Basic question about orbital speed 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?
 A: 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.
A: I will try to explain it mathematically. Our orbital velocity is given by formula:
$$v_{\text{orbital}}=\sqrt{\frac{GM_{\text{star}}}{R_{\text{orbit}}}}$$
It’s obvious that numerator terms are not going to change. So, if the orbital velocity changes, the radius of the orbit changes. Since they are inversely proportional, if one increases, the other decreases and vice versa. So you get the result you are looking for.
In an elliptical orbit itself, the velocity changes as the celestial body revolves around the star (present at the ellipse's focus). Here, also:
$$L=mv_{\text{orbital}}a=\text{constant}$$
And you can see that perpendicular distance ($a$) and orbital velocity are inversely proportional.
Or think another way. We are stuck to our planet's gravitational field. But if we increase our velocity, to escape velocity, we can leave the planet altogether.
Calculating the new radius by altering velocity can be easily found by conservation of energy.
A: This is the well-known satellite orbit paradox. The key point is HOW you thrust in a cicular orbit. When you do a so-called impulsive thrust, i.e a thrust for a short time (small relative to the orbital period), you do it at that given position and the thrust energy goes only into the kinetic energy, i.e. you locally increase speed, which brings you from a circular orbit into an elliptic orbit. On this orbit the speed lowers the further you get out owing to the gravitational potential.
On the other hand, if you do a continuous small thrust along-track (this is where the paradox comes in), then you actually do NOT increase orbital speed. The thrust energy then goes into the orbit, which is determined by its orbital radius: The bigger the radius the bigger the orbital energy, because you work against the gravitational potential. In summary, by the type of thrusting you determine how the thrust energy is transfered into orbital energy.
