Moons perigee mass as a missing component of the Earths precession-nutation theory

In this work, the nutation momentum acting upon the Earth from the Moon’s perigee mass that has not been taken into account in the Earth’s precession-nutation theory was revealed. This missing momentum exhibits itself in the so-called “local latitude variation” with the Chandler’s period. The results of our work raise the question of updating the Earth’s precession-nutation theory and revising some postulates of the time service, astronomy, geophysics, satellite navigation, etc.

💡 Research Summary

The paper attempts to introduce a previously neglected component into the Earth’s precession‑nutation theory: the gravitational effect of the Moon’s perigee, modeled as a fictitious “perigee mass.” The authors argue that this missing momentum manifests as the observed local latitude variation with the Chandler period (≈433 days). They begin with a historical overview of zenith‑distance (latitude) observations dating back to the 18th century, emphasizing the discovery of a 400‑440 day periodicity (the Chandler wobble) and the traditional interpretation that it represents a free or residual motion of the Earth’s rotation axis within the solid Earth.

In the problem definition, the authors simplify the Earth–Moon system by treating both bodies as rigid spheres and focusing exclusively on the variation of the Moon’s gravity field at a fixed surface point A. They introduce a local coordinate system (A‑xyz) whose z‑axis points toward the Earth’s centre, and define the angle γ(t) between the local vertical (plumb line) and the direction of the net gravitational force. The net force is taken as the sum of Earth’s own gravity and an additional force due to the Moon’s perigee.

The core of the paper is the construction of a “perigee mass” mΠ. Starting from the integral of the Moon’s gravitational attraction over one lunar month (≈28 days), they derive an average force directed toward the perigee. By equating this average force to the Newtonian attraction of a point mass located at the perigee, they obtain

 mΠ = Mmoon e (1+e)²,

where e is the instantaneous orbital eccentricity. For the present lunar eccentricity (≈0.055) this yields a perigee mass of roughly 1/20 of the Moon’s actual mass. The authors then assume that the eccentricity varies harmonically with a period equal to the perigee‑to‑perigee cycle (≈8.85 years), thereby making mΠ a slowly varying quantity.

Next, they describe the observer’s reference frames. Two orthogonal Cartesian systems are defined: one fixed to the ecliptic (O‑xeyez) and one fixed to the Earth’s equator (O‑x′y′z′). The observer’s position on Earth is expressed in terms of latitude ϕA, longitude λA, and the Earth‑Moon distance RA, which depends on the ellipsoidal shape of the Earth. The authors emphasize that the time step of the observation series must be an integer multiple of the solar day (24 h) rather than the sidereal day (≈23 h 56 m) because the observer rotates with respect to the Sun, not the stars. This choice, they claim, introduces an extra angular velocity ωyear = dλsun/dt that must be accounted for when extracting external perturbation periods.

The Moon’s orbit is modeled with six time‑dependent elements: inclination i(t), nodal longitude ψ(t), argument of perigee ϕ(t), eccentricity e(t), semi‑major axis a(t), and the epoch of perigee passage t*. Approximate analytical expressions for ψ(t) and ϕ(t) are taken from a 1900 epoch solution, yielding nodal precession period Tψ ≈ –18.6 years and perigee precession period Tϕ ≈ 6 years. The combined perigee cycle is then calculated as

 Tperigee = (Tψ · Tϕ)/(Tψ + Tϕ) ≈ 8.85 years.

With the perigee mass positioned in the ecliptic frame via a sequence of rotation matrices (inclination, nodal, and perigee rotations), the authors compute the vector rΠ(t) of the perigee mass and the vector rA(t) of the observer. The total gravitational force at the observer is

 f(t) = G mA Mearth rA/|rA|³ + G mA mΠ (rΠ – rA)/|rΠ – rA|³.

The direction of this force defines the plumb line, and the angle γ(t) = arccos