MULTIWAVELENGTH OBSERVATIONS OF MARKARIAN 421 IN 2005–2006

All the TeV gamma-ray observations presented here were made with the 10 m Gamma-ray Telescope at the Fred Lawrence Whipple Observatory (FLWO; Kildea et al. 2007). Although sensitive in the energy range from 200 GeV to 20 TeV, the peak response energy of the telescope to a Crab-like spectrum during the observations reported upon here was approximately 400 GeV. This is the energy at which the telescope is most efficient at detecting gamma rays and, as is typical for IACTs such as Whipple, is subject to a 20% uncertainty.

The Mrk 421 data were analyzed using the imaging technique and analysis procedures developed by the Whipple Collaboration (Hillas 1985; Reynolds et al. 1993). A Quicklook analysis was performed at the conclusion of each observation and the results were posted daily on a public Web site.³⁶ Three offline analyses (independent but using the standard Supercuts methodology) were used to derive the TeV light curves presented here.

Two different modes of observation are employed at the Whipple Telescope, "On–Off" and "Tracking" (Catanese et al. 1998). In both modes, the data are usually taken in 28 minute scans. Unlike data taken in the On–Off mode, the scans taken in the Tracking mode do not have independent control data which can be used to establish the background level of gamma-ray like events during the scan. These control data are essential in order to estimate the number of events passing all cuts that would have been detected during the scan in the absence of the candidate gamma-ray source. In order to perform this estimate, a tracking ratio is calculated by analyzing "darkfield data" (Horan et al. 2002). These consist of Off-Source data taken in the On–Off mode and of observations of objects found not to be sources of gamma rays. A large database of these scans is analyzed and, in this way, the background level of events passing all gamma-ray selection criteria can be characterized as a function of zenith angle. The data presented here were mostly taken in the Tracking mode.

During the period MJD 53676 to MJD 53908, a total of 144.1 hr of data were taken on Mrk 421. Since IACTs can only operate during moonless conditions, there is a period centered on full moon each month when gamma-ray observations are not possible. From the total of 328 runs on the source (On and Tracking modes), 275 runs (84%), comprising 122.57 hr, survived the quality selection process. Apart from any problems reported by observers in the nightly logs, the main parameter used to determine whether the data run should be included in the final analysis was the stability of the raw trigger rate. Any data run whose raw trigger rate deviated significantly from being steady for its duration was discarded.

When possible, the AGN data were interspersed with observations of the Crab Nebula, the standard candle in this energy regime. A total of 44.41 hr of these were taken and subjected to the same analysis procedure. In addition, nightly runs were taken with the telescope pointing to the zenith; these data were used to calculate the telescope throughput (Lebohec & Holder 2003) so that data could be corrected for inter-nightly changes in atmospheric transparency. Both the AGN and the Crab data were corrected for variations in the throughput. In addition to this, a correction was applied to compensate for elevation effects in the data due to the increasing volume of atmosphere through which the Cherenkov light propagates as the zenith angle of the observations increases. These corrections ensured that data taken on different nights, under different atmospheric conditions and at different elevations could be compared. The Mrk 421 rates were then converted into equivalent Crab rates. It should be noted that this simplistic scaling is strictly only valid for a TeV spectrum near that of the Crab Nebula (spectral index of −2.6). Although Mrk 421 has been known to display spectral variability in the past, the measurement uncertainties on the flux points are such that the effect of spectral variability should not be significant. The gamma-ray light curve containing the rate from each approximately 28 minute scan on Mrk 421 is shown in Figure 2.

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The X-ray data for this MWL campaign span the energy range from 0.2 to 50 keV. These data come from four different instruments, two on the Rossi X-ray Timing Explorer (RXTE) and two on the Swift satellite. The X-ray light curves are shown together with the gamma-ray data in Figure 3. The details of the analysis of each X-ray data set are given in the following subsections.

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2.2.1. The All Sky Monitor (ASM)

2.2.2. The Proportional Counter Array (PCA)

2.2.3. The X-ray Telescope (XRT)

2.2.4. The Burst Alert Telescope (BAT)

Eight optical observatories contributed data sets to this campaign. They are the FLWO 1.2 m telescope (located adjacent to the Whipple 10 m Gamma-ray Telescope on Mt. Hopkins), the Tenagra 0.8 m telescope in Tenagra, Arizona, USA, the Bradford Robotic Telescope in Tenerife, Canary Islands, Spain, the WIYN 0.9 m telescope on Kitt Peak, Arizona, USA, the 0.7 m telescope at Abastumani Observatory in Abastumani, Georgia, the 0.6 m Bell Observatory at Western Kentucky University, USA, Bordeaux Observatory in Floriac, France, and the 1.05 m REOSC telescope at Osservatorio Astronomico di Torino, Italy. Table 2 gives a brief description of each telescope, its filter system and bandpass, and its contribution to the optical data set.

The data from the various observatories were reduced and the photometry performed independently by different analysts using different strategies. The FLWO, Bradford, and Tenagra data, for example, usually consisted of a single or a few images per filter per night. Relative aperture photometry was performed using standard routines in IRAF.³⁸ Magnitudes were calculated with respect to star 1 of Villata (Villata et al. 1998) with a photometry aperture of 10 arcsec. The host galaxy light was not subtracted at this stage of the analysis. Other optical analysts used slightly different strategies: the photometry for the Bell, WIYN, and Abastumani data were quoted with respect to stars 1–3 of Villata, and the Torino photometry was performed using Gaussian fitting rather than by counting within a fixed aperture.

Combining the various optical data to produce a composite light curve for each spectral band is complicated by the fact that different observatories use different photometric systems. Furthermore, photometric apertures and the definition of the reported measurement error for each nightly averaged flux is inconsistent across data sets. Therefore, we have adopted a simple approach for the construction of the composite light curves whereby a unique flux offset is found for each spectral band of every instrument based on overlapping observations (Steele et al. 2008).

In the R band, for example, we first find the initial average flux offset for each R-band light curve with respect to that of the FLWO (since it spans the largest range of dates) based on observations overlapping by less than 0.8 days (except for the Bell and Torino light curves, whose coincidence windows were widened to 1.8 days to ensure a sufficient number of overlaps). Then, for each light curve, a comparison light curve is constructed using the light curves from the other observatories with the initial flux offsets applied. A new offset is then found for each light curve based on overlapping observations in the respective comparison light curve. The process is repeated iteratively until the variance of the offset residuals for each light curve is minimized. The final composite light curve for each spectral band is then made by simply combining the nightly averaged flux from each observatory with the final offsets applied and these are shown in Figure 4. Of the three wavelengths, the R band is the best sampled, comprising 166 data points on 97 nights spanning a period of 260 days. Figure 5 shows the very well-sampled optical light curves, after all the data from different instruments have been combined and normalized, along with the gamma-ray data during this campaign. The mean magnitude of Mrk 421 during this campaign was 12.95 in the R band, 13.40 in the V band, and 13.82 in the B band.

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Typically, the variations in AGN are slower at radio frequencies than at the higher frequency bands with the most variability being observed at the higher radio frequencies (see, e.g., Kovalev et al. 2002). In general, because of this slower long-term variability, AGNs are not observed with as high temporal coverage in this regime. The radio data presented here were taken at eight frequencies at three different radio observatories but, for four of these frequency bands the radio coverage was very sparse. The fluxes and their associated standard errors are given in janskys, so they have already been normalized for the bandwidth of their receivers. Generally, Mrk 421 is found to vary on monthly timescales. The radio light curves are shown along with the gamma-ray light curve in Figure 6. In the subsections that follow, the radio data and analysis from the participating observatories are described.

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2.4.1. Metsähovi Radio Observatory

2.4.2. University of Michigan Radio Astronomy Observatory

2.4.3. Radio Astronomical Telescope of the Academy of Sciences

2.4.4. VLBA: MOJAVE Program

In addition to these radio light curves, the data from the 2 cm Very Long Baseline Array (VLBA)/MOJAVE monitoring program (Kellermann et al. 2004; Lister & Homan 2005) for the epoch 2006 April 5 (MJD 53830) were used (see the 15 GHz VLBA image in Figure 7). The total flux density (Stokes I) at 15 GHz integrated over this image is 336 mJy. The flux density from the core region, when it is modeled using a circular Gaussian, is 288 mJy so Mrk 421 was very core dominated at this time implying that the bulk of the radio-band variability is associated with the VLBI-imaged core. The angular size of the modeled core on the half power level is 0.046 mas. The core size resolution limit was estimated for this data set following (Kovalev et al. 2005) and appeared to be less than the measured core size value. The linearly polarized flux density is 6 mJy. This polarized flux is detected from the core region only. VLBI core brightness temperature in the source frame is estimated to be 8 × 10¹¹ K.

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The fractional rms variability amplitude, F_var, and the point-to-point fractional rms variability amplitude, F_pp (Zhang et al. 2005), were calculated for each of the energy bands where a significant amount of data were gathered. They are defined as

$$\begin{equation} F_{\rm var} = \sqrt{ \frac{S^2 - {\overline{\sigma ^2}}}{{\bar{F}^2} }} \end{equation} \tag{ 1 }$$

and

$$\begin{equation} F_{pp} = \frac{1}{\bar{F}} \sqrt{\frac{1}{2(N-1)} \sum _{i=1}^{N-1} (F_{i+1} - F_i)^2 - \overline{\sigma ^2}}, \end{equation} \tag{ 2 }$$

where each flux measurement F_i has a measurement error σ_i, $\bar{F}$ is the arithmetic mean of the counts, and

$$\begin{equation*} S^2 = \frac{1}{N-1} \sum _{i=1}^{N} (F_i - \bar{F})^2, \end{equation*}$$

and $\overline{\sigma ^2}$ is the mean error squared:

$$\begin{equation*} \overline{\sigma ^2} = \frac{1}{N} \sum _{i=1}^{N} \sigma _i^2. \end{equation*}$$

These quantities were calculated using 1-day binning for each of the data sets. These results can be seen for all wavebands apart from the 37 GHz radio data set in Table 3. For the 37 GHz data, the difference between the terms to be subtracted in both Equations (1) and (2) was negative, indicating that the measurement errors were larger than the variability observed, and therefore the calculation could not be completed. The fractional rms variability amplitude, F_var, quantifies the integrated level of variability present in a particular waveband, while the point-to-point fractional rms variability amplitude, F_pp, probes the short-timescale variability by measuring the variations between adjacent points in the light curve. Their ratio, taken as a function of energy, provides information on the dependence of the power spectral density slope on energy (Zhang et al. 2005). Figure 8 shows F_var and the ratio of F_var to F_pp for 10 of the 12 wavebands. The values for the BAT data are not plotted because, as will be discussed in Section 4, the significance level of the BAT detection was less than 3σ on most nights (∼58%) during this campaign.

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Throughout the observations presented here, the gamma-ray rate from Mrk 421 was found to be variable. This is evident from the light curves shown in Figure 9, in which the average nightly gamma-ray rate is shown, and in Figure 2, where the gamma-ray rate is plotted for each observing scan (typically of 28 minute duration). Mrk 421 was observed by the TACTIC Gamma-ray Telescope (Yadov et al. 2007) during a time period overlapping with these Whipple observations and, although the energy ranges and sensitivities of the two instruments are slightly different, similar overall trends are present in the two data sets. The maximum gamma-ray rate for one of the Whipple 28 minute exposures occurred on MJD 53733 when a rate of 4.38 ± 0.49 Crab was recorded. This is a factor of 3.74 increase over the mean rate for a 28 minute exposure during this campaign. The light curve for MJD 53733 is shown in Figure 9. The mean gamma-ray rate for this night was the second highest recorded during this campaign at 2.50 ± 0.32 Crab. The flux from Mrk 421 can be seen to vary throughout this night, starting off at 1.91 ± 0.42 Crab before reaching its peak value and then decreasing to 1.67 ± 0.26 Crab. The maximum average nightly gamma-ray rate of 2.59 ± 0.37 Crab, a factor of 2.49 higher than the mean nightly rate, was recorded on MJD 53884. A summary of the mean, minimum, and maximum gamma-ray rate from Mrk 421 during this campaign is given in Table 3. The fractional rms variability amplitude in the gamma-ray band was found to be 0.511 with a point-to-point fractional rms variability amplitude of 0.246. The ratio of these two quantities, at ∼2 (see Figure 8), indicates that red-noise variability is present (Zhang et al. 2005).

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The X-ray flux from Mrk 421, was monitored from 0.2 to 50 keV during this observing campaign. Many episodic outbursts are seen in all X-ray bands reported on here with the maximum emission reaching 1.7 times the mean emission level in that band for the XRT data, 2.9 times the mean emission level for that band in the ASM data, 2.7 times the mean emission level for that band in the PCA data, and 5.3 times the mean emission level for that band in the BAT data. These data are summarized in Table 3. The data are plotted in one-night bins in Figure 3 where inter-night variability in addition to overall trends are evident in the X-ray light curve. The X-ray emission in the ASM, PCA, and BAT regimes is well correlated. The night of maximum emission was MJD 53877 for the ASM data. No PCA data were taken this night and the maximum in its emission occurred on MJD 53875, which was the second brightest night for the BAT data. There is a trend toward increasing emission around this time with the previous ∼30 days showing elevated emission levels in the ASM, PCA, and BAT data. For the XRT light curve, which is not as well sampled as those in the other wavebands, evidence for heightened emission over a period of two nights was seen from MJD 53739 to MJD 53740. The rms variability amplitude is found to increase with energy in the X-ray regime, from 0.269 for the XRT data to 0.529 for the PCA data. As can be seen in Figure 8, the ratio of F_var to F_pp is found to be ∼2 for all X-ray data, indicative of red-noise variability.

To further study the variability properties of the X-ray light curves, the normalized, first-order structure function, SF⁽¹⁾(τ), was computed for each light curve. SF⁽¹⁾(τ) is defined as

$$\begin{equation} SF^{(1)}(\tau) = \frac{1}{N} \sum \left(\frac{F(t) - F(t+\tau)}{\overline{F(t)}}\right)^2. \end{equation} \tag{ 3 }$$

For a discrete time series, we calculate the structure function in bins of width Δτ such that the value of the structure function for a given bin is found by summing over pairs of observations separated by a time difference Δt satisfying τ − Δτ/2 < Δt < τ + Δτ/2. The structure functions for each of the X-ray bands in which substantial data were collected and for the gamma-ray data are shown in Figure 10. In all cases, the structure function is found to rise approximately linearly when plotted in log–log representation. The data from the ASM show evidence for a break between τ ∼ 20 and 25 days with the slope of the structure function remaining approximately constant before and after this break. The structure function of the PCA data shows evidence for a break between τ ∼ 5 and 8 days. Again, no significant change in the slope is seen after this break. There is clear evidence for a break in the structure function of the BAT data between τ ∼ 55 and 71 days after which it rises linearly once more but with a steeper slope. There are no strong features in the structure function of the gamma-ray data.

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The well-sampled optical light curves show evidence for variability during this campaign. The mean magnitudes for each band are shown in Table 3. When converted to linear fluxes, the optical flux was found to peak at 1.43 times its mean value in both the R band and the V band and to peak at 1.83 its mean flux in the B band.

Several R-band flare features with amplitudes of 0.2 mag are resolved in the particularly well-sampled range of dates spanning 53,800 < MJD < 53,900. The fractional rms variability amplitude and point-to-point fractional rms variability amplitude are 0.13 and 0.04, respectively, for the R-band light curve, 0.14 and 0.05, respectively, for the V-band light curve and 0.19 and 0.06, respectively, for the B-band light curve. In all cases, the ratio of F_var to F_pp is ∼3, indicating that, like the gamma-ray and X-ray bands, red-noise processes are responsible for the optical variability. The measured optical flux includes contributions from the host galaxy, estimated to be ∼15% (Nilsson et al. 1999). Since it affects only the mean flux, not the variability, the optical fractional rms variability amplitude should be about 15% higher. The level of optical variability seen in this analysis is slightly lower but is consistent with the level of variability seen in other recent MWL studies of this source (Rebillot et al. 2004).

Variations in flux among the three optical bands are highly correlated. There is weak evidence in this data set supporting a trend of flattening of the optical spectrum with increasing R-band flux: a trend seen more definitively by other authors in the long-term light curves of Mrk 421 and other BL Lac objects (Vagnetti & Trevese 2003; Hu et al. 2006). This behavior, predicted by several authors (Li & Kusunose 2000; Spada et al. 2001; Vagnetti et al. 2003), is consistent with the expected spectral variability behavior of a jet with emission dominated by the SSC processes.

The normalized, first-order structure function of the composite R-band light curve has been computed and is shown in Figure 11. The structure function rises approximately linearly at short timescales up to a time lag of between about 40 and 60 days where a clear break is seen, above which the structure function rises linearly again. The break most likely indicates a characteristic timescale of shotlike emission events. We note that structure functions with similarly shaped breaks have been observed in other wavebands with different characteristic break frequencies, for example, the EUV emission from Mrk 421 recorded by the Extreme Ultraviolet Explorer during a 1998 MWL campaign (Takahashi et al. 2000).

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This is the waveband in which the light curves are least sampled and also, the waveband in which the least amount of variability is seen. This is typical for blazars; the variations observed in the radio regime tend to occur over longer timescales than those at shorter wavelengths (Kovalev et al. 2002; Błażejowski et al. 2005; Rebillot et al. 2006; Hovatta et al. 2007). This is supported by the fact that the lowest values of the fractional rms variability amplitude, 0.04–0.27, were found for the radio data. The fact that these data were less well sampled than the data at other frequencies would most likely lead to an underestimation of F_var in this waveband. The ratio of F_var to F_pp for each of the four radio bands plotted in Figure 8 is found to be ∼1, indicative of white-noise variability.

All of the X-ray data taken during this campaign are plotted with the gamma-ray data in Figure 3. For this plot, the XRT and PCA data are plotted between 2 and 10 keV, their counts having been converted to fluxes using a log-parabolic model to fit the data. The length of the exposure acquired varies from night to night, even for a single instrument. There is evidence for correlated variability throughout the campaign, in particular, around the period of MJD 53875 to MJD 53885. The mean rate over the entire observing campaign for each waveband is plotted as a horizontal, solid red line on these plots. The data can be seen to follow similar trend in terms of their mean flux level. Figure 12 shows flux–flux diagrams (for 1-day timescales) for each X-ray band with the gamma-ray data. Each data set was first normalized by dividing each flux point by the mean flux in that waveband so that the flux–flux correlations between the gamma-ray data and each X-ray band could be more easily compared. Using a weighted total least-squares algorithm (Anton 2007), which took into account uncertainties in the X-ray and the gamma-ray data, a straight line was fitted to each of the flux–flux plots. The slopes of these lines are found to increase with X-ray energy and are 0.73, 1.00, 1.51, and 3.01 for the XRT, ASM, PCA, and BAT data, respectively. As would be expected due to their overlap in energy range, the slopes of the flux–flux relations between the gamma-ray and the XRT (0.2–10 keV), ASM (2–10 keV), and PCA (3–25 keV) data are quite close to each other, their differences presumably arising due to the slightly different energy ranges and measurement uncertainties. Although the lower energy X-ray data (i.e., not those from the BAT) and gamma-ray data do look somewhat correlated, there are outliers on all plots. In general, the gamma-ray and the lower energy X-ray data exhibit low and high fluxes at similar times. To compare the relationship between these X-ray and gamma-ray data with that derived by Fossati et al. (2008), the data were replotted on a log scale, with the gamma-ray data on the Y-axis (Figure 13). The slopes of the lines were found to be 1.03 for the XRT (0.2–10 keV), 0.98 for the ASM (2–10 keV), 0.48 for the PCA (3–25 keV), and 0.22 for the BAT (15–50 keV) data. In the energy range between 2 and 10 keV, Fossati et al. (2008) found a slope of 0.88, similar to that found here in that energy range, for the relationship between the PCA and gamma-ray data from Mrk 421. When only the significant detections (defined here as >3σ and highlighted in green in Figure 13) were included in the slope calculation, the higher flux data remained and the slopes of the XRT, ASM, and PCA data decreased slightly to 0.94, 0.78, and 0.33, respectively, while the relationship between the BAT and the gamma-ray data became inverted (slope of −0.18), with little evidence for correlation between the two bands. Flux–flux diagrams were also made for weekly and monthly timescales. The ASM data, in particular, were found to show evidence for correlated emission with the gamma-ray data on weekly and on monthly timescales as shown in Figure 14. The slopes of the lines are 1.0, 1.0, and 0.5 for the daily, weekly, and monthly light curves, respectively.

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Figure 15 shows the flux divided by the mean flux in that band for each of the X-ray data sets and for the gamma-ray data. A high degree of correlation can be seen, in particular, between the gamma-ray data and those from the ASM and the PCA.

We searched for correlations between fluxes in the X-ray bands and the gamma-ray band using the DCF. The DCF, introduced by Edelson & Krolik (1988), is an approximation to the classical correlation function which is applicable to discrete time series. It gives the linear correlation coefficient (R) for two light curves as a function of the time lag between them. The DCF is well suited to sparsely sampled light curves and is less likely to give rise to spurious results than a traditional correlation analysis with interpolated light curves. To construct the DCF, we first collect the set of unbinned discrete correlations, UDCF_ij,

$$\begin{equation} {\rm UDCF}_{ij} = \frac{(a_i -\overline{a})(b_j - \overline{b})}{\sqrt{\big(\sigma ^2_a - e^2_a\big)\big(\sigma ^2_b - e^2_b\big)}} \end{equation} \tag{ 4 }$$

for all pairs of observations a_i, b_j, with measurement errors σ_a, σ_b, and mean measurement errors of e_a, e_b, respectively. Each pair is associated with a time lag Δt_ij = t_j − t_i. The DCF for a given time lag, τ, is then constructed as

$$\begin{equation} {\rm DCF}(\tau) = \frac{1}{M} \sum {\rm UDCF}_{ij}, \end{equation} \tag{ 5 }$$

where the sum runs over the M pairs of observations separated by τ − δt/2 < Δt_ij < τ + δt/2, where δt is the chosen bin width. The uncertainty on the value of the DCF in a given bin is calculated as the rms variance of all the contributing UCDF_ij about the value DCF(τ).

The DCF for each of the X-ray bands with the gamma-ray data is shown in Figure 16. In these plots, the x-axis is defined such that in the gamma–PCA DCF for example, a positive lag means that the gamma-ray data lagged the X-ray data. For the XRT–gamma-ray DCF, 4-day binning is used due to the sparsity of the XRT data. For the ASM–gamma-ray DCF, 1-day binning is used and, for the PCA- and BAT–gamma-ray DCF, 2-day binning is used. According to both the SSC and EC models for blazar emission, the same population of electrons is responsible for the X-ray and the gamma-ray emission. Therefore, the X-ray and gamma-ray light curves should be correlated. For the well sampled ASM, PCA, and BAT light curves, a zero-lag correlation is seen clearly, with peak values of the DCF at zero lag of 0.44, 0.56, and 0.39, respectively. The shape of the DCF looks similar for the BAT and ASM data, as their light curves are similar. The PCA data have the smallest uncertainties and also show the highest DCF value at zero lag.

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To measure the chance probability of accidental correlations in the DCF, the Monte Carlo method described in Jordan (2004) was employed. Taking a correlation at zero lag as the a priori hypothesis, the chance probability of obtaining the measured value of the DCF at zero lag, with the assumption that the times of the flares are randomly sequenced, was calculated. 100,000 light curves were simulated with the same characteristic timescale as the PCA, determined by the SF analysis (Section 3.3). The DCF was calculated for each of these simulated light curves and the gamma-ray light curve. The chance probability of obtaining a DCF value at or above 0.56 from this Monte Carlo method was found to be 0.1%. We can state, therefore, with reasonable confidence that the gamma-ray and PCA X-ray data are correlated within 1.5 days.

Figure 17 shows the flux–flux correlation for the gamma-ray and optical data (linear fluxes). Each data set was first normalized by dividing each flux point by the mean flux in that waveband so that the flux–flux correlations between the gamma-ray and optical data could be more easily compared. Using a weighted total least-squares algorithm (Anton 2007), which took into account uncertainties in the optical and the gamma-ray data, a straight line was fitted to each of the flux–flux plots. Although not well correlated, the data do show a somewhat negative correlative trend, with the high-flux gamma-ray points tending to coincide with the low-flux optical points and vice versa. The slopes of the lines shown in Figure 17 are found to decrease marginally as energy increases, with a slope of −0.60 in the R band, −0.43 in the V band, and −0.40 in the B band.

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We searched for correlations between fluxes in the optical R band, which was the best sampled of the optical light curves, and all other nonoptical wavebands presented in this investigation using the DCF. Of all the pairs of wavebands investigated, only the DCF between the optical and TeV light curves, presented in Figure 18, shows noteworthy features. The DCF indicates a possible optical-TeV correlation with the optical flux lagging the TeV flux by about 7 days. An elevated level of correlation is also seen with the optical leading the TeV by 25 to 45 days (Steele et al. 2008).

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We investigated the significance of these possible correlations using simulated optical light curves with the same variability properties as the observed R-band light curve. The simulated optical light curves were generated deterministically as a superposition of harmonic oscillators with random phases whose frequencies are integer multiples of a fundamental frequency corresponding to the total length of the campaign. The amplitude of an oscillator at a given frequency is drawn from a power spectral density (PSD) function derived from the first-order structure function of the observed R-band light curve. Assuming the process(es) responsible for the optical variability are stationary over the period of observation, the slope of the PSD, β, at a given frequency is related to the slope of the first-order structure function, α, by β = 1 + α. Using this relation, we derive the PSD of our simulated light curves by a fit to the first-order structure function that rises linearly with a slope α = 0.568 until the beginning of the break at t = 39.8 days, is flat (α = 0) between 39.8 days <t < 63.0 days, and then rises again with slope α = 1.700.

After multiple realizations of the campaign using simulated optical light curves which are, by definition, not correlated with the TeV light curve, we can assess the probability to have seen the observed correlation features due to chance. For each realization, we compute the DCF between the simulated optical and the real TeV light curves. For each bin of the DCF, we record the values of the correlation coefficient, R, its error, σ_R, and the test statistic S = R/σ_R. After many realizations, we build the distribution of S seen in each bin of the DCF and compute the chance probability to have found the observed value of S = S_obs or higher by integrating the distribution above S_obs. This gives us the chance probability of having observed a given correlation in a particular bin, but it does not speak to the likelihood of correlation features involving more than one bin. To assess the likelihood of the observed DCF features as a whole, we construct two more distributions, $\mathcal {L}_{\pm } = \prod _{i} p_{i}$ where for $\mathcal {L}_{-}$ the product runs over all DCF bins, i, with τ < 0, and for $\mathcal {L}_{+}$ the product runs over bins with τ ⩾ 0. Using these two distributions, we find the likelihood to have observed the optical precursor (postcursor) features by chance to be 20% (60%). Further investigation reveals that the DCF feature seen at τ = 7 days probably arises due to contamination from the R-band autocorrelation function, which also exhibits features at multiples of 7 days. The possible reality of the optical precursor emission is more difficult to assess since our method assesses all the correlation features on a given side of the DCF at once. Unfortunately, it would not have been possible to narrow the likelihood computation to smaller range of τ in order to assess a particular feature without subjecting the result to an unknown trials penalty.

SEDs are shown on three representative dates in Figure 19 when the VHE emission was at a high (MJD 53763; filled blue squares), medium (MJD 53852; open green circles), and low (MJD 53820; closed red triangles) emission level; these dates were chosen because there were MWL data available over the full spectrum. They were not chosen where there was the best correlation with the flux at X-ray wavelengths and hence the SEDs are more typical. These nights are marked on the gamma-ray light curve in Figure 2. On MJD 53763, 135 minutes of noncontiguous gamma-ray data were taken; on MJD 53852, 138 minutes of noncontiguous gamma-ray data were taken, while on MJD 53820, 90 minutes of noncontiguous gamma-ray data were taken. The optical data shown on the SED has the galaxy contribution subtracted. There was no significant variation in the radio fluxes over the three dates. The X-ray spectrum for the PCA data was obtained using XSPEC,³⁹ while the TeV spectrum was obtained using the forward-folding method described in Rebillot et al. (2006). Mrk 421 was in the field of view of the BAT during parts of each of the three observations. It was detected on MJDs 53763 and 53852 and upper limits were obtained for MJD 53820. A spectral analysis of the BAT survey mode data on Mrk 421 was performed for two nights, MJDs 53763 and 53852. The BAT data from these nights comprised 74 and 44 minutes, respectively, and were processed using standard Swift FTOOLS⁴⁰ to remove the diffuse background and the effects of other bright point sources in the field of view. Eight-channel spectral and response files were produced for the position of Mrk 421 for each observation and were fit using XSPEC 11 to a simple power-law model to provide fluxes in the energy range of 14–195 keV. The highest energy at which a significant detection was made with the BAT was 75 keV.

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The data from the nights chosen to represent the high-state and medium-state of Mrk 421 are plotted in Figures 20 and 21 along with archival Mrk 421 data (Buckley 2000). The dashed purple lines show the synchrotron and self-Compton distributions for the parameters given in Table 4. The black dot-dashed curves show a hypothetical black-body component peaked at 1 μm and the corresponding EC component. The red solid line shows the sum of the SSC and EC models fitting results, which, in both cases, is in good agreement with the simultaneous optical, X-ray, and gamma-ray data.

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