Weyl conformastatic perihelion advance

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.

Effective Apsidal Precession in Oblate Coordinates

“…Moreover, this metric is diffeomorphic to the Schwarzschild's one, and it does not lose its asymptotes and is asymptotically flat (Weyl, 1917;Rosen, 1949;Zipoy 1966;Gautreau, Hoffman and Armenti, 1972;Stephani et al, 2003). Differently from the works of (González, Gutiérrez-Piñeres and Ospina, 2008;Gutiérrez-Piñeres, González and Quevedo, 2013;Ujevic andLetelier, 2004, 2007) and (Vogt and Letelier, 2008) where the authors use a mass distribution with Weyl's exact of Einstein equations, we studied approximate solutions of this metric for a test particle by expanding the metric coefficient functions (or potentials) into a Taylor's series and as a result the obtained perihelion shift was about 43.105 arcsec/century (Capistrano, Roque and Valada, 2014). To obtain the oblate coordinates, a change of variable can be applied in such a form ρ = a cosh v cos θ and z = a sinh v sin θ, and a is a length parameter.…”

“…We use 9 data points concerning observations on the perihelion advance of Mercury in units of arcsecond per century ( .cy −1 ) as shown in table 01. We denote δφ sch for standard (Einstein) perihelion precession and δφ W eyl for the resulting perihelion advance using the Weyl conformastatic solution (Capistrano, Roque and Valada, 2014), which comes from an axially-symmetric motion of a test particle in Weyl's line element (Weyl, 1917). To control the systematics, we use GnuPlot 5.2 software to compute non-linear least-squared fitting by using the Levenberg-Marquardt algorithm for the goodness-of-fitting to data.…”

Effective apsidal precession from a monopole solution in a Zipoy spacetime

“…Differently from the works of [35][36][37][38] and [39], where the authors use a mass distribution to model galactic relativistic disks with Weyl's exact solution of Einstein equations, we investigated in [40] approximated solutions of this metric for a test particle in the perihelion precession by expanding the coefficient functions (or potentials) of the metric into a Taylor's series. As a result, e.g., we obtained the perihelion shift of Mercury about 43.105 arcsec/century in accordance with observations.…”

“…In the fourth column, some observational values of perihelion precession are available. The first data point was adapted from [61] by adding a supplementary precession calibrated with the Ephemerides of the Planets and the Moon (EPM2011) [62,63] δφ 43.20 ± 0.86 [64] 43.11 ± 0.22 [65] 43.11 ± 0.22 [66] 42.98 ± 0.09 [67] 43.13 ± 0.14 [68] 42.98 ± 0.04 [69,70] 43.03 ± 0.00 [71] 43.11 ± 0.45 [72,73] for the resulting perihelion advance using the Weyl conformastatic solution [40,76], which comes from an axiallysymmetric motion of a test particle in Weyl's line element [31]. To control the systematics, we use GnuPlot 5.2 software to compute non-linear least-squared fitting by using the Levenberg-Marquardt algorithm for the goodness-of-fitting to data.…”

On Nearly Newtonian Potentials and Their Implications to Astrophysics

“…Instead of making an application of approximation methods of GRT, the nearly Newtonian approximation may provide a simpler option to study astrophysical phenomena. In recent years, we have been focused on the apsidal precession (perihelion) problem applied for several celestial bodies including planets, comets and asteroids with satisfactory results [27,28].…”