Abstract
A first introduction is presented to the comparison between classical and relativistic gravitational effects related to planetary shape characterization. The Earth and the giant planets are the examples considered. The analysis is mainly devoted to relativistic and classical predictions of periastron shifts for equatorial or almost equatorial orbits around the Earth and the giant planets, which can be used as tools for determinations of the shape and density distribution. The ratios between relativistic (up to the Lense–Thirring order correction) and classical (resulting from the harmonic expansion) effects and their dependence on the orbit parameters are analyzed in order to identify the conditions improving the possibility to resolve mixed effects. In a complementary approach, predictions for freely falling test particles from relativistic corrections and classical harmonic expansions of the Earth and other planets are compared within the same shape characterization framework.
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Notes
Recall that \(P_{2}(\cos(\pi/2))=-1/2\), \(P_{3}(\cos(\pi/2))=0\), and \(P_{4}(\cos(\pi/2))=3/8\).
This mean density should result from a local density invariant under translations in the directions \(x\) and \(y\).
In this case we assume that the actual density leading to the mean value \(\rho\) only depends on the distance from the center of the planet.
We adopt the signature \((+---)\).
There is also a precession of the orbital plane which we shall not discuss here; see below.
A fluid planet could have a differential rotation rate, with a radial dependence.
For example, on Earth we would have \(v^{2}/c^{2}\sim 10^{-11}\) and \(Gm^{\prime}/(c^{2}R)\sim 10^{-9}\).
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Fazzito, S.Y., Simeone, C.M. A Preliminary Study on Earth and Other Planets Shape Determination: Comparison of Classical and Relativistic Gravitational Effects. Gravit. Cosmol. 30, 172–188 (2024). https://doi.org/10.1134/S0202289324700075
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DOI: https://doi.org/10.1134/S0202289324700075