Binet Equation

Binet Equation

The shape of an orbit is often described in terms of relative distance \(r\) as a function of \(\theta\) . By Binet Equation, we can solve for the force in terms of the function \(u(\theta)\) , where \(u = \frac{1}{r}, h = L/m\) is the special angular momentum.

\[ F=-mh^2u^2(\frac{d^2u}{d\theta^2}+u) \]

Derivation

Newton's Second law for purely central force is

\[ F(r)=m(\ddot{r}-r\dot{\theta}^2) \]

The conservation of momentum requires that

\[ r^2\dot{\theta}=h=\text{constant} \]

Rewrite the derivative of \(r\) with respect to time as derivatives of \(u=1/r\) with respect to angle:

\[ \frac{du}{d\theta}=\frac{d}{dt}\bigg(\frac1r\bigg)\frac{dt}{d\theta}=\frac{-\dot{r}}{r^2\dot{\theta}}=-\frac{\dot{r}}{h} \]

\[ \frac{d^2u}{d\theta^2}=\frac{d}{d\theta}\bigg(-\frac{\dot{r}}{h}\bigg)=-\frac{1}h\frac{d\dot{r}}{dt}\frac{dt}{d\theta}=-\frac{\ddot{r}}{h\dot{\theta}}=-\frac{\ddot{r}}{h^2u^2} \]

Thus,

\[ \dot{r}=-h\frac{du}{d\theta} \]

\[ \ddot{r}=-h^2u^2\frac{d^2u}{d\theta^2} \]

Replace it back into the original equation for our central force, we get the Binet Equation as desired.