THE TWO STATES OF Sgr A* IN THE NEAR-INFRARED: BRIGHT EPISODIC FLARES ON TOP OF LOW-LEVEL CONTINUOUS VARIABILITY

Our 2004–2009 Ks-band data set consists of ≈12,000 images.⁹ The data were reduced in the standard way by applying a sky subtraction, a flat-field correction, and a hot/dead pixel correction (see, e.g., Trippe et al. 2007).

We found it necessary to apply a quality cut to eliminate the worst data and ensure we obtained a homogeneous data set on which we could perform accurate photometry. This quality cut was carried out by eye, and the criteria for elimination included the following:

• 1.  
  the two calibration stars were not in the image (this happened rarely, and only in the polarimetric data),
• 2.  
  image ghosts close to Sgr A* or PSF artifacts, and
• 3.  
  simply a bad quality image mostly corresponding to low Strehl ratio data resulting from poor atmospheric conditions. The quality of images for this criterion was judged by the general visibility of S-stars, not by the visibility of Sgr A* to avoid bias toward bright fluxes from Sgr A*.

We additionally eliminated 854 images from 2009 July which were taken in a triggered mode, so that our data set remains unbiased and representative of the overall variability of Sgr A*. The remaining data set totaled 6774 images which were used to obtain a light curve for Sgr A*.

To extract fluxes for Sgr A* as well as for several control stars, we carried out aperture photometry on each image. However, if we used a standard circular aperture centered on Sgr A*, we would have a varying contribution from S17 (the brightest star to be confused with Sgr A* 2004–2009) to the light curve as the star moves through the aperture during 2006–2008. A second star within the aperture, but not centered within the aperture also increases the error on our measurements of the flux of Sgr A*. We would obtain more accurate results if S17 could somehow also be well centered in the aperture at all times, as well as Sgr A*.

To solve the problem of S17, we thus used a two-aperture method with two circular sub-apertures, one centered on Sgr A* and one centered on S17, and measure their combined flux, averaging the results where the two sub-apertures are each 40, 53, and 66 mas in size. We used this method for all data 2004–2009, so that we always measure the combined flux of S17 + Sgr A*. While there may be additional effects due to confusion with other, fainter stars, at first order our method ensures that we measure the fainter fluxes of Sgr A* with similar measurement errors from year to year than we would have if we neglected to include a sub-aperture about S17.

To further ensure the self-consistency of our data set, we used the same set of calibrator stars for calibration of each image. This restricted us to only two suitable calibration stars, S30 and S65, due to the small field of view of some (especially polarimetric) images. To measure the raw counts of control stars and the calibration stars, we used the same aperture as for Sgr A* and S17, but centered the star in just one of the two sub-apertures. For each image, the raw counts of Sgr A*/S17 and the control stars are then divided by the counts of the two calibrator stars and the calibrated counts are then flux calibrated by scaling relative to the median of S65 for the observation night, using m_Ks = 13.7 mag for S65 (e.g., Gillessen et al. 2009). Our photometric calibration is also consistent with Schödel et al. 2010 (S65: m_Ks = 13.64 ± 0.02 ± 0.06 mag). To convert magnitudes to fluxes (mJy), we used the zero point from Tokunaga (2000; m_Ks = 0 mag corresponds to 667 Jy).

We computed the expected positions of Sgr A* and of the control stars using the orbital (polynomial) fits of Gillessen et al. (2009). For polarimetric data, we added the raw photon counts recorded in the apertures for all targets in both ordinary and extraordinary images first in order to recover the unpolarized intensities,¹⁰ before the flux calibration was carried out. We dereddened the final flux values using A_Ks = 2.5 (Schödel et al. 2010; T. K. Fritz 2011, in preparation). Note that the absolute photometric calibration and extinction correction are not important for the analysis of relative fluxes presented in this paper, where we have focused on maximizing the long-term relative accuracy of our photometric measurements. However, where we compare with previous publications, we additionally scale the reported fluxes to the photometric calibration (as best as possible) and extinction of this paper.

The light curve we produced in this way for Sgr A* is shown in Figure 1. The data presented here are representative of the overall variability of Sgr A* since it is not biased toward the presence of obvious "flare events" and the only data selection carried out concerned data quality and photometric consistency as explained above. It is by far the most extensive data set for Sgr A* that has been published: the observations we present here effectively comprise a ∼184 hr long light curve (if one subtracts from the total time all gaps larger than about 20 minutes, i.e., gaps not due to sky observations). The total amount of exposure time (i.e., the time spent with the shutter open, excluding overheads) is smaller and totals ≈72 hr, though 184 hr is a better indication of the amount of continuous coverage.

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Figure 1 shows the 2004–2009 light curve of Sgr A* and comparison star S7. On top of the more rapid variability, there is a general trend for the lowest fluxes of the light curve to wander from year to year. It is at its lowest in 2006. In 2004, the extra flux appears to be due to a combination of a faint star,¹¹ a contribution from the halo of S2, which was much closer to Sgr A* during that year, and S19, which was confused with S17. The light curve also shows some increased flux in 2008 and 2009; and as we will show in Section 2.1.1, there is at least one additional (previously unidentified) faint star which must also contribute to the increased flux in those years.

Since we can generally identify that stars are responsible there is good justification for subtracting these long (i.e., year) timescale trends from the light curve. For want of a better method of determining the stellar contribution to the light curves, we subtracted the difference between the median of the data from each year and the minimum median value (in 2006). This brings the offset in the light curve to roughly 3.5–4 mJy, much of which can be reasonably attributed to S17 (between 2.9 and 3.3 mJy in the light curve can be attributed to S17; see Section 2.2).

It is apparent from Figure 1 that Sgr A* is much more variable than the comparison source S7. Some of the flux excursions are much more dramatic than others, with fluxes above 8 mJy (∼5 mJy intrinsic upon subtraction of S17). Many of these have been previously published—for example, the second brightest peak of the data set, in 2006 May, was published previously in Meyer et al. (2006) and Trippe et al. (2007) and has 16.7 mJy (∼13.5 mJy intrinsic; see Table 1).

On 2008 August 5, we saw a particularly extreme event which can be seen as the most dramatic flux excursion in the light curve in Figure 1. This event is also shown in higher time resolution in Figure 2. On this particular night, the source at the position of Sgr A* brightened by a factor >4 in a period of ∼40 minutes, reaching a peak flux of 30.7 mJy (intrinsic flux of ≈27.5 mJy minus S17, a factor increase over the upper limits on the lowest fluxes from Sgr A* of ≈27). An increase in Ks-band emission of this brightness is unparalleled in the literature for Sgr A*: the second brightest that has been published is a flare from 2003 October (Meyer et al. 2007) which reached an intrinsic flux of ∼20.1 mJy (see Table 1).

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2.1.1. Faint Stellar Confusion 2008–2009

To create the 2004–2009 flux distribution of Sgr A*, we subtracted a trend from the data that we suspected to be due to faint stars. However, the trend for higher fluxes in 2008–2009 is not explained by the presence of any known (previously identified) stars.

Here, however, we are able to present evidence that there was indeed one to two faint stars confused with Sgr A* in the years 2007 to 2009. In Figure 3, we show that there are deviations in the source position of Sgr A* measured from imaging mosaics taken between 2007 and 2009. The systematic deviations imply that there was at least one, previously unidentified, faint star confused with Sgr A*. In fact, in new, very high quality observations of early 2010, it is possible to identify two new faint stars very close to Sgr A*. One or both of these stars may have undergone a close pericenter passage.

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From Figure 3, it appears that one of the stars was present in 2007 also, while our offset for 2007 does not argue for any extra star. However, as we discuss in Section 4.2 there appears to be a relatively steady (on the order of ∼0.5 mJy) stellar contribution to the flux of Sgr A* (including the years 2006–2007), which, together with the subtracted offsets, is probably the result of exactly such faint stars moving slowly in and out of confusion with Sgr A*: probably one of the newly discovered stars does indeed already contribute to the flux in 2007 (with the additional increase in flux in 2009 due to the second star).

It is somewhat a mystery why these stars have not been identified previously. The stars responsible for the observed deviations in the source position have not yet been identified in data prior to 2007 when they should have been resolved separately. Perhaps they have until now always been confused with other stellar sources. The star close to Sgr A* in 2004 (identified as S62 in Sabha et al. 2010, though we disagree with this identification) may be a candidate.

We additionally carried out PSF photometry on the Ks-band data from 2009 (13 mas pixel scale imaging only) using Starfinder (Diolaiti et al. 2000), which we use especially for investigating the question of whether Sgr A* is continuously variable at low fluxes (Section 3.2). We use an automated program for running Starfinder on many images developed by Rank (2007). The measured fluxes were again calibrated on the S-star S65 (m_Ks = 13.7; Gillessen et al. 2009) and the results are shown in Figure 4. We note that due to the presence of the two previously unidentified faint stars (Section 2.1.1), we improve our chances of detecting a source in most images, while at the same time the faintness of this additional stellar contribution ensures that our flux measurements have very small error, such that this data set is optimum for investigating the variability of Sgr A* at low fluxes.

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To produce the light curve, we first required that a faint S-star (S21, of ∼1.3 mJy) was detected in a given image as a basic data quality cut. From these images, we then examined the detected positions of sources in the near neighborhood (∼100 mas) of the nominal position of Sgr A* (see Figure 4) and selected sources that were within 20 mas of the mean position of this detected source (by this method a source was detected in 87% of the good quality selected images). Due to this method of source selection, the data set we present here is not completely unbiased, due to the 13% of images in which no source was detected, but the fraction in which Sgr A* was not detected is small enough that the data set still serves well to assess whether Sgr A* is in general more variable than stars of comparable flux.

From the figure, Sgr A* certainly appears most of the time more variable than the comparison stars of similar flux (S21 and S60 in Figure 4). S60, in particular, provides a very good comparison since it is also close to S31, a star of similar brightness to S17; the two sources S60 and S31 also have a very similar separation to Sgr A* and S17 during 2009 (see Figure 4). The flux distribution of S60 is fit by a Gaussian with μ = 1.309 and σ = 0.098 which does not give any indication that the photometric accuracy is significantly worse for sources of this separation and flux ratio (the more isolated star S21 is fit by a Gaussian with μ = 1.333 and σ = 0.090).

In general, the Starfinder and aperture photometry fluxes agree very well, with a constant offset of ∼2.9 mJy. However, there is a systematic increase in the offset between the measured fluxes from the aperture photometry method and the Starfinder method toward the end of 2009, when the source(s) confused with Sgr A* has(have) moved the furthest to the south (see Section 2.1.1). This is likely explained by Starfinder's one-PSF fit to the distorted PSF of two sources (which has the effect of decreasing the flux measured with the Starfinder method); the sources are even beginning to become separated at the end of the year (Figure 4).

This method also provides us with an estimate for the flux contribution of S17 to the 2004–2009 flux distribution. The offset of ∼2.9 mJy is slightly fainter than the flux of S17 (3.3 ± 0.2 mJy when separated from Sgr A* in the 2009 data using Starfinder). The lower total fluxes obtained with the aperture photometry method are probably a result of the higher stellar background surrounding Sgr A* (i.e., in the background aperture) compared to the more isolated flux calibration star and/or a side effect of the double aperture method. 2.9 mJy is then a low estimate for the contribution of S17 to the combined Sgr A*+S17 2004–2009 light curve, while 3.3 mJy from Starfinder photometry is a high estimate. Therefore, we estimate the contribution of the flux of S17 to our 2004–2009 light curve produced via aperture photometry to be in the range 2.9–3.3 mJy.

For the flux distribution, we define the detection frequency in bin k as

$$\begin{equation} \mbox{freq}_{{\rm det},k} = \frac{\Sigma _{i=1}^{N} \Delta t_i(F_k<F_i<F_{k+1}) }{(F_{k+1}-F_k)\Sigma _{i=1}^{N} \Delta t_{i}}. \end{equation} \tag{ 1 }$$

where Δt_i and F_i are the exposure time and flux of the ith image, respectively, there are N images and F_k and F_k+1 denote the bin edges. The detection frequency in each bin is divided by the bin width so that the area under the measured flux distribution is equal to 1. In the flux distribution, each light-curve data point was weighted by its exposure time so that the flux distribution is not biased toward observations with shorter time samplings. We chose a logarithmic binning with bins spaced at intervals separated by a factor of 10^0.05. The errors for each bin are computed as the square root of the exposure times in the bin added in quadrature (with the same normalization as in Equation (1)), i.e.,

$$\begin{equation} \sigma (\mbox{freq}_{{\rm det},k}) = \frac{\sqrt{\Sigma _{i=1}^{N} (\Delta t_i(F_k<F_i<F_{k+1}))^2} }{(F_{k+1}-F_k)\Sigma _{i=1}^{N} \Delta t_{i}}. \end{equation} \tag{ 2 }$$

Figures 5 and 6 present the flux distribution of Sgr A* for each of the two data sets described in Section 2.2 and Section 2.1, respectively. Figure 5 presents a comparison of the PSF and aperture photometric methods, demonstrating the effect of observational errors on the flux distribution, and Figure 6 shows various model fits, including the effects of observational errors, for the 2004–2009 aperture photometry dataset, as well as for a comparison star. The parameters and fit results are given in Table 2.

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3.1.1. Accounting for Observational Errors

The main difficulty with fitting the distribution of fluxes is the influence of observational errors, which are important enough at low fluxes to widen the peak of the distribution from its intrinsic shape. This effect is demonstrated in Figure 5 where the flux distributions for the two photometric methods (with different error properties) on the same data (2009 subset, see Section 2.2) are compared.

We characterized the observational errors by fitting Gaussian functions to the flux distributions of 22 nearby stars of fluxes between ≈0.6 and 25 mJy. Fitting a power law, we obtain

$$\begin{equation} \log _{10}\sigma _{\rm obs}(F) = (-0.76\pm 0.08)+ {(0.5\pm 0.1)}\log _{10}F \end{equation} \tag{ 3 }$$

similar to that of Do et al. (2009b) who found a typical dependence of σ_obs(F) ≈ 0.2F^0.3mJy for their data. The ≈0.5 power-law dependence of the errors is consistent with a photon noise origin (Fritz et al. 2010). We do not find any flattening of the power-law dependence of the noise toward low fluxes with statistical significance.

The 13 mas pixel data from 2009 are of much higher quality and for the Starfinder photometry have noise properties

$$\begin{equation} \log _{10}\sigma _{\rm obs}(F) = (-1.2\pm 0.1)+ {(0.6\pm 0.2)}\log _{10}F. \end{equation} \tag{ 4 }$$

The noise dependence was determined in the same way as for the 2004–2009 data set through comparison of 22 stars of fluxes 0.5–25 mJy.¹²

We include the effect of observational errors in our model fits by convolving the intrinsic flux distribution with a Gaussian of width σ_obs(F), which has the dependence on flux of Equation (3):

$$\begin{equation} P_{0+{\rm err}}(F) = \int P_{0}(F^{\prime })\frac{1}{\sqrt{2\pi } \sigma _{\rm obs}(F)}\exp \left(\frac{-(F-F^{\prime })^2}{2{\sigma _{\rm obs}(F)}^2}\right)dF^{\prime }, \end{equation} \tag{ 5 }$$

where P₀ is the intrinsic flux distribution (without observational errors). The effect is only important at low fluxes ≲6 mJy (see Figure 5) and is responsible for broadening the low flux peak in the observed flux distribution for Sgr A*.

3.1.2. Lognormal Model

A lognormal fit to the 2004–2009 flux distribution from aperture photometry is shown in Figure 6, with the best-fit parameters given in Table 2. This model has intrinsic probability distribution P₀ = P_logn, where

$$\begin{equation} P_{\rm logn}(F) = \frac{1}{\sqrt{2\pi } \sigma _* (F-F_b)} \exp \left({- \frac{(\ln (F-F_b) - \mu _*)^2}{2{\sigma _*}^2}}\right), \end{equation} \tag{ 6 }$$

where F represents the observed flux (measured in mJy here and elsewhere in this paper), and F > F_b with F_b a flux offset due to some constant contribution (e.g., S17 + contaminating stars). The parameters μ_* and σ_* of the lognormal distribution have a natural analogy to the normal distribution when exponentiated: the source can be thought of as having a median flux of exp(μ_*) with a multiplicative standard deviation of exp(σ_*) (i.e., the interval exp(μ_*)/exp(σ_*) to exp(μ_*) × (exp σ_*) contains ≈68% of the probability).

3.1.3. Lognormal Model with Flatter High Flux Tail

The residuals at low fluxes of the lognormal model are probably just due to the low flux wing of the error distribution which is not perfectly Gaussian (see star S7 in Figure 6). The residuals for the lognormal model at high fluxes, on the other hand, cannot arise from measurement error, and we obtain a much better fit to the flux distribution with a lognormal model which makes a transition to a flatter tail at high fluxes. The intrinsic probability distribution in this case is

$$\begin{equation} P_{\rm logn+tail}(F) = \left\lbrace \begin{array}{@{}ll}k P_{\rm logn}(F) & : F \le F_{b}+F_{t}\\ k P_{\rm logn}(F_t) \left(\frac{F-F_b}{F_t}\right)^{-s} & : F > F_{b}+F_{t}, \end{array} \right. \end{equation} \tag{ 7 }$$

where P_logn(F) is the lognormal distribution of Equation (6), F_t is the flux at which the distribution makes the transition to the flatter tail, s is the power-law slope of the tail, and k is a renormalizing factor¹³ such that the total probability is 100%.

The flattening of the distribution toward high fluxes in the 2004–2009 aperture photometry flux distribution is significant at >6σ. However, the fits are not formally "good fits." This is for the most part due to residuals in the lowest flux bins, most likely caused by the deviation of the observational errors from the plain Gaussian description used in our model. If we instead only fit only to fluxes >3.5 mJy (ignoring the lowest three bins), we obtain a more reasonable χ²/dof for the lognormal plus tail model of 17.4/13 (while the best-fit lognormal model for the same has 37.6/15). In this case, the flatter tail of the distribution is still significant at >4σ.

Interestingly, in the 2009 Starfinder photometric data set the tail is also present with similar best-fit parameters, although in this smaller data set (which only reaches fluxes of ∼10 mJy) the tail has a significance of only 1.4σ.

What is not clear in the 2004–2009 data set is whether it is appropriate to fit only continuously emitting, continuously variable models to the flux distribution. A plausible alternative is a model in which the source of near-infrared emission emits only sporadically (i.e., exhibits "on" and "off" periods). If this is the case, observational errors alone (on the measurement of a constant component to the flux) should be responsible for observed variations in the flux level during "off" periods. The goal of this section is to determine whether there is too much variation at low levels for the data to be explained by observational errors in "off periods" of a sporadic model.

We use the following prescription for a sporadic model for the flux distribution:

$$\begin{eqnarray} \nonumber P_{0+\mbox{err}}(F) &=& p_{\rm var} \int P_{\rm 0}(F^{\prime }) P_{\rm gauss}(F-F^{\prime }-F_b)dF^{\prime }\\ &&+ (1-p_{\rm var})P_{\rm gauss}(F-F_b), \end{eqnarray} \tag{ 8 }$$

where p_var is the fraction of time the source spends in an "on" state.

Using P₀(F) = P_logn(F) (Equation (6)) and fitting to both the 2004–2009 aperture photometry flux distribution and to the 2009 Starfinder distribution with the measured noise laws of Equations (3) and (4), we find in both cases the best fits to the distributions require p_var > 92% and p_var > 96% (at 3σ), respectively, i.e., the measured noise laws cannot account for the observed amount of variability at low fluxes.

To look at the question from a different angle, we investigate what level of observational noise would be required to produce the observed variation at low fluxes. To do this, we allow the normalization of the noise law to be a free parameter in the model fits, i.e.,

$$\begin{equation} \log _{10} \sigma _{\rm obs}(F) = A + 0.5\log _{10}F. \end{equation} \tag{ 9 }$$

We find A = −0.66 ± 0.02 and A = −0.79 ± 0.04 for the 2004–2009 aperture photometric data set and the 2009 Starfinder data set, respectively. We also refit the noise laws to the stars (see Equations (3) and (4)) fixing the slope of the error law to 0.5 and obtain A_stars = −0.84 ± 0.05 and A_stars = −1.12 ± 0.07 for the 2004–2009 aperture photometric data set and the 2009 Starfinder data set, respectively. The best-fit noise law to the 2004–2009 flux distribution from aperture photometry allows p_var = 68 ± 7 but the normalization of the noise law is inconsistent with that of the stars at a level of 3.3σ. For the 2009 flux distribution from Starfinder photometry, the required noise law is even more unlikely and is ruled out at a level of 4.5σ. We conclude that the observed behavior of Sgr A* at low fluxes is best consistent with a continuously emitting, continuously varying source.

We also detect longer timescale trends in the low flux level of Sgr A* in the 2009 data (Figure 4), such as an ∼0.8 mJy increase in the minimum level from the fourth to the fifth data sections, with a gap in time of only 12 days, and a similar size decrease from the second to the fourth data sections with a gap in time of only 11 days.

It might be that a passing star (such as the two new stars discovered in 2010, see Section 2.1.1) is responsible for these trends, either passing in and out of confusion very quickly, or even just moving within the Sgr A* PSF and distorting the flux: the further apart the two sources the lower the fitted flux. However, these stars cannot move quickly enough for this: the change in flux is so large that in either case the star must move by at least ∼50 mas within a period of 12 days to distort the flux as much as observed, which at the 8 kpc distance of the Galactic Center requires an average speed of at least 0.1c over a two-week timescale. Even assuming the most extreme case—a star moving only in the plane of the sky and falling toward Sgr A* in a parabolic orbit at the escape speed—a star should take at least ∼110 days (0.02 c in average) to cover such a distance. A larger centroid shift between the data sets would also be expected if it were the case that a star were distorting the source PSF (∼20 mas instead of the observed ∼5–8 mas). Furthermore, even if this could explain the trend between the fourth and fifth data sets, the same explanation would have to be invoked to again explain the variation between the second and fourth data sets, also on a timescale of only ∼11 days.

We therefore conclude that the observed trends on timescales of a few weeks in the light curve are most likely intrinsic. Though they cannot be due to direct flux contamination from passing stars, the trends are possibly still related to the feeding of the accretion flow from stellar winds of the passing stars, which may trigger increased activity as the star passes orbital pericenter (e.g., Loeb 2004).

We have shown in Section 3 that the flux distribution of Sgr A* is best described by a continuously emitting, continuously variable component with a lognormal flux distribution at low fluxes, but additionally that there is a significantly flatter tail above about 5 mJy, implying that there is something different about the high flux emission from Sgr A*. Put together, it seems justified to identify the following:

• 1.  
  emission below 5 mJy as quiescent state emission and
• 2.  
  emission above 5 mJy as flaring emission.

This is the (phenomenological) definition by which we hereafter refer to either "flares" or "quiescent" emission in Sgr A*. We present how the two states may fit into a possible physical picture below.

4.1.1. Quiescent State Emission

4.1.2. Flaring Emission

Our results are reasonably consistent with the median flux of 2.0 mJy over six separate observation nights from 2006 to 2007 found by Do et al. (2009b; see Table 1 for the scaling factors applied to scale the reported values to our calibration/extinction). We find a median flux of ≈1.6 mJy for our 2004–2009 light curve with the long-timescale trend and S17 (assuming 3.1 mJy) removed (note however that this still includes some constant contribution: the median flux of the lognormal component alone is 1.1 mJy).

A number of previous measurements have put upper limits on any long-term steady quiescent state at near-infrared wavelengths which have been on the ∼1–1.5 mJy level (Hornstein et al. 2002; Schödel et al. 2007; Sabha et al. 2010; see Table 1). These measurements are consistent with the picture we have put forward here for Sgr A* at low levels, of a continuously emitting, continuously varying source. For a lognormally varying quiescent state with the properties of our best-fit model to the 2004–2009 flux distribution, an upper limit of 1–1.4 mJy in selected observations is not surprising: the source is expected to be at ≲1 mJy roughly 50% of the time.

However, more likely the upper limits of 1–1.4 mJy include a stellar contribution to the flux. Sabha et al. (2010) estimated an upper limit on the intrinsic flux from Sgr A* of F≲0.5 mJy in the presence of a likely ∼0.8–1 mJy stellar contribution. In the context of our best-fit lognormal model, this low flux measurement is a rarer, but not unlikely occurrence, with a probability of ∼16%.¹⁵ The 0.8–1.0 mJy diffuse component reported by Sabha et al. (2010) might be associated with the ∼0.3–0.7 mJy offset we find in our fits to the flux distribution of Sgr A*, between the fitted constant component (3.6 mJy) and the expected contribution of S17 (2.9–3.3 mJy). Do et al. (2009b) also estimated from the color of Sgr A* at low fluxes, that they most likely had ∼35% contribution from stellar contamination (0.7 mJy) for their observations in 2006. These results all seem to point toward an additional stellar component to the flux of Sgr A* with a constant flux of ∼0.5 mJy (our data), though it does not necessarily always consist of the same stars from year to year.

Lognormal distributions of fluxes are also seen in other astrophysical objects, for example, X-ray binaries (prototype Cyg X-1 in the low-hard state; Uttley et al. 2005), implying a connection between Sgr A* other low-accretion rate black holes (Falcke et al. 2004; McHardy et al. 2006). The physical picture of the low-hard state of X-ray binaries is that—compared to higher luminosity flows—the inner optically thick accretion disk disappears and is replaced by a hot inner flow (Done et al. 2007). Similar pictures, with hot inner flows, exist for Sgr A* (e.g., Quataert 2003). Sgr A* is, however, at the extreme low end of the accretion rate range of observable sources (the "quiescent" state). Connections between Sgr A* and low-hard state X-ray binaries of higher accretion rate such as Cygnus X-1 can shed light on whether or not the quiescent state is indeed a distinct state or whether there is simply a smooth continuation of the hard state down to such low-accretion rates (Markoff 2010).

If timescale scales with mass, then the flares in Sgr A* of ∼10–100 minute duration correspond to ≈ millisecond timescales in a stellar mass black hole binary. It is also true that millisecond-timescale flares are seen in Cyg X-1, though it was shown by Uttley et al. (2005) that the majority of these (and all those in the low/hard state) were not more frequent than expected in the context of the extension of the low-level lognormal distribution in this source to higher fluxes. There was one flare, however, a 12.4σ event that occurred (in fact when Cyg X-1 was in the high/soft state) that could not be accounted for in the context of the underlying variability and was posited by the authors to have a different physical origin. In this vein, it could be that the flares in Sgr A* are due to the source's "flickering attempts at outburst activity from out of quiescence," as suggested by Markoff (2010).

A connection between the near-infrared variability of Sgr A* and the variability of X-ray binaries has also been made previously from a long-timescale break in the power spectral density of Sgr A* which can be associated with a break timescale in the low-hard state of X-ray binaries (Meyer et al. 2009), though the single data set used to constrain the high-frequency power spectrum in this case may not be representative. The lognormal distribution for low-level fluxes makes however an independent case for an analogy between the variability of Sgr A* and low-hard state X-ray binaries.

The flux distributions of Yusef-Zadeh et al. (2006) and Yusef-Zadeh et al. (2009) at 1.6 μm and 1.7 μm display a qualitatively similar shape to our Ks-band distribution, although these data sets are still much smaller than ours, and the low-level activity appears to be dominated by the observational noise. Nevertheless, with more data at 1.6 and 1.7 μm and modeling of the observational errors, it might be possible to obtain constraints on the colors of the low level and high flux components. In future work, we intend to use the same techniques as we have used in this paper at the Ks band, in other bands (L' and H) to investigate in detail the dependence of the flux distribution with wavelength.

Our work is not the first to point out that the flux distribution of Sgr A* is not well described by a Gaussian: Yusef-Zadeh et al. (2006, 2009) and Do et al. (2009b) have all noted that the flux distribution has a high flux tail. However, the timing analyses which have been carried out to date (Do et al. 2009b; Meyer et al. 2008, 2009) in which Sgr A* was compared to simulated red noise light curves have nevertheless used simulated light curves which have had, by construction, a Gaussian flux distribution. Sgr A*'s intrinsic mean and variance are such that if one were to plot the simulated light curves scaled to Sgr A*'s mean and variance (for the 2004–2009 data sets μ = 1.3 mJy subtracting 3.8 mJy for S17 plus other faint stars, and σ = 1.9 mJy), the model red noise light curves would have negative intrinsic fluxes much of the time which is rather unphysical.

In Do et al. (2009b), the high flux tail was excluded when fitting the flux distribution, and only fluxes ≲3.2 mJy were fitted with a Gaussian (see Table 1 for scaling factors; the reported value in Do et al. 2009b was a reddened flux of 0.3 mJy). Thus, their model light curves best resemble the low-level emission of Sgr A*. Indeed the entire data set used by Do et al. (2009b) does not include much bright emission ≳5 mJy, and thus their results really apply to the low-level emission of Sgr A*. That the low-level emission from Sgr A* is consistent with red noise fits together with our finding of the lognormal distribution of fluxes at low fluxes. Both of these aspects of the variability are seen in X-ray binaries such as Cyg X-1 (Uttley et al. 2005). However, given that the flux distribution is not Gaussian, it remains questionable that the power spectrum has yet been reliably constrained from the Monte Carlo simulations of previous timing studies.

In particular, high and low flux states of Sgr A* may require separate timing analysis. In this respect, it is interesting that Meyer et al. (2008) find a quasi-periodic signal with false alarm probability of only 2 × 10⁻⁵ (4.2σ significance in Gaussian equivalent terms), if one considers just the window 385–445 minute subset from 2005 July 30 to 31, though the signal was deemed insignificant when the entire light curve from the night was analyzed.