Newton's theorem of revolving orbits

In classical mechanics, Newton's theorem of revolving orbits identifies the type of central force needed to multiply the angular speed of a particle by a factor k without affecting its radial motion (Figures 1 and 2).

Newton applied his theorem to understanding the overall rotation of orbits (apsidal precession, Figure 3) that is observed for the Moon and planets.

Isaac Newton derived this theorem in Propositions 43–45 of Book I of his Philosophiæ Naturalis Principia Mathematica, first published in 1687.

This theorem remained largely unknown and undeveloped for over three centuries, as noted by astrophysicist Subrahmanyan Chandrasekhar in his 1995 commentary on Newton's Principia.

Any orbit can be described with a sufficient number of judiciously chosen epicycles, since this approach corresponds to a modern Fourier transform.

[6] Roughly 350 years later, Claudius Ptolemaeus published his Almagest, in which he developed this system to match the best astronomical observations of his era.

To explain the epicycles, Ptolemy adopted the geocentric cosmology of Aristotle, according to which planets were confined to concentric rotating spheres.

The modern understanding of planetary motion arose from the combined efforts of astronomer Tycho Brahe and physicist Johannes Kepler in the 16th century.

Arguing from his laws of motion, Newton showed that the orbit of any particle acted upon by one such force is always a conic section, specifically an ellipse if it does not go to infinity.

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.

Newton applied this approximation to test models of the force causing the apsidal precession of the Moon's orbit.

Newton's theorem simplifies orbital problems in classical mechanics by eliminating inverse-cube forces from consideration.

If k2 is greater than one, F2 − F1 is a negative number; thus, the added inverse-cube force is attractive, as observed in the green planet of Figures 1–4 and 9.

The path of the particle ignores the time dependencies of the radial and angular motions, such as r(t) and θ1(t); rather, it relates the radius and angle variables to one another.

For this purpose, the angle variable is unrestricted and can increase indefinitely as the particle revolves around the central point multiple times.

If such an inverse-cube force is introduced, Newton's theorem says that the corresponding solutions have a shape called Cotes's spirals[clarification needed].

On the other hand, when k is greater than one, the range of allowed angles increases, corresponding to an attractive force (green, cyan and blue curves on left in Figure 7); the orbit of the particle can even wrap around the center several times.

Thus, Poinsot spiral motion only occurs for repulsive inverse-cube central forces, and applies in the case that L is not too large for the given μ.

Such curves result when the strength μ of the repulsive force exactly balances the angular momentum-mass term Two types of central forces—those that increase linearly with distance, F = Cr, such as Hooke's law, and inverse-square forces, F = C/r2, such as Newton's law of universal gravitation and Coulomb's law—have a very unusual property.

In other words, the path of a bound particle is always closed and its motion repeats indefinitely, no matter what its initial position or velocity.

As shown by Bertrand's theorem, this property is not true for other types of forces; in general, a particle will not return to its starting point with the same velocity.

This method for producing closed orbits does not violate Bertrand's theorem, because the added inverse-cubic force depends on the initial velocity of the particle.

[2] 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.

Therefore, the observed slow rotation of the apsides of planetary orbits suggest that the force of gravity is an inverse-square law.

If an elliptical orbit is stationary, the particle rotates about the center of force by 180° as it moves from one end of the long axis to the other (the two apses).

[38] The currently accepted explanation for this precession involves the theory of general relativity, which (to first approximation) adds an inverse-quartic force, i.e., one that varies as the inverse fourth power of distance.

The second term, so Newton reasoned, might represent the average perturbing force of the Sun's gravity of the Earth-Moon system.

For every revolution, the long axis would rotate 1.5°, roughly half of the observed 3.0°[34] Isaac Newton first published his theorem in 1687, as Propositions 43–45 of Book I of his Philosophiæ Naturalis Principia Mathematica.

However, as astrophysicist Subrahmanyan Chandrasekhar noted in his 1995 commentary on Newton's Principia, the theorem remained largely unknown and undeveloped for over three centuries.

To find the magnitude of F2(r) from the original central force F1(r), Newton calculated their difference F2(r) − F1(r) using geometry and the definition of centripetal acceleration.

Figure 1: An attractive force F ( r ) causes the blue planet to move on the cyan circle. The green planet moves three times faster and thus requires a stronger centripetal force , which is supplied by adding an attractive inverse-cube force. The red planet is stationary; the force F ( r ) is balanced by a repulsive inverse-cube force. A GIF version of this animation is found here .
Figure 2: The radius r of the green and blue planets are the same, but their angular speed differs by a factor k . Examples of such orbits are shown in Figures 1 and 3–5.
Retrograde motion of Mars as viewed from the Earth.
Figure 3: Planets revolving the Sun follow elliptical (oval) orbits that rotate gradually over time ( apsidal precession ). The eccentricity of this ellipse is exaggerated for visualization. Most orbits in the Solar System have a much smaller eccentricity, making them nearly circular. A GIF version of this animation is found here .
Figure 4: All three planets share the same radial motion (cyan circle) but move at different angular speeds. The blue planet feels only an inverse-square force and moves on an ellipse ( k = 1). The green planet moves angularly three times as fast as the blue planet ( k = 3); it completes three orbits for every orbit of the blue planet. The red planet illustrates purely radial motion with no angular motion ( k = 0). The paths followed by the green and blue planets are shown in Figure 9 . A GIF version of this animation is found here .
Figure 5: The green planet moves angularly one-third as fast as the blue planet ( k = 1/3); it completes one orbit for every three blue orbits. The paths followed by the green and blue planets are shown in Figure 10 . A GIF version of this animation is found here .
Figure 6: For the blue particle moving in a straight line, the radius r from a given center varies with angle according to the equation b = r cos(θ − θ 0 ) , where b is the distance of closest approach ( impact parameter , shown in red).
Figure 7: Epispirals corresponding to k equal to 2/3 (red), 1.0 (black), 1.5 (green), 3.0 (cyan) and 6.0 (blue). When k is less than one, the inverse-cube force is repulsive, whereas when k is greater than one, the force is attractive.
Figure 8: Poinsot spirals ( cosh spirals) corresponding to λ equal to 1.0 (green), 3.0 (cyan) and 6.0 (blue).
Figure 9: Harmonic orbits with k = 1 (blue), 2 (magenta) and 3 (green). An animation of the blue and green orbits is shown in Figure 4.
Figure 10: Subharmonic orbits with k = 1 (blue), 1/2 (magenta) and 1/3 (green). An animation of the blue and green orbits is shown in Figure 5.
The Moon's motion is more complex than those of the planets, mainly due to the competing gravitational pulls of the Earth and the Sun.
Diagram illustrating Newton's derivation. The blue planet follows the dashed elliptical orbit, whereas the green planet follows the solid elliptical orbit; the two ellipses share a common focus at the point C . The angles UCP and VCQ both equal θ 1 , whereas the black arc represents the angle UCQ, which equals θ 2 = k θ 1 . The solid ellipse has rotated relative to the dashed ellipse by the angle UCV, which equals ( k −1) θ 1 . All three planets (red, blue and green) are at the same distance r from the center of force C .