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.

Highlights

  • At the dawn of modern astronomy, Hevelius and Galileo established that the optical libration of the Moon in longitude leads to a small (50–80) seeming oscillations of the figure of our satellite in the East-West direction with a period in the anomalistic month

  • In the era of the differentiation of the Moon, it was this difference in tidal forces (15) that caused the displacement of the center of mass of the Moon toward the Earth

  • In the first variant of the evolution of the Moon’s orbit, the modern Earth’s observer, in accordance with Figure 3b, will see the Moon’s center of mass displaced to the left from the direction to the center of the figure. It is this location of the center of mass of the Moon relative to the center of its figure that we observe in our era

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Summary

Introduction

At the dawn of modern astronomy, Hevelius and Galileo established that the optical libration of the Moon in longitude leads to a small (50–80) seeming (for terrestrial observer) oscillations of the figure of our satellite in the East-West direction with a period in the anomalistic month. Due to the proximity of the Moon in our time, the movement of our satellite is studied with such high accuracy that even a small asymmetry of its internal structure must be taken into account This asymmetry is manifested in that the center of the Moon’s mass COM is offset relative to the geometric center of the lunar figure COF. Despite the seemingly geometric simplicity of the problem, the offset of the center of the Moon’s mass remains an unexplored problem in the lunar science The importance of this problem is that the Moon is close enough to the Earth and the accuracy of observations of its spin-orbital motion by the method LLR is so much high that for correct interpretation of these movements it is necessary to take into account many celestial mechanical disturbances, including the indicated internal asymmetry of the Moon’s body. We consider two geometric mechanisms that allow us to explain this important feature of the internal structure of the Moon and shed light on some of the currently controversial features of its evolution and origin (see [13–15])

Optical libration of the Moon for the observer from the second focus
À e cos E
À e2 sin χ
On the difference on tidal forces from the Earth in near and far lunar hemispheres
The first version: the evolution of the lunar orbit with increase in its eccentricity
The second version of the evolution: from more flattened to less flattened lunar orbit
Correction factor to mechanism of orbit evolution
Differential equation for evolution of the angle α
Findings
10. Discussion and conclusions
Full Text
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