Stephen A. Lawrence wrote:
The secret is that the planet slowed down, just a little. You stole energy from the planet. This becomes apparent when you consider conservation of momentum: The spaceship's momentum vector is now pointing in the opposite direction, so the planet must have picked up the difference.
When considering anything involving orbits and spaceships, it's worth keeping in mind that in classical Newtonian mechanics, gravity involves action at a distance, and the rate of propagation of the gravitational field of a moving body is assumed to be infinite. When dealing with things like slow spaceships on close approaches to planets, this approximation is adequate.
When you start dealing with very fast moving bodies or with very long distances the propagation delay of the gravitational field becomes important and it's necessary to use a more accurate model. And as soon as you assume a finite propagation rate for the G-field, you also either need to ascribe momentum and energy to the field itself, or you need to find some other way to "make the books balance" (assuming you want to keep conservation of momentum and energy in the model).
Sorry -- I've wandered rather far afield here; my fingers got away from me.
