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It's well known that the planets don't orbit the sun in perfect circles and the characteristics of the elliptical orbits which serve as better approximations to their motion have been calculated fairly accurately.

How accurately do elliptical orbits model their actual paths?

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You'll want to read up on "precession of Mercury's perihelion" and "gravitational perturbations" by other planets (particularly Jupiter). –  dmckee Jun 9 '12 at 15:37

1 Answer 1

dmckee's comment has basically answered your question: this is really just adding a bit more flesh to his bones.

If you look at a single orbit of any of the planets then it is almost exactly an ellipse i.e. it's so closely an ellipse that it's quite hard to measure any deviation. However as described in dmckee's second link, the presence of the other planets causes the ellipse to precess and change in eccentricity. The effect is biggest with Mercury (see dmckee's first link), so although any single orbit of mercury is almost exactly an ellipse, if you watch for a lot of orbits you'll find that over time the ellipse precesses.

An ellipse is the orbit you get in a central inverse square law field. The term "central" means the force always point to the centre of the system, and if all forces in the Solar System were central then the orbits would be perfect ellipses. In practice the force on, for example, the Earth is almost central because the Sun is by far the biggest object in the Solar System. However the Earth also feels the gravitational attraction of the other planets, and these add small but significant off centre contributions. It's these off centre contributions that cause the ellipses to precess and change eccentricity.

Neptune was discovered because of the perturbations of the orbit of Uranus. Have a look at http://www.scribd.com/doc/76294844/1/The-strange-motion-of-Uranus for an excellent description of the perturbation. Over one orbit (about 80 years) the position of Uranus moved from it's expected position by about an arc-second i.e. 1/3600 degrees or about 1/1800 of the width of the moon.

Although the perturbations of the planetary orbits are small the orbit of the Moon is so strongly perturbed by the Sun that the ancient Greeks were able to measure the deviations. This is because at the Moon the gravitational attraction of the Sun is comparable to the gravitational attraction of the Earth.

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I don't consider "go read this stuff" as an answer myself, and I couldn't actually tell you the scale of either of those corrections. This is much better because you have numbers in it. –  dmckee Jun 9 '12 at 16:03

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