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Burns at perigee have best efficiency to make in-plane changes (raise or lower orbit) due to the Oberth effect.

Burns at apogee have best efficiency for out-of-plane maneuvers (change orbital inclination plane) due to lowest velocity of the spacecraft.

Highly elliptical orbits can do both at a few hours' notice.

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Burns at perigee can only raise or lower part of the orbit though, right? A burn at perigee lowers or raises the apogee. To lower or raise the perigee, you need to burn at apogee.

Correct.

Burns don't have to happen only at perigee/apogee so there's more flexibility in practice. And counter intuitive stuff like bi-elliptic transfers.

Ahhhh, bi-elliptic transfers!

It's funny, I remember first learning to use such arcane maneuvers in a game called Children of a Dead Earth, in order to get into a terribly low orbit of Neptune to strike a methane refinery... My poor brain was always used to simple Hohmann transfers in games like KSP!

Thanks, I've got only a vague grasp of this.

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