Orbital Motion – Basic Question About Orbital Speed

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I was reading a Sci-Fi book recently and had a weird thought:

I know that objects closer to a gravitational well need to move faster to stay in orbit and objects further away move slower. But if you want to increase your orbit/escape the gravitational well you have to speed up while if you want to lower your orbit you have to decrease your speed.

In my mind this seems like a paradox. I'm sure I'm just thinking about it the wrong way but I can't figure out how to solve this. Can someone explain it to me, please?

Best Answer

Assume a high circular orbit above a planet. If you want to drop to a lower orbit, you have to do a small retro "burn" (fire your rocket engine to provide thrust opposite the direction that you are traveling) to reduce your tangential velocity a bit. If you don't slow down too much, you will go into an elliptical orbit, gain kinetic energy as you drop lower in altitude due to a decrease in gravitational potential energy, and approach the lowest point in the new orbit at high speed. The speed at lowest approach (perigee) will be too high to remain at that distance from the planet, and you will eventually rise back up to the point where you fired the retro rocket. To prevent this, you have to do another retro burn at perigee to go into a circular orbit. Once this happens, the orbital speed will be higher than it was at a higher altitude, with that increase in speed coming from the decrease in gravitational potential energy minus the decrease in speed from the second retro burn.