Binet equation

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

$${\displaystyle -ku^{2}=-mh^{2}u^{2}\left({\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u\right)}$$

$${\displaystyle {\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u={\frac {k}{mh^{2}}}\equiv {\text{constant}}>0.}$$

If the angle ${\displaystyle \theta }$ is measured from the periapsis, then the general solution for the orbit expressed in (reciprocal) polar coordinates is

$${\displaystyle lu=1+\varepsilon \cos \theta .}$$

The above polar equation describes conic sections, with ${\displaystyle l}$ the semi-latus rectum (equal to ${\displaystyle h^{2}/\mu =h^{2}m/k}$) and ${\displaystyle \varepsilon }$ the orbital eccentricity.

The relativistic equation derived for Schwarzschild coordinates is[2]

$${\displaystyle {\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u={\frac {r_{s}c^{2}}{2h^{2}}}+{\frac {3r_{s}}{2}}u^{2}}$$

where ${\displaystyle c}$ is the speed of light and ${\displaystyle r_{s}}$ is the Schwarzschild radius. And for Reissner–Nordström metric we will obtain

$${\displaystyle {\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u={\frac {r_{s}c^{2}}{2h^{2}}}+{\frac {3r_{s}}{2}}u^{2}-{\frac {GQ^{2}}{4\pi \varepsilon _{0}c^{4}}}\left({\frac {c^{2}}{h^{2}}}u+2u^{3}\right)}$$

where ${\displaystyle Q}$ is the electric charge and ${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity.

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

$${\displaystyle l\,{\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}=-\varepsilon \cos \theta .}$$

The force law is therefore

$${\displaystyle F=-mh^{2}u^{2}\left({\frac {-\varepsilon \cos \theta }{l}}+{\frac {1+\varepsilon \cos \theta }{l}}\right)=-{\frac {mh^{2}u^{2}}{l}}=-{\frac {mh^{2}}{lr^{2}}},}$$

which is the anticipated inverse square law. Matching the orbital ${\displaystyle h^{2}/l=\mu }$ to physical values like ${\displaystyle GM}$ or ${\displaystyle k_{e}q_{1}q_{2}/m}$ reproduces Newton's law of universal gravitation or Coulomb's law, respectively.

The effective force for Schwarzschild coordinates is[3]

$${\displaystyle F=-GMmu^{2}\left(1+3\left({\frac {hu}{c}}\right)^{2}\right)=-{\frac {GMm}{r^{2}}}\left(1+3\left({\frac {h}{rc}}\right)^{2}\right).}$$

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[4]).

In the parameterized post-Newtonian formalism we will obtain

$${\displaystyle F=-{\frac {GMm}{r^{2}}}\left(1+(2+2\gamma -\beta )\left({\frac {h}{rc}}\right)^{2}\right).}$$

where ${\displaystyle \gamma =\beta =1}$ for the general relativity and ${\displaystyle \gamma =\beta =0}$ in the classical case.

An inverse cube force law has the form

$${\displaystyle F(r)=-{\frac {k}{r^{3}}}.}$$

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

$${\displaystyle {\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u={\frac {ku}{mh^{2}}}=Cu.}$$

The differential equation has three kinds of solutions, in analogy to the different conic sections of the Kepler problem. When ${\displaystyle C<1}$, the solution is the epispiral, including the pathological case of a straight line when ${\displaystyle C=0}$. When ${\displaystyle C=1}$, the solution is the hyperbolic spiral. When ${\displaystyle C>1}$ the solution is Poinsot's spiral.

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 ${\displaystyle D}$ is

$${\displaystyle D\,u(\theta )=\sec \theta .}$$

Differentiating ${\displaystyle u}$ twice and making use of the Pythagorean identity gives

$${\displaystyle D\,{\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}=\sec \theta \tan ^{2}\theta +\sec ^{3}\theta =\sec \theta (\sec ^{2}\theta -1)+\sec ^{3}\theta =2D^{3}u^{3}-D\,u.}$$

The force law is thus

$${\displaystyle F=-mh^{2}u^{2}\left(2D^{2}u^{3}-u+u\right)=-2mh^{2}D^{2}u^{5}=-{\frac {2mh^{2}D^{2}}{r^{5}}}.}$$

Note that solving the general inverse problem, i.e. constructing the orbits of an attractive ${\displaystyle 1/r^{5}}$ force law, is a considerably more difficult problem because it is equivalent to solving

$${\displaystyle {\frac {\mathrm {d} ^{2}u}{\mathrm {d} \theta ^{2}}}+u=Cu^{3}}$$

which is a second order nonlinear differential equation.