| Eccentricity | a | b | Cartesian Form | |
|---|---|---|---|---|
| Ellipse | $0 \leq e<1$ | $a>0$ | Semi-minor axis: $b=a\sqrt{1-e^2}$ | $x^2/a^2+y^2/b^2=1$ |
| Parabola | $e=1$ | $a=\infty$ | $b=0$ | $y=x^2/4r_p$ |
| Hyperbola | $e>1$ | $a<0$ | Impact parameter: $b=-a\sqrt{e^2-1}$ | $x^2/a^2-y^2/b^2=1$ |
Any conic section can be represented by two numbers.
This can be any two of these: $l$, $r_p$ and $e$.
$a$ should work most of the time, but it isn’t defined for parabolas
$r_a=a(1+e)$, the aphelion distance, only works for ellipses
$b=|a|\sqrt{|1-e^2|}$ only works if you know what it represents (semi-minor axis or impact parameter)
For all orbital types:
Total energy:
$$ E=-\frac{GMm}{2a} $$
Polar equation:
$$ r = \frac{l}{1+e \cos \theta} $$
Where $\theta$ is the true anomaly ($\theta=0$ at perihelion)
$l$ is the semi latus rectum and $r_p$ is the perihelion distance
$$ l= \begin{dcases} a(1-e^2) & \textrm{for } e \neq 1\\ 2 r_p & \textrm{for } e=1\end{dcases} $$
$$ r_p= a(1-e) \textrm{ for } e \neq 1 $$
$$ e=\frac{c}{a} \textrm{ for } e \neq 1 $$