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Newton's theorem of revolving orbits was his first attempt to understand apsidal precession quantitatively. According to this theorem, the addition of a particular type of central force—the inverse-cube force—can produce a rotating orbit; the angular speed is multiplied by a factor k , whereas the radial motion is left unchanged.
At the top of the diagram, a satellite in a clockwise circular orbit (yellow spot) launches objects of negligible mass: (1 - blue) towards Earth, (2 - red) away from Earth, (3 - grey) in the direction of travel, and (4 - black) backwards in the direction of travel. Dashed ellipses are orbits relative to Earth.
Newton's style of demonstration in all his writings was rather brief in places; he appeared to assume that certain steps would be found self-evident or obvious. In 'De Motu', as in the first edition of the Principia , Newton did not specifically state a basis for extending the proofs to the converse.
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General relativity is a more exact theory than Newton's laws for calculating orbits, and it is sometimes necessary to use it for greater accuracy or in high-gravity situations (e.g. orbits near the Sun).
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An extension of Newton's theorem was discovered in 2000 by Mahomed and Vawda. [29] Assume that a particle is moving under an arbitrary central force F 1 (r), and let its radius r and azimuthal angle φ be denoted as r(t) and φ 1 (t) as a function of time t.
Circular orbits are possible when the effective force is zero: = = [+] =; i.e., when the two attractive forces—Newtonian gravity (first term) and the attraction unique to general relativity (third term)—are exactly balanced by the repulsive centrifugal force (second term).