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

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.

• Coming from a Hohmann transfer, the Red Dragon would enter the Mars sphere of influence at about 3 km/s, give or take. By the time it reaches the atmosphere it would be moving about 6 km/s. The way you put the question seems to indicate you have wrong assumptions. May 24, 2017 at 21:19

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.

• Holy hell. I am irrationally angry at you for pointing out that Vinf is 3 km/s, not 1 km/s. Did the energy independently and it checks out. I am in your debt. Thanks for the numbers as well, super useful. May 26, 2017 at 1:19

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.

• Jettisoning can be anywhere after launch, and not, obviously, right before landing, since the butt of the trunk is always open. We want to "orbit" (QSO) the moons, so while capturing into an elliptical orbit is okay, we'll need to circularize anyways, into a ~25,000 km orbit.Thanks for the Mystic advice, and any numbers would be helpful - still in the mission-concept-planning phase, so we live off of reference missions and estimates. May 26, 2017 at 1:23