Families of three-dimensional periodic solutions in the circular restricted four-body problem
In this paper we study the 3D symmetric periodic orbits of the circular restricted four-body problem, through their bifurcation from plane orbits. To this end, we calculate the special generating planar periodic orbits, i.e. the vertical-critical orbits, from five basic planar families of symmetric periodic orbits of the problem. We study the simplest 3D periodic orbits and so we restrict our calculations to simple vertical bifurcation orbits leading to families of one or two revolutions three-dimensional symmetric periodic orbits. We found 21 such families of simple 3D symmetric periodic orbits and typical orbits of all symmetry type 3D orbits are presented. The stability of each calculated 3D periodic orbit is also studied. Characteristic curves as well as stability diagrams of families of 3D periodic orbits are illustrated.
KeywordsFour-body problem Stability Three-dimensional orbits Vertical-critical orbits
- Andoyer, M.H.: Sur les solutions periodiques voisines des positions d’equilibre relatif, dans le probleme des n corps. Bull. Astron. 23, 129–146 (1906) Google Scholar
- Gascheau, M.: Examen d’une classe d’equations differentielles et application a un cas particulier du probleme des trois corps. Compt. Rend. 16, 393–394 (1843) Google Scholar
- Markellos, V.V., Goudas, C., Katsiaris, G.: In: Investigating the Universe, vol. 319. Reidel, Dordrecht (1981) Google Scholar
- Palmore, J.: Relative equilibria of the n-body problem. Thesis, University of California, Berkeley, CA (1973) Google Scholar