delta _{\rm q})^4$. Table 10 shows reasonable agreement between L^LE and L^syn, and L^HE and L^IC. However, while for outburst III L^HE and L^IC are very consistent, L^syn is larger than L^LE by a factor of 5. The opposite occurs for outbursts I and II: there is good agreement between L^syn and L^LE, while L^IC is underestimated by a factor of five with respect to L^HE. Taking into account that the mm-wave emission region is larger than the γ-ray emission region by a factor of ∼100, the discrepancies suggest slight variations of the Doppler factor within the mm-wave emission region at a given epoch, with a higher δ than the mean value in the volume where the γ-ray emission originates. Therefore, it appears that a change in Doppler factor can explain differences in the amplitudes of the outbursts, as was proposed previously by Villata et al. (2007) and Raiteri et al. (2011).

The spectropolarimetric observations performed at Steward Observatory provide spectra of the normalized linear polarization Stokes parameters q and u in the range of 4000–7550 Å with a dispersion of 4 Å. We use the Stokes spectra to calculate the degree of polarization as a function of wavelength, $P(\lambda)=\sqrt{q(\lambda)^2+u(\lambda)^2}$, within the range of 4500–7000 Å, which avoids noisy edges of the q and u spectra. Since not all spectropolarimetric observations were accompanied by photometric measurements in V band, we used the light curve in R band to associate polarization and photometric measurements if they were performed within 3 hr of each other. Figure 26 shows three examples of the P(λ) dependence that appear to be representative: (1) the degree of polarization increases with wavelength when the optical emission is weak and P_opt is moderately high (e.g., on 2009 May 1, S_R = 1.51 ± 0.02 mJy and P_opt = 7.94% ± 0.09%); (2) the degree of polarization does not depend on wavelength when the optical emission is sufficiently bright and P_opt is high (e.g., on 2010 November 15, S_R = 10.51 ± 0.18 mJy and P_opt = 13.00% ± 0.04%); and (3) the degree of polarization decreases with wavelength when the optical emission is very bright and highly polarized (e.g., on 2010 November 10, S_R = 18.63 ± 0.36 mJy and P_opt = 18.84% ± 0.05%). We use a linear fit, P(λ) = A + Bλ, to approximate the dependence of the fractional polarization on wavelength for each spectrum obtained from 2009 April to 2011 August (244 spectra). Examples of the fits are shown in Figure 26 by red solid lines with slope B equal to (0.112 ± 0.011) × 10⁻⁴ Å⁻¹, (0.38 ± 0.44) × 10⁻⁶ Å⁻¹, and (−0.661 ± 0.050) × 10⁻⁵ Å⁻¹ for 2009 May, 2010 November 15, and 2010 November 10, respectively. Figure 27 plots derived values of slope B versus brightness in R band for all observations, while Table 6 lists the average values of slope B for different brightness intervals. Figure 27 and Table 6 show that there is a change in the P(λ) dependence with brightness: for S_R ⩽ 4.5 mJy the coefficient B is positive-definite despite significant scatter, while for S_R > 4.5 mJy B is close to zero. This agrees very well with the finding discussed in Section 5.2 that the optical continuum consists of two components, blue (BBB) and red (synchrotron). The contribution of the BBB to the optical emission is significant in the blue part of the spectrum, especially at low flux levels. Dilution by this component leads to a degree of polarization that increases at longer wavelengths (Figure 27), as found previously by Smith et al. (1988). The latter trend suggests that the emission of the BBB is unpolarized. The P(λ) dependence disappears when the nonthermal component dominates the total optical emission, as expected for the synchrotron emission over a relatively narrow wavelength range. Figure 27 and Table 6 also show that at very bright flux levels (S_R > 10 mJy) B is negative, which implies a higher degree of polarization at shorter wavelengths. Although the number of such observations is small, such a tendency supports models in which electrons are accelerated at a front and then lose energy to radiation. In this case, higher-energy electrons occupy a smaller volume beyond the front—with a more uniform magnetic field—than do lower-energy electrons that radiate at longer wavelengths (e.g., Marscher & Gear 1985; Marscher et al. 1992; Marscher & Jorstad 2010). Note that we have investigated dependence of position angle of polarization on wavelength and found that χ_opt does not depend on λ, either at high or moderate degrees of polarization, although at a low level of polarization uncertainties of χ_opt(λ) increase significantly.

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Figure 28 (left) shows the dependence between the degree of optical polarization and flux of 3C 454.3 in R band. The fractional polarization changes from <0.5% to 30%, while the flux varies from ∼1 mJy to 20 mJy. The Spearman rank correlation coefficient, ρ = 0.565, gives a statistical significance of 98.7% that the values are related, with the degree of polarization rising along with the flux. We have shown above that the optical continuum consists of two components: unpolarized thermal (BBB) and polarized synchrotron (jet) emission. According to Section 5.2, the contribution of the BBB in R band is $S_{\rm R}^{\rm disk}\sim 0.91$ mJy. If we assume that the thermal component was constant during our observations and completely unpolarized, we can derive the flux and degree of polarization of the variable synchrotron component: $S_{\rm R}^{\rm var}=S_{\rm R}^{\rm obs}-S_{\rm R}^{\rm disk}$ and $P_{\rm opt}^{\rm var}= P_{\rm opt}\times S_{\rm R}^{\rm obs}/S_{\rm R}^{\rm var}$ with $S_{\rm R}^{\rm disk}=0.91\pm 0.16$ mJy (Section 5.2). Figure 28 (right) shows the dependence between the degree of optical polarization and flux for the variable component. In this case an increase of the degree of polarization is observed also at a very small flux level of the variable source, with $P_{\rm opt}^{\rm var}$ rising up to ∼40% while S_var < 1 mJy, although the uncertainties in $P_{\rm opt}^{\rm var}$ are significant owning to an uncertainty in the derived BBB flux. An increase of the degree of polarization along with the flux level might be a signature of shock processes owing to ordering of the magnetic field in the shocked region if the quiescent jet has a chaotic magnetic field. However, Figure 28 (right) suggests that in a completely quiescent state the synchrotron component originates in a region with a very well-ordered magnetic field. This implies that the optical synchrotron emission during quiescent and active states arises from different regions in the jet. These two synchrotron components (quiescent and active) perhaps possess different polarization properties that can explain the minimum values of $P_{\rm opt}^{\rm var}$ at fluxes of 2–3 mJy when a quiescent synchrotron component has a flux comparable to that of an active synchrotron component at a moderate stage of activity. A "competition" between the thermal and different synchrotron components at a moderate flux level might be responsible also for the largest scatter in the slope of the P(λ) dependence seen at flux level between 2 and 4 mJy (Table 6).

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If the optical synchrotron emission originates in different regions during quiescent and active states, we can expect that the magnetic field configuration of the regions is different. Figures 29 show the distributions of the position angle of optical polarization with respect to the jet direction for S_var < 1 mJy (left) and S_var > 5 mJy (right). It is clear that the position angle of polarization differs during quiescent and active states, with χ_opt tending to align nearly along the jet direction during quiescent states, while being oblique, or nearly perpendicular to the jet direction, during active states. This suggests that the optical synchrotron emission during a quiescent state originates in a region with a well-ordered toroidal magnetic field. Such a region is most likely located in the magnetically dominated part of the jet, relatively close to the black hole (within several thousand Schwarzschild radii; Meier et al. 2000; McKinney 2006). On the other hand, the optical synchrotron emission during an active state originates in a region with a chaotic magnetic field, although a further increase of the flux leads to ordering the magnetic field along the jet axis. The latter implies that the region of flaring synchrotron emission is located farther downstream the jet where either spiral loops of the toroidal magnetic field are very loose (Lyutikov et al. 2005) or the effects of velocity shear align the magnetic field with the jet axis (Laing 1980; D'Arcangelo et al. 2009).

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We compare the position angle of the polarization (EVPA) at optical wavelengths and in the VLBI core at 43 GHz for simultaneous observations. Figure 30 shows the distribution of differences between the optical EVPA, χ_opt, and EVPA in the VLBI core, χ_core. The distribution is bimodal, with χ_opt either similar to χ_core or different by >50°. Note that a good agreement between the position angles, |χ_opt − χ_core| < 20° (20 cases), is observed when the source is brighter than 2 mJy in R band.

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The distribution therefore suggests that there may be a relationship between the properties of optical and VLBI core polarization when the source is in an active state. We find stronger evidence for such a connection in observed similarities between optical polarization parameters and those in the core during outbursts. Figure 31 shows the parameters of the optical and core polarization during outbursts I, II, and III (we also use the polarization data obtained in V band by Sasada et al. 2012 during outburst I). The data for each outburst are plotted relative to the corresponding time of γ-ray flux maximum, $T_\gamma ^{\rm max}$, listed in Table 1. In general, there is an increase of P_opt during the outbursts. However, measurements at the peak of outbursts I and II reveal a significant drop of the degree of optical polarization (down to 2%–3%) over a period of ∼3–4 days centered at $T_\gamma ^{\rm max}$. The degree of polarization in the core reveals similar behavior: P_core increases during outbursts up to 4%, but it drops below 1% close to $T_\gamma ^{\rm max}$ for outburst II, for which there are observations within 2 days of $T_\gamma ^{\rm max}$.

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The behavior of the position angle of polarization during outbursts is extremely interesting. Figure 31 shows that: (1) near the beginning of an outburst, χ_opt is relatively stable at ∼ − 25°, while χ_core differs from χ_opt by ∼90° for outburst III, which indicates that the core is most likely optically thick at 43 GHz; (2) ∼10 days before $T_\gamma ^{\rm max}$, χ_opt starts to rotate, although it does so in opposite directions for outbursts I and III; (3) at the peak of a γ-ray outburst χ_opt fluctuates on a timescale of several hours; (4) ∼5–10 days after $T_\gamma ^{\rm max}$, χ_opt starts a new cycle of rotation with the same counter-clockwise direction for both outbursts I and III, and at a similar rate of ∼9° day⁻¹ over at least 10–15 days; (5) despite similarities in the rotation of χ_opt, there is a constant offset of ∼40° between the rotation lines (the dashed lines in Figure 31, bottom panel); this corresponds to a shift in the directions of the trajectories of knots K09 and K10 within 0.2 mas of the core (see Figure 12); (6) ∼35 days after $T_\gamma ^{\rm max}$, the optical EVPA stabilizes at χ_opt ∼ 0°, which is the EVPA of the core as well as that of K09 and K10 when they first appear in the jet (see Figure 14). Therefore, although the optical polarization varies dramatically during γ-ray outbursts, the behavior of the optical polarization maintains a tight connection with the properties of the mm-wave core and superluminal knots K09 and K10.