Impact of the Quadrupole Moment of the Sun on the Dynamics of the Earth-Moon System

Abstract

Range of values of the Sun’s mass quadrupole moment of coefficient J ₂ arising both from experimental and theoretical determinations enlarge across literature on two orders of magnitude, from around 10⁻⁷ until to 10⁻⁵. The accurate knowledge of the Moon’s physical librations, for which the Lunar Laser Ranging data reach an outstanding precision level, prove to be appropriate to reduce the interval of J ₂ values by giving an upper bound of J ₂. A solar quadrupole moment as high as 1.1 10⁻⁵ given either from the upper bounds of the error bars of the observations, or from the Roche’s theory, is not compatible with the knowledge of the lunar librations accurately modeled and observed with the LLR experiment. The suitable values of J ₂ have to be smaller than 3.0 10⁻⁶.

As a consequence, this upper bound of 3.0 10⁻⁶ is accepted to study the impact of the Sun’s quadrupole moment of mass on the dynamics of the Earth-Moon system. Such an effect (with J ₂ = 5.5 ± 1.3 × 10⁻⁶) has been already tested in 1983 by Campbell & Moffat using analytical approximate equations, and thus for the orbits of Mercury, Venus, the Earth and Icarus. The approximate equations are no longer sufficient compared with present observational data and exact equations are required. As if to compute the effect on the lunar librations, we have used our BJV relativistic model of solar system integration including the spin-orbit coupled motion of the Moon. The model is solved by numerical integration. The BJV model stems from general relativity by using the DSX formalism for purposes of celestial mechanics when it is about to deal with a system of n extended, weakly self-gravitating, rotating and deformable bodies in mutual interactions.

The resulting effects on the orbital elements of the Earth have been computed and plotted over 160 and 1600 years. The impact of the quadrupole moment of the Sun on the Earth’s orbital motion is mainly characterized by variations of ̇Ω, ̇ω, and ̇Ė. As a consequence, the Sun’s quadrupole moment of mass could play a sensible role over long time periods of integration of solar system models.