Periapsis shift in the Zipoy-Voorhees spacetime

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 $γ$, 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, we show that the non-dimensional quadrupole moment M̃₂ of the supermassive compact object at Sagittarius A* is constrained to M̃₂ 9.7 × 10² or equivalently γ 1.9 × 10⁻² from observational data on S2, whereas -1/3 ≤ M̃₂ 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.