Orbits and Energy

Again we have two masses, m and M, with m << M. The smaller mass will be placed at a particular distance from the larger one and given an initial velocity directed perpendicular to the line joining the masses. We'll examine a few different cases, giving the mass initial velocities of various speeds and seeing what kind of orbit we get in each case.

Case 1: A circular orbit. Let's say this happens to require an initial velocity of 1 unit.

Case 2: v < 1.0. With even less kinetic energy, the mass follows an elliptical path. The starting point is the aphelion, the point furthest from the Sun.

Case 3: v = 0.0. The object simply gets sucked in to the large mass.

Case 4: v > 1.0 but the total energy is still negative. We still have a bound system. The orbit is elliptical again, but this time the starting point is the perihelion - the point closest to the Sun.

Case 5: v is larger by a factor of the square root of two than the speed needed to go in a circle. This is actually the escape speed - the orbit is parabolic, and the object never comes back.

Case 6: v is larger than the escape speed, so the total energy is positive. The orbit is hyperbolic - note that it's much straighter than the parabolic curve.