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Astron. Astrophys. 342, 69-86 (1999)

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5. Study of the shortest time scales of variability

The time scales of the Mkn 501 TeV flux variability have been studied in two ways. First, the time gradients of the daily flux amplitudes [FORMULA] have been analyzed. The analysis uses the exponential increase/decay-constant [FORMULA], as computed for each pair of two daily flux amplitudes according to:


where [FORMULA] is the time difference between the two measurements and [FORMULA] is the difference between the logarithms of the differential fluxes at 2 TeV. This formula has been derived by assuming a time dependence of [FORMULA] according to [FORMULA]. In the case of small changes in the flux amplitude [FORMULA], where [FORMULA] is the time averaged flux amplitude, the "doubling time" is commonly used to characterize the variability time scale. It is defined as the time in which the flux would have increased or decreased by 100%, assuming a linear increase or decrease of the flux:


For small changes in the flux amplitude the increase/decay-constant [FORMULA] computed with Eq. 6 equals [FORMULA] computed with Eq. 7.

In Fig. 15 the [FORMULA]-values computed for adjacent nights are shown. The most rapid [FORMULA]-values are in the order of 15 h. In Table 5 the [FORMULA]-values more rapid than 24 h are listed. The distribution of the [FORMULA]-values is to first order approximation symmetric under time reversal (see Fig. 16), i.e. the daily data does not give obvious evidence for a different rising and falling behavior.

[FIGURE] Fig. 15. The increase/decay constants [FORMULA] computed for the flux amplitudes [FORMULA] of adjacent nights plotted against the mean MJD of the two nights (MJD 50550 corresponds to April 12th, 1997).


Table 5. The most rapid increase- and decay-times .

[FIGURE] Fig. 16. The distribution of increase/decay constants [FORMULA] computed for the flux amplitudes [FORMULA] of adjacent nights. Only [FORMULA]-values between -250 h and +250 h are shown.

In a second approach a dedicated search for variability on sub-day time scales has been performed. In order to search on time scales well below one hour, where the determination of a differential spectrum is plagued by large statistical uncertainties, the search directly uses the [FORMULA]-ray excess rates. Using only data taken under zenith angles below [FORMULA] and applying a cut on the distance r of the shower axis from the center of the telescope array [FORMULA] m, the zenith angle dependence of the [FORMULA]-ray rate is negligible, i.e. less than [FORMULA] for [FORMULA]-ray spectra with differential spectral indices between -2.5 and -1.

In Fig. 17 the excess-rate histories computed with 10 minute temporal bins are shown for two typical nights, i.e. May 6th/7th and May 9th/10th. Here, as well as in the analysis below, the cuts [FORMULA] and [FORMULA] have been applied. No strong variability can be seen and a method is needed to decide on the statistical significance of the observed rate variations.

[FIGURE] Fig. 17. The [FORMULA]-ray detection rate of two individual nights: MJD = 50574/50575 (May 6th/7th), and 50577/50578 (May 9th/10th). A 10 minutes binning has been used.

For the systematic search, a method introduced in (Collura & Rosner 1987) has been used. It is based on the [FORMULA]-fitting technique and permits a search on multiple time scales. It provides easy and straightforward computation of the significances of the detected variabilities. For each night a variability is searched for, using different time bin sizes, by computing the [FORMULA]-value of a constant model fit to the data, and performing an averaging procedure over the relative location of the data points within these binning schemes. We use time bins with durations [FORMULA] between 10 minutes and 2.24 hours, i.e. with [FORMULA] minutes) for [FORMULA] to 15. The lower limit on the bin duration is given by the requirement of an expected number of recorded events per bin [FORMULA] @ The upper limit is determined by the maximum duration of the Mkn 501 observations per night which is in the order of 4 h. For each night the bin duration which yields the most significant variability detection is determined and the chance probability [FORMULA] for a constant flux to yield a more significant variability is computed using an analytic approximation.

The chance probabilities, actually the [FORMULA]-values, for the 51 nights for which sufficient data with zenith angles smaller than [FORMULA] is available, are shown in Fig. 18 together with the distribution of large [FORMULA]-values expected in the absence of flux variability. The [FORMULA]-values distribute as expected in the absence of any variability, except for values near 0 (chance probabilities near 1) and except for values larger than 3 (chance probabilities smaller than 1 per mill). The deviation at small values derives from the analytical computation of the chance probabilities, and is thus a pure artifact of the method (see also Collura & Rosner 1987). The deviation at large values correspond to two nights, the night of May 8th/9th (MJD 50576/50577) and the night of June 7th/8th, 1997 (MJD 50606/50607), for which sub-day variability is indicated.

[FIGURE] Fig. 18. The results of the search for sub-day variability. For each night, the chance probability [FORMULA], computed with an analytic approximation, with which a constant signal would yield a more significant variability as the observed one has been computed. Shown is the distribution of the [FORMULA]-values (histogram) together with distribution expected for small chance probabilities in the absence of variability (dotted line).

In the case of the first night, the most significant variability has been found with 20 minute bins; in the case of the second night with 67 min bins. The excess rates during the two nights are shown in Fig. 19. In the night from May 8th to May 9th, 1997, the [FORMULA]ray rate oscillated between 3 [FORMULA]/min and 5.4 [FORMULA]/min on a time scale of 1.5 hours. In the night from June 7th to June 8th, 1997, the [FORMULA]ray rate continuously increased from 4.9 [FORMULA]/min to 7 [FORMULA]/min within 2.6 h. The chance probability for a more significant apparent variability at constant flux is computed to be [FORMULA] for the first night and [FORMULA] for the second night. Taking into account that 51 nights have been searched for variability, the chance probabilities increase to 0.4% for the first night and to 1% for the second night. The variability detected in these two nights corresponds to an increase/decay constant of about 3 h for the first night and 7 h for the second night.

[FIGURE] Fig. 19. The [FORMULA]-ray detection rate of the two nights: MJD = 50576/50577 (May 8th/9th) and 50606/50607 (June 7th/8th). For the first night, a 20 minutes binning and for the second night, a 67 minutes binning has been used.

To summarize, the study of the time gradients of the diurnal fluxes yields smallest increase/decay times of the order of 15 h. The dedicated search for flux variability within individual nights on time scales between 10 minutes and several hours yielded weak evidence for variability on time scales of around 5 hours.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998