Binet equation

The Binet equation, derived by Jacques Philippe Marie Binet, provides the form of a central force given the shape of the orbital motion in plane polar coordinates. The equation can also be used to derive the shape of the orbit for a given force law, but this usually involves the solution to a second order nonlinear ordinary differential equation. A unique solution is impossible in the case of circular motion about the center of force.
Equation.
The shape of an orbit is often conveniently described in terms of relative distance formula_1 as a function of angle formula_2 . For the Binet equation, the orbital shape is instead more concisely described by the reciprocal formula_3 as a function of formula_2. Define the specific angular momentum as formula_5 where formula_6 is the angular momentum and formula_7 is the mass. The Binet equation, derived in the next section, gives the force in terms of the function formula_8:
Derivation.
Newton's Second Law for a purely central force is
The conservation of angular momentum requires that
Derivatives of formula_1 with respect to time may be rewritten as derivatives of formula_3 with respect to angle:
Combining all of the above, we arrive at
Examples.
Kepler problem.
Classical.
The traditional Kepler problem of calculating the orbit of an inverse square law may be read off from the Binet equation as the solution to the differential equation
If the angle formula_2 is measured from the periapsis, then the general solution for the orbit expressed in (reciprocal) polar coordinates is
The above polar equation describes conic sections, with formula_20 the semi-latus rectum (equal to formula_21) and formula_22 the orbital eccentricity.
Relativistic.
The relativistic equation derived for Schwarzschild coordinates is
where formula_24 is the speed of light and formula_25 is the Schwarzschild radius. And for Reissner–Nordström metric we will obtain
where formula_27 is the electric charge and formula_28 is the vacuum permittivity.
Inverse Kepler problem.
Consider the inverse Kepler problem. What kind of force law produces a noncircular elliptical orbit (or more generally a noncircular conic section) around a focus of the ellipse?
Differentiating twice the above polar equation for an ellipse gives
The force law is therefore
which is the anticipated inverse square law. Matching the orbital formula_31 to physical values like formula_32 or formula_33 reproduces Newton's law of universal gravitation or Coulomb's law, respectively.
The effective force for Schwarzschild coordinates is
where the second term is an inverse-quartic force corresponding to quadrupole effects such as the angular shift of periapsis (It can be also obtained via retarded potentials).
In the parameterized post-Newtonian formalism we will obtain
where formula_36 for the general relativity and formula_37 in the classical case.
Cotes spirals.
An inverse cube force law has the form
The shapes of the orbits of an inverse cube law are known as Cotes spirals. The Binet equation shows that the orbits must be solutions to the equation
The differential equation has three kinds of solutions, in analogy to the different conic sections of the Kepler problem. When formula_40, the solution is the epispiral, including the pathological case of a straight line when formula_41. When formula_42, the solution is the hyperbolic spiral. When formula_43 the solution is Poinsot's spiral.
Off-axis circular motion.
Although the Binet equation fails to give a unique force law for circular motion about the center of force, the equation can provide a force law when the circle's center and the center of force do not coincide. Consider for example a circular orbit that passes directly through the center of force. A (reciprocal) polar equation for such a circular orbit of diameter formula_44 is
Differentiating formula_46 twice and making use of the Pythagorean identity gives
The force law is thus
Note that solving the general inverse problem, i.e. constructing the orbits of an attractive formula_49 force law, is a considerably more difficult problem because it is equivalent to solving
which is a second order nonlinear differential equation.