Abstract
From the observations of the gravitational field and the figure of the Moon, it is known that its center of mass (briefly COM) does not coincide with the center of figure (COF), and the line “COF/COM” is not directed to the center of the Earth, but deviates from it to the South–East. Here we study the deviation of the lunar COM to the East from the mean direction to Earth.
At first, we consider the optical libration of a satellite with synchronous rotation around the planet for an observer at a point on second (empty) orbit focus. It is found that the main axis of inertia of the satellite has asymmetric nonlinear oscillations with amplitude proportional to the square of the orbit eccentricity. Given this effect, a mechanism of tidal secular evolution of the Moon’s orbit is offered that explains up to \(20\%\) of the known displacement of the lunar COM to the East. It is concluded that from the alternative—evolution of the Moon’s orbit with a decrease or increase in eccentricity—only the scenario of evolution with a monotonous increase in orbit eccentricity agrees with the displacement of lunar COM to the East. The precise calculations available confirm that now the eccentricity of the lunar orbit is actually increasing and therefore in the past it was less than its modern value, \(e = 0.0549\).
To fully explain the displacement of the Moon’s COM to the East was deduced a second mechanism, which is based on the reliable effect of tidal changes in the shape of the Moon. For this purpose the differential equation which governs the process of displacement of the Moon’s COM to the East with inevitable rounding off its form in the tidal increase process of the distance between the Earth and the Moon is derived. The second mechanism not only explains the Moon’s COM displacement to the East, but it also predicts that the elongation of the lunar figure in the early epoch was significant and could reach the value \(\varepsilon\approx0.31\). Applying the theory of tidal equilibrium figures, we can estimate how close to the Earth the Moon could have formed.
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26 February 2019
Correction to: Astrophys Space Sci (2018) 363:186 https://doi.org/10.1007/s10509-018-3405-z
After publication of the original article, the author made a series of checks on the main results. In general, the results of calculations were confirmed. Nevertheless, in numerical calculations in the second of the mechanisms presented there, an error was made (the deflection angle ‘alpha’ was taken with the opposite sign), which eventually led to an incorrect result for ‘math expression’. After correction, the formula (48) will have the form ‘math expression’. After amendment in the sign of the angle ‘alpha’, Eq. (57) gives the value of ‘math expression’ (instead of the previous ‘math expression’). The solution to Eq. (57) is shown in Fig. 1.
Fig. 1 The dependence of the normalized to ‘math expression’ distance ‘math expression’ from the Earth, on which the Moon could have been formed in the early era of the elongation ‘varepsilon’ of its figure. The strokes show the results of calculations
According to the corrected result, the Moon was formed in the annular zone at a distance of 3 to 4 average radii of the modern Earth. This result seems to be consistent with the modern idea that the Moon was formed as a result of a giant impact in the immediate vicinity of Proto-Earth.
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Kondratyev, B.P. On the deviation of the lunar center of mass to the East. Two possible mechanisms based on evolution of the orbit and rounding off the shape of the Moon. Astrophys Space Sci 363, 186 (2018). https://doi.org/10.1007/s10509-018-3405-z
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DOI: https://doi.org/10.1007/s10509-018-3405-z