Abstract
In this paper, we examine a static gravitational field with axial symmetry over probe particles in the solar system. Using the Weyl conformastatic solution as a model, we find a non-standard expression to perihelion advance due to the constraints imposed by the topology of the local gravitational field. We show that the application of the slow motion condition to the geodesic equations without altering Einstein's equations does not necessarily lead to the Newtonian limit; rather it leads to an intermediate nearly-newtonian gravitational stage, which can be applied to astrophysical problems in solar scale. We apply the model to the perihelion advance of inner planets and minor objects (NEO asteroids and the comets). As a result, we obtain an expression of a non-standard relativistic precession that reveals a close agreement to observational data calibrated with the Ephemerides of the Planets and the Moon (EPM2011). The study of perihelion advance of eight small celestial bodies (asteroids and comets) is also considered.
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