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The semi-analytic planetary theory VSOP (French: ''Variations Séculaires des Orbites Planétaires'', abbreviated as ''VSOP'') was developed and is maintained (updating it with the results of the latest and most accurate measurements) by the scientists at the Bureau des Longitudes in Paris, France. The first version, VSOP82, computed only the orbital elements at any moment. An updated version, VSOP87, besides providing improved accuracy, computed the positions of the planets directly, as well as their orbital elements, at any moment. The ''secular variations of the planetary orbits'' is a concept describing long-term changes (secular variation) in the orbits of the planets Mercury to Neptune. If one ignores the gravitational attraction between the planets and only models the attraction between the Sun and the planets, then with some further idealisations, the resulting orbits would be Keplerian ellipses. In this idealised model the shape and orientation of these ellipses would be constant in time. In reality, while the planets are at all times roughly in Keplerian orbits, the shape and orientation of these ellipses does change slowly over time. Over the centuries increasingly complex models have been made of the deviations from simple Keplerian orbits. In addition to the models, efficient and accurate numerical approximation methods have also been developed. At present the difference between computational predictions and observations is sufficiently small that the observations do not support the hypothesis that the models are missing some fundamental physics. Such hypothetical deviations are often referred to as post-Keplerian effects. ==History== Predicting the position of the planets in the sky was already performed in ancient times. Careful observations and geometrical calculations produced a model of the motion of the solar system known as the Ptolemaic system, which was based on an Earth-centered system. The parameters of this theory were improved during the Middle Ages by Indian and Islamic astronomers. The work of Tycho Brahe, Kepler, and Isaac Newton in early modern Europe laid a foundation for a modern heliocentric system. Future planetary positions continued to be predicted by extrapolating past observed positions as late as the 1740 tables of Jacques Cassini. The problem is that, for example, the Earth is not only gravitationally attracted by the Sun, which would result in a stable and easily predicted elliptical orbit, but also in varying degrees by the Moon, the other planets and any other object in the solar system. These forces cause perturbations to the orbit, which change over time and which cannot be exactly calculated. They can be approximated, but to do that in some manageable way requires advanced mathematics or very powerful computers. It is customary to develop them into periodic series which are a function of time, e.g. ''a''+''bt''+''ct''2+...×cos(''p''+''qt''+''rt''2+...) and so forth one for each planetary interaction. The factor ''a'' in the preceding formula is the main amplitude, the factor ''q'' the main period, which is directly related to an harmonic of the driving force, that is a planetary position. For example: ''q''= 3×(length of Mars) + 2×(length of Jupiter). (The term 'length' in this context refers to the ecliptic longitude, that is the angle over which the planet has progressed in its orbit, so ''q'' is an angle over time too. The time needed for the length to increase over 360° is equal to the revolution period.) It was Joseph Louis Lagrange in 1781, who carried out the first serious calculations, approximating the solution using a linearization method. Others followed, but it was not until 1897 that George William Hill expanded on the theories by taking second order terms into account. Third order terms had to wait until the 1970s when computers became available and the vast amounts of calculations to be performed in developing a theory finally became manageable. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「VSOP (planets)」の詳細全文を読む スポンサード リンク
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