16.346 Astrodynamics
Spring 2012
Instructors: Emilio Frazzoli, Jeffrey A Hoffman, Mark D Van de Loo, Sheila E Widnall, David W Miller, Thomas M Coffee, Olivier L de Weck
Lecture: MW1-2.30 (33-419)
Announcements
Possible sign issue with coordinate velocities
Note that if you decided to use the CR3BP coordinate system in the Koon et al. paper rather than the one we used in lecture, you may encounter a sign issue if you express the x' or y' velocity components of your initial conditions in terms of the Jacobi constant.The formula derived for dy0 in the lecture demos uses the positive square root, which is what you will want for initial conditions with positive y' velocity, like those near a Lyapunov orbit's y' > 0 crossing of the y = 0 surface of section. However, if you want to compute initial conditions with y' < 0 in terms of the Jacobi constant, you will need the negative square root. The same applies to surfaces of section with fixed x: the sign of the root must match the condition of the Poincare section.
If you use the coordinate system from lecture, I expect you will only need positive roots for x' or y', so you should not run into this issue.
Announced on 20 May 2012 1:23 a.m. by Thomas M Coffee
Project 2 final submissions
For those still doing work on Project 2, please note that Tuesday is the last day we can look at anything you turn in before submitting term grades.If there is work you've done that did not lead to the intended results, please explain what you were trying to do so we can understand your approach.
Announced on 18 May 2012 10:39 p.m. by Thomas M Coffee
Project 2 part (k)
A couple of you asked whether, for part (k), you were expected to examine an injection into a lunar parking orbit; the answer is no, but in comparing your final trajectory with the direct (high-energy) trajectory from Problem Set 3, keep in mind that this maneuver is a significant part of the difference in propulsive requirements between the two.Note that lunar orbit is not the only applicable destination for the low-energy lunar transfer: late last year, the ARTEMIS spacecraft used this type of transfer to become the first spacecraft to enter Earth-Moon libration point orbits: http://www.universetoday.com/87044/artemis-spacecraft-curlicuing-their-way-to-lunar-orbit/
Announced on 18 May 2012 5:57 p.m. by Thomas M Coffee
No office hours today
Since the issues from Monday's and Wednesday's office hours seem to have been resolved and I'm not aware of any new ones, I have not planned any additional office hours. For those still doing work on the project, I will do my best to respond to specific questions by email.Announced on 18 May 2012 4:30 a.m. by Thomas M Coffee
Additional notes on periodic orbits and invariant manifolds
https://stellar.mit.edu/S/course/16/sp12/16.346/courseMaterial/topics/topic16/lectureNotes/lecture_18-19_posted/lecture_18-19_posted.pdf
To generate the invariant manifolds of your periodic orbits, keep in mind the following:
* smaller initial offsets will give you more accurate results, but if your offsets are too small, it will be difficult to judge the range of offsets necessary to fully "wrap around" the manifold, since your offsets will be overshadowed by integration error; you should probably make your smallest offset somewhere around 10^-8 or 10^-7 in magnitude, but you can adjust it as necessary based on the results you get
* the eigenvalue of the monodromy matrix associated with the (un)stable direction tells you the sensitivity of the flow after one revolution to offsets in that direction: if your initial offset along the (un)stable direction has magnitude 10^-7 and the corresponding eigenvalue is 10^4, then to first order you would expect the offset to grow to magnitude ~10^-3 after one revolution, though this will not be exact since the offsets are finite
* the offset along the (un)stable direction after one revolution from your minimum offset indicates the range of initial offsets you need to fully "wrap around" the manifold (when you reach this offset you will have fully cycled around the phase parameter)
* since the divergence from the periodic orbit is exponential, in order to "evenly" sample the manifold, you would (to first order) space your initial offsets logarithmically in magnitude: instead of a sequence like 10^-7, 2*10^-7, 3*10^-7, etc., rather a sequence like 10^-7, 10^-6.9, 10^-6.8, etc.; the more savvy alternative is to define a function that returns the Poincare map onto the desired surface of section from a given offset along the (un)stable direction, and ParametricPlot this function over a range of offset values, which will adaptively sample the offsets to generate a smooth curve on the surface of section.
Announced on 10 May 2012 5:17 p.m. by Thomas M Coffee
