Calculating low-thrust capture delta-V with high $V_{inf}$

Question

I'm designing a CubeSat mission to the Martian moons, and am having trouble calculating capture delta-V. Once it gets into a circular orbit, I can do the rest, but the capture itself is giving me trouble.

The CubeSat would launch with SpaceX's Red Dragon mission, and would be ejected at some point during the cruise to Mars. According to a paper, Red Dragon will hit the Martian atmosphere at 6 km/s, which would mean it enters the Martian system at 1 km/s.

Most papers on this topic are not especially useful to me, as they assume transfer to Mars using low-thrust propulsion, which enters the Mars system at a much smaller relative velocity.

The CubeSat's thruster would provide an acceleration of about 1 mm/s^2. I can give more details if required. Ideally I'd like to capture into a flat, circular orbit at 25,000 – 30,000 km, but getting the numbers for capture into any orbit would be invaluable.

Answer 1

It sounds like the Red Dragon is relying on aerobraking to exit the Hohmann transfer ellipse and soft land on Mars. I don't think the cube sat would want to use aerobraking to shed velocity. For one thing to get a good enough alpha for a 1 mm/s^2 acceleration, you would need large and fragile solar arrays.

You can't exploit the Oberth effect with 1 mm/s^2 acceleration. So you would need to gain the full 2.7 km/s Vinfinity before arriving at Mars sphere of influence. At 1 mm/s^2, it would take about 31 days to do 2.7 km/s delta V. So the cube sat would need to separate from the Dragon and start accelerating about a month before Mars rendezvous.

Once the ship is loosely captured at the Sun Mars L1 (SML1) it is moving about .1 km/s wrt to Mars. Deimos is moving about 1.35 km/s wrt to Mars. So I would say about 1.2 km/s to get from SML1 to Deimos.

Answer 2

Disclaimer: I am not sure I understand your problem, so let me know if I misunderstood and I'll edit my answer.

The main problem with low thrust engines is that they are ... low thrust: you'll need to solve an optimal control problem (or a sub-optimal one) in order to orbit either moon of Mars. The design of your mission will depend on the following criteria:

1. When you are jettisoned from the Red Dragon, are you on a collision course with the Mars system? To know this precisely, you need to calculate the B-Plane parameters of your spacecraft: after jettison, propagate (without thrusting) your orbit until a distance of three times the sphere of influence of Mars and compute the projection. To get a rough idea, look at the mission design of the Red Dragon (supposing that is already done and available to you) and check if you are jettisoned after all deep space trajectory correction maneuvers or not.

2. Do you need to target one of the Moons directly? My initial hunch is that the easiest would be to target a highly elliptical orbit around Mars. Once in orbit, you can target one of the two moons. The reason here is that it'll be much easier to perform an orbital injection around Mars than around either Moon because of their low mass and size and the precision you'll need to get your ephemerides right. If you inject around Mars, you can more easily compute the injection parameters. The reason why you may want a highly elliptical injection orbit is because you need less $Delta v$ to reach that from a hyperbolic orbit.

3. How precise of a solution do you want? Low thrust involves solving a control problem. There are a few good sub-optimal solution (cf. Q-Law by Petropoulos or the Naasz control laws) and some global optimal solvers. The latter are much harder to implement (and require knowledge of the math field of convex optimization). However, if you are a US citizen and affiliated to a US university, you may ask for access to NASA Mystic (which uses a "static differential dynamical programming" method, and it is currently used for their Dawn mission). If computer RAM isn't much of a limitation, you should be able to apply to sub-optimal control laws by using simple differential programming where the solution space is split up and you build a graph of all possible solutions from an initial orbit to a final orbit (and then traverse the graph using Bellman's algorithm to find the optimal path).

I hope this helps a bit. Again, I may have misunderstood your question, and if so, let me know so I can update my answer.

Answer 3

Mars SOI radius: 576,000km src

Mars escape velocity: 5.03 km/s src

You're coming in at 6km/s so you need to shed 1km/s to capture. At 1mm/s^2 that's 1mln seconds, or some 11 days.

In 1mln seconds, at 5.5km/s average, you'll travel 5.5mln km, so you definitely can't just happily center your maneuver on Mars. You need to start braking at least 10 days, or 5mln km before closest approach. That way, you will capture into a very long elliptical orbit before exiting Mars SOI through the other side. It's wasteful (not benefiting from Oberth effect) but it's hard to do any other way.

Once in any orbit, you can try making this into the orbit you want. Perform inclination changes when passing equatorial plane as far from the planet as possible, then reduce periapsis by a burn near apoapsis, finally bring your cubesat into low orbit through multiple burns near periapsis.