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,


\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. Cite error: Closing </ref> missing for <ref> tag

[9]

[10]

[11]

[5]

[2]

[12]

[6]

[7]

[13]

[3]

[1]

[8]

[14]

[15]


  1. 1.0 1.1 Wikipedia (2015, 02 08). Lunar distance (astronomy). Retrieved 02 09, 2015, from Wikipedia: http://en.wikipedia.org/wiki/Lunar_distance_(astronomy)
  2. 2.0 2.1 2.2 Kaula, W.M. (1964, November). Tidal Dissipation by Solid Friction and the Resulting Orbital Evolution. Reviews of Geophysics, 2(4), 661-685
  3. 3.0 3.1 Thompson, T. (1999, December). The recession of the Moon and the Age of the Earth-Moon System. Retrieved 02 09, 2015, from The TalkOrigins Archive: http://www.talkorigins.org/faqs/moonrec.html
  4. Cite error: Invalid <ref> tag; no text was provided for refs named Bowden00
  5. 5.0 5.1 Goldreich, P., & Soter, S. (1965, November 17). Q in the Solar System. Icarus, 375-389. Henry, J. (2006, August). The moon’s recession and age. Journal of Creation, 65-70.
  6. 6.0 6.1 Pogge, R. (2007, October 14). Astronomy 161, Lecture 20:Tides. Retrieved 02 09, 2015, from Ohio State: http://www.astronomy.ohio-state.edu/~pogge/Ast161/Unit4/tides.html
  7. 7.0 7.1 Poulsen, S. K. (2009). Tidal Deformatoin of the Solid Earth - A Finite Difference Discretization. Faculty of Science. Niels Bohr Institute, University of Copenhagen.
  8. 8.0 8.1 Wikipedia (2014, 11 28). Spherical Harmonics. Retrieved 02 09, 2015, from Wikipedia: http://en.wikipedia.org/wiki/Spherical_harmonics
  9. Brown, W. (2008). In the Beginning: Compelling evidence for Creation and the Flood (8th edition). Center for Scientific Creation.
  10. Caputo, M. (1967). Linear Models of Dissipation whose Q is almost Frequency Independent-II. Geophysics Journal International, 529-539.
  11. Denis, C., Schreider, A., Varga, P., & Zavoti, J. (2002). Despinning of the earth rotation in the geological past and. Journal of Geodynamics, 667-685. Gladman, B., Quinn, D., Nicholson, P., & Rand, R. (1996). Synchornous Locking of Tidally Evolving Satellites. Icarus, 166-192.
  12. Maslov, L. (1991). Geodynamics of the Pacific segment of the Earth. Nauka (Science) in Russian.
  13. Su, Y., Fu, H., & Hu, H. (2012). Study on Correlation of Tidal Forces with Global Great. International Journal of Geosciences, 373-378.
  14. Wikipedia (2015, January 15). Bulk Modulus. Retrieved 02 10, 2015, from Wikipedia: http://en.wikipedia.org/wiki/Bulk_modulus
  15. Wikipedia (2014, December 31). Lunar Laser Ranging experiment. Retrieved 02 10, 2015, from Wikipedia: http://en.wikipedia.org/wiki/Lunar_Laser_Ranging_experiment
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