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This article has an erratum: [erratum]

Issue
A&A
Volume 573, January 2015
Article Number A46
Number of page(s) 15
Section Galactic structure, stellar clusters and populations
DOI https://doi.org/10.1051/0004-6361/201424682
Published online 15 December 2014

Online material

Appendix A: Calibration of the ncp_prior relation

thumbnail Fig. A.1

Simulations of point measurements (Poisson signal of average 0.1) to determine ncp_prior = −log (γ). Top: false positive fraction p1 vs. value of ncp_prior with separate curves for the values N = 1000,2000,3000,4000,5000, and 6000 (left to right). The points at which the rate becomes unacceptable (here 0.05; dashed line) determine the recommended values of ncp_prior shown as a function of N in the bottom panel. Bottom: calibration of ncp_prior as a function of the number of counts (N) for a value of p1 (here: 0.05). The dashed line is the linear fit of the simulation points.

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We cannot use the scaling relation given in Scargle et al. (2013a) for our data set because it has different statistical properties than the simulated data set used by Scargle et al. (2013a). First, our events are affected by Poissonian noise and not by Gaussian noise. Second, our event lists with about ~4000 counts is longer than the published simulation limited to 1024 counts. To calibrate the relation between ncp_prior (the prior of the number of block) and the false positive rate (p1), we simulate 100 constant light curves with Poisson noise around a level of 0.1 count s-1, which is the typical non-flaring level measured by XMM-Newton/EPIC pn during our observations. For each sequence of 100 simulations, we increase the ncp_prior value from 2 to 9 by a step of 0.5 and we compute the number of change points detected. The percentage of change points detected in 100 simulations determines the p1. We repeat this operation for different numbers of count N in the light curve (from 1000 counts to 6000 counts by step of 1000 counts). With the p1 values and the corresponding ncp_prior, we can create the graph presented in Fig. A.1. Then, we can take different values of p1 and report the relation between the count number and ncp_prior that satisfied p1. An example with p1 = 0.05 is given in the bottom graph of Fig. A.1. The dashed line is the linear fit of the curve. Thus, we have the same number of relations between N and ncp_prior as the number of value of p1 that we choose. By combining these relations, which relies p1, N, and ncp_prior, we find our calibration: (A.1)with N the number of events in a range of [1000:6000] counts. For N lower than 1000, the last term is lower than 0.01, which is negligible. For a probability of false detection equals exp(−3.5) and N = 4000, ncp_prior = 7.0099.

Appendix B: Detection rate vs. flare peak and duration

thumbnail Fig. B.1

Detection level for different values of Gaussian amplitude and p1 = exp(−3.5). The solid line corresponds to FWHM = 56.62 s, the dotted line corresponds to FWHM = 318.49 s, and the dashed line corresponds to FWHM = 1104 s.

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thumbnail Fig. B.2

Light curves of Sgr A* in the 210 keV energy range obtained during the flare on 2011 March 30 binned on 100 s. Top: the total XMM-Newton/EPIC light curve. The horizontal dashed line represents the non-flaring level calculated as the sum of the non-flaring level in each instrument found by the Bayesian blocks. The vertical dashed lines represent the beginning and the end of the flare calculated by the Bayesian-blocks algorithm on pn camera. The solid line is the smoothed light curve that is the sum of the smoothed light curve for each instrument (calculated on the same time range). The gray curve shows the errors associated with the smoothed light curve. In all panels, the time period during which the camera did not observe is shown by a light gray box. Second panel: the XMM-Newton/EPIC pn light curve of Sgr A*. Third panel: the XMM-Newton/EPIC MOS1 light curve of Sgr A*. The vertical dashed lines represent the beginning and the end of the flare calculated by the Bayesian-blocks algorithm on MOS1 camera. Bottom panel: the XMM-Newton/EPIC MOS2 light curve of Sgr A*. The vertical dashed lines represent the beginning and the end of the flare calculated by the Bayesian-blocks algorithm on MOS2 camera.

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thumbnail Fig. B.3

Light curves of Sgr A* in the 210 keV energy range obtained during the flare on 2011 April 3 binned on 100s. Top: the total XMM-Newton/EPIC light curve. The horizontal dashed line represents the non-flaring level calculated as the sum of the non-flaring level in each instrument found by the Bayesian blocks. The vertical dashed lines represent the beginning and the end of the flare calculated by the Bayesian-blocks algorithm on pn camera. The solid line is the smoothed light curve that is the sum of the smoothed light curve for each instrument (calculated on the same time range). The gray curve shows the errors associated with the smoothed light curve. In all panels, the time period during which the camera did not observe is shown by a light gray box delimited by vertical solid lines. Second panel: the XMM-Newton/EPIC pn light curve of Sgr A*. The dark gray box is the time during which pn did not observe. Third panel: the XMM-Newton/EPIC MOS1 light curve of Sgr A*. The light gray vertical line shows the time during which MOS1 did not observe. The vertical dashed lines represent the beginning and the end of the flare calculated by the Bayesian-blocks algorithm on MOS1 camera. Bottom panel: the XMM-Newton/EPIC MOS2 light curve of Sgr A*. The vertical dashed lines represent the beginning and the end of the flare calculated by the Bayesian-blocks algorithm on MOS2 camera.

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To evaluate our detection level, we simulate light curves with a Poisson signal of average 0.1 count s-1 for EPIC pn and 0.04 count s-1 for EPIC MOS corresponding to the non-flaring level of these cameras. This difference in the non-flaring level between the two cameras implies a difference in the Poisson noise (the higher the non-flaring level, the higher the Poisson noise), hence in the detection rate. On these constant light curves, we add a Gaussian with a FWHM equal to 1104 s, 318.49 s, and 56.62 s, which correspond to the maximum, the median, and the minimum, respectively, of the FWHM of the X-ray flares from Sgr A* detected by Chandra and reported by Neilsen et al. (2013). We vary the amplitude of the Gaussian between 0 and 0.17 count s-1 above the non-flaring level. For each amplitude, we perform 100 simulations and compute the number of flare (two change points) found by the Bayesian-blocks algorithm for a false positive rate equal to exp(−3.5). The results are shown in Fig. B.1. We can see that the higher the amplitude and the FWHM of the flare, the higher the detection rate. We can also see that the main difference between the detection rate in the XMM-Newton/EPIC MOS and pn camera (the former has a non-flaring level that is two times lower than in pn) is that the small flares with large FWHM are more detected in MOS than in pn.

Figures B.2 and B.3 show the flare light curves obtained with of XMM-Newton/EPIC observed on 2011 March 30 and April 3. We can see that the first and second subflares on 2011 March 30 are distinguishable on XMM-Newton/EPIC pn and MOS1 but not in MOS2 even if a flare is detected by the Bayesian-blocks algorithm. The flare on 2011 April 3 is not detected by the Bayesian-blocks algorithm in XMM-Newton/EPIC MOS1. This is because the algorithm allows us to find a flare whose FWHM ≈ 900 s in EPIC MOS camera with a probability of 95% if its amplitude above the non-flaring level is higher than 0.07 count s-1 with a probability of false detection equal to exp(−3.5), but in XMM-Newton/EPIC MOS1, the flare amplitude is about 0.06 count s-1. Since XMM-Newton/EPIC MOS1 and MOS2 have lower number counts than XMM-Newton/EPIC pn because of the RGS, it is on XMM-Newton/EPIC pn that the flare will have higher amplitude and thus higher accuracy on the determination of the beginning and end of the flare.

Appendix C: Time dilatation around a Kerr black hole

We use the Kerr metric in Boyer-Lindquist coordinates: (C.1)with τ the proper time, t the observed time, r the radial distance in gravitational radius, a the dimensionless spin parameter, Σ = r2 + a2 cos2θ, Δ = r2r + a2, and θ = 0 defining the spin

axis (Bardeen et al. 1972). For a direct circular orbit in the equatorial plane, we have , , and (Bardeen et al. 1972). Thus, the relation between the proper time and the observed time is (C.2)Figure C.1 shows the time dilatation as a function of the radial distance plotted from the innermost boundary of the circular orbit, i.e., rg for a = 1.

thumbnail Fig. C.1

Ratio between the proper time and the observed time close to a Kerr black hole with a dimensionless spin parameter of 1.

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© ESO, 2014

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