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+ct2+...×cos(p+qt+rt2+...) 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.
Read more about this topic: VSOP (planets)
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