### 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.

1. Planets move in elliptical orbits with the sun at one focus of the ellipse.
2. Equal areas are swept out in equal times (angular momentum conservation).
3. T2 ∝ a3, 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.