Discovery
The variation was discovered by Tycho Brahe, who noticed that, starting from a lunar eclipse in December 1590, at the times of syzygy (new or full moon), the apparent velocity of motion of the Moon (along its orbit as seen against the background of stars) was faster than expected. On the other hand, at the times of first and last quarter, its velocity was correspondingly slower than expected. (Those expectations were based on the lunar tables widely used up to Tycho's time. They took some account of the two largest irregularities in the Moon's motion, i.e. those now known as the equation of the center and the evection, see also Lunar theory - History.)Variation
The main visible effect (in longitude) of the variation of the Moon is that during the course of every month, at the octants of the Moon's phase that follow the syzygies (i.e. halfway between the new or the full moon and the next-following quarter), the Moon is about two thirds of a degree farther ahead than would be expected on the basis of its mean motion (as modified by the equation of the centre and by the evection). But at the octants that precede the syzygies, it is about two thirds of a degree behind. At the syzygies and quarters themselves, the main effect is on the Moon's velocity rather than its position. image:Variation (astronomy).gif, frame, Variational orbit: nearly an ellipse, with the Earth at the center. The diagram illustrates the perturbing effect of the Sun on the Moon's orbit, using some simplifying approximations, e.g. that in the absence of the Sun, the Moon's orbit would be circular with the Earth at its center In 1687 Newton published, in the 'PhilosophiƦ Naturalis Principia Mathematica, Principia', his first steps in the gravitational analysis of the motion of Lunar theory#Newton, three mutually-attracting bodies. This included a proof that the Variation is one of the results of the perturbation of the motion of the Moon caused by the action of the Sun, and that one of the effects is to distort the Moon's orbit in a practically elliptical manner (ignoring at this point the eccentricity of the Moon's orbit), with the centre of the ellipse occupied by the Earth, and the major axis perpendicular to a line drawn between the Earth and Sun. The Variation has a period of half aElliptical distortion
Thus the (central) elliptical distortion of the Moon's orbit caused by the variation should not be confused with an undisturbed eccentric elliptical motion of an orbiting body. The variational effects due to the Sun would still occur even if the hypothetical undisturbed motion of the Moon had an eccentricity of zero (i.e. even if the orbit would be circular in the absence of the Sun). Newton expressed an approximate recognition that the real orbit of the Moon is not exactly an eccentric Keplerian ellipse, nor exactly a central ellipse due to the variation, but "an oval of another kind".D T Whiteside (ed.) (1973), ''The Mathematical papers of Isaac Newton, Volume VI: 1684-1691'', Cambridge University PressSee also
* EvectionReferences
{{ReflistBibliography
* Brown, E.W. ''An Introductory Treatise on the Lunar Theory.'' Cambridge University Press, 1896 (republished by Dover, 1960). * Celestial mechanics *