Abstract
A supermassive black hole, Sagittarius A* (Sgr A*), at the center of the Milky Way galaxy, is surrounded by the S-Star Cluster, a small cluster of high-velocity stars. Through years of observation and measurement, scientists calculated the black hole’s mass and location. Astronomers have also proposed several explanations of the formation of the S-Star Cluster, including the “Stellar Masquerade” theory, which argues that the S-Stars are old stars masking as young stars after violent collisions stripped their outer layers away. However, none of the existing proposed scenarios of these stars’ formation is supported by sufficient evidence. One significant discovery regarding the S-Star Cluster is the confirmation of Einstein’s Theory of General Relativity. We use the calculated orbital data of three S-Stars, S2, S38, and S55, to map out the stars’ revolution and compare the situations under Newtonian Gravity and General Relativity. The orbits under General Relativity demonstrate a rosette shape, a rose-shaped arrangement of orbits. Such observation indicates that the spacetime around the supermassive black hole is curved, as stated in Einstein’s Theory of General Relativity, throwing the S-Stars’ orbits off a little in every loop.
Introduction
Sgr A* and S-Star Cluster’s Properties
At the center of the Milky Way galaxy lurks a supermassive black hole, Sagittarius A* (Sgr A*). Dozens of stars orbit around the black hole, which together are called the S-Star Cluster. The three most well-known S-Stars are S2, S38, and S55. The first discovered one is S2, which has a mass of about 14
. Its peak velocity at the pericenter – the closest position to Sgr A* – is almost 2.5% the speed of light, enabling S2 to complete its full orbit in less than 16 years.1 Utilizing near-infrared imaging and spectroscopy, Parsa and coworkers recorded the orbit of S2 and measured its orbital period, angle, and radial velocity.2 Based on Kepler’s Laws, they derived the mass of Sgr A* and predicted its location – around 17 light hours to the pericenter of S2. The proper motion of S38 also helped the scientists to determine the mass of Sgr A* with greater accuracy, which is essential for testing General Relativity. A portion of its orbit could be clearly observed, while the rest of the trajectory could be mapped out through stimulation. S38 was significant to the research due to its particular location near the supermassive black hole. The star is located in the central region of the S-Star Cluster, but most of its orbit is to the west of the Galactic Center, where is less crowded; therefore, in this region, S38 is less prone to perturbations from other masses. Moreover, uncertainty about the north-south motion of Sgr A* arose from the observation of S2, which can only be observed from one angle on the Earth. However, since the orbit of S38 is perpendicular to the orbit of S2, the problem was easily solved. Together, the two stars’ orbits could constrain the gravitational potential of the Galactic Center. The last important star to the research, S55, is located closer to the black hole than S2 and S38, and it has the shortest orbital period, approximately 12.8 years. Then, further observations and analysis helped scientists to utilize the Trigonometric Parallax and calculate that Sgr A* is approximately 8.19 kiloparsecs away from the Earth. Then, with Kepler’s Third Law, the scientists derived that the mass of Sgr A* is about
.
S-Star Cluster’s Formation Mystery
One main question arises from the study of the S-Star Cluster: how can the young stars orbit the supermassive black hole in the region with such an intense magnetic field and strong gravitational forces that usually prevent stars’ formation? There are a few proposed explanations; however, they can be countered by certain observation. One proposed explanation is that the S-Stars are old stars masquerading as young stars.4 This proposal suggests that even though the S-Stars look like young stars, they may be the cores of old stars that collided and merged together. During the collisions, the cool outer layers of the stars could be stripped away and leave only the hot interiors, which can be mistaken for young stars. The result is that the S-Stars may be much older than they appeared to be. However, the problem is that violent collisions that could strip away the outer layers would also destroy stars and leave only a cluster of hot gases. A competing scenario is that the S-Stars formed elsewhere in the Milky Way and migrated close to the Galactic Center due to the supermassive black hole’s substantial gravity. The problem with this scenario is that most stars in the Galaxy form in its spiral arms and a migration to the black hole would take too long compared to the stars’ age. As main sequence stars, the S-Stars cannot be older than 100 million years; thus, the stars’ orbital motion should be more disturbed than they are now because of the gravitational interaction with other stars near Sgr A*.3 One alternative scenario argues that the star might form in the dense dust clouds closer to the black hole than the spiral arms. After their formation, the stars then migrated close to the Galactic Center in less than 10 million years, which is certainly possible. Though, a problem also exists with this proposal: as the stars got closer to the Galactic Center, they should lose angular momentum to prevent themselves from falling straight into the black hole.4 The stars could achieved this through collisions with neighboring stars, yet it is hard to conceive how the stars could endure such collisions and still remain in a safe migration and orbits. Moreover, the collisions would leave a trail of stars towards the black hole, which scientists have not yet been able to detect; in fact, the S-Star Cluster has a clear outer edge. Another possibility of the stars’ formation is that the stars formed with the accretion disk around the black hole. This proposal is supported by the fact that most stars in the central cluster are roughly in the same plane. However, not all scientists agree that the cluster is a disk-like structure, and some argue that the disk needs to be dense enough to hold itself against the black hole’s tidal forces. Yet, it is conceivable that the collisions in the accretion disk could trigger the stars’ formation. While the newly formed stars lose angular momentum to settle in an orbit plane around the black hole, the black hole could sweep away the leftover dust and materials. The last possible proposal suggests that collisions between some young stars far away from the Galactic Center could trigger the formation of an intermediate-size black hole, which later pulled these stars close to Sgr A*. This explanation can be supported by a discovery of any small black hole nearby, which the scientists have yet found evidence. No scenario proposed so far is conclusive; thus, scientists still have much work left to in order to understand the S-Star Cluster formation and the Galactic Center.
Results
Confirmation of Einstein’s Theory of General Relativity
After deriving the mass of Sgr A*, scientists realized that S2’s orbit is not fixed in position; instead, the intense gravity of Sgr A* throws S2 off in a minor shift in every loop. The star’s motion indicates a Schwarzschild precession: the location of the orbit’s pericenter to Sgr A* rotates every time, drawing a rosette, which is a circular arrangement of orbits that makes a rose shape. The orbit of one mass around another is not fixed, but rather precesses forward in a plane motion, confirming Einstein’s Theory of General Relativity.5 This effect is similar to Mercury’s precession around the Sun, which was the first evidence of curved spacetime due the gravitational force of a substantial mass.1 Moreover, the light wavelengths are stretched longer as S2 approaches Sgr A*.6 After concluding that the light from S2 experiences curving, scientists demonstrated that the star itself experiences the effect too. However, Einstein’s Theory of General Relativity does not solve all the problems. It cannot explain the gravity inside the black hole, which requires more advanced theories to be fully described.
With the aim of understanding more about astrophysical black holes, we present various models which are used to study the Sgr A* gravitational effect. Given the mass of Sgr A* and the orbital elements of S2, S38, and S5 in Table 1, the first model based on Newtonian gravity is shown in Figure 2.2 From 2003 to 2017, S2 and S38 completed only a proportion of their orbits in 14 years. S55, the closest S-Star with the shortest orbital period, already completed one full orbit. As Newtonian gravity depicts, S55 has a closed elliptical orbit. This is because Newtonian gravity does not explain the curved spacetime around the supermassive black hole – a star’s next orbit is always identical to its previous one.
When we set the model’s time range from 2003 to 3000, Figure 3a demonstrates that all three stars finish their full orbits and stay in fixed orbits for the 1000 years without any deviation. The three orbits are perfect eclipses, and each star’s the next orbit always strictly overlaps its previous one. In the next model, we show the result of changing the three S-Stars’ masses from 1 to 0.001. The stars’ orbits stay the same, illustrated in Figure 3b. This is because Sgr A* is a supermassive black hole, more than a million. The masses of S-Stars are too small to have any noticeable effect. Here, even though their masses are set to much lower, the stars’ orbits, determined predominantly on the black hole’s mass, remain the same.
. Figure 3b (right). The masses of S2, S38, and S55, under Newtonian Gravitational Law from 2003 to 3000, are set to 0.001. The orbits are all closed eclipses. Changing the stars’ masses does not affect their orbits.The model changes when Einstein’s Theory of General Relativity is instead considered. Setting the time range back to 2003 to 2017 in Figure 4, we saw that the S2’s and S38’s orbits remain similar to their orbits under the condition of Newtonian gravity. However, the S55 orbit behaves differently. Its apocenter – the furthest point to the Sgr A*– shifts by a small amount, so its next orbit is not identical to the previous one anymore. At the cross of the first and second orbits, the S55’s orbital rosette indicates a precession.
After the time is changed to the range from 2003 to 3500, the effect of precession becomes much more evident in Figure 5. All three S-Stars’ orbits make a full rosette around the Galactic Center that can be observed from the year of 2003 to 3500, and their precession is clearly shown. Thus, the model confirms Einstein’s Theory of General Relativity: the spacetime around Sgr A* is curved, causing the S-Stars’ orbits to make a shift every time they come close to the black hole.
S2 plays a more significant role than S55 in the confirmation mainly because S2’s orbit is more readily observable and was discovered earlier. Even though S55 is much closer to Sgr A* than S2 (and thus experiences a strong gravitational effect described by Einstein’s Theory of General Relativity), S2 is the primary factor for detecting Schwarzschild precession due its accurate orbital path and radius determined from more adequate observation. In addition, because of its proximity to the Galactic Center, S55’s orbit can be affected by the black hole’s other properties, such as its rotation; therefore, the confirmation of Einstein’s Theory of General Relativity cannot be easily tested by primarily observing S55.
Conclusions
Through the Kepler’s Laws and the observation of the proper motion of S2, S38, and S55, scientists derived the mass of Sgr A* and its distance to Earth. However, the S-Star Cluster’s formation still remains unsolved, and further studies are required for supporting any conclusion of the work involved at the Galactic Center. In this work, we see how under the condition of Newtonian gravity, the orbits of S2, S38, and S55 are all closed eclipses. The next orbit is always identical to the previous one. Sgr A* has a substantial mass compared to the S-Stars, so even if we set the S-Stars’ masses as “1” /”1000″ of their original masses, their orbits remain the same. The model indicates that Sgr A* is the primary factor that determines the stars’ trajectories. When we change to Einstein’s Theory of General Relativity, the orbits experience a Schwarzschild precession. They are not fixed in place because the supermassive black hole’s mass curves the spacetime, throwing the stars’ orbits off in every loop. Our models and supported insights demonstrate that the S-Stars’ orbits confirm Einstein’s Theory of General Relativity.
References
1. Abuter, R. et al. Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole. arXiv 5, 1–14 (2020).
2. Parsa, M. et al. Investigating the relativistic motion of the stars near the supermassive black hole in the galactic center. arXiv 22, (2017).
3. Canadian Astronomical Society. The origin of the s-star cluster at the galactic center. phys.org (2013).
4. Dvorak, J. Secrets Of The Strange Stars That Circle Our Supermassive Black Hole. Discover Magzine (2018).
5. “Tracing the Edge of Doom” –Strange Star S2 Orbits Milky Way’s Black Hole. The Daily Galaxy | Great Discoveries Channel (2020).
6. Do, T. et al. Relativistic redshift of the star S0-2 orbiting the Galactic Center supermassive black hole. Science (80-. ). 365, 664–668 (2019).
LaTeX codes
