Collinear Lagrangian Points Research Articles

Restricted Full Three-Body Problem: Applicationto Binary System 1999 KW4

DOI: 10.2514/1.30937 The motion of a satellite in the gravitational field of a binary system is investigated. The two bodies of the binary systemaremodeledas asphere andatriaxialellipsoid andthesatelliteisassumed tohavenoinfluenceonthemotion of the two primaries. The case of relative equilibria in the full two-body problem is assumed. In earlier papers, we looked at equilibrium solutions in the restricted full three-body problem, especially for L4;5. In the present work, we investigate energy constraints on the motion of the spacecraft using zero-velocity curves, collinear Lagrangian points, and the Jacobi constant for varying parameters of the system. We study transit and nontransit trajectories between the components. The methods are applied in the study of the binary asteroid system 1999 KW4, and some results are shown.

Stability and control of spacecraft formation flying in trajectories of the restricted three-body problem

This paper addresses the problem of relative position control of spacecraft formation flying (SFF) utilizing the framework of the circular restricted three-body problem (CR3BP) with the Sun and Earth as the primary gravitational bodies. Particularly, the results are not confined to the close vicinity of the collinear Lagrangian libration points. Rather, a linearization is performed relative to an arbitrary non-Keplerian reference trajectory, so that linear time-varying differential equations result. It is rigorously proved that the open-loop linearized SFF dynamics is unstable but controllable. Uncontrolled formations with bounded relative separation that constitute the stable subspace of the dynamics are found using the dual system method. This constitutes a powerful observation, since bounded formations in three-body trajectories have been found thus far only at the vicinity of the Lagrangian equilibrium points. Assuming ideal state measurements, a time-varying continuous linear-quadratic control law is subsequently developed. A complete internal disturbance model is used, rendering a robust disturbance rejection performance. An illustrative example is used to show that the propulsion for deep-space formationkeeping requires a considerable dynamic range. It is concluded that plasma electric propulsion having a micro-thrusting capability is a most suitable means for deep-space formationkeeping.

Hill stability of a triple system with an inner binary of large mass ratio

We determine the maximum dimensionless pericentre distance a third body can have to the barycentre of an extreme mass ratio binary, beyond which no exchange or ejection of any of the binary components can occur. We calculate this maximum distance, q′/a, where q′ is the pericentre of the third mass to the binary barycentre and a is the semimajor axis of the binary, as a function of the critical value of L2E of the system, where L is the magnitude of the angular momentum vector and E is the total energy of the system. The critical value is obtained by calculating L2E for the central configuration of the system at the collinear Lagrangian points. In our case we can make approximations for the system when one of the masses is small. We compare the calculated values of the pericentre distance with numerical scattering experiments as a function of the eccentricity of the inner orbit, e, the mutual inclination i and the eccentricity of the outer orbit, e′. These show that the maximum observed value of q′/a is indeed the critical q′/a, as expected. However, when e′→1, the maximum observed value of q′/a is equal to the critical value calculated when e′=0, which is contrary to the theory, which predicts exchange distances several orders of magnitude larger for nearly parabolic orbits. This does not occur because changes in the binding energy of the binary are exponentially small for distant, nearly parabolic encounters.

Tidal Effects on Globular Clusters

It was von Hoerner (1957) who pointed out that the tidal field of the Galaxy imposed a boundary condition on the size of globular clusters, a fact that had observational consequences when the first deep star counts became available (King 1962). More recently, though, dispute (cf Innanen, et al. 1983) has arisen over the identification of the observed limiting radius from star counts with that predicted by simple dynamical theory (King 1962, 1966; Szebehely 1967). Straightforward application of the classical three body problem seemed to imply that no stable stellar orbits would be found beyond the collinear Lagrangian points. Thus, to first order, the observed limiting radius would be just the radius of the inner Langrangian point L2 from the cluster center. For clusters in eccentric orbits, the tidal radius would be set at perigalacticon, where the tidal force is strongest. Numerical models of stellar orbits about a cluster in a realistic galactic potential by several authors (Jefferys 1976; Keenan and Innanen 1975; Keenan 1981) showed that stable orbits can exist with apocluster distances greater than rL2, due to the presence of non algebraic integrals of the motion that constrained the stars to move in a restricted region about the cluster rather than escaping.

Equilibrium solutions of the restricted problem of 2+2 bodies

Fourteen equilibrium solutions of the restricted problem of 2+2 bodies are shown to exist. Six of these solutions are located about the collinear Lagrangian points of the classical restricted problem of three bodies. Eight solutions are found in the neighborhood of the triangular Lagrangian points. Linear stability analysis reveals that all of the equilibrium solutions are unstable with the exception of four solutions; two in the vicinity of each of the triangular Lagrangian points. These four solutions are found to be stable provided the mass parameter of the primary masses is less than a critical value which depends also on the mass of the minor bodies.

Almost rectilinear halo orbits

Numerical studies over the entire range of mass-ratios in the circular restricted 3-body problem have revealed the existence of families of three-dimensional ‘halo’ periodic orbits emanating from the general vicinity of any of the 3 collinear Lagrangian libration points. Following a family towards the nearer primary leads, in 2 different cases, to thin, almost rectilinear, orbits aligned essentially perpendicular to the plane of motion of the primaries. (i) If the nearer primary is much more massive than the further, these thin L3-family halo orbits are analyzed by looking at the in-plane components of the small osculating angular momentum relative to the larger primary and at the small in-plane components of the osculating Laplace eccentricity vector. The analysis is carried either to 1st or 2nd order in these 4 small quantities, and the resulting orbits and their stability are compared with those obtained by a regularized numerical integration. (ii) If the nearer primary is much less massive than the further, the thin L1-family and L2-family halo orbits are analyzed to 1st order in these same 4 small quantities with an independent variable related to the one-dimensional approximate motion. The resulting orbits and their stability are again compared with those obtained by numerical integration.

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

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.

Low Energy Transit Orbits in the Restricted Three-Body Problems

Low Energy Transit Orbits in the Restricted Three-Body Problems

Doubly symmetric orbits about the collinear Lagrangian points

view Abstract Citations (22) References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Doubly symmetric orbits about the collinear Lagrangian points Bray, T. A. ; Gouclas, C. L. Abstract The forms and stability of three families of periodic solutions in the three-dimensional case of the re- stricted problem are discussed using the two-point boundary value theory and computing the eigenvalues of the Jacobians of the solutions. It is found that stable members of low inclinations in the case of L2 and L8 indeed exist. The question of a possible new integral in the neighborhood of these points is discussed, to- gether with the equivalent case of galactic motions. It is concluded that a new integral with a single-valued gradient cannot exist unless its gradient is collinear to the gradient of the energy integral along all "singular" closed solutions. "Ordinary" solutions, on the other hand, are never found after extensive search by many investigators; and thus the above conclusion seems to be general. Publication: The Astronomical Journal Pub Date: March 1967 DOI: 10.1086/110218 Bibcode: 1967AJ.....72..202B full text sources ADS |

A note concerning the collinear libration centers

The synodical barycentric abscissas of the collinear libration centers L 1 and L 2 are expanded in power series of the cubic root of the mass ratio up to the seventh order. A proof given by Wintner concerning an essential property of the characteristic exponents at the collinear Lagrangian points is shown to be wrong and is corrected. Through numerical computation for different values of the mass ratio, it is found that the absolute value of both characteristic exponents at L 1 and L 3 is a strictly increasing function of the mass ratio on the interval (0, 1/2), while, at L 2, it is a strictly decreasing function.