Claudius Ptolemy (100 AD - 170 AD)

Copernicus (1473-1543)

Kepler (1571-1630)

- A planet orbits the sun in an ellipse. The Sun is at one focus of that ellipse.
- A line connecting a planet to the Sun sweeps out equal areas in equal times
- The square of a planet's orbital period is proportional to the cube of the average distance between the planet and the sun: $P^2 \propto a^3$.

Mars' orbit [$e = 0.0934$] looks a lot like a circle.

The eccentricity $\epsilon = 0$, and the radius $r$ is constant.

Both foci are located at the center.

These don't really exist in nature but are just mathematical possibilities

Are bound and closed orbits where the eccentricity is $0 \lt \epsilon \lt 1$

Have the gravitational center at the principle focus

$r_p = a(1 - \epsilon)$ and $r_a = a(1+ \epsilon)$

Shape is determined by any of the following pairs:

$E$ | $l$ |

$\epsilon$ | $r_p$ |

$a$ | $b$ |

$\epsilon = 1$

Like circules, these don't actually occur but are just mathematical possibilities.

$\epsilon > 1$

The only central-force potentials $U(r)$ for which all bounded orbits are closed are:

- Gravity: $U \propto \frac{1}{r}$
- Springs: $U \propto r^2$

The semi-major axis of the transfer orbit: \begin{equation} a_\textrm{transfer} = \frac{r_e+r_m}{2} \end{equation}