References for constant low-thrust spiral up-coast-spiral down phasing maneuvers for circular and non-circular orbits?

Question

I am trying to find papers that describe constant low-thrust spiral up-coast-spiral down phasing maneuvers for circular and non-circular orbits. The thrust is assumed to be fixed at $T_{const}$ when the engine is turned on and $0$ when the engine is turned off during the coast phase.

Circular Orbits

The related question Phasing maneuvers; methods to travel between satellites with very similar circular orbits contains an excellent answer that describes a low-thrust spiral up/down maneuver for a perfectly circular orbit.

Q1: Is there a reference to learn more about this method for low-thrust circular orbit phasing?

Non-circular Orbits

The closest I could find were these two papers:

• Analytic Orbital Averaging Technique for Computing Tangential-Thrust Trajectories by Gao and Kluever (2005)
• Small Satellite LEO Maneuvers with Low-Power Electric Propulsion by King, Walker, and Kluever (2008)

Q2: Is there a different reference that contains a simpler description of a spiral up-coast-spiral down maneuver that can be used for low-thrust non-circular phasing with $e \approx 0.2$ ?

I'm basically trying to find the papers that people in the industry refer to for low-thrust phasing in the preliminary design stage. Analytic methods are preferred. Any references and insights would be deeply appreciated!

Answer 1

Not the reference you are looking for, but a response to this:

Any references and insights would be deeply appreciated!

I would like to note that low-thrust phasing in elliptical orbits is "boring", in the sense that the optimal strategy is conceptually simple.

For sufficiently low thrust, the phasing orbit does not have time to noticeably deviate from the initial orbit before the phasing angle has been covered. Said in another way, these manoeuvres nudge the orbit a tiny bit, and this tiny difference is iterated over a great number of orbits.

It is therefore just a problem of maximising the response in orbital period as a result of small changes in velocity, and therefore just a problem of maximising local velocity, which means prograde and retrograde burns.

The strategy is therefore:

1. If the target is ahead of you, apply retrograde thrust until half the distance has been covered, and then turn around and do prograde thrust until orbits match again.

2. If the target is behind you, do the same, but swap prograde and retrograde thrust.

The rest of an analytic solution is then "just" calculus, like for instance starting from the response in semi-major axis per change in velocity, expressed in terms of orbital radius:

$$\frac{da}{dv} = \frac{\mu}{r^2\sqrt{\mu\left(\frac{2}{r} - \frac{1}{a}\right)}}$$

And then combining it with how that again affects the orbital period:

$$\frac{dT}{da} = \frac{3\pi\sqrt{\frac{a^3}{\mu}}}{a}$$

... and then a it's a lot of equation wrangling, which is what these papers are generally dealing with