Newton's Theorem of Revolving Orbits - Limit of Nearly Circular Orbits

Limit of Nearly Circular Orbits

In Proposition 45 of his Principia, Newton applies his theorem of revolving orbits to develop a method for finding the force laws that govern the motions of planets. Johannes Kepler had noted that the orbits of most planets and the Moon seemed to be ellipses, and the long axis of those ellipses can determined accurately from astronomical measurements. The long axis is defined as the line connecting the positions of minimum and maximum distances to the central point, i.e., the line connecting the two apses. For illustration, the long axis of the planet Mercury is defined as the line through its successive positions of perihelion and aphelion. Over time, the long axis of most orbiting bodies rotates gradually, generally no more than a few degrees per complete revolution, because of gravitational perturbations from other bodies, oblateness in the attracting body, general relativistic effects, and other effects. Newton's method uses this apsidal precession as a sensitive probe of the type of force being applied to the planets.

Newton's theorem describes only the effects of adding an inverse-cube central force. However, Newton extends his theorem to an arbitrary central forces F(r) by restricting his attention to orbits that are nearly circular, such as ellipses with low orbital eccentricity (ε ≤ 10%), which is true of seven of the eight planetary orbits in the solar system. Newton also applied his theorem to the planet Mercury, which has an eccentricity ε of roughly 21%, and suggested that it may pertain to Halley's comet, whose orbit has an eccentricity of roughly 97%.

A qualitative justification for this extrapolation of his method has been suggested by Valluri, Wilson and Harper. According to their argument, Newton considered the apsidal precession angle α (the angle between the vectors of successive minimum and maximum distance from the center) to be a smooth, continuous function of the orbital eccentricity ε. For the inverse-square force, α equals 180°; the vectors to the positions of minimum and maximum distances lie on the same line. If α is initially not 180° at low ε (quasi-circular orbits) then, in general, α will equal 180° only for isolated values of ε; a randomly chosen value of ε would be very unlikely to give α = 180°. Therefore, the observed slow rotation of the apsides of planetary orbits suggest that the force of gravity is an inverse-square law.