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 $\gamma$, 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 to observational data on S2, a star orbiting closely to Sagittarius A* (Sgr A*), we show that the parameter $\gamma$ is constrained to $\gamma \gtrsim 1.9 \times 10^{-2}$ 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.