Correlation Analysis of Delays between Variations of Gamma-Ray and Optical Light Curves of Blazars

Karen E. Williamson 1,*, Svetlana G. Jorstad 1,2, Alan P. Marscher 1, Valeri M. Larionov 2,3,4, Iván Agudo 1,5, Arkady A. Arkharov 3, Dmitry A. Blinov 2,6, Carolina Casadio 5, José L. Gómez 5, Vladimir A. Hagen-Thorn 2,4, Manasvita Joshi 1, Tatiana S. Konstantinova 2, Evgenia N. Kopatskaya 2, Elena G. Larionova 2, Liudmilla V. Larionova 2, Michael P. Malmrose 1, Ian M. McHardy 7, Sol N. Molina 5, Daria A. Morozova 2, Brian W. Taylor 1,8 and Ivan S. Troitsky 2 1 Institute for Astrophysical Research, Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA; jorstad@bu.edu (S.G.J.); marscher@bu.edu (A.P.M.); iagudo@iaa.es (I.A.); mjoshi@bu.edu (M.J.); mmalmros@bu.edu (M.P.M.); taylor.2112@gmail.com (B.W.T.) 2 Astronomical Institute, St. Petersburg State University, Universitetskij Pr. 28, Petrodvorets, 198504 St. Petersburg, Russia; vlar2@yandex.ru (V.M.L.); dmitriy.blinov@gmail.com (D.A.B.); hth-home@yandex.ru (V.A.H.-T.); azt8@mail.ru (T.S.K.); enik1346@rambler.ru (E.N.K.); sung@mail.ru (E.G.L.); lliudmila@yandex.ru (L.V.L.); comitcont@gmail.com (D.A.M.); void@star.math.spbu.ru (I.S.T.) 3 Main (Pulkovo) Astronomical Observatory of RAS, Pulkovskoye shosse, 60, 196140 St. Petersburg, Russia; arkadi@arharov.ru 4 Isaac Newton Institute of Chile, St. Petersburg Branch, 198504 St. Petersburg, Russia 5 Instituto de Astrofísica de Andalucía, CSIC, Apartado 3004, 18080 Granada, Spain; casadio@iaa.es (C.C.); jlgomez@iaa.es (J.L.G.); smolina@iaa.es (S.N.M.) 6 Department of Physics, University of Crete, 71003 Heraklion, Greece 7 Department of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK; I.M.Mchardy@soton.ac.uk 8 Lowell Observatory, Flagstaff, AZ 86001, USA * Correspondence: kwilliam@bu.edu


Introduction
We present preliminary results from a correlation analysis on the 13 BL Lac objects and 21 quasars monitored by the Boston University blazar team and collaborators since August 2008.Here, we focus on correlations between γ-ray and optical light curves.The γ-ray light curves at 0.1-200 GeV are constructed from data provided by the Fermi Large Area Telescope (LAT).Optical light curves are represented by data in R-band, which has the best time coverage in observations combined from different telescopes (Table 1 presents the legend for the observatories.)Most correlation studies of blazars have focused on singular events of contemporaneous multi-wavelength outbursts.With a 7-year accumulation of data, we examine correlations over multiple periods of high activity in a number of blazars, testing for evidence of consistency.
Prevailing models for γ-ray production (e.g., [1][2][3]) explain the origin of γ-rays by inverse Compton upscattering of infrared to ultraviolet photons by relativistic electrons that also emit synchrotron optical-ultraviolet photons.The sources of seed photons and locations of the emission zones continue to be debated (e.g., [4]).Therefore, correlation analysis-especially determination of delays between variations-between γ-ray and optical light curves is key to understanding where and how high-energy emission is produced in blazar jets.It is particularly interesting to determine whether there are consistent correlations among different blazars and for different events in an individual blazar.

Results
In the external-radiation inverse Compton model, we expect no optical/high-energy lag, while in the synchrotron self-Compton model, light-travel delays of the seed photons can cause the high-energy flux to lag [8].A delay of optical variations with respect to γ-rays suggests stratification of the emission region with respect to energy.For this analysis, we limit our lag times to ±50 days, since for longer delays aliasing is problematic.
For each blazar, we classify the results as follows: (1) The object displays a statistically significant, single correlation when tested over the entire 7-year period ("Overall Correlation", OC); (2) The majority of individual epochs of high activity display a similar correlation ("Consistent Individual Correlations", CIC).
Table 2 displays the preliminary results of our analysis.We quote the time lag as determined by the z-transformed discrete correlation function (ZDCF) maximum likelihood method [9] if the bootstrap analysis also gives a significance exceeding a 2-σ probability.Five sources (0528+134, 1127-145, 1406-076, 1611+343, and 3C446) did not have sufficient coverage of data to determine correlations.From each classification, we display one object's light curves and correlation results in Figures 1-4.Although the ZDCF algorithm has been shown to effectively determine correlations of unevenly sampled data, both the binning of the Fermi data and the gaps in optical observations affect the resolution of lag times.Most of our Fermi data during periods of high activity have been binned over 1-3 days, low activity over 7 days.The uncertainty of a time lag is derived as the FWHM of the ZDCF peak and cannot be less than the binning interval.

Method
We perform the correlation analysis using the ZDCF and Maximum Likelihood PLIKE algorithm [10].We verify the significance of the correlation by comparing with the statistics of correlations of 3000 pairs of bootstrapped artificial light curves (ALC).

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Each object's active periods are identified based on the light curve behavior (details can be found in [11]).

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Most γ-ray active periods with fluxes exceeding F ν + 3σ w ν are re-reduced, allowing the photon index to vary and binning on a shorter time interval.Optical data are binned into 1-day periods.

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For each source, the ZDCF is calculated for the entire time span of observations and for each active period.

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Each ALC is built by randomly selecting and randomly placing active periods, preserving the observational dates by either using the closest observed point, if within 7 days, or interpolating the data.

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After the active periods have been placed in the ALC, the remaining observational fluxes are randomly selected and randomly placed on the remaining observational dates.

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The ALCs are randomly paired and sent through the ZDCF for analysis.

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Results of the ZDCF analysis of ALCs are used to derive 1-, 2-, and 3-σ probabilities to obtain a given coefficient of correlation by chance.

Summary
Our preliminary results reveal a statistically significant correlation for 7-year light curves in seven BL Lacs and 10 FSRQs.Generally, the 17 sources exhibiting a consistent OC correspond to either γ-ray leading optical outbursts or zero time lag within the uncertainty.The six sources consistent in both OC and CIC display lag times of 0 ± 7 days, while the 11 sources consistent in OC but not CIC generally have longer lag times.
In many cases, the classification is somewhat uncertain.For example, 1510-089 exhibits statistically significant correlations at two lag times (−12 days and 1 day), although one also could classify it as a single, broad correlation.However, the individual-period plots display patterns that support each peak.OJ287 displays an apparent correlation of −1 day, but it is not statistically significant according to our bootstrap analysis.Many objects (e.g., Mrk421) have prolonged periods of high γ-ray activity, causing broad correlations over a range of lag times.No significant difference in behavior is seen between the BL Lacs and the quasars.

Figure 1 .
Figure 1.Sample light curves and correlation plots of 3C454.3.Panels (a) contain the light curves of the data across energy bands.The sources of the data are indicated by symbols and colors identified in Table1.Vertical dashed lines indicate the epochs selected for analysis.Panels (b) contain the results of the correlation analysis over the entire period of available data and Panels (c) display a series of correlation results analyzed over shorter periods (as indicated in Panels (a)).The highest significantly correlated time lags from the bootstrap analysis are labeled in red if greater than 3σ, black if exceeding 2σ.Dotted lines are ±2σ.

Figure 2 .Figure 3 .
Figure 2. Sample light curves and correlation plots of 3C279.See Figure 1 for details.

Figure 4 .
Figure 4. Sample light curves and correlation plots of Mkn501.See Figure 1 for details.

Table 1 .
List of Observatories Providing Measurements for this Study.

Table 2 .
Correlation Classification.Time lags in days are in parentheses; a negative value denotes γ-ray leading optical variations.