John Breakwell, the restricted problem, and halo orbits

Celestial Mechanics, 1984

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 thegeneral 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 Ist 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 Ist 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.

Archive of Applied Mechanics

In this paper, the effect of radiation pressure force on formation of the nominal halo orbit around the collinear point L 1 is estimated in the framework of Sun-Earth system. Using Lindstedt-Poincaré technique, the periodic solutions up to third-order approximation are computed. Further, the Fourier series solutions of nominal halo orbit are also evaluated. With considering the radiation pressure force effect, the proposed analysis in this work can be used to design a trajectory for spacecraft moving in more realistic system. Keywords Restricted three-body problem • Radiation pressure • Halo orbit • Fourier series 1 Introduction The three-body problem is one of the most dynamical systems, which have been repeatedly and deliberated studied in celestial mechanics. Of specific interest are the infinitesimal body trajectory motion under the gravitational effect of the other two finite bodies. This problem has been received considerable numerical and analytical analysis [1-8]. In general, the restricted three-body problem has five libration points. Two of them are called triangular points, L 4 and L 5 [9,10], while the other are called collinear points, L 1 , L 2 and L 3 , which take locations on the joining line between the primaries [11-16]. The libration points in the Earth-Moon or Sun-Earth systems are equilibrium points in the gravitational field, where the spacecrafts can preserve stationary without consumption of extra-fuel [17,18]. The dynamics around libration points furnishes periodic orbit families, which are serviceable for placing a spacecraft, and has an extraordinary advantageous for solar observation and astrophysics missions as well as communication links, etc. In fact, these points have a considerable significance in the Sun-Earth system. In particular, the point

2019

As evidenced by the Global Exploration Roadmap, international interest exists in a new era of human exploration of the solar system. Such an effort is commencing with the examination of options for maintaining a facility-at times crewed-in an orbit nearby the Moon. Thus, the key objectives in advancing colonization of interplanetary space include positioning and maintaining an inhabited facility in a long-term and relatively stable orbit in the lunar vicinity. At this time, one orbit of interest for a habitat spacecraft in cislunar space is a Near Rectilinear Halo Orbit (NRHO). Near Rectilinear Halo Orbits are members of the L 1 or L 2 halo orbit families and are characterized by favorable stability properties. As such, they are strong candidates for a future habitat facility in cislunar space. This type of trajectory was first identified in cislunar space-the Earth-Moon CR3BP. However, for arrival to and departure from this Earth-Moon region, the impact of Solar gravity cannot be ignored. Thus, the orbital characteristics and stability properties are examined within the context of the bicircular restricted four-body problem.

2021

The paper considers the problem of calculating the low-energy high-and low-thrust trajectories to the libration points of the Earth-Moon system and to the halo orbits. An approach for solving the problem is proposed. It consists in calculating stable manifolds of libration points or halo orbits and calculating impulsive or low-thrust trajectory from an initial circular earth orbit to a given point of this manifold. Numerical examples are given for the calculation of direct and low-energy trajectories to libration points and to halo orbits and the optimization of entry points to stable manifolds for low-energy trajectories.

INTERNATIONAL CONFERENCE ON ADVANCES IN MULTI-DISCIPLINARY SCIENCES AND ENGINEERING RESEARCH: ICAMSER-2021

In the determination of location of Lagrangian points, the mass ratio of primaries in Restricted Three Body Problem (RTBP) plays an important role. Due to change in the value of mass ratio, the corresponding variations in different parameters of Halo Orbits (HO) are studied. It has been observed that halo orbits around L 1 move towards the first primary, X altitude, A X , corresponding to a fixed Z altitude, A Z , and period around L 1 decreases whereas X altitude, A X , corresponding to a fixed Z altitude, A Z , and period around L 2 increases with the increase in mass ratio. Further, orbits around L 2 move away from the second primary until μ ≈ 0.17894 and then move towards the primaries with the increase in mass ratio. Variations in other parameters such as the initial distance from the origin of a spacecraft and its initial velocity in a halo orbit around L 1 and L 2 are also obtained corresponding to variation in mass ratio. The results are validated by comparing with the data obtained for various Sun-Planet systems. Also, different parameters of halo orbits around the Sun-Earth and the Sun-Earth+Moon system are compared.

Celestial Mechanics and Dynamical Astronomy, 2021

This paper deals with direct transfers from the Earth to Halo orbits related to the translunar point. The gravitational influence of the Sun as a fourth body is taken under consideration by means of the Bicircular Problem (BCP), which is a periodic time dependent perturbation of the Restricted Three Body Problem (RTBP) that includes the direct effect of the Sun on the spacecraft. In this model, the Halo family is quasi-periodic. Here we show how the effect of the Sun bends the stable manifolds of the quasi-periodic Halo orbits in a way that allows for direct transfers.

Celest Mech Dynam Astron, 1993

The purpose of this paper is to study a transfer strategy from the vicinity of the Earth to a halo orbit around the equilibrium point L1 of the Earth-Sun system. The study is done in the real solar system (we use the DE-118 JPL ephemeris in the simulations of motion) although some simplified models, such as the restricted three body problem (RTBP) and the bicircular problem, have been also used in order to clarify the geometrical aspects of the problem. The approach used in the paper makes use of the hyperbolic character of the halo orbits under consideration. The invariant stable manifold of these orbits enables the transfer to be achieved with, theoretically, only one manoeuvre: the one of insertion into the stable manifold. For the total Av required, the figures obtained are similar to the ones given by the standard procedures of optimization.

Celest Mech Dynam Astron, 1993

The purpose of this paper is to study a transfer strategy from the vicinity of the Earth to a halo orbit around the equilibrium point L1 of the Earth-Sun system. The study is done in the real solar system (we use the DE-118 JPL ephemeris in the simulations of motion) although some simplified models, such as the restricted three body problem (RTBP) and the bicircular problem, have been also used in order to clarify the geometrical aspects of the problem. The approach used in the paper makes use of the hyperbolic character of the halo orbits under consideration. The invariant stable manifold of these orbits enables the transfer to be achieved with, theoretically, only one manoeuvre: the one of insertion into the stable manifold. For the total Av required, the figures obtained are similar to the ones given by the standard procedures of optimization.

2012

The coupled orbit and attitude motion of a spacecraft in three dimensional (halo) reference orbits is explored in this investigation within the context of the Circular Restricted Three Body Problem (CR3BP). The motion of a spacecraft, comprised of two rigid bodies connected by a single degree of freedom joint, is examined. The nonlinear variational form of the equations of motion is employed to mitigate numerical effects caused by large discrepancies in the length scales. Several nonlinear, periodic reference orbits in the vicinity of the collinear libration points are selected for case studies and the effects of the orbit on the orientation, as well as the orientation on the orbit, are examined. Additionally, the model is extended to include solar radiation pressure and the motion of a solar sail type spacecraft is considered.

We develop and illustrate techniques to obtain periodic orbits around the second Lagrangian point L2 in the Sun-Earth system based on the Re- stricted Three-Body Problem. In the case of Lyapunov (planar) orbits, the solutions to the linearized equations of motion allow the generation of the entire family of orbits. For Halo orbits, however, the method of strained coordinates is applied to generate higher-order approximate ana- lytic solutions. Subsequent application of Newton's method improves the initial conditions to obtain periodic solutions to the equations of motions. A graphical user interface has been developed to implement these numeri- cal tools in a user-friendly computational environment.