# Modeling the Orbital Evolution of the Moon-Earth System

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## Abstract

Dissipation of tidal energy in the Earth’s surface and interior as well as the pull of the Earth’s tidal bulge on the Moon causes the Moon’s orbit to recede away from the Earth at a rate that is currently about 3.8 cm/year.[1] This is consistent with model calculations such as Kaula, 1964[2]. The rate of the lunar orbital recession has been used as an argument by Young-Earth creationists to place an upper limit on the age of the Earth-Moon system, which has brought about controversy due to doubts about the accuracy of the model used [3][4]. Such model calculations are based on estimations of the specific energy dissipation function per unit volume $Q$, the bulk modulus $k$, and the rigidity modulus $\mu$ throughout the Earth’s surface and interior, which do not lend themselves easily to direct measurements. In this paper we propose a method for computing these constitutive parameters by using ICME principles and performing simulations at multiple scales.

## Background

Figure 1. The force of attraction between the satellite m and the nearer tidal bulge A exceeds that between m and B; a component of the net torque retards the rotation of the planet M and accelerates the satellite in its orbit. [Included from Goldreich & Soter, 1965][5]

The Moon’s gravity causes the Earth to bulge approximately along the Earth-Moon orientation and to depress in the perpendicular plane. This applies not only to oceans, but also to the Earth’s crust and interior. However, because the Earth spins around its axis in one day, while it takes one month for the Moon to rotate in the same direction, the tidal bulge ends up slightly misaligned with the Earth-Moon direction as shown on Figure 1. Due to this asymmetry, the Earth would experience a body moment in the opposite direction from its rotation, i.e. in the direction of slowing down the Earth’s rotation, while at the same time the Moon would experience a body force component that would accelerate the Moon’s orbital velocity thus raising it up to a higher orbit. Effectively, a part of the Earth’s spin energy is being transferred to the Moon’s rotational energy. [6]

From the view point of someone on Earth near the equator, it would appear that the tidal bulge moves at a velocity of ≈1000 "miles/h" across the face of the Earth opposite to the direction of the Earth’s spin. The bending, compressing and tension of the crust and mantle would not be elastic and so some energy would be dissipated in the process. The bulge and energy dissipation are what ultimately determine the rate of the lunar orbital recession. These in turn depend on material specific parameters that can potentially be derived using ICME techniques.

Figure 2. Overview of ICME scales, simulations and bridges

The equation for the rate of lunar orbital recession as given by Kaula, 1964[2] is the following,

$a\dot\approx -$ [/itex]

## References

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