The Apparent Tail of the Galactic Center Object G2/DSO

SINFONI was previously mounted at the unit telescope (UT) 4 (VLT). It was afterwards relocated to UT3 (VLT). It is now decommissioned. The instrument uses a slicer to create pseudo longslits. Then, a spectral dispersion creates groups of wavelength-dependent longslits. ¹⁰ After this process, a 3d data cube is reconstructed with two spatial dimensions and one spectral dimension. For a fraction of the data, a laser guide star (LGS) was used. Since UT3 does not support LGS-guidance, the data that were observed after the relocation exclusively used a natural guide star (NGS). Typically, this NGS is located 1554 north and 885 south of Sgr A*. Since SINFONI uses an optical wave front sensor, the selection of possible bright (14–15 mag) NGS-sources is limited. The location of the bright radio source Sgr A*, which can be associated with the SMBH (see Eckart et al. 2002, 2017), can be found at R.A. (R.A.) 17:45:40.05 and decl. (decl.) −29° 00' 28120 (J2000). In the following subsections, we will explain the procedure that was used to derive the position of Sgr A* in the individual data cubes. We list the investigated SINFONI cubes in Appendix A, including the quality, exposure time of the individual observations, the related IDs, and the publications where we already used the data.

For the observations, the smallest available plate scale was used (12.5 mas) and the wavelength range was set to the H + K-band (1.45–2.45 μm). The exposure time for a single data cube was set between 400 and 600 s. We use the standard object-sky-object nod cadence. The sky corrections can then be applied to the individual data cubes. Because of the background noise and the small field of view (FOV) of 08 × 08, the GC/S-cluster observations are dithered around the position of Sgr A* or S2. After the usual reduction steps were applied with the ESO pipeline (Modigliani et al. 2007), the single data cubes of several nights (see Appendix A for the used data) are stacked to create a mosaic for each year between 2005 and 2019 with not only a higher signal-to-noise (S/N) ratio but also an increased FOV. Furthermore, the SINFONI pipeline automatically applies a barycentric and heliocentric correction.

From the well-observed orbit of the brightest (in K-band) S-cluster member S2 (see, e.g., Parsa et al. 2017; Ali et al. 2020), we derive the position of Sgr A* with an uncertainty of less than 12.5 mas. This uncertainty already contains a linear transformation and it also contains corrections for a distorted FOV. While this is applied in a satisfying procedure for the data discussed in Parsa et al. (2017) and Ali et al. (2020), the SINFONI data suffers from image motion as a function of wavelength. While this effect is certainly suppressed in single-band observations, the H + K-band observations with SINFONI do show a non-negligible movement of the stars between the H- and K-band. For example, the position of S2 does change by over 1 pixel (=12.5 mas) by comparing individual line maps. While Jia et al. (2019) addresses some of the GC observation problems for the KECK telescope that can also be applied to the VLT data (stellar confusion, variable PSF, artificial PSF-wing sources), we do not agree with the 1 mas uncertainty given by Gillessen et al. (2017) because it underestimates general crowding problems (e.g., blend stars, see Sabha et al. 2012) and does not reflect the noise character of the SINFONI data. As pointed out by the SINFONI manual, ¹¹ the shape and intensity of the point-spread function is a function of the source position on the detector. This results in inaccurate positions of stellar sources. Furthermore, Eisenhauer et al. (2003) discussed the issue of image motion for high-exposure observations (above several hours). Since this effect is nonlinear and depends on the total integration time, as well as the weather conditions, we will adapt a conservative uncertainty of 12.5 mas for the position of Sgr A* in the SINFONI data. In the following, we will elaborate on this issue in more detail.

2.3.1. Image Motion of SINFONI Long-time Exposures

A line-map and the related channel of a data cube represents a specific wavelength. If this wavelength is Doppler-shifted, then a LOS velocity v_z with v_z ≠ 0 can be derived. A SINFONI 3d data cube consists of 2000+ single line maps that can also be called channels. To isolate a single line, one has to subtract the underlying continuum. Typically, a polynomial fit (here, 2nd degree) will help to get rid of the continuum and partially the background emission (see Figure 22, Appendix B). However, the emission line itself can be heavily influenced by a variable background or atmospheric OH emission lines. For the Doppler-shifted Paα, He i, and Brγ lines that have been used for the analysis of G2/DSO (see e.g., Gillessen et al. 2013a), one has to consider the OH vibrational transition states 7-5, 8-6, and 9-7. As pointed out by Davies (2007), these OH lines have a non-negligible influence on the shape, intensity, position, and consequently a velocity of the object of interest.

In this work, we study position–position–velocity (PPV) maps instead of position–velocity (PV) diagrams because in this way we can preserve more accurate information. For this purpose, we transform the Doppler-shifted wavelength information of a single pixel (i.e., spaxel) to a LOS velocity. For the analysis, we will not use a smoothing kernel because this will have an impact on the result (see Table 6 and Figure 23, Appendix C).

Given the distance of the G2/DSO from Sgr A* and its LOS velocity evolution, we apply a Keplerian model fit to reconstruct the trajectory of the object. For the Keplerian fit, we use a SMBH mass of M_{Sgr A*} = 4.15 × 10⁶ M_⊙ with a distance of d = 8.3 kpc. For the MCMC simulations, we leave the boundaries for the different parameters open, except for the pericenter passage. Considering the uncertainty of the position of Sgr A*, as well as the sensitive emission lines of the G2/DSO, we categorize different non-Keplerian interpretations of the orbit that involve more parameters—in particular ,the magnetohydrodynamic drag force analyzed by Gillessen et al. 2019, or the fermionic dark matter dense core-diluted halo model of Becerra-Vergara et al. 2020—as challenging and unnecessary given the current data quality.

As described in Peißker et al. (2019, 2020a, 2020c, 2020d), a high-pass filter can be used to access information which is suppressed by overlapping PSF wings. Since the natural SINFONI PSF (NPSF) shows irreparable imperfections because no source is isolated in the small and crowded SINFONI FOV (∼08 × 08), we construct an artificial PSF (APSF) with comparable parameters (x- and y-FWHM, angle with respect to north) with respect to the NPSF. For that, we fit a PSF (∼6 pixel) sized Gaussian to S2, which is the brightest source in the FOV. From this, we derive the necessary input parameter to construct the NPSF. The fit uncertainties are in the range of about 1%.

We then place the input file in a 256 × 256 array. We use 10,000 iterations to minimize the chance of a false positive. Furthermore, we do a background subtraction, which is of the order of 15%–30%. With this procedure, the background noise can be suppressed. Subsequently, we apply the Lucy–Richardson algorithm (Lucy 1974) and a delta map is created. By convolving this map with a suitable Gaussian kernel (around 70% of the size of the input PSF), we get the final high-pass filtered image. We verify the robustness of the resulting image by comparing known stellar positions to it. In every step, the input files are normalized to the peak intensity. In most cases, the peak intensity is associated with S2. In every other case, the peak intensity is related to S35.

The OH emission lines in the H + K band and the related correction are widely discussed by Davies (2007) and were more recently discussed by Ulmer-Moll et al. (2019). Since the emission lines of G2/DSO are Doppler-shifted, they coincide with several vibrational transitions of OH (Rousselot et al. 2000). Furthermore, given that Davies (2007) not only provides the sky correction for the SINFONI pipeline but also describes an under- and over-subtraction of various OH/sky lines in the NIR, we want to investigate the influence on the high-exposure observations carried out in the GC. The typical observation scheme for these observations is object (o)-sky (s)-object (o). However, several observational programs that are available in the ESO archive do show a nontypical observational scheme. Instead of o-s-o, these programs use o-o-s-o-o with a long exposure time of 600 s. This results in nonmatching sky/OH emission from the science object. Because of the exposure time, the effect is maximized. As shown in Figure 1, the Paα line suffers from strong telluric emission/absorption features. In contrast, the OH emission lines do not influence the spectral range between 1.85 and 1.95 μm at a noticeable level because the relative flux is below 10%. This is not valid for the Doppler-shifted He i and Brγ regime. Since the latter line is more prominent than the He i line by about 40%–60%, we will emphasize the analysis of the spectral range around the Brγ rest wavelength of 2.1661 μm in Section 3. This is consistent with the analysis and statements given by Gillessen et al. (2013a).

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In Table 3, we list some prominent OH lines that could impact the line shape of the G2/DSO (see the following section).

The line maps (Figure 3) are extracted from the final mosaic data cube of the related year. We emphasize the investigation of the redshifted (until 2014.3) and blueshifted (after 2014.4) Brγ line. After the Doppler-shifted Brγ line is selected, we subtract the underlying continuum. In every dataset, the G2/DSO can be identified without confusion. The local background, the distance to the surrounding stars, weather conditions, and the on-source integration time have a major impact on the shape of the source (see Figure 3). The line maps show the obvious periapse of the G2/DSO in 2014. Since the source can exclusively be observed in the redshifted Brγ regime until 2014.35 and subsequently in the blueshifted domain after 2014.45, the periapse passage must have happened in between. In the following, we use the identification of the G2/DSO in the line maps to derive an exact value for the time of periapse from the Keplerian model fit.

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3.1.1. Brγ Line Evolution for G2/DSO, OS1, and OS2

Based on the clear line-map detection of the G2/DSO between 2005 and 2019, we extract the source spectrum to derive several properties (Figure 4). To that goal, we use an aperture with a radius of 2 pixel (25 mas). ¹² We subtract an aperture with an inner radius of 3 pixels and an outer radius of 10 pixels because of the dominant background/continuum. Then, we fit a polynomial function to the spectrum and fit a Gaussian to the Brγ line (see Table 4). With this procedure, we derive the LOS velocity for the G2/DSO, as well as the line width σ. By studying σ as a function of time, we find a quadratic behavior of the Brγ line width (Figures 5 and 25, Appendix D). We note that the gradient is expected to be the largest around the time of periapse because of the viewing angle change and the foreshortening factor close to unity (see the following subsection). Throughout the years, the PSF size of the source is ∼75–100 mas in the x- and y-direction, therefore the G2/DSO can be described as compact (Table 4).

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Table 4. Spectral and Line Map Properties of the G2/DSO

Epoch	Wavelength	Velocity	Standard Deviation σ	Line Map Size (mas)
	(μm)	(km s−1)	(km s−1)	x (R. A.)	y (Decl.)
2005.5	2.1740	1094.69	48.97	65	40
2006.4	2.1744	1159.96	102.51	94	78
2007.5	2.1753	1282.89	96.60	71	60
2008.2	2.1753	1287.36	87.06	74	73
2009.3	2.1762	1405.98	151.64	68	69
2010.3	2.1773	1555.53	165.31	89	74
2011.3	2.1783	1693.99	174.92	85	94
2012.4	2.1796	1881.95	180.40	71	76
2013.4	2.1822	2231.68	229.72	90	98
2014.3	2.1857	2726.20	355.02 (234.31)	51	59
2014.6	2.1425	−3265.23	126.74	56	74
2015.4	2.1490	−2355.72	369.12 (243.61)	60	86
2016.5	2.1534	−1752.34	122.32	31	55
2017.5	2.1559	−1407.56	104.67	69	86
2018.4	2.1572	−1228.13	119.71	56	64
2019.4	2.1579	−1126.27	98.87	61	85

Note. Please see Figure 4 for the related fit. Here, the standard deviation σ is a measure of the line width. The size is measured by using a 6 pixels Gaussian fit. Usual uncertainties for the size of the G2/DSO are in the range of 12.5 mas. For the values in 2014.3 and 2015.4, please see Section 2.8 and Section 4. Here, the format of the time is given in decimal years.

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The related spectral line fits and tables for OS1 and OS2 can be found in the Appendix E (see Figures 26, 27, and Table 14).

As mentioned in Section 2 and shown in Figure 2, we use the orbit of S2 to derive the position of Sgr A* with a positional uncertainty of 12.5 mas (= 1 pixel). We list the orbital elements that are adapted from Peißker et al. (2020a) in Table 5. For the Keplerian orbital fit, we extract the G2/DSO position with a Gaussian fit that simultaneously provides a positional uncertainty. Since this value would underestimate the unstable character of the data, we adapt the common uncertainty of 12.5 mas for the spatial position of the G2/DSO with respect to Sgr A*. Please consider Table 15, Appendix F for the derived relative position with respect to Sgr A*. The related LOS velocities can be found in Table 4.

In Figure 6, we show the outcome of the Keplerian model fit. This solution underlines the Keplerian trajectory of the G2/DSO. The best-fit orbital elements are listed in Table 5. Based on the orbital elements, we find that a periapse passage of the G2/DSO occurred in ∼2014.38, which is in agreement with the line-map detection (see the following subsection and Figure 3), as well as with the earlier calculation of the pericenter passage presented in Valencia-S. et al. (2015), where the authors report t_closest = 2014.39 ± 0.14. In Figure 7, we show the LOS velocity evolution of the G2/DSO as a function of time. The absolute value of the LOS velocity of the G2/DSO reaches 3338 km s⁻¹. The common uncertainty for the velocities is ± 35 km s⁻¹ (see Peißker et al. 2020d). We use the orbital elements given in Table 5 to investigate the statistical robustness (Figure 29, Appendix G). The MCMC simulations agree with the initial input parameters. Since the uncertainty does not reflect the character of the data nor the observations, we again choose a more conservative approach and round up the range of possible values. The final uncertainties for the orbital elements (which are based on the Keplerian fit) are given in Table 5. Additional results of this analysis are the mass M_{Sgr A*} of Sgr A* of about M_{Sgr A*} = 4.15 × 10⁶ M_⊙ and the distance d to the GC of d = 8.3 kpc. Based on the orbital elements, we find a pericentre distance r_p of r_p = a(1 − e) ≈ 0.66 mpc.

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Since the literature suffers from inconsistencies regarding the nature of the G2/DSO or the general periapse sequence (Gillessen et al. 2012; Witzel et al. 2014; Pfuhl et al. 2015; Valencia-S. et al. 2015), we focus on the observations that cover the epoch of 2014. It was predicted and shown that the gas cloud G2/DSO can be observed simultaneously before and after its periapse, therefore we will investigate the region of interest (see Figure 4 shown in Gillessen et al. 2019). In combination with the velocity (and hence wavelength) information given in Pfuhl et al. (2015), we show in Figure 8 the reported area of the blueshifted part of G2/DSO in 2014.3 (216 single exposures), 2014.6 (94 single exposures), and 2014.5 (310 single exposures). Based on the information of the published data (see Figure 15 in Pfuhl et al. 2015), we select the wavelength range 2.145–2.155 μm, which corresponds to the blueshifted velocities 1537–2922 km s⁻¹. As shown in Figure 8, the line maps suffer from noise and artefacts. This is consistent with Valencia-S. et al. (2015), where the authors encounter the same situation. However, we do not agree with the smoothed and artificially enhanced emission (the authors call it "scaling adjustment") that is shown in Pfuhl et al. (2015) for 2014.3. The G2/DSO in Figure 8 cannot be observed because the line maps are noise dominated, due to the large selected spectral range. We advise the interested reader to compare the significance of the Brγ emission in Figure 3 with Figure 8. We will discuss this result in detail in Section 4.

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According to Pfuhl et al. (2015), a clear velocity gradient should be observable, especially in the pre-pericentre phase of the G2/DSO. To maintain the overall shape of the data and hence provide a certain level of comparability with the pipeline results (i.e., the output mosaic data cube), we use PPV diagrams. We favor this approach because the spatial dimensions of a PV diagram are reduced to one parameter, which can result in confused detections. Since the outcome of a PV diagram purely rely on the derived orbit, we question the capability of this technique in the first place. However, for the PPV plots that are presented here, the spectral pixel information is transformed to a velocity. In Figure 9, we present a selected overview of the pre-periapse time of the G2/DSO. We divide the investigated spectral range of each year into equal-sized sections (i.e., spectral slices) and we then subtract the neighboring channels. Following this procedure, we preserve the information and avoid a disintegration of the emission. By comparing the PPV plots, no prominent velocity gradient is present. Furthermore, the data again suffer from artefacts and noise. We underline that we do not use any Gaussian-smoothing kernel to enhance nonlinear parts of the G2/DSO. Since the background subtraction influences the shape of the Doppler-shifted Brγ emission line (see Appendix C), the velocity distribution may contain artefacts. Please see a smoothed version of Figure 9 and a possible tail emission in Appendix C, Figure 23. In the following, we will investigate the influence of the sky emission on the data.

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3.4.1. Artificial Velocity Gradient

As previously discussed in Section 2, we analyze the efficiency of the sky correction in case the object- and sky-observations contain irregularities. For that purpose, we manipulate the sky correction with an error of −10% of the maximum flux of the input files; i.e., we subtract 10% of the peak emission of the sky frames. Furthermore, we smooth the original position–position–velocity map with a one-pixel Gaussian. The results of this analysis are listed in Table 6. While combining the observational data of an object for each year leads to a natural velocity gradient because of the intrinsic motion ("expected gradient"), we find lower values for the Brγ emission of the G2/DSO. Since the observations do not cover a complete year, this result is expected. However, measuring the velocity gradient of the G2/DSO in the sky manipulated data shows increased values that are close to or over the expected values. Smoothing the data increases the velocity gradient by about 30%–50%. As shown in Figure 9, the noisy character of the data is responsible for this increased velocity gradient. Hence, smoothing data impacts the outcome of the velocity gradient analysis.

Table 6. Influence of a Poor Sky Correction and Gaussian Smoothing on the Velocity Gradient

	2008	2010	2012
	$\left(\mathrm{km}\,{{\rm{s}}}^{-1}\right)$	$\left(\mathrm{km}\,{{\rm{s}}}^{-1}\right)$	$\left(\mathrm{km}\,{{\rm{s}}}^{-1}\right)$
Expected gradient	±45.58	±78.77	±101.2
No error	±19.97	±37.01	±63.77
10% error	±35.99	±88.27	±138.58
Smoothed, 1 px	±80.46	±143.94	±200.5
Plewa et al. (2017)	±238	±261	±464

Note. Details are given in the text.

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To investigate the possibility of a stellar source that is associated with the observed Doppler-shifted Brγ line emission, we apply the Lucy–Richardson algorithm to the data (see also Peißker et al. 2020d). We select the K-band in the related data cube and apply a background emission of 15%–30% of the peak intensity emission depending on the variable background. With this approach, we eliminate the chance of a false positive.

In Figure 10, we present the resulting convolved images. In agreement with the position of the Doppler-shifted Brγ line emission, we detect a source that is approaching and passing Sgr A* on a trajectory comparable to the G2/DSO. By comparing the K-band continuum position with the Brγ detection shown in Figure 3, we find a small offset of ≤12.5 mas (∼100 au) between the centroid of the stellar source and the line emission. For this comparison, please consider Figure 28, Appendix F. In 2016 and 2017, S31 and S23 (see Figure 2) coincide with the blueshifted G2/DSO Brγ line position.

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To investigate the K-band continuum magnitude and the flux density evolution of the G2/DSO along the orbit, we use the detection of the high-pass filter analysis shown in Figure 10. For the analysis, the zero magnitude flux is set to 653 Jy and an effective wavelength of 2.180 μm is applied. These settings are related to the ESO K-band filter (for comparable values, see also Tokunaga & Vacca 2007). For the magnitude, we use the extinction-corrected S2 magnitude of 14.15 mag. In the following, we adopt the bolometric magnitude relation with

$$\begin{eqnarray}&&{m}_{{\rm{G}}2/\mathrm{DSO}}=-14.15+2.5\times \mathrm{log}(\mathrm{ratio})\end{eqnarray} \tag{ 1 }$$

where "ratio" is defined as the counts of the object of interest (here G2/DSO) and the reference star. Since we normalize the data to the flux of S2, "ratio" simplifies to the counts of the G2/DSO. Because of confusion, we exclude the data points of 2005, 2016, and 2017. While the uncertainty is based on the standard deviation, we find an average magnitude of the G2/DSO of 18.48 ± 0.22 mag. For the averaged flux f, we adapt

$$\begin{eqnarray}&&{f}_{{\rm{G}}2/\mathrm{DSO}}={f}_{{\rm{S}}2}\,\times \,{10}^{-0.4({\mathrm{mag}}_{{\rm{G}}2/\mathrm{DSO}}-{\mathrm{mag}}_{{\rm{S}}2})}\end{eqnarray} \tag{ 2 }$$

from Sabha et al. (2012) to calculate the related flux density of G2/DSO with f_S2 = 14.73 mJy and mag_S2 = 14.1. We get 0.26 ± 0.06 mJy, which is fully consistent with the literature. Comparing the pre- and post-periapse epochs, we find a slightly decreasing magnitude toward Sgr A*. This is expected because Sabha et al. (2012) shows that the density of (old) faint stars increases toward Sgr A*. To eliminate the chance that the magnitude that is presented here is correlated to background fluctuations or nearby stars, we measure its intensity close to G2/DSO with a one-pixel aperture at a distance (∼40 mas) larger than the radius of the SINFONI PSF (31.25–37.5 mas). Based on Figure 11, we find that neither the pre- nor post-periapse data is correlated to background fluctuations or surrounding stars. It is reasonable to assume that the K-band magnitude of G2/DSO would increase because of close-by stars or the dominant background light toward Sgr A*. In Section 4, we will elaborate on these points.

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As was previously shown and described in Peißker (2018), we created a series of PV diagrams inspired by Gillessen et al. (2012) where the authors show the gas cloud G2/DSO and its tail component with a Keplerian orbit approaching Sgr A*. Since Gillessen et al. (2012) smooth parts of the presented image (see the dotted box in the left-hand panel of Figure 12), we investigate the nonsmoothed emission between 2.1661 and 2.182 μm in contrast. Without an imperious smoothing kernel that is applied to the data, we confirm emission at the position of the so-called tail that is above the noise level. However, we cannot agree with the interpretation of this emission because we detect isolated sources that show temporary close distance to the G2/DSO (see the right-hand panel of Figure 12). In the following subsections, we will investigate these sources, which we refer to as OS1 and OS2, in detail. Additional material covering the so-called tail can be found in Appendix C, see Figures 23 and 24.

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3.7.1. OS1

As indicated in Figure 12 (right-hand panel), we identify OS1 in several epochs following the G2/DSO on a similar orbit (see Figures 13 and 14). In several epochs, the identification of OS1 suffers from a decreased data quality and nearby stellar sources.

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Using the extracted positions, we derive an orbit for OS1 (see Figure 15). The uncertainties reflect the distance to nearby stellar sources and include the discussed Sgr A* position range.

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In Table 7, we list the related orbital elements.

Table 7. Orbital Elements for the Sources OS1 and OS2

Source	a (mpc)	e	i (deg)	ω (deg)	Ω (deg)	tclosest (yr)
OS1	16.75 ± 0.50	0.762 ± 0.075	71.39 ± 20.68	93.79 ± 8.42	271.52 ± 16.50	2020.67 ± 0.02
OS2	12.48 ± 0.19	0.605 ± 0.019	84.62 ± 4.98	140.26 ± 4.92	245.28 ± 2.17	2029.87 ± 0.05
G2/DSO	17.45 ± 0.20	0.962 ± 0.004	58.72 ± 2.40	92.81 ± 1.60	295.64 ± 1.37	2014.38 ± 0.01

Note. For comparison, we additionally list the orbital elements of the G2/DSO (Table 5). The pericenter distance of OS1 and OS2 is 3.99 mpc/822 au and 4.93 mpc/1017 au, respectively. For comparison, the pericenter distance of the G2/DSO is 0.66 mpc/137 au.

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With r_p = a(1 − e), we derive a pericenter distance of 1201.56 au (100.08 mas) for OS1.

Please see Figure 26 in Appendix E for the spectral line evolution, and Tables 14 and 15 for the related positions and LOS velocities.

3.7.2. OS2

An inspection of Figure 12 (right-hand panel) implies that OS2 can be observed with an increased intensity count compared to OS1 in 2008. Hence, the object could be less prone to confusion due to nearby stellar sources. Given the fluctuating background in the S-cluster and the changing distance to nearby stars, the intensity difference in 2008 may not be true for every other epoch. However, using the analysis tools that we already applied to the G2/DSO and OS1, we find OS2 throughout the data approaching Sgr A* (Figure 16). We use a Keplerian orbital fit and find with a satisfying agreement with the data a plausible solution for the trajectory of OS2 (Figure 17). The related orbital elements can be found in Table 7. Using the relation for the pericenter distance, r_p = a(1 − e), we calculate a pericenter distance of 1485.82 au or 123.75 mas for OS2. In Appendix E, in Figure 27, we show the Doppler-shifted LOS velocity evolution (Brγ based) with the related fit. Furthermore, we list the positions and LOS velocities in Tables 14 and 15, respectively.

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By inspecting and analyzing the SINFONI line maps, we find G2/DSO on its Keplerian trajectory around Sgr A* between 2005 and 2019 in the Doppler-shifted Brγ regime. The Brγ line is less affected by tellurics but coincides with OH emission lines with a relative flux between 20% and 50%. As we have shown in Sections 2 and 3, not only image motion but also the sky emission variability influences the analysis of G2/DSO. While the distortions of long-time SINFONI exposure data cubes show nonlinear inconsistencies regarding object positions (and hence impact positional uncertainties), the shape of the emission lines are affected by an insufficient sky correction. By introducing an error to the sky correction files, we find values for the velocity gradient that are twice as prominent as the observed value. Arguably, our naive approach of the sky variability may not cover every aspect of the topic because this is beyond the purpose of this analysis. Hence, it is safe to assume that this introduced error could be increased or decreased in reality.

Since the final data cubes cover a wide range of data of the related epochs, a "natural" velocity gradient is expected. However, smoothing the data leads to a velocity gradient that is twice as large as the expected value. This is expected because of the noisy character of the SINFONI data. Nevertheless, the confusion-free detection of the Doppler-shifted Brγ emission line implies a Keplerian orbital evolution. The periapse of G2/DSO can be dated to 2014.38 with a pericenter distance of about 137 au. Comparing the line emission before and after the pericenter passage reveals a preserved shape of the observed envelope (Figure 3), which implies that the gravitational influence of Sgr A* is in the uncertainty domain (Eckart et al. 2013). The LOS velocity follows the evolution expected for the Keplerian trajectory (Figure 14). Based on the Doppler-shifted G2/DSO line analysis, we find the maximum values of v_z = 3000 km s⁻¹ (pre-periapse) and v_z = 3200 km s⁻¹ (post-periapse) which is about 1% of the speed of light. Furthermore, we investigated the blueshifted line maps in 2014 that should show a prominent structure, as shown in Pfuhl et al. (2015). As was already indicated in Valencia-S. et al. (2015), the line maps do not show comparable structures, as presented in Pfuhl et al. (2015) and Plewa et al. (2017). Using a slit along the orbit in combination with a smoothing kernel and enhancing only parts of the presented image may lead to a false interpretation of the data (see Figure 12 and Table 6). To underline this point, we presented nonsmoothed PPV diagrams of G2/DSO of 2008, 2010, and 2012. In these years, the source is rather isolated in relation to nearby stars but should also exhibit a noticeable velocity gradient. However, the analysis shows a rather compact source that suffers from background noise. In Appendix C, in Figure 23, we smooth the data with a 3 px Gaussian kernel. The results indeed show some structures that could be interpreted as a possible tail structure (marked in Figure 23, Appendix C). Unfortunately, the nonsmoothed data do not support this interpretation because of the noise. Hence, placing a slit along the orbit and smoothing the emission will most likely produce artefacts. We will investigate this point separately in detail in an upcoming publication because it would exceed the scope of this work.

As was first proposed by Murray-Clay & Loeb (2012), the presence of a stellar counterpart surrounded by a gaseous-dusty photoevaporating protoplanetary disk can explain both the ionized gas of ∼ 10⁴ K traced by broad Brγ emission, as well as the dust component revealed by the prominent excess toward longer infrared wavelengths, in particular L- and M-bands. The color–color diagram (H − K versus K − L) of Eckart et al. (2013), the foreshortening factor temporal evolution presented in Valencia-S. et al. (2015), the detected polarized continuum light by Shahzamanian et al. (2016), as well as the derived SED based on the 3D dusty model by Zajaček et al. (2017) and also Peißker et al. (2020d) all support the dust-enshrouded star model of G2/DSO. These findings are also consistent with Scoville & Burkert (2013), who suggested a supersonic low-mass T-Tauri star with a stellar wind that produces a two-layer bow-shock while interacting with the ambient hot X-ray emitting gas. In this scenario, a Brγ emission line is produced in the colder and denser stellar-wind shock via the collisional ionization at the shock front and/or the cooling X-ray/UV radiation of the post-shock gas. Additionally, Ciurlo et al. (2020) used a stellar model for the findings of the so-called G-sources which is compatible with the discussion of the same objects in Peißker et al. (2020d). Witzel et al. (2014, 2017), who also favor a dust-embedded star scenario, came to the same conclusion by analyzing the L-band flux density before and after the periapse of G2/DSO and G1, respectively. They found a constant flux density for G2/DSO within uncertainties, which implies a compact dust-enshrouded stellar source, while for G1 they detected a drop by nearly 2 magnitudes that can be interpreted by the tidal truncation of an extended envelope. The constant flux behavior of G2/DSO can also be confirmed for its pericenter passage in this work. We note a slight flux increase for the data between 2018 and 2019. This can be explained by the partial removal of an envelope material during the pericenter passage where the G2/DSO host star is more revealed and as a result the overall K-band flux density increases. Eckart et al. (2013) suggest that the location of the Lagrange point L1 hinders a complete disruption of the envelope because the denser component that is inside approximately one astronomical unit is bound to the star, see the tidal (Hill)radius estimate given by Equation (3). Detailed numerical models are beyond the scope of this work but should be investigated in future works based on our findings. Since we used a high-pass filter, we minimized the contaminating influence of overlapping PSF (-wings) and maximize the accessible information. We find a stellar counterpart at the expected position in agreement with the Brγ emission line that is moving on the same orbit as G2/DSO. Note that we have already investigated the broad spectrum of results that can be derived by using a high-pass filter, see in particular the results presented in Peißker et al. (2020a). We find that a high number of iterations (∼10,000 iterations) in combination with a solid PSF can result in robust detections. As discussed in Eckart et al. (2013) and Peißker et al. (2021a), and also shown in Sabha et al. (2012), the possibility for a side-by-side flyby becomes negligible after about 3 yr. Even though the orbit of S23 and S31 interfered with the trajectory of the G2/DSO during the years 2015–2017 and 2019, we confirm the robust detection of a stellar source at the position of the G2/DSO for most of the investigated years.

Several publications claim the existence of a tail component of G2/DSO that is supposed to be created because of the gravitational and hydrodynamical interaction with the environment of Sgr A*. Unfortunately, it is not clear why the tail moves on a different orbit than the head component (see Figure 3 in Pfuhl et al. 2015). If there is a responsible process for this orbit discrepancy, then we ask why the head is unaffected? Assuming a much higher density for the head compared to the so-called tail, it is controversially reported that G2/DSO was supposed to be destroyed during the periapse. However, Gillessen et al. (2019) reports that G2 is rather compact again after the periapse, due to tidal focusing, and moves on a drag-force influenced orbit. Assuming that material would have been accreted by Sgr A* during the periapse of G2/DSO, the flare observed by Do et al. (2019b) could be a speculative link. However, it is also reasonable to assume that the periapse of S2 in 2018 (Schödel et al. 2002; Do et al. 2019a) could have created instabilities in the accretion disk of Sgr A* (Suková et al. 2021), which result in a bright flare. In contrast to the drag-force influenced orbit as proposed by Gillessen et al. (2019), we show in this work that G2/DSO follows a Keplerian trajectory where the source is not significantly affected by the tidal field of Sgr A* (see, e.g., Figure 3). Furthermore, we show in Figure 24 that the ionized gas which is associated with the tail (Gillessen et al. 2013b) was in the S-cluster before G2/DSO passed by. By investigating the tail we find that it consists rather of isolated sources that can be, like G2/DSO, detected in the Doppler-shifted Brγ line (see Figure 18). Considering the observed number of S-stars (about 40), the amount of line-emitting objects is of the same order of magnitude (almost 20). Hence, the presence of OS1 and OS2 contributes to the overall distribution of line-emitting objects (for a complete overview, see Ciurlo et al. 2020; Peißker et al. 2020d).

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Due to the Brγ compactness of the object in combination with the photometric detection of a K-band counterpart at the position of the line-emitting source (Figures 10 and 28, Appendix F), we find strong support for a possible young stellar object. As shown in Lada (1987), a young protostar (<1 × 10⁷ yr) consists of a blackbody stellar emission, as well as a cooler disk/envelope component. Hence, a two-component SED fit as presented in Peißker et al. (2020d) provides a suitable explanation for the observed continuum emission and is in agreement with the predictions by Murray-Clay & Loeb (2012). Because of the ongoing accretion processes, the emission lines can show a nonsymmetric profile, which is amplified by the interaction with the ambient medium, in particular by the formation of a bow-shock layer (Shahzamanian et al. 2016; Zajaček et al. 2016). As pointed out by Zajaček et al. (2017), the sum of the observational results underlines the stellar nature of the object that consists of a non-spherical gaseous-dusty envelope shaped by the bow shock, as well as by bipolar cavities. Hence, it is expected that the ionized gas is not centered at the source itself but exhibits an offset. Considering the mentioned bow-shock layer (Zajaček et al. 2016) in combination with the polarized continuum detection (Shahzamanian et al. 2016), the proposed nature of G2/DSO as a young T-Tauri star by Scoville & Burkert (2013) and Eckart et al. (2013) seems reasonable. Furthermore, the Brγ width variation (Figure 25, Appendix D) of G2/DSO is in line with observations of YSOs by Stock et al. (2020) and emphasizes the proposed classification.

The pericenter passage of G2/DSO is dated to 2014.38, OS1 and OS2 follow in 2020.67 and 2029.87, respectively. Even though the orbital elements for OS1 and OS2 are different, the former source shows similarities in inclination and the argument of periapse with G2/DSO. Because of the compactness and the Doppler-shifted Brγ line emission in combination with the missing [Fe iii] detection in contrast to the D-/G-sources (see Ciurlo et al. 2020; Peißker et al. 2020d), which are located west of Sgr A*, we hypothesize if G2/DSO, OS1, and OS2 share a common history. If so, then they should have been formed in the same dynamical process. Following this speculative scenario, the combination with the detected reservoir of fast moving molecular cloudlets (Moser et al. 2017; Goicoechea et al. 2018; Hsieh et al. 2021) and the simulation of an infalling cloud presented in Jalali et al. (2014) might provide a suitable explanation for the in situ star formation event. The authors of Jalali et al. (2014) model a 100 M_⊙ cloud that crosses the Bondi radius of Sgr A* (for observations, see also Tsuboi et al. 2018). As an initial setup, Jalali et al. (2014) assume the loss of an angular momentum via the clump–clump collisions within the CND (see also Scoville et al. 1986; Tan 2000; Tartėnas & Zubovas 2020; Tress et al. 2020, for studies where molecular cloud–cloud collisions were considered). Because of the gravitational potential of Sgr A*, the initial cloud gets stretched and triggers, because of a compression force acting perpendicular to the orbital motion, the creation of YSO associations, similar to IRS13N. We will elaborate on this in more detail in Section 4.7.

It is rather unlikely that the G2/DSO itself was formed in the clockwise stellar disk located further out (CW disk, see Figure 19) as, for example, claimed by Burkert et al. (2012) because of a large difference of inclinations (i_CSD ≈ 115° ≠ 60° ≈ i_G2/DSO). The nondetected destruction process of the source (see, e.g., Burkert et al. 2012; Schartmann et al. 2012) and the previously derived inclination of about 110°–120° for G2/DSO may be explained by the lower data baseline.

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By analyzing the K-band emission of the continuum counterpart of the Doppler-shifted Brγ line source, we find an averaged magnitude of 18.48 ± 0.22 mag with a correlated flux of 0.26 ± 0.06 mJy, which matches the values presented in Eckart et al. (2013) and Shahzamanian et al. (2016). In agreement with Sabha et al. (2012), we observe a background magnitude close to G2/DSO and as a function of the distance toward Sgr A* with a slope of 0.13. This underlines the robust observation of the K-band magnitude observation of G2/DSO in pre-/post-periapse epochs which is not correlated to the increasing background light toward Sgr A*.

The G2/DSO passed Sgr A* in 2014.38 at a pericenter distance r_p of about 137 au. The true size of the G2/DSO can be inferred from the foreshortening factor, which is maximized during the periapse (Valencia-S. et al. 2015). By applying a Gaussian fit to the Brγ line map in 2014.38 (see Figure 3), we find a symmetrical shaped FWHM of about 37.5 mas (i.e., 3 px). Over 85% of the total emission of the G2/DSO is concentrated in a very compact area with a radius of 12.5 mas in 2014.38 (see Figure 3). In combination with the foreshortening factor, the character of the G2/DSO can be classified as compact. Furthermore, the authors of Scoville & Burkert (2013), Eckart et al. (2013), and Valencia-S. et al. (2015) derive a mass of 1–2 M_⊙ for the G2/DSO. From the observed averaged K-band magnitude in this work of mag_K = 18.48 in combination with the H- and L'-band magnitude (see Eckart et al. 2013; Peißker et al. 2020d), we derive colors of K − L = 4.08 mag and H − K = 1.48 mag for the G2/DSO (see Figure 20). These are common values not only for Herbig Ae/Be stars with ice features but also for the IRS13N sources (see also Eckart et al. 2004; Moultaka et al. 2005). Recently, Cheng et al. (2020) reported matching values for observed YSOs with a circumstellar disk. With the upcoming Mid-Infrared Instrument (see Bouchet et al. 2015; Ressler et al. 2015; Rieke et al. 2015) for the James Webb Space Telescope, we will be able to confirm the presence of ice features (Moultaka et al. 2015).

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The estimated radius of 12.5 mas close to the pericenter passage corresponds to the upper limit on the physical length scale of ∼103 au. However, the true physical size of the G2/DSO envelope is plausibly one or two orders of magnitude smaller. Using the derived pericenter distance and the G2/DSO mass estimate, we obtain the tidal radius at the pericenter of

$$\begin{eqnarray}\begin{array}{rcl}{r}_{{\rm{t}}}^{\mathrm{per}} & = & a(1-e){\left(\displaystyle \frac{{m}_{\star }}{3{M}_{\bullet }}\right)}^{1/3}\\ & \sim & 0.6{\left(\displaystyle \frac{{m}_{\star }}{1\,{M}_{\odot }}\right)}^{1/3}\,\mathrm{au}.\end{array}\end{eqnarray} \tag{ 3 }$$

Given the orbital period of the G2/DSO, P_orb ∼ 105 yr, the source likely went through several pericenter passages, possibly thousands in case of a stellar nature. Therefore, the length scale of the G2/DSO is likely to be small, of the order as expressed by Equation (3), which makes the source interesting from the point of view of stellar evolution and extreme star formation. The 3D MCMC radiative transfer simulations performed by Zajaček et al. (2017) showed that the basic continuum properties of the source can be reproduced with the compact gaseous and dusty envelope of the order of an astronomical unit. The constant L-band (Witzel et al. 2014) and also K-band flux density (this work) imply that the tidal prolongation and truncation of the envelope has been rather small, which is in contrast to the G1 source (Witzel et al. 2017) that has exhibited profound SED changes in the post-pericenter phase. The recent increase in the K-band flux density after 2017, see Figure 11, may indicate changes in the envelope morphology. However, this will need to be clarified with more post-pericenter L- and K-band data. In conclusion, the length scale of the G2/DSO has been ∼1 au, both before and after the pericenter passage.

4.5.1. Brγ Line Width of the G2/DSO

Pfuhl et al. (2015) discussed and showed a possible drag force acting on the G2/DSO. Within this model, the authors combined the data of G1 and G2/DSO to demonstrate the effect of the predicted inspiraling cloud toward Sgr A* after the pericenter passage (see, e.g., Gillessen et al. 2012). In Figure 21, we compare the predicted trajectory of G2/DSO that is following G1 as part of a gas streamer (Pfuhl et al. 2015) with the observed data (black data points) and the derived orbit. The comparison indicates that no measurable drag force as proposed by Pfuhl et al. (2015) can be observed and its assumption that G2/DSO is part of a gas streamer cannot be confirmed. In comparison to Pfuhl et al., the magnetohydrodynamic drag-force analysis by Gillessen et al. (2019) discusses a smaller drag force with respect to the pre-pericenter orbit of G2/DSO. However, the hardly recognizable presented effect is underlined by shortening the observational available pre-pericenter data points in the related publication of Gillessen et al. (2019) (see their Figure 2). When we consider all available SINFONI data up to 2019, the position- and velocity-data can simultaneously be fitted well with a simple Keplerian orbital solution, see Figures 6 and 7. Following Occam's razor (see, e.g., Ariew 1976), the inclusion of an additional drag term appears redundant. Moreover, we have shown that the influence of image motion (Table 1) and sky variability can have a significant impact on the data by up to 40%. Based on the data presented in Gillessen et al. (2019), we conclude that the arguably small offset (< few percent) of the Brγ source in combination with the presented line shape might not be explained by a drag force and can rather be explained by the effects discussed in this work. By carefully inspecting the provided Kepler- and Drag Force-fits in Gillessen et al., we note that neither model perfectly matches the shown Brγ emission that underlines the noisy character of the SINFONI data. Based on these points, we do not find a strong need for a drag force to explain the orbital evolution of G2/DSO (Figure 3). Similarly, Witzel et al. (2017) found that the evolution of G1 is consistent with the Keplerian motion on a different highly eccentric orbit to that of the G2/DSO, which supports the stellar nature of both sources. In this regard, the tests of alternative gravitational theories, such as the fermionic dark matter compact core–diffuse halo (Becerra-Vergara et al. 2020), that has made use of the G2/DSO orbit should be updated accordingly.

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In Zajaček et al. (2014), the authors propose the possibility that G2/DSO can be associated with a binary or multiple-star dynamics close to the Galactic center. In particular, they showed that if G2/DSO is a binary system, then it would lead to the disruption event at the pericenter. This model was soon followed by scenarios that explain the G2/DSO and related objects as binary merger products (Perets et al. 2007; Stephan et al. 2016, 2019). Following the binary merger fraction discussion of Ciurlo et al. (2020), we adapt

$$\begin{eqnarray}&&R=\displaystyle \frac{1}{2}\displaystyle \frac{{N}_{B}}{{N}_{m}}\end{eqnarray} \tag{ 4 }$$

from this paper, and assume that line-emitting and dusty objects are binary merger products. By using this equation, the binary fraction R is calculated with the number of binaries N_B and the number of low-mass stars N_m . To provide a certain degree of comparability, we adapt N_m = 478 from Ciurlo et al. and consider the additional sources from Peißker et al. (2020d), which results in N_B = 85. For R, we derive ∼9%, which is almost twice as much as the binary fraction of ∼5% calculated in Ciurlo et al. (2020). Since we assumed that all objects are binary mergers, it is safe to conclude that the assumption is not justified. We obtain a low-mass binary fraction that is almost twice than that derived by Ciurlo et al. and also larger than the overall predicted value of 6%–7% (Raghavan et al. 2010; Stephan et al. 2019; Ciurlo et al. 2020). Hence, we conclude that a considerable fraction of the overall population of line and dust emitting sources are not necessarily binary mergers but also YSOs.

Assuming a speculative binary disruption scenario, the pericenter passage at an unspecified time of the G2/DSO would have been responsible for the tidal break-up of the binary components that we denote here as G2/DSO₁ and G2/DSO₂. In this regard, G2/DSO₂ is a runaway star, while G2/DSO₁ is a highly eccentric component that follows a slightly modified Keplerian trajectory (Hills 1988) with a smaller eccentricity and a semimajor axis (Zajaček et al. 2014). Following this, the post-pericenter trajectory of G2/DSO₁ could mimic the inspiral due to the drag force (Gillessen et al. 2019). If we consider the second star (G2/DSO₂), the magnitude of this component is at most 18.5 mag. Because of the preserved Brγ shape in the post-periapse years of the observed G2/DSO₁, we are able to speculate that G2/DSO₂ did not capture most of the surrounding stellar material. This implies a mass estimate of G2/DSO₁ > G2/DSO₂. With the derived mass for the G2/DSO of about 1–2 M_⊙, we assume that an upper limit for the mass of G2/DSO₂ is closer to ${m}_{{\rm{G}}2/{\mathrm{DSO}}_{2}}\,\lt \,1\,{M}_{\odot }$.

We note that the disruption scenario also applies to the theory of in situ star formation, where an infalling cloud that is stretched and compressed forms associations of young stellar objects (Jalali et al. 2014). These associations are first bound systems with a certain velocity dispersion. If they orbit the SMBH on an eccentric orbit, then the tidal radius of the YSO cluster is time-variable depending on the distance from the SMBH. In particular, the tidal (Hill) radius during the pericenter passage is

$$\begin{eqnarray}\begin{array}{rcl}{r}_{{\rm{t}}}^{\mathrm{cluster}} & = & \overline{a}(1-\overline{e}){\left(\displaystyle \frac{{m}_{\mathrm{cluster}}}{3{M}_{\bullet }}\right)}^{1/3}\,\\ & \sim & 14.4\left(\displaystyle \frac{\overline{a}}{15.6\,\mathrm{mpc}}\right)\left(\displaystyle \frac{\overline{e}}{0.78}\right){\left(\displaystyle \frac{{m}_{\mathrm{cluster}}}{100\,{M}_{\odot }}\right)}^{1/3}\,\mathrm{au},\end{array}\end{eqnarray} \tag{ 5 }$$

where we considered the mean values of the semimajor axis and the eccentricity based on G2/DSO, OS1, and OS2 (see Table 7). The cluster mass was scaled to the order of magnitude considered in the simulations of Jalali et al. (2014) for an infalling molecular cloud. Since the YSO cluster length scale approximately given by the Jeans length scale for a given critical density for self-gravitation is larger than ${r}_{{\rm{t}}}^{\mathrm{cluster}}$, the cluster is effectively dissolved at the distances where dusty orbits are now seen to orbit. Hence, during the pericenter passage, the cluster will tend to dissociate and lose its members, which will afterwards orbit around Sgr A* on independent orbits with initially similar orbital elements (i.e., inclination, longitude of the ascending node, and argument of the pericenter). Due to the resonant relaxation and the perturbative effects of the S cluster, as well as Newtonian (mass) and Schwarzschild precession, the inclination and also the other orbital elements will start to deviate. In this regard, G2/DSO, OS1, OS2, and the other dusty objects can share a common origin, despite certain offsets in the orbital elements; see Table 7 and Figure 19 for comparison.

To estimate how much individual dynamical processes can alter orbital elements of interest (i.e., inclination, longitude of the ascending node, and argument of the pericenter), we compare their fundamental timescales with the assumed lifetime of the dusty sources of ∼ 10⁵ yr. In particular, the argument of pericenter shift between mainly G2/DSO and OS2 could be explained by the Schwarzschild precession, which per orbit can be estimated as follows,

$$\begin{eqnarray}&&{\rm{\Delta }}\phi =\displaystyle \frac{6\pi {{GM}}_{\bullet }}{{c}^{2}{a}_{{\rm{G}}2}(1-{e}_{{\rm{G}}2}^{2})}\end{eqnarray} \tag{ 6 }$$

which for the G2/DSO yields ${\rm{\Delta }}\phi =9\buildrel{\,\prime}\over{.} 52$ per orbital period (P_G2 ∼ 108 yr). The difference for the argument of periapse of ∼50° between OS2 and G2/DSO (see Figure 19) could thus be achieved in ∼35,000 yr, due to the much faster prograde relativistic precession of G2/DSO than OS2 (∼157 per its orbital period). The Newtonian (retrograde) mass precession counter-balances the relativistic precession and acts in an opposite direction. The mass coherence timescale ${T}_{{\rm{c}}}^{M}$, on which the arguments of the periapse would be randomized, can be estimated as

$$\begin{eqnarray}\begin{array}{rcl}{T}_{{\rm{c}}}^{M} & \sim & \displaystyle \frac{{M}_{\bullet }}{\langle {M}_{\star }\rangle }\displaystyle \frac{{P}_{{\rm{G}}2}}{{N}_{\star }}\\ & \sim & 4\times {10}^{5}{\left(\displaystyle \frac{\langle {M}_{\star }\rangle }{10\,{M}_{\odot }}\right)}^{-1}\left(\displaystyle \frac{{P}_{{\rm{G}}2}}{100\,\mathrm{yr}}\right){\left(\displaystyle \frac{{N}_{\star }}{100}\right)}^{-1}\,\mathrm{yr},\end{array}\end{eqnarray} \tag{ 7 }$$

where 〈M_⋆〉 is the mean stellar mass, P_G2 is the orbital period of G2/DSO, and N_⋆ is the number of stars inside its orbit. If the lifetime of G2/DSO, OS1, and OS2 is at most ∼ 10⁵ yr, then the mass precession has not significantly altered their orbital orientations.

The resonant relaxation process, which is characteristic of highly symmetric potentials, such as inside the sphere of influence of Sgr A*, proceeds in two modes: a faster vector resonant relaxation (VRR) and a slower scalar resonant relaxation (SRR). The VRR relaxation timescale is

$$\begin{eqnarray}\begin{array}{rcl}{T}_{\mathrm{VRR}} & \sim & \displaystyle \frac{{M}_{\bullet }}{\langle {M}_{\star }\rangle }\displaystyle \frac{{P}_{{\rm{G}}2}}{\sqrt{{N}_{\star }}}\\ & \sim & 4{\left(\displaystyle \frac{\langle {M}_{\star }\rangle }{10\,{M}_{\odot }}\right)}^{-1}\left(\displaystyle \frac{{P}_{{\rm{G}}2}}{100\,\mathrm{yr}}\right){\left(\displaystyle \frac{{N}_{\star }}{100}\right)}^{-1/2}\,\mathrm{Myr},\end{array}\end{eqnarray} \tag{ 8 }$$

while the SRR relaxation, which also changes the magnitude of the angular momentum is about 10 times slower because of the definition of the scalar relaxation time, T_SRR ∼ P_G2 M_•/〈M_⋆〉 (see Hopman & Alexander 2006; Alexander 2017, for details). If G2/DSO, OS1, and OS2 were formed approximately in the same orbital plane ∼ 10⁵ yr ago, then the VRR has not had enough time to randomize their orbits but it could have contributed to the ∼20° spread in orbital inclinations over time.

In terms of the deviations in the longitude of the ascending node, the orbital precession can be relevant; especially when the stellar motion is perturbed by the presence of an inclined massive stellar disk. Ali et al. (2020) revealed that the S cluster consists of at least two perpendicular stellar disks, which indicates a non-randomized stellar distribution. Beyond the S cluster, at the radius of R_d ∼ 0.1 pc, there is a stellar disk of about hundred OB stars, with a potential total mass of M_d ∼ 10,000 M_⊙ (Paumard et al. 2006; Bartko et al. 2009). If the G2/DSO infrared source is inclined by β ∼ 60° with respect to the disk plane, then the stellar disk induces torques on the misaligned dust-enshrouded objects, which then precess with a certain period T_prec around the symmetry axis, effectively shifting the line of nodes. The rate of this shift depends on the semimajor axis a_⋆. Because a_G2 = 0.01745 pc < R_d ∼ 0.1 pc, one can use the following analytical formula to estimate the stellar precession period with respect to the disk (Nayakshin 2005; Löckmann et al. 2008),

$$\begin{eqnarray}&&{T}_{\mathrm{prec}}\simeq \displaystyle \frac{8\pi {M}_{\bullet }}{3{M}_{{\rm{d}}}\cos \beta }\sqrt{\displaystyle \frac{{a}_{\star }^{3}}{{{GM}}_{\bullet }}}\displaystyle \frac{{({a}_{\star }^{2}+{R}_{{\rm{d}}}^{2})}^{5/2}}{{a}_{\star }^{3}{R}_{{\rm{d}}}^{2}}.\end{eqnarray} \tag{ 9 }$$

For the G2/DSO, the precession period can be estimated to be T_p ∼ 2.34 × 10⁷ yr; hence it is a slow process, which leads to the shift of ∼15 in terms of the argument of the ascending node during 10⁵ yr or ∼108 in 700,000 yr. For OS2, the precession shift is slightly smaller than for G2/DSO and OS1, ∼1 degree per 100,000 yr, which could have contributed to the ascending node offset of ∼6° during the last million years. Hence, most of the difference in the ascending node is potentially attributable to the intrinsic dispersion during the formation process.

Finally, the presence of a perturbing stellar disk at the scale of R_d ∼ 0.1 pc with the mass of M_d ∼ 10⁴ M_⊙ induces a Kozai–Lidov (KL) oscillations of eccentricity and inclination. The period of the KL cycle can be estimated as (Šubr & Karas 2005),

$$\begin{eqnarray}&&{T}_{\mathrm{KL}}=2\pi \left(\displaystyle \frac{{M}_{\bullet }}{{M}_{{\rm{d}}}}\right){\left(\displaystyle \frac{{R}_{{\rm{d}}}}{{a}_{{\rm{G}}2}}\right)}^{3}{P}_{{\rm{G}}2},\end{eqnarray} \tag{ 10 }$$

which yields T_KL ∼ 4.73 × 10⁷ yr. Since the timescale is about two orders of magnitude longer than the assumed lifetime of dust-embedded sources, the KL effect can slightly contribute to the above-mentioned effects to account for the overall offset of orbital elements.

Since all of the relevant dynamical processes operate on longer timescales than the assumed lifetime of dust-enshrouded stars, most of the offset among orbital elements reflects the way they formed—i.e., from a turbulent fragmenting molecular cloud with an intrinsic offset due to a velocity dispersion, or G2/DSO and OS1 formed initially as a binary that disrupted, with OS2 forming separately as a single star. More detailed numerical dynamical studies are beyond the scope of this study.

In the binary merger scenario, the two components of a binary star initially orbit the SMBH, which acts as a more distant perturber. The SMBH as a perturber would induce Kozai–Lidov (Kozai 1962; Lidov 1962; Naoz 2016) resonances of the two components, where during one cycle the eccentricity growth is exchanged for the inclination decrease, and vice versa. Finally, at large orbital eccentricities, the two components are tidally distorted, induce torques on each other, and both stars are finally driven to merge. Such a merger product contracts on a Kelvin–Helmholtz timescale and is often associated with optically thick dusty outflows that give rise to the NIR-excess (Stephan et al. 2016, 2019). Hence, because of the resemblance to young stellar objects, it is difficult to distinguish between binary mergers and pre-main-sequence stars merely based on the photometry and the spectroscopy of the sources. Moreover, as mentioned earlier, the SINFONI data suffers from noisy behavior. Hence, the observational indications to verify a merger scenario are rather challenging to determine.

One distinguishing feature between YSOs that formed in situ in a single star formation event and dust-enshrouded merger products could be their distribution of orbital elements. While for YSOs we expect comparable orbital elements on timescales less than the resonant relaxation timescale within the S cluster, binary mergers should not follow such a condition because they form continuously and on different orbits. Since G2/DSO and OS1 share comparable orbital elements in terms of their inclination, semimajor axis, and the argument of the pericenter, see also Figure 19, the common origin in the same star formation event is plausible. Following this argument implies that OS2 might be a binary merger. This would be fully compatible with the discussed infalling-cloud scenario because Jalali et al. (2014) predicts that a certain fraction of the resulting sources are binaries, and single low- and high-mass YSOs (Yusef-Zadeh et al. 2013; Ciurlo et al. 2020; Peißker et al. 2020d).

Combining some of the mechanisms discussed in the previous subsection, we hypothesize that G2/DSO, OS1, and OS2, as well as other dust-enshrouded objects could be remnant YSOs captured by the SMBH during a nearly parabolic infall of a young star cluster that formed further away at the scales of ∼1 pc and beyond.

The advantage of this scenario is that larger distances from the SMBH put only moderate restrictions on the critical Roche density necessary for the molecular cloud to withstand disruptive tidal forces. The lower limit on the number density of the self-gravitating cloud at the distance r from Sgr A* is,

$$\begin{eqnarray}\begin{array}{rcl}{n}_{{\rm{c}}{\rm{l}}{\rm{o}}{\rm{u}}{\rm{d}}} & \gtrsim & \displaystyle \frac{3{M}_{\bullet }}{2\pi \mu {m}_{{\rm{p}}}{r}^{3}}\,\\ & \sim & 7.7\times {10}^{7}\left(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}\,{M}_{\odot }}\right){\left(\displaystyle \frac{r}{1{\rm{p}}{\rm{c}}}\right)}^{-3}\,{{\rm{c}}{\rm{m}}}^{-3}.\end{array}\end{eqnarray} \tag{ 11 }$$

In comparison, the Roche limit according to Equation (11) within the S cluster (∼0.01 pc) gives as much as n_cloud > 10¹⁴ cm⁻³.

The critical density for the star formation to take place could be reached via cloud–cloud collisions (Scoville et al. 1986; Tan 2000; Tan & McKee 2004; Hobbs & Nayakshin 2009) within the circumnuclear disk (CND), where individual clumps have ∼ 10⁶–10⁸ cm⁻³ (Hsieh et al. 2021). A further enhancement in the density can be provided by UV radiation pressure originating in NSC OB stars at the inner rim of the CND (Yusef-Zadeh et al. 2013). Moreover, fast stellar winds and occasional supernova explosions can be another source of a star formation trigger. External pressure is necessary for individual clumps to overcome the turbulent pressure, which prevents them from collapsing (Hsieh et al. 2021). Clump–clump collisions can partially remove the angular momentum, which helps to set the resulting self-gravitating cloud on the infalling trajectory toward Sgr A* with a small impact parameter (Jalali et al. 2014; Tartėnas & Zubovas 2020; Tress et al. 2020). However, the overall hydrodynamics of clump–clump interactions is rather complex and only a fraction of such collisions may end up with a self-gravitating and fragmenting cloud complex falling radially toward Sgr A*. First, this is due to the complex internal structure of molecular clumps, in particular their turbulent field (Salas et al. 2021), and hence the "hard ball" approximation does not apply to them. Second, the clumps at larger distances from Sgr A*, beyond ∼2 pc, follow the Galactic rotation and thus have a larger angular momentum with respect to Sgr A*, which needs to be removed for the cloud to fall in with a sufficiently small impact parameter. These two points imply that shearing likely takes place, which can result in a formation of a new cloud without a sufficient loss of the angular momentum or a sheared gaseous streamer. The formation of shearing gaseous streamers from a set of clumps was demonstrated in the 3D N-body/smoothed particle hydrodynamics (SPH) simulations by Salas et al. (2021). In their work, the turbulence was continually injected to mimic the effect of supernovae and stellar winds. In this way, the high dispersion of the gas within the Central Molecular Zone (CMZ) can be effectively reproduced. As shown by Salas et al. (2021), the injected turbulence results in the accretion to smaller scales down to Sgr A* in the form of turbulent accretion flows (Salas et al. 2020) or high-density spiral streamers (Dinh et al. 2021). Regardless of this complex behavior within the CMZ, for simplicity we assume here that at least once in every ∼ 10⁶ yr (Wardle & Yusef-Zadeh 2008; Jalali et al. 2014) a fragmenting star-forming cloud can reach the vicinity of Sgr A* with the impact parameter at the length scale of the S cluster, where it is expected to tidally disintegrate, with a fraction of YSOs being captured by Sgr A* (Gould & Quillen 2003), while the remaining fraction is unbound on hyperbolic orbits (Fragione et al. 2017). If the molecular cloud size is comparable or larger than its impact parameter, then it can completely engulf Sgr A* and leave behind a compact star-forming disk (Wardle & Yusef-Zadeh 2008), which may help to explain the multi-disk configuration of the S-cluster (Ali et al. 2020).

The crucial point of the "infalling-cloud" model is that YSOs of the age of ∼ 10⁵ yr can already be formed during the infall phase. Thus, we require that the infall timescale of the cloud toward Sgr A*, which is half of the orbital timescale, t_infall = P_orb/2, is longer than the freefall timescale of the clump with the critical density n_cloud, ${t}_{\mathrm{ff}}={[3\pi /(32G\mu {m}_{{\rm{p}}}{n}_{\mathrm{cloud}})]}^{1/2}$. Hence, we obtain the lower limit on the initial distance of an infalling cloud,

$$\begin{eqnarray}\begin{array}{rcl}{d}_{\mathrm{cloud}} & \gtrsim & {\left(\displaystyle \frac{3{M}_{\bullet }}{32\pi \mu {m}_{{\rm{p}}}{n}_{\mathrm{cloud}}}\right)}^{1/3}\\ & \sim & 0.4{\left(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}\,{M}_{\odot }}\right)}^{1/3}{\left(\displaystyle \frac{{n}_{\mathrm{cloud}}}{{10}^{8}\,{\mathrm{cm}}^{-3}}\right)}^{-1/3}\,\mathrm{pc},\end{array}\end{eqnarray} \tag{ 12 }$$

for the freefall timescale of t_ff ∼ 5000 yr, which can, however, get smaller as the density within the fragmenting clumps increases beyond the limit given by Equation (11). The outer distance range can be inferred from the G2/DSO lifetime of t_G2/DSO ∼ 10⁵ yr and from the condition t_infall ≲ t_G2/DSO,

$$\begin{eqnarray}\begin{array}{rcl}{d}_{\mathrm{out}} & \lesssim & {({{GM}}_{\bullet })}^{1/3}{\left(\displaystyle \frac{{t}_{{\rm{G}}2/\mathrm{DSO}}}{\pi }\right)}^{2/3}\\ & \sim & 2.6{\left(\displaystyle \frac{{M}_{\bullet }}{4\times {10}^{6}\,{M}_{\odot }}\right)}^{1/3}{\left(\displaystyle \frac{{t}_{{\rm{G}}2/\mathrm{DSO}}}{{10}^{5}\,\mathrm{yr}}\right)}^{2/3}\,\mathrm{pc}.\end{array}\end{eqnarray} \tag{ 13 }$$

Given the distance range between 0.4 and 2.6 pc, it is quite plausible that the self-gravitating cloud was formed or rather triggered toward star formation by external pressure within the CND, which is located between ∼1.5 pc and ∼3–4 pc (Christopher et al. 2005)

In the further discussion, we analyze the scenario where the self-gravitating cloud from the CND fragmented into a cluster of pre-main-sequence stars on its way toward Sgr A*. We assume the mass of this minicluster or rather a young stellar association of m_⋆ = 100 M_⊙, which is in the range of masses of 0.05–0.2 pc clumps within the CND (Hsieh et al. 2021). The stellar velocity dispersion is adopted from the IRS 13N association of young stars, σ_⋆ ∼ 50 km s⁻¹ (Mužić et al. 2008). Then, from the virial theorem, we obtain the stellar association radius of

$$\begin{eqnarray}&&{r}_{\star }=0.2\left(\displaystyle \frac{{m}_{\star }}{100\,{M}_{\odot }}\right){\left(\displaystyle \frac{{\sigma }_{\star }}{50\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)}^{-2}\,\mathrm{mpc}.\end{eqnarray} \tag{ 14 }$$

The young stellar cluster on a highly eccentric orbit will dissociate at the tidal disruption radius. This can be expressed as

$$\begin{eqnarray}\begin{array}{rcl}{r}_{\mathrm{dis}} & = & {r}_{\star }{\left(\displaystyle \frac{{M}_{\bullet }}{{m}_{\star }}\right)}^{1/3}\\ & = & \displaystyle \frac{{{GM}}_{\bullet }^{1/3}{m}_{\star }^{2/3}}{{\sigma }_{\star }^{2}}\\ & = & \sim 1214{\left(\displaystyle \frac{{m}_{\star }}{100\,{M}_{\odot }}\right)}^{2/3}{\left(\displaystyle \frac{{\sigma }_{\star }}{50\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)}^{-2}\,\mathrm{au},\end{array}\end{eqnarray} \tag{ 15 }$$

which is close to the pericenter distances of OS1 and OS2, as well as other dusty sources (Peißker et al. 2020d). A few stars from an infalling cluster will remain bound to the SMBH (Gould & Quillen 2003), while others will escape the NSC altogether as hypervelocity stars (Fragione et al. 2017). The high eccentricity of G2/DSO and also G1 implies that they might have been constituents of binary systems within the cluster, while the other components escaped (Zajaček et al. 2014). However, given a nearly radial infall of the cluster and its velocity dispersion, a fraction of YSOs should reach the pericenter distance of G2/DSO (∼137 au) during the disruption event (see also Gould & Quillen 2003 for statistical estimates of a similar scenario for S2 star). The detailed numerical modeling of such a cluster dissociation is beyond the scope of this paper.

Hence, once two clumps within the CND collide or there is another external trigger (intense UV radiation, supernova blast wave, collision with a jet), the clump of several 100 M_⊙ will start collapsing and fragmenting into protostars on the scale of several thousand years, which is a comparable timescale or shorter than the infall time from the CND. The collisional timescale of clumps within the CND was estimated by Jalali et al. (2014) to be τ_col ∼ 10⁵ yr, which is comparable to the lifetime of G2/DSO-like objects. In this sense, G2/DSO, OS1, OS2, and similar dusty objects in the Galactic center could be remnants of the previous infall of a fragmented star-forming cloud from the CND.