Simple mathematical demonstration solves one of the problems of lunar motion, the regression of lunar nodes, observed more than 2000 years ago. Although their motions draw similar traces (such as retrograde motion in Ptolemy’s and Copernicus’s models of the universe) we show that celestial bodies do not rotate around their common centre of mass, but, by unified law, one around the other. Due to the rigidity of principle, regardless of the calculation of rounded up values, the range of discrepancy between predicted and observed cycle of regression is in level of magnitude of only 3.4×10-5 (due to lunar trajectory perturbation, its perceived values also on a small-scale vary cyclically). Besides the constant π and Terrestrial measure of time, the only variables used in the solution of this dual orbiting system problem are radiuses and surface accelerations of observed bodies.
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The ancient Greeks observed that the positions of ascending and descending nodes at which the Moon passes through the fixed plane of the Earth’s orbit around the Sun, the ecliptic, decrease, i.e. orbit the Earth in the opposite direction to the Moon, in such a rate that the cycle of that regression amounts almost exactly 18.6 Earth’s years. In other words, if the Moon, during the spring or autumn equinox, when viewed from stationary point on Earth, ascends at a certain position on the east horizon, describes the curve of its path and descends at another particular point on the west horizon, it would take 18.6 years for this trajectory to be repeated. In past centuries, developing lunar theory, many famous mathematicians and astronomers have dealt with described problem (Newton, Clairaut, D’Alembert, Euler, Laplace, Damoiseau, Plana, Poisson, Hansen, De Pontécoulant, J. Herschel, Airy, Delaunay, G.W. Hill, E.W. Brown) indicating its inherent difficulty and the theoretical and practical importance. Presented geometry accurately predicts the described cycle. Relative to Earth, Moons trail draws torus. For artistic purposes, in this video, geometry is adapted to exactly mach 18 years cycle.
The background sound is a clip from the soundtrack “Symphonies of planets” recorded by NASA’s Voyager.