Consistency of the Infrared Variability of SGR A* over 22 yr

For this project, we have developed a new implementation of speckle holography. This builds on the work of Schödel et al. (2013), which has been applied to the GCOI data sets presented in Meyer et al. (2012; version 2_0) and Boehle et al. (2016; version 2_1) to study the short-period stars. In theory, the speckle holography technique uses the instantaneous PSF, which is measured from a set of reference sources, to deconvolve, in Fourier space, the distorted images and realize the contribution of all speckle information to the final diffraction-limited core, as follows:

$$\begin{eqnarray}&&O=\displaystyle \frac{\langle {I}_{m}{P}_{m}^{* }\rangle }{\langle | {P}_{m}^{2}| \rangle }\end{eqnarray} \tag{ 1 }$$

where O is the Fourier transform of the object, I_m and P_m are the Fourier transforms of the mth short-exposure image and of its instantaneous PSF, respectively, and the brackets denote the mean over N frames. ${P}_{m}^{* }$ is the conjugate complex of P_m (Primot et al. 1990).

In practice, speckle holography images are constructed through an iterative process. A key component of this analysis uses the PSF fitting program StarFinder (Diolaiti et al. 2000; also see Section 3.2). Below we detail the steps to construct the speckle holography images (version 2_2), and show how they differ from the implementation used in Boehle et al. (2016; version 2_1).

• 1.  
  Shift all short-exposure frames to align the brightest speckle of IRS 16C. Subtract a constant background, which is estimated for each individual frame, from each short-exposure frame for version 2_2.
• 2.  
  Rebin the speckle frames from original 20 mas pixel⁻¹ scale down to 10 mas pixel⁻¹ scale. Bilinear and cubic interpolation are used in version 2_1 and version 2_2, respectively.
• 3.  
  Combine and construct an SAA image from all datacubes per observing epoch.
• 4.  
  Extract astrometry and photometry of stars in the SAA images with StarFinder to identify potential PSF reference stars for the speckle holography analysis.
• 5.  
  Select the brightest isolated sources as PSF reference stars for speckle holography. Each datacube typically has two to five reference sources (IRS 16NE, IRS 16C, IRS 16NW, IRS 16SW, IRS 33N), depending on the centering and image quality of the datacube.
• 6.  
  Estimate the instantaneous PSF for each speckle frame from the median of the aligned and flux-normalized images of the reference stars. For each PSF, we subtract a constant value of b_g + n × σ (n = 3 for speckle images), where b_g is the background and σ is the noise. All resulting negative values in the PSF are set to 0. As a final step, a circular mask is applied to the PSF and the PSF is normalized to a total flux of 1. In version 2_2, we fixed a bug in StarFinder in which secondary stars that are not PSF reference stars were not being subtracted from the primary reference stars.
• 7.  
  Improve the PSF estimate by subtracting all known secondary contaminating sources near the reference stars in each frame, using the preliminary PSFs from step (6) and information from step (4).
• 8.  
  Estimate the object Fourier transform O by applying Equation (1).
• 9.  
  Apodize O with a model for the optical transfer function (OTF) of the telescope. Here we use a Gaussian function for 10 m aperture.
• 10.  
  Reconstruct the image with inverse Fourier transform.
• 11.  
  Repeat of the process from step (4) with the holographically reconstructed image, which has significantly higher quality than the initial SAA image.
• 12.  
  Create multiple images for error estimation. In version 2_1, the data set for each epoch is divided into three equal-quality subsets to produce three speckle holography maps. In version 2_2, we use a bootstrapping method to produce 100 bootstrap data sets using sampling with replacement resulting in 100 speckle holography images produced from the same number of frames as the original data set.

The most important improvements of version 2_2 compared to version 2_1 are listed here.

• 1.  
  Subtracting a constant sky background from each frame results in less background variation in the combined images, and in significantly suppressed edge effects near the edge of the FoV (Step 1).
• 2.  
  Version 2_1 only used IRS 16C as PSF reference source. Version 2_2 uses up to five stars as PSF reference sources, depending on the instantaneous FoV (Step 5).
• 3.  
  The bootstrapping method results in a more robust estimate of the astrometric and photometric uncertainties (Step 12).

Figure 1 shows the central region of the final reconstructed image from 2005 July. Compared to the earlier implementation of speckle holography and SAA analysis, the new analysis (version 2_2) has both improved the image quality and reduced the edge-effect artifacts.

Zoom In Zoom Out Reset image size

Point sources are extracted from each epoch's final reconstructed image using StarFinder. Here, like in Section 3.1, we use the version of StarFinder utilized in Boehle et al. (2016) setting the cross-correlation threshold to 0.8, and a slightly lower minimum signal-to-noise ratio cut (3σ, versus 5σ). In order to estimate the astrometric and photometric uncertainties for the sources extracted, we perform StarFinder on 100 bootstrap images for each epoch and then calculate the standard deviation.

In order to minimize the impact of edge effects of speckle holography images, we restricted our analysis in the central 2'' × 2'' region center around Sgr A* for all epochs in the rest of this work. Owing to the observing strategy, in which stationary mode was used (see Section 2), the final image has significant variations in the number of individual frames that contributes to the each pixel toward the outer edge of the image. See contours in Figure 2.

Zoom In Zoom Out Reset image size

Because speckle holography is a Fourier deconvolution technique, point-like artifacts can be produced in the middle of the FoV from edge effects, background, and PSF extraction. We therefore require sources to be detected above a minimum threshold number of bootstrap images to be considered real. This threshold is set by demanding that the probability of fake detections be less than 1% (see Appendix B.1). The bootstrap threshold for each epoch is reported in Table 2 and with an average of 10% ± 7%.¹²

The list of extracted sources from speckle holography images is photometrically calibrated using a two-step procedure described in detail in Appendix A. This process results in an average photometric systematic zero-point uncertainty of 0.14 mag and an average relative photometric uncertainty of 0.04 mag respectively in NIRC K bandpass (see Table 1). While we use the standard photometry in Blum et al. (1996) as the initial system calibration, we find that our photometry of stars in the GC is consistent with both the photometric systems of Witzel et al. (2012) and Schödel et al. (2010) for the AO measurements of Sgr A* obtained between 2004 and 2017. The overall photometric difference between Schödel et al. (2010) and us is ∼1% with an average difference of only 0.01 ± 0.09 mag and a zero-point uncertainty of 0.16 mag. See Figure 3.

Zoom In Zoom Out Reset image size

This work has introduced an improved implementation of speckle holography. Appendix B.2 compares in detail the performances between the current and the old versions and shows the clear improvements of version 2_2. In particular, version 2_2 is 0.4 mag deeper on average but can be as much as a factor of three more sensitive in the extreme. From here on, we only consider the analysis based on the new speckle holography (version 2_2), which has an average detection limit of 16.9 mag (see Table 1).

We use the following steps to identify Sgr A* in the source list.¹³

• 1.  
  Determine the position of Sgr A* in each epoch from the offsets between Sgr A* and IRS 16C and S0-2 because they are bright enough (IRS 16C: K = 9.8 mag; S0-2: K = 14.2 mag) to always be identified and obtain the accurate positions in the image. The offsets were generated by aligning all of our speckle holography and AO data sets together. See Ghez et al. (2008), Gautam et al. (2019), Jia et al. (2019), and Sakai et al. (2019) for more details.
• 2.  
  Search Sgr A* in the source list using the expected positions estimated in step (1). Sgr A* detected candidates are extracted if they are within the search radius of 10 mas. The search radius was determined by exploring the median astrometric error for all real detections in speckle epochs (Appendix B.1) in the central 2'' × 2'' region. See Figure 4. Empirically for sources with K ∼ 17 mag (average speckle holography detection limit for all epochs, see Appendix B.2), the astrometric uncertainty is typically 10 mas. Based on this, we do not expect any real Sgr A* detections beyond 10 mas search radius.
• 3.  
  Identify epochs where there is confusion with a known star that is passing within a 40 mas radius (Jia et al. 2019) away from Sgr A*.

Zoom In Zoom Out Reset image size

Results on the detections of Sgr A* in our 27 epochs of speckle holography imaging data fall into four categories: detections without source confusion (13); non-detections (5); confusion with brighter sources (8); and confusion with fainter sources (1). See Table 2 for details.

4.1.1. Detections without Source Confusion

Figure 5 presents the images of epochs with Sgr A* detections that are free from source confusion. The average observed magnitude of Sgr A* as obtained from these 13 epochs, which span the 7 yr period from 1998 to 2005, is K = 16.0 ± 0.4 (standard deviation, std) with average relative photometric uncertainty of 0.1 mag, corresponding to the average observed flux density of 0.35 ± 0.13 mJy with average relative photometric uncertainty of 0.04 mJy.

Zoom In Zoom Out Reset image size

4.1.2. Non-detections

4.1.3. Detections Confused with Brighter Sources

In eight epochs, Sgr A* was confused with a source brighter than its average value. As shown in Figure 6 confusion occurred with the following brighter stars: S0-2, K_ave = 13.6 mag in 2002 April, May, and July; S0-19, K_ave = 15.0 mag in 1995 June; S0-16, K_ave = 15.1 mag in 2000 April, May, July, and October. These epochs were removed from further analysis.

Zoom In Zoom Out Reset image size

4.1.4. Detections Confused with Fainter Sources

4.2.1. Average Brightness

Figure 7 shows all our detections and detection limits of Sgr A* from 1996 to 2005. We convert the observed values into dereddened flux densities using the relationship ${F}_{{K}_{s}}=\,6.67\times {10}^{5}\times {10.0}^{-0.4\times ({K}_{s}-{A}_{\mathrm{ext},{{\rm{K}}}_{s}})}$ mJy (Tokunaga 2000) and assuming ${A}_{\mathrm{ext},{{\rm{K}}}_{s}}=2.46$ mag extinction (Schödel et al. 2010, 2011). Then we use the filter transformation ${F}_{K}=1.29{F}_{{K}_{s}}$, which is computed for the observed color of Sgr A* (H–K = 2.6 mag), to convert from K_s fluxes to K fluxes (see Table 3). See Appendix A.1 for more discussion of the filter transformations. Here, in order to present the absolute dereddened fluxes all uncertainties contain both the photometric systematic uncertainties and relative uncertainties (see Appendix A). The average detected dereddened flux density (ignore the brightness limit) is 3.4 ± 1.2 mJy (standard deviation) with average uncertainty of 0.6 mJy (0.4 mJy of relative photometric uncertainty only). If including the brightness limit, (a) treat brightness limit as a value: the upper limit of the dereddened flux density is 2.9 mJy; (b) treat brightness limit as zero: the lower limit is 2.3 mJy. The average from detections and the variance (see following Section 4.2.2) are consistent with the expectations from simulations based on more recent AO observations (2006–2017) as modeled from Witzel et al. (2018; see Section 5.2.2, and Figures 13 and 14).

Table 3.  Filter Transformation for Sgr A*

Name	${K}_{\mathrm{NIRC}2}^{{\prime} }-{K}_{\mathrm{NIRC}}$	${K}_{s,\mathrm{NACO}}-{K}_{\mathrm{NIRC}}$
Sgr A*	${0.367}_{-0.02}^{+0.01}$	${0.275}_{-0.02}^{+0.01}$

Download table as:  ASCIITypeset image

Zoom In Zoom Out Reset image size

4.2.2. Long Timescale Variance (40 days–7 yr)

We used the first-order structure function to characterize the variability of Sgr A* over the 7 yr. This approach is similar to the analysis of the timescale and the intrinsic variability of active galactic nucleus (AGN) light curves (e.g., Simonetti et al. 1985; Hughes et al. 1992; Paltani 1999) and Sgr A* short time variability (e.g., Do et al. 2009; Witzel et al. 2012, 2018). For the set of flux measurements shown in the light curve, s(t), the first-order structure function V(τ) measures the flux density variance for a given time separation τ:

$$\begin{eqnarray}&&V(\tau )\equiv \langle {\left[s(t+\tau )-s(t)\right]}^{2}\rangle .\end{eqnarray} \tag{ 2 }$$

We calculated [s(t + τ) − s(t)]² using Sgr A* dereddened fluxes reported in this work for all possible pairs of time lags from real speckle observational time series. Then we put the variances into bins with a bin size of 100,000 minutes (∼70 days). This yields nine bins covering the timescale between 42 days and 7.3 yr, and each bin contains at least five and as many as 19 data points. In each bin, we assigned the median lag time to be the lag time for that bin, and the average of the V(τ) values to be the value of the structure function at that lag. The error of the structure function for each bin is calculated from ${\sigma }_{\mathrm{bin}}/\sqrt{{N}_{\mathrm{bin}}}$. Here σ is the standard deviation of the V(τ) values and N is the number of points in that bin. The structure functions calculated with Sgr A* dereddened fluxes from observations are presented in Figure 8. The structure function is flat over timescale from 42 days to 7.3 yr, and has an average value of 3.1 mJy² with the standard deviation of 1.1 mJy².

Zoom In Zoom Out Reset image size

Based on 7 yr of speckle holography data, we can use the variability of Sgr A* as the indication of the accretion activity between 1998 and 2005. During this time, the dusty source G1 went through the closest approach in 2001. This object has similar observational properties to G2, which is the first example of a spatially resolved object tidally interacting with Sgr A*. The hypotheses for the nature of G1 are similar to those proposed to explain G2. These predictions range from compact gas clouds to the product of binary mergers (Gillessen et al. 2012; Murray-Clay & Loeb 2012; Phifer et al. 2013). Predictions of compact gas clouds near Sgr A* suggest that they may increase the accretion flow onto Sgr A*. For G2, one prediction (Schartmann et al. 2012) is that if it is a gas cloud, it may be tidally disrupted and accrete onto Sgr A*, increasing the black hole's luminosity by up to a factor of 80. Because G1's tidal radius is even smaller than that of G2, it would be more influenced by the black hole. If G1 was a gas cloud and some of the gas had been accreted onto the black hole, we may expect the additional accretion at the periapse. We marked the time following of G1's periapse passage in Figure 9. Between 2001 and 2005, there is no increase in flux observed in Sgr A*. Sgr A* was quite steady with no evidence of large variations in brightness. The result is consistent with G1 being a self-gravitating object (such as a merger of two stars) as suggested by Witzel et al. (2017).

Zoom In Zoom Out Reset image size

While the short-term variability of Sgr A* in the NIR is well characterized as a red-noise process, the long timescale variation has not been well probed. In order to explore if the observed long timescale variability shown in this work is consistent with models derived from shorter timescales, we simulate the NIR Sgr A* light curves with the model presented in Witzel et al. (2018; see Section 5.2.1). Section 5.2.2 presents the comparison of the simulations to the observations. See Section 5.2.3 for further discussion of the characteristic break timescale.

5.2.1. Light Curve Simulations

This model contains two key components. The first component describes the temporal characteristics using the PSD, which is modeled as a broken power law

$$\begin{eqnarray}\mathrm{PSD}(f)\propto \left\{\begin{array}{ccc}{f}^{-{\gamma }_{0}} & \mathrm{for} & f\lt {f}_{b}\\ {f}^{-{\gamma }_{1}} & \mathrm{for} & f\geqslant {f}_{b},\end{array}\right.\end{eqnarray} \tag{ 3 }$$

where γ₀ = 0 (assumed), γ₁ = 2.1 ± 0.1, f_b = (4.1 ± 0.7) ∗ 10⁻³ minute⁻¹ (which corresponds to a timescale of τ ∼ 245 minutes). See Figure 10 for the modeled PSD.

Zoom In Zoom Out Reset image size

The second component describes the distribution of fluxes with a log-normal probability density function (PDF)

$$\begin{eqnarray}&&\begin{array}{l}{ \mathcal P }[F| ({\mu }_{\mathrm{log}n},{\sigma }_{\mathrm{log}n})]\\ =\,{\left(\sqrt{2\pi }F{\sigma }_{\mathrm{log}n}\right)}^{-1}\cdot \exp \left(-\displaystyle \frac{{\left[\mathrm{ln}\left(\tfrac{F(K)}{\mathrm{mJy}}\right)-{\mu }_{\mathrm{log}n}\right]}^{2}}{\sqrt{2}{\sigma }_{\mathrm{log}n}^{2}}\right),\end{array}\end{eqnarray} \tag{ 4 }$$

where Sgr A* flux density $F\in [0,\infty ]$, log-normal mean in K-band ${\mu }_{\mathrm{log}n}\in [-\infty ,+\infty ]$, and log-normal standard deviation in K-band ${\sigma }_{\mathrm{log}n}\in [0,\infty ]$.

Following the method in Timmer & Koenig (1995) and using the modeled parameters in Witzel et al. (2018), we are able to generate the simulated light curves of Sgr A* using the same time series sampling as the real observations following the time of datacubes. Here we assigned the time of each datacube to be the time of first frame in that cube. We have tested the effects of only using the datacube time series instead of frame time series if considering time delay between each frame. No significant differences were found. We therefore use the datacube sampling for computational efficiency. To create each simulated light curve (with observed flux density), we added Gaussian-distributed noise (σ = 0.035 mJy, average uncertainty from speckle observations; see Section 4.1.1). In order to convert from observed fluxes at Kp (NIRC2 AO instrument used in this model from Witzel et al. 2018) to our observations at K (NIRC speckle instrument), we did filter transformation for Sgr A* of NIRC2 Kp–NIRC $K=0.367\,{\mathrm{mag}}_{-0.02}^{+0.01}$ (see Table 3), similar to the bandpass correction described in Section 4.2.1 and Appendix A.1. As a final step, the dereddened flux of a simulated light curve was obtained following the process presented in Section 4.2.1. See Figure 11 for the examples of the final simulated light curve with dereddened flux.

Zoom In Zoom Out Reset image size

5.2.2. Comparison of the Simulations to the Observations

We extracted the average flux density of every single simulated light curve (see Section 5.2.1), in order to imitate each real observational image that combines and averages the fluxes from all datacubes. For each epoch, we repeated the simulation 10,000 times using the posterior values from Witzel et al. (2018). See Figure 12 for the distribution of the average dereddened flux density of simulated light curves in 19 available epochs,¹⁴ and Sgr A* observations for comparison.

Zoom In Zoom Out Reset image size

Then we used simulated Sgr A* light curves to calculate the expectation of Sgr A* average flux and average structure function of our available epochs based on the model described above. In order to take into account the effect of detection limit of observations in simulations, we calculate the expectation of Sgr A* average flux and average structure function only with the simulated fluxes that are higher than the detection limit in that epoch. The steps are as follows.

• 1.  
  Probe 19 available epochs that have either a detection or a detection limit.
• 2.  
  Among all 10,000 simulations, for each one set of simulated flux densities from all 19 epochs, mark the epoch if the simulated flux density passes the corresponding Sgr A* detection limit. Calculate one average flux density and one series of structure functions (for all possible pairs of time lags) with only marked epochs. Then generate one Sgr A* average variance.
• 3.  
  Repeat step (2) for all 10,000 sets of simulations. The number of epochs used to generate each average flux density and series of structure functions varies depending on how many simulated flux densities pass the detection limit in that round.

Figure 13 presents the comparison of Sgr A* average dereddened flux from 13 detected epochs and its expectation calculated above from simulations based on the model in Witzel et al. (2018). Figure 14 presents average structure function of Sgr A* detections and its expectation from simulations. The observed Sgr A* average flux density and average structure function are consistent with the predictions that are modeled from Witzel et al. (2018) with a power-law PSD and log-normal flux distribution. These results show that Sgr A* long-term variability status in the past (1998–2005) is well consistent with the extrapolation from shorter timescale AO-based observations at later time. Sgr A* has had similar brightness and variability characteristics over two decades.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

5.2.3. Characteristic Break Timescale

We perform photometric system calibration using four initial calibration stars (IRS 16C, IRS 33E, S2-16 and S2-17). These stars are the only ones that have both reference flux measurements as reported by Blum et al. (1996) and are located within our FoV. As Figure 15 shows, IRS 16C is ideally located close to the center of the FoV and therefore measured in every epoch. S2-17 and S2-16 are measured in almost every epoch, and in each epoch typically have more than half of the frames obtained. In contrast, IRS 33E is much closer to the edge of the final FoV and detected in only two-thirds the epochs, and in these epochs it typically has one-third of the frames.

Zoom In Zoom Out Reset image size

There are three considerations made to convert Blum's measurements into flux predictions for the speckle holography measurements made with NIRC. First and most importantly, we applied an aperture correction to the Blum's measurements to account for the low resolution of their measurements. With ∼1'' seeing, Blum's measurements include neighboring stars that are resolved in our speckle holography observations. Therefore, we did aperture correction by subtracting the fluxes of nearby sources within the radius of ∼05 aperture. The correction ranges from 0 to 0.3 mag. Second, owing to the slight differences between the bandpass used in Blum et al. (1996) and our NIRC K instrument (see Figure 16), there are photometric offsets between the filters. These offsets were calculated by convolving the extincted stellar model atmospheres with the filter functions (see Appendix A in Gautam et al. 2019 for more details). The offsets range from 0.06 to 0.11 mag. Third, the uncertainties in the predicted brightness of the four calibrators are increased by the known level of variability from the work in Gautam et al. (2019). The additional uncertainties range from less than 0.03–0.07 mag, which for each star is less than the uncertainty in the original Blum's measurements. Table 4 summarizes all these considerations and provides the final predictions.

Zoom In Zoom Out Reset image size

Table 4.  Calibration Stars

Star	Calib.	Na	Varia.b	KBlum+96	Aperture	KNIRC-	KNIRC,Pred.d	KNIRC	KNIRC
Name	Type	Epoch	(mag)		Corr.c (δmag)	KBlum+96		Init.e	Rel.f
IRS 16C	Phot.	27	0.040	9.86 ± 0.05	0.07 ± 0.001	−0.06 ± 0.01	9.87 ± 0.06g ± 0.04h	9.81 ± 0.04 ± 0.14	9.79 ± 0.08i
IRS 33E	Phot.	18	<0.035	10.02 ± 0.05	0 ± 0	−0.06 ± 0.01	9.96 ± 0.06 ± 0.04	10.14 ± 0.04 ± 0.14	10.04 ± 0.07
S2-17	Phot.	26	<0.034	10.03 ± 0.07	0.32 ± 0.007	−0.06 ± 0.01	10.29 ± 0.09 ± 0.04	10.62 ± 0.04 ± 0.14	10.64 ± 0.06
S2-16	Phot.	25	0.070	11.90 ± 0.22	0.26 ± 0.001	−0.11 ± 0.01	12.05 ± 0.23 ± 0.04	11.79 ± 0.09 ± 0.14	11.79 ± 0.10
IRS 16NW	Rel.	27	<0.031	⋯	⋯	⋯	⋯	10.00 ± 0.04 ± 0.14	10.00 ± 0.05
S1-17	Rel.	26	<0.030	⋯	⋯	⋯	⋯	12.24 ± 0.04 ± 0.14	12.24 ± 0.04
S1-34	Rel.	26	<0.029	⋯	⋯	⋯	⋯	12.92 ± 0.04 ± 0.14	12.93 ± 0.04
S1-1	Rel.	27	<0.029	⋯	⋯	⋯	⋯	12.85 ± 0.04 ± 0.14	12.86 ± 0.04
S1-21	Rel.	25	<0.030	⋯	⋯	⋯	⋯	13.26 ± 0.05 ± 0.14	13.23 ± 0.05
S0-14	Rel. & Veri.	26	<0.028	⋯	⋯	⋯	13.50 ± 0.03 ± 0.07	13.50 ± 0.04 ± 0.14	13.51 ± 0.05
S0-13	Veri.	27	<0.030	⋯	⋯	⋯	13.22 ± 0.04 ± 0.07	13.21 ± 0.04 ± 0.14	13.23 ± 0.04
S0-6	Veri.	27	<0.030	⋯	⋯	⋯	13.99 ± 0.03 ± 0.07	13.95 ± 0.04 ± 0.14	13.95 ± 0.04
S0-12	Veri.	27	<0.034	⋯	⋯	⋯	14.20 ± 0.04 ± 0.07	14.15 ± 0.04 ± 0.14	14.13 ± 0.05
S0-4	Veri.	27	<0.032	⋯	⋯	⋯	14.17 ± 0.04 ± 0.07	14.22 ± 0.04 ± 0.14	14.20 ± 0.05
S1-10	Veri.	27	<0.029	⋯	⋯	⋯	14.67 ± 0.04 ± 0.07	14.69 ± 0.04 ± 0.14	14.71 ± 0.04
S0-31	Veri.	27	<0.037	⋯	⋯	⋯	15.08 ± 0.05 ± 0.07	14.86 ± 0.06 ± 0.14	14.89 ± 0.07
S1-33	Veri.	27	<0.030	⋯	⋯	⋯	14.82 ± 0.04 ± 0.07	14.92 ± 0.04 ± 0.14	14.89 ± 0.04
S0-11	Veri.	27	<0.035	⋯	⋯	⋯	15.00 ± 0.04 ± 0.07	15.06 ± 0.05 ± 0.14	15.07 ± 0.06
S1-6	Veri.	27	0.069	⋯	⋯	⋯	15.25 ± 0.08 ± 0.07	15.35 ± 0.08 ± 0.14	15.44 ± 0.08
S0-27	Veri.	26	0.049	⋯	⋯	⋯	15.45 ± 0.05 ± 0.07	15.35 ± 0.07 ± 0.14	15.37 ± 0.10
S1-31	Veri.	26	<0.036	⋯	⋯	⋯	15.50 ± 0.05 ± 0.07	15.51 ± 0.05 ± 0.14	15.51 ± 0.09

Notes.

^aNumber of epochs detected in the speckle holography images. ^bAdditional uncertainty from variability (Gautam et al. 2019), which for each star is less than the uncertainty of the original Blum's measurements. Reported in Gautam et al. (2019). ^cAperture correction with a radius of 05 (see Appendix A.1). ^dPredicted NIRC K mag for four initial calibrators obtained from Blum et al. (1996) after aperture and bandpass correction; predicted NIRC K mag for 12 verification calibrators obtained from Schödel et al. (2010) after bandpass correction. See details in Appendix A.1. ^eMeasured NIRC K magnitude after initial photometric system calibration (see Appendix A.1). ^fMeasured NIRC K relative magnitude after relative photometric calibration (see Appendix A.2). ^gPhotometric uncertainty with additional variability uncertainty. ^hAverage photometric systematic zero-point uncertainty. ⁱRelative photometric uncertainty (see Appendix A.2). The average zero-point uncertainty for relative photometry is 0.04 mag.

References. Blum et al. (1996), Gautam et al. (2019).

Download table as:  ASCIITypeset image

The zero-point for each epoch, zp, was calculated as a weighted mean of the ratios between the calibrators' measured instrumental flux and reference flux (d_i = f_i,reference/f_i,measured):

$$\begin{eqnarray}&&\mathrm{zp}=\displaystyle \frac{\sum {w}_{i}\ast {d}_{i}}{\sum {w}_{i}}.\end{eqnarray} \tag{ 5 }$$

Here, w_i indicates the weight for calibrator i, derived from uncertainty in its reference flux (σ_f,i): w_i = (σ_f,i)⁻². Zero-point uncertainties in the photometric calibration for each observation epoch, σ_zp, were derived from the weighted standard deviation of flux differences (between the calibrator stars' reference magnitudes and measured magnitudes) then divided by $\sqrt{{N}_{\mathrm{Calibs}.}-1}$ (here N_Sys.Calibs. = 4).

$$\begin{eqnarray}&&{\sigma }_{\mathrm{zp}}=\sqrt{\displaystyle \frac{\sum \,{w}_{i}{\left({d}_{i}-\mathrm{zp}\right)}^{2}}{\sum \,{w}_{i}}}/\sqrt{{N}_{\mathrm{Calibs}.}-1}.\end{eqnarray} \tag{ 6 }$$

Each epoch's zero-point uncertainty is reported in Table 1. Overall, we achieved an average zero-point uncertainty ${\overline{\sigma }}_{\mathrm{zp}}$ for the photometric system calibration of 0.14 mag in NIRC K bandpass (see Figure 17).

Zoom In Zoom Out Reset image size

As a final step, we verify that our speckle holography measurements are at the same photometric system as Witzel et al. (2018), whose model is used to simulated the NIR Sgr A* light curves in Section 5.2. Witzel et al. (2012) used and reported 13 stars as photometric calibrators, 12 of which are contained in our studies (see Appendix B). These measurements were tied to the absolute Ks observations reported in Schödel et al. (2010). Therefore, we transformed from the VLT NACO Ks to NIRC K (see Figure 16 for different transmissions) with bandpass corrections that are similar to that process described above. Figure 3 shows that our measured K magnitudes (after photometric system calibration) are highly consistent with the predicted K magnitudes (corrected from Ks photometric system), with an average difference of only 0.01 ± 0.09 mag (see Table 4).

We perform relative photometric calibration using the secondary calibrator stars identified by Gautam et al. (2019) that are detected in the FoV of our observations. These calibrator stars, IRS 16NW, S1-17,S1-34, S1-1, S0-14, and S1-21 (see the left panel of Figure 15), are selected to be non-variable and well-distributed in the FoV. The reference fluxes of these calibrators were obtained from the photometric system calibration described in Appendix A.1 (see Table 4). See Figure 18 for the light curve with measured relative K magnitude for each of the calibrator after relative photometric calibration. The derivation of the zero-point and uncertainties in the relative calibration is the same as for the photometric system calibration procedure (see Appendix A.1). We achieved an average uncertainty ${\overline{\sigma }}_{\mathrm{zp}}$ for the relative photometric calibration of 0.04 mag in the NIRC K bandpass. See Table 1 for the details of a single epoch.

Zoom In Zoom Out Reset image size

In order to define criteria for real detections, we can use the stellar photometric and astrometric information from the analysis of our extensive AO imaging data (e.g., Jia et al. 2019), which are on the order of three magnitudes deeper than the speckle images. We define a reference set of 88 sources (real stars) in the central 2'' × 2'' region with K < 17.6 mag (deepest speckle data sets limit) and detected as the same source in at least one-third of 39 AO epochs. The real sources typically have high bootstrap fractions, while the remaining detections have quite low bootstrap fractions. The bootstrap fraction of any given object is defined as the portion of bootstrap images (among overall 100 bootstraps) in which the object can be detected (see Step 4 in Section 3.5). We use the remaining sources to estimate the surface density of likely spurious detections in the central 2'' × 2'' region at each bootstrap fraction, below which the detection is treated as likely spurious. Assuming a random process we can thus compute the probability of obtaining a false detection within 10 mas search radius around its nominal position, which is a function of the bootstrap fraction threshold that is applied to each epoch (see Figure 19). To avoid false detections, we require that the bootstrap fraction cut of a real source detection in each epoch is obtained and has a probability of a false detection within a 10 mas radius that is always <1%. See Table 2 for the summary of the bootstrap fraction cut.

Zoom In Zoom Out Reset image size

Then the source detection limit for each epoch is determined to be the 95th percentile of all K magnitudes in the sample, which includes all sources with a bootstrap fraction that is higher than the threshold in the central 2'' × 2'' FoV. The median of the detection limit for all epochs is 16.9 mag at K, which corresponds to observed flux of 0.15 mJy and dereddened flux of 1.4 mJy, respectively. See Table 2 for details.

We performed the real source analysis (see Appendix B.1) for both speckle holography version 2_1 and version 2_2, and then obtained the real source lists for two data sets, respectively. Here we compare speckle holography analysis version 2_1 and version 2_2 based on the real detection list in the central 2'' × 2'' region.

• 1.  
  The speckle holography technique version 2_2 results in deeper detections. The average magnitude limit for speckle data has been increased from K = 16.5 (version 2_1) to K = 16.9 (version 2_2). See Table 5 for details of each epoch. See Figure 20.
• 2.  
  In >80% epochs, the speckle holography technique version 2_2 results in more real detections. The average number of real detected stars in the central 2'' × 2'' region has been increased from N = 46 (version 2_1) to N = 59 (version 2_2). See Table 5 for details of each epoch.
• 3.  
  The version 2_2 increases the completeness of detection between K = 15 ∼ 17 mag, where Sgr A* lies (see Figure 20).
• 4.  
  The halos around sources, especially bright ones, are reduced in the speckle holography version 2_2 data sets (see Section 3.1 and Figure 1).
• 5.  
  For the speckle holography version 2_1 data sets, we obtain the uncertainties from running StarFinder on three sub-maps (standard deviation divided by square root 3). For the version 2_2 data sets, we calculate the uncertainties from running StarFinder on up to 100 bootstraps. This new bootstrapping technique gives mathematically more accurate uncertainties.

Zoom In Zoom Out Reset image size

Table 5.  Comparison between Speckle Holography version 2_1 and version 2_2

Date		Klim (mag)		NReal Stars		Npixb		Max Frames		Nrefc
(U.T.)	(Decimal)	Valued	Δ me	Value	Ratio	Value	Ratio	Value	Ratio	Value	Ratio
1995 Jun 9–12	1995.439	17.0	1.15	41	1.21	108042	0.90	5286	1.24	19	0.95
1996 Jun 26–27	1996.485	15.8	0.36	49	1.40	82505	0.79	2336	0.52	22	1.10
1997 May 14	1997.367	16.8	0.48	51	1.21	92467	0.74	3486	2.99	25	0.83
1998 Apr 2–3	1998.251	15.8	0.25	39	1.00	95816	0.78	1730	0.83	24	0.81
1998 May 14–15	1998.366	16.8	−0.02	45	0.92	102328	0.82	7685	0.77	24	0.89
1998 Jul 3–5	1998.505	16.4	0.57	43	1.08	116557	0.83	2053	0.81	24	0.83
1998 Aug 4–6	1998.590	17.1	0.16	47	0.94	109269	N/Af	11047	0.46	23	0.77
1998 Oct 9, 11	1998.771	16.6	0.41	45	1.22	97215	0.81	2015	0.87	24	0.80
1999 May 2–4	1999.333	17.2	0.10	52	0.96	107882	0.77	9427	0.96	22	0.81
1999 Jul 24–25	1999.559	17.4	0.73	54	1.02	100567	0.76	5776	0.99	23	0.79
2000 Apr 21	2000.305	15.7	0.13	56	1.81	96248	0.78	662	0.21	21	0.84
2000 May 19–20	2000.381	17.5	0.39	55	0.89	96853	0.80	15591	0.98	23	0.79
2000 Jul 19–20	2000.584	17.0	0.32	63	1.29	86452	0.78	10678	0.98	23	0.82
2000 Oct 18	2000.797	16.2	0.51	52	1.73	82315	0.84	2247	0.88	17	0.74
2001 May 7–9	2001.351	17.2	0.53	64	1.28	85028	0.91	6678	0.85	21	0.84
2001 Jul 28–29	2001.572	17.4	0.22	74	1.21	96872	0.78	6654	0.99	23	0.85
2002 Apr 23–24	2002.309	17.5	0.65	74	1.30	96953	0.79	13469	0.98	23	0.82
2002 May 23–24	2002.391	17.6	0.51	72	1.22	98552	0.83	11860	0.99	21	0.78
2002 Jul 19–20	2002.547	16.8	0.59	69	1.73	99994	0.79	4192	0.72	22	0.81
2003 Apr 21–22	2003.303	16.4	0.32	58	1.49	90963	0.93	3715	0.89	23	0.96
2003 Jul 22–23	2003.554	16.8	0.25	65	1.41	87265	0.79	2914	0.96	24	0.86
2003 Sep 7–8	2003.682	17.1	0.60	74	1.57	95367	0.79	6324	1.00	20	0.77
2004 Apr 29–30	2004.327	16.8	0.15	58	1.07	125423	0.99	6212	0.51	26	1.00
2004 Jul 25–26	2004.564	17.4	0.48	80	1.45	99819	0.78	13085	0.99	22	0.85
2004 Aug 29	2004.660	16.8	0.60	63	1.54	96172	0.96	2299	0.75	25	0.93
2005 Apr 24–25	2005.312	17.1	0.24	70	1.46	105715	0.81	9644	0.88	24	0.89
2005 Jul 26–27	2005.566	16.8	0.81	84	2.33	108360	0.79	5642	0.96	23	0.92

Notes.

^aK_lim is the magnitude that corresponds to the 95th percentile of all K magnitudes in the sample of real stars in the central 2'' × 2'' region (see Appendix B.1). ^bN_pix refers to the number of pixels in a given image that meet a .8 of maximum frames used criteria. ^cN_ref refers to the number of reference stars used to align the epoch of data. ^dAll values given in the table are the results of version 2_2. ^e ${\rm{\Delta }}m={K}_{\mathrm{lim},2\_2}-{K}_{\mathrm{lim},2\_1}$, is the detection limit difference between the version 2_2 and the version 2_1. The average magnitude limit has been increased from K = 16.5 (version 2_1) to K = 16.5 (version 2_2). ^fThe number of pixels in 1998 August is removed due to an artifact in the old holography image.

Download table as:  ASCIITypeset image