Vertical stability of periodic solutions around the triangular equilibrium points

Abstract

Abstract. The vertical stability character of the families of short and long period solutions around the triangular equilibrium points of the restricted three-body problem is examined. For three values of the mass parameter less than equal to the critical value of Routh (/~R) i.e. for # = 0.000953875 (Sun-Jupiter), # = 0.01215 (Earth-Moon) and/~ =/1R = 0.038521, it is found that all such solutions are vertically stable. For # > PR vertical stability is studied for a number of 'limiting' orbits extended to/~ = 0.45. The last limiting orbit computed by Deprit for # = 0.044 is continued to a family of periodic orbits into which the well known families of long and short period solutions merge. The stability characteristics of this family are also studied.

Equilibrium Points and Related Periodic Motions in the Restricted Three-Body Problem with Angular Velocity and Radiation Effects

“…Our aim here is to reproduce that schematic representation of Henrard [28] by precise numerical computations, so as to present the mechanism of the long period family's evolution in more detail. For the vertical stability of the long and short period families, we may mention here the works by Perdios and Zagouras [34] and Hou and Liu [35]. In Figure 7, we present the evolution of this family by showing some of its members in the Oxy-plane.…”

Formation-Keeping Strategies At The Earth-Moon L4 Triangular Libration Point

“…It was generally agreed upon that a projected spacecraft's trajectory could be greatly altered with only slight variations to initial conditions, and that stability within the orbital plane of the primaries was heavily dependent on their respective masses [12,13]. Motion perpendicular to the plane of orbit of the primaries were found to be stable [ 14] and is generally not disputed. Differences in opinion emerge regarding what particular mass values could give rise to bounded motions, and if the triangular points of the Earth-Moon system are naturally stable.…”

“…The gain matrix K is solved from Equation (3)(4), and the relative control angles can be described by (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) where i is the state error vector. This can then be substituted into the Equations (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) and (3)(4)(5)(6)(7)(8)(9)(10)(11) to yield more accurate simulation results. It should be noted that the assumption of a constant B matrix was only necessary to determine the constant K matrix.…”

“…Consider an nth order system. The sliding mode surface is defined via the relation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) where x = x-xd are the state errors. The symbols x and xd denote the states and the desired states here.…”

Vertical bifurcation families from the long and short period families around the equilateral equilibrium points

“…These families are often called bridges and have been widely studied (Deprit and Henrard 1968;Henrard 2002;Bruno and Varin 2007). However, seldom work has been done about the vertical bifurcation orbits (Perdios and Zagouras 1991;Bruno and Varin 2006). We wonder whether these vertical bifurcation orbits are connected with some bifurcation orbits in a special periodic family, as what happens in the planar case.…”