📝 SummarXiv

Abstract

We study the periapsis shift of timelike bound orbits in the Zipoy-Voorhees spacetime, which is an exact, static, axisymmetric, and vacuum solution characterized by the deformation parameter $\gamma$, including the Schwarzschild spacetime as $\gamma = 1$. 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, we show that the non-dimensional quadrupole moment $\tilde{M}_2$ of the supermassive compact object at Sagittarius A* is constrained to $\tilde{M}_2 \lesssim 9.7 \times 10^2$ or equivalently $\gamma \gtrsim 1.9 \times 10^{-2}$ from observational data on S2, whereas $-1/3 \leq \tilde{M}_2$ is required from this 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.