Vicinity Of Collinear Libration Points Research Articles

Space debris in vicinity of collinear libration points of the Earth–Moon system

Practical space exploration has a priority the issues of safety of space missions. And an important fundamental problem arises — the study of the evolution and the stability of a space debris cloud, consisting of artificial objects remnants, or debris of extra-terrestrial origin.We discuss a new behavior of mechanical system near an unstable equilibrium position — the trajectories located in the selected neighborhood of unstable equilibrium. As an important application, we consider perturbed motion in collinear libration points vicinity of the restricted circular three-body problem.Numerical simulations for the parameters of the Earth–Moon system convincingly illustrate our theoretical study. This mathematical model allows us predicting how long the debris with certain energy can fly in vicinity of unstable collinear libration point, being close to the localized trajectory. This must be taken under consideration when planning the space missions. And also the localized trajectories may be useful when spaceship flies around the collinear libration point.

Station-keeping error analysis for halo orbits around Libration point L1 using linear control logic

In this article, we perform the computation of halo orbits around libration point L1 in the Sun–Jupiter system. Third-order approximation, Lindstedt–Poincare method is used to select the initial guess for libration point orbits. Furthermore, Newton’s differential correction scheme is utilized to correct the initial guess obtained from an analytical approach. Halo orbits are exceptionally unstable, the station-keeping maneuver is required to maintain the nominal halo orbit in the vicinity of collinear libration points. We have applied linear control logic in the non-linear equation of motion for the station-keeping error analysis of halo orbits and also computed the control input for halo orbits. Numerical results illustrate the viability of the proposed control technique for the exact station-keeping error at libration point L1.

Periodic and quasi-periodic orbit design based on the center manifold theory

This paper proposes a new numerical method for finding libration point orbits in the vicinity of collinear libration points in the circular restricted three-body problem. The main advantage of this method is that it requires neither an initial guess nor complex algebraic manipulations for finding both quasi-periodic and periodic orbits. The proposed method consists of two steps: center manifold design and differential correction. The first step provides a quasi-periodic orbit parametrized by a single parameter vector. In the second step, the parameter vector in the first step is used to obtain an exact periodic orbit. This method is applied to find periodic and quasi-periodic orbits in the Sun-Earth restricted three-body problem around the L1 and L2 libration points.

Station-keeping of libration point orbits by means of projecting to the manifolds

There are many different types of (quasi)periodic orbits in the vicinity of collinear libration points. These libration point orbits (LPOs) are inherently unstable and must be controlled. This paper studies general station-keeping strategies that are suitable for various types of LPOs. Two complementary methods are proposed by exploiting the hyperbolic dynamics of the collinear libration points. One main advantage of these methods is that no information of the nominal orbit is needed in order to do the station-keeping. The first is a semi-analytical projection method which is based on the high order polynomial approximation of the center manifold. After the transformation from synodic coordinates to center manifold coordinates is derived, the error state can be projected to center manifold or its stable manifold by introducing a station-keeping Delta-V, effectively eliminating the unstable component. The second is a fully numerical method. An escape time algorithm is designed to find the intersection of stable and unstable manifolds, effectively projecting the error state to the center manifold. By applying these techniques to the collinear libration points of Earth–Moon system, it has been demonstrated that these proposed methods can achieve better maneuver cost performance and they can handle station-keeping problems of various types of LPOs in a unified manner.

An improved numerical method for constructing Halo/Lissajous orbits in a full solar system model

An improved numerical method that can construct Halo/Lissajous orbits in the vicinity of collinear libration points in a full solar system model is investigated. A full solar system gravitational model in the geocentric rotating coordinate system with a clear presentation of the angular velocity relative to the inertial coordinate system is proposed. An alternative way to determine patch points in the multiple shooting method is provided based on a dynamical analysis with Poincaré sections. By employing the new patch points and sequential quadratic programming, Halo orbits for L1, L2, and L3 points as well as Lissajous orbits for L1 and L2 points in the Earth-Moon system are generated with the proposed full solar system gravitational model to verify the effectiveness of the proposed method.

About regions of convergence of expansions of differential equations of the three-dimensional restricted three-body problem in the vicinity of collinear libration points

The analysis of regions of convergence of expansions of the right-hand sides of the differential equations of motion in the vicinity of the collinear libration points in the circular restricted three-body problem in powers of coordinates of the infinitesimal body due to F. R. Moulton (Moulton, 1920) is shown to be erroneous and his results are corrected. The generalisation of Moulton's results to analogous expansions of the equations in the elliptic problem of three bodies made by R. W. Farquhar (Farquhar, 1968) is shown to be groundless.