2.3 Reconstruction and Particle Identification
3.1.5 Estimates of the Frequency of Galactic Center Flares
There are various estimates of the frequency of X-ray flares for Sgr A* due to regular monitoring programs like those of Swift and Chandra. Swift recently published an estimate for the recurrence time of 5–10 days which means a frequency of 0.1–0.2 X-ray flares per day. Another census of Sgr A* flares which was performed using Chandra data claims roughly one flare per day. The average duration of an X-ray flare has been given with 2600 seconds [35] which is a bit longer than a H.E.S.S. run. Assuming that these processes also reach the VHE γ-ray regime, one can obtain two insights for planning a study based on H.E.S.S. data from these numbers when assuming that the underlying mechanism also causes VHEγ-ray emission: Taking into account that the whole usable dataset is about 350 hours at the moment (including 250 hours from H.E.S.S. I) one would expect that H.E.S.S. has observed about 1.5 to 15 of these flares by chance during its lifetime. The lower limit is based on the Swift population estimate, while the upper limit uses the population estimated by Chandra. More details about the H.E.S.S. dataset follow in Chapter 4. An important point here is that one would expect a variability at a timescale of H.E.S.S. runs (28 minutes) and below due to the average duration of these X-ray flares which is in the order of the duration of H.E.S.S. runs. Therefore, it is important to focus also on such short timescales in an analysis of H.E.S.S. data. IR flares even occur with a larger frequency: A census by Genzel et al. gave an estimate
of as much as 2–6 near-infrared flares per day [40] which is confirmed independently in Ref. [110].
3.2 Discussion of H.E.S.S. I Results on the Search for a Variability of HESS J1745-290 from 2006
In 2009 the H.E.S.S. Collaboration published a study on the spectrum and the variability of the source HESS J1745-290 using the dataset from the GC campaign from the years 2004, 2005 and 2006. The total livetime taken into account for this analysis was 48.7 hours in the “wobble observation mode”. With respect to a possible variability of Sgr A*, the study could not reveal any hint for an effect. It included a search for QPOs at frequencies ranging from 100 s to 2250 s which are known from X-ray observations [16]
and a χ2 fit of a night-wise light curve with a constant. This fit with a constant shown in Fig. 3.2.1a gave a χ2 of 233 for 216 degrees of freedom (d.o.f.) which corresponds to a significance for variability of less than 1σ. Concerning the search for a periodicity, the distribution of the mean Rayleigh power as a function of time was investigated. As one can see in Fig. 3.2.1b all values are close to one, which means that also here no evidence for a variability could be found [101]. The theory behind the Rayleigh test will be explained in more detail in Sec. 4.1.2.2.
Although the first search for variability of HESS J1745-290 using H.E.S.S. data did not lead to a positive result it is not yet ruled out completely for various reasons:
1. Observation time: The dataset taken into account for the published analysis was only 3 years of the 10 years which are available now. If the variability is a weak effect more data is required in order to find a hint for it.
2. Occasional flares: There will always be the possibility that Sgr A* undergoes occasional flares reaching the GeV and TeV regime, which is difficult to rule out without permanent monitoring of this source.
3. QPOs: In X-ray and IR data QPOs only have been observed during flare activity.
Even if there had been such a flare in the dataset which was taken into account for the analysis it would not affect the distribution of the mean Rayleigh powers significantly if the remaining runs do not contain flares.
4. Cuts and statistics: For the analysis the STD cuts were used, which lead to relatively low statistics compared to the Loose cuts and an energy threshold of
(a) A three years light curve of the integrated flux >1 TeV using the H.E.S.S.
GC data taken until 2006 and STD cuts for the analysis.
(b) The mean Rayleigh power as a function of frequency.
Figure 3.2.1: Light curve and mean RL Power from the analysis of the H.E.S.S. 2004-2006 dataset with the STD cuts. The two plots were originally published in H.E.S.S.
paper about the GC from 2006 (Ref. [101]).
300 GeV. Especially for the search for variability at the timescale of minutes, this lack of statistics could be a problem, since it implies large statistical errors. On the other hand if the γ-ray signal observed from the direction of Sgr A* is a superposition of constant contribution of a DC source (e.g. the pulsar wind nebula) and a variable part from the black hole itself this might be leptonic process with a cut-off at some 100 GeV, which would mean that the variability would not have been detected in the previous study. Although this is somewhat speculative, it is possible.
5. Binning of the light curve: The light curve for this analysis was created on a night-wise base. In case the variability of Sgr A* takes place at a shorter timescale, it would get lost with such a binning.
In summary one can say that a variability of the GC source could not entirely be ruled out by this study due to the reasons above and therefore a (re-)analysis of the complete H.E.S.S. dataset, which consists of about 250 h of H.E.S.S. I observations and about 350 h in total, may still give interesting insights.
3.3 Summary
The flaring behavior and tentative long-term variability of Sgr A* from the radio to the hard X-ray band have already given some interesting insights into the nature of Sgr A*
and started a discussion about the origin of the frequent and short flares. The flares vary dramatically in their characteristics like luminosity and duration. Therefore, it is unlikely that they are produced by a single effect but rather an interplay of different mechanisms should be expected. Their similar time structure over different wavelength bands and some synchronous IR and X-ray flares, which have been observed, indicate that there might be a common underlying mechanism for IR and X-ray flares but this needs more detailed investigation. This assumption is further backed by reports of similar QPOs seen for both types of flares. Since none of the current models is completely excluding that these effects also reach the VHE γ-ray band, further searches and studies are urgently needed. Although a first search for variability of HESS J1745-290 with H.E.S.S. data for different timescales lead to a negative result, the presence of such an effect could not yet be fully excluded for the reasons discussed above. Therefore, it is necessary to repeat this study with the full dataset available to date, including all the information collected by other experiments during the last 10 years.
Search for Variability of the Galactic Center Source HESS J1745-290
The following chapter is dedicated to the search for variability of the γ-ray source HESS J1745-290. While the first part is explaining the methods in use, the second part presents the results of their application to the available HESS J1745-290 dataset.
The analysis takes into account data which were taken between 2004 and 2014.
4.1 Methods: Variability and Periodicity Tests
First the methods, which will be used for the data analysis, are presented and tested with toy data. Since the χ2 fit of a H.E.S.S. light curve with a constant is a rather common method the focus here was set to the periodicity tests which will be used later, namely the Lomb-Scargle (L-S) test and the Rayleigh (RL) test.
4.1.1 Light Curves: A Definition
Light curves are a common tool to investigate the flux or intensity of an astronomical object as a function of time. Most light curves refer to a certain energy range. By applying different statistical tests to a light curve, one can investigate if the object has a constant flux, is variable or even shows periodic behavior. A prominent use case of light curves is the search for extrasolar planets: There the variation of the brightness of a star can give hints for orbiting planets, which periodically absorb a fraction of the light emitted by their parent star. Furthermore, light curves are a useful tool for studying systems with periodic behavior like binary systems or black holes with a rotating accretion disk. Their periodicity usually can be observed across a large range
of wavelengths.
4.1.2 Periodicity Tests
Since Sgr A* has shown periodic and quasi-periodic behavior in radio, X-ray and IR data it is an interesting question if such periodic behavior is also observable for the VHE γ-ray source HESS J1745-290. In order to answer this question, the L-S test and the RL test will be applied to the HESS J1745-290 dataset after a brief introduction and discussion of these two periodicity tests. While the L-S test can be applied to the flux values of a light curve with irregular sampling, the RL test is an event-wise periodicity test, which will be used for the search for a short-term variability during single runs.
4.1.2.1 The Lomb-Scargle Test
The L-S test is a powerful statistical test to find weak periodic signals within unevenly sampled datasets, which for example could be time series with variable gaps due to irregular observations. The method ignores the non-equal spacing and calculates the regular Fourier power spectrum as if the data had an equal spacing [44,55]. Furthermore, it gives a solid statistical significance in case a signal is present. The formalism of this test has been developed by Barning [19] and Lomb [67] who added the correct normalization.
Scargle [93] gave a statistical interpretation in 1982 and found a way to calculate the significance of a signal if present. Alternative methods to search for a periodic signal within a dataset are the classical Fourier Transform (FT) or Fast Fourier Transform (FFT) algorithms. The problem with these methods is that periodic gaps in time often produce large powers for periods comparable to these gaps. The L-S method is an extension of these classical periodograms, which is able to minimize these undesired effects by evaluating the data only for time-windows during which the measurements actually took place and in principle is equivalent to a fit with sine and cosine functions.
The L-S power is defined as follows: In case one has a series of N data pointshi =h(ti), which were actually measured at timesti, one can calculate a mean ¯h and the standard deviationσ by the classical formulas
The normalized L-S power can then be defined as The constant τ is called time-offset and is defined as
tan(2ωτ) =
P
jsin(2ωtj)
P
jcos(2ωtj) (4.1.3)
and makes the obtained L-S power independent from shifting all thetisby an arbitrary constant. Therefore, time translation invariance is a useful property of the L-S peri-odogram. The time offset has to be calculated for each investigated angular frequency ω. As shown by Lomb, the time offset makes the expression for the normalized L-S power equivalent to a linear least square fit with the function
h(t) = Acos(ωt) +Bsin(ωt) (4.1.4) which is an estimate of the harmonic content of the dataset under consideration. The maximum L-S power occurs exactly at the period which minimizes the sum of squares of a fit of Eq. 4.1.4 to the data. In case the data are evenly sampled Eq. 4.1.2 reduces to the classical periodogram which is directly derived from the discrete FT.
4.1.2.1.1 The Statistical Interpretation Considering the null distribution of L-S powers which are obtained by chance, it can be shown that Ph(ω) for any given frequencyωis following an exponential distribution. One can write down the cumulative distribution function (CDF) of thePh(ω):
F(z) = Prob[Ph(ω)<=z] = 1−exp(−z). (4.1.5) This means it gets exponentially unlikely to obtain a power like the observed one or a larger one by chance. However, this holds in the case of scanning a single frequency only. In case more frequencies are scanned the probability of obtaining a large Ph(ω) by chance increases and the statistical significance has to be corrected accordingly. For strictly uncorrelated periods the false alarm probability for getting the observed or a larger L-S power by chance can be written as
p= 1−(1−exp(−z))N (4.1.6)
where N is the number of frequencies which were scanned. From this formula the detec-tion threshold z0 for a certain p-value p0 can be derived as
z0 =−log[1−(1−p0)N1]. (4.1.7) In case one wants to detect a periodic signal at the 5σ level doing 10 trials one would need to observe a peak with Ph(ω) larger than 17.32. The classical L-S method does not take any statistical measurement errors into account. However, one can assume that in case of large statistical measurement errors a weak signal would rather be destroyed than artificially added to the dataset.
4.1.2.1.2 Application to Toy Monte Carlo Data In order to test the implemen-tation of the L-S method, it was first applied to toy MC data. Different sets of random data were simulated with and without the superposition with a sinusoidal signal. For the test a toy light curve covering 1000 days was created which contains 200 measure-ments with uneven spacing and gaps. Fig. 4.1.1a shows the noise-only light curve and periodogram where the red line marks the 5σ detection threshold for the number of trials in the test. The L-S powers which were encountered in this test, stay below the detection threshold as one would expect. In Fig. 4.1.1b a weak periodical sinusoidal signal with a period of 40 days was added which is optically not visible in the light curve. Its amplitude was chosen to be in the same order of magnitude as the standard deviation of the Gaussian noise. The distribution of the L-S powers for the noise-only case also correctly showed the expected mean value of one.
4.1.2.1.3 Application to Data from the Swift X-Ray Telescope The L-S test has also been applied to data from Swift BAT, which is publicly available. As example for a source with known periods the Galactic Center microquasar 1E 1740.7-2942 [97]
was chosen, which is also known under the popular name “Great Annihilator” and is the brightest electron (positron) source in the sky. In order to test the L-S method, the data of a day wise Swift light curve were used which were taken from February 2004 until December 2014. The result of this first application of the analysis code to real astronomical data is presented in Fig. 4.1.2a where the L-S power is plotted versus the period in days. The resulting power spectrum shows 5 significant peaks for 1E 1740.7-2942 at frequencies of 140 d, 360 d, 500 d, 710 d and 1010 d and one slightly crossing the 5σ detection threshold at 420 d. The 5σ threshold is marked by a red line, taking into account the correct number of trials which was 221 for the analysis: Periods from 100 d to 1200 d were investigated with a step-width between neighboring periods of 5 d.
(a) A toy Monte Carlo light curve containing Gaussian noise only (left) with the resulting L-S periodogram (right).
(b) A toy Monte Carlo light curve with a sinusoidal signal (left) with the resulting L-S periodogram (right). At the simulated period of 40 days a clear and significant peak can be observed.
Figure 4.1.1: Test of the L-S method with toy MC data. The red lines in the peri-odograms mark the 5σ detection threshold for the number of trials which were used to create the power spectrum.
(a) The L-S periodogram for 1E 1740.7-2942 is showing five clear peaks at periods of140 d, 360 d, 500 d, 710 d and 1010 d.
(b) The LS-periodicity test applied to a scrambled light curve of 1E 1740.7-2942.
Figure 4.1.2: The L-S periodicity test applied to a light curve of the “Great Annihilator”
1E 1740.7-2942. Both figures plot the L-S power versus the period in days.
In order to rule out that the periodic behavior is produced by sampling effects (this should not be the case since the LS test is designed to be robust to that but in any case it is better to test) the flux values of the light curve of 1E 1740.7-2942 were exchanged in a random fashion keeping the time stamps untouched. The L-S periodogram obtained with that manipulated light curve is shown in Fig. 4.1.2b: Exchanging the fluxes makes the periodic behavior disappear completely, which rules out that sampling effects are fooling the L-S test here for the original light curve. The method of randomizing the fluxes, which was successfully tested here, will also be later applied for the H.E.S.S. GC data analysis.
Both the period at around 140 days and the very large period around 1010 days have not been found in the literature and therefore seem to be unpublished at the time of this writing. Although not being directly part of the topic of this thesis, this is certainly the most interesting side result of this thesis. There is no room for further investigation here but these periods seem to be an interesting topic for further studies. Another test with a Swift light curve of the object IGR J17475-2822 [88] which is a molecular cloud shows a stable zero result as expected. The L-S periodogram obtained in this second test with real X-ray data can be found in Fig. 4.1.3. These additional tests show that the method works also with real data and only detects periodic behavior where it is present.
Figure 4.1.3: L-S periodogram of the molecular cloud IGR J17475-2822. As expected all L-S powers stably stay below the 5 sigma detection threshold which is indicated by the red line.
4.1.2.2 The Rayleigh Test
The RL test is a powerful event-wise periodicity test, which can easily be applied to a series of event arrival times for example. It is the special casen= 1 for the Zn2 statistics which is defined as and is summing over the first n Fourier components of the series. The θi are the event phases, which are taken from the interval [0,2π) and are calculated using an assumed period T. The indexiruns from 1 to N, which is the number of events. For the simplest case setting n = 1 the variable Z212 is equivalent to the normalized RL power P:
P = Z12
The overall distribution of normalized RL powers follows an exponential distribution with mean equal to one in the absence of a signal, therefore the false alarm probability for a power obtained can be calculated in the same way as discussed previously for the L-S test (Eq. 4.1.5 - Eq. 4.1.7). Applying the RL test to data one has to be careful to apply it to a sample which has a length of an exact multiple of the length of the period of interest, which means that for most applications of the test the sample needs to be truncated. Otherwise the result will be biased towards larger powers. The RL test is
very sensitive for simple sine waves. For more complex pulse shapes the Z22 statistics could be taken into account [95].
4.1.2.2.1 The Distribution of Rayleigh Powers As mentioned before the pure background distribution of the normalized RL powers S is equivalent to an exponential with mean one and slope minus one. In case one is searching for a weak signal which may be only present in parts of the dataset Ref. [59] gives a good overview how one could do so taking into account the shape of the distribution of all RL powers obtained when applying the test. The basic idea here is that the presence of a weak, periodic signal would slightly change the shape of the whole distribution of RL powers from the theoretical distribution. A possible way to quantify the difference from the theoretical exp(−P) distribution is the following: In case one has N data segments, e.g. H.E.S.S.
runs, one can calculate the sum over all RL powers for a particular period over all segments. This sum is called Ptot in the following and can be defined as:
Ptot =
k=nseg
X
k=1
Pk (4.1.10)
where nseg is the total number of data segments. It can be easily shown that for the pure background 2Pk follows a χ22 distribution (the index in subscript stands for the
where nseg is the total number of data segments. It can be easily shown that for the pure background 2Pk follows a χ22 distribution (the index in subscript stands for the