# Convert the impulsive solution to a low-thrust

The spacecraft transfers from an orbit with the following parameters:

• Perigee - 700 km
• Apogee - 6000 km
• Inclination - 64 deg
• Argument of perigee - 250 deg

to and orbit with the parameters:

• Perigee - 800 km
• Apogee - 30000 km
• Inclination - 64 deg
• Argument of perigee - 280 deg

That's to say, this is an in-plane maneuver (the plane doesn't change). I have calculated the solution considering impulsive maneuvers (using the Lambert solver found 2 impulses).

Now, I have to calculate the optimal trajectory considering low-thrust maneuvers for 2 cases: minimum time and minimum fuel consumption.

Is it possible and would it be correct to convert the impulsive solution to a low-thrust? How to calculate the mass change (would the rocket equation work?)? Should I limit the velocity change value for each low-thrust impulse?

Would appreciate for links/papers. I found this paper.

• Radius or altitude? Of course, in case of non-Earth, it could be a radius. But If You say "Apogee" - it is about Earth. Commented Sep 2, 2020 at 14:32

The trouble with optimizing low-thrust trajectories is there are so many different possible maneuver profiles that it is very hard to tell whether there might be a better answer hiding behind a slightly different parameterization of the motion. You can find the best choice out of all the options you considered in your model (much easier in some cases than others), but there are always other options you didn't make available to the solver, and you can't know how good they might be.

You might want to read a few of these:

Avanzini, Palmas, and Vellutini, “Solution of Low-Thrust Lambert Problem with Perturbative Expansions of Equinoctial Elements”

Markopoulos, “Analytically Exact Non-Keplerian Motion for Orbital Transfers”

Markopoulos, “Non-Keplerian Manifestation of the Keplerian Trajectory Equation and a Theory of Orbital Motion Under Continuous Thrust,”

Petropoulos and Longuski, “Automated Design of Low-Thrust Gravity-Assist Trajectories”

Petropoulos and Sims, “A Review of Some Exact Solutions to the Planar Equations of Motion of a Thrusting Spacecraft”

Quarta and Mengali, “New Look to the Constant Radial Acceleration Problem”

• Is it possible to convert an impulsive solution to a correct low-thrust? Commented Sep 3, 2020 at 6:02
• Not immediately, but yes with some experimentation. That is, I don't know of a formula to plug into, and changing the impulse into a low-thrust maneuver with the same delta-V won't get you exactly where you want to be, but you can plug it into a propagator, see where you end up, and iteratively adjust the maneuver until you end up in the right place. Any optimizer, even if it's just you eyeballing it and changing numbers by hand, is going to need a starting guess, which the simple but not quite right conversion of the impulsive solution can supply. Commented Sep 3, 2020 at 15:35
• This may work, but I'm not sure I can use the rocket equation for mass change calculation. Also, considering that the specific impulse is relatively high, probably, I have to somehow limit the velocity change for each low-thrust impulse. Would you be so kind to edit your answer, adding some information on that? I found also a paper arc.aiaa.org/doi/10.2514/6.2020-1691 Commented Sep 4, 2020 at 7:18
• There are tons of papers on this topic. Most are not directly comparable, since they each solve different pieces of the problem. You are asking for a canonical answer, but I don't think there is one yet: this is an active research area. Various authors have been able to show their solution was the best out of all those fulfilling various additional constraints they needed to assume in order to get any solutions at all. The link you posted looks very interesting, but it is only a few months old! I think the main reason no one else has commented is that they don't know the answer, either. Commented Sep 10, 2020 at 12:23