Constraints on Graviton Mass from Schwarzschild Precession in the Orbits of S-Stars around the Galactic Center
Abstract
:1. Introduction
2. Orbital Precession in Yukawa-like Gravitational Potential
3. Stellar Orbits in Extended/Modified PPN Formalisms
4. Results: Constraints on the Compton Wavelength and Mass of the Graviton
PPN Fit of the Observed Orbit of the S2 Star
- 1.
- First, a simulated orbit of the S2 star in the extended formalism is calculated by numerical integration of the equations of motion (7), starting from initial conditions , where the first two components specify the initial position and the last two the initial velocity in the orbital plane. In our simulations, the initial conditions correspond to the time of apocenter passage, , preceding the first astrometric observation at : , where T is the time of pericenter passage, P is the orbital period, and k is the smallest positive integer (number of periods) for which . Then, the initial conditions are: , , , and where is the apocenter distance and is the corresponding orbital velocity at the apocenter.
- 2.
- The true orbit obtained in the first step, which depends only on , was then projected to the observer’s sky plane using the remaining geometrical orbital elements , in order to obtain the corresponding positions along the apparent orbit,
- 3.
- Finally, is obtained according to Equation (16), taking into account only those apparent positions that are calculated at the same epochs as the astrometric observations .
5. Conclusions
- 1.
- We found the condition for parameter of the phenomenological Yukawa-like gravitational potential (3) under which the orbital precession in this potential deviates from the Schwarzschild precession in GR by a factor ;
- 2.
- 3.
- Both of the studied PPN formalisms result in close and very similar simulated orbits of S-stars, which practically overlap during the first orbital period and then begin to slowly diverge over time due to some assumed theoretical approximations;
- 4.
- In most cases, the constraints on the upper bound on graviton mass, , and its absolute error, , obtained using the phenomenological potential (3), were improved by ∼30% in respect to our previous corresponding estimates from [55], obtained using a slightly different Yukawa-like potential derived in the frame of theories of gravity, although the relative errors in both cases remained the same;
- 5.
- These results were also confirmed in the case of the S2 star by fitting its observed orbit into the frame of the extended formalism, which resulted in the best-fit value for the graviton Compton wavelength, , within the error intervals of its corresponding estimates obtained according to Equation (14) from the detected values of ;
- 6.
- The least reliable constraints with unrealistically high uncertainties were only obtained from estimates for that were very close to its value predicted by GR, being only slightly larger than 1;
- 7.
- If one compares the results from Table 2 with those from Table 3, it can be seen that the upper bounds on graviton mass, , are very similar. In the case of the S2 star, eV and the relative error is approximately 50%. We can conclude that more precise future observations are required in order to further improve the upper graviton mass bounds.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
GC | Galactic Center |
GR | General Relativity |
SMBH | Supermassive Black Hole |
PPN formalism | Parameterized Post-Newtonian formalism |
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R.E. | ||||||||
---|---|---|---|---|---|---|---|---|
(AU) | (%) | |||||||
1.100 | 0.190 | 66361.5 | ± | 63890.7 | 124.9 | ± | 120.2 | 96.3 |
1.010 | 0.160 | 209853.4 | ± | 1681506.5 | 39.5 | ± | 316.5 | 801.3 |
1.141 | 0.144 | 55886.4 | ± | 29251.3 | 148.3 | ± | 77.6 | 52.3 |
Star | R.E. | R.E. | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
(AU) | (%) | (AU) | (%) | |||||||||||
S1 | 1.6e+06 | ± | 1.6e+06 | 5.1 | ± | 5.1 | 100.7 | 1.2e+06 | ± | 6.6e+05 | 7.0 | ± | 3.9 | 55.7 |
S2 | 6.6e+04 | ± | 6.4e+04 | 124.9 | ± | 120.2 | 96.3 | 4.8e+04 | ± | 2.5e+04 | 172.1 | ± | 88.3 | 51.3 |
S4 | 8.8e+05 | ± | 8.5e+05 | 9.4 | ± | 9.1 | 96.7 | 6.4e+05 | ± | 3.3e+05 | 13.0 | ± | 6.7 | 51.7 |
S6 | 1.0e+06 | ± | 9.5e+05 | 8.3 | ± | 7.9 | 95.2 | 7.2e+05 | ± | 3.6e+05 | 11.5 | ± | 5.8 | 50.2 |
S8 | 5.5e+05 | ± | 5.4e+05 | 15.0 | ± | 14.7 | 98.0 | 4.0e+05 | ± | 2.1e+05 | 20.6 | ± | 10.9 | 53.0 |
S9 | 4.5e+05 | ± | 4.4e+05 | 18.6 | ± | 18.6 | 99.7 | 3.2e+05 | ± | 1.8e+05 | 25.7 | ± | 14.0 | 54.7 |
S12 | 2.4e+05 | ± | 2.3e+05 | 34.9 | ± | 33.6 | 96.4 | 1.7e+05 | ± | 8.9e+04 | 48.1 | ± | 24.7 | 51.4 |
S13 | 5.5e+05 | ± | 5.2e+05 | 15.1 | ± | 14.5 | 95.5 | 4.0e+05 | ± | 2.0e+05 | 20.9 | ± | 10.5 | 50.5 |
S14 | 7.3e+04 | ± | 7.8e+04 | 114.1 | ± | 122.4 | 107.3 | 5.3e+04 | ± | 3.3e+04 | 157.2 | ± | 98.0 | 62.3 |
S17 | 8.7e+05 | ± | 8.5e+05 | 9.5 | ± | 9.2 | 97.1 | 6.3e+05 | ± | 3.3e+05 | 13.1 | ± | 6.8 | 52.1 |
S18 | 4.5e+05 | ± | 4.3e+05 | 18.4 | ± | 17.8 | 96.5 | 3.3e+05 | ± | 1.7e+05 | 25.4 | ± | 13.1 | 51.5 |
S19 | 9.4e+05 | ± | 1.1e+06 | 8.8 | ± | 10.2 | 116.4 | 6.8e+05 | ± | 4.9e+05 | 12.1 | ± | 8.7 | 71.4 |
S21 | 2.5e+05 | ± | 2.5e+05 | 33.3 | ± | 33.2 | 99.6 | 1.8e+05 | ± | 9.9e+04 | 45.9 | ± | 25.1 | 54.6 |
S22 | 5.9e+06 | ± | 6.7e+06 | 1.4 | ± | 1.6 | 114.1 | 4.3e+06 | ± | 3.0e+06 | 1.9 | ± | 1.3 | 69.1 |
S23 | 4.5e+05 | ± | 5.2e+05 | 18.5 | ± | 21.4 | 115.6 | 3.2e+05 | ± | 2.3e+05 | 25.5 | ± | 18.0 | 70.6 |
S24 | 1.3e+06 | ± | 1.3e+06 | 6.6 | ± | 6.8 | 103.2 | 9.2e+05 | ± | 5.3e+05 | 9.1 | ± | 5.3 | 58.2 |
S29 | 7.4e+05 | ± | 8.1e+05 | 11.1 | ± | 12.2 | 109.1 | 5.4e+05 | ± | 3.5e+05 | 15.4 | ± | 9.8 | 64.1 |
S31 | 1.1e+06 | ± | 1.0e+06 | 7.8 | ± | 7.5 | 96.4 | 7.8e+05 | ± | 4.0e+05 | 10.7 | ± | 5.5 | 51.4 |
S33 | 1.8e+06 | ± | 1.9e+06 | 4.7 | ± | 5.0 | 107.0 | 1.3e+06 | ± | 7.9e+05 | 6.5 | ± | 4.0 | 62.0 |
S38 | 1.1e+05 | ± | 1.0e+05 | 76.9 | ± | 73.3 | 95.4 | 7.8e+04 | ± | 3.9e+04 | 106.0 | ± | 53.4 | 50.4 |
S39 | 2.5e+05 | ± | 2.5e+05 | 33.2 | ± | 32.8 | 98.8 | 1.8e+05 | ± | 9.7e+04 | 45.8 | ± | 24.6 | 53.8 |
S42 | 3.2e+06 | ± | 4.0e+06 | 2.6 | ± | 3.1 | 122.7 | 2.4e+06 | ± | 1.8e+06 | 3.5 | ± | 2.7 | 77.7 |
S54 | 1.9e+06 | ± | 3.5e+06 | 4.4 | ± | 8.4 | 188.3 | 1.4e+06 | ± | 1.9e+06 | 6.1 | ± | 8.8 | 143.3 |
S55 | 9.6e+04 | ± | 9.3e+04 | 86.5 | ± | 84.5 | 97.6 | 6.9e+04 | ± | 3.7e+04 | 119.3 | ± | 62.7 | 52.6 |
S60 | 6.6e+05 | ± | 6.4e+05 | 12.6 | ± | 12.3 | 97.7 | 4.8e+05 | ± | 2.5e+05 | 17.4 | ± | 9.2 | 52.7 |
S66 | 8.5e+06 | ± | 8.6e+06 | 1.0 | ± | 1.0 | 101.4 | 6.2e+06 | ± | 3.5e+06 | 1.3 | ± | 0.8 | 56.4 |
S67 | 5.2e+06 | ± | 5.2e+06 | 1.6 | ± | 1.6 | 100.1 | 3.8e+06 | ± | 2.1e+06 | 2.2 | ± | 1.2 | 55.1 |
S71 | 1.3e+06 | ± | 1.4e+06 | 6.4 | ± | 6.8 | 107.3 | 9.4e+05 | ± | 5.9e+05 | 8.8 | ± | 5.5 | 62.3 |
S83 | 7.6e+06 | ± | 8.4e+06 | 1.1 | ± | 1.2 | 110.3 | 5.5e+06 | ± | 3.6e+06 | 1.5 | ± | 1.0 | 65.3 |
S85 | 2.3e+07 | ± | 4.9e+07 | 0.4 | ± | 0.8 | 211.0 | 1.7e+07 | ± | 2.8e+07 | 0.5 | ± | 0.8 | 166.0 |
S87 | 2.0e+07 | ± | 2.1e+07 | 0.4 | ± | 0.4 | 102.4 | 1.5e+07 | ± | 8.5e+06 | 0.6 | ± | 0.3 | 57.4 |
S89 | 3.6e+06 | ± | 3.8e+06 | 2.3 | ± | 2.5 | 107.8 | 2.6e+06 | ± | 1.6e+06 | 3.2 | ± | 2.0 | 62.8 |
S91 | 1.2e+07 | ± | 1.2e+07 | 0.7 | ± | 0.7 | 101.9 | 8.4e+06 | ± | 4.8e+06 | 1.0 | ± | 0.6 | 56.9 |
S96 | 8.4e+06 | ± | 8.4e+06 | 1.0 | ± | 1.0 | 100.0 | 6.1e+06 | ± | 3.3e+06 | 1.4 | ± | 0.7 | 55.0 |
S97 | 1.5e+07 | ± | 1.9e+07 | 0.6 | ± | 0.7 | 125.9 | 1.1e+07 | ± | 8.8e+06 | 0.8 | ± | 0.6 | 80.9 |
S145 | 4.5e+06 | ± | 6.1e+06 | 1.9 | ± | 2.5 | 136.7 | 3.2e+06 | ± | 3.0e+06 | 2.6 | ± | 2.4 | 91.7 |
S175 | 8.2e+04 | ± | 9.0e+04 | 101.4 | ± | 111.8 | 110.3 | 5.9e+04 | ± | 3.9e+04 | 139.7 | ± | 91.2 | 65.3 |
R34 | 7.6e+06 | ± | 9.2e+06 | 1.1 | ± | 1.3 | 120.5 | 5.5e+06 | ± | 4.2e+06 | 1.5 | ± | 1.1 | 75.5 |
R44 | 3.3e+07 | ± | 5.2e+07 | 0.2 | ± | 0.4 | 156.2 | 2.4e+07 | ± | 2.7e+07 | 0.3 | ± | 0.4 | 111.2 |
Star | R.E. | R.E. | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
(AU) | (%) | (AU) | (%) | |||||||||||
S1 | 1.3e+06 | ± | 7.2e+05 | 6.4 | ± | 3.6 | 55.7 | 1.4e+06 | ± | 7.6e+05 | 6.1 | ± | 3.4 | 55.7 |
S2 | 5.2e+04 | ± | 2.7e+04 | 158.0 | ± | 81.0 | 51.3 | 5.5e+04 | ± | 2.8e+04 | 149.9 | ± | 76.8 | 51.3 |
S4 | 7.0e+05 | ± | 3.6e+05 | 11.9 | ± | 6.1 | 51.7 | 7.4e+05 | ± | 3.8e+05 | 11.3 | ± | 5.8 | 51.7 |
S6 | 7.9e+05 | ± | 4.0e+05 | 10.5 | ± | 5.3 | 50.2 | 8.3e+05 | ± | 4.2e+05 | 10.0 | ± | 5.0 | 50.2 |
S8 | 4.4e+05 | ± | 2.3e+05 | 18.9 | ± | 10.0 | 53.0 | 4.6e+05 | ± | 2.4e+05 | 17.9 | ± | 9.5 | 53.0 |
S9 | 3.5e+05 | ± | 1.9e+05 | 23.6 | ± | 12.9 | 54.7 | 3.7e+05 | ± | 2.0e+05 | 22.3 | ± | 12.2 | 54.7 |
S12 | 1.9e+05 | ± | 9.7e+04 | 44.1 | ± | 22.7 | 51.4 | 2.0e+05 | ± | 1.0e+05 | 41.8 | ± | 21.5 | 51.4 |
S13 | 4.3e+05 | ± | 2.2e+05 | 19.2 | ± | 9.7 | 50.5 | 4.6e+05 | ± | 2.3e+05 | 18.2 | ± | 9.2 | 50.5 |
S14 | 5.7e+04 | ± | 3.6e+04 | 144.3 | ± | 89.9 | 62.3 | 6.1e+04 | ± | 3.8e+04 | 136.9 | ± | 85.3 | 62.3 |
S17 | 6.9e+05 | ± | 3.6e+05 | 12.0 | ± | 6.3 | 52.1 | 7.3e+05 | ± | 3.8e+05 | 11.4 | ± | 5.9 | 52.1 |
S18 | 3.6e+05 | ± | 1.8e+05 | 23.3 | ± | 12.0 | 51.5 | 3.8e+05 | ± | 1.9e+05 | 22.1 | ± | 11.4 | 51.5 |
S19 | 7.4e+05 | ± | 5.3e+05 | 11.1 | ± | 8.0 | 71.4 | 7.8e+05 | ± | 5.6e+05 | 10.6 | ± | 7.5 | 71.4 |
S21 | 2.0e+05 | ± | 1.1e+05 | 42.2 | ± | 23.0 | 54.6 | 2.1e+05 | ± | 1.1e+05 | 40.0 | ± | 21.8 | 54.6 |
S22 | 4.7e+06 | ± | 3.2e+06 | 1.8 | ± | 1.2 | 69.1 | 4.9e+06 | ± | 3.4e+06 | 1.7 | ± | 1.2 | 69.1 |
S23 | 3.5e+05 | ± | 2.5e+05 | 23.4 | ± | 16.5 | 70.6 | 3.7e+05 | ± | 2.6e+05 | 22.2 | ± | 15.7 | 70.6 |
S24 | 1.0e+06 | ± | 5.8e+05 | 8.3 | ± | 4.8 | 58.2 | 1.1e+06 | ± | 6.1e+05 | 7.9 | ± | 4.6 | 58.2 |
S29 | 5.9e+05 | ± | 3.8e+05 | 14.1 | ± | 9.0 | 64.1 | 6.2e+05 | ± | 4.0e+05 | 13.4 | ± | 8.6 | 64.1 |
S31 | 8.5e+05 | ± | 4.3e+05 | 9.8 | ± | 5.0 | 51.4 | 8.9e+05 | ± | 4.6e+05 | 9.3 | ± | 4.8 | 51.4 |
S33 | 1.4e+06 | ± | 8.6e+05 | 6.0 | ± | 3.7 | 62.0 | 1.5e+06 | ± | 9.1e+05 | 5.6 | ± | 3.5 | 62.0 |
S38 | 8.5e+04 | ± | 4.3e+04 | 97.3 | ± | 49.0 | 50.4 | 9.0e+04 | ± | 4.5e+04 | 92.3 | ± | 46.5 | 50.4 |
S39 | 2.0e+05 | ± | 1.1e+05 | 42.0 | ± | 22.6 | 53.8 | 2.1e+05 | ± | 1.1e+05 | 39.8 | ± | 21.4 | 53.8 |
S42 | 2.6e+06 | ± | 2.0e+06 | 3.2 | ± | 2.5 | 77.7 | 2.7e+06 | ± | 2.1e+06 | 3.1 | ± | 2.4 | 77.7 |
S54 | 1.5e+06 | ± | 2.1e+06 | 5.6 | ± | 8.0 | 143.3 | 1.6e+06 | ± | 2.2e+06 | 5.3 | ± | 7.6 | 143.3 |
S55 | 7.6e+04 | ± | 4.0e+04 | 109.5 | ± | 57.6 | 52.6 | 8.0e+04 | ± | 4.2e+04 | 103.8 | ± | 54.6 | 52.6 |
S60 | 5.2e+05 | ± | 2.7e+05 | 16.0 | ± | 8.4 | 52.7 | 5.5e+05 | ± | 2.9e+05 | 15.2 | ± | 8.0 | 52.7 |
S66 | 6.7e+06 | ± | 3.8e+06 | 1.2 | ± | 0.7 | 56.4 | 7.1e+06 | ± | 4.0e+06 | 1.2 | ± | 0.7 | 56.4 |
S67 | 4.1e+06 | ± | 2.3e+06 | 2.0 | ± | 1.1 | 55.1 | 4.4e+06 | ± | 2.4e+06 | 1.9 | ± | 1.0 | 55.1 |
S71 | 1.0e+06 | ± | 6.4e+05 | 8.1 | ± | 5.0 | 62.3 | 1.1e+06 | ± | 6.8e+05 | 7.6 | ± | 4.8 | 62.3 |
S83 | 6.0e+06 | ± | 3.9e+06 | 1.4 | ± | 0.9 | 65.3 | 6.4e+06 | ± | 4.2e+06 | 1.3 | ± | 0.8 | 65.3 |
S85 | 1.8e+07 | ± | 3.0e+07 | 0.5 | ± | 0.8 | 166.0 | 1.9e+07 | ± | 3.2e+07 | 0.4 | ± | 0.7 | 166.0 |
S87 | 1.6e+07 | ± | 9.3e+06 | 0.5 | ± | 0.3 | 57.4 | 1.7e+07 | ± | 9.8e+06 | 0.5 | ± | 0.3 | 57.4 |
S89 | 2.8e+06 | ± | 1.8e+06 | 3.0 | ± | 1.9 | 62.8 | 3.0e+06 | ± | 1.9e+06 | 2.8 | ± | 1.8 | 62.8 |
S91 | 9.1e+06 | ± | 5.2e+06 | 0.9 | ± | 0.5 | 56.9 | 9.6e+06 | ± | 5.5e+06 | 0.9 | ± | 0.5 | 56.9 |
S96 | 6.6e+06 | ± | 3.6e+06 | 1.2 | ± | 0.7 | 55.0 | 7.0e+06 | ± | 3.8e+06 | 1.2 | ± | 0.7 | 55.0 |
S97 | 1.2e+07 | ± | 9.6e+06 | 0.7 | ± | 0.6 | 80.9 | 1.2e+07 | ± | 1.0e+07 | 0.7 | ± | 0.5 | 80.9 |
S145 | 3.5e+06 | ± | 3.2e+06 | 2.4 | ± | 2.2 | 91.7 | 3.7e+06 | ± | 3.4e+06 | 2.2 | ± | 2.0 | 91.7 |
S175 | 6.5e+04 | ± | 4.2e+04 | 128.2 | ± | 83.7 | 65.3 | 6.8e+04 | ± | 4.4e+04 | 121.6 | ± | 79.4 | 65.3 |
R34 | 6.0e+06 | ± | 4.6e+06 | 1.4 | ± | 1.0 | 75.5 | 6.4e+06 | ± | 4.8e+06 | 1.3 | ± | 1.0 | 75.5 |
R44 | 2.6e+07 | ± | 2.9e+07 | 0.3 | ± | 0.3 | 111.2 | 2.8e+07 | ± | 3.1e+07 | 0.3 | ± | 0.3 | 111.2 |
Parameter | Value | Fit Error | Unit |
---|---|---|---|
82,175.7 | 9828.05 | AU | |
M | 4.15 | 0.27 | |
R | 8.33 | 0.198 | kpc |
a | 0.1229 | 0.00430 | arcsec |
e | 0.8797 | 0.01597 | |
i | 134.89 | 1.984 | ° |
224.57 | 5.208 | ° | |
62.78 | 4.562 | ° | |
P | 15.98 | 0.362 | yr |
T | 2018.12219 | 0.696709 | yr |
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© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
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Jovanović, P.; Borka Jovanović, V.; Borka, D.; Zakharov, A.F. Constraints on Graviton Mass from Schwarzschild Precession in the Orbits of S-Stars around the Galactic Center. Symmetry 2024, 16, 397. https://doi.org/10.3390/sym16040397
Jovanović P, Borka Jovanović V, Borka D, Zakharov AF. Constraints on Graviton Mass from Schwarzschild Precession in the Orbits of S-Stars around the Galactic Center. Symmetry. 2024; 16(4):397. https://doi.org/10.3390/sym16040397
Chicago/Turabian StyleJovanović, Predrag, Vesna Borka Jovanović, Duško Borka, and Alexander F. Zakharov. 2024. "Constraints on Graviton Mass from Schwarzschild Precession in the Orbits of S-Stars around the Galactic Center" Symmetry 16, no. 4: 397. https://doi.org/10.3390/sym16040397