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If there were a satellite in an elliptical orbit, at what point in its orbit should a booster rocket be fired to kick it to its max radius? I was thinking whennit was closest to the plant/object orbited around for max velocity or the farthest since the potential energy would be the least, but im not sure

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To raise the apoapsis (maximum radius) as high as possible, thrust should be delivered at the periapsis (minimum radius). This takes advantage of the Oberth effect, whereby a given impulse results in a larger change in kinetic energy if it's delivered when the object has a higher speed. Because the satellite has its highest velocity at the periapsis, it is the most efficient point to deliver an impulse.

You can also consider the fact that an impulse delivered at apoapsis will raise the periapsis, but not the maximum radius (at least until the periapsis has been raised past the current apoapsis). Firing a rocket at apoapsis doesn't do anything to raise the apoapsis, until the orbit is circular - at that point, you're just raising the original apoapsis, so you're better off starting there, by firing the rocket at the periapsis.

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