Modeling the Orbital Evolution of the Moon-Earth System

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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.


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,

\dot{a} = - {{3 k_2 a_e^5 G m}\over{[G (M+m)]^{1/2} a^{11/2}}} sin{\epsilon_{2200}}

In the above equation, a is the semi-major axis of the Moon’s orbit; a_e is the mean radius of the Earth; M and m are the masses of the Earth and Moon, respectively, while G is the gravitational constant. The parameters \epsilon_{2200} and k_2 deserve more elaboration as explained below:

  • k_2 is a so called Love number (named after Augustus Edward Hough Love), which represents the correction that needs to be applied to the gravitational tidal potential as a result of the Earth’s own deformation in response to that potential. Thus, k_2 is also a kind of a measure of the rigidity of the Earth. For a perfectly rigid planet, k_2=0, but for Earth the actual value is k_2 \approx 0.3 [7]. Actually, this is an averaged value; in reality the value would vary for each point on the Earth’s globe and interior depending on its depth and location, such as land versus water or mantle versus core.
  • \epsilon_{2200} is the (0,0) component of the strain tensor in spherical coordinates. The subscripts (2,2) refer to the 2^nd degree and 2^nd order spherical harmonics (l=2,m=2). [8] That is, the \epsilon_{22}) strain tensor is one of the components of Fourier series expansion of the overall strain tensor, and that furthermore the (l=2,m=2) component has been determined to be the dominant one. [2]

In his paper Kaula, 1964[2], shows how one can calculate \epsilon_{2200} starting from the rigidity (a.k.a. shear) modulus \mu, bulk modulus k, of Earth, and the specific energy dissipation function Q. In the same paper, these values have been estimated and tabulated at different depths. The values for k_2 that Kaula uses are also based on estimates at different depths. Note that the specific energy dissipation function Q is defined as follows [5],

Q^{-1}={1\over {2 \pi E_0}} \oint {-{dE \over dt}}\,dt

where E_0 is the maximum energy stored in the tidal distortion and the integral over -dE/dt that is the rate of dissipation, is the energy lost during the complete cycle. Furthermore in Caputo, 1967 [9] the author offers an analytical method for computing Q based on the density \rho, shear modulus \mu, and Lamé’s first parameter \lambda, the latter being related to the shear modulus and the bulk modulus according to the following formula: \lambda=k-2\mu/3 [10]. Despite the proposed approximating formula for computing Q and k_2, the values of these parameters have been largely unknown for most bodies in the Solar system. Except for the Moon, for which the ratio k_2/Q=0.0011 is available and also for the Earth for which there are available estimates [2], for most other bodies the values for Q are arbitrarily chosen to be Q=100. [11][2]. The need to choose arbitrary values for Q shows the difficulty in obtaining these values just based on observation. Fundamentally, these are material dependent parameters that are internal state variables (ISVs).












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