Checking the possibility of determining the relative orbits of stars rotating around the center body of the Galaxy

1 Introduction

Astrometric observations of binary stars have been carried out at the Pulkovo observatory for many decades, beginning with the work of V.Ya. Struve. The purpose of these observations is to study motions, determine orbits and masses, and solve many other problems associated with stellar astronomy and astrophysics. This article reflects our experience in observing and determining the orbits of binary stars with the methods developed at Pulkovo, used, among other things, to determine the orbits of stars from the S-cluster in the center of our Galaxy. We have tried to determine by the astrometric method the orbits of possible “squeezars” – stars orbiting the center of the Galaxy at close distances. An attempt was also made to estimate some of the relativistic effects that are theoretically possible for these stars. The studied objects were the stars S02, S102, S27, and the recently discovered star S4711.

Earlier, at the suggestion of Yu.N. Gnedin, using the methods used in Pulkovo to determine the orbits of binary stars, the possibility of obtaining these orbits for stars orbiting close to the center of our Galaxy was investigated (Kisselev et al. 2007). As an example, the star S02 was studied, the orbital period of which – 16 years – was the shortest at that time. For this, the results of high-precision observations with the Keck and VLT telescopes were used.

By means of apparent motion parameters (AMP) method and direct geometric method (DGM) developed in Pulkovo by Kisselev (Kisselev 1997, Kisselev et al. 2007), the orbit of the star S02 was constructed and the mass of the supermassive black hole (SMBH) was estimated. The efficiency is shown of the method of the parameters of apparent motion, based on measuring the curvature of a sufficiently short orbital arc in the presence of observational data of the relative radial velocity and parallax.

Later, publications became available with the exact positions for a number of stars, namely, the relative coordinates Δ R.A., Δ Decl., see Meyer et al. (2012), Gillessen et al. (2009, 2017), Parsa et al. (2017), Peissker et al. (2020), etc.

In addition, the most probable values of the mass M of the central body Sgr A ∗ equal to 4.15 million solar mass and its distance from the Sun Ro equal to 8.19 kpc were used. Sgr A ∗ – adopted name of hot plasma radio source in the center of which a compact massive object is located, presumably a black hole. This source is a part of more extended source Sgr A (without “ ∗ ”) with nonthermal radiation.

Information appeared about the motion of closer stars, such as S102 with an orbital period of about 12 years, as well as about the so-called “squeezars” with even shorter orbital periods. These stars, located in a dense cluster around the SMBH in the center of our Galaxy, are considered ideal candidates for observing gravitational effects such as the periapse shift.

It was of interest to determine the orbit in an independent way for such objects, as well as for stars with exact positions relative to the central body, but ambiguous results obtained at different telescopes (Keck and VLT). In addition, it is known that for a number of cluster stars there are problems with the construction of orbits due to uncertainly determined curvature, acceleration, radial velocity, and other parameters of motion.

2 Choice of stars

We have chosen four stars from the S-cluster.

“Sgr A ∗ cluster” or S-cluster is the accepted designation for a cluster of S-stars which, according to their observed motion, are gravitationally bound to a massive compact object, a possible black hole, at the center of this cluster. The maximum radius of this cluster is 1 ″ on the celestial sphere. In the paper of Gillessen et al. (2017), orbits were obtained for 40 cluster stars, the elements of which are given in the publication. Also a list of other 54 stars is given in this article without orbits, but with the moments of observations and the relative motion Δ R.A. Δ Decl. depending on time.

In this article, some examples of star exploration S55/102, S02, S27, and the newly discovered star S4711 are given. The stars S02 and S55 (S102), which we have chosen, are conventionally referred to as “classical” objects for which there are many published positions. Therefore, we were able to select a number of fairly short intervals from which the orbits were determined for comparison.

We recalculated the orbit of S02 using relatively short orbital arc from VLT observations. Then we considered the star S102, which, according to literary data, has some discrepancies in the parameters of the orbits. The orbit from the combined Keck and VLT series has been determined by our methods.

The next star on our list is S27, and it is classified as problematic one by the authors. It possibly possesses unusual, unexpected movement parameters. There are no exact locations published for the star S27. Its positions are given on the diagram – R.A. and Decl. versus time, see Gillessen et al. (2009). Orbital elements are also given. In the article by Gillessen et al. (2017), there is no orbit for this star. The authors explain this by the uncertainty in its orbital motion due to the facts that (a) it was not possible to estimate the curvature and (b) may be an error was made in estimating the acceleration of motion.

Then the recently discovered star S4711 was considered (with the shortest orbital period of 7.6 years, which fell under the definition of “squeezars,” see Peissker et al. 2020). As well as for S27, we did not find the exact relative positions for this star and therefore used the diagrams of the movement data. The resulting orbits are shown in Table 1.

3 Applied methods

Three methods were applied for the study of these stars, of which two were developed by Kisselev et al. (2007): (1) the method of the AMP, which is based on the use of a high-precision dense series of relative positions, as well as trigonometric parallax and radial velocity of the components.

The method AMP uses the following initial parameters:

1. ρ , θ – visible distance and position angle of component B relative to A .

2. μ – apparent proper motion of B relative to A along the tangent T -axis,

3. ψ – its position angle,

4. ρ c – radius of curvature of the visible arc.

   These parameters determine the apparent movement of B relative to A at the time of observation.

   Then we used additionally:

5. π – trigonometric parallax.

6. V r = V r ( B ) − V r ( A ) – relative radial velocity of component B with respect to A .

7. M ( A + B ) . The sum of the masses of the system and the relative radial velocity are additional parameters.

(2) Direct geometrical method (DGM) is the geometric method, useful for determining the orbits of stars with relatively short orbital periods and where a necessary condition is the construction of a visible ellipse for the entire period of rotation. The method was present for the double star orbits (Kisselev 1997), tested on constructing the orbit of a dark satellite of ADS 15571 (Grosheva 2006) and a satellite of the asteroid Kalliope (Sokova et al. 2014). This method proposes a way for constructing a visible ellipse and determining the coordinates of the center of the areas from observations. Formulas for determining the elements of the true orbit are derived. This method was applied by us for the orbit of S02 in Kisselev et al. (2007).

(3) The third method was developed by Izmailov (IZM), see: http://izmccd.puldb.ru/PIA35.pdf, by using elements of Thiele-Innes (Izmailov 2019). A fundamental feature of the approach used is that along with the central orbit, a set of possible orbits is calculated by varying the initial data by the Monte Carlo method.

This set makes it possible to determine both the errors of the orbital elements and all quantities derived from elements. The most important quantity is the mass of the system. In complex cases, only this approach allows one to adequately estimate the errors of the derived quantities, since the elements of the calculated orbit contain complex, nonlinear correlations with each other.

4 The orbits

Tables 1 and 2 show the dynamic parameters of the true orbit: P is the period of rotation, a is the semi-major axis, e is the eccentricity, and T p is the moment of passage through the periastron. Geometric elements are given that determine the orientation of the true orbit in relation to the picture plane, which touches the celestial sphere at the point where the central body is located. These elements are as follows: i is the inclination of the orbit or the angle between the line of sight plane and the plane of the true orbit, ω is the position angle that determines the position of the line of nodes – the line of intersection of the visible and true orbital planes, Ω is the angle between the line of nodes and the direction to the periastron, measured in the plane of the true orbit.

The symbol i is the inclination of the orbit to the line of sight plane, measured from the plane of the sky to the plane of the orbit in the direction of the minimum inclination angle in the range of ± 9 0 ∘ . Since we take the “star – central body” system for a binary star system, our elements are given in the orientation system adopted for visual binary stars. In order to obtain ephemeris (relative positions) and compare apparent orbits, we took i from the cited works and used the values of 180° minus i for our calculation algorithm. These cases are marked with an asterisk “ ∗ ” in Table 1.

Tables 1 and 2 give orbital elements from the literature as well as those orbits obtained by method using the elements of Thiele-Innes, which was developed by Izmailov (2019) (IZM) and preliminary orbits using the AMP method.

For all compared orbits, we used, Ro = 8.19 kpc, except for Meyer et al. (2012), where it is 7.7 kpc. For all orbits, 4.15 × 1 0 6 solar masses are used, except for S02 in Table 2 where it is 2.5 × 1 0 6 and Meyer 4.1 × 1 0 6 solar masses. The last two lines give the orbits of the star S4711, with the shortest orbital period around the galactic center. The large value of the O–C error is due to the fact that we had the opportunity to obtain its positions only from the graph in the work of Peissker et al. (2020). Where the inclination i has ∗ , it means that for our algorithm of calculating the ephemeris, we used the value 180° − i, which is done in the table.

We give two variants of orbit S02 (Table 1). They were determined with different observation arcs. For comparison, we give the values from some works, which is given in the last column of the table.

S02 in Table 2 is different from the orbits in Table 1 because here we used the mass sum 2.5 × 1 0 6 obtained as a free parameter.

S27 in the last and penultimate row in Table 2 differs between themselves due to the use of different methods. The data in Table 3 for the star S27 correspond to its parameters in Table 2.

Some of our results and comparison with other orbits are given in figures. In Figure 1, the orbits for S02 obtained by the AMP method (left) and the orbit (Parsa et al. 2017 (right) method are given. The solid line is orbit (IZM) from Table 2.

Figure 1

Orbits of S2.

Figure 2 shows the orbits of S102 constructed with observational data of Keck telescope (Meyer, 2012) (left panel, black squares) and VTL (Parsa, 2017) (central panel, white circles). Right panel shows the orbit constructed in the present work with the united data of Keck and VLT. See Table 1. Figure 3 shows orbit of S4711 obtained by the AMP method using graphic with median values given in the previous work (Peissker et al. 2020). Our orbit is shown by dotted line and orbit of Peissker et al. (2020) is shown by solid line.

Figure 2

Orbits of S102.

Figure 3

Orbits of S4711.

Figures 4 and 5 show two orbits for S27, see Table 2.

Figure 4

Orbits of S27.

Figure 5

Orbit of S27.

As our analysis has shown that, when applying the method developed by Izmailov, the positions of S27 may contain errors. Presumably, even errors in identification arising from the presence of close images of overlapping stars. For S27, we used the diagram from Gillessen et al. (2009). We measured the positions Δ R.A., Δ Decl. Using the AMP method, a new orbit of the star S27 was obtained using a short arc (13 years of observations, and the orbital period up to 112 years). Table 1 gives our orbit and the orbit from Gillessen et al. (2009). We also recalculated the orbit of S02 using a relatively short arc from the VLT observations. The O–C errors calculated from the orbits we obtained are comparable to the O–C orbital errors obtained by observers at the Keck and VLT telescopes.

5 Relativistic parameters

At the next stage, we made an attempt to estimate the expected relativistic parameters for stars by orbital elements. We used the corresponding formulas.

Here Γ is the ratio of the Schwarzschild radius r s to the distance to the periastron r p = a ( 1 − e ) where a and e are orbital elements. r s = 0.081 AU. Δ φ – Schwarzschild precession, i.e., change in direction of the line of nodes due to gravitational influence of supermassive body. M is the mass of the central body, G is the gravitational constant, a is the semi-major axis, e is the eccentricity of the orbit of a star rotating around the central body, c is the speed of light. V is the velocity in the periastron in AU in day.

In Table 3 additional orbital and relativistic parameters for S4711, S27 stars in comparison with S2 star are given. From the left to the right column, we list star name, its pericenter distance r p (in AU), its apocenter distance r a (in AU), the pericenter velocity V p (in km/s), the relativistic parameter defined as Γ , the gravitational redshift Z gr c (in km/s), and the Schwarzschild precession Δ φ (in arcmin).

Results of theoretical values of parameters are listed in Table 3.

The relativistic components were theoretically estimated from our obtained orbital elements and assuming that the mass of the primary component is equal to 4.15 million solar mass and its distance from the Sun Ro equal to 8.19 kpc. We obtained these elements for control purposes, to check, as a first approximation, (a) whether they are comparable in order of value with the same stars in the cited publications and (b) thus assess the possibility of obtaining them from observations with the accumulation of observational material.

6 Comments

Let us note the features in the results obtained, which can be explained on the basis of more extensive material during further research. When applying the AMP method, the orbits are obtained close to the orbits published by the observers Keck and VLT. When applying the modified Thiele–Innes method to the star S27, the elements were obtained with large errors, the orbital period P is more longer (705 years instead of 103 years). It should also be noted that among our stars, only S02 has a distance to the periastron r p < 120 AU (namely, 119 AU) as one of the conditions for being a squeezars, see Alexander & Morris (2003). r p for S27 depends on method by which the orbit was calculated, see the last lines in Table 2.

As the authors of the cited works note: after many years of high-precision astrometric observations and many years of Doppler measurements of radial velocities, the accuracy of the available data has reached a level at which one could hope to detect deviations from Keplerian orbits. Such deviations may be a consequence of relativistic effects. There are examples of detecting the magnitude of Γ = 0.00088 ± 0.00080 from observations of the star S02, with an expected value of 0.00065 (Parsa et al. 2017). Also, as it was shown in Gravity (2018), there is an estimate of the gravitational redshift. But the periastron shift Δ φ is the cumulative effect, that accumulate from one orbital period to another. Therefore, most likely, it can be determined more precisely for stars that have made more than one orbital revolution during the observation time. Given the astrometric position accuracy (0.1 mas for S02 and 1 mas for S102), such effects are likely to be detected only with next-generation telescopes such as the 30 m telescope.

7 Conclusion

It is concluded that it is possible to use Pulkovo methods – AMP and the geometric method developed by Kisselev and colleagues, as well as the method proposed by Izmailov – to obtain preliminary orbits of S-stars of the Sgr A ∗ cluster (in particular, “squeezars”) in accordance with the relative coordinates.

We also tried to improve our orbit for S02 and calculate an orbit for S27. The orbits S02 and S27 were determined on the basis of a short arc: S02 – at six average annual points with an orbital period of 16 years and observations of more than 16 years, S27 – at 13 observation points and with an orbital period of 112 years. We made an attempt to calculate theoretical relativistic parameters for some stars according to their orbital elements (Table 3). These parameters are of the same order of value as those expected by observers at the Keck and VLT telescopes.