Akihito Katsumata | Searcharxiv

arXiv2025

Periapsis shift in the Zipoy-Voorhees spacetime

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.

gr-qc↗

arXiv2024

New series expansion method for the periapsis shift

We propose a new series expansion method for the periapsis shift. The method formulates the periapsis shift in various spacetimes analytically without using special functions and provides simple and highly accurate approximate formulae. We derive new series representations for the periapsis shift in the Kerr and the Chazy-Curzon spacetimes by using the method, where the expansion parameter is defined as the eccentricity divided by the non-dimensional quantity that vanishes in the limit of the innermost stable circular orbit. That is to say, the expansion parameter denotes how eccentric the orbit is and how close it is to the innermost stable circular orbit. The smaller the eccentricity, the higher the accuracy of the formulae that are obtained by truncating the new series representations up to a finite number of terms. If the eccentricity is sufficiently small, the truncated new representations have higher accuracy than the post-Newtonian expansion formulae even in strong gravitational fields where the convergence of the post-Newtonian expansion formula is not guaranteed. On the other hand, even if the orbit is highly eccentric, the truncated new representations have comparable or higher accuracy than the post-Newtonian expansion formulae if the semi-major axis is sufficiently large. An exact formula for the periapsis shift of the quasi-circular orbit in the Chazy-Curzon spacetime is also obtained as a special case of the new series representation.

gr-qc↗