In previous studies, Lissajous trajectories associated with the collinear libration points in the restricted three-body problem have been successfully computed analytically to at least third-order. Those approximations are utilized to determine such trajectories numerically for an arbitrary, predetermined number of revolutions in the rotating frame, for the case of circular primary motion. The numerical approach first identifies target positions at specified intervals along the trajectory and locates a continuous path through those points with velocity discontinuities. Then the Δv are simultaneously reduced in an iterative process. Such trajectories have been constructed in various primary systems, for a wide range of orbit sizes and a large number of revolutions.

Numerical determination of Lissajous trajectories in the restricted three-body problem

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The dynamics around the collinear equilibrium points of the RTBP

Quasihalo orbits are Lissajous trajectories librating about the well known halo orbits. The main feature of these orbits is that they keep an exclusion zone in the same way that halo orbits do. As a result, the knowledge of this type of orbit gives more flexibility to the mission analysis design about collinear libration points of any pair of primaries in the solar system. This paper is devoted to the semianalytical computation of quasihalo orbits in the circular restricted three-body problem by means of an ad hoc Lindstedt-Poincare method. The study of the practical convergence of the procedure and the extension of the orbits to suitable locations in the solar system using Jet Propulsion Laboratory (JPL) ephemerides is also discussed.

Quasihalo Orbits Associated with Libration Points

A global grid search is performed to find axi- and doubly-symmetric periodic orbits in the restricted three-body problem using the dimensioned parameters associated with the Jupiter-Europa system. Local differential correctors are applied to regions of the initial condition phase space that appear to be near solutions. A three-dimensional initial condition mesh with billions of nodes is evaluated, and over 600,000 periodic solutions are identified. Families of direct and retrograde solutions, both new and previously published, are identified and discussed. Stability is analyzed for each solution and general regions of stability are noted. Of the most promising results is the observation and characterization of a large class of stable yet highly-inclined direct orbits. Finally, all of the solutions and associated properties are archived. The resulting database is a practical reference for preliminary design of missions to Europa.

Global search for planar and three-dimensional periodic orbits near Europa

In recent years, three-dimensional periodic and quasi-periodic orbits near the collinear libration points in the Sun-Earth/Moon three-body problem have been the focus of great interest for space mission design. Thus, more effort in the astrodynamics community has been directed toward analysis and computation of families of such orbits. But families of periodic orbits in the three-body problem have been the subject of much study for many decades. The first such orbit to be employed for a spacecraft trajectory (ISEE-3) was a member of a particular type of simply symmetric three-dimensional family, i.e., a halo orbit. Thus, attention shifted to further analysis of these periodic halo families. Some of the significant work in the development of periodic solutions in the three-body problem has been reviewed and a number of the highlights from the analysis and eventual numerical computation of halo families is presented here. The halo families of periodic orbits extend from each of the libration points to the nearest primary; they appear to exist for all values of the mass ratio. Thus, this further understanding may serve to support future spacecraft mission planning as well.

Families of orbits in the vicinity of the collinear libration points

Axisymmetric periodic orbits of the restricted problem in three dimensions.

The regions of stability for the ‘out of plane’ equilibrium points of the photogravitational restricted three-body problem are given. Second order expansions of periodic solutions around these points are constructed and the corresponding families are computed. It is found that two such families exist. One of them originates and terminates on the same equilibrium point while the other terminates by flattening on the orbital plane.

Periodic solutions about the out of plane equilibrium points in the photogravitational restricted three-body problem

The three families of three-dimensional periodic oscillations which include the infinitesimal periodic oscillations about the Lagrangian equilibrium pointsL1,L2 andL3 are computed for the value μ=0.00095 (Sun-Jupiter case) of the mass parameter. From the first two vertically critical (|a v |=1) members of the familiesa, b andc, six families of periodic orbits in three dimensions are found to bifurcate. These families are presented here together with their stability characteristics. The orbits of the nine families computed are of all types of symmetryA, B andC. Finally, examples of bifurcations between families of three-dimensional periodic solutions of different type of symmetry are given.

Three-dimensional periodic oscillations generating from plane periodic ones around the collinear Lagrangian points

The evolution of the periodic orbits around the collinear equilibrium positions, belonging to the Stromgren families a, b and c, with the radiation pressure parameter of the more massive body is studied in the Sun-Jupiter system. These families are determined for a single value of the radiation pressure parameter and particularly when the radiation force of the more massive body is equal to one half of the gravitational attraction. Then the critical stability orbits of each family are transferred with the radiation parameter. The stability of each periodic solution is also studied.

Periodic solutions around the collinear langrangian points in the photo gravitational restricted three-body problem: Sun-Jupiter case

The third-order parametric expansions given by Buck in 1920 for the three-dimensional periodic solutions about the triangular equilibrium points of the restricted Problem are improved by fourthorder terms The corresponding family of periodic orbits, which are symmetrical wrt the (x, y) plane, is computed numerically for μ=000095 It is found that the family emanating from L4 terminates at the other triangular point L5 while it bifurcates with the family of three-dimensional periodic orbits originating at the collinear equilibrium point L3 This family consists of stable and unstable members A second family of nonsymmetric three-dimensional periodic orbits is found to bifurcate from the previous one It is also determined numerically until a collision orbit is encountered with the computations

C. G. Zagouras

Papers

Axisymmetric periodic orbits of the restricted problem in three dimensions.

Periodic solutions about the out of plane equilibrium points in the photogravitational restricted three-body problem

Three-dimensional periodic oscillations generating from plane periodic ones around the collinear Lagrangian points

Periodic solutions around the collinear langrangian points in the photo gravitational restricted three-body problem: Sun-Jupiter case

Three-dimensional periodic orbits about the triangular equilibrium points of the restricted problem of three bodies