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Assuming one wants to go from a 400x800 to a 1000x1000 orbit. The solution that require the least D/v is a Hohmann transfer, with one burn at the perigee and one burn at the new apogee (or vice versa).

Obviously no spacecraft performs an instant burn, and the real life solution is to start burning slightly before the optimal point.

However, I'm not sure which is the best heading to take for the burn:

  • Prograde (adjusting the heading while progressing into the orbit)

  • In the prograde direction of the ideal point (apogee/perigee), keeping a fixed heading for the whole burn.

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  • $\begingroup$ Too short to be an answer: Fire parallel to the velocity vector, and do so near periapsis. That said, this is not necessarily optimal. (Periapsis with respect to what??) $\endgroup$
    – David Hammen
    Sep 30, 2018 at 15:24

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Turn and burn is more efficient. You get the most energy change in the orbit by burning in the velocity direction.

The next question is what is the optimal time to start the burn ...

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  • $\begingroup$ This one is really complicated because your velocity and acceleration change over time... $\endgroup$
    – Antzi
    Oct 1, 2018 at 0:49

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