4.2 Data Analysis
4.2.4 Search for a long-term Variability
When searching for a long-term variability at a timescale larger than the duration of a H.E.S.S. run, which is ∼ 28 minutes, two questions arise: The first is if there is a long-term variability at all and in case hints for such an effect are found the second question would be if it is rather a random effect like irregular flares or if it also shows a recurring or even periodic pattern. For answering the first question a simpleχ2 fit with a constant can be applied to the HESS J1745-290 light curve and in case evidence of a variability is found, the L-S test is the correct tool to test whether a possible effect also shows periodic behavior. Therefore, after discussing the application of the χ2 fit with a constant to the H.E.S.S. I and H.E.S.S. II light curves the L-S test will be applied to search for a possible periodicity in the next step. The search for a long-term variability contains all runs which passed the spectral data quality selection criteria. No additional selection criteria were introduced at this point.
4.2.4.1 Analyzing the run-wise Light Curve of HESS J1745-290
Two independent run-wise light curves of HESS J1745-290 were created for the inte-grated flux above 100 GeV using the standard method for light curves at H.E.S.S. which is implemented within the HAP data analysis software package for both the H.E.S.S.
I and the H.E.S.S. II dataset. This procedure requires the spectral index to be set as input parameter. For the H.E.S.S. I light curve which will be discussed in this section,
the empirical index of 2.3±0.2 syst, which was derived with the Loose cuts before, was used but it will also be shown that the level of variability of the integrated flux does not depend on a 10% variation of the spectral index. The χ2 fit of the light curve was performed separately for the H.E.S.S. I light curve including the years 2004 to 2012 and H.E.S.S. II using data from the years 2013 and 2014. There are two possible ways of combining these two independent results: The first is directly combining the light curves (by scaling them to a comparable level of integrated flux) and fitting the resulting com-bined light curve with a constant. A second approach is to simply combine the results of the independent fits by adding theχ2 and d.o.f. of the two independent fits according to the theorem of additivity of χ2 variables [86]. In the following the second approach will be used.
4.2.4.1.1 Light curve from H.E.S.S. Phase I The light curve derived from the H.E.S.S. I dataset is shown in Fig. 4.2.4a. It was created using an integration radius of 0.2°. Although the light curve does not show any obvious flares which could be detected
“by eye”, theχ2fit with a constant shows evidence for variability. The fit resulted in aχ2 of 693.1 which was obtained for 516 d.o.f. This corresponds to a p-value of only 2.8·10-7 in favor of the null-hypothesis of the light curve being flat, implying a significance level for variability of 5.0σ for the H.E.S.S. I light curve.
As a control source, which is located in the same FoV, the SNR G 0.9+0.1 was used to cross-check the result for HESS J1745-290. The G 0.9+0.1 data were taken exactly under the same instrumental and atmospheric conditions like the HESS J1745-290 data.
The χ2 fit of the G 0.9+0.1 light curve with a constant did not show any sign of a significant variability with a p-value of 0.19 which implies that the G 0.9+0.1 fluxes are in good agreement with the null-hypothesis of a flat light curve. Comparing the integrated flux > 100 GeV for the integration radius of 0.2° HESS J1745-290 shows an integrated flux of 4.9±0.1·10−11cm-2s−1 while G 0.9+0.1 only has an integrated flux of 1.3±0.1·10-11cm−2s−1, which means that the integrated flux of G 0.9+0.1 is only 26% of that of HESS J1745-290.
In order to investigate if the observed variability is a methodical artifact of the algorithm which is used to create H.E.S.S. light curves, several parameters were varied during a cross-check. For example one could argue that the lower integration threshold for the light curve of 100 GeV was chosen too low, since the mean energy threshold of the Loose cuts is at about 150 GeV, although the algorithm for the light curve at H.E.S.S.
should extrapolate the integrated flux correctly to the given integration threshold. The spectral index which also needs to be set as parameter for creating a H.E.S.S. light curve
Date in MJD
53000 53500 54000 54500 55000 55500 56000 56500
]-1s-2 Integrated Flux > 100 GeV [cm
(a)χ2fit of the run-wise fluxes of HESS J1745-290 with a constant.
Date in MJD
53000 53500 54000 54500 55000 55500 56000 56500
]-1s-2 Integrated Flux > 100 GeV [cm
(b)χ2 fit of the run-wise fluxes of the SNR G 0.9+0.1 with a constant.
Figure 4.2.4: Fit of the run-wise fluxes with a constant using H.E.S.S. I data only. The fit was applied to a light curve of HESS J1745-290 and also to a light curve of G 0.9+0.1 as a control source from the same FoV.
Source θ Eth α p σ
Table 4.3: The fit results in terms ofp-valuepand significance for variabilityσ of theχ2 fit of different HESS J1745-290 light curves, which were built based on different spectral indicesα, different energy thresholds Eth and integration radii θ.
might also have an effect on the behavior of the light curve with respect to variability. In order to rule out systematic effects due to these parameters the light curve was created and fit assuming two different spectral indices, two different energy thresholds and also using different integration radii. The results of these cross-checks can be found in Table 4.3.
By this cross-check it could be shown that the variability for HESS J1745-290 can be observed independently of the input parameters for the light curve at a significance level between 4.1σ and 5.2σ which mainly depends on the integration radius. On the other hand, G 0.9+0.1 shows a zero result for all parameter settings. Therefore, the effect can be considered not to be an artifact of the method or its predefined parameters respectively.
In order to investigate if the variability is only observed for certain pointing positions which would be a hint for a systematic effect, which could be introduced by particular, problematic background regions, the light curve was also split into two different subsets with respect to the pointing position. The first partial curve only consists of pointings in the north of the Galactic plane (b >0), while for the second partial curve only pointings below the Galactic plane with b <0 were taken into account. The pointings with b > 0 take the background mainly from the north of the GC while the pointings withb <0 take the background mainly from the south. This is due to the radial symmetry of acceptance and the fact that the background regions are located along circles with a constant radial acceptance. Since the Galactic plane itself is excluded for the background, for pointings north (south) of the GC the northern (southern) part of this circle is taken into account for the background, since its segments in the south (north) of the GC (if geometrically
Figure 4.2.5: The NSB level along the Galactic plane. The position of the GC is marked by a red dot.
possible) are mostly excluded.
One could argue that the variability is caused by the NSB, a light pollution caused by bright stars etc. which is at a different level in the south of the GC than in the north as shown in Fig. 4.2.5 where the NSB along the Galactic plane is plotted. The GC is marked by a red dot in this plot. If one were to observe a significant difference of the mean flux between the two light curves or the variability were only be observed for one of them one could argue that the variability is likely to be a systematic artifact, which is either introduced by the NSB or problematic background regions in a particular region of the sky. On the other hand, if the flux level and the level of variability are comparable between the two partial light curves it is unlikely that such systematic explanations can account for the observed variability.
The first observation from this cross-check was, that the number of pointings in the south of the GC is larger than the number of pointings in the north of the GC with 290 versus 227 pointings. The mean integrated flux >100 GeV for the pointings in the north of the GC is 4.90±0.15×10−11cm−2s−1 while for the pointings in the south of the GC it is 5.06±0.14×10−11cm−2s−1. This implies that the mean flux values of the two independent partial light curves agree within statistical errors. The difference of the mean values is only 3%. With respect to variability, the northern pointings show a χ2/d.o.f. of 300.1/226 in aχ2 fit with a constant corresponding to ap-value of 7.0·10−4, while the pointings in the south show a χ2/d.o.f. of 392.4/289 which is equivalent to a p-value of 4.8·10−5. When combining the fit results for the two independent light curves again, one obtains aχ2/d.o.f. of 692.5 to 515. The p-value corresponding to this result is 2.6·10−7 or 5.0σ in units of sigma. Both independent fits show signs of variability and the combined p-value is comparable to that which was obtained from the fit of the full light curve which was 2.8·10−7. The two different values for the significance level can be explained by the fact that performing the fit two times to independent, partial light curves introduces an additional degree of freedom since the mean flux may differ for each of the independent fits.
By this cross-check it could be shown that the observed signs of variability are
inde-Subset Index
2 4 6 8 10
]-1s-2 Integrated Flux > 100 GeV [cm
40 45 50 55 60
−12
×10 χ2 / ndf 14.51 / 9
p0 5.061e−11 ± 9.837e−13 / ndf
χ2 14.51 / 9
p0 5.061e−11 ± 9.837e−13
Figure 4.2.6: The mean integrated flux>100 GeV for 10 subsets of the H.E.S.S. I light curve consisting of 50 consecutive runs each.
pendent of pointings north and south of the GC. Therefore, it is unlikely that the NSB or a problematic background region are causing the observed variability.
After it was shown that the long-term variability of the integrated flux > 100 GeV is observed independently of the Galactic latitude where the background is taken from, another type of possible systematic effects was investigated: The H.E.S.S. I light curve was built upon 8 years of GC observations. It is possible that changes in the performance of the detector, aging of the mirrors of the telescopes or an increasing number of broken pixels have an effect on the integrated flux, although some of these effects like the decrease of the mirror reflectivity are corrected in the calibration. If such systematic effects influence the integrated flux, one would observe an increase or decrease of the integrated flux as a function of the time. Of course, such an increase could also be due to physics but it is worth studying if it is present in the data at all. In order to investigate if the integrated flux changes as a function of the time, the mean flux was calculated for 10 subsets of 100 consecutive flux points. The values which were obtained for the mean flux of each sub-set in this test are plotted in Fig. 4.2.6.
It could be shown that the mean flux does not change significantly between the
subset p-value σvar
Table 4.4: The p-values and significance levels for variability for subsets 1–10.
different subsets by fitting the 10 data points with a constant. The χ2/d.o.f of 14.5/9 corresponds to a 1.3σ significance level for variability. Furthermore, the last data point from the year 2012 shows a comparable flux level like the first data point from 2004.
This implies that the variability is likely not to be caused by aging effects of the detector.
Another interesting question is which of the subsets show an internal variability by discussing the p-values of the fit for each of the subsets. These p-values and their corresponding significance levels for variability can be found in Table4.4. Although the mean significance level for variability of the first 5 fits (2.2σ) is larger than that of the last 5 fits (0.7σ), the significance for variability shows a tendency of an increase for the subsets 9 and 10. The lesson which could be learned from these cross-checks is that the mean flux for subsets of 50 runs does not vary significantly as a function of time and systematic effects related to the age of the detector are unlikely to affect the integrated flux since the mean flux value of the first subset agrees with that of the last subset within statistical errors.
Furthermore, the observed variability seems to occur on a timescale of a few runs rather than a timescale>50 runs since it averages out when fitting the mean values of the 10 subsets with a constant. However, particular subsets still show signs of variability internally when they are fit with a constant. Since the sampling in this test is not correlated with the calender date, it is not possible to make a strict statement about the development of the observed variability as a function of time.
4.2.4.1.2 Light Curve from Phase II In the following also a H.E.S.S. II light curve based on Subset B is analyzed. Since the data quality of the H.E.S.S. II data from 2013 and 2014 cannot be considered to be at the same level than that of the H.E.S.S.
Date in MJD
(a) Theχ2 fit of the run-wise fluxes of HESS J1745-290 with a constant.
Date in MJD
(b) Theχ2 fit of the run-wise fluxes of G 0.9+0.1 with a constant.
Figure 4.2.7: Fit of the H.E.S.S. II run-wise fluxes with a constant for both HESS J1745-290 and G 0.9+0.1.
I, dataset this part of the analysis rather has the character of a cross-check in order to investigate if the tendency observed for the H.E.S.S. I light curve is also continuing for the H.E.S.S. II data. For the fit with a constant shown in Fig. 4.2.7a the χ2 is 211.4 for 141 d.o.f. corresponding to 3.7σ in terms of a Gaussian significance. The mean integrated flux of (2.6±0.1·10−11) cm-2s−1 >100 GeV differs from the value which was obtained from the H.E.S.S. I dataset. As discussed previously this difference can be accounted for by the different integration radii which are used by the H.E.S.S. I and H.E.S.S. II configurations. This implies that there also is evidence for a variability of the HESS J1745-290 fluxes in the H.E.S.S. II dataset. The cross-check with data from G 0.9+0.1 which can be found in Fig. 4.2.7b did not show any signs of variability confirming the H.E.S.S. I data.
4.2.4.1.3 Combining the Results from H.E.S.S. I and H.E.S.S. II Summing over theχ2 from the two independent fits of the H.E.S.S. I and H.E.S.S. II data sample one obtains aχ2 of 904.5 at 657 d.o.f, which is corresponding to a p-value of 1.1·10−10 . In terms of a Gaussian significance this is equivalent to 6.1σ.
The variability which was observed in the χ2 fit of the light curve is a promising result which requires a careful discussion of systematic effects. Previously it has already been shown that the effect for the H.E.S.S. I dataset does not depend on the spectral index and the energy thresholds which were used as input parameters for the light curve. Also using a tighter integration radius of 0.15° did not reduce the effect for the H.E.S.S. I light curve. Furthermore, the cross-check with G 0.9+0.1 can be used to estimate possible systematic effects. The constant γ-ray flux of G 0.9+0.1 still rules out most of the possible systematic effects which could cause the observed variability of HESS J1745-290, since the two sources are located in the same FoV at a distance of only 1.0°. If problems with the background rates caused the variability, this effect would have affected both regions with the same probability due to different pointing offsets which were used for the observations. If such an effect existed on average both regions should be affected with the same probability. Also possible gradients in the background are unlikely to cause the observed variability, since in case such gradients caused the variable flux which was observed for HESS J1745-290 there would also be an effect for an object at 1° offset. Also taking the background only from the north or only from the south of the Galactic plane did not eliminate the observed variability. Therefore, effects induced by the background can be most likely excluded as systematic origin of the variability, since the G 0.9+0.1 fluxes do not fluctuate beyond noise level.
On the other hand, due to the different flux levels of both sources systematic
vari-ations of the rates of γ-like events by atmospheric irregularities cannot be entirely ex-cluded. However, the systematic influence of variations of the atmosphere on the inte-grated flux at H.E.S.S. is only at the 5% level [45]. Different systematic errors on the integrated flux will be discussed later in this section.
4.2.4.1.4 The Excess Variance The normalized excess variance is often used to characterize the variability of light curves of astronomical sources [14]. It is defined as
σN XV = 1
whereN stands for the number of flux measurements and ¯xis the unweighted arithmetic mean of the flux values of the light curve. The quantities xi and σ2i are the flux value and variance of a particular data point. In case a light curve shows variability beyond the noise level, the quantityσN XV is expected to be>0. The normalized excess variance is often given in population studies, for example, to compare different AGNs. It cannot be translated directly into a significance level for variability, since it also depends on parameters like the length of the light curve. In order to translate it into a significance, detailed simulations of the light curve under consideration are necessary. Nevertheless, it will be given here for completeness and as a further cross-check for the previous results, since obtaining a normalized excess variance which is <0 would contradict the variability which was revealed in theχ2 fit of the light curve with a constant. The excess variance was calculated for both the H.E.S.S. I and the H.E.S.S. II light curve separately again. For the H.E.S.S. I light curve an excess variance of 0.097 was obtained, while for the H.E.S.S. II light curve this quantity could be determined to be 0.063. As expected both values are>0 and furthermore of the same order like excess variances which were quoted for several low mass X-ray binaries in Ref. [82]. Although one has to be careful with such comparisons, the excess variance>0 which was found during this cross-check is consistent with the variability, which was reported for the χ2 fit.
4.2.4.1.5 A Systematic Error on the Flux Values In order to further investigate the robustness of the result, different systematic errors in percent have been added to the data points of the light curves. The systematic error was increased in steps of 5%
up to 15%. Figure 4.2.8 shows the development of the significance level for variability from the χ2 fit of the light curve with a constant as a function of the systematic error in percent.
The initial significance level for variability drops from 6.1σ to 2.9σ after applying a
Systematic Error in Percent
0 2 4 6 8 10 12 14 16
Significance on Variability
3 3.5 4 4.5 5 5.5
6 HESS J1745-290
level σ 5
Figure 4.2.8: The dependence of the significance level for variability as a function of the systematic error on the fluxes in percent.
15% systematic error, which is the systematic error which was given for the integrated flux of the Crab Nebula when comparing the fluxes between different nights [8]. The question is if this systematic error of 15% also correctly accounts for the systematic difference of the integrated fluxes from two different runs of the HESS J1745-290 light curve. There are several reasons for the full 15% error being in fact be too large for the
15% systematic error, which is the systematic error which was given for the integrated flux of the Crab Nebula when comparing the fluxes between different nights [8]. The question is if this systematic error of 15% also correctly accounts for the systematic difference of the integrated fluxes from two different runs of the HESS J1745-290 light curve. There are several reasons for the full 15% error being in fact be too large for the