It is well known that there are periodical three-dimensional orbits around L1 and L2 libration points in circular restricted three-body problem called halo orbits. Existence of these orbits is justified numerically: anybody can state a system of nonlinear equations (conditions of symmetry and orthogonality to the xz plane) and solve it numerically to obtain a solution with high precision. But is there any analytical proof that these periodical orbits exist, mathematically?
As I know, existence of the Lyapunov orbits in CR3BP is a consequence of the Lyapunov's centre theorem:
- Meyer, K. R. and Hall, G. R. (1992). Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Applied Mathematical Sciences, vol. 90. Springer-Verlag, New York.
But why halo orbits exist? Why there is an energy level at which there is a bifurcation from the planar Lyapunov orbits that gives rise to halo orbits?
Hi, I think a part of the answer can be found in:
V. Szebehely, Theory of Orbits, Academic Press, 1967.
There are some recent research on the topics :
Article Non-integrability of the three-body problem
or
https://uma.ensta-paristech.fr/conf/gravasco/abstracts/P1b_abs_Combot.pdf
using JP Ramis and Morales-Ruiz results.
Sincerely,
G. Thomas, PhD
Dear Dr. Epenoy,
Thank you for the recommendation to look in the monography of Szebehely. Indeed, there is much information about how to prove the existence of some orbits in CR3BP. Unfortunately, the corresponding chapters pay attention primarily to orbits around the main bodies, and contain some thoughts are about the planar Lyapunov orbits. 3-dimensinal halo orbits (as I understand) are out of the scope of this work.
Dear Maksim,
You might be interested in the recent preprint:Article Spatial periodic orbits in the equilateral circular restrict...
which will appear in Celestial Mechanics and Dynamical Astronomy. It deals with the 4Body Problem, but the same techniques could be used in the three body problem. I think you would definitely also be interested in contacting Daniel Wilczak in Poland, who has been doing work on the CRFBP which is exactly what you are asking for.
Daniel Wilczak
I think his paper is not on the archive yet but I would expect it soon.
As for analytical results I suggest the paper by Angel Jorba:
http://www.maia.ub.edu/dsg/2009/0904farres.pdf
This paper contains many other interesting reference about this question.
They take the approach of computing a center manifold at the libration point which allows them to predict numerically the existence of the halo orbits. The manifold is computed via a formal series expansion and it is very difficult to show that this expansion converges. But at least it gives a good explanation for why the orbits are there. On the other hand the difficulty of analyzing the global convergence of center manifold expansions and normal forms is part of the reason people have been developing direct methods of computer assisted proof like those mentioned above.
Cheers,
Jay
Dears,
I guess one of the best theoretical results in this direction are
M. Ceccaroni, A. Celletti and G. Pucacco, Halo orbits around the collinear points of the restricted three-body problem, Physica D 317, (2016) 28–42.
A. Celletti, G. Pucacco and D. Stella, Lissajous and Halo orbits in the restricted three-body problem, J. Non. Sci. 25, (2015) 343–370.
Computer-assisted proofs of the existence of halo orbits for wide range of mass parameters, as well as continuation and bifurcations of halo orbits for selected mass parameters are given in our recent paper submitted to CNSNS
http://ww2.ii.uj.edu.pl/~wilczak/papers/halo/cnsns-ww-halo.pdf
Kind regards,
Daniel
Dear Dr. James and Dr. Wilczak,
Thank you very much for your assistance! I will study the papers carefully.
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