# Optimum delta-v burn to change periapsis or apoapsis at an arbitrary point on an elliptical orbit?

This question is explicitly not about doing burns at the apoapsis or periapsis in order to raise/lower the other apsis. Of course those are going to be the most efficient maneuvers, and the burns should be prograde or retrograde to the orbit at those points. The question is also not about Hohmann transfers, or circular starting orbits.

But, given a point which is closer to midway between the apses, what is the optimum burn $\Delta V$ and direction to modify one of the apses? For a more concrete example, lets say that we are "roughly" halfway on the return from the apoasis to the periapsis and wish to raise or lower the periapsis. Also this problem could be a hyperbolic entry orbit and we're trying to raise or lower the periapsis. The change in the final apoapsis is only going to be constrained by the requirement that the $\Delta V$ of the burn is minimized.

I know of one solution to this problem which assumes that a good enough burn direction will be tangent to the surface we are orbiting (not tangent to the orbit) and then does a binary search to find the burn that produces the right periapsis. Is there a better closed-form solution to this problem, and one which finds the minimal-$\Delta V$ burn?

Burning prograde or retrograde can also raise or lower the periapsis, but when away from the apoapsis this direction of the burn is less efficient.

• this seems to be sort of similar to the lambert problem, but the absolute location of the periapsis is going to change in response to the burn? Commented May 15, 2016 at 22:13
• Both apses will change location and height. With an elongated orbit, a burn perpendicular to the orbit roughly "halfway between" will either make the orbit "slimmer" (lower periapsis, raise apoapsis) if done towards the center of the ellipse, or tend to circularize (lower apoapsis, raise periapsis) if done towards the "outside". Of course as the orbit trajectory changes, so does the "perpendicular" direction and so it should be performed somewhere between the initially perpendicular, and finally perpendicular direction - but where exactly - I can't tell.
– SF.
Commented May 15, 2016 at 23:05
• You'll first need to make some choices. If you want to maximize the periapsis change, then you'll need to accept changes in both the apoapsis and the orbit orientation ($\omega$). If you want to not rotate the orbit, you will need to accept a smaller change in the periapsis, and still a change in apoapsis. If you want to hold the apoapsis, you'll need to accept an orbit rotation. You can't change only one. Commented May 16, 2016 at 2:40
• If you're trying to develop your understanding rather than solve a specific problem, you could do worse than spending some time playing with maneuver nodes in Kerbal Space Program. Commented May 16, 2016 at 14:49
• @RussellBorogove I'm writing my own maneuver node planner, and the the solution I know of with the "good enough" solution that uses the burn tangent to the surface is the algorithm that MechJeb uses. Commented May 17, 2016 at 0:46

I am not aware of a closed form solution, and I'm pretty sure there isn't one. You need to optimize the direction for each $\Delta V$, and find the pair that gets you to where you want to be optimally.
Here is a simple example, where the direction was optimized for a fixed $\Delta V$ to maximize the periapsis increase: