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Issue
A&A
Volume 532, August 2011
Article Number A26
Number of page(s) 43
Section Astrophysical processes
DOI https://doi.org/10.1051/0004-6361/201016134
Published online 14 July 2011

© ESO, 2011

1. Introduction

Stellar dynamics and strongly variable emission allow us to firmly associate Sagittarius A* (SgrA*) at the center of the Milky Way with a 4 × 106 M⊙ supermassive black hole (SMBH; see Eckart & Genzel 1996; Genzel et al. 1997, 2000; Ghez et al. 1998, 2000, 2003, 2005a, 2008; Eckart et al. 2002; Schödel et al. 2002, 2003, 2009; Eisenhauer et al. 2003, 2005; Gillessen et al. 2009).

Recent observations at wavelengths from the radio to the X-ray domain give detailed insight into the physical emission mechanisms at work in SgrA*, such as synchrotron, SSC, and bremsstrahlung emission (e.g. Baganoff et al. 2001, 2002, 2003; Eckart et al. 2003, 2004, 2006a,b, 2008a,b, 2009; Porquet et al. 2003, 2008; Goldwurm et al. 2003; Genzel et al. 2003; Ghez et al. 2004b,a; Eisenhauer et al. 2005; Bélanger et al. 2006; Hornstein et al. 2007; Yusef-Zadeh et al. 2006a,b, 2007, 2008, 2009; Marrone et al. 2008; Dodds-Eden et al. 2009; Sabha et al. 2010). The luminosity associated with SgrA* is of the order of 10-9 to 10-10 times lower than the Eddington luminosity LEdd and many orders of magnitudes below that of SMBHs in active galactic nuclei (AGN) with comparable masses.

In a similar way to results derived for individual flare events, the NIR flare emission has observed spectral indices Fν(∝να)\hbox{$\left(F_{\nu} \propto \nu^{\alpha}\right)$} of α ~ −0.6  ±  0.2 (Ghez et al. 2005a,b; Hornstein et al. 2007) or even steeper (Eisenhauer et al. 2005; Gillessen et al. 2006). However, the determination of the spectral index is complicated by the correction of the flux contribution of the surrounding blue stars to the emission extracted for SgrA*. Here we present a new method to calculate the expectation value of the H/Ks-band spectral index for the SgrA* flare emission from the statistics of bright flares in the H- and Ks-band. This method is based on the comparison of the histograms of flare number versus flare flux. In Sect. 2, we describe the observations and data reduction, and in Sect. 3 present details of the spectral index calculation. Results and conclusions are summarized in Sects. 4 and 5.

2. Observations and data reduction

The NIR observations were carried out with the NIR camera CONICA and the adaptive optics (AO) module NAOS (briefly “NACO”) at the ESO VLT unit telescope 4 (YEPUN) on Paranal, Chile. We restricted our analysis to all Ks-band (2.18 ± 0.35)   μm data taken on observations between 13 June 2003 and 18 May 20091 and all H-band (1.66 ± 0.33)   μm data taken between 29 August 2002 and 22 June 2008. The VLT data sets used in this investigation were all reduced in an identical way to ensure a homogeneous data reduction quality. A detailed analysis of the Ks-band light curves will be given in a forthcoming paper by Witzel et al. (see also Dodds-Eden et al. 2011). Here we concentrate on the H-band data that are essential to understand the short wavelength NIR spectra of SgrA*. We also include an H-band ((1.63 ± 0.30)   μm) light curve taken with NIRC at the W. M. Keck telescope on Mauna Kea, Hawaii, USA. Detailed information about the observations and publications used can be taken from Table A.1. Since the H-band central filter wavelengths of the different instruments agree with each other to within 10% of their bandwidths, no transformation between them was applied.

A total of 312 out of 45 available VLT observation blocks3 in H-band from August 2002 until June 2008 were used for the present analysis. We discarded 14 data sets because of their low quality (strongly variable or insufficient sky transmission, bad seeing, or insufficient AO correction), missing sky or flat exposures, or having observation lengths significantly below 40 min. All H-band data cover a total observing time of  ~34 h. Additional statistical data on the H-band observations can be taken from Table 1. The same criteria were used for VLT Ks-band observations, covering a total of  ~100 h.

Table 1

Statistical information on the NACO H-band flares.

For all observations with the VLT, the infrared wavefront sensor of NAOS was used to lock the AO loop on the NIR bright super-giant IRS 7, located about 5.6′′ north of SgrA* (Ks-band magnitude  ~6.5, H-band magnitude  ~8.9 (e.g. Rafelski et al. 2007; Ott et al. 1999; Blum et al. 1996). The images were taken in dither mode. The images of each observation were corrected for sky contributions by subtracting the median of the respective stack of dithered exposures of a dark cloud located about 400′′ and 713′′ west of SgrA*. Furthermore, they were flat-fielded and corrected for dead or bad pixels. These steps were performed with the DPUSER software for astronomical image analysis (T. Ott, MPE; see also Eckart & Duhoux 1990). The PSFs were extracted from these images with StarFinder (Diolaiti et al. 2000) and deconvolved with a Lucy-Richardson (Lucy 1974) algorithm (LR). The beam was subsequently restored with a Gaussian of a FWHM near the diffraction limit. These data reduction and deconvolution steps were also taken for polarimetric data (Zamaninasab et al. 2010; Witzel et al. 2011, in prep.). In these cases, flux densities of observations made with the Wollaston prism were obtained by summing the two orthogonal channels. Therefore a total flux for each retarder position can be derived.

The flux densities of the sources were measured by aperture photometry. These apertures were chosen to be small, 2.5 pixels for SgrA* corresponding to 67.7 mas for S27 optics), to minimize any flux contamination by nearby stars. With better seeing, the AO correction improved and the S13 optics was used, enabling us to apply smaller apertures (2.5 pixels corresponding to 33 mas) and to minimize any contamination. The calibration and reference stars were measured with an aperture of 4 pixels in both optics. Appropriate corrections were applied to compensate for different aperture sizes (e.g. in the case of background contributions). A correction for extinction was applied, using AK = 2.8 and AH = 4.3 for K- and H-band (Rieke & Lebofsky 1985; Scoville et al. 2003) i.e. AH/AK = 1.536. Within the uncertainties, this is close to the values (i.e. AH/AKs = 1.598) used by Hornstein et al. (2007), which we later use as a comparison reference for the NIR spectral index of SgrA*. More recent values of the extinction along the line of sight towards SgrA* of AKs = 2.54  ±  0.12 and AH = 4.48  ±  0.13 (i.e. AH/AKs = 1.764) (Schödel et al. 2010) will result in weaker 2.2 μm intrinsic dereddened flux densities and on average bluer spectral indices.

The flux density calibration was carried out with the known Ks- and H-band flux densities of IRS16C, IRS16NE, and IRS21 by Blum et al. (1996), (see also Schödel et al. 2010). Subsequently precise relative photometry for SgrA* was performed using up to 14 sources within 1.5′′ of SgrA* as secondary calibrators (S67, S27, S26, S96, S6, S7, S8, S35, S51, S83, S86, S30, S100, S98; Gillessen et al. 2010). This results in a flux density of  ~18  ±  1 mJy in the H-band of the high velocity secondary reference stars S10 and  ~22  ±  1 mJy in Ks-band of S2 (see Ghez et al. 2005b; Eckart et al. 2008b). The measurement uncertainties for SgrA* were obtained from the standard deviation in the flux density of S10 in H-band and S2 in Ks-band. Images in which the secondary reference stars S10 and S2 deviate by more than 1.5σ from their mean were discarded. The background flux in the immediate vicinity of SgrA* was identified with the average of the measurements at six random locations in a field centered about 0.6′′ west of SgrA* that is free of obvious stellar sources. We estimated the systematic error introduced by both the uncertainty in the zero point and extinction to be about  ~10%.

thumbnail Fig. 1

Low state flux density vs. PSF FWHM for VLT observations in H-band (left panel) and Ks-band (right panel).

3. Calculation of the spectral index

3.1. Flare definition

The flux density variations of Sgr A* can be explained using either a disk or jet model (see e.g. discussion in Eckart et al. 2006a,b, 2008a), or they could be seen as a consequence of an underlying physical process that can be mathematically described as red-noise with β = 1 (P(ν)  ∝  ν−β), where P denotes the power spectral density, ν is the frequency, and β is the power spectral index (Mauerhan et al. 2005; Eckart et al. 2008a; Do et al. 2009; Zamaninasab et al. 2010). However, Zamaninasab et al. (2010) and Witzel et al. (2011) have demonstrated that highly polarized flare events are statistically significant compared to the randomly polarized red-noise. The brightest events in these variations can be considered as flares. In our present analysis, we investigate the distribution of the peak flare flux densities only. This ensures that the influence by any flux density contamination of the stellar background towards SgrA* is minimized. However, before the properties of these flare events can be examined, one has to establish a definition that assigns an event within the light curve to the group of flares. This definition should include preferably all events, but exclude random statistical fluctuations in the light curve that may be due to the measuring process. A definition based on the rising/decaying flank of an event seems appropriate. To ensure that relatively weak events are counted as flares, a minimal deviation of SgrA*’s light curve of 4σ within a time no longer than 15 min was chosen, here σ denotes the standard deviation of the reference star. A second, more restrictive definition was set up to account for clear, strong events only, with flux density excursions of at least 5σ within a time shorter than 30 min. It is emphasized that almost all of the 5σ flares are also covered by the 4σ definition. Furthermore, the flux density of a data point, potentially marking the beginning of a flare, has to be higher than or equal to the background flux density. If no rising flank is available, the decaying flank has to meet the selection criteria. A flare is completed when the flux density drops to the initial level at the beginning of the flare or if the end of the observing run terminates the light curve. The 4σ criterion is quite sensitive to weak events that occur in phases of low flux density at the position of SgrA*. In the following, we refer to this phase as the quiescent phase (see also Eckart et al. 2004). These variations are considered as valid events since the quiescent phase can be regarded as a sequence of frequent low-amplitude flares (Eckart et al. 2006b,c). We found 28 (12)\hbox{$\left(12\right)$} events in the H-band and 101 (63) in the Ks-band fulfilling the 4σ (5σ) criterion. Further statistical data is displayed in Table 1, where the flare frequency corresponds to the number of flares during all H-band observations normalized to 24 h. The flare duration is defined by the median FWZP measured by the time difference between the data point, at which the light curve leaves the low flux level, and the one, at which it returns to it. If we follow the definition of flare duration by Eckart et al. (2006a), defining periods of increased activity (when the flux density exceeds the average flux plus its standard deviation (5.8 ± 2.2) mJy) with a common duration of 100 min, we derive 1.6  ±  0.4 periods per day.

3.2. Measured flux density during the quiescent phase

The origin of the quiescent state mentioned before has not yet been revealed. Eckart et al. (2004) proposed to consider it as a sequence of frequent low amplitude flares but nearby, predominantly blue stars may contribute up to 35% of the flux density at the position of SgrA* (Sabha et al. 2010). Both mechanisms possibly have to be considered as being relevant. One could expect the flux contamination from nearby stellar sources to depend on the seeing conditions, because with decaying seeing the AO correction becomes less effective and the FWHM of the PSF, including its seeing foot, becomes broader. In this case, the flux density contribution from neighboring sources at the position of SgrA* will increase and small imperfections in the PSF estimate will alienate the flux density of SgrA* retrieved from the LR-algorithm. Consequently the low flux density phase, if it is dominated by the flux from nearby stars, should correlate with the seeing estimate for the FWHM of the PSF. To investigate this effect in more detail, we monitor the width of the PSF of stars close to SgrA* for each data set through its median FWHM value and also estimate the low flux density. Figure 1 show that there is no correlation between the two quantities. A similar result was obtained by Gillessen et al. (2006), who did not find a correlation between the spectral index and the seeing and therefore excluded there being any contamination caused by seeing (i.e. the width of the PSF). This leads to the conclusion that the variable intrinsic flux density contribution of SgrA* to the flux density measured at its position is also dominant during the low flux density phase. In this case, a constant flux contribution must not be subtracted from the flux density measured during the quiescent phase of SgrA*. The exact amount of a possible constant contribution of SgrA* to the quiescent state of a flare is unknown. Dodds-Eden et al. (2011), Sabha et al. (2010), Morris et al. (2007), and Hornstein et al. (2007) stated that SgrA* may exhibit a quiescent phase with a flux density at or just below 2 mJy in Ks-band. The following analysis consequently is performed without any flux density subtraction.

3.3. Deriving the spectral index

thumbnail Fig. 2

The methodology used in this work to derive the spectral index.

We now describe the method used to derive the spectral index statistically. The complete methodology is depicted in Fig. 2. With the definitions presented in Sect. 3.1, we can construct flux density histograms of SgrA* in the K- and H-band (see Figs. 3 and 4). The median flux density for flares that fulfill the 4σ criterion is (3.61 ± 1.62) mJy and (6.03 ± 1.85) mJy for the H- and Ks-band, respectively. Flares fulfilling the 5σ criterion have median flux densities of (5.94 ± 0.92) mJy in the H-band and (6.72 ± 1.95) mJy in the Ks-band. The uncertainties are calculated as the median absolute differences from the median flux density value.

thumbnail Fig. 3

Flux densities and number of occurrence of 4σH-band (left panel) and Ks-band (right panel) flares.

thumbnail Fig. 4

Flux densities and number of occurrence of 5σH-band (left panel) and Ks-band (right panel) flares.

We expect the H- and the Ks-band flares to be due to the same broad-band emission mechanism that can be described by a power-law distribution. Therefore the shape of the flux density histograms should be identical. Observations indicate that the spectral index is predominantly negative and consequently the H-band counterparts of weaker Ks-band flares cannot be detected since they fall below the H-band detection limit. For this reason, the Ks-band flare was matched to the H-band distribution by systematically reducing the number of Ks-band flares in the following way. We assume several intrinsic spectral indices αlimit = {0,−0.5,−0.6,−1,−1.5,−2,−2.5,−3,−3.5} (Fν ∝ ν+α). For each value of αlimit, we create a ten bin histogram with identical bin sizes and then determine the H-band flare “detection limit” as the lower border of the FWHM interval of the H-band flare flux histogram. In combination with αlimit we then determine the corresponding weakest detectable Ks-band flux density. All Ks-band flares below this introduced limit are then discarded. Following this criterion has three advantages:

  • 1.

    we concentrate on the brighter peak flare fluxes that give a morereliable measurement of the spectral index;

  • 2.

    we minimize the influence of measurement uncertainties in the peak flux densities of the weakest Ks-band and corresponding H-band flares; and

  • 3.

    we ensure that the spectral index determination is only done using flares that might statistically have been detected in both bands.

To derive an expectation value of the H-Ks-band NIR spectral index, not only the median values and their median deviations are of interest but also the shapes of the peak flare flux density histograms. To take the shape into account, we normalize the total number of Ks-band flares to the total number of H-band flares. We then shift the H-band flux densities to higher values by multiplying them with a shift factor. The shift factor SF)(\hbox{$\left(\textit{SF}\right)$} changes the H-band flux density by F(Hnew)=SF·F(Horiginal),\begin{eqnarray} F\left(H_{\rm new}\right) = \textit{SF} \cdot F\left(H_{\rm original}\right), \end{eqnarray}(1)where FHoriginal()\hbox{$F\left(H_{\rm original}\right)$} denotes the median H-band flux density of the original distribution and FHnew()\hbox{$F\left(H_{\rm new}\right)$} the flux density after shifting FHoriginal()\hbox{$F\left(H_{\rm original}\right)$}. We then determine how well this modified, normalized H-band distribution matches the shape of the Ks-band histogram. The Ks-band histogram is also arranged using ten bins with equidistant bin borders. In the ideal case of perfectly matching H- and Ks-band number distributions, FHnew()\hbox{$F\left(H_{\rm new}\right)$} will be centered close to the median Ks-band flux density.

To compare the normalized and scaled histograms independently of binning effects, we allow the Ks-band bin width to change. As for SF, a resize factor (RF)\hbox{$\left(RF\right)$} changes the bin widths of the Ks-band distribution. If Bi is the original upper border of bin i, Bi,new is the one after resizing, and i = (1...10) is the bin number, then the resize algorithm implies that Bi,new=RF·Bi.\begin{eqnarray} B_{i,{\rm new}} = RF \cdot B_i. \end{eqnarray}(2)In this algorithm, the deviations between the H- and the Ks-band distributions are calculated for shift factors 1 ≤ SF ≤ SFmax in steps of 0.001, were SFmax is the maximum shift factor to be taken into account calculated as SFmax=median(F(K))+median(ΔF(K))median(F(Horiginal))−median(ΔF(Horiginal)),\begin{eqnarray} \textit{SF}_{\rm max} = \frac{{\rm median}(F(K)) + {\rm median}(\Delta F(K))}{{\rm median}(F(H_{\rm original})) - {\rm median}(\Delta F(H_{\rm original}))}, \end{eqnarray}(3)where ΔF denotes the deviation of the peak flux density of a flare for the median peak flux from all flares (median(F)). The value for RF is altered for each SF in steps of 0.001 to a maximum of RFmax=median(Δ(F(K))median(F(K))·\begin{eqnarray} RF_{\rm max} = \frac{{\rm median}(\Delta(F(K))}{{\rm median}(F(K))}\cdot \end{eqnarray}(4)The standard deviation between the new H-band and the Ks-band distribution gives a measure of the uncertainty in the match. It is calculated by picking the number of H-band flares (NH) and Ks-band (NK) flares in bin x and calculating (NHx − NKx)2. The sum of deviations of all bins in a histogram forms the total deviation. We therefore search for the minimum deviation in the SF − RF-plane to find the closest match. For example, the standard deviation for the cases where αlimit equals αflux (see Sect. 4, first paragraph) is 6.939 for 4σ flares and 0.443 for 5σ flares.

The new “median” flux density of the H-band distribution is now recovered by dividing the median flux density of the Ks-band (FK)()\hbox{$\left(F\left(K\right)\right)$} distribution by the SF. The spectral index is calculated as αflux=log(F(H)F(K))/log(ν(H)ν(K))·\begin{eqnarray} \alpha_{\rm flux} = \log \left(\frac{F\left(H\right)}{F\left(K\right)}\right)/\log \left(\frac{\nu\left(H\right)}{\nu\left(K\right)}\right)\cdot \end{eqnarray}(5)To determine the uncertainty in the relative shift between the two distributions, we assumed that the positive peak flux density values of the flare emission follow a log-normal distribution. We then calculated the logarithms of the fluxes to establish their logarithmic distributions, rebin them, over-resolve each bin by a factor of nine, and perform a cross-correlation between the distributions. To obtain the uncertainty in the relative shift between the two distributions, we fit a Gaussian to the central component of the cross-correlation result (see lower part of Figs. 5 and 6). The standard deviation in this Gaussian then determines the uncertainty and is denoted as Σ (to avoid confusion with the excursion criterion in Sect. 3.1). The uncertainty in α is now obtained from the spectral indices calculated for both FK)(+10Σ\hbox{$F\left(K\right) + 10^{\Sigma}$} and FK)(−10Σ\hbox{$F\left(K\right) - 10^{\Sigma}$}.

4. Results

The most reliable result for a spectral index between the H- and Ks-band is achieved if the assumed spectral index, which takes the H-band detection limit into account and modifies the lower end of the H-band distribution (see previous section), equals the spectral index calculated from the median flux densities, i.e. αlimit = αflux. For a lower value of αlimit, the value of αflux will be systematically too small because the lower end of the K-band distribution will be over-corrected, resulting in too high a median K-band flux density. For a higher value of αlimit, the value of αflux will be severely overestimated because an increasing number of weak flare events will be left in the K-band distribution that cannot lead to H-band flares above the detection limit. This will lead to too low a median K-band flux density.

The closest correspondence is obtained for 4σ flares with αlimit = −2.5 and αflux=−2.52±0.791.01\hbox{$\alpha_{\rm flux} = -2.52 \pm ^{1.01}_{0.79}$}. For 5σ flares, the uncertainties are even smaller with αlimit = −0.5 and αflux=−0.50±0.630.76\hbox{$\alpha_{\rm flux} = -0.50 \pm ^{0.76}_{0.63}$}.

As an example we show in the upper part of Fig. 5 the logarithmic and re-binned diagram of the closest match of H- (black) and Ks-band (red) distributions after shifting and resizing the 4σ flare distributions. The shift factor is 1.989 and the resize factor is 1.250. The number of Ks-band flares was reduced by assuming a spectral index of αlimit = −2.5 for the H-band flux density, which is equivalent to the distribution maximum minus FWHM/2, so this H-band flux is taken as the flare detection limit. In the lower part of Fig. 5 we present the cross-correlation of the distributions. The logarithmic and re-binned results for the 5σ flare distributions are shown in Fig. 6. For these distributions, the derived shift factor is 1.145, and the resize factor is 1.065. A spectral index of αlimit = −0.5 was assumed. The data calculated for all introduced αlimit is shown in Tables 2 and 3.

thumbnail Fig. 5

Upper panel: logarithmic and re-binned diagram of the closest match of the H- (black) to the Ks-band (red) distributions after shifting and resizing the 4σ flare distributions (see text for explanation). Lower panel: cross-correlation of both distributions.

thumbnail Fig. 6

Same as in Fig. 5, but for the 5σ flare distributions.

Table 2

Details on data used and derived for the calculation of the spectral index αflux for 4σ flares.

Table 3

Details of the data used and derived for the calculation of the spectral index αflux for 5σ flares.

The equality of αlimit and αflux is represented in Fig. 7 by the bisecting line (green solid) of the angle defined by the axes. The black (red) solid line is the regression line of the data points for the group of 4σ (5σ) flares. The dotted lines show their respective standard deviation. These deviations depend on the flux density weight of the removed Ks-band data points relative to the assumed common shape of the Ks- and H-band flare flux histograms used to calculate αflux. Thus, these deviations are not independent of αlimit and the uncertainties are underestimated values. We therefore form an uncertainty corridor by the parallel displacement of the regression lines such that they represent the median trend of the positive and negative excursions described by the error bars that stem from the cross-correlation. The intersections of the bisecting line with the regression lines are marked with blue squares and have spectral indices of αflux,4σ=−2.66-0.83+1.03\hbox{$\alpha_{\rm flux,4\sigma} = -2.66^{+1.03}_{-0.83}$} and αflux,5σ=−0.66-0.58+0.63\hbox{$\alpha_{\rm flux,5\sigma} = -0.66^{+0.63}_{-0.58}$}. We drew the median line through both the positive and the negative excursions corresponding to the uncertainties in the cross-correlation, i.e. we illustrated the uncertainties in the uncertainties. Compared to the earlier given values of αflux, these are now interpolated values, independent of the discretely sampled αlimit values. The interpolated values are our more accurate ones. We derived their corresponding Ks-band flux densities F as the weighted average of the Ks-band fluxes determined from the neighboring αlimit. The flux uncertainties were calculated the same way. This results in F(αflux,4σ) = (6.47 ± 1.56) mJy and F(αflux,5σ) = (7.87 ± 1.47) mJy for the 4σ and 5σ flares, respectively. The effect of systematic errors (see Sect. 2) on the spectral index is well within the error bars. Data for both flare criteria can be taken from Tables 2 and 3.

thumbnail Fig. 7

αlimit vs. αflux. Black asterisks (red diamonds): values for 4σ (5σ) flares. Green solid line: bisecting line of the axes’ angle. Black (red) solid line: regression line of data points for the set of 4σ (5σ) flares. The dotted lines are their respective standard deviation. The dashed lines form the uncertainty corridor, as described in the text. The blue squares mark the positions of the intersections of the regression lines with the bisecting line.

The derived values including some of those taken from the literature are compared in Fig. 8. Here the red star with dashed error bars marks the spectral index αflux,4σ=−2.66±0.831.03\hbox{$\alpha_{\rm flux,4\sigma} = -2.66\pm^{1.03}_{0.83}$} for the amount of flares complying with the 4σ criterion. The data point agrees with the values of Krabbe et al. (2006) (blue star) within its uncertainties. The red star with solid error bars marks the spectral index for the set of flares having an excursion steeper than 5σ: αflux,5σ=−0.66±0.580.63\hbox{$\alpha_{\rm flux,5\sigma} = -0.66\pm^{0.63}_{0.58}$}. The value agrees very well with those found by Ghez et al. (2005a). Within its uncertainties, the spectral index αflux derived from the 5σ flares confirms the previously obtained values of α = −0.5  ±  0.3 by Ghez et al. (2005a), α = −1.2  ±  0.4 by Hornstein et al. (2007), as well as the values for bright flares from Gillessen et al. (2006). For clarity the latter two are shown in Fig. 9 rather than Fig. 8. This supports the assumption that either pure synchrotron radiation or a mixed contribution of synchrotron and synchrotron self-Compton (Eckart et al. 2006a) are the underlying emission mechanisms. Pure synchrotron radiation has an expected spectral index of −0.7 (e.g. Moffet 1975).

The H-Ks-spectral index for our set of 4σ flares agrees with the value of −2.6  ±  0.9 found by Krabbe et al. (2006). The spectral index for the 4σ flares may also be regarded as being consistent with the flatter of the three otherwise very steep spectral index values of  ~−4  ±  1 for faint flares reported by Eisenhauer et al. (2005). Hence, our results suggest that the spectral index might depend on the flare flux density, because our 4σ flare definition is more dominated by weak flares and variations in the low state flux than the flares obeying the 5σ definition.

thumbnail Fig. 8

Our derived spectral index values including data taken from the literature (see text).

5. Discussion and conclusions

The uncertainty in our spectral index determination for 5σ flare events can almost exclude the existence of blue flares (with positive spectral indices) and we consider them as unlikely. Our procedure is not biased against blue flares, as blue flares would populate the high flux tail of the H-band distribution, which is not modified. In the case of thermal bremsstrahlung, a blue flux density contribution can only occur if a sufficient amount of gas with temperatures  ≪108 K) is mixed into the central accretion flow or disk. This scenario appears to be unlikely in the case of SgrA* (see discussions in Yuan et al. 2003; Nayakshin & Sunyaev 2003; Page et al. 2004; Sabha et al. 2010). Blue synchrotron emission must come from the optically thick part of the spectrum (see discussion in Eckart et al. 2009). Synchrotron components with such a high degree of compactness have not yet been reported in the context of SgrA*.

The FWHM values of the H- and Ks-band flare histograms are similar and the expectation value of the NIR H-Ks-band spectral index equals the spectral indices derived from spectroscopic and multicolor data of individual flare events (see references below). Both suggest that the uncertainties in the estimates for individual flares depend more on the statistical variations in the flux densities in the H- and Ks-bands rather than intrinsic variations in the spectral index. This strongly supports the previous finding that – at least for the bright flares – the NIR emission is produced by pure optically thin synchrotron radiation.

The quiescent level has possibly a different relativistic electron distribution that is responsible for the synchrotron emission, e.g. a variable exponential cutoff in the energy of the electron population may also lead to variable and red spectral indices (Eckart et al. 2006a). The steep spectral indices observed for SgrA* in the NIR wavelength domain may be due to synchrotron losses, i.e. they may reflect the presence of a cutoff in the relativistic electron distribution (e.g. Eckart et al. 2006a; Liu et al. 2006). This will result in a modulation of the otherwise intrinsically flat spectra with an exponential cutoff proportional to exp [− (ν/ν0)0.5]  (see e.g. Bregman 1985; and Bogdan & Schlickeiser 1985) and a cutoff frequency of ν0 between the infrared and sub-mm wavelength domain. In several extragalactic jets, these cutoffs have also been observed to be relevant for the NIR emission (3C 293: Floyd et al. 2006; M 87: Perlman et al. 2001; Perlman & Wilson 2005; 3C 273: Jester et al. 2001, 2005). In these extragalactic sources the cutoff frequency at which the synchrotron losses become dominant is νjet,break = 4 × 1013 Hz (corresponding to 7.5 μm wavelength). This is remarkably similar to what may be required in the case of SgrA*, for which an exponential cutoff in the NIR/MIR wavelength range of 4−20 μm is suggested by Eckart et al. (2006a). For SgrA*, this cutoff can quite naturally explain the observed NIR spectral indices reported by several authors (Eisenhauer et al. 2005; Ghez et al. 2005a,b; Hornstein et al. 2007; Gillessen et al. 2006; Krabbe et al. 2006), despite the low upper flux density limits in the MIR (Schödel et al. 2007).

Table 4

K-band spectral indices for pre-flare, dim, and bright states of SgrA*.

We note that there is a number of flare events that begin before the start or that stop after the end of the light curve. These flares will result in an overestimate of flares with lower peak fluxes. However, we find that the percentage number of these events is, within the uncertainties, the same in the Ks- and H-band: in the H-band 25% of the 4σ/50% of the 5σ flares and in the Ks-band 27% of the 4σ/43% of the 5σ flares are affected. These flares will therefore alter the peak flare flux histograms in the same way, such that this effect will not have any influence on the result.

In Fig. 9, we show the relation between the H-Ks-band spectral index and the Ks-band flux density calculated for a power law with an exponential cutoff as described above. The thick solid black curves represent different model calculations that cover the expected spectral indices for the synchrotron power law and a flare strength at 1 THz that corresponds well to the degree of variability observed at high frequencies (e.g. Mauerhan et al. 2005). Model α is based on a power law spectrum with a spectral index of α = −0.4 and a flux density of 0.071 Jy at 1 THz. Curves with higher flux densities would look similar. They would, however, level out to the same value of α at lower Ks-band fluxes. For model β, the corresponding values are α = −0.8 and 0.5 Jy, for model γ they are α = −0.6 and 0.76 Jy, and for model δ they are α = −0.4 and 1.4 Jy. Above 8.5 mJy, model β, and above a Ks-band flux density of 12 mJy, models α, γ, and δ have 8.6 μm flux densities that are above the 22 mJy flux density limit (Schödel et al. 2007) at which SgrA* has not yet been detected.

In addition, we show in Fig. 9 light blue rectangular boxes that cover the flux density and NIR spectral index range obtained from the broad-band NIR spectroscopy of Gillessen et al. (2006) (their table in Sect. 4 of the paper) and from the H-/K′-imaging of Hornstein et al. (2007) (their right graph in Fig. 5 of the paper). The values are shown in Table 4.

thumbnail Fig. 9

The H-Ks-band spectral index versus the de-reddened Ks-band magnitude. The individual data points are described and labeled in Fig. 8. The black dots are the logarithms of the cutoff frequencies and the black solid lines are theoretical curves α to δ explained in the text. These curves are parameterized and labeled with the logarithm of the cutoff frequency. The light blue boxes represent different flux density states using the combined data by Gillessen et al. (2006) and Hornstein et al. (2007) as described in Table 4.

The data from Hornstein et al. (2007) were treated with a background subtraction method that mostly resembles the off-state subtraction described by Gillessen et al. (2006), although, the data points were originally not attributed to different activity states. On the basis of Gillessen et al. (2006) three states of activity were defined, which can be attributed to flux densities F with F < 2 mJy, 2 ≤ F ≤ 5.5 mJy, and F > 5.5 mJy for the pre-flare, the dim state and the bright state, respectively. These values were used by ourselves to dedicate each data point from Hornstein et al. (2007) to an activity state of SgrA*. From these values, based on both publications including their 2σ uncertainty, the median values of their upper and lower limits for each activity state were derived. These median values span areas in the α − F-plane (rectangular boxes in Fig. 9) that represent the possible combinations of spectral index and flux density for each activity level.

The flux density limit between the pre-flare and the dim state placed at the Ks-band flux density of 2 mJy seems reasonable. This value is currently discussed as the upper limit or actual flux density of a possible low- or quiescent-state of SgrA* (Do et al. 2009; Sabha et al. 2010; Dodds-Eden et al. 2011). Figure 9 shows that within the model basically all observed spectral indices measured at their corresponding Ks-band flux densities can be explained. For the NIR bright states, all intrinsically weaker flares with flux densities of below about half a Jansky at 1 THz and a cutoff frequency for the synchrotron losses at or above about 1014 Hz (3 μm wavelength) are responsible. There is the general tendency that the fainter Ks-band flares with very steep H-K-band spectra must also be comparatively brighter at cutoff frequencies below 1014 Hz. In addition, there is the possibility that a disputed low- or quiescent state may influence the NIR spectral index measurements at the lowest flux densities. An overcorrection of the SgrA* flux especially in the H-band may lead to steeper spectral indices. This is especially a danger for flux densities at or below 2 mJy. The data by Krabbe et al. (2006) who inferred an additional background calibration step actually falls below the measurements of Ghez et al. (2005a,b), Hornstein et al. (2007), and this work.

Finally we also note that the spectral index based on the statistics of the peak values of bright flares equals the spectral indices derived for the bright phases of time-resolved flares (Hornstein et al. 2007; Gillessen et al. 2006). The latter measurements indicate that the spectral index may change on timescales of the individual flare length. These changes could occur if the cutoff frequency ν0 at which high frequency synchrotron losses become important varies during a single flare as a function of time. If the flare is correlated with an increase in ν0, one would expect steeper spectra at the beginning and towards the end of the flare. A more detailed investigation of the infrared spectral index of SgrA*, especially in the low flux density state, requires a higher angular resolution and sensitivity to discriminate the infrared counterpart of SgrA* from the neighboring stars. This will be possible in the near future using NIR interferometers (VLTI, the Keck interferometer or interferometry with the LBT) or the planned large aperture telescopes such as the E-ELT.

Online material

Appendix A: Observation in the H-band

Table A.1

Details of the analyzed observation runs.

Appendix B: H-band lightcurves used in this work

In the following, the H-band lightcurves used are displayed (see Table A.1). In general in the upper panel of each figure the lower black datapoints mark the background measurements, the upper black curve shows the reference star S10 (if another reference star is used it is explicitly mentioned in the caption), and the green curve represents the flux density development of SgrA*. The lower panel shows the observational conditions by the FWHM of the 2-dimensional PSF (red in the direction of right ascension, purple in the direction of declination).

thumbnail Fig. B.1

Observation from 19 March 2003.

thumbnail Fig. B.2

Observation from 9 May 2003.

thumbnail Fig. B.3

Observation from 14 June 2003.

thumbnail Fig. B.4

Observation from 15 June 2003.

thumbnail Fig. B.5

Observation from 16 June 2003.

thumbnail Fig. B.6

Observation from 21 July 2003.

thumbnail Fig. B.7

Observation from 5 September 2003.

thumbnail Fig. B.8

Observation from 6 September 2003.

thumbnail Fig. B.9

Observation from 29 April 2004.

thumbnail Fig. B.10

Observation from 11 June 2004.

thumbnail Fig. B.11

Observation from 12 June 2004.

thumbnail Fig. B.12

Observation from 6 July 2004.

thumbnail Fig. B.13

Observation from 8 July 2004.

thumbnail Fig. B.14

Observation from 29 July 2004.

thumbnail Fig. B.15

Observation from 16 May 2005.

thumbnail Fig. B.16

Observation from 29 April 2006.

thumbnail Fig. B.17

Observation from 29 April 2006.

thumbnail Fig. B.18

Observation from 31 May 2006.

thumbnail Fig. B.19

Observation from 13 June 2006. S2 is used as reference star for this lightcurve.

thumbnail Fig. B.20

Observation from 29 June 2006.

thumbnail Fig. B.21

Observation from 24 July 2006.

thumbnail Fig. B.22

Observation from 27 July 2006.

thumbnail Fig. B.23

Observation from 28 August 2006.

thumbnail Fig. B.24

Observation from 16 September 2006.

thumbnail Fig. B.25

Observation from 2 October 2006.

thumbnail Fig. B.26

Observation from 15 October 2006.

thumbnail Fig. B.27

Observation from 17 March 2007.

thumbnail Fig. B.28

Observation from 4 April 2007.

thumbnail Fig. B.29

Observation from 19 July 2007.

thumbnail Fig. B.30

Observation from 21 July 2007.

thumbnail Fig. B.31

Observation from 21 June 2008.

thumbnail Fig. B.32

Observation from 22 June 2008.

1

For references see Witzel et al. (in prep.).

2

Based on observations made with ESO Telescope UT4 at Paranal Observatories under program IDs 70.B-0649(B), 71.B-0077(A), 71.B-0077(C), 71.B-0077(D), 71.B-0078(A), 073.B-0084(A), 073.B-0085(A), 073.B-0085(D), 073.B-0775(A), 077.B-0014(A), 077.B-0014(B), 077.B-0014(C), 077.B-0014(E) 077.B-0014(F), 078.B-0136(A), 078.B-0136(B), 179.B-0261(A), 179.B-0261(D), 179.B-0261(T), 273.B-5023(C).

3

Observations separated by at least 30 min are regarded as independent observation blocks.

Acknowledgments

Part of this work was supported by COST Action MP0905 Black Hole in a violent Universe. We are grateful to all members of the NAOS/CONICA and the ESO PARANAL team. During this work, M. Zamaninasab was a member of the International Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the MPIfR and the Universities of Bonn and Cologne. R. Schödel acknowledges support by the Ramón y Cajal programme by the Ministerio de Ciencia e Innovación of the government of Spain. Macarena García Marín is supported by the German federal department for education and research (BMBF) under the project numbers 50OS502 and 50OS0801. We thank the referee for helpful and constructive comments.

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All Tables

Table 1

Statistical information on the NACO H-band flares.

In the text
Table 2

Details on data used and derived for the calculation of the spectral index αflux for 4σ flares.

In the text
Table 3

Details of the data used and derived for the calculation of the spectral index αflux for 5σ flares.

In the text
Table 4

K-band spectral indices for pre-flare, dim, and bright states of SgrA*.

In the text
Table A.1

Details of the analyzed observation runs.

In the text

All Figures

thumbnail Fig. 1

Low state flux density vs. PSF FWHM for VLT observations in H-band (left panel) and Ks-band (right panel).
In the text
thumbnail Fig. 2

The methodology used in this work to derive the spectral index.
In the text
thumbnail Fig. 3

Flux densities and number of occurrence of 4σH-band (left panel) and Ks-band (right panel) flares.
In the text
thumbnail Fig. 4

Flux densities and number of occurrence of 5σH-band (left panel) and Ks-band (right panel) flares.
In the text
thumbnail Fig. 5

Upper panel: logarithmic and re-binned diagram of the closest match of the H- (black) to the Ks-band (red) distributions after shifting and resizing the 4σ flare distributions (see text for explanation). Lower panel: cross-correlation of both distributions.
In the text
thumbnail Fig. 6

Same as in Fig. 5, but for the 5σ flare distributions.
In the text
thumbnail Fig. 7

αlimit vs. αflux. Black asterisks (red diamonds): values for 4σ (5σ) flares. Green solid line: bisecting line of the axes’ angle. Black (red) solid line: regression line of data points for the set of 4σ (5σ) flares. The dotted lines are their respective standard deviation. The dashed lines form the uncertainty corridor, as described in the text. The blue squares mark the positions of the intersections of the regression lines with the bisecting line.
In the text
thumbnail Fig. 8

Our derived spectral index values including data taken from the literature (see text).
In the text
thumbnail Fig. 9

The H-Ks-band spectral index versus the de-reddened Ks-band magnitude. The individual data points are described and labeled in Fig. 8. The black dots are the logarithms of the cutoff frequencies and the black solid lines are theoretical curves α to δ explained in the text. These curves are parameterized and labeled with the logarithm of the cutoff frequency. The light blue boxes represent different flux density states using the combined data by Gillessen et al. (2006) and Hornstein et al. (2007) as described in Table 4.
In the text
thumbnail Fig. B.1

Observation from 19 March 2003.
In the text
thumbnail Fig. B.2

Observation from 9 May 2003.
In the text
thumbnail Fig. B.3

Observation from 14 June 2003.
In the text
thumbnail Fig. B.4

Observation from 15 June 2003.
In the text
thumbnail Fig. B.5

Observation from 16 June 2003.
In the text
thumbnail Fig. B.6

Observation from 21 July 2003.
In the text
thumbnail Fig. B.7

Observation from 5 September 2003.
In the text
thumbnail Fig. B.8

Observation from 6 September 2003.
In the text
thumbnail Fig. B.9

Observation from 29 April 2004.
In the text
thumbnail Fig. B.10

Observation from 11 June 2004.
In the text
thumbnail Fig. B.11

Observation from 12 June 2004.
In the text
thumbnail Fig. B.12

Observation from 6 July 2004.
In the text
thumbnail Fig. B.13

Observation from 8 July 2004.
In the text
thumbnail Fig. B.14

Observation from 29 July 2004.
In the text
thumbnail Fig. B.15

Observation from 16 May 2005.
In the text
thumbnail Fig. B.16

Observation from 29 April 2006.
In the text
thumbnail Fig. B.17

Observation from 29 April 2006.
In the text
thumbnail Fig. B.18

Observation from 31 May 2006.
In the text
thumbnail Fig. B.19

Observation from 13 June 2006. S2 is used as reference star for this lightcurve.
In the text
thumbnail Fig. B.20

Observation from 29 June 2006.
In the text
thumbnail Fig. B.21

Observation from 24 July 2006.
In the text
thumbnail Fig. B.22

Observation from 27 July 2006.
In the text
thumbnail Fig. B.23

Observation from 28 August 2006.
In the text
thumbnail Fig. B.24

Observation from 16 September 2006.
In the text
thumbnail Fig. B.25

Observation from 2 October 2006.
In the text
thumbnail Fig. B.26

Observation from 15 October 2006.
In the text
thumbnail Fig. B.27

Observation from 17 March 2007.
In the text
thumbnail Fig. B.28

Observation from 4 April 2007.
In the text
thumbnail Fig. B.29

Observation from 19 July 2007.
In the text
thumbnail Fig. B.30

Observation from 21 July 2007.
In the text
thumbnail Fig. B.31

Observation from 21 June 2008.
In the text
thumbnail Fig. B.32

Observation from 22 June 2008.
In the text

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