On distributions law of planetary rotations and revolutions as a function of aphelia, following Lagrange's formulation

TL;DR

This paper investigates the relationships between planetary rotations, revolutions, and aphelia using Lagrange's mechanics, revealing power-law behaviors and energy transfer mechanisms that differ between terrestrial and giant planets.

Contribution

It introduces a novel analysis of planetary motions based on Lagrange's formalism, highlighting power-law dependencies and energy exchanges linked to aphelia.

Findings

Giant planets' gravitational constants decrease with aphelia.

Perihelion delays follow a (-5/2) power law of aphelia.

Energy transfer between planetary revolutions and rotations follows a linear power law.

Abstract

Earth rotation is determined by polar motion (PM) and length of day (lod). The excitation sources of PM are torques linked to fluid circulations ("geophysical excitations"), and those of lod to luni-solar tides ("astronomical excitations"). We explore the links between the rotations and revolutions of planets, following Lagrange's (1853) presentation of mechanics. The energy of a planet in motion in a central field is the sum of kinetic, centrifugal (planet dependent) and centripetal (identical for all planets) energies. For each planet, one can calculate a "constant of gravitation" Gp . For the giant planets, Gp decreases as a function of aphelia. There is no such organized behavior for the terrestrial planets. The perturbing potential of other planets generates a small angular contribution to the displacement : this happens to be identical to Einstein's famous formula for precession. Delays in the planet's perihelia follow a (-5/2) power law of a. This is readily understood in the Lagrange formalism (the centrifugal term takes over for small distances). The telluric planets have lost energy, probably transferred to the planets rotations. The ratio of areal velocities to rotation obeys a -5/2 power law of a. The ratio of areal velocity to integrated period R also fits a -5/2 power dependence, implying linearity of the energy exchange between revolution and rotation. For Einstein deformation of space-time by the Sun is the origin of the field perturbation. For Lagrange the perturbation could only be due to the interactions of torques. The perihelion delays, the areal velocities and the planetary rotations display power laws of aphelia, whose behavior contrasts with that of the kinetic moment. The areal velocity being linearly linked to the kinetic moment of planets, this must be the level at which the transfer is achieved.