# 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 $r$ as a function of angle $\theta$ . For the Binet equation, the orbital shape is instead more concisely described by the reciprocal $u=1/r$ as a function of $\theta$. Define the specific angular momentum as $h=L/m$ where $L$ is the angular momentum and $m$ is the mass. The Binet equation[1] is

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

## Derivation

Newton's Second Law for a purely central force is

$F(r)=m(\ddot{r}-r\dot{\theta }^{2}).$

The conservation of angular momentum requires that

$r^{2}\dot{\theta }=h=\text{constant}.$

Derivatives of $r$ with respect to time may be rewritten as derivatives of $u$ with respect to angle.

\begin{align} &\frac{\mathrm{d}u}{\mathrm{d}\theta }=\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{r}\right)\frac{\mathrm{d}t}{\mathrm{d}\theta }=-\frac{{\dot{r}}}{r^{2}\dot{\theta }}=-\frac{{\dot{r}}}{h} \\ & \frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}=-\frac{1}{h}\frac{\mathrm{d}\dot{r}}{\mathrm{d}t}\frac{\mathrm{d}t}{\mathrm{d}\theta }=-\frac{{\ddot{r}}}{h\dot{\theta }}=-\frac{{\ddot{r}}}{h^{2}u^{2}} \\ \end{align}

Combine all of the above and we have

$F=m(\ddot{r}-r\dot{\theta }^2)=-m\left(h^{2}u^{2}\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}+h^{2}u^{3}\right)=-mh^{2}u^{2}\left(\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}+u\right)$

## Examples

### 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

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

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

$l u =1 + \varepsilon \cos\theta.$

The above polar equation describes conic sections, with $l$ the semi-latus rectum and $\varepsilon$ the orbital eccentricity.

The relativistic equation derived for Schwarzschild coordinates is[2]

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

where $c$ is the speed of light and $r_s$ is the Schwarzschild radius. And for Reissner–Nordström metric we will obtain

$\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}+u=\frac{r_s c^{2}}{2 h^{2}}+\frac{3 r_s}{2}u^{2}-\frac{G Q^{2}}{4 \pi \varepsilon_0 c^{4}}\left(\frac{c^{2}}{h^{2}}u+2u^3\right)$

where $Q$ is the electric charge and $\varepsilon_0$ 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

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

The force law is therefore

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

which is the anticipated inverse square law. Matching the orbital $h^2/l$ to physical values like $GM$ or $k_e q_1 q_2/m$ reproduces Newton's law of universal gravitation or Coulomb's law, respectively.

The effective force for Schwarzschild coordinates is[3]

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

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

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

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

### Cotes spirals

An inverse cube force law has the form

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

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

$\frac{\mathrm{d}^{2}u}{\mathrm{d}\theta ^{2}}+u=\frac{k u}{m h^2} = C u.$

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

$D \, u(\theta)= \sec \theta.$

Differentiating $u$ twice and making use of the Pythagorean identity gives

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

The force law is thus

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

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

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

which is a second order nonlinear differential equation.