Effective apsidal precession from a monopole solution in a Zipoy spacetime

Abstract

In this work, we examine the orbit equations originated from Zipoy's oblate metric. Accordingly, the solution of Einstein's vacuum equations can be written as a linear combination of Legendre polynomials of positive definite integers l. Starting from the zeroth order l = 0, in a nearly newtonian regime, we obtain a non-trivial formula favoring both retrograde and advanced solutions for the apsidal precession, depending on parameters related to the metric coefficients. Using a Chi-squared statistics, we apply the model to the apsidal precessions of Mercury and asteroids (1566 Icarus and 2-Pallas). As a result, we show that the obtained values favor the oblate solution as a more adapted approach as compared to those results produced by Weyl's cylindric and Schwarzschild solutions. Moreover, it is also shown that the resulting solution converges to the integrable case γ = 1 in the sense of the Zipoy-Voorhees metric.

Effective Potentials and Orbits in Weyl Conformastatic Slender Disk

“…The conformastatic solution is used to emphasize the restriction of axisymmetric symmetry by using just one Weyl potential function σ(r, z). This basic idea was applied in previous works to dierent themes such as apsidal precession in dierent systems [13,14,15,16] and to the dark matter problem [17].…”

Constraining deviations from spherical symmetry using $ γ$-metric

“…Using the method as shown in [20,21], we work with the previous orbit equation. The deviation angle δφ can be found analytically using…”