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 derive both the exact formula for the periapsis shift of a quasi-circular orbit and the formula for the periapsis shift by the post-Newtonian (PN) expansion. We show that the periapsis shift in the Zipoy-Voorhees spacetime for $1/11 < γ\leq 1/5$ is the same as in the Schwarzschild spacetime to the 2PN order if $e=\sqrt{(γ^{-1}-5)/6}$, where $e$ is the eccentricity of the orbit. Furthermore, we show that the parameter $γ$ and the corresponding quadrupole moment $\tilde{M}_2$ of the supermassive compact object at Sagittarius A* are constrained to $γ\gtrsim 1.7 \times 10^{-2}$ and $\tilde{M}_2 \lesssim 1.2 \times 10^3$, respectively, from observational data on S2 using the obtained PN expansion formula. Finally, we derive a new series representation for the periapsis shift in the Zipoy-Voorhees spacetime using a recently proposed prescription, which shows fast convergence not only in the weak-field regime but also for small eccentricity.