Free Access
Issue
A&A
Volume 520, September-October 2010
Article Number A83
Number of page(s) 16
Section Extragalactic astronomy
DOI https://doi.org/10.1051/0004-6361/201014484
Published online 07 October 2010
A&A 520, A83 (2010)

VHE ${\gamma}$-ray emission of PKS 2155-304: spectral and temporal variability

HESS Collaboration - A. Abramowski4 - F. Acero15 - F. Aharonian1,13 - A. G. Akhperjanian2 - G. Anton16 - U. Barres de Almeida8,[*] - A. R. Bazer-Bachi3 - Y. Becherini12 - B. Behera14 - W. Benbow1 - K. Bernlöhr1,5 - A. Bochow1 - C. Boisson6 - J. Bolmont19 - V. Borrel3 - J. Brucker16 - F. Brun19 - P. Brun7 - R. Bühler1 - T. Bulik29 - I. Büsching9 - T. Boutelier17 - P. M. Chadwick8 - A. Charbonnier19 - R. C. G. Chaves1 - A. Cheesebrough8 - L.-M. Chounet10 - A. C. Clapson1 - G. Coignet11 - J. Conrad31 - L. Costamante1,34 - M. Dalton5 - M. K. Daniel8 - I. D. Davids22,9 - B. Degrange10 - C. Deil1 - H. J. Dickinson8 - A. Djannati-Ataï12 - W. Domainko1 - L. O'C. Drury13 - F. Dubois11 - G. Dubus17 - J. Dyks24 - M. Dyrda28 - K. Egberts1,30 - P. Eger16 - P. Espigat12 - L. Fallon13 - C. Farnier15 - S. Fegan10 - F. Feinstein15 - M. V. Fernandes4 - A. Fiasson11 - A. Förster1 - G. Fontaine10 - M. Füßling5 - S. Gabici13 - Y. A. Gallant15 - L. Gérard12 - D. Gerbig21 - B. Giebels10 - J. F. Glicenstein7 - B. Glück16 - P. Goret7 - D. Göring16 - D. Hampf4 - M. Hauser14 - S. Heinz16 - G. Heinzelmann4 - G. Henri17 - G. Hermann1 - J. A. Hinton33 - A. Hoffmann18 - W. Hofmann1 - P. Hofverberg1 - M. Holleran9 - S. Hoppe1 - D. Horns4 - A. Jacholkowska19 - O. C. de Jager9 - C. Jahn16 - I. Jung16 - K. Katarzynski27 - U. Katz16 - S. Kaufmann14 - M. Kerschhaggl5 - D. Khangulyan1 - B. Khélifi10 - D. Keogh8 - D. Klochkov18 - W. Kluzniak24 - T. Kneiske4 - Nu. Komin7 - K. Kosack7 - R. Kossakowski11 - G. Lamanna11 - J.-P. Lenain6 - T. Lohse5 - C.-C. Lu1 - V. Marandon12 - A. Marcowith15 - J. Masbou11 - D. Maurin19 - T. J. L. McComb8 - M. C. Medina6 - J. Méhault15 - R. Moderski24 - E. Moulin7 - M. Naumann-Godo10 - M. de Naurois19 - D. Nedbal20 - D. Nekrassov1 - N. Nguyen4 - B. Nicholas26 - J. Niemiec28 - S. J. Nolan8 - S. Ohm1 - J.-F. Olive3 - E. de Oña Wilhelmi1 - B. Opitz 4 - K. J. Orford8 - M. Ostrowski23 - M. Panter1 - M. PazArribas5 - G. Pedaletti14 - G. Pelletier17 - P.-O. Petrucci17 - S. Pita12 - G. Pühlhofer18 - M. Punch12 - A. Quirrenbach14 - B. C. Raubenheimer9 - M. Raue1,34 - S. M. Rayner8 - O. Reimer30 - M. Renaud12 - R. de los Reyes1 - F. Rieger1,34 - J. Ripken31 - L. Rob20 - S. Rosier-Lees11 - G. Rowell26 - B. Rudak24 - C. B. Rulten8 - J. Ruppel21 - F. Ryde32 - V. Sahakian2 - A. Santangelo18 - R. Schlickeiser21 - F. M. Schöck16 - A. Schönwald5 - U. Schwanke5 - S. Schwarzburg18 - S. Schwemmer14 - A. Shalchi 21 - I. Sushch5 - M. Sikora24 - J. L. Skilton25 - H. Sol6 - \L. Stawarz23 - R. Steenkamp22 - C. Stegmann16 - F. Stinzing16 - G. Superina10 - A. Szostek23,17 - P. H. Tam14 - J.-P. Tavernet19 - R. Terrier12 - O. Tibolla1 - M. Tluczykont4 - K. Valerius16 - C. van Eldik1 - G. Vasileiadis15 - C. Venter9 - L. Venter6 - J. P. Vialle11 - A. Viana7 - P. Vincent19 - M. Vivier7 - H. J. Völk1 - F. Volpe1,10 - S. Vorobiov15 - S. J. Wagner14 - M. Ward8 - A. A. Zdziarski24 - A. Zech6 - H.-S. Zechlin4

1 - Max-Planck-Institut für Kernphysik, PO Box 103980, 69029 Heidelberg, Germany
2 - Yerevan Physics Institute, 2 Alikhanian Brothers St., 375036 Yerevan, Armenia
3 - Centre d'Étude Spatiale des Rayonnements, CNRS/UPS, 9 Av. du Colonel Roche, BP 4346, 31029 Toulouse Cedex 4, France
4 - Universität Hamburg, Institut für Experimentalphysik, Luruper Chaussee 149, 22761 Hamburg, Germany
5 - Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489 Berlin, Germany
6 - LUTH, Observatoire de Paris, CNRS, Université Paris Diderot, 5 place Jules Janssen, 92190 Meudon, France
7 - CEA Saclay, DSM/IRFU, 91191 Gif-Sur-Yvette Cedex, France
8 - University of Durham, Department of Physics, South Road, Durham DH1 3LE, UK
9 - Unit for Space Physics, North-West University, Potchefstroom 2520, South Africa
10 - Laboratoire Leprince-Ringuet, École Polytechnique, CNRS/IN2P3, 91128 Palaiseau, France
11 - Laboratoire d'Annecy-le-Vieux de Physique des Particules, Université de Savoie, CNRS/IN2P3, 74941 Annecy-le-Vieux, France
12 - Astroparticule et Cosmologie (APC), CNRS, Université Paris 7 Denis Diderot, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France [*]
13 - Dublin Institute for Advanced Studies, 5 Merrion Square, Dublin 2, Ireland
14 - Landessternwarte, Universität Heidelberg, Königstuhl, 69117 Heidelberg, Germany
15 - Laboratoire de Physique Théorique et Astroparticules, Université Montpellier 2, CNRS/IN2P3, CC 70, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France
16 - Universität Erlangen-Nürnberg, Physikalisches Institut, Erwin-Rommel-Str. 1, 91058 Erlangen, Germany
17 - Laboratoire d'Astrophysique de Grenoble, INSU/CNRS, Université Joseph Fourier, BP 53, 38041 Grenoble Cedex 9, France
18 - Institut für Astronomie und Astrophysik, Universität Tübingen, Sand 1, 72076 Tübingen, Germany
19 - LPNHE, Université Pierre et Marie Curie Paris 6, Université Denis Diderot Paris 7, CNRS/IN2P3, 4 place Jussieu, 75252 Paris Cedex 5, France
20 - Charles University, Faculty of Mathematics and Physics, Institute of Particle and Nuclear Physics, V Holesovickách 2, 180 00 Prague 8, Czech Republic
21 - Institut für Theoretische Physik, Lehrstuhl IV: Weltraum und Astrophysik, Ruhr-Universität Bochum, 44780 Bochum, Germany
22 - University of Namibia, Department of Physics, Private Bag 13301, Windhoek, Namibia
23 - Obserwatorium Astronomiczne, Uniwersytet Jagiellonski, ul. Orla 171, 30-244 Kraków, Poland
24 - Nicolaus Copernicus Astronomical Center, ul. Bartycka 18, 00-716 Warsaw, Poland
25 - School of Physics & Astronomy, University of Leeds, Leeds LS2 9JT, UK
26 - School of Chemistry & Physics, University of Adelaide, Adelaide 5005, Australia
27 - Torun Centre for Astronomy, Nicolaus Copernicus University, ul. Gagarina 11, 87-100 Torun, Poland
28 - Instytut Fizyki Jadrowej PAN, ul. Radzikowskiego 152, 31-342 Kraków, Poland
29 - Astronomical Observatory, The University of Warsaw, Al. Ujazdowskie 4, 00-478 Warsaw, Poland
30 - Institut für Astro- und Teilchenphysik, Leopold-Franzens-Universität Innsbruck, 6020 Innsbruck, Austria
31 - Oskar Klein Centre, Department of Physics, Stockholm University, Albanova University Center, 10691 Stockholm, Sweden
32 - Oskar Klein Centre, Department of Physics, Royal Institute of Technology (KTH), Albanova, 10691 Stockholm, Sweden
33 - Department of Physics and Astronomy, The University of Leicester, University Road, Leicester, LE1 7RH, UK
34 - European Associated Laboratory for Gamma-Ray Astronomy, jointly supported by CNRS and MPG

Received 22 March 2010 / Accepted 14 May 2010

Abstract
Context. Observations of very high-energy ${\gamma}$-rays from blazars provide information about acceleration mechanisms occurring in their innermost regions. Studies of variability in these objects lead to a better understanding of the mechanisms in play.
Aims. To investigate the spectral and temporal variability of VHE ( ${>}100~{\rm GeV}$) ${\gamma}$-rays of the well-known high-frequency-peaked BL Lac object PKS 2155-304 with the HESS imaging atmospheric Cherenkov telescopes over a wide range of flux states.
Methods. Data collected from 2005 to 2007 were analyzed. Spectra were derived on time scales ranging from 3 years to 4 min. Light curve variability was studied through doubling timescales and structure functions and compared with red noise process simulations.
Results. The source was found to be in a low state from 2005 to 2007, except for a set of exceptional flares that occurred in July 2006. The quiescent state of the source is characterized by an associated mean flux level of $(4.32\pm0.09_{\rm stat}\pm0.86_{\rm syst})$ $\times $ $10^{-11}~{\rm cm^{-2}}~{\rm s^{-1}}$ above  $200~{\rm GeV}$, or approximately $15\%$ of the Crab Nebula, and a power-law photon index of $\Gamma=3.53\pm0.06_{\rm stat}\pm0.10_{\rm syst}$. During the flares of July 2006, doubling timescales of ${\sim}2~{\rm min}$ are found. The spectral index variation is examined over two orders of magnitude in flux, yielding different behavior at low and high fluxes, which is a new phenomenon in VHE ${\gamma}$-ray emitting blazars. The variability amplitude characterized by the fractional rms $F_{\rm var}$ is strongly energy-dependent and is ${\propto}E^{0.19\pm0.01}$. The light curve rms correlates with the flux. This is the signature of a multiplicative process that can be accounted for as a red noise with a Fourier index of ${\sim}2$.
Conclusions. This unique data set shows evidence of a low-level ${\gamma}$-ray emission state from PKS 2155-304 that possibly has a different origin than the outbursts. The discovery of the light curve lognormal behavior might be an indicator of the origin of aperiodic variability in blazars.

Key words: gamma rays: general - galaxies: active - galaxies: jets - BL Lacertae objects: individual: PKS 2155-304

1 Introduction

The BL Lacertae (BL Lac) category of active galactic nuclei (AGN) represents the vast majority of the population of energetic and extremely variable extragalactic very high-energy ${\gamma}$-ray emitters. Their luminosity varies in unpredictable, highly irregular ways, by orders of magnitude and at all wavelengths across the electromagnetic spectrum. The very high-energy (VHE, $E\geq 100~{\rm GeV}$) ${\gamma}$-ray fluxes vary often on the shortest timescales that can be seen in this type of object, with large amplitudes that can dominate the overall output. It thus indicates that the understanding of this energy domain is the most important one for understanding the underlying fundamental variability and emission mechanisms in play in high flux states.

It has been difficult, however, to ascertain whether ${\gamma}$-ray emission is present only during high flux states or also when the source is in a more stable or quiescent state but with a flux that is below the instrumental limits. The advent of the current generation of atmospheric Cherenkov telescopes with unprecedented sensitivity in the VHE regime gives new insight into these questions.

The high-frequency-peaked BL Lac object (HBL) PKS 2155-304, located at redshift z=0.117, initially discovered as a VHE ${\gamma}$-ray emitter by the Mark 6 telescope (Chadwick et al. 1999), has been detected by the first HESS telescope in 2002-2003 (Aharonian et al. 2005b). It has been frequently observed by the full array of four telescopes since 2004, either sparsely during the HESS monitoring program or intensely during dedicated campaigns, such as described in Aharonian et al. (2005c), showing mean flux levels of ${\sim}20\%$ of the Crab Nebula flux for energies above $200~{\rm GeV}$. During the summer of 2006, PKS 2155-304 exhibited unprecedented flux levels accompanied by strong variability (Aharonian et al. 2007a), making temporal and spectral variability studies possible on timescales on the order of a few minutes. The VHE ${\gamma}$-ray emission is usually thought to originate from a relativistic jet, emanating from the vicinity of a supermassive black hole (SMBH). The physical processes in play are still poorly understood, but the analysis of the ${\gamma}$-ray flux spectral and temporal characteristics is well-suited to providing deeper insight.

For this goal, the data set of HESS observations of PKS 2155-304 between 2005 and 2007 is used. After describing the observations and the analysis chain in Sect. 2, the emission from the ``quiescent'', i.e. nonflaring, state of the source is characterized in Sect. 3. Section 4 explains the spectral variability related to the source intensity. Section 5 describes the temporal variability during the highly active state of the source and its possible energy dependence. Section 6 illustrates the observed variability phenomenon by a random stationary process, characterized by a simple power density spectrum. Section 7 shows how limits on the characteristic time of the source can be derived. The multi-wavelength aspects from the high flux state will be presented in a second paper.

2 Observations and analysis

HESS is an array of four imaging atmospheric Cherenkov telescopes situated in the Khomas Highland of Namibia ( $23^\circ16\hbox{$^\prime$ }18\hbox{$^{\prime\prime}$ }$ South, $16^\circ30\hbox{$^\prime$ }00\hbox{$^{\prime\prime}$ }$ East), at an elevation of 1800 m above sea level (see Aharonian et al. 2006). PKS 2155-304 was observed by HESS each year after 2002; results of observations in 2002, 2003, and 2004 can be found in Aharonian et al. (2005b), Aharonian et al. (2005c) and Giebels et al. (2005). The data reported here were collected between 2005 and 2007. In 2005, 12.2 h of observations were taken. A similar observation time was scheduled in 2006, but following the strong flare of July 26 (Aharonian et al. 2007a), it was decided to increase this observation time significantly. Ultimately, from June to October 2006, this source was observed for 75.9 h, with a further 20.9 h in 2007.

\begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg1.eps}\vspace*{-1mm}\end{figure} Figure 1:

Zenith angle distribution for the 202 4-telescopes observation runs from 2005 to 2007. The inset shows, for each zenith angle, the energy threshold associated with the analysis presented in Sect. 2.

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Table 1:   Summary of observations for each year.

The data were recorded during runs of 28 min nominal duration, with the telescopes pointing at $0.5^\circ$ from the source position in the sky to enable a simultaneous estimate of the background. This offset was taken alternatively in both right ascension and declination (with both signs), in order to minimize systematics. Only the runs passing the HESS data-quality selection criteria were used for the analyses presented below. These criteria imply good atmospheric conditions and checks that the hardware state of the cameras is satisfactory. The number of runs thus selected is 22 for 2005, 153 for 2006, and 35 for 2007, corresponding to live-times of 9.4, 66.1, and 13.8 h respectively. During these observations, zenith angles were between 7 and 60 degrees, resulting in large variations in the instrument energy threshold ( $E_{\rm th}$, see Fig. 1) and sensitivity. This variation has been accounted for in the spectral and temporal variability studies presented below.

Table 2:   The various data sets used in the paper, referred to in the text by the labels presented in this table.

The data have been analyzed following the prescription presented in Aharonian et al. (2006), using the loose set of cuts that are well adapted for bright sources with moderately soft spectra, and the Reflected-Region method for the definition of the on-source and off-source data regions. A year-wise summary of the observations and the resulting detections are shown in Table 1. A similar summary is given in Appendix A for the 67 nights of data taken, showing that the emission of PKS 2155-304 is easily detected by HESS almost every night. For 66 nights out of 67, the significance per square root of the live-time ( $\sigma/\!\sqrt{T}$, where T is the observation live-time) is at least equal to $3.6\sigma/\!\sqrt{h}$, the only night with a lower value - MJD 53705 - corresponding to a very short exposure. In addition, for 61 nights out of 67 the source emission is high enough to enable a detection of the source with 5$\sigma$ significance in one hour or less, a level usually required in this domain to firmly claim a new source detection. In 2006 the source exhibits very strong activity (38 nights, between MJD 53916-53999) with a nightly  $\sigma/\!\sqrt{T}$ varying from 3.6 to 150, and being higher than $10\sigma/\!\sqrt{h}$ for 19 nights. The activity of the source climaxes on MJD 53944 and 53946 with statistical significances that are unprecedented at these energies, the rate of detected ${\gamma}$-rays corresponding to 2.5 and $1.3~{\rm Hz}$, with 150 and  $98\sigma/\!\sqrt{h}$ respectively.

For subsequent spectral analysis, an improved energy reconstruction method with respect to the one described in Aharonian et al. (2006) was applied. This method is based on a look-up table determined from Monte-Carlo simulations, which contains the relation between an image's amplitude and its reconstructed impact parameter as a function of the true energy, the observation zenith angle, the position of the source in the camera, the optical efficiency of the telescopes (which tend to decrease due to the aging of the optical surfaces), the number of triggered telescopes and the reconstructed altitude of the shower maximum. Thus, for a given event, the reconstructed energy is determined by requiring the minimal $\chi^2$ between the image amplitudes and those expected from the look-up table corresponding to the same observation conditions. This method yields a slightly lower energy threshold (shown in Fig. 1 as a function of zenith angle), an energy resolution that varies from 15% to 20% over all the energy range, and biases in the energy reconstruction that are smaller than 5%, even close to the threshold. The systematic uncertainty in the normalization of the HESS energy scale is estimated to be as large as 15%, corresponding for such soft spectrum source to 40% in the overall flux normalization as quoted in Aharonian et al. (2009).

All the spectra presented in this paper have been obtained using a forward-folding maximum likelihood method based on the measured energy-dependent on-source and off-source distributions. This method, fully described in Piron et al. (2001), performs a global deconvolution of the instrument functions (energy resolution, collection area) and the parametrization of the spectral shape. Two different sets of parameters, corresponding to a power law and to a power law with an exponential cut-off, are used for the spectral shape, with the following equations:

$\displaystyle %
\phi(E) = \phi_0 \left(\frac{E}{E_0}\right)^{-\Gamma}$   (1)
$\displaystyle \phi(E) = \phi_0 \exp\left({\frac{E_0}{E_{\rm cut}}}\right)\left(\frac{E}{E_0}\right)^{-\Gamma}\exp\left(-{\frac{E}{E_{\rm cut}}}\right)$   (2)

$\phi_0$represents the differential flux at E0 (chosen to be 1 TeV), $\Gamma $ is the power-law index and  $E_{\rm cut}$ the characteristic energy of the exponential cut-off. The maximum likelihood method provides the best set of parameters corresponding to the selected hypothesis, and the corresponding error matrix.

Finally, various data sets have been used for subsequent analyses. These are summarized in Table 2.

3 Characterization of the quiescent state

\begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg2.eps} \vspace*{3mm}
\end{figure} Figure 2:

Monthly averaged integral flux of PKS 2155-304 above $200~{\rm GeV}$ obtained from data set D (see Table 2). The dotted line corresponds to 15% of the Crab Nebula emission level (see Sect. 3.2).

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As can be seen in Fig. 2, with the exception of the high state of July 2006 PKS 2155-304 was in a low state during the observations from 2005 to 2007. This section explores the variability of the source during these periods of low-level activity, based on the determination of the run-wise integral fluxes for the data set  $D_{\rm QS}$, which excludes the flaring period of July 2006 and also those runs whose energy threshold is higher than $200~{\rm GeV}$ (see 3.1 for justification). As for Sects. 5 and 6, the control of systematics in such a study is particularly important, especially because of the strong variations of the energy threshold throughout the observations.

3.1 Method and systematics

The integral flux for a given period of observations is determined in a standard way. For subsequent discussion purposes, the formula applied is given here:

$\displaystyle %
\Phi = N_{\rm exp}\frac{\int_{E_{\rm min}}^{E_{\rm max}}S(E){\r...
..._{0}^{\infty}\int_{E_{\rm min}}^{E_{\rm max}} A(E)R(E,E')S(E){\rm d}E'{\rm d}E}$     (3)

where T represents the corresponding live-time, A(E) and R(E,E') are, respectively, the collection area at the true energy E and the energy resolution function between E and the measured energy E', and S(E) the shape of the differential energy spectrum as defined in Eqs. (1) and (2). Finally, $N_{\rm exp}$ is the number of measured events in the energy range [ $E_{\rm min}$$E_{\rm max}$].

\begin{figure}
\par\includegraphics[width=15.5cm,clip]{14484fg3nv.eps}
\end{figure} Figure 3:

Distributions of the logarithms of integral fluxes above $200~{\rm GeV}$ in individual runs. Left: from 2005 to 2007 except the July 2006 period (data set  $D_{\rm QS}$), fitted by a Gaussian. Right: all runs from 2005 to 2007 (data set D), where the solid line represents the result of a fit by the sum of 2 Gaussians (dashed lines). See Table 4 for details.

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In the case that S(E) is a power law, an important source of systematic error in the determination of the integral flux variation with time comes from the value chosen for the index $\Gamma $. The average 2005-2007 energy spectrum yields a very well determined power-law index[*]. However, in Sect. 4 it will be shown that this index varies depending on the flux level of the source. Moreover, in some cases the energy spectrum of the source shows some curvature in the TeV region, giving slight variations in the fitted power-law index depending on the energy range used.

For runs whose energy threshold is lower than $E_{\rm min}$, a simulation performed under the observation conditions corresponding to the data shows that an index variation of $\Delta\Gamma=0.1$ implies a flux error at the level of $\Delta\Phi\sim1\%$, this relation being quite linear up to $\Delta\Gamma\sim0.5$. However, this relation no longer holds when the energy threshold is above  $E_{\rm min}$, as the determination of $\Phi $ becomes much more dependent on the choice of $\Gamma $. For this reason, only runs whose energy threshold is lower than $E_{\rm min}$ will be kept for the subsequent light curves. The value of  $E_{\rm min}$ is chosen as $200~{\rm GeV}$, which is a compromise between a low value which maximizes the excess numbers used for the flux determinations and a high value which maximizes the number of runs whose energy threshold is lower than  $E_{\rm min}$.

3.2 Run-wise distribution of the integral flux

From 2005 to 2007, PKS 2155-304 is almost always detected when observed (except for two nights for which the exposure was very low), indicating the existence, at least during these observations, of a minimal level of activity of the source. Focussing on data set  $D_{\rm QS}$ (which excludes the July 2006 data where the source is in a high state), the distribution of the integral fluxes of the individual runs above $200~{\rm GeV}$ has been determined for the 115 runs, using a spectral index $\Gamma =3.53$ (the best value for this data set, as shown in 3.4). This distribution has an asymmetric shape, with mean value $(4.32\pm0.09_{\rm stat})$ $\times $ $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $ and root mean square (rms) $(2.48\pm0.11_{\rm stat})$ $\times $ $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $, and is very well described with a lognormal function. Such a behavior implies that the logarithm of fluxes follows a normal distribution, centered on the logarithm of $(3.75\pm0.11_{\rm stat})$ $\times $ $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $. This is shown in the left panel of Fig. 3, where the solid line represents the best fit obtained with a maximum-likelihood method, yielding results independent of the choice of the intervals in the histogram. It is interesting to note that this result can be compared to the fluxes measured by HESS from PKS 2155-304 during its construction phase, in 2002 and 2003 (see Aharonian et al. 2005b,c). As shown in Table 3, these flux levels extrapolated down to $200~{\rm GeV}$ were close to the value corresponding to the peak shown in the left panel of Fig. 3.

Table 3:   Integral fluxes and their statistical errors from 2002 and 2003 observations of PKS 2155-304 during the HESS construction phase.

The right panel of Fig. 3 shows how the flux distribution is modified when the July 2006 data are taken into account (data set D in Table 2): the histogram can be accounted for by the superposition of two Gaussian distributions (solid curve). The results, summarized in Table 4, are also independent of the choice of the intervals in the histogram. Remarkably enough, the characteristics of the first Gaussian obtained in the first step (left panel) remain quite stable in the double Gaussian fit.

Table 4:   The distribution of the flux logarithm.

This leads to two conclusions. First, the flux distribution of PKS 2155-304 is well described considering a low state and a high state, for each of which the distribution of the logarithms of the fluxes follows a Gaussian distribution. The characteristics of the lognormal flux distribution for the high state are given in Sects. 5-7. Secondly, PKS 2155-304 has a level of minimal activity that seems to be stable on a several-year time-scale. This state will henceforth be referred to as the ``quiescent state'' of the source.

3.3 Width of the run-wise flux distribution

In order to determine if the measured width of the flux distribution (left panel of Fig. 3) can be explained as statistical fluctuations from the measurement process a simulation has been carried out considering a source that emits an integral flux above $200~{\rm GeV}$ of 4.32 $\times $ $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $ with a power-law spectral index $\Gamma =3.53$ (as determined in the next section). For each run of the data set  $D_{\rm QS}$ the number  $n_{\gamma}$ expected by convolving the assumed differential energy spectrum with the instrument response corresponding to the observation conditions is determined. A random smearing around this value allows statistical fluctuations to be taken into account. The number of events in the off-source region and also the number of background events in the source region are derived from the measured values  $n_{\rm off}$ in the data set. These are also smeared in order to take into account the expected statistical fluctuations.

10 000 such flux distributions have been simulated, and for each one its mean value and rms (which will be called below RMSD) are determined. The distribution of RMSD thus obtained, shown in Fig. 4, is well described by a Gaussian centred on 0.98 $\times $ $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $ (which represents a relative flux dispersion of 23%) and with a $\sigma_{\rm RMSD}$ of 0.07 $\times $  $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $.

It should be noted that here the effect of atmospheric fluctuations in the determination of the flux is only taken into account at the level of the off-source events, as these numbers are taken from the measured data. But the effect of the corresponding level of fluctuations on the source signal is very difficult to determine. If a conservative value of 20% is considered[*] that is added in the simulations as a supplementary fluctuation factor for the number of events expected from the source, a RMSD distribution centred on 1.30 $\times $ $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $ with a  $\sigma_{\rm RMSD}$ of 0.09 $\times $ $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $ is obtained. Even in this conservative case, the measured value for the flux distribution rms ( $(2.48\pm0.11_{\rm stat})$ $\times $ $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $) is very far (more than 8 standard deviations) from the simulated value. All these elements strongly suggest the existence of an intrinsic variability associated with the quiescent state of PKS 2155-304.

\begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg4.eps}
\end{figure} Figure 4:

Distribution of RMSD obtained when the instrument response to a fixed emission ( $\Phi=4.32 \times 10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $ and $\Gamma =3.53$) is simulated 10 000 times with the same observation conditions as for the 115 runs of the left part of Fig. 3.

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3.4 Quiescent-state energy spectrum

The energy spectrum associated with the data set $D_{\rm QS}$, shown in Fig. 5, is well described by a power law with a differential flux at 1 TeV of $\phi_0=(1.81\pm0.13_{\rm stat})$ $\times $ $10^{-12}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}~{\rm TeV}^{-1}} $ and an index of $\Gamma =3.53$ $\pm $ $0.06_{\rm stat}$. The stability of these values for spectra measured separately for 2005, 2006 (excluding July), and 2007 is presented in Table 5. The corresponding average integral flux is $(4.23\pm0.09_{\rm stat})$ $\times $ $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $, which is as expected in very good agreement with the mean value of the distribution shown in the left panel of Fig. 3.

\begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg5.eps}
\end{figure} Figure 5:

Energy spectrum of the quiescent state for the period 2005-2007. The green band correponds to the 68% confidence-level provided by the maximum likelihood method. Points are derived from the residuals in each energy bin, only for illustration purposes. See Sect. 3.4 for further details.

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Table 5:   Parametrization of the differential energy spectrum of the quiescent state of PKS 2155-304, determined in the energy range 0.2-10 TeV, first for the 2005-2007 period and also separately for the 2005, 2006 (excluding July) and 2007 periods.

Bins above 2 TeV correspond to $\gamma-$ray excesses lower than 20$~\gamma$ and significances lower than $2\sigma$. Above 5 TeV excesses are even less significant ( ${\sim}1\sigma$ or less) and 99% upper-limits are used. There is no improvement of the fit when a curvature is taken into account.

  4 Spectral variability

4.1 Variation of the spectral index for the whole data set 2005-2007

The spectral state of PKS 2155-304 has been monitored since 2002. The first set of observations (Aharonian et al. 2005b), from July 2002 to September 2003, shows an average energy spectrum well described by a power law with an index of $\Gamma=3.32$ $\pm $ $0.06_{\rm stat}$, for an integral flux (extrapolated down to $200~{\rm GeV}$) of $(4.39\pm0.40_{\rm stat})$ $\times $ $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $. No clear indication of spectral variability was seen. Consecutive observations in October and November 2003 (Aharonian et al. 2005c) gave a similar value for the index, $\Gamma =3.37$ $\pm $ $0.07_{\rm stat}$, for a slightly higher flux of $(5.22\pm0.54_{\rm stat})$ $\times $ $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $. Later, during HESS observations of the first (MJD 53944, Aharonian et al. 2007a) and second (MJD 53946, Aharonian et al. 2009) exceptional flares of July 2006, the source reached much higher average fluxes, corresponding to $(1.72\pm0.05_{\rm stat})$ $\times $ $10^{-9}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $ and $(1.24\pm0.02_{\rm stat})$ $\times $ $10^{-9}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $[*] respectively. In the first case, no strong indications for spectral variability were found and the average index $\Gamma=3.19$ $\pm $ $0.02_{\rm stat}$ was close to those associated with the 2002 and 2003 observations. In the second case, clear evidence of spectral hardening with increasing flux was found.

The observations of PKS 2155-304 presented in this paper also include the subsequent flares of 2006 and the data of 2005 and 2007. Therefore, the evolution of the spectral index is studied for the first time for a flux level varying over two orders of magnitude. This spectral study has been carried out over the fixed energy range 0.2-1 TeV in order to minimize both systematic effects due to the energy threshold variation and the effect of the curvature observed at high energy in the flaring states. The maximal energy has been chosen to be at the limit where the spectral curvature seen in high flux states begins to render the power law or exponential curvature hypotheses distinguishable. As flux levels observed in July 2006 are significantly higher than in the rest of the data set (see Fig. 6), the flux-index behavior is determined separately first for the July 2006 data set itself ( $D_{\rm JULY06}$) and secondly for the 2005-2007 data excluding this data set ( $D_{\rm QS}$).

\begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg6.eps}
\end{figure} Figure 6:

Integral flux above $200~{\rm GeV}$ measured each night during late July 2006 observations. The horizontal dashed line corresponds to the quiescent state emission level defined in Sect. 3.2.

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On both data sets, the following method was applied. The integral flux was determined for each run assuming a power law shape with an index of $\Gamma =3.37$ (average spectral index for the whole data set), and runs were sorted by increasing flux. The set of ordered runs was then divided into subsets containing at least an excess of 1500 ${\gamma}$ above $200~{\rm GeV}$ and the energy spectrum of each subset was determined[*].

The left panel of Fig. 7 shows the photon index versus integral flux for data sets  $D_{\rm QS}$ (grey crosses) and  $D_{\rm JULY06}$ (black points). Corresponding numbers are summarized in Appendix B. While a clear hardening is observed for integral fluxes above a few $10^{-10}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $, a break in this behavior is observed for lower fluxes. Indeed, for the data set  $D_{\rm JULY06}$ (black points) a linear fit yields a slope $\rm d\Gamma$/ $\rm d\Phi=(3.0\pm0.3_{\rm stat})$ $\times $ $10^{8}~{\rm cm}^{2}~{\rm s}$, whereas the same fit for data set  $D_{\rm QS}$ (grey crosses) yields a slope $\rm d\Gamma$/ $\rm d\Phi=(-3.4\pm1.9_{\rm stat})$ $\times $ $10^{9}~{\rm cm}^{2}~{\rm s}$. The latter corresponds to a $\chi^2$ probability $P(\chi^2)=71\%$; a fit to a constant yields $P(\chi^2)=33\%$ but with a constant fitted index incompatible with a linear extrapolation from higher flux states at a 3$\sigma$ level. This is compatible with conclusions obtained either with an independent analysis or when these spectra are processed following a different prescription. In this prescription the runs were sorted as a function of time in contiguous subsets with similar photon statistics, rather than as a function of increasing flux.

\begin{figure}
\par\includegraphics[width=16.5cm,clip]{14484fg7.eps}
\end{figure} Figure 7:

Evolution of the photon index $\Gamma $ with increasing flux $\Phi $ in the 0.2-1 TeV energy range. The left panel shows the results for the July 2006 data (black points, data set  $D_{\rm JULY06}$) and for the 2005-2007 period excluding July 2006 (grey points, data set  $D_{\rm QS}$). The right panel shows the results for the four nights flaring period of July 2006 (black points, data set  $D_{\rm FLARES}$) and one point corresponding to the quiescent state average spectrum (grey point, again data set  $D_{\rm QS}$). See text in Sects. 4.1 ( left panel) and 4.2 ( right panel) for further details on the method.

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\begin{figure}
\par\includegraphics[width=17.5cm,clip]{14484f8.eps}
\end{figure} Figure 8:

Integrated flux versus time for PKS 2155-304 on MJD 53944-53947 for four energy bands and with a 4-min binning. From top to bottom: ${>}0.2~{\rm TeV}$, 0.2- $0.35~{\rm TeV}$, 0.35- $0.6~{\rm TeV}$ and 0.6- $5~{\rm TeV}$. These light curves are obtained using a power law spectral shape with an index of $\Gamma =3.37$, also used to derive the flux extrapolation down to  $0.2~{\rm TeV}$ when the threshold is above that energy in the top panel (grey points). Because of the high dispersion of the energy threshold of the instrument (see Sect. 2, Fig. 1), and following the prescription described in 3.1, the integral flux has been determined for a time bin only if the corresponding energy threshold is lower than  $E_{\rm min}$. The fractional rms for the light curves are respectively, 0.86 $\pm $ $0.01_{\rm stat}$, 0.79 $\pm $ $0.01_{\rm stat}$, 0.89$\pm $ $0.01_{\rm stat}$ and 1.01 $\pm $ $0.02_{\rm stat}$. The last plot shows the variation of the photon index determined in the 0.2-1 TeV range. See Sect. 4.2 and Appendix B.4 for details.

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The form of the relation between the index versus integral flux is unprecedented in the TeV regime. Prior to the results presented here, spectral variability has been detected only in two other blazars, Mrk 421 and Mrk 501. For Mrk 421, a clear hardening with increasing flux appeared during the 1999/2000 and 2000/2001 observations performed with HEGRA (Aharonian et al. 2002) and also during the 2004 observations performed with HESS (Aharonian et al. 2005a). In addition, the Mrk 501 observations carried out with CAT during the strong flares of 1997 (Djannati-Ataï et al. 1999) and also the recent observation performed by MAGIC in 2005 (Albert et al. 2007) have shown a similar hardening. In both studies, the VHE peak has been observed in the $\nu F_\nu$ distributions of the flaring states of Mrk 501.

4.2 Variation of the spectral index for the four flaring nights of July 2006

In this section, the spectral variability during the flares of July 2006 is described in more detail. A zoom on the variation of the integral flux (4-min binning) for the four nights containing the flares (nights MJD 53944, 53945, 53946, and 53947, called the ``flaring period'') is presented in the top panel of Fig. 8. This figure shows two exceptional peaks on MJD 53944 and MJD 53946 that climax respectively at fluxes higher than 2.5 $\times $ $10^{-9}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $ and 3.5 $\times $ $10^{-9}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $ (${\sim}9$ and ${\sim}12$ times the Crab Nebula level above the same energy), both about two orders of magnitude above the quiescent state level.

The variation with time of the photon index is shown in the bottom panel of Fig. 8. To obtain these values, the ${\gamma}$ excess above $200~{\rm GeV}$ has been determined for each 4-min bin. Then, successive bins have been grouped in order to reach a global excess higher than 600 ${\gamma}$. Finally, the energy spectrum of each data set has been determined in the 0.2-1 TeV energy range, as before (corresponding numbers are summarized in Appendix Table B.4). There is no clear indication of spectral variability within each night, except for MJD 53946 as shown in Aharonian et al. (2009). However, a variability can be seen from night to night, and the spectral hardening with increasing flux level already shown in Fig. 7 is also seen very clearly in this manner.

It is certainly interesting to directly compare the spectral behavior seen during the flaring period with the hardness of the energy spectrum associated with the quiescent state. This is shown in the right panel of Fig. 7, where black points correspond to the four flaring nights; these were determined in the same manner as for the left panel (see 4.1 for details). A linear fit here yields a slope $\rm d\Gamma$/ $\rm d\Phi=(2.8\pm0.3_{\rm stat})$ $\times $ $10^{8}~{\rm cm}^{2}~{\rm s}$. The grey cross corresponds to the integral flux and the photon index associated with the quiescent state (derived in a consistent way in the energy range from 0.2-1 TeV), showing a clear rupture with the tendancy at higher fluxes (typically above  $10^{-10}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $).

These four nights were further examined to search for differences in the spectral behavior between periods in which the source flux was clearly increasing and periods in which it was decreasing. For this, the first 16 min of the first flare (MJD 53944) are of special interest because they present a very symmetric situation: the flux increases during the first half, and then decreases to its initial level. The averaged fluxes are similar in both parts (${\sim}1.8$ $\times $ $10^{-9}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $), and the observation conditions (and thus the instrument response) are almost constant - the mean zenith angle of each part being respectively 7.2 and 7.8 degrees. Again, the spectra have been determined in the 0.2-1 TeV energy range, giving indices of $\Gamma=3.27$ $\pm $ $0.12_{\rm stat}$ and $\Gamma=3.27$ $\pm $ $0.09_{\rm stat}$ respectively. To further investigate this question and avoid potential systematic errors from the spectral method determination, the hardness ratios were derived (defined as the ratio of the excesses in different energy bands), using for this the energy (TeV) bands [0.2-0.35], [0.35-0.6] and [0.6-5.0]. For any combination, no differences were found beyond the $1\sigma$ level between the increasing and decreasing parts. A similar approach has been applied - when possible - for the rest of the flaring period. No clear dependence has been found within the statistical error limit of the determined indices, which is distributed between 0.09 and 0.20.

Finally, the persistence of the energy cut-off in the differential energy spectrum along the flaring period has been examined. For this purpose, runs were sorted again by increasing flux and grouped into subsets containing at least an excess of 3000 ${\gamma}$ above $200~{\rm GeV}$[*]. For the seven subsets found, the energy spectrum has been determined in the 0.2-10 TeV energy range both for a simple power law and a power law with an exponential cut-off. This last hypothesis was found to be favoured systematically at a level varying from 1.8 to 4.6$\sigma$ compared to the simple power law and is always compatible with a cut-off in the 1-2 TeV range.

5 Light curve variability and correlation studies

This section is devoted to the characterization of the temporal variability of PKS 2155-304, focusing on the flaring period observations. The high number of ${\gamma}$-rays available not only enabled minute-level time scale studies, such as those presented for MJD 53944 in Aharonian et al. (2007a), but also to derive detailed light curves for three energy bands (Fig. 8): 0.2-0.35 TeV, 0.35-0.6 TeV and 0.6-5 TeV.

The variability of the energy-dependent light curves of PKS 2155-304 is in the following quantified through their fractional rms  $F_{\rm var}$ defined in Eq. (4) (Nandra et al. 1997; Edelson et al. 2002). In addition, possible time lags between light curves in two energy bands are investigated.

5.1 Fractional rms Fvar

All fluxes in the energy bands of Fig. 8 show a strong variability that is quantified through their fractional rms  $F_{\rm var}$ (which depends on observation durations and their sampling). Measurement errors  $\sigma_{i,{\rm err}}$ on each of the N fluxes $\phi_i$ of the light curve are taken into account in the definition of  $F_{\rm var}$:

\begin{displaymath}%
F_{\rm var}=\frac {\sqrt {S^2 - \sigma^{2}_{\rm err} }}{\overline{\phi}}
\end{displaymath} (4)

where S2 is the variance

\begin{displaymath}%
S^2=\frac{1}{N-1}\sum_{i=1}^{N}(\phi_i-\overline{\phi})^2,
\end{displaymath} (5)

and where $\sigma^2_{\rm err}$ is the mean square error and  $\overline{\phi}$ is the mean flux.

The energy-dependent variability $F_{\rm var}(E)$ has been calculated for the flaring period according to Eq. (4) in all three energy bands. The uncertainties on  $F_{\rm var}(E)$ have been estimated according to the parametrization derived by Vaughan et al. (2003b), using a Monte Carlo approach which accounts for the measurement errors on the simulated light curves.

Figure 9 shows the energy dependence of $F_{\rm var}$ over the four nights for a sampling of 4 min where only fluxes with a significance of at least 2 standard deviations were considered. There is a clear energy-dependence of the variability (a null probability of  ${\sim}10^{-16}$). The points in Fig. 9 are fitted according to a power law showing that the variability follows $F_{\rm var}(E)\propto E^{0.19\pm 0.01}$.

\begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg9nv.eps}
\end{figure} Figure 9:

Fractional rms $F_{\rm var}$ versus energy for the observation period MJD 53944-53947. The points are the mean value of the energy in the range represented by the horizontal bars. The line is the result of a power law fit where the errors on $F_{\rm var}$ and on the mean energy are taken into account, yielding $F_{\rm var}(E)\propto E^{0.19\pm 0.01}$.

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This energy dependence of $F_{\rm var}$ is also perceptible within each individual night. In Table 6 the values of  $F_{\rm var}$, the relative mean flux and the observation duration, are reported night by night for the flaring period. Because of the steeply falling spectra, the low-energy events dominate in the light curves. This lack of statistics for high energy prevents to have a high fraction of points with a significance more than 2 standard deviation in light curves night by night for the three energy bands previously considered. On the other hand, the error contribution dominates, preventing the estimation of the  $F_{\rm var}$ in all these three energy bands. Therefore, only two energy bands were considered: low (0.2-0.5 TeV) and high (0.5-5 TeV). As can be seen in Table 6 also night by night the high-energy fluxes seem to be more variable than those at lower energies.

Table 6:   Mean Flux and the fractional rms $F_{\rm var}$ night by night for MJD 53944-53947.

5.2 Doubling/halving timescale

While $F_{\rm var}$ characterizes the mean variability of a source, the shortest doubling/halving time (Zhang et al. 1999) is an important parameter in view of finding an upper limit on a possible physical shortest time scale of the blazar.

If $\Phi_i$ represents the light curve flux at a time Ti, for each pair of $\Phi_i$ one may calculate $T_2^{i,j}=\vert\Phi \Delta T/ \Delta \Phi\vert$, where $\Delta T$ = Tj-Ti, $\Delta \Phi$ = $\Phi_j-\Phi_i$ and $\Phi=(\Phi_j+\Phi_i)/2$. Two possible definitions of the doubling/halving are proposed by Zhang et al. (1999): the smallest doubling time of all data pairs in a light curve (T2), or the mean of the 5 smallest T2i,j (in the following indicated as  $\tilde{T_2}$). One should keep in mind that, according to Zhang et al. (1999), these quantities are ill defined and strongly depend on the length of the sampling intervals and on the signal-to-noise ratio in the observation.

This quantity was calculated for the two nights with the highest fluxes, MJD 53944 and MJD 53946, considering light curves with two different binnings (1 and 2 min). Bins with flux significances more than $2\sigma$ and flux ratios with an uncertainty smaller than 30% were required to estimate the doubling time scale. The uncertainty on T2 was estimated by propagating the errors on the $\Phi_i$, and a dispersion of the 5 smallest values was included in the error for  $\tilde{T_2}$.

In Table 7, the values of T2 and $\tilde{T_2}$ for the two nights are shown. The dependence with respect to the binning is clearly visible for both observables. In this table, the last column shows that the fraction of pairs in the light curves that are kept in order to estimate the doubling times is on average $\sim$45%. Moreover, doubling times T2 and  $\tilde{T_2}$ have been estimated for two sets of pairs in the light curves where $\Delta \Phi$ = $\Phi_j-\Phi_i$ is increasing or decreasing respectively. The values of the doubling time for the two cases are compatible within $1\sigma$, therefore no significant asymmetry has been found.

Table 7:   Doubling/Halving times for the high intensity nights MJD 53944 and MJD 53946 estimated with two different samplings, using the two definitions explained in the text.

It should be noted that these values are strongly dependent on the time binning and on the experiment's sensitivity, so that the typical fastest doubling timescale should be conservatively estimated as being less than ${\sim}2~{\rm min}$, which is compatible with the values reported in Aharonian et al. (2007a) and in Albert et al. (2007), the latter concerning the blazar Mrk 501.

5.3 Cross-correlation analysis as a function of energy

Time lags between light curves at different energies can provide insight into acceleration, cooling and propagation effects of the radiative particles.

The discrete correlation function (DCF) as a function of the delay (White & Peterson 1984; Edelson & Krolik 1988) is used here to search for possible time lags between the energy-resolved light curves. The uncertainty on the DCF has been estimated using simulations. For each delay, 105 light curves (in both energy bands) have been generated within their errors, assuming a Gaussian probability distribution. A probability distribution function (PDF) of the correlation coefficients between the two energy bands has been estimated for each set of simulated light curves. The rms of these PDF are the errors related to the DCF at each delay. Figure 10 shows the DCF between the high and low-energy bands for the four-night flaring period (with 4 min bins) and for the second flaring night (with 2 min bins). The gaps between each 28 min run have been taken into account in the DCF estimation.

The position of the maximum of the DCF has been estimated by a Gaussian fit that shows no time lag between low and high energies for either the 4 or 2 min binned light curves. This sets a limit of 14 $\pm $ $41~{\rm s}$ from the observation of MJD 53946. A detailed study on the limit on the energy scale on which quantum gravity effects could become important, using the same data set, are reported in Aharonian et al. (2008a).

\begin{figure}
\par\includegraphics[width=8cm,clip]{14484fg10.eps}
\end{figure} Figure 10:

DCF between the light curves in the energy ranges 0.2-0.5 TeV and 0.5-5 TeV and Gaussian fits around the peak. Full circles represent the DCF for MJD 53944-53947 4-min light curve and the solid line is the Gaussian fit around the peak with mean value of 43 $\pm $ $51~{\rm s}$. Crosses represent the DCF for MJD 53946 with a 2-min light curve binning, and the dashed line in the Gaussian fit with a peak centred at 14 $\pm $  $41~{\rm s}$.

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5.4 Excess rms-flux correlation

Having defined the shortest variability time scales, the nature of the process that generates the fluctuations is investigated, using another estimator: the excess rms. It is defined as the variance of a light curve (Eq. (5)) after subtracting the measurement error:

\begin{displaymath}%
\sigma_{\rm xs}= {\sqrt{(S^2-\sigma^2_{\rm err})}}.
\end{displaymath} (6)

Figure 11 shows the correlation between the excess rms of the light curve and the flux, where the flux here considered are selected with an energy threshold of  $200~{\rm GeV}$. The excess variance is estimated for 1- and 4-min binned light curves, using 20 consecutive flux points $\Phi_i$ that are at least at the $2\sigma$ significance level (81% of the 1 min binned sample). The correlation factors are r1=0.60+0.21-0.25 and r4=0.87+0.10-0.24for the 1 and 4 min binning, excluding an absence of correlation at the $2\sigma$ and $4\sigma$ levels respectively, implying that fluctuations in the flux are probably proportional to the flux itself, which is a characteristic of lognormal distributions (Aitchinson & Brown 1963). This correlation has also been investigated extending the analysis to a statistically more significant data set including observations with a higher energy threshold in which the determination of the flux above $200~{\rm GeV}$ requires an extrapolation (grey points in the top panel in Fig. 8). In this case the correlations found are compatible ( $r_1=0.78^{\rm +0.12}_{-0.14}$ and $r_4=0.93^{\rm +0.05}_{-0.15}$ for the 1 and 4 min binning, respectively) and also exclude an absence of correlation with a higher significance ($4\sigma$ and $7\sigma$, respectively).

Such a correlation has already been observed for X-rays in the Seyfert class AGN (Edelson et al. 2002; Vaughan et al. 2003a,b; M$^{\rm c}$Hardy et al. 2004) and in X-ray binaries (Uttley & M$^{\rm c}$Hardy 2001; Uttley 2004; Gleissner et al. 2004), where it is considered as evidence for an underlying stochastic multiplicative process (Uttley et al. 2005), as opposed to an additive process. In additive processes, light curves are considered as the sum of individual flares ``shots'' contributing from several zones (multi-zone models) and the relevant variable that has a Gaussian distribution (namely Gaussian variable) is the flux. For multiplicative (or cascade) models the Gaussian variable is the logarithm of the flux. Therefore, this first observation of a strong rms-flux correlation in the VHE domain fully confirms the log-normality of the flux distribution presented in Sect. 3.2.

\begin{figure}
\par\includegraphics[width=8cm,clip]{14484fg11_nv.eps}
\end{figure} Figure 11:

The excess rms $\sigma _{\rm xs}$ vs. mean flux  $\bar{\Phi}$ for the observation in MJD 53944-53947 (Full circles). The open circles are the additional points obtained when also including the extrapolated flux points - see text). Top: $\sigma _{\rm xs}$ estimated with 20 min time intervals and a 1 min binned light curve. Bottom: $\sigma _{\rm xs}$estimated with 80 min time intervals and a 4 min binned light curve. The dotted lines are a linear fit to the points, where $\sigma _{\rm xs}\propto 0.13$ $\times $ $\bar{\Phi}$ for the 1 min binning and $\sigma _{\rm xs}\propto 0.3$ $\times $ $\bar{\Phi}$ for the 4 min binning. Fits to the open circles yield similar results.

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6 Characterization of the lognormal process during the flaring period

This section investigates whether the variability of PKS 2155-304 in the flaring period can be described by a random stationary process, where, as shown in Sect. 5.4, the Gaussian variable is the logarithm of the flux. In this case the variability can be characterized through its Power Spectral Density (PSD) (van der Klis 1997), which indicates the density of variance as function of the frequency $\nu$. The PSD is an intrinsic indicator of the variability, usually represented in large frequency intervals by power laws ( ${\propto}\nu^{-\alpha}$) and is often used to define different ``states'' of variable objects (see e.g., Paltani et al. 1997; and Zhang et al. 1999, for the PSD of PKS 2155-304 in the optical and X-rays). The PSD of the light curve of one single night (MJD 53944) was given in Aharonian et al. (2007a) between 10-4 and $10^{-2}~{\rm Hz}$, and was found to be compatible with a red noise process ( $\alpha\geq2$) with ${\sim}10$ times more power as in archival X-ray data (Zhang et al. 1999), but with a similar index. This study implicitely assumed the ${\gamma}$-ray flux to be the Gaussian variable. In the present paper, the PSD is determined using data from 4 consecutive nights (MJD 53944-53947) and assuming a lognormal process. Since direct Fourier analysis is not well adapted to light curves extending over multiple days and affected by uneven sampling and uneven flux errors, the PSD will be further determined on the basis of parametric estimation and simulations.

In the hypothesis where the process is stationary, i.e., the PSD is time-independent, a power-law shape of the PSD was assumed, as for X-ray emitting blazars. The PSD was defined as depending on two parameters and as follows: $P(\nu) = K(\nu_{\rm ref}/\nu)^{\alpha}$, where $\alpha $ is the variability spectral index and K denotes the ``power'' (i.e., the variance density) at a reference frequency  $\nu_{\rm ref}$. This latter was conventionally chosen to be $10^{-4}~{\rm Hz}$, where the two parameters $\alpha $ and K are found to be decorrelated. Since a lognormal process is considered, $P(\nu)$ is the density of variance of the Gaussian variable $\ln~\Phi$. The natural logarithm of the flux is conveniently used here, since its variance over a given frequency interval[*] is close to the corresponding value of  $F_{\rm var}^2$, at least for small fluctuations. For a given set of $\alpha $ and K, it is possible to simulate many long time series and to modify them according to experimental effects, namely those of background events and of flux measurement errors. Light curve segments are further extracted from this simulation, with exactly the same time structure (observation and non-observation intervals) and the same sampling rates as those of real data. The distributions of several observables obtained from simulations for different values of $\alpha $ and K will be compared to those determined from observations, thus allowing these parameters to be determined from a maximum-likelihood fit.

The simulation characteristics will be described in Sect. 6.1. Sections 6.2-6.4 will be devoted to the determination of $\alpha $ and K by three methods, each of them based on an experimental result: the excess rms-flux correlation, the Kolmogorov first-order structure function (Rutman 1978; Simonetti et al. 1985) and doubling-time measurements.

6.1 Simulation of realistic time-series

For practical reasons, simulated values of $\ln\Phi$ were calculated from Fourier series, thus with a discrete PSD. The fundamental frequency $\nu_{0}=1/T_{0}$ that corresponds to an elementary bin $\delta \nu \equiv \nu_0$ in frequency, must be much lower than 1/T if T is the duration of the observation. The ratio T0/T was chosen to be of the order of 100, in such a way that the influence of a finite value of T0 on the average variance of a light curve of duration T would be less than about 2%. Taking T0 = 9 $\times $ $10^5~{\rm s}$, this condition is fulfilled for the following studies. With this approximation, the simulated flux logarithms are given by:

\begin{displaymath}%
\ln~\Phi(t)=a_{0}+\sum_{n=1}^{N_{\rm max}}a_{n}~\cos(2n\pi\nu_{0}t+\varphi_{n})
\end{displaymath} (7)

where $N_{\rm max}$ is chosen in such a way that $T_{0}/N_{\rm max}$ is less than the time interval between consecutive measurements (i.e., the sampling interval). According to the definition of a Gaussian random process, the phases  $\varphi_{n}$ are uniformly distributed between 0 and 2$\pi$ and the Fourier coefficients an are normally distributed with mean 0 and variances given by $P(\nu) ~ \delta \nu /2$ with $\nu= n ~ \delta \nu = n ~ \nu_0$.

From the long simulated time-series, light curve segments were extracted with the same durations as the periods of continuous data taking and with the same gaps between them. The simulated fluxes were further smeared according to measurement errors, according to the observing conditions (essentially zenith angle and background rate effects) in the corresponding data set.

6.2 Characterization of the lognormal process by the excess rms-flux relation

For a fixed PSD, characterized by a set of parameters $ \left\{\alpha, K \right\}$, 500 light curves were simulated, reproducing the observing conditions of the flaring period (MJD 53944-53947), according to the procedure explained in Sect. 6.1.

For each set of simulated light curves, segments of 20 min duration sampled every minute (and alternatively segments of 80 min duration sampled every 4 min) were extracted and, for each of them, the excess rms  $\sigma _{\rm xs}$ and the mean flux  $\bar{\Phi}$ were calculated as explained in Sect. 5.4. For a wide range of values of $\alpha $ and K, simulated light curves reproduce well the high level of correlation found in the measured light curves. On the other hand, the fractional variability  $F_{\rm var}$ and  $\bar{\Phi}$ are essentially uncorrelated and will be used in the following. A likelihood function of $\alpha $ and K was obtained by comparing the simulated distributions of  $F_{\rm var}$ and  $\bar{\Phi}$ to the experimental ones. An additional observable that is sensitive to $\alpha $ and K is the fraction of those light curve segments for which $F_{\rm var}$ cannot be calculated because large measurement errors lead to a negative value for the excess variance. The comparison between the measured value of this fraction and those obtained from simulations is also taken into account in the likelihood function. The $95\%$ confidence contours for the two parameters $\alpha $ and K obtained from the maximum likelihood method are shown in Fig. 12 for both kinds of light curve segments. The two selected domains in the $ \left\{\alpha, K \right\}$ plane have a large overlap which restricts the values of $\alpha $ to the interval (1.9, 2.4).

\begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg12.eps}
\end{figure} Figure 12:

95% confidence domains for $\alpha $ and K at $\nu_{\rm ref}=10^{-4}~{\rm Hz}$ obtained by a maximum-likelihood method based on the $\sigma _{\rm xs}$-flux correlation from 500 simulated light curves. The dashed contour refers to light curve segments of 20 min duration, sampled every minute. The solid contour refers to light curve segments of 80 min duration, sampled every 4 min. The dotted contour refers to the method based on the structure function, as explained in Sect. 6.3.

Open with DEXTER

6.3 Characterization of the lognormal process by the structure function analysis

Another method for determining $\alpha $ and K is based on Kolmogorov structure functions (SF). For a signal X(t), measured at N pairs of times separated by a delay $\tau$, $\left\{ t_i, t_i
+ \tau \right\}$ (i=1,...,N), the first-order structure function is defined as (Simonetti et al. 1985):

                       $\displaystyle %
{\rm SF}(\tau)$ = $\displaystyle \frac{1}{N}\sum_{i=1}^{N}[X(t_i)-X(t_i+\tau)]^2$  
  = $\displaystyle \frac{1}{N}\sum_{i=1}^{N}[\ln \Phi(t_i)-\ln \Phi(t_i+\tau)]^2.$ (8)

In the present analysis, X(t) represents the variable whose PSD is being estimated, namely $\ln\Phi$. The structure function is a powerful tool for studying aperiodic signals (Rutman 1978; Simonetti et al. 1985), such as the luminosity of blazars at various wavelengths. Compared to the direct Fourier analysis, the SF has the advantage of being less affected by ``windowing effects'' caused by large gaps between short observation periods in VHE observations. The first-order structure function is adapted to those PSDs whose variability spectral index is less than 3 (Rutman 1978), which is the case here, according to the results of the preceding section.

Figure 13 shows the first-order SF estimated for the flaring period (circles) for $\tau < 60$ h. At fixed $\tau$, the distribution of ${\rm SF}(\tau)$ expected for a given set of parameters $ \left\{\alpha, K \right\}$ is obtained from 500 simulated light curves. As an example, taking $\alpha =2$ and $\log_{10} (K/{\rm Hz}^{-1})=2.8$, values of ${\rm SF}(\tau)$ are found to lie at 68% confidence level within the shaded region in Fig. 13.

\begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg13_v2_nv.eps}
\end{figure} Figure 13:

First order structure function SF for the observations carried out in the period MJD 53944-53947 (circles). The shaded area corresponds to the 68% confidence limits obtained from simulations for the lognormal stationary process characterized by $\alpha =2$ and $\log_{10} (K/{\rm Hz}^{-1})=2.8$. The superimposed horizontal band represents the allowed range for the asymptotic value of the SF as obtained in Sect. 7.

Open with DEXTER

In the case of a power law PSD with index $\alpha $, the SF averaged over an ensemble of light curves is expected show a variation as  $\tau^{\alpha-1}$ (Kataoka et al. 2001). However, this property does not take into account the effect of measurement errors, nor of the limited sensitivity of Cherenkov telescopes at lower fluxes. For the present study, it was preferable to use the distributions of  ${\rm SF}(\tau)$ obtained from realistic simulations including all experimental effects. Using such distributions expected for a given set of parameters  $ \left\{\alpha, K \right\}$, a likelihood function can be obtained from the experimental SF and further maximized with respect to these two parameters. Furthermore, the likelihood fit was restricted to values of $\tau$ lower than $10^4~{\rm s}$, for which the expected fluctuations are not too large and are well-controlled. The 95% confidence region in the $ \left\{\alpha, K \right\}$ plane thus obtained is indicated by the dotted line in Fig. 12. It is in very good agreement with those based on the excess rms-flux correlation and give the best values for $\alpha $ and K:

\begin{displaymath}%
\alpha = 2.06 \pm 0.21 \: \: \: \mbox{and} \: \: \:
\log_{10}(K/{\rm Hz}^{-1}) = 2.82 \pm 0.08.
\end{displaymath} (9)

The variability index $\alpha $ at VHE energies is found to be remarkably close to those measured in the X-ray domain on PKS 2155-304, Mrk 421, and Mrk 501 (Kataoka et al. 2001).

6.4 Characterization of the lognormal process by doubling times

Simulations were also used to investigate if the estimator T2 can be used to constrain the values of $\alpha $ and K. However, for $\alpha $ less than 2, no significant constraints on those parameters are obtained from the values of T2. For higher values of $\alpha $, doubling times only provide loose confidence intervals on K that are compatible with the values reported above. This can be seen in Table 8, showing the 68% confidence intervals predicted for T2 and  $\tilde{T_2}$ for a lognormal process with $\alpha =2$ and $\log_{10}(K/(\rm Hz^{-1}))=2.8$, as obtained from simulation. Therefore, the variability of PKS 2155-304 during the flaring period can be consistently described by the lognormal random process whose PSD is characterized by the parameters given by Eq. (9).

Table 8:   Confidence interval at 68% c.l. for T2 and $\tilde{T_2}$ predicted by simulations for $\alpha $ = 2 and $\log_{10}(K/(\rm Hz^{-1}))=2.8$ for the two high-intensity nights MJD 53944 and MJD 53946, with two different sampling intervals (1 and 2 min).

7 Limits on characteristic time of PKS 2155-304

In Sect. 5.2 the shortest variability time scale of PKS 2155-304 using estimators like doubling times have been estimated. This corresponds to exploring the high-frequency behavior of the PSD. In this section the lower ( ${<}10^{-4}~{\rm Hz}$) frequency part of the PSD will be considered, aiming to set a limit on the timescale above which the PSD, characterized in Sect. 6, starts to steepen to $\alpha $ > 2. A break in the PSD is expected to avoid infrared divergences and the time at which this break occurs can be considered as a characteristic time, from which physical mechanisms occurring in AGN could be inferred.

Clearly the description of the source variability during the flaring period by a stationary lognormal random process is in good agreement with the flux distributions shown in Fig. 3. Considering the second Gaussian fit in the right panel of Fig. 3, the excess variance in the flaring regime reported in Table 9, although affected by a large error, is an estimator of the intrinsic variance of the stationary process. It has been demonstrated that  $2\sigma^2_{\rm xs}$ represents the asymptotic value of the first-order structure function for large values of the delay $\tau$ (Simonetti et al. 1985). On the other hand, as already mentioned, a PSD proportional to  $\nu^{-\alpha}$ with $\alpha \approx 2$ cannot be extrapolated to arbitrary low frequencies; equivalently, the average structure function cannot rise as  $\tau^{\alpha-1}$ for arbitrarily long times. Therefore, by setting a 95% confidence interval on $\log_{10} {\rm SF}_{\rm asympt} =
\log_{10}(2\sigma^2_{\rm xs})$ of  $\left[0.38, 0.66 \right]$ from Table 9, it is possible to evaluate a confidence interval on a timescale above which the average value of the structure function cannot be described by a power law. Taking account of the uncertainties on $\alpha $ and K given by Eq. (9), leads to the 95% confidence interval for this characteristic time  $T_{\rm char}$ of the blazar in the flaring regime:

\begin{eqnarray*}3 \: \mbox{h} < T_{\rm char} < 20 \: \mbox{h}.
\end{eqnarray*}


This is compatible with the behavior of the experimental structure function at times $\tau > 10^4$ s (Fig. 13), although the large fluctuations expected in this region do not allow a more accurate estimation. In the X-ray domain, characteristic times of the order of one day or less have been found for several blazars including PKS 2155-304 (Kataoka et al. 2001). The results presented here suggest a strong similarity between the PSDs for X-rays and VHE ${\gamma}$-rays during flaring periods.

8 Discussion and conclusions

This data set, which exhibits unique features and results, is the outcome of a long-term monitoring program and dedicated, dense, observations. One of the main results here is the evidence for a VHE ${\gamma}$-ray quiescent-state emission, where the variations in the flux are found to have a lognormal distribution. The existence of such a state was postulated by Stecker & Salamon (1996) in order to explain the extragalactic ${\gamma}$-ray background at 0.03- $100~{\rm GeV}$ detected by EGRET (Fichtel 1996; Sreekumar et al. 1998) as coming from quiescent-state unresolved blazars. Such a background has not yet been detected in the VHE range, as it is technically difficult with the atmospheric Cherenkov technique to find an isotropic extragalactic emission and even more to distinguish it from the cosmic-ray electron flux (Aharonian et al. 2008b). In addition, the EBL attenuation limits the distance from which $\sim$TeV ${\gamma}$-rays can propagate to ${\sim}1~{\rm Gpc}$ (Aharonian et al. 2007b). As pointed out by Cheng et al. (2000), emission mechanisms might be simpler to understand during quiescent states in blazars, and they are also the most likely state to be found observationally. In the X-ray band, the existence of a steady underlying emission has also been invoked for two other VHE emitting blazars (Mrk 421, Fossati et al. 2000; and 1ES 1959+650, Giebels et al. 2002). Being able to separate, and detect, flaring and nonflaring states in VHE ${\gamma}$-rays is thus important for such studies.

Table 9:   Variability estimators (definitions in Sect. 5.1) relative to $\ln\Phi$ both for the ``quiescent'' and flaring regime, as defined in Sect. 3.2.

The observation of the spectacular outbursts of PKS 2155-304 in July 2006 represents one of the most extreme examples of AGN variability in the TeV domain, and allows spectral and timing properties to be probed over two orders of magnitude in flux. For the flaring states a clear hardening of the spectrum with increasing flux above a few $10^{-10}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $ is observed, as was seen for the blazars Mrk 421 and Mrk 501. In contrast, for the quiescent state of PKS 2155-304 an indication of a softening is noted. If confirmed, this is a new and intruiging observation in the VHE regime of blazars. The blazar PKS 0208-512 (of the FSRQ class) also shows such initial softening and subsequent hardening with flux in the MeV range, but no general trend could be found for ${\gamma}$-ray blazars (Nandikotkur et al. 2007). In the framework of synchrotron self-Compton scenarios, VHE spectral softening with increasing flux can be associated with, for example, an increase in magnetic field intensity, emission region size, or the power law index of the underlying electron distribution, keeping all other parameters constant. A spectral hardening can equally be obtained by increasing the maximal Lorentz factor of the electron distribution or the Doppler factor (see e.g. Fig. 11.7 in Kataoka 1999). A better understanding of the mechanisms in play would require multi-wavelength observations to be taken over a similar time span and with similar sampling density as the data set presented here.

It is shown that the variability time scale $t_{\rm var}$ of a few minutes are only upper limits for the intrinsic lowest characteristic time scale. Doppler factors of $\delta \geq 100$ of the emission region are derived by Aharonian et al. (2007a) using the $\sim$ $10^9~M_\odot$ black hole (BH) Schwarzschild radius light crossing time as a limit, while Begelman et al. (2008) argue that such fast time scales cannot be linked to the size of the BH and must occur in regions of smaller scales, such as ``needles'' of matter moving faster than average within a larger jet (Ghisellini & Tavecchio 2008), small components in the jet dominating at TeV energies (Katarzynski et al. 2008), or jet ``stratification'' (Boutelier et al. 2008). Levinson (2007) attributes the variability to dissipation in the jet coming from radiative deceleration of shells with high Lorentz factors.

The flaring period allowed the study of light curves in separated energy bands and the derivation of a power law dependence of  $F_{\rm var}$ with the energy ( $F_{\rm var}\propto E^{\sim 0.2}$). This dependence is comparable to that reported in Giebels et al. (2007), Lichti et al. (2008), Maraschi et al. (2002), where $F_{\rm var}(E)\propto E^{\sim0.2}$ between the optical and X-ray energy bands was found for Mrk 421 and PKS 2155-304, respectively. An increase with the energy of the flux variability has been found for Mrk 501 (Albert et al. 2007) in VHE ${\gamma}$-rays on timescales comparable to those observed here.

The flaring period showed for the first time that the intrinsic variability of PKS 2155-304 increases with the flux, which can itself be described by a lognormal process, indicating that the aperiodic variability of PKS 2155-304 could be produced by a multiplicative process. The flux in the ``quiescent regime'', which is on average 50 times lower than in the flaring period and has a 3 times lower  $F_{\rm var}$, also follows a lognormal distribution, suggesting similarities between these two regimes.

It has been possible to characterize a power spectral density of the flaring period in the frequency range 10-4- $10^{-2}~{\rm Hz}$, resulting in a power law of index $\alpha=2.06$ $\pm $ 0.21 valid for frequencies down to ${\sim}1/{\rm day}$. The description of the rapid variability of a TeV blazar as a random stationary process must be taken into account by time-dependent blazar models. For PKS 2155-304 the evidence of this log-normality has been found very recently in X-rays (Giebels & Degrange 2009) and as previously mentioned, X-ray binaries and Seyfert galaxies also show lognormal variability, which is thought to originate from the accretion disk (M$^{\rm c}$Hardy et al. 2004; Lyubarskii et al. 1997; Arévalo & Uttley 2006), suggesting a connection between the disk and the jet. This variability behavior should therefore be searched for in existing blazar light curves, independently of the observed wavelength.

Acknowledgements
The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of HESS is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the UK Science and Technology Facilities Council (STFC), the IPNP of the Charles University, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment.

Appendix A: Observations summary

The journal of observations for the 2005-2007 is presented in Table A.1.

Table A.1:   Summary of the 2005 to 2007 observations.

Appendix B: Spectral variability

The numerical information associated with Fig. 7 is given in Tables B.1 (left panel, grey points), B.2 (left panel, black points) and B.3 (right panel). In addition, numerical information associated with Fig. 8 is given in Table B.4.

Table B.1:   Integral flux ( $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $) in the 0.2-1 TeV energy range versus photon index corresponding to grey points in the left panel of Fig. 7.

Table B.2:   Integral flux ( $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $) in the 0.2-1 TeV energy range versus photon index corresponding to black points in the left panel of Fig. 7.

Table B.3:   Integral flux ( $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $) in the 0.2-1 TeV energy range versus photon index corresponding to the right panel of Fig. 7.

Table B.4:   MJD, integral flux ( $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $) in the 0.2-1 TeV energy range, and photon index corresponding to the entries of Fig. 8.

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Footnotes

...[*]
Supported by CAPES Foundation, Ministry of Education of Brazil.
... France[*]
UMR 7164 (CNRS, Université Paris VII, CEA, Observatoire de Paris)
... index[*]
The resulting spectral index is $\Gamma =3.37$ $\pm $ $0.02_{\rm stat}$. The alternative hypothesis with a curvature in the spectrum (Eq. (2)) is favored at $8.4\sigma$, yelding a harder index ( $\Gamma=3.05$ $\pm $ $0.05_{\rm stat}$) with an exponential cut-off at $E_{\rm cut}=1.76$ $\pm $ $0.27_{\rm stat}~{\rm TeV}$. As the integral flux is dominated by the low-energy part of the spectrum, the choice of the model has a little effect on the integral flux values above $200~{\rm GeV}$.
... considered[*]
 A similar procedure has been carried out on the Crab Nebula observations. Assuming this source is perfectly stable, it allows the derivation of an upper limit to the fluctuations of the Crab signal due to the atmosphere. Nonetheless, this value, ${\sim}10\%$, is linked to the observations' epoch and zenith angles, and to the source spectral shape.
...$10^{-9}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $[*]
Corresponding to data set T200 in Aharonian et al. (2009).
... determined[*]
Even for lower fluxes, the significance associated with each subset is always higher than 20 standard deviations.
...$200~{\rm GeV}$[*]
To be significant, the determination of an energy cut-off needs a higher number of ${\gamma}$ than for a power-law fit.
... interval[*]
If $\sigma^2$ is the variance of $\ln~\Phi$, $F_{\rm var} = \sqrt{\exp (\sigma^2)-1}$.

All Tables

Table 1:   Summary of observations for each year.

Table 2:   The various data sets used in the paper, referred to in the text by the labels presented in this table.

Table 3:   Integral fluxes and their statistical errors from 2002 and 2003 observations of PKS 2155-304 during the HESS construction phase.

Table 4:   The distribution of the flux logarithm.

Table 5:   Parametrization of the differential energy spectrum of the quiescent state of PKS 2155-304, determined in the energy range 0.2-10 TeV, first for the 2005-2007 period and also separately for the 2005, 2006 (excluding July) and 2007 periods.

Table 6:   Mean Flux and the fractional rms $F_{\rm var}$ night by night for MJD 53944-53947.

Table 7:   Doubling/Halving times for the high intensity nights MJD 53944 and MJD 53946 estimated with two different samplings, using the two definitions explained in the text.

Table 8:   Confidence interval at 68% c.l. for T2 and $\tilde{T_2}$ predicted by simulations for $\alpha $ = 2 and $\log_{10}(K/(\rm Hz^{-1}))=2.8$ for the two high-intensity nights MJD 53944 and MJD 53946, with two different sampling intervals (1 and 2 min).

Table 9:   Variability estimators (definitions in Sect. 5.1) relative to $\ln\Phi$ both for the ``quiescent'' and flaring regime, as defined in Sect. 3.2.

Table A.1:   Summary of the 2005 to 2007 observations.

Table B.1:   Integral flux ( $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $) in the 0.2-1 TeV energy range versus photon index corresponding to grey points in the left panel of Fig. 7.

Table B.2:   Integral flux ( $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $) in the 0.2-1 TeV energy range versus photon index corresponding to black points in the left panel of Fig. 7.

Table B.3:   Integral flux ( $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $) in the 0.2-1 TeV energy range versus photon index corresponding to the right panel of Fig. 7.

Table B.4:   MJD, integral flux ( $10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $) in the 0.2-1 TeV energy range, and photon index corresponding to the entries of Fig. 8.

All Figures

  \begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg1.eps}\vspace*{-1mm}\end{figure} Figure 1:

Zenith angle distribution for the 202 4-telescopes observation runs from 2005 to 2007. The inset shows, for each zenith angle, the energy threshold associated with the analysis presented in Sect. 2.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg2.eps} \vspace*{3mm}
\end{figure} Figure 2:

Monthly averaged integral flux of PKS 2155-304 above $200~{\rm GeV}$ obtained from data set D (see Table 2). The dotted line corresponds to 15% of the Crab Nebula emission level (see Sect. 3.2).

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=15.5cm,clip]{14484fg3nv.eps}
\end{figure} Figure 3:

Distributions of the logarithms of integral fluxes above $200~{\rm GeV}$ in individual runs. Left: from 2005 to 2007 except the July 2006 period (data set  $D_{\rm QS}$), fitted by a Gaussian. Right: all runs from 2005 to 2007 (data set D), where the solid line represents the result of a fit by the sum of 2 Gaussians (dashed lines). See Table 4 for details.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg4.eps}
\end{figure} Figure 4:

Distribution of RMSD obtained when the instrument response to a fixed emission ( $\Phi=4.32 \times 10^{-11}~\ensuremath{{\rm cm}^{-2}~{\rm s}^{-1}} $ and $\Gamma =3.53$) is simulated 10 000 times with the same observation conditions as for the 115 runs of the left part of Fig. 3.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg5.eps}
\end{figure} Figure 5:

Energy spectrum of the quiescent state for the period 2005-2007. The green band correponds to the 68% confidence-level provided by the maximum likelihood method. Points are derived from the residuals in each energy bin, only for illustration purposes. See Sect. 3.4 for further details.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg6.eps}
\end{figure} Figure 6:

Integral flux above $200~{\rm GeV}$ measured each night during late July 2006 observations. The horizontal dashed line corresponds to the quiescent state emission level defined in Sect. 3.2.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=16.5cm,clip]{14484fg7.eps}
\end{figure} Figure 7:

Evolution of the photon index $\Gamma $ with increasing flux $\Phi $ in the 0.2-1 TeV energy range. The left panel shows the results for the July 2006 data (black points, data set  $D_{\rm JULY06}$) and for the 2005-2007 period excluding July 2006 (grey points, data set  $D_{\rm QS}$). The right panel shows the results for the four nights flaring period of July 2006 (black points, data set  $D_{\rm FLARES}$) and one point corresponding to the quiescent state average spectrum (grey point, again data set  $D_{\rm QS}$). See text in Sects. 4.1 ( left panel) and 4.2 ( right panel) for further details on the method.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=17.5cm,clip]{14484f8.eps}
\end{figure} Figure 8:

Integrated flux versus time for PKS 2155-304 on MJD 53944-53947 for four energy bands and with a 4-min binning. From top to bottom: ${>}0.2~{\rm TeV}$, 0.2- $0.35~{\rm TeV}$, 0.35- $0.6~{\rm TeV}$ and 0.6- $5~{\rm TeV}$. These light curves are obtained using a power law spectral shape with an index of $\Gamma =3.37$, also used to derive the flux extrapolation down to  $0.2~{\rm TeV}$ when the threshold is above that energy in the top panel (grey points). Because of the high dispersion of the energy threshold of the instrument (see Sect. 2, Fig. 1), and following the prescription described in 3.1, the integral flux has been determined for a time bin only if the corresponding energy threshold is lower than  $E_{\rm min}$. The fractional rms for the light curves are respectively, 0.86 $\pm $ $0.01_{\rm stat}$, 0.79 $\pm $ $0.01_{\rm stat}$, 0.89$\pm $ $0.01_{\rm stat}$ and 1.01 $\pm $ $0.02_{\rm stat}$. The last plot shows the variation of the photon index determined in the 0.2-1 TeV range. See Sect. 4.2 and Appendix B.4 for details.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg9nv.eps}
\end{figure} Figure 9:

Fractional rms $F_{\rm var}$ versus energy for the observation period MJD 53944-53947. The points are the mean value of the energy in the range represented by the horizontal bars. The line is the result of a power law fit where the errors on $F_{\rm var}$ and on the mean energy are taken into account, yielding $F_{\rm var}(E)\propto E^{0.19\pm 0.01}$.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=8cm,clip]{14484fg10.eps}
\end{figure} Figure 10:

DCF between the light curves in the energy ranges 0.2-0.5 TeV and 0.5-5 TeV and Gaussian fits around the peak. Full circles represent the DCF for MJD 53944-53947 4-min light curve and the solid line is the Gaussian fit around the peak with mean value of 43 $\pm $ $51~{\rm s}$. Crosses represent the DCF for MJD 53946 with a 2-min light curve binning, and the dashed line in the Gaussian fit with a peak centred at 14 $\pm $  $41~{\rm s}$.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=8cm,clip]{14484fg11_nv.eps}
\end{figure} Figure 11:

The excess rms $\sigma _{\rm xs}$ vs. mean flux  $\bar{\Phi}$ for the observation in MJD 53944-53947 (Full circles). The open circles are the additional points obtained when also including the extrapolated flux points - see text). Top: $\sigma _{\rm xs}$ estimated with 20 min time intervals and a 1 min binned light curve. Bottom: $\sigma _{\rm xs}$estimated with 80 min time intervals and a 4 min binned light curve. The dotted lines are a linear fit to the points, where $\sigma _{\rm xs}\propto 0.13$ $\times $ $\bar{\Phi}$ for the 1 min binning and $\sigma _{\rm xs}\propto 0.3$ $\times $ $\bar{\Phi}$ for the 4 min binning. Fits to the open circles yield similar results.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg12.eps}
\end{figure} Figure 12:

95% confidence domains for $\alpha $ and K at $\nu_{\rm ref}=10^{-4}~{\rm Hz}$ obtained by a maximum-likelihood method based on the $\sigma _{\rm xs}$-flux correlation from 500 simulated light curves. The dashed contour refers to light curve segments of 20 min duration, sampled every minute. The solid contour refers to light curve segments of 80 min duration, sampled every 4 min. The dotted contour refers to the method based on the structure function, as explained in Sect. 6.3.

Open with DEXTER
In the text

  \begin{figure}
\par\includegraphics[width=8.5cm,clip]{14484fg13_v2_nv.eps}
\end{figure} Figure 13:

First order structure function SF for the observations carried out in the period MJD 53944-53947 (circles). The shaded area corresponds to the 68% confidence limits obtained from simulations for the lognormal stationary process characterized by $\alpha =2$ and $\log_{10} (K/{\rm Hz}^{-1})=2.8$. The superimposed horizontal band represents the allowed range for the asymptotic value of the SF as obtained in Sect. 7.

Open with DEXTER
In the text


Copyright ESO 2010

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