Periapsis shift in the Zipoy-Voorhees spacetime
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arXiv
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| Natura: | Preprint |
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2025
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| author | Katsumata, Akihito Harada, Tomohiro |
| author_facet | Katsumata, Akihito Harada, Tomohiro |
| contents | We study the periapsis shift of timelike bound orbits in the equatorial plane in the Zipoy-Voorhees spacetime, which is an exact, static, axisymmetric, and vacuum solution characterized by the deformation parameter $γ$, including the Schwarzschild spacetime as $γ= 1$. We newly derive the formula for the periapsis shift by the post-Newtonian (PN) expansion and show that the quadrupole moment contributes to the periapsis shift apparently as in the 2PN order. Applying this formula to observational data on S2, a star orbiting closely to Sagittarius A* (Sgr A*), we show that the parameter $γ$ is constrained to $γ\gtrsim 1.9 \times 10^{-2}$ for the gravitational field of Sgr A*. This is equivalent to $\tilde{M}_2 \lesssim 9.7 \times 10^2$ in terms of the dimensionless quadrupole moment $\tilde{M}_2$, that is, the supermassive compact object at Sgr A* cannot be very prolate, whereas $\tilde{M}_2 \geq -1/3$ is required from the solution. Finally, we derive a new series representation in this spacetime using a recently proposed prescription, which shows fast convergence not only in the weak-field regime with not necessarily small eccentricity but also in the strong-field regime with small eccentricity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04280 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Periapsis shift in the Zipoy-Voorhees spacetime Katsumata, Akihito Harada, Tomohiro General Relativity and Quantum Cosmology Astrophysics of Galaxies We study the periapsis shift of timelike bound orbits in the equatorial plane in the Zipoy-Voorhees spacetime, which is an exact, static, axisymmetric, and vacuum solution characterized by the deformation parameter $γ$, including the Schwarzschild spacetime as $γ= 1$. We newly derive the formula for the periapsis shift by the post-Newtonian (PN) expansion and show that the quadrupole moment contributes to the periapsis shift apparently as in the 2PN order. Applying this formula to observational data on S2, a star orbiting closely to Sagittarius A* (Sgr A*), we show that the parameter $γ$ is constrained to $γ\gtrsim 1.9 \times 10^{-2}$ for the gravitational field of Sgr A*. This is equivalent to $\tilde{M}_2 \lesssim 9.7 \times 10^2$ in terms of the dimensionless quadrupole moment $\tilde{M}_2$, that is, the supermassive compact object at Sgr A* cannot be very prolate, whereas $\tilde{M}_2 \geq -1/3$ is required from the solution. Finally, we derive a new series representation in this spacetime using a recently proposed prescription, which shows fast convergence not only in the weak-field regime with not necessarily small eccentricity but also in the strong-field regime with small eccentricity. |
| title | Periapsis shift in the Zipoy-Voorhees spacetime |
| topic | General Relativity and Quantum Cosmology Astrophysics of Galaxies |
| url | https://arxiv.org/abs/2507.04280 |