%% ekn-cs.tex - cheat sheet for planet orbits
%% Author: Markus Rost <https://www.math.uni-bielefeld.de/~rost/>
%% Created: Aug 26 2026

%% Cf. Appendix 1. Conic sections in: Milnor, John.  On the geometry
%% of the Kepler problem.  Amer. Math. Monthly 90 (1983), no. 6,
%% 353-365.  MR 707149, Zbl 0518.70008.

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\centerline{\bf CHEAT SHEET FOR PLANET ORBITS}

\bigbreak
\bigbreak

\noindent %
Given real numbers $k>0$ and $P$, $Q$ with
\begin{displaymath}
  L=PQ\neq0 ,\qquad E=\frac12Q^2-\frac{k}{P}<0
\end{displaymath}
the solution to the Newton differential equation
\begin{displaymath}
  \frac{d}{d\tau}\frac{dz}{d\tau}=-\frac{k}{|z|^2}\frac{z}{|z|}
  ,\qquad z(0)=P ,\qquad \frac{dz}{d\tau}(0)=iQ
\end{displaymath}
is
\begin{displaymath}
  z(s) = c+a\cos s + ib\sin s ,\qquad \tau(s)
  =\sqrt{\frac{a}{k}}(as+c\sin s)
\end{displaymath}
where
\begin{displaymath}
  a=-\frac{k}{2E} ,\qquad b=\frac{L}{\sqrt{-2E}} ,\qquad
  c=P+\frac{k}{2E}
\end{displaymath}
$P$ is the longest or shortest distance to the sun, resulting in
$c\geq0$ resp.~$c\leq0$.  One has
\begin{displaymath}
  |z(s)|=a+c\cos s
\end{displaymath}
and the polar ellipse equation of the orbit is
\begin{displaymath}
  |z|=\frac{L^2/k}{1+(1+2PE/k)\cos\theta}=\frac{b^2}{a-c\cos\theta}
  \qquad\quad (z=|z|e^{i\theta})
\end{displaymath}
The eccentricity and the semi-latus rectum of the ellipse are
\begin{displaymath}
  \varepsilon=|1+2PE/k|=|c|/a ,\qquad \Lambda=L^2/k=b^2/a
\end{displaymath}
The constants of motion (angular momentum and energy) are
\begin{displaymath}
  L=z\times\frac{dz}{d\tau}=|z|^2\frac{d\theta}{d\tau} ,\qquad
  E=\frac12\left|\frac{dz}{d\tau}\right|^2-\frac{k}{|z|}
\end{displaymath}
and the period time is
\begin{displaymath}
  T=\frac{2\pi k}{(-2E)^{3/2}}=2\pi\sqrt{\frac{a^3}{k}}
\end{displaymath}

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\begin{displaymath}
  \begin{tikzpicture} [scale=1/2]
    \draw (0,0)--(0,4) (-3,0)--(5,0) ;
    \node[right] at (0,4/2) {$b$} ;
    \node[above] at (-3/2,0) {$c$} ;
    \node[above] at (5/2,0) {$a$} ;
    \draw[thick] (0,0) ellipse (5 and 4) ;
    \fill[red] (-3,0) circle[radius=5pt] ;
    \fill[blue] (5,0) circle[radius=3pt] ;
  \end{tikzpicture}
\end{displaymath}

\end{document}

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%% Created:
%% 2026-08-26
%% Latest change:
%% 2026-08-31