Gravity and Orbits

Basic Orbits

Claudius Ptolemy

Claudius Ptolemy (100 AD - 170 AD)

Why circles?

Copernicus - woodcut attributed to Christoph Murer, from Nicolas Reusner's Icones (1587)

Copernicus (1473-1543)

The Copernican System consisted of circular orbits centered around the sun.

Tycho Brahe

tycho brahe

Danish astronomer Tycho Brahe [1546-1601]. Made many very good measurements of the stars and planets.

The compromise.

Johannus Kepler

German mathematician, astronomer, and astrologer. [1571-1630] Kepler was Tycho's assistant and was believed in the Copernican system.

Kepler (1571-1630)

The Platonic Solid Model

Kepler's 3 laws of orbiting bodies

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

1st Law: Ellipses

The orbit of Mars compared to a circle.

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

2nd Law: Equal Areas in Equal Times

The blue areas in these figures will be the same if $t_2 - t_1$ is the same.

2nd Law: Proof

An orbiting object moves through angle $\Delta \theta$ in a given time: $\Delta t$

Characteristics of Different Orbits

Circular:

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

Elliptical

The basic parts of an elliptical orbit.

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$

Parabolic

$\epsilon = 1$

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

Hyperbolic

$\epsilon > 1$

Bertrand's Theorem

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

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

Hohmann Transfer Orbit

A typical Homann Transfer Orbit

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

Special Cases

2 -> 3 bodies

Lagrange Points

The basic FBD and L1 point

All 5 Lagrange points

Find L1

Find L1 →