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Say you are in a simple circular orbit, $r = 1$ and want to raise your apoapsis to some higher altitude, say an orbit $r_P = 1, r_A = 2$

A high thrust spacecraft would to this in one impulse, at $\Delta v = \frac{2}{\sqrt{3}} - 1 \approx 0.15$.

But how would a low thrust spacecraft quickly achieve this?

One could of course "emulate" an impulse by only using the engine a short while for each revolution at periapsis. But this does not sound very time-efficient, as the engine isn't in use most of the time. To illustrate this, a circular-circular transfer from $r=1$ to $r=2$ can be achieved by a spiralling low-thrust craft much faster than by "emulating" a Hohmann transfer.

Is there some similar strategy to "spiralling" when the target orbit is not circular?

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Different methods of low thrust optimization are discussed in Chapter 2 of "Low Thrust Trajectory Optimization in Cislunar and Translunar space", a dissertation published in 2018 by Dr. Parrish.

In short, there is no analytical solution. There are several methods for low thrust optimization depending on the problem you'd like to solve. For orbits with lots of revolutions, it's usually recommended to do a Sims-Flanagan spiral method. Otherwise, a multiple shooting method is recommended, either direct or indirect (e.g. using costates).

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    $\begingroup$ "What does a low thrust apoapsis raising look like?" and "But how would a low thrust spacecraft efficiently achieve this?" ask for an example. So far this only provides information that someone intending to answer the question might find helpful. Are there any examples there that you can at least briefly summarize? $\endgroup$ – uhoh Oct 4 at 6:34
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    $\begingroup$ Are there good heuristic for the initial conditions of the costates, namely I played around with it and the reduced optimization problem does not look convex, so depending on the initial conditions one might not find the global optimum. $\endgroup$ – fibonatic Oct 21 at 2:06
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    $\begingroup$ One method is to use a direct optimization (without costates) to seed the indirect optimization (with the costates). A direct optimization converges well even on random initial conditions. @uhoh, you're right, I should add some examples, but I'm currently developing some code for that, so I don't have anything to show yet :-( $\endgroup$ – ChrisR Oct 22 at 4:57
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    $\begingroup$ @ChrisR I think of life as a process; the fun is in getting there, not being there. $\endgroup$ – uhoh Oct 22 at 5:02

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