[NOTES/CM-05009] Keplar Orbit Parameters

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For an attractive \(-k/r\) potential, the orbit is an ellipse. The equation of the orbit is given by \begin{equation}\Label{EQ101} \frac{\ell_0}{r} = \frac{\mu k}{L^{2}} 1 + \epsilon \cos(\phi - \phi_{0}) \end{equation} where \(\ell_0=\frac{L^{2}}{\mu k}\), and \(L\) is angular momentum, $\epsilon$, and $ \phi_{0}$ are constants of integration.

We shall now relate maximum and minimum values of \(r\), \(r_1\), and (r_2,\), with the eccentricity \(\epsilon\) and energy \(E\). If $ r_{1}$ and $ r_{2}$ are the minimum and maximum values of \(r\), then from the equation of the orbit,\eqref{EQ101}, we have \begin{equation}\Label{EQ20} \frac{1}{r_{1,2}} = \frac{\mu k}{L^{2}} ( 1 \pm \epsilon) \end{equation}

We relate these values \(r_{1,2}\) in terms to \(E\). At the maximum of \(r\), and also at minimum, $ \dot{r} = 0$. The energy conservation equation, \begin{equation} E = \frac{1}{2} \mu \dot{r}^{2} + \frac{L^{2}}{2 \mu r^{2}} + V(r), \end{equation} with \(\dot{r}=0\),becomes \begin{equation} E = \frac{L^{2}}{2 \mu r^{2}} - \frac{k}{r} \qquad \qquad \ \ \ \ \ (\because \dot{r} = 0), \end{equation} or \begin{equation} \frac{1}{r^{2}} - \frac{2 \mu k}{L^{2} r} - \frac{2 \mu E}{L^{2}} = 0. \end{equation} The values $ r_{1}$ and $r_{2}$ are roots of this equation, hence \begin{eqnarray} \frac{1}{r_{1}} + \frac{1}{r_{2}} &=& \frac{2 \mu k}{L^{2}}\\ \frac{1}{r_{1}} \frac{1}{r_{2}} &=& \frac{- 2 \mu E}{L^{2}}\Label{EQ25} \end{eqnarray} \eqref{EQ20} and \eqref{EQ25} give us the desired expressions for $\epsilon$ in terms of energy \begin{equation} \frac{\mu^{2} k^{2}}{L^{4}} ( 1 - \epsilon^{2}) = \frac{-2 \mu E}{L^{2}} \end{equation} or \begin{eqnarray} \epsilon^{2} - 1 &=&\frac{2 E L^{2}}{\mu k^{2}}\\ \epsilon &=& \sqrt{1+\frac{2 E L^{2}}{\mu k^{2}}} \end{eqnarray} Therefore, the final form of the equation of the orbit is \begin{equation} \frac{1}{r} = \frac{\mu k }{L^{2}} ( 1 + \epsilon \cos(\phi - \phi_{0})) \end{equation} $\phi_{0}$ is the value of $\phi$ at the turning point $r = r_{min}$. This equation represents a conic section with eccentricity = $\epsilon$. Types of orbits traversed, corresponding to different values of \(\epsilon\), are summarised below