Study of the Xray activity of Sagittarius A* during the 2011 XMMNewton campaign^{⋆}
^{1}
Observatoire Astronomique de Strasbourg, Université de Strasbourg, CNRS,
UMR 7550, 11 rue de l’Université,
67000
Strasbourg,
France
email:
enmanuelle.mossoux@astro.unistra.fr
^{2}
Nicolaus Copernicus Astronomical Center, ul. Bartycka
18, 00716
Warszawa,
Poland
Received: 25 July 2014
Accepted: 19 September 2014
Context. At the dynamical center of the Milky Way, there is the closest supermassive black hole: Sgr A*. Its nonflaring luminosity is several orders of magnitude lower than the Eddington luminosity, but flares can be observed in the infrared and Xrays. This flaring activity can help us to understand radiation mechanisms from Sgr A*.
Aims. Our aim is to investigate the Xray flaring activity of Sgr A* and to constrain the physical properties of the Xray flares and their origin.
Methods. In March and April 2011, we observed Sgr A* with XMMNewton with a total exposure of ≈226 ks in coordination with the 1.3 mm VeryLongBaseline Interferometry array. We performed timing analysis of the Xray emission from Sgr A* using a Bayesianblocks algorithm to detect Xray flares observed with XMMNewton. Furthermore, we computed Xray smoothed light curves observed in this campaign in order to have better accuracy on the position and the amplitude of the flares.
Results. We detected two Xray flares on March 30 and April 3, 2011, which for comparison have a peak detection level of 6.8 and 5.9σ in the XMMNewton/EPIC (pn+MOS1+MOS2) light curve in the 2−10 keV energy range with a 300 s bin. The former is characterized by two subflares: the first one is very short (~458 s) with a peak luminosity of L^{unabs}_{210~keV} ~9.4 × 10^{34} erg s^{1}, whereas the second one is longer (~1542 s) with a lower peak luminosity (L^{unabs}_{210~keV} ~6.8 × 10^{34} erg s^{1}). The comparison with the sample of Xray flares detected during the 2012 Chandra XVP campaign favors the hypothesis that the 2011 March 30 flare is a single flare rather than two distinct subflares. We model the light curve of this flare with the gravitational lensing of a simple hotspotlike structure, but we cannot satisfactorily reproduce the large decay of the light curve between the two subflares with this model. From magnetic energy heating during the rise phase of the first subflare and assuming an Xray photons production efficiency of 1 and a magnetic field of 100 G at 2 r_{g}, we derive an upper limit to the radial distance of the first subflare of 100^{+19}_{29} r_{g}. We use the decay phase of the first subflare to estimate a lower limit to the radial distance of 4 r_{g} from synchrotron cooling in the infrared.
Conclusions. The Xray emitting region of the first subflare is located at a radial position of 4−100^{+19}_{29} and has a corresponding radius of 1.8−2.87 ± 0.01 in r_{g} unit for a magnetic field of 100 G at 2 r_{g}.
Key words: Galaxy: center / Xrays: individuals: Sgr A* / radiation mechanisms: general
Appendices are available in electronic form at http://www.aanda.org
© ESO, 2014
1. Introduction
Our Galaxy hosts Sgr A* at its dynamical center. It is the closest supermassive black hole (SMBH) at a distance of about 8 kpc (Genzel et al. 2010; Falcke & Markoff 2013). Sgr A* has a mass, M_{BH}, of about 4 × 10^{6} M_{⊙}, which was determined thanks to the measurements of star motions (Schödel et al. 2002; Ghez et al. 2008; Gillessen et al. 2009). The Galactic center SMBH is usually in a steady state, emitting predominately at radio to submillimeter wavelengths. Its bolometric luminosity is about 10^{36} erg s^{1} (Yuan et al. 2003), which corresponds to ≈2 × 10^{9} L_{Edd} with L_{Edd} = 3.3 × 10^{4}(M_{BH}/M_{⊙}) L_{⊙}. To explain this low luminosity, researchers have developed various massaccretion flow models, such as the advectiondominated accretion flows (ADAF; Narayan et al. 1998) and jetdisk models like the ejection of magnetized plasma (Falcke et al. 1993).
Wang et al. (2013) have recently inferred the temperature and density profile of the Xray emitting gas around Sgr A* with the help of deep Chandra observations. They have shown that ≤1% of the gas initially captured by the SMBH at the Bondi radius reaches the innermost region around Sgr A*; i.e, ≥99% of the gas is ejected, which is consistent with the predictions of radiatively inefficient accretion flow (RIAF) models. Therefore, Sgr A* is the ideal astronomical target for investigating the physics of mass accretion and ejection onto SMBH in the regime of a low massaccretion rate, a state where they are supposed to spend most of their lifetime (Ho et al. 2008). This physical understanding could then be extended to the normal galaxies that dominate the population of galaxies in the local Universe.
The detections of flares from Sgr A* (first discovered in Xrays; Baganoff et al. 2001) have provided a valuable way to scrutinize accreting matter close to the event horizon. The Xray flare frequency is 1.1 (1.0−1.3) flare per day with L_{2−8 keV} ≥ 10^{34} erg s^{1} (Neilsen et al. 2013), though episodes of higher Xray flaring activity can also be observed (Porquet et al. 2008; Neilsen et al. 2013). The bulk of Xray flares detected so far have fainttomoderate amplitudes with factors of about 2 to 45 compared to the nonflaring luminosity (L_{2−8 keV} ≈ 3.6 × 10^{33} erg s^{1}; Baganoff et al. 2003; Neilsen et al. 2013), and three very bright flares (factors of 100−160 times the nonflaring luminosity) have been observed to share very similar spectral properties (Porquet et al. 2003, 2008; Nowak et al. 2012). The light curves of the Xray flares can exhibit deep drops with short duration indicating that the Xray emission comes from a region as compact as seven Schwarzschild radii (R_{S} ≡ 2GM_{BH}/c^{2} = 1.2 × 10^{12} cm for Sgr A*, i.e., ≈0.6 AU; Porquet et al. 2003).
When nearinfrared (NIR) and Xray flares are detected simultaneously, their light curves have similar shapes, and there is no apparent delay (<3 min) between the peaks of flare emission (e.g., YusefZadeh et al. 2006; DoddsEden et al. 2009; Eckart et al. 2012). The current interpretation is that both Xray/NIR flares come from a region close to the event horizon, while delayed submm (e.g., ≈100 min; Marrone et al. 2008) and mm peaks (up to 5 h; YusefZadeh et al. 2009) have been interpreted as the adiabatic cooling of an expanding relativistic plasma blob. While NIR flares are known to be due to synchrotron emission (Eisenhauer et al. 2005; Eckart et al. 2006), the Xray flare emission mechanism has not been settled yet, with arguments for synchrotron (DoddsEden et al. 2009; Barrière et al. 2014), inverse Compton (YusefZadeh et al. 2012), and synchrotron selfCompton (Eckart et al. 2008) models.
We report here the results of our Sgr A* observation campaign performed with XMMNewton from March 28 to April 5, 2011 in coordination with the 1.3 mm VeryLongBaseline Interferometry array (VLBI). In Sect. 2 we describe the XMMNewton observations and data processing. In Sect. 3 we present our timing analysis of Sgr A*. In Sect. 4 we describe the spectral analysis of the two flares from Sgr A* detected during this 2011 campaign. In Sect. 5 we compare these flares with those detected in the 2012 Chandra XVP campaign. We also try to model the first subflare with a simple hotspot model and estimate a lower and upper limit to the radial distance of this subflare. Finally, in Sect. 6 we summarize our main results.
2. XMMNewton observations and data processing
2.1. Observation setup
These Xray observations of Sgr A* with XMMNewton (AO8, 5 × 33 ks; PI: D. Porquet) were designed to perform the first simultaneous observational campaign in Xrays and at 1.3 mm with the VLBI (Doeleman et al. 2008), in order to constrain the location Xray flares. Five observing nights with the 1.3 mm VLBI were planned in 2011 between March 28 and April 5, using the weather forecast each day at noon for the final optimized scheduling (PI: S. Doeleman). The merged visibility window of the 1.3 mm VLBI array formed by the Atacama Pathfinder Experiment (APEX) in Chile, the Submillimeter Telescope (SMT) in Arizona, the Combined Array for Research in Millimeterwave Astronomy (CARMA) in California, and the Submillimeter Array (SMA) in Hawaii is 10:45−15:45 UT. Since Xray flare peaks appear to occur before submillimeter peak (Marrone et al. 2008) and since they can last up to three hours (Baganoff et al. 2001), our XMMNewton observations started about three hours before the VLBI visibility window. We observed Sgr A* with XMMNewton continuously from about 07:40 UT to about 16:00 universal time (UT), which is a duration of 30 ks. The XMMNewton visibility windows finally constrained the five following dates: 2011 March 28 and 30 and April 1, 3, and 5. The 1.3 mm VLBI observations were obtained on 2011 March 29 and 31 and April 1 (simultaneous with XMMNewton), 2, and 4; the results of these observations will be reported elsewhere. Two complementary Chandra observations were obtained to extend the Xray coverage on 2011 March 29 and March 31 from 10:29 UT to 15:29 UT (Cycle 12; PI: F. Baganoff), the former being simultaneous with VLBI. The results of these observations will be reported elsewhere.
The two XMMNewton/EPIC MOS cameras (Turner et al. 2001) and the XMMNewton/EPIC pn camera (Strüder et al. 2001) were operated in the full frame window mode with the medium filter. EPIC pn camera starts to observe after EPIC MOS cameras and stops before them. The effective starting and end times of each observation are reported in Table 1. These times are the time of the beginning and the stop of the observation with EPIC MOS cameras in the terrestrial time (TT) referential. For this observation, the relation between terrestrial time and UT is UT = TT−66.18s (NASA’s HEASARC Tool: xTime^{1}).
XMMNewton observation log for the Spring 2011 campaign.
2.2. Data processing
We observed Sgr A* five times with XMMNewton in early 2011 for a total effective exposure of ≈226 ks. We use the version 13.5 of the Science Analysis Software (SAS) package for the data reduction and analysis, with the latest release of the current calibration files (CCF; as of 04/04/2014). The MOS and pn event lists were produced using the SAS tasks emchain and epchain, respectively. The full detector light curves in the 2−10 keV energy range computed by these tasks reveal that the observation was only slightly affected by weak soft proton flares. The count rate of these soft protons was high only during the last four, three, one, and four hours of the 1^{st}, 2^{nd}, 3^{rd}, and 4^{th} observations, respectively.
We concentrate on analysis of the central point source, Sgr A*and, in particular, on the search for the variability of its Xray emission. To do this, we define the source+background region as a 10″radius disk around the VLBI radio position of Sgr A*: RA(J2000) , Dec(J2000) (Reid et al. 1999). We do not register the EPIC coordinates again because the absolute astrometry for the EPIC cameras is about (Guainazzi 2013). To create the light curves, we selected the events for MOS and pn with PATTERN≤ 12 and #XMMEA_SM, and PATTERN≤ 4 and FLAG==0, respectively. The contribution of the background proton flares was estimated using a ≈3′ × 3′ area with a low level of Xray extended emission, located on the same CCD at ≈4′ north of Sgr A*, where the Xray emission of point sources were subtracted. This data reduction is the same as in Porquet et al. (2008).
For each observation and detector, we first built the source+background (extracted from the 10″radius region) and the background (extracted from the 3′ × 3′ region) light curves in the 2−10 keV energy range with 300 s time bins. During this operation, we used the epiclccorr task to apply relative corrections to those light curves. The relative corrections specify the good time intervals (GTI) of the event list according to the corresponding CCD and compute the livetime, i.e., select the time inside each CCD frame where the events were collected effectively (no FIFO reset/overflow, minimum ionizing particles, or readouttime). Then, this task subtracts the background light curve (scaled to the same source extraction area) from the source+background light curve and scales up count rates and errors affected by the lost of exposure. Finally, the backgroundsubtracted light curves of the three detectors were summed to produce the EPIC light curves. Any missing count rate in a detector was inferred by the one observed by the other detectors using a scaling factor between them. To do this, we calculated the scaling factor between the detectors during a time period where all three cameras were turned on. The pn count rate is 1.31 times the sum of the MOS count rates.
3. Timing analysis
3.1. Bayesian blocks analysis
To identify the flaring and nonflaring levels under a certain probability using the unbinned event arrival time, we used the Bayesian blocks analysis proposed by Scargle (1998) and recently improved by Scargle et al. (2013a). The Bayesian blocks analysis of an event arrival time list from one of the EPIC cameras allowed us to segment the observing period with statistically different count rate levels and created a succession of constant count rate blocks. The time defining two successive blocks is called a change point. The count rate within each block is simply the number of events it contains divided by its effective exposure (livetime). The nonflaring and flaring levels are identified as the lowest and higher blocks, respectively. The duration of the flares are determined as the time range of the Bayesian block corresponding to the elevated count rate. This algorithm gives us the duration of the flaring and nonflaring levels with better accuracy than in a binned light curve since it uses the best temporal resolution available.
The number of change points is controlled by two input parameters: the false detection probability (p_{1}) and the prior estimate of the number of change points, ncp_prior. We use p_{1} = exp(−3.5) (Neilsen et al. 2013; Nowak et al. 2012), i.e., the probability that a found change point is a real change point is 1−exp(−3.5) = 96.8% and the probability that a flare (at least two change points) is a real flare is . We cannot use the geometric prior of Scargle et al. (2013a) since our data contain more events and our nonflaring level is lower than in the simulations used by Scargle et al. (2013a), see Appendix A for a detailed explanation.
Fig. 1 XMMNewton/EPIC (pn+MOS1+MOS2) light curves of Sgr A* in the 2−10 keV energy range obtained in Spring 2011. The time interval used to bin the light curve is 300 s. The Xray flares are labeled from 1 to 2. The horizontal lines below these labels indicate the flare durations. The nonflaring level of Sgr A* corresponds to only 10% of the nonflaring level of these light curves (Porquet et al. 2008). 

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Fig. 2 XMMNewton light curve of the 2011 March 30 flare from Sgr A* in the 2−10 keV energy range. Top panel: the XMMNewton/EPIC pn light curve binned on 100 s. The crosses are the data points of the light curve. The horizontal dashed lines represent the nonflaring level found by the Bayesianblocks algorithm. The vertical dashed lines show the start and stop of the Bayesian block. The solid line is the smoothed light curve. The gray curves are the errors associated with the smoothed light curve. Bottom panel: the XMMNewton/EPIC (pn+MOS1+MOS2) light curve binned on 100 s. The horizontal dashed line represents the nonflaring level calculated as the sum of the nonflaring 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 Bayesianblocks algorithm on pn camera. The solid line is the smoothed light curve, which is the sum of the smoothed light curve for each instrument (calculated on the same time range). The gray curves are the errors associated with the smoothed light curve. 

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Fig. 3 XMMNewton light curve of the 2011 April 3 flare from Sgr A* in the 2−10 keV energy range. Top panel: the XMMNewton/EPIC pn light curve binned on 100s. The crosses are the data points of the light curve. The horizontal dashed lines represent the nonflaring level found by the Bayesianblocks algorithm. The vertical dashed lines show the start and stop of the Bayesian block. The solid line is the smoothed light curve. The gray curve are the errors associated with the smoothed light curve. The time period during which the camera did not observe is shown with a light gray box. Bottom panel: the XMMNewton/EPIC (pn+MOS1+MOS2) light curve binned on 100 s. The horizontal dashed line represents the nonflaring level calculated as the sum of the nonflaring 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 Bayesianblocks algorithm on pn camera. The solid line is the smoothed light curve, which is the sum of the smoothed light curve for each instrument (calculated on the same time range). The gray curve are the errors associated with the smoothed light curve. 

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Characteristics of the Xray flares observed by XMMNewton/EPIC in 2011.
We used the EXPOSU## extension (## corresponding to the CCD number where the source extraction region is located) of the event list to compute the detector live time from the nominal TIMEDEL of the corresponding instrument, i.e., the integration time without the time of the shift of a CCD line to the readout node, which is about 0.0687022 and 2.59 s for pn and MOS, respectively. The Bayesianblock algorithm is used on the list of event of the source+background and of the background in which we selected the GTI (i.e., we reject the time where the camera did not observe). It allows us to correct the light curve source+background from the flaring background following the recipe of Scargle et al. (2013b). Indeed, thanks to Bayesian blocks, we know what the background count rate is and where the highbackground levels are. We can correct the source+background event list of any background contribution by applying a weight to each event, which is w = CR_{src + bkg}/ (CR_{src + bkg} + CR_{bkg}) with CR_{src + bkg} the count rate of the Bayesian blocks of the source+background event list and CR_{bkg} the count rate of the corresponding Bayesian block of the background that is surfacecorrected^{2}. Then, the Bayesianblocks algorithm is applied a second time to this corrected source+background event lists. This method is used on the three cameras separately.
3.2. Smoothed light curve
We compute a smoothed light curve by applying a density estimator (Silverman 1986; Feigelson & Babu 2012) on the unbinned event arrival times using GTI to suppress camera switchoff. The density estimator improves the characterization of light curve features, e.g., the amplitude and the time of a local maximum or minimum. The density is computed using quantreg in R package, which convolves the event arrival times with a smoothing kernel. We modify quantreg to use the Epanechnikov kernel, which is defined as for  x  ≤ 1 and K(x) = 0 for  x  > 1. We chose the Epanechnikov kernel since it has “good performance” (Feigelson & Babu 2012); moreover, it is defined on a finite support, which allows us to control any boundary effects. The density estimator can be expressed as (1)with h the width of the kernel window, N the number of count in the event list, t_{i} the arrival time of the event i and t the time at which we compute the smoothed light curve, w(t) the weight that corrects the density at the time t from the flaring background thanks to the Bayesianblocks algorithm (see above Sect. 3.1) and livetime is the live time in the time interval [t_{i} − h/ 2, t_{i} + h/ 2]. The time t is chose by the user. Here, we take an even time grid with point interval of 5 s^{3}. We choose a constant window width of the kernel h = 100 s. The smoothed count rate (CR) is obtained from the density by . The error of the smoothed light curve is assumed to be Poissonnian (= with n the number of count in the kernel window). Then, we insert observing gaps using GTIs and combine the light curves of the three instruments.
3.3. Results
The EPIC (pn+MOS1+MOS2) backgroundsubtracted light curves of Sgr A* in the 2−10 keV energy range, with a time bin interval of 300 s, are shown in Fig. 1. Our Bayesianblocks analysis of the event list for individual detectors shows that during the first exposure, no flares were detected and the activity of the source region is constant. The first flare (#1) was observed on 2011 March 30 and the last flare (#2) on 2011 April 3. The nonflaring level is determined as the count rate of the longest time interval of the Bayesian blocks of the nonflaring level, which allows very good accuracy on the count rate of the nonflaring level. On 2011 March 28, March 30, April 1, April 3, and April 5, the total nonflaring level was equal to 0.179 ± 0.003, 0,185 ± 0.004, 0.177 ± 0.003, 0.183 ± 0.004 count s^{1}, and 0.179 ± 0.003 count s^{1}, respectively. It is consistent with the one previously observed with XMMNewton (e.g., in 2007, Porquet et al. 2008). This nonflaring emission is a combination of emission coming from the complex of stars IRS 13, the candidate pulsar wind nebula G359.950.04, and a diffuse component, which all contribute 90% of this nonflaring level in the 2−10 keV energy range (Baganoff et al. 2003; Porquet et al. 2008) and the emission from Sgr A* which contribute only 10%. Figures 2 and 3 focus on the flare light curves obtained with EPIC (pn+MOS1+MOS2) and EPIC pn with a bin time interval of 100 s. The comparison with the EPIC MOS1 and MOS2 light curves can be found in Appendix B. We also show the Bayesian block corresponding to each camera with a dashed line. Table 2 gives the characteristics of these Xray flares.
The first flare has two components: a short (~458 s) and symmetrical subflare and a longer (~1542 s) and fainter symmetrical subflare. Between these two subflares, the smoothed light curve returns at 17.87 h and during less than 100 s to a lower level, which is consistent with the nonflaring state. The first flare is seen in EPIC MOS1 camera with a shift of ≈75 s of its maximum at the first peak but the double subflare configuration is not seen in the EPIC MOS2 camera. The amplitude of the flare in the smoothed light curve corresponds to 6.8σ (the standard deviation of the nonflaring level in the 300 s binned light curve) after subtracting the nonflaring level computed by the Bayesianblocks algorithm.
The second flare is seen by the Bayesianblocks algorithm in pn and EPIC MOS2 cameras but not in MOS1. This can be explained by the detection limit of the algorithm and the lower sensitivity of the MOS cameras (see details in Appendix B). The time start of this flare is lower than or equal to the start of the observation. In EPIC MOS2 camera we can see an enhancement of the countrate level after 8.25 h on April 3 but the Bayesian block algorithm detected also an enhancement at the beginning of the observation (before 8 h). Because of the time delay of observation start of EPIC pn camera, we caught with this camera only the end of this flare^{4}. The amplitude of the flare subtracted from the nonflaring level corresponds to 5.9σ.
We also computed the hardness ratio using the 2−4.4 keV and 4.4−10 keV energy bands for all observations, but we found no significant spectral change during the flare interval. The peak count rates of the first and second flares are three and eight times less than that of the bright flare reported in Porquet et al. (2003). The durations of theses flares are 1.4 and 1.8 times shorter than this bright flare.
4. Spectral analysis
We did a spectral analysis of the first flare. The extraction region is the same as the one we used to construct the light curves, i.e., a circle of 10″ radius centered on the Sgr A* radio position. The spectrum analysis was only done on the pn instrument since the flare in MOS1 and MOS2 has a number of counts that is too small to constrain the spectral properties. The Xray photons were selected with PATTERN≤ 4 and FLAG==0. The time interval of the flare was constrained by the results of the Bayesianblocks algorithm (see Table 2). The background time interval is composed of two subintervals: the first one began at the start of the March 30 observation (see Table 1) and ended 300 s before the beginning of the flare to avoid any bias. The second one started 300 s after the end of the flare and stopped at the end of the March 30 observation.
The spectrum, response matrices, and ancillary files were computed with the SAS task especget. We used ISIS version 1.6.2−27 (Houck & Denicola 2000; Houck 2002) to fit the spectrum with Xray emission models. The model that we used is an absorbed power law with dust scattering (dustscat; Predehl & Schmitt 1995). NuSTAR observations of Sgr A* confirm that this model is still a good description of the flare spectra above 10 keV (Barrière et al. 2014). We used TBnew for the absorption model, the interstellar medium abundances developed by Wilms et al. (2000), and the cross sections from Verner et al. (1996). These lower metal abundances and updated cross sections imply decreasing the column density (Nowak et al. 2012) input to the dustscat model (Predehl & Schmitt 1995) by a factor of 1.5 times. It uses the N_{H} vs. τ_{scatt} relation obtained with wabs (Anders & Ebihara 1982’s abundances, Morrison & McCammon 1983).
The results of the fit using 90% confidence level are hydrogen column density (N_{H}) of , photon index (Γ) of , absorbed flux between 2 and 8 keV of 2.5 × 10^{12} erg s^{1} cm^{2} and unabsorbed flux between 2 and 10 keV of . The extracted spectrum and best fit are shown in Fig. 4.
We can compare the spectral parameters of this flare with those of the two brightest flares detected with XMMNewton, which have the better constrained spectral parameters thanks to the high throughput and no pileup. The very bright flare of 2002 October 3 was fitted using the same modeling with Γ = 2.3 ± 0.3, , and (Porquet et al. 2003; Nowak et al. 2012). The bright flare of 2007 April 4 was also fitted using the same modeling with , and (Porquet et al. 2008; Nowak et al. 2012). In Fig. 5, the confidence contours of these two bright flares show that these N_{H} and Γ parameters are well constrained. However, those of the 2011 March 30 flare are not, since there are few events collected from this flare, which implies that the number of spectral bins with a minimum signaltonoise ratio of 4 is small. The photon index and hydrogen column of the flare of 2011 March 30 agree with those of the flare of 2007 April 4 and 2002 October 3 within the confidence levels for two parameters of 90% and 99%, respectively.
Fig. 4 XMMNewton/EPIC pn spectrum of the 2011 March 30 flare. The data are denoted by crosses. The vertical bars is the 1σ error in the count rate and horizontal bars show the spectral bin in energy. The events have been grouped with a minimum signaltonoise ratio of 4. Top: the result of the fit is shown by the continuous solid line. Bottom: the χ^{2} residual in units of σ. 

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The second flare on 2011 April 3 does not contain enough counts to constrain the spectral parameters. Its unabsorbed luminosity given in Table 2 is calculated with ISIS by fixing the photon index Γ to 2 and column density to N_{H} = 14.3 × 10^{22} cm^{2}, i.e., to the spectral values of the 2002 October 3 flare (Porquet et al. 2003; Nowak et al. 2012). Thus, the only free parameter is the unabsorbed flux, which is . The unabsorbed luminosity between 2 to 10 keV is for a 8 kpc distance.
5. Discussions
5.1. The 2011 March 30 flare vs. the 2012 Chandra XVP campaign flares
We compared the spectral properties of the 2011 March 30 flare with the ones reported by Neilsen et al. (2013) from the 2012 Chandra XVP campaign. In this paper, the spectral properties of all Chandra flares have been derived by assuming the spectral parameters of the brightest flares obtained by Nowak et al. (2012): Γ = 2 and N_{H} = 14.3 × 10^{22} cm^{2}. We use two physical quantities given in the Table 1 of Neilsen et al. (2013): the unabsorbed 2−10 keV luminosity and the duration of the flare. We also derived two other physical quantities that are independent of the instrumental characteristics: the unabsorbed 2−10 keV fluence in erg (the product of the unabsorbed 2−10 keV luminosity with the duration) and the unabsorbed 2−10 keV peak luminosity. To compute the peak luminosity of the Chandra flares, we first derived the mean count rate in each flare as the pileup corrected fluence in counts (see Eq. (1) Neilsen et al. 2013) divided by the flare duration, and then we computed the linear relation between the unabsorbed 2−10 keV luminosity and the mean count rate (higher the mean count rate, higher the luminosity). We obtained (CR/count s^{1}) with a correlation parameter r of 0.9997. Then, we applied this relation to the peak count rate given in Table 1 of Neilsen et al. (2013) to obtain the peak luminosity for each flare. The relations between these four physical quantities are shown in Fig. 6.
Fig. 5 Confidence contours for Sgr A* spectral parameters. Contours are the confidence levels of 68% (dotted line), 90% (solid line), and 99% (dashed line) for the two parameters in the graph. The three sets of confidence contours represent the 2011 March 30 flare (black lines and asterisk), the 2007 April 4 flare (light gray lines and X point), and the 2002 October 3 flare (dark gray lines and cross point). 

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Fig. 6 2011 March 30 flare vs. the 2012 Chandra XVP campaign flares. The Xray flares from the Chandra XVP campaign (Neilsen et al. 2013) are shown by crosses, the two brightest flares seen be XMMNewton are triangles, the 2011 March 30 flare is represented by a diamond, and the two subflares are shown with squares. 

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Since we used quantities that are independent of the instrument, we can compare flares observed with Chandra and XMMNewton. First, we place the two brightest flare seen by XMMNewton in the three diagrams. The unabsorbed 2−10 keV fluence, duration, and unabsorbed 2−10 keV luminosity are reported in Nowak et al. (2012). The unabsorbed 2−10 keV peak luminosity are computed as the ratio between the peak count rate and the mean count rate multiplied by the unabsorbed 2−10 keV luminosity in the flares (Porquet et al. 2008). We can see that these flares have high luminosities and fluences. They are thus in the upper righthand corner of the diagrams representing the unabsorbed 2−10 keV peak luminosity and the unabsorbed 2−10 keV fluence.
Characteristics of the 2011 March 3 flare and its two subflares assuming Γ = 2 and N_{H} = 14.3 × 10^{22} cm^{2}.
We also represent our first flare as a single flare (diamond) and as two distinct subflares (squares) defined as follows. The first subflare starts at the beginning of the Bayesian blocks corresponding to the 2011 March 30 flare and stops at the time corresponding to the minimum of the smoothed light curve between the two subflares. The beginning of the second subflare is the end of the first one, and its end corresponds to the end of the Bayesian block. The live time of the single flare (Table 3) is shorter than the flare duration reported in Table 2. The mean rate of the flare is given by the Bayesianblocks algorithm. The mean rate in each subflare is the number of counts in each subflare divided by their live time. To be consistent with Neilsen et al. (2013) for direct comparison purpose, the unabsorbed 2−10 keV luminosity is computed with the same spectral parameter (Γ = 2 and N_{H} = 14.3 × 10^{22} cm^{2}), which implies that the luminosity of the flare is slightly different than those computed with our present best fit spectral analysis (see Sect. 4). The derived quantities for the first flare (#1) and the first (#1.1) and second (#1.2) subflares are reported in Table 3.
In Fig. 6, we can see that the unabsorbed 2−10 keV luminosity of the total flare and the two subflares are nearly the same since they have more or less the same mean count rate, but the fluence of the first subflare is small compared to the second subflare owing to the shorter duration. Thus, the first subflare lies within the shortest and less energetic flares detected by Chandra, but the apparent lower detection limit of 400 s in the flare duration is probably due to the method used by Nowak et al. (2012) to identify flares in Chandra light curves. In fact, they use a Gaussian fit on the light curve binned with 300 s, which implies that they might missed flares whose duration is below 300 s.
We can see that, in all the diagrams, if we assume a single flare, it lies in the mean of the flares seen by Chandra and can then be considered as a genuine medium luminosity flare. Furthermore, if we consider the minimum waiting time between flares in the 2012 Chandra XVP campaign shown in Fig. 1 of Neilsen et al. (2013), we can see that the nearest flares are separated by ~3500 s. This waiting time can be considered as a lower limit for observing two distinct flares. The two subflare peaks of our first flare are separated with only 1000 s, which favors a single flare.
5.2. Gravitational lensing of a hotspotlike structure
We modeled the light curve of the 2011 March 30 flare with a single mechanism in order to explain the two subflares. Indeed, the very short (~458 s) first subflare and the second much longer (~1542 s) one peaking ~1000 s later but with lower amplitude can be the signature of a gravitational lensing of a hotspotlike structure. We used a hotspot model and a raytracing code to compute the observed intensity (Karas et al. 1992; Schnittman & Bertschinger 2004; Broderick & Loeb 2005; Hamaus et al. 2009; Dexter & Agol 2009).
5.2.1. The hotspot model
We call a hotspot a spherical, optically thin structure, orbiting around the black hole with Keplerian angular velocity. The sphere is initially assumed to be in solid rotation around the black hole. No shearing or expansion of the sphere is taken into account. Such a hotspot is thus only defined by its radius R and its orbital radius r in gravitational radius unit (r_{g} ≡ 0.5R_{S}). The black hole inclination i is assumed to be close to an edgeon view, i.e., i ≈ 90°. Its actual value is a parameter of the model. The emitted spectrum of the hotspot is assumed to follow a power law, , where α is a constant number, related to the photon index Γ through Γ = 2 − α. It is then straightforward to show that the observed intensity integrated over a range of frequency Δν_{obs} is (2)where g ≡ ν_{obs}/ν_{em} is the redshift factor.
Maps of ℐ^{obs} were computed by using the opensource raytracing code GYOTO^{5} (Vincent et al. 2011). We computed maps of 300 × 300 pixels over one orbital period with a time step of about δt ≈ 10 s, which is close to the time sampling of the smoothed light curve. The light curve is obtained by summing each of these maps over all pixels, which boils down to integrating over all solid angles, i.e., to computing a flux.
The light curve of a hotspot seen edgeon shows a typical doublebump feature (see Fig. 7). The primary maximum (t = 0 s) is due to the gravitational lensing of the light emitted by the hotspot when it is on the opposite side of the black hole with respect to the observer. The secondary maximum (t ≈ 1000 s) is due to the relativistic beaming effect: light emitted when the source is moving toward the observer is boosted.
5.2.2. Constraining the orbital radius of the hotspot
The orbital radius is easy to constrain because it is directly linked to the time interval between the two local maxima of the light curve, as illustrated in Fig. 7. The variation of this time interval as a function of the orbital radius r evolved like this: δt ≈ 860, 960, 1040, 1130, and 1230s for r/r_{g} = 10.5, 11, 11.5, 12, and 12.5. It is clear that if the hotspot model is correct, then r/r_{g} ≈ 11−12.
Fig. 7 Normalized hotspot light curves obtained for an orbital radius equal to the value of the best fit. The flux is in arbitrary units. Left: hotspot radius equal to the best fit value. Inclination i varies over all grid values. (The range of these parameters are defined in Sect 5.2.3.) The closer the inclination to 90°, the higher the ratio between the two local maxima. Right: inclination fixed at its best fit value and R varies over all grid values. The smaller R, the bigger the ratio between the two subflare peaks. The time interval between the two local maxima is the same for all curves in these two figures. 

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It is not obvious to constrain the remaining parameters (R and i) by a quick comparison to the observed data. Both of them have a strong impact on the flux ratio between the two local maxima, as well as on the flux ratio between the primary maximum and the local minimum between the two bumps.
5.2.3. Fitting the parameters of the hotspot model
Our hotspot model is defined by five parameters: the orbital radius r, the hotspot radius R, the black hole inclination i, the temporal additive shift dt, and the flux multiplicative scaling df. The two last parameters are defined according to the following. The smoothed light curve defines the zero of time: it is by definition the time of its primary maximum. Then, each theoretical light curve is first shifted so that its zero of time corresponds to its own primary maximum. The parameter dt allows the fitting of any time shift between the theoretical and the observed light curve. Each theoretical light curve is also scaled vertically. Each of them is first divided by the maximum of all fluxes computed by GYOTO (then all GYOTO fluxes are between 0 and 1). Each theoretical light curve is then again multiplied by the maximum value of the smoothed light curve, from which the nonflaring ground level was subtracted. (Then all GYOTO flux values are between 0 and M, the maximum of the smoothed light curve, in observed unit.) The multiplicative df fitting parameter is applied to these rescaled theoretical light curves.
The spin parameter has a low impact on the light curve, thus it is fixed to a = 0.99 (high spins lead to slightly smaller χ^{2} in the fit) and not fitted. The photon index is fixed to Γ = 2 (Porquet et al. 2003, 2008; Nowak et al. 2012; Barrière et al. 2014). Here we are interested in determining whether the hotspot model is viable or not, not in fitting in detail all the parameters.
The fitting is performed by determining the minimum of the following χ^{2} on a grid of parameters (3)where f_{Gyoto} is the theoretical light curve, f_{non−flaring} is the nonflaring level of the observed data (determined from the Bayesianblocks analysis), f_{smooth} the pn smoothed light curve, σ_{smooth} the error on the smoothed flux, and the sum is performed over a subset of the range of observed times taken into account in the smoothing procedure, with a time step of about 10 s. We use conservatively only the pn smoothed light curve since pn is the most sensitive instrument. The grid that we use for the three physical parameters is r ∈ [ 10.5,11,11.5,12,12.5 ], R/r_{g} ∈ [ 1.2,1.4,1.6,1.8,2,2.2 ] , and i ∈ [ 81.93,83.08,84.22,85.37,86.52,87.66 ] where radii are in units of GM/c^{2}, the inclination is in degrees, with i = 90° being an exact edgeon view (i.e., maximum lensing effect). For each set of parameter values, the theoretical light curve corresponding to (r,R,i) is read. It is rescaled as described above. Then the parameters dt and df are fitted using the lmfit routine of the Yorick software. The set of parameters that gives the smallest χ^{2} following this procedure is the bestfitting set. For the fitting, the theoretical light curve is fitted to the smoothed data, interpolating linearly to determine the theoretical value at the smoothed times.
Figure 8 shows the best fit that is found for the following values of the parameters: r = 12r_{g},R = 1.4r_{g},i = 86.5°,dt = 11.1 ± 4.0s,df = 1.40 ± 0.02. The 1σ error on the two last parameters being computed by the lmfit routine. The final reduced χ^{2} is 0.85.
Fig. 8 Modeling of the 2011 March 30 flare pn light curve with a rotating hotspot. Bestfitting theoretical light curve (dotdashed line) plotted over the smoothed light curve (solid line, with 1σ error in gray). The nonflaring level is given by the horizontal dashed line. The vertical axis is in observed units, horizontal axis is in seconds. The lower panel gives the residual in units of σ. 

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5.2.4. Viability of the hotspot model
The best fit illustrated in Fig. 8 clearly shows that one part of the smoothed data is not well fit by the hotspot model: the local minimum of the light curve, in between the two bumps at around 17^{h}52^{m}34^{s}. At this point, the observed data reach the nonflaring level while the model stays much higher, around 2.5σ distant. The remaining data is explained well by the model at the 1σ level. However, this 2.5σ inadequacy of the model at the local minimum is sufficient to reject the model. Indeed, a hotspotlike model will always produce a local minimum at a higher level than the nonflaring level. Indeed, this part of the light curve is associated to the part of the trajectory where the hotspot is moving from behind the black hole to the approaching side of the orbit. At this position, the relativistic beaming effect will always be significantly greater than at the receding side of the orbit, which corresponds to the minimum flux level.
To be quantitative, we compare the flux ratio between the lensing maximum and the nonflaring level and the lensing maximum and the local minimum flux level (in between the two bumps), for all light curves computed in our grid. The first ratio is always greater than ten, while the second ratio varies between 1.5 and 4. In conclusion, no set of parameters can give the same ratio for these two quantities.
5.2.5. Refining the hotspot model
One may wonder whether, by adding more physics to the hotspot model, this local minimum problem could be solved. To investigate this, we considered the two most natural ways of making our model more sophisticated: considering an elongated hotspot due to the shearing of the sphere by the differential Keplerian rotation and allowing the hotspot to vary in radius (R) along its trajectory. To model an elongated hotspot, we first computed the effect of elongation over a hotspot of initial radius R = 1.8 over the time elapsed between the triggering of the hotspot and the local minimum. The precise triggering time of the hotspot is not constrained, thus we assume the hotspot is created at the time that corresponds to the minimum of the theoretical light curve (thus at about −1500 s when t = 0 is set at the primary maximum). Under this assumption, the elapsed time between the creation of the hotspot and the local minimum is of ΔT ≈ 1800 s for the bestfitting values of parameters. This is equivalent to one third of the period. It is now straightforward to compute the difference of angular distance Δθ covered by the most distant (in terms of radial coordinate r) and least distant parts of a sphere with radius R = 1.8 whose center is at a radius r = 11.5 from the central black hole. Explicitly, (4)where r_{1} is the shortest distance to the black hole and r_{2} the largest (the spin is neglected here). For the bestfitting hotspot, Δθ ≈ π/ 3. This elongation has approximately multiplied the angular extension of the initially spherical bestfit hotspot by three. Thus, we model the elongated hotspot in a simple way by considering three spherical bestfit hotspots tangent one to the next, orbiting the same orbit. The intensity emitted by each of the spheres is divided by three with respect to the standard single hotspot case in order to allow a simple comparison. Figure 9 shows the light curve associated to this elongated hotspot. Here, the hotspot is always elongated and does not change shape as a function of time.
To determine the effect of volume changing on the light curve, we have modeled a swelling, single hotspot. The swelling hotspot is modeled by requiring that the initial radius of the hotspot is the bestfitting value, R_{0} = 1.8 and that it will increase linearly with time until it reaches 2 R_{0} at the time corresponding roughly to the local minimum observation. The emitted intensity is inversely proportional to the sphere volume. Figure 9 shows the light curve associated to this swelling hotspot.
Fig. 9 Comparison of the bestfit light curve (solid line) with the light curve of an elongated (dashed line) and swelling (dotted line) hotspot defined by the same parameters (see text for details). The flux is in arbitrary units. 

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Both toy models show that changing the shape of the hotspot will not solve the central problem of the model: the local minimum is always significantly higher than the nonflaring level. We cannot formally exclude that a more sophisticated model, such as a nonconstant density hotspot, or a trajectory not confined within the equatorial plane is able to fit our data. However, it is important to note that our hotspot model is ruled out precisely because the minimum of the flare light curve goes down to the quiescent level. Without adding some ad hoc new components to a simple hotspot model (like an obscuring component), it is clear that the flare light curve will always have a minimum above the quiescent level, since the hotspot will always be visible and make a nonzero contribution to the total intensity. As a consequence, we believe that fitting a hotspotlike model to our data would require some fine tuning using extra parameters, which would make the model less reliable.
5.3. Constraining the radial distance of the first 2011 March 30 subflare from magnetic energy heating and synchrotron cooling
We consider that the short duration of the rise phase of the first subflare places a limit to the size of the flaring region (e.g., DoddsEden et al. 2009). Following Barrière et al. (2014), we assume that the energy released during the flare is powered during the rise phase by the magnetic energy available inside the flaring region, which constrains the radial distance of the flare. By identifying the decay phase of the first subflare with synchrotron cooling, we derive a lower limit to the radial position of the first subflare.
5.3.1. Timescales of the first subflare
We define the start time of the rise phase as the time when the count rate of the smoothed light curve is too high to have been produced by the Poissonnian fluctuation of the nonflaring level at the 99.87% of confidence level (corresponding to Gaussian singlesided confidence level of 3σ). This threshold level is defined by CR_{0} = N/h, with h the width of the kernel window (h = 100 s) and N the lowest integer solution of the following equation: (5)with CDF the cumulative distribution function of the Poissonnian distribution, λ the nonflaring rate (i.e., 0.107 ± 0.001 pn counts s^{1}). We find CR_{0} = 0.23 pn counts s^{1} at t_{start} = 17h44m59s on 2011 March 30. The end time of the rise phase is the maximum of the smoothed light curve that is reached at t_{max} = 17h46m54s. Thus, the rise phase duration is Δt_{rise} = t_{max} − t_{start} = 115 s. The end time of the first subflare is the time of the minimum between the two subflares: t_{end} = 17^{h}52^{m}34^{s}. This leads to Δt_{flare} = t_{end} − t_{start} = 455 s.
The proper duration of any event around an SMBH is always longer than the observed duration due to time dilation in strong gravity field. Therefore, we compute the propertoobserved time ratio versus radial position (see Appendix C). Hereafter we use a dimensionless spin parameter of one.
5.3.2. Magnetic energy heating
We constrain the radius of the spherical flaring region by considering that the Alfven velocity cannot be higher than the speed of light (DoddsEden et al. 2009): R<cΔτ_{rise}, where Δτ_{rise} is the proper duration of the rise phase. This leads to the upper limit to the volume of the flaring region: . The magnetic energy contained inside this volume is with B = B_{1RS}2r_{g}/r the magnetic field vs. the radial distance r (see Barrière et al. 2014,and references therein).
We define η, the Xray photon production efficiency, as the ratio of the flare fluence in Xrays to the available magnetic energy. The flare fluence in Xrays is the product of the unabsorbed Xray luminosity with the duration of the first subflare (i.e., Δτ_{flare}). Indeed, we have to compute the fluence released during the whole first subflare since all the Xray emission from this event is powered by the magnetic heating of the emitting region. We compute this luminosity with the parameters that were fitted to the flare spectrum, i.e., N_{H} = 6.7 × 10^{22} cm^{2} and Γ = 1.5 (see first part of Sect. 4). The average luminosity of the first subflare is . As a result, . Therefore, the upper limit to the radial distance can be computed by the relation (6)If we assume a maximum efficiency (η = 1), the upper limit to the radial distance is (see Fig. 10). The corresponding radius of the flaring region at this distance is R = 2.87 ± 0.01 r_{g}.
Fig. 10 Determination of the radial distance of the flaring region. Top panel: magnetic energy vs. radial distance for a magnetic field of 100 G at 2 r_{g} and an Xray photon production efficiency and dimensionless spin parameter of 1. The solid line is the distribution of the magnetic energy (see left yaxis) vs. the radial distance. The dashed and dotted lines represent the central value of the Xray fluence and its errors with 90% confidence level, respectively. The vertical lines are the corresponding upper limits to the distance. The dasheddotted line represents the radius of the emitting region (see right yaxis). Bottom panel: Xray photons production efficiency vs. radial distance for the fluence and its upper and lower limit. The solid and dotted lines represent the efficiency for the central value of the fluence and its errors within 90% confidence level, respectively. 

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Fig. 11 Synchrotron cooling time vs. the radial distance. The solid line represents the proper duration of the decay phase. The dashed inclined line represents the synchrotron cooling time for infrared photons. The dotteddashed inclined line is the synchrotron cooling time for Xray photons. The corresponding vertical lines are the lower limit to the radial distance for each cooling timescale. 

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We can neglect any magnification of the observed luminosity compared to the proper luminosity at this radial distance. Indeed in the hotspot model, the combined effects of the beaming and the gravitational redshift on the proper luminosity are small at r = 100 r_{g} since the corresponding orbital period is ~1.5 days, which implies that any magnification has a long timescale and a very small amplitude (Broderick & Loeb 2005; Hamaus et al. 2009). In the jet geometry, the Doppler factor is small owing to the small inclination and the mild velocity of the Sgr A* jet; therefore, the beaming factor, varying as the square of the Doppler factor, is also small (Barrière et al. 2014).
5.3.3. Synchrotron cooling
The electrons that were accelerated by the release of the magnetic energy will cool by emitting synchrotron radiation with the following timescale: τ_{sync} = 8 × (B/ 30 G)^{− 3/2} × ^{(}ν/ 10^{14} Hz^{)}^{−1/2} min (DoddsEden et al. 2009). If the Xray photons at 10^{18} Hz are the primary source of synchrotron cooling, then s. From , we derive r > 114 r_{g}, which is not consistent with the previously derived upper limit. Therefore, if the Xrays are the primary source of synchrotron cooling in this subflare, sustained heating must also be present during the decay phase.
We know that Xray flares are always associated with NIR flares (e.g., DoddsEden et al. 2009), which have powerlaw spectra consistent with synchrotron process (Eisenhauer et al. 2005). Thus, we consider the synchrotron cooling time of NIR photons (ν = 10^{14} Hz)– s – which leads to r > 4 r_{g} with the flaring region outside the event horizon. The evolution of these synchrotron cooling times with the radial distance is shown in Fig. 11.
We conclude that in this subflare for η = 1 and B_{1RS} = 100 G. The corresponding radii of the flaring region at these distances are 1.8 r_{g}<R< 2.87 ± 0.01 r_{g}. The minimum distance of r > 1.9 r_{g} is required to have the flaring region well outside the event of horizon.
5.3.4. Comparison with previous works
The upper limit to the radial distance of the first subflare on 2011 March 30 is five times more than the one derived for the flare detected by NuSTAR on 2012 July 21 (Barrière et al. 2014). The latter was longer (1896 s) and about ~3.5 times more luminous (mean luminosity of 21 × 10^{34} erg s^{1}) than the former. Moreover, the 2012 July 21 NuSTAR flare was characterized by a plateau phase of ≈1700 s between the rise and decay phases of 100 s.
Barrière et al. (2014) assume that the radius of the emitting region is constant after the rise phase. But for a radial position lower than 20 r_{g} and B_{1RS} = 100 G, the synchrotron cooling time of NIR photons is lower than 2500 s, which implies that the heating process is still required after the rise phase to produce the observed plateau phase. Therefore, the radius of the emitting region of this NuSTAR flare is likely much larger than those computed by Barrière et al. (2014) from only the rise phase. If we remove the plateau phase of this NuSTAR flare, only the rise and decay phases remain, so, we can use the same method as for our first subflare.
We revise the upper limit to the radial distance of the 2012 July 21 NuSTAR flare to at least 64 r_{g}. Including the likely increase in the radius of the flaring region during the plateau phase due to sustained heating leads to an even higher value for the upper limit to the radial distance.
6. Summary
We have reported the data analysis of the XMMNewton 2011 campaign observation of Sgr A* (five observations with a total of exposure of ≈226 ks). We used the Bayesianblocks algorithm developed by Scargle (1998) and a density estimator with an Epanechnikov kernel to constrain the duration, the position, and the amplitude of the Xray flares with better accuracy. The Bayesianblocks algorithm uses the unbinned event arrival time on the EPIC cameras to identify the flaring and nonflaring period and their corresponding countrate levels. This analysis of the event’s arrival time increases the accuracy on the time of the beginning and the end of a flare in comparison with a detection above a given threshold of a binned light curve. The algorithm uses a Bayesian statistic to find the time when the countrate level is statistically different under a given probability. We worked with a false detection probability of exp(−3.5), which implies that the detected flare is a real flare with a probability of 99.9%. We corrected the contribution of the flaring background by applying twice this algorithm on the source and the background regions. We used a density estimator to improve the determination of the characteristics of the flares. The density estimator applies a convolution between the event list corrected from the GTI and a kernel defined on a finite support in order to control any boundary effects. Thanks to the Bayesianblocks algorithm, we could also correct the resulting smoothed light curves from the flaring background.
We observed two Xray flares during these observations. The former occurred on 2011 March 30 and the latter on 2011 April 03. For comparison, these flares have a peak detection level of 6.8 and 5.9σ in the XMMNewton/EPIC (pn+MOS1+MOS2) light curve in the 2−10 keV energy range with a 300 s bin. The first flare is composed of two subflares: a very shortduration (~458 s) one with a peak luminosity of and a longer (~1542 s) and less luminous one ( at the peak). The spectral analysis of this flare allowed us to derive these parameters: , , , and . These spectral parameters are consistent with those previously found by Porquet et al. (2003, 2008) and Nowak et al. (2012) but are not really constrained.
A comparison of the physical characteristics of this flare with those reported by Neilsen et al. (2013) from the 2012 Chandra XVP campaign shows that it lies in the mean of the Xray flares detected by Chandra, but the first subflare is one of the shortest and less luminous Xray flares. The distribution of the minimum waiting time between two successive flares in the Chandra XVP campaign favors the hypothesis of a single flare.
We modeled its two subflares with a single physical phenomenon using the gravitational lensing of a hotspotlike structure. However, the consistency of the flux level between the two subflare peaks with the nonflaring one led us to conclude that the light curve of this Xray flare cannot satisfactorily be reproduced by a gravitational lensing event.
We also constrained the radial position of the emitting region of the first 2011 March 30 subflare by assuming that the heating energy is provided by the magnetic field available in the spherical emitting region whose radius is determined by the duration of the rise phase of this first subflare. A comparison of the duration of the decay phase of this subflare and the synchrotron cooling timescale allowed us to determine a lower limit to the radial distance. We conclude that the Xray emitting region of the first subflare is located at a radial position of and has a corresponding radius of 1.8−2.87 ± 0.01 in r_{g} unit for a magnetic field of 100 G at 2 r_{g}.
The website of xTime is: http://heasarc.gsfc.nasa.gov/cgibin/Tools/xTime/xTime.pl
With this recipe we keep all the source+background events, sowe do not have to remove (arbitrarily) some individual events from the event list as proposed by Stelzer et al. (2007) to subtract the background events.
The position of local extrema can be easily computed with required accuracy directly from the first derivative of Eq. (1).
A coordinated nearinfrared observation was obtained with VLT/NACO on 2011 April 3 from 06:30 to 08:18 UT (ESO’s archive), which detected the rise of this flare (S. Gillessen 2011, priv. comm.), but lead to only 22 min of simultaneous observation with EPIC MOS before the flare peak that we observed in Xrays.
This code can be freely downloaded at the URL http://gyoto.obspm.fr
Acknowledgments
E.M. acknowledges the Université de Strasbourg for her IdEx PhD grant. The XMMNewton project is an ESA Science Mission with instruments and contributions directly funded by ESA Member States and the USA (NASA). This work has been financially supported by the Programme National Hautes Energies (PNHE). The research leading to these results received funding from the European Union Seventh Framework Program (FP7/2007−2013) under grant agreement N° 312789.
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Online material
Appendix A: Calibration of the ncp_prior relation
Fig. A.1 Simulations of point measurements (Poisson signal of average 0.1) to determine ncp_prior = −log (γ). Top: false positive fraction p_{1} 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 p_{1} (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 (p_{1}), we simulate 100 constant light curves with Poisson noise around a level of 0.1 count s^{1}, which is the typical nonflaring level measured by XMMNewton/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 p_{1}. 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 p_{1} values and the corresponding ncp_prior, we can create the graph presented in Fig. A.1. Then, we can take different values of p_{1} and report the relation between the count number and ncp_prior that satisfied p_{1}. An example with p_{1} = 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 p_{1} that we choose. By combining these relations, which relies p_{1}, 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
Fig. B.1 Detection level for different values of Gaussian amplitude and p_{1} = 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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Fig. B.2 Light curves of Sgr A* in the 2−10 keV energy range obtained during the flare on 2011 March 30 binned on 100 s. Top: the total XMMNewton/EPIC light curve. The horizontal dashed line represents the nonflaring level calculated as the sum of the nonflaring 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 Bayesianblocks 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 XMMNewton/EPIC pn light curve of Sgr A*. Third panel: the XMMNewton/EPIC MOS1 light curve of Sgr A*. The vertical dashed lines represent the beginning and the end of the flare calculated by the Bayesianblocks algorithm on MOS1 camera. Bottom panel: the XMMNewton/EPIC MOS2 light curve of Sgr A*. The vertical dashed lines represent the beginning and the end of the flare calculated by the Bayesianblocks algorithm on MOS2 camera. 

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Fig. B.3 Light curves of Sgr A* in the 2−10 keV energy range obtained during the flare on 2011 April 3 binned on 100s. Top: the total XMMNewton/EPIC light curve. The horizontal dashed line represents the nonflaring level calculated as the sum of the nonflaring 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 Bayesianblocks 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 XMMNewton/EPIC pn light curve of Sgr A*. The dark gray box is the time during which pn did not observe. Third panel: the XMMNewton/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 Bayesianblocks algorithm on MOS1 camera. Bottom panel: the XMMNewton/EPIC MOS2 light curve of Sgr A*. The vertical dashed lines represent the beginning and the end of the flare calculated by the Bayesianblocks 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 nonflaring level of these cameras. This difference in the nonflaring level between the two cameras implies a difference in the Poisson noise (the higher the nonflaring 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 Xray 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 nonflaring level. For each amplitude, we perform 100 simulations and compute the number of flare (two change points) found by the Bayesianblocks 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 XMMNewton/EPIC MOS and pn camera (the former has a nonflaring 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 XMMNewton/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 XMMNewton/EPIC pn and MOS1 but not in MOS2 even if a flare is detected by the Bayesianblocks algorithm. The flare on 2011 April 3 is not detected by the Bayesianblocks algorithm in XMMNewton/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 nonflaring level is higher than 0.07 count s^{1} with a probability of false detection equal to exp(−3.5), but in XMMNewton/EPIC MOS1, the flare amplitude is about 0.06 count s^{1}. Since XMMNewton/EPIC MOS1 and MOS2 have lower number counts than XMMNewton/EPIC pn because of the RGS, it is on XMMNewton/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 BoyerLindquist 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 = r^{2} + a^{2} cos^{2}θ, Δ = r^{2} − r + a^{2}, 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., r_{g} for a = 1.
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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All Tables
Characteristics of the 2011 March 3 flare and its two subflares assuming Γ = 2 and N_{H} = 14.3 × 10^{22} cm^{2}.
All Figures
Fig. 1 XMMNewton/EPIC (pn+MOS1+MOS2) light curves of Sgr A* in the 2−10 keV energy range obtained in Spring 2011. The time interval used to bin the light curve is 300 s. The Xray flares are labeled from 1 to 2. The horizontal lines below these labels indicate the flare durations. The nonflaring level of Sgr A* corresponds to only 10% of the nonflaring level of these light curves (Porquet et al. 2008). 

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In the text 
Fig. 2 XMMNewton light curve of the 2011 March 30 flare from Sgr A* in the 2−10 keV energy range. Top panel: the XMMNewton/EPIC pn light curve binned on 100 s. The crosses are the data points of the light curve. The horizontal dashed lines represent the nonflaring level found by the Bayesianblocks algorithm. The vertical dashed lines show the start and stop of the Bayesian block. The solid line is the smoothed light curve. The gray curves are the errors associated with the smoothed light curve. Bottom panel: the XMMNewton/EPIC (pn+MOS1+MOS2) light curve binned on 100 s. The horizontal dashed line represents the nonflaring level calculated as the sum of the nonflaring 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 Bayesianblocks algorithm on pn camera. The solid line is the smoothed light curve, which is the sum of the smoothed light curve for each instrument (calculated on the same time range). The gray curves are the errors associated with the smoothed light curve. 

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In the text 
Fig. 3 XMMNewton light curve of the 2011 April 3 flare from Sgr A* in the 2−10 keV energy range. Top panel: the XMMNewton/EPIC pn light curve binned on 100s. The crosses are the data points of the light curve. The horizontal dashed lines represent the nonflaring level found by the Bayesianblocks algorithm. The vertical dashed lines show the start and stop of the Bayesian block. The solid line is the smoothed light curve. The gray curve are the errors associated with the smoothed light curve. The time period during which the camera did not observe is shown with a light gray box. Bottom panel: the XMMNewton/EPIC (pn+MOS1+MOS2) light curve binned on 100 s. The horizontal dashed line represents the nonflaring level calculated as the sum of the nonflaring 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 Bayesianblocks algorithm on pn camera. The solid line is the smoothed light curve, which is the sum of the smoothed light curve for each instrument (calculated on the same time range). The gray curve are the errors associated with the smoothed light curve. 

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In the text 
Fig. 4 XMMNewton/EPIC pn spectrum of the 2011 March 30 flare. The data are denoted by crosses. The vertical bars is the 1σ error in the count rate and horizontal bars show the spectral bin in energy. The events have been grouped with a minimum signaltonoise ratio of 4. Top: the result of the fit is shown by the continuous solid line. Bottom: the χ^{2} residual in units of σ. 

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In the text 
Fig. 5 Confidence contours for Sgr A* spectral parameters. Contours are the confidence levels of 68% (dotted line), 90% (solid line), and 99% (dashed line) for the two parameters in the graph. The three sets of confidence contours represent the 2011 March 30 flare (black lines and asterisk), the 2007 April 4 flare (light gray lines and X point), and the 2002 October 3 flare (dark gray lines and cross point). 

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In the text 
Fig. 6 2011 March 30 flare vs. the 2012 Chandra XVP campaign flares. The Xray flares from the Chandra XVP campaign (Neilsen et al. 2013) are shown by crosses, the two brightest flares seen be XMMNewton are triangles, the 2011 March 30 flare is represented by a diamond, and the two subflares are shown with squares. 

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In the text 
Fig. 7 Normalized hotspot light curves obtained for an orbital radius equal to the value of the best fit. The flux is in arbitrary units. Left: hotspot radius equal to the best fit value. Inclination i varies over all grid values. (The range of these parameters are defined in Sect 5.2.3.) The closer the inclination to 90°, the higher the ratio between the two local maxima. Right: inclination fixed at its best fit value and R varies over all grid values. The smaller R, the bigger the ratio between the two subflare peaks. The time interval between the two local maxima is the same for all curves in these two figures. 

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In the text 
Fig. 8 Modeling of the 2011 March 30 flare pn light curve with a rotating hotspot. Bestfitting theoretical light curve (dotdashed line) plotted over the smoothed light curve (solid line, with 1σ error in gray). The nonflaring level is given by the horizontal dashed line. The vertical axis is in observed units, horizontal axis is in seconds. The lower panel gives the residual in units of σ. 

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In the text 
Fig. 9 Comparison of the bestfit light curve (solid line) with the light curve of an elongated (dashed line) and swelling (dotted line) hotspot defined by the same parameters (see text for details). The flux is in arbitrary units. 

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In the text 
Fig. 10 Determination of the radial distance of the flaring region. Top panel: magnetic energy vs. radial distance for a magnetic field of 100 G at 2 r_{g} and an Xray photon production efficiency and dimensionless spin parameter of 1. The solid line is the distribution of the magnetic energy (see left yaxis) vs. the radial distance. The dashed and dotted lines represent the central value of the Xray fluence and its errors with 90% confidence level, respectively. The vertical lines are the corresponding upper limits to the distance. The dasheddotted line represents the radius of the emitting region (see right yaxis). Bottom panel: Xray photons production efficiency vs. radial distance for the fluence and its upper and lower limit. The solid and dotted lines represent the efficiency for the central value of the fluence and its errors within 90% confidence level, respectively. 

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In the text 
Fig. 11 Synchrotron cooling time vs. the radial distance. The solid line represents the proper duration of the decay phase. The dashed inclined line represents the synchrotron cooling time for infrared photons. The dotteddashed inclined line is the synchrotron cooling time for Xray photons. The corresponding vertical lines are the lower limit to the radial distance for each cooling timescale. 

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In the text 
Fig. A.1 Simulations of point measurements (Poisson signal of average 0.1) to determine ncp_prior = −log (γ). Top: false positive fraction p_{1} 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 p_{1} (here: 0.05). The dashed line is the linear fit of the simulation points. 

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In the text 
Fig. B.1 Detection level for different values of Gaussian amplitude and p_{1} = 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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In the text 
Fig. B.2 Light curves of Sgr A* in the 2−10 keV energy range obtained during the flare on 2011 March 30 binned on 100 s. Top: the total XMMNewton/EPIC light curve. The horizontal dashed line represents the nonflaring level calculated as the sum of the nonflaring 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 Bayesianblocks 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 XMMNewton/EPIC pn light curve of Sgr A*. Third panel: the XMMNewton/EPIC MOS1 light curve of Sgr A*. The vertical dashed lines represent the beginning and the end of the flare calculated by the Bayesianblocks algorithm on MOS1 camera. Bottom panel: the XMMNewton/EPIC MOS2 light curve of Sgr A*. The vertical dashed lines represent the beginning and the end of the flare calculated by the Bayesianblocks algorithm on MOS2 camera. 

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In the text 
Fig. B.3 Light curves of Sgr A* in the 2−10 keV energy range obtained during the flare on 2011 April 3 binned on 100s. Top: the total XMMNewton/EPIC light curve. The horizontal dashed line represents the nonflaring level calculated as the sum of the nonflaring 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 Bayesianblocks 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 XMMNewton/EPIC pn light curve of Sgr A*. The dark gray box is the time during which pn did not observe. Third panel: the XMMNewton/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 Bayesianblocks algorithm on MOS1 camera. Bottom panel: the XMMNewton/EPIC MOS2 light curve of Sgr A*. The vertical dashed lines represent the beginning and the end of the flare calculated by the Bayesianblocks algorithm on MOS2 camera. 

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In the text 
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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In the text 