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.
Contents
Equation
The shape of an orbit is often conveniently described in terms of relative distance
Derivation
Newton's Second Law for a purely central force is
The conservation of angular momentum requires that
Derivatives of
Combine all of the above and we have
Kepler problem
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
The above polar equation describes conic sections, with
The relativistic equation derived for Schwarzschild coordinates is
where
where
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
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
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
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
Differentiating
The force law is thus
Note that solving the general inverse problem, i.e. constructing the orbits of an attractive
which is a second order nonlinear differential equation.