### Orbits and Energy

For a planet of mass m in orbit around a sum of mass M >> m, the
properties of the orbit are given by **Kepler's Laws**.

- Planets move in elliptical orbits with
the sun at one focus of the ellipse.
- Equal areas are swept out in equal times
(angular momentum conservation).
- T
^{2} ∝ a^{3}, where
T is the orbital period and a is the semi-major axis length.

Let's examine the orbits corresponding to several initial velocities:

Case 1: Circular orbit. We choose the
radius so that this initial velocity v = 1.

Case 2: v < 1. The mass follows an elliptical orbit. The starting point is the
aphelion, the point furthest from the Sun.

Case 3: v = 0. The object gets sucked in
to the Sun along a straight line.

Case 4: √ 2 > v > 1 but total energy is still
negative. The orbit is again elliptical, but
this time the starting point is the perihelion - the point closest to the
Sun.

Case 5: v = √ 2. This is the escape
speed where the total energy equals zero. The orbit is parabolic and never returns.

Case 6: v > √ 2, so the total energy
is positive. The orbit is hyperbolic -
straighter than the parabola.