1 INTRODUCTION

In the accepted classification of binary stars (BSs) based on their semimajor axes, they are divided into close (\(a < 1000 {{R}_{ \odot }}\)) and wide (\(a \sim 100\) AU) pairs [1]. According to modern GAIA observations, the upper limit for the semimajor axis of wide BS can be extended to 10 000 AU (0.0485 pc). In the case of close BS, these limits are associated with the phase of deep collapse of the initial rotating gas-dust clouds, and in the case of wide systems, it involves the process of star cluster disintegration or the capture of neighboring stars due to close encounters of binary or single stars.

In the systems we are studying, the distance between their components can be comparable to the radius of the Galaxy and even larger. It would be more accurate to say that in the past the components of these pairs could belong to a single parent BS, the connection with which was severed. Scenarios in which the links of BSs can be broken are well-known: the disruption of unstable triple systems, the double collisions of close BSs, their collisions with single stars, supernova explosions in close BSs, and the dynamic capture of a BS by a supermassive black hole (SMBH). Let us focus on the latter scenario.

Due to the redistribution of angular momentum, which is inevitably encountered in the three-body problem, both binary components can end up in new statuses: a central near-nuclear S-star and a hypervelocity star (HVS) whose velocity during ejection allows it not only to escape from the vicinity of the SMBH but also to become unbound from the parent galaxy. Depending on the time elapsed since ejection and the ejection velocity, an HVS may remain within the bounds of the Galaxy, reach its periphery, and even escape beyond its boundaries. Jack Hills was the first to propose such a scenario, providing model estimates of the escape velocity of one of the BS components from the vicinity of the SMBH to several thousand km/s [2], in order to draw attention to the phenomenon of HVSs, which, with their anomalous kinematics, would serve as indirect evidence of the existence of SMBHs.

However, history took its course, and S-stars [3] were discovered first. Thanks to the technological revolution in adaptive optics and high-angular-resolution spectroscopy, there was an avalanche of S-star discoveries. The triumphant outcome of many years of (starting from 1989) observations of S-stars and non-trivial methods of their analysis, proving the presence of a massive central body—SMBH—led to the award of the Nobel Prize for 2020 to Andrea Ghez and Re-inhard Genzel. At the moment, there are about 10 000 S‑stars in the search map of the Galactic Center, and routine work is still needed to recover the elements of their orbits [4]. Despite the constantly increasing number of discovered central stars, no more than several dozen S-stars remain informatively valuable, which surprisingly coincides with the statistics of discovered HVSs up to the present time.

The phenomenon of HVSs allowed a reevaluation of the kinematic structure of the Galaxy and formulated the problem of identifying an object with anomalous kinematics due to central ejection. Since the first HVS, SDSS J090745.0+024507, was discovered in 2005 by Warren Brown and his colleagues [5], the statistics have grown to 40 stars over the years. Two approaches have been developed in the search for HVSs during this time: to search among bright halo stars, which contrast with the local population of old globular clusters (and therefore likely arriving in the halo with high velocities, since the flight time should be consistent with their age), and among nearby low-mass disk stars, which are the majority. Both approaches have their “uncertainty ceiling” due to, on the one hand, the problem of determining the proper motion of faint, distant halo objects and, on the other hand, the abundance of low-mass disk stars, among which the search involves the routine process of data analysis. Initially enthusiastic about the possibility of an accelerated search for HVSs in its observation program, the GAIA team encountered difficulties in identifying stars with anomalous kinematics. GAIA gives three times larger errors for stars fainter than \({{18}^{m}}\) compared to HST, so it is more effective to use it for the second approach. Conversely, for stars brighter than \({{18}^{m}}\), GAIA provides 3–4 times more accurate measurements of distances and proper motions compared to HST.

Thus, one could say that by chance, at the present time, observational statistics for both classes (S-stars and HVS) coincide. And from comparing these statistics, i.e., from the individually observed HVS [6] and S-stars [7], we can begin to solve the problem of reconstructing S-HVS pairs. It should be understood that these data are equally subject to the effects of observational selection due to the faint magnitude of HVS due to their distance from us by tens and even hundreds of kiloparsecs and the “light pollution” of S‑stars from the densely populated Galactic Center (GC).

A detailed investigation of this problem can be conducted based on the possibility of numerical modeling of the capture of BSs into the vicinity of supermassive black holes (SMBHs) in the context of the Hills scenario [2]. The capture modeling is carried out in several stages. First, a simplified layered model of star density in the Galactic Center (0.1–0.01 parsecs) is tested in the mode of paired random encounters [8]. To effectively study the conditions for the capture of BS with subsequent ejection of a star as a HVS, the principle of ensemble representation of initial data is used. This allows for a wide range of parameters for the initial configurations of BS (orbital phase of BS and the inclination of its own orbit relative to the orbit around the SMBH) to be varied in a single calculation [9]. Variation of the parameters of the approach of BSs to the SMBH allows the extraction of the maximum and average ejection velocities of stars as HVSs depending on the pericentric distance. The reliability of the obtained values is tested during the modeling of BS in the N-body approach, where each of the BS components is defined by a set of structurally gravitating elements that interact with each other [10]. This approach provides an objective view of the survival of a star as it approaches the SMBH.

Through numerical experiments posed in both the three-body and N-body scenarios, obtained statistics of HVS and S-star populations, considering the segregation of their masses, proved to be consistent with their lifetimes, including the time of capture of S stars into the vicinity of the central SMBH [11]. Thus, the Hills scenario [2] is capable of explaining both the phenomenon of HVSs and the accumulation of S‑stars, although it is not the only channel for the formation of S-stars.

The problem of direct identification of S–HVS pairs remains technically unsolved today, posing a challenge to modern astrometry and spectroscopy. Theoretical identification algorithms also face difficulties, involving aspects such as synchronizing the evolutionary scales of previously related components. A star that remains in the vicinity of the SMBH may be partially or completely disrupted by its tidal field, while the other companion will continue to evolve over the course of its flight. We can attempt to address these questions based on the analysis of observational data, following the principle of complementing the S‑star catalog with HVS data.

2 RECONSTRUCTION OF S–HVS PAIRS BASED ON OBSERVATIONAL DATA

Over nearly thirty years of continuous monitoring of the central region of the Galaxy, the orbits of 39 S‑stars from the Gillessen et al. catalog [7] have been reconstructed. Using this data, it is interesting to investigate whether these S-stars could have a companion in a parent BS system in the past. To begin with, the specific binding energy of the S-star in the central field of the SMBH can be estimated within the framework of the three-body problem in order to convert it into the kinetic energy of the possible companion’s ejection, assuming that the total energy of the BS system at infinity becomes zero. The ejection velocity can be then derived from the kinetic energy of the ejection. The obtained velocity spectrum of the presumed ejected companions of S-stars should be compared with the spatial velocities of observed HVSs from the Brown et al. catalog [6]. This approach to the reconstruction of S–HVS pairs adheres to the principle of complementarity, where data from one catalog are tested against data from another.

2.1 S-Star Catalog and Reconstructed HVSs

From the Gillessen et al. catalog [7], 39 S-stars with known orbital periods \(P\) and angular sizes of the semimajor axis of their orbits \(a\) around the SMBH were selected. Using estimates of the distance from the Sun to the Galactic Center, \({{R}_{0}} \approx 8.32\) kpc, and the mass of the SMBH, \({{M}_{{{\text{SMBH}}}}} = 4.28 \times {{10}^{6}} {{M}_{ \odot }}\), derived from multi-star orbital fitting of 17 S-stars selected based on the significance criterion of gravitational acceleration according to [7], the linear sizes of the S-star orbits \(A = \frac{{a''}}{{206 265}}{{R}_{0}}\) and their average orbital velocities \({{v}_{{{\text{orb}}}}} = 2\pi A{\text{/}}P\) were calculated (Table 1). Based on this data, it is possible to calculate the specific binding energy of the S-star, \({{e}_{{\text{b}}}}\), located in the central field of the SMBH:

$${{e}_{{\text{b}}}} = \frac{{v_{{{\text{orb}}}}^{2}}}{2} - G \frac{{{{M}_{{{\text{SMBH}}}}}}}{A} .$$
(1)

Neglecting the interaction between the components of BSs compared to their interaction with the SMBH, and assuming that the total energy of the former S‑HVS pair is zero at infinity, the ejection velocity of the ejected component can be estimated within the Hills scenario [2] as \({{v}_{{{\text{eject}}}}} = (2{{e}_{{\text{b}}}}{{)}^{{1/2}}}\) (Table 1). Using the kinematic criterion for HVSs as stars gravitationally unbound from the Galaxy, which, for the Galactic Center region, corresponds to \({{v}_{{{\text{eject}}}}} > 750\) km/s according to Wu et al. [12], it turns out that 22 out of 39 S-stars could have had a companion component that was subsequently ejected as an HVS during the dynamic capture of the parent BS.

Table 1.   Orbital parameters of observed S-stars from the catalog of Gillessen et al. [7] and the ejection rates of potential evolutionarily paired objects derived from them with an analysis of their status as HVSs under the condition \({{{v}}_{{{\text{eject}}}}} > 750\) km/s [12]
Full size table

The HVS ejection velocity spectrum (light histogram in Fig. 1), reconstructed from the observed distribution of S-stars based on their average orbital velocities (blue histogram in the same figure), maintains a correlation with it and also aligns well with the observed velocity spectrum of HVSs from the Brown et al. catalog (pink histogram) [6]. Due to observational selection effects, the distribution of observed HVSs lies within a narrow range of ejection velocities (600–800 km/s). It is likely that the fastest HVSs with velocities exceeding 1000 km/s escape the Galaxy in less than 50 Myr. In this distribution, one notable object stands out with a spatial velocity of ~1755 km/s, known as Koposov’s star [13], discovered within the Southern Spectroscopic Survey S\(^{5}\) at a distance of \( \sim {\kern 1pt} 8.6\) kpc from the Galactic Center. It is now used as an etalon test for the scenario of central origin of HVSs.

Fig. 1.
figure 1

A comparison of the observed distribution of S-stars from the Gillessen et al. catalog [7] based on their mean orbital velocities (blue histogram) with the calculated (cyan histogram) distribution of ejection velocities for the presumed paired with S-stars and with the observed (pink histogram) distribution of spatial velocities of HVS from the Brown et al. catalog [6].

Full size image

The analysis of HVS ejection velocities \({{v}_{{{\text{eject}}}}}\) (Table 1), reconstructed from the observations of S-stars from the catalog [7], confirms the plausibility of the scenario of their capture by a parent BS into the vicinity of the SMBH, followed by the ejection of the second component: in 22 cases as HVSs and in 17 cases while remaining gravitationally bound to the Galaxy.

2.2 HVS Catalog and Reconstructed S-Stars

Similarly, it is possible to perform the reverse reconstruction, i.e., reconstruct the orbital parameters of the captured companion, such as the semimajor axis of its orbit around the SMBH, based on observations of HVSs. To date, the most complete catalog of HVSs is Brown et al. [6], which contains information on 39 objects, including their spatial velocity values \({{v}_{{{\text{obs}}}}}\). Assuming that the kinetic energy of the ejected HVS from the vicinity of the SMBH was balanced by the gravitational energy of the interaction of the presumably remaining companion (S-star) in the central field of the SMBH, it is possible to estimate the semimajor axis of its orbit using the formula \(a_{{\text{S}}}^{{{\text{calc}}}} = G {{M}_{{{\text{BH}}}}}{\text{/}}{v}_{{{\text{obs}}}}^{2}\).

The observed distribution of S-stars from the catalog [7] based on the semimajor axes of their orbits (blue histogram in Fig. 2) overlaps with a similar distribution reconstructed from the observational data of the HVS catalog [6] (cyan histogram in Fig. 2), as if they were pairs in parent BSs. These two distributions overlap in the range of \({\kern 1pt} \sim {\kern 1pt} 9300{-} 32 600\) AU (\((2{-} 7) \times {{10}^{6}} {{R}_{ \odot }}\)),which clearly demonstrates the correlation between the data from the Brown et al. [6] catalog and the Gillessen et al. [7] catalog.

Fig. 2.
figure 2

The observed distribution of S-stars from the Gillessen et al. catalog [7] based on their major semi-axes (blue histogram) and the reconstructed data from HVS observations from the Brown et al. catalog [6] (cyan histogram).

Full size image

Figure 2 illustrates the effect of observational selection in the data from both catalogs, manifested in the presence of opposite (mirror-symmetric) maxima. The peak of the observed distribution of S-stars [7] corresponds to the value of the semimajor axis \( \sim {\kern 1pt} 1 \times {{10}^{6}} {{R}_{ \odot }}\). S-stars on wider orbits (with semimajor axes \(a > 7 \times {{10}^{6}} {{R}_{ \odot }}\)) have not been discovered yet; monitoring of the central region has been conducted for just over thirty years, whereas longer observations are required to determine the orbits of S-stars from images.

Let us return to Fig. 1. It shows the reconstructed distribution of HVSs based on the observed S-stars from the catalog [7] (cyan histogram) with an extended maximum in the range of 500–1300 km/s. In reality, the observed HVSs from the catalog [6] have a maximum in the velocity distribution in the range of 600–700 km/s (pink histogram). In cases of low-impulse ejections of one of the components of a BS that ended up in the vicinity of the SMBH, the second component remains on a wider orbit around the SMBH. This means that the peak of the pink histogram (600–700 km/s) in the distribution of observed HVSs (Fig. 1) shifts to the value of \( \sim {\kern 1pt} 6 \times {{10}^{6}} {{R}_{ \odot }}\) in the cyan histogram (Fig. 2), which shows the reconstructed distribution of S-stars based on their semimajor axes. This is why the effect of inverse symmetry is observed.

The model distribution of S-stars obtained within the framework of the Hills scenario [2] shows the presence of compact orbits with semimajor axes two orders of magnitude smaller, \( \sim {\kern 1pt} 139{-} 330\) AU (\((3{-} 7) \times {{10}^{4}} {{R}_{ \odot }}\)) (Fig. 2 from [11]), compared to the observed ones. This discrepancy is primarily associated with the effects of observational selection. S-stars on small orbits are more difficult to detect; so far, the most compact orbit among the detected S-stars is for S 4716 with a semimajor axis\(A \sim 400\) AU [14]. Another possible reason is that the gravitational potential of the Galaxy, which is still not precisely known, decelerates HVSs, causing their velocities to be lower than expected from the observed orbits of S-stars. Additionally, the deviation from the model distribution of S-stars in terms of semi-major axes could be related to the conditions for a star to be ejected as an HVS. Allowing for ejections with velocities lower than those corresponding to the HVS criterion would result in larger semimajor axes for captured S-stars in the Hills scenario, ensuring agreement with the observed distribution.

Other causes of the discrepancy between observed and calculated values of semimajor axes may reflect different scenarios for the origin of S-stars unrelated to the disruption of BSs through the Hills mechanism [2]. According to calculations by Loose et al. [15], S‑stars could have originated from natural recurrent star formation associated with gas accumulation in the vicinity of the Galactic Center. The origin of S-stars may also be linked to tidal “stripping” of young star clusters during their approach to the central SMBH, as indicated by Fragione et al. [16]. An example of such a cluster is the young and dense Arches cluster, located 25 pc away from the SMBH.

A detailed kinematic analysis conducted by Ali et al. [17] for 112 S-stars revealed 3D structures of their orbits, which belong to two closely adjacent, oppositely rotating disks inclined at 45° to the Galactic plane. The distribution of orbit inclinations and proper motion vectors contradicts the initial data on thermal eccentricity distribution and chaotic orbit distribution of S-stars. Ordered orbital structures could result from resonant processes in the nuclear disk.

One should consider the tidal disruption of S-cluster stars in the gravitational field of the SMBH. The theoretical estimate of their frequency obtained in the homogeneous bulge model [18] is comparable to the frequency of flashes attributed to the accretion activity of the SMBH. According to numerical modeling of tidal disruptions of observed BSs in the central disk cluster at distances from the center of 0.05–0.5 pc, Generozov [19] predicts that a conical stream of ejected HVSs with an angular resolution of 30° was produced in the vicinity of the Galactic Center in the recent past (\( \sim 5\) Myr ago). These ejected HVSs are currently at galactocentric distances of several tens of kpc. This distribution, along with the results of kinematic analysis of S-cluster stars [17], can facilitate the search for HVSs in the GAIA data. It is worth mentioning the star S5-HVS1 (Koposov’s star) [13], which was considered part of the conical stream in the modeling [19], and its observational data were used as a reference for predictions.

Thanks to the development of image filtering technologies, such as the Lucy–Richardson algorithm, the following S-cluster stars have been discovered to date from the closest to the Sgr A* source: S 4715 (a ~ 1190 AU), S 4714 (\(a \sim 840\) AU), S 62 (\(a \sim 740\) AU), S 4711 (\(a \sim 620\) AU), S 4716 (\(a \sim 400\) AU) [20, 14]. It is with such stars, primarily, that the problem of reconstructing S–HVS pairs should be addressed.

3 SEARCH FOR S–HVS PAIRS

The main question of the paper remains: is it possible to “assemble” S–HVS pairs from direct observations? Let us attempt to outline an algorithm for “assembling” S–HVS pairs.

1. For S-stars closest to the Galactic center with known orbital elements, reconstruct, within the framework of the Hills scenario [2], the ejection velocities of their potential companions based on their binding energy with the SMBH. Selecting the closest S-stars will satisfy the condition for releasing evolutionarily linked companions as HVSs.

2. For the same S-stars, analyze their known orbital elements that define the orientation of their orbital plane in space. Additionally, estimate their spectral class, mass, and age from spectroscopic observations.

3. The search for a candidate among observed HVSs to pair with the selected S-star should consider several necessary conditions:

—similar spectral classes between the S-star and the HVS, considering for the “spectral twins” effect, which follows from the conditions of star formation in the parent binary system;

—matching ages between the S-star and its potential companion (candidate HVS), which follows from the youth of the near-nuclear S-cluster;

—knowledge of the HVS’s velocity and its current distance from the Galactic Center, which will constrain the establishment of the pair with the selected S-star;

—meeting the requirement of the central ejection of the chosen HVS as a candidate for pairing with the S-star, which can be determined by reverse-time integration of the HVS’s trajectory in the Galaxy’s gravitational potential (the knowledge of which is a separate major challenge!). The HVS’s total (radial plus tangential) velocity vector at the time \(t = 0\) should point toward the Galactic Center;

—the spatial velocity vector of the HVS at time \(t = 0\), reconstructed through the method of reverse integration of its motion, should lie in the orbital plane of the S-star to which the considered HVS is assumed to be a companion. However, the current spatial velocity vector of the HVS does not necessarily have to lie in the orbital plane of the S-star due to possible deviations in the galactic potential during its journey.

As for the absence of sufficiency conditions, perhaps they cannot exist in probabilistic problem formulations, except in rare cases when it becomes possible to directly observe the process of breaking the BS connection with the ejection of an HVS.

In perspective, this appears to be an exciting task of reconstructing genetically related components of BSs disrupted by the tidal gravity of a SMBH.

4 CONCLUSIONS

The problem of reconstructing S-HVS pairs no longer seems fantastical today. The technical progress of the 21st century has shown the smallest gap in history between the development and implementation of methods and technologies that have proven their effectiveness and unveiled new possibilities in a long list of discoveries: HVSs, SMBHs, gravitational waves, and quantum entanglement.

Disentaglement S–HVS pairs is a task of the macro world, and much has already been achieved to address it. The discovery of Koposov’s star [13] provided compelling evidence for the Hills scenario [2]. Numerical modeling of dynamic captures of BSs in the vicinity of the SMBH allows us to predict the population of the Galactic center with S-stars, imposing restrictions on capture cross-sections and the time, upper limit of which is constarined by the age of our Galaxy (~13.6 Gyr). This prediction is conditioned on the assumption that a capture event will always result in the ejection of an HVS, and the statistics of such ejections are consistent with the statistics of S-stars [11]. Modeling tidal deformations of stars in the near-nuclear S-cluster and the ejections of some of them during their pericenter passages near the SMBH provides constraints on the selection of preferential ejection directions in the form of conical “fountains.” These fountains are caused by the disk-like structure of the S-cluster and its inclination to the Galactic plane [16, 17]. All of these refined models will contribute to solving this problem.