How can i calculate the initial vy0 component for a distant retrograde orbit in Sun-Earth CR3BP around Earth?
is to choose an Earth apoapsis distance, then simply try many of them, where many = thousands or even millions, and see which ones result in something that is close enough to periodic to meet your currently unstated criterion for what would be considered a sufficiently "distant retrograde orbit around Earth".
I believe (someone please correct me if not) that a much faster way would be to use some kind of "shooting method" and converge much more quickly by checking successive tests to see how close to zero vx is at periapsis. vx0 = 0 when x=0 in the rotating synodic frame, so vx a half-period later at periapsis should be at or very close to zero as well.
Shooting Method: (in no particular order)
- Chapter 3 in Robert E. Pritchett's Masters Thesis (Purdue) Numerical Methods for Low-Thrust Trajectory Optimization
- Single Shooting Handout, Interplanetary Mission Design Spring 2010, Kate Davis.
- DESCANSO Book Series; Deep Space Communications and Navigation Series Volume 12, Low-Energy Lunar Trajectory Design Section 2.6.
- Quasi-Periodic Orbits of the Restricted Three-Body Problem Made Easy
Would be to read all of the really excellent answers to How to best think of the State Transition Matrix, and how to use it to find Halo orbits? and to use them to build a fast algorithm to converge on a periodic orbit.
A Third Way
would be to read up on Planar Lyapunov Orbits and find one that is suitable. This would also require a numerical technique.
A Fourth Way
would be to "go deep" and dig into all the crazy possible CR3BP planar and nearly-planar subsets of the huge variety possible periodic orbits. See for example the papers mentioned below, and also see a sampling of that paper that I show in this answer.
The extremely cool and colorful paper E. J. Doedel et al, (2007) Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem International Journal of Bifurcation and Chaos 17, 2625 (2007). https://doi.org/10.1142/S0218127407018671 builds a system of illustrations that show all of the known, periodic, orbits in the CR3BP (Circular Restricted Three-Body Problem). This includes many kinds or classes of orbits as shown in the table, but excludes Lissajous Orbits because they are not in general periodic. (note: ignore the drawing in the Wikipedia article!)
You can and probably should also download the paper from its non-paywalled ResearchGate site, make some coffee, then spend six months enjoying it.
There is also an un-paywalled copy of their earlier paper available: The Computation of Periodic Solutions of the 3-Body Problem Using the Numerical Continuation Software AUTO D. J. Dichmann, E. J. Doedel, and R. C. Paffenroth Int. Conf. on Libration Point Orbits and Applications, Aiguablava, Spain, 10-14 June, 2002
Even laptops are gigaflop machines
these days, so this is not as challenging (computationally) as it would be even a decade ago. Of course depending on the method you choose, it can still be mathematically challenging.
So I'd start with the first method.
- Write a simple integrator for a 2D implementation of the CR3BP and run it with a bunch of values for vy0 and see what happens!
- Stop each one after it passes through x=0
- Interpolate to get it's vx at x=0 (what you are trying to zero out)
- Make a scatter plot of vx(x=0) versus vy0 and look for zero crossings
- Find a way to automate this or converge on the zero
- Have fun!
- If it doesn't work well, post your results in a New Question. Hopefully with more than one sentence this time, and you'll likely get further assistance. The more details you add to your question about what you've tried, the more willingness there will be to help you solve it.