Newton's theorem of revolving orbits

From Wikipedia, the free encyclopedia

All three planets share the same radial motion, as shown by the cyan circle, but they 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), due to an additional attractive inverse-cube central force; it completes three orbits for every blue orbit. The red planet illustrates purely radial motion with no angular motion (k=0), due to an additional repulsive inverse-cube central force.
All three planets share the same radial motion, as shown by the cyan circle, but they 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), due to an additional attractive inverse-cube central force; it completes three orbits for every blue orbit. The red planet illustrates purely radial motion with no angular motion (k=0), due to an additional repulsive inverse-cube central force.

In classical mechanics, Newton's theorem of revolving orbits states that the addition of an inverse-cube central force can change the angular motion of a particle without changing its radial motion. The addition of such a central force does not affect the conservation of energy or the conservation of angular momentum. Isaac Newton and Alexis Clairaut used this theorem to investigate the motion of the Moon. The theorem can also be used to understand the anomalous precession of the orbit of the planet Mercury about the Sun, due to an effective inverse-cube force resulting from Einstein's theory of general relativity. However, the theorem is more general than the inverse-square force of gravity; by adding an inverse-cube central force to any central force, the angular speed can be accelerated or decelerated by an arbitrary factor k without affecting the radial motion.

Contents

[edit] Quantitative statement

Harmonics of the original orbit are produced if k is an integer. Here, k equals 2 (blue) and 3 (red), while the original elliptical trajectory is shown as a dotted black curve.
Harmonics of the original orbit are produced if k is an integer. Here, k equals 2 (blue) and 3 (red), while the original elliptical trajectory is shown as a dotted black curve.

Let there be a particle acted upon by the radial force F1(r) of a distance-dependent potential energy V1(r)

F_{1}(r) = -\frac{dV_{1}}{dr}

Such central forces produce planar orbits and have a constant angular momentum, sweeping out equal areas in equal times. Let the coordinates of the particle in this plane be denoted by polar coordinates (r,θ1), both of which are functions of time. Let there be a second orbit with an identical radial time dependence r(t), but whose angular speed is always k-fold faster, where k is any constant, so that their azimuthal angles are related by the equation θ2(t) = k θ1(t). In Propositions 42-44 of his Principia, Newton showed that such an orbit can be produced by adding an inverse-square potential energy to the first potential energy

V_{2}(r) = V_{1}(r) + \frac{L_{1}^{2}}{2mr^{2}} \left( 1 - k^{2} \right)

Expressed another way, such an orbit can be produced by adding an inverse-cube central force to the first radial force

F_{2}(r) = F_{1}(r) + \frac{L_{1}^{2}}{mr^{3}} \left( 1 - k^{2} \right)
Here, the green planet moves angularly one-third as fast as the blue planet (k=1/3), due to an additional repulsive inverse-cube central force; it completes one orbit for every three blue orbits. Thus, the green orbit is a third subharmonic of the blue orbit.
Here, the green planet moves angularly one-third as fast as the blue planet (k=1/3), due to an additional repulsive inverse-cube central force; it completes one orbit for every three blue orbits. Thus, the green orbit is a third subharmonic of the blue orbit.
Subharmonic orbits with k = 1/2 (blue), 1/3 (green) and 1/6 (red). The green path here is followed by the green planet in the animation above.
Subharmonic orbits with k = 1/2 (blue), 1/3 (green) and 1/6 (red). The green path here is followed by the green planet in the animation above.

Isaac Newton and Alexis Clairaut used this theorem in their separate treatments of the Moon's orbit, but it is more generally applicable, since the theorem holds regardless of the form of the original potential V1(r). If k is greater than one, the added inverse-cube force is attractive, whereas if k is less than one, it is repulsive. The ratio of angular speeds k may have any value, integer or non-integer; if k is an integer or inverse integer, the orbits are harmonics and subharmonics of the original orbits. If k is very close, but not equal, to one, the second orbit resembles the first, but revolves gradually about the center of force; this is known as orbital precession, which was central to the first experimental validation of general relativity.

[edit] Historical applications

Newton used this theorem to treat the motion of the Moon in the three-body problem of Sun-Earth-Moon. Clairaut revisited this problem later, and suggested that Newton’s inverse-square force law for gravitation be augmented with another radial force, most likely an inverse-cube force, to account for the rapid precession of the lunar perigee.[1] This was ultimately shown to be unnecessary to account for the Moon’s motion. However, exactly such an inverse-cube radial force results from the theory of general relativity, and accounts for the unexpectedly rapid precession of the planet Mercury.

[edit] Derivation

By assumption, the angular speeds are related by the equation

\omega_{2} = \frac{d\theta_{2}}{dt} = k \frac{d\theta_{1}}{dt} = k \omega_{1}

Since the two radii have the same behavior with time, r(t), the conserved angular momenta are related by the same factor

L_{2} = m r^{2} \omega_{2} = m r^{2} k \omega_{1} = k L_{1} \,\!

The equation of motion for a radius r of a particle of mass m moving in a central potential V(r) is given by Lagrange's equations

m\frac{d^{2}r}{dt^{2}} - mr \omega^{2} = m\frac{d^{2}r}{dt^{2}} - \frac{L^{2}}{mr^{3}} = F(r)

For circular orbits, the first term on the left-hand side is zero; the remaining equation states that the applied inwards force F(r) equals the centripetal force requirement mrω2, as expected for a circular orbit. Applying the general formula to the two orbits yields the equation

m\frac{d^{2}r}{dt^{2}} = F_{1}(r) + \frac{L_{1}^{2}}{mr^{3}} = F_{2}(r) + \frac{L_{2}^{2}}{mr^{3}} = F_{2}(r) + \frac{k^{2} L_{1}^{2}}{mr^{3}}

which can be re-arranged to the form

F_{2}(r) = F_{1}(r) + \frac{L_{1}^{2}}{mr^{3}} \left( 1 - k^{2} \right)

This equation relating the two radial forces can be understood qualitatively as follows. The difference in angular speeds (or equivalently, in angular momenta) causes a difference in the centripetal force requirement; to offset this, the radial force must be altered with an inverse-cube force.

Precession of an elliptical orbit when k is close to one. The orbit is not rotating as a whole, since that would only add to the orbital speed; instead, the orbital speed is multiplied by a constant factor k.
Precession of an elliptical orbit when k is close to one. The orbit is not rotating as a whole, since that would only add to the orbital speed; instead, the orbital speed is multiplied by a constant factor k.

The radial force equation can also be written as

- \frac{dV_{2}}{dr} = - \frac{dV_{1}}{dr} + \frac{L_{1}^{2}}{mr^{3}} \left( 1 - k^{2} \right)

A single integration yields the second potential energy

V_{2}(r) = V_{1}(r) + \frac{L_{1}^{2}}{2mr^{2}} \left( 1 - k^{2} \right)

[edit] Precession

The precession of orbits when k is close to one cannot be derived by re-expressing the equations of motion in a uniformly rotating reference frame. A reference frame rotating uniformly with angular speed Ω would lead to an addition to the angular speed, ω2 = ω1 + Ω. However, by this theorem, the angular speeds are related by multiplication ω2 = k ω1.

[edit] References

  1. ^ Clairaut, AC (1745). "Du Système du Monde dans les principes de la gravitation universelle". Histoire de l'Académie royale des sciences avec les mémoires de mathématique et de physique 1749: 329–364. 

[edit] Further reading

  • Whittaker, ET (1937). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies, 4th ed., New York: Dover Publications, p. 83. ISBN 978-0-521-35883-5. 
  • D’Eliseo, MM (2007). "The first-order orbital equation". American Journal of Physics 75: 352–355. doi:10.1119/1.2432126. 
  • Lynden-Bell, D; Lynden-Bell RM (1997). "On the Shapes of Newton’s Revolving Orbits". Notes and Records of the Royal Society of London 51: 195–198. doi:10.1098/rsnr.1997.0016.