PC1672 Advanced dynamics

3.10 Orbital energies

The fact the total energy of an object

$\begin{displaymath}\eta={1\over 2}v^2-{GM\over r}=-{GM\over 2a}\end{displaymath}$

is a constant can be used to find the speed of the object at any point in its orbit

$\begin{displaymath}v^2=GM\left({2\over r}-{1\over a}\right)\end{displaymath}$

At pericentre $r_1=a(1-\epsilon)$ the object has maximum speed

$\begin{displaymath}v_1^2={GM\over a}{1+\epsilon\over 1-\epsilon}\end{displaymath}$

while at apocentre $r_2=a(1+\epsilon)$ it has minimim speed

$\begin{displaymath}v_2^2={GM\over a}{1-\epsilon\over 1+\epsilon}\end{displaymath}$

The angular momentum of the object

$\begin{displaymath}\lambda=\vert{\bf r}\times{\bf v}\vert\end{displaymath}$

is also a constant. It is most easily calculated at pericentre or apocentre, when the velocity is perpendicular to the radius and so

$\begin{displaymath}\lambda=r_1v_1\end{displaymath}$

Textbook references

Mike Birse
17th May 2000