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Old 05-04-2012, 12:17 AM   #1
Default Manual calcultion of lunar transfer

For better understanding of orbital mechanics I'm trying to perform the transfer from the Earth to the Moon and back to the Earth using only Keplerian orbital elements shown on Orbit MFD and have some problems with that.
I usually start with DeltaGliderIV docked to ISS. If I initially align my orbital plane with Moon's one using Align planes MDF to make the transfer coplanar, calculating a point to burn engine isn't a problem. If I perform Hohmann transfer, it will take exactly one half of period of the transfer orbit. So the angle of arc which Moon will cover during transfer is 180 degrees multiplied by ratio of period of transfer orbit to the Moon's one. According to the Kepler's third law, it equals (Moon_SMa / Transfer_SMa) ^ 1.5. To make things simpler I neglect SMa of parking orbit (about 6.6M vs. 385M) and end up with angle of 180 * 0.5^1.5. So the Moon's true anomaly in rendezvous point will be Moon_TrA + (Moon_SMa / Transfer_SMa) ^ 1.5. Then I project this point to my orbit by reduction to common periapsis (for both orbits having very low eccentricity it's OK, I suppose): Moon_AgP - Parking_AgP + Moon_TrA + (Moon_SMa / Transfer_SMa) ^ 1.5. And the point to burn engines will be on the opposite side of the orbit: Moon_AgP - Parking_AgP + Moon_TrA + (Moon_SMa / Transfer_SMa) ^ 1.5 + 180. If I end up with value larger than 360 or lesser than 0, I just subtract or add 360. Despite all neglecting, this formula sometimes leads me directly to the Moon's surface without mid-course correction.
But the whole point is to do this without Align planes MDF and without initial planes alignment at all for it requires much more fuel than to slow down when approaching Moon. So I need to calculate non-coplanar transfer. I understand that rendezvous point will be in the point of intersection between parking orbital plane and the Moon's one, and the point to burn engine is right on the opposite intersection. Now I need to calculate true anomalies of these points in term of both parking and Moon's orbits and I failed with it. I think I solved it for some particular cases such as Moon's orbit being equatorial, but it's not the one. And I have no idea in how to solve it in general case. Can you help please?
The second problem being even more challenging is transfer back to the Earth from lunar orbit, for it involves leaving lunar Hill sphere on hyperbolic trajectory. I know that far enough from celestial body hyperbola could approximated with it's asymptote. And I suspect that in a case of coplanar transfer the best way from the Moon to the Earth is to have this asymptote pointed to direction opposite to Moon's own velocity and to fly with speed close enough to its one. But how do I do it? Well, I looked on the picture of the Moon's orbit around the Earth and my orbit around the Moon in the same ecliptic frame and concluded that desired direction is "somewhere over here". And experimentally came up whit idea of burning engine about 45 before it. OK, I actually returned back to the Earth but I want to know the right mathematical way to do it. And I don't know what to do in the case of non-coplanar transfer.
I'd also be grateful if you recommend me a book to explain this stuff. Previously I have read some of them but for some reason while dealing with complex problems as orbiting aspherical bodies they didn't explain neither how to calculate anomaly of point to burn engines (they just say something like "in the point of desired periapsis) nor how to deal with hyperbolic transfers.
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