Spatial Restricted Three-body Problem Research Articles

Quaternion Methods and Regular Models of Celestial Mechanics and Space Flight Mechanics: Local Regularization of the Singularities of the Equations of the Perturbed Spatial Restricted Three-Body Problem Generated by Gravitational Forces

Quaternion Methods and Regular Models of Celestial Mechanics and Space Flight Mechanics: Local Regularization of the Singularities of the Equations of the Perturbed Spatial Restricted Three-Body Problem Generated by Gravitational Forces

Powered lunar flyby for transfers between non-coplanar orbits around Earth

This work studies transfer between non-coplanar circular orbits around Earth with the space vehicle performing a powered lunar flyby maneuver. The complete transfer trajectory is accomplished by an application of two or three impulsive velocity increments. First and final velocity increments are applied tangentially, respectively, to the departing and the arrival orbits around Earth. An optional second velocity increment is applied at the perilune in order to increase the effects of the flyby maneuver. Despite many works consider the powered lunar flyby instead of a natural lunar flyby, it is important to compare both maneuvers in the context of the complete trajectory. In this direction, the present work formulates and solves multiple point boundary value problems that determine the transfer trajectories considering three models: a three-dimensional patched-conic approximation, a model based on the spatial restricted three-body problem, and, a model based on the spatial bi-circular restricted four-body in which the influence of the Sun is included. The transfer trajectory solutions are compared with classical maneuvers and with transfers that perform a natural flyby maneuver. An interesting result shows that a decelerating propulsion during the flyby maneuver can provide a transfer trajectory with a fuel consumption smaller than the one of bi-parabolic maneuver even if the Sun’s attraction is considered. Moreover, the influence of the Sun can decrease the time of flight and the apogee of the trajectory and it can save fuel consumption if the Sun’s initial phase angle is properly chosen.

AVERAGED CIRCULAR SPATIAL RESTRICTED THREE-BODY PROBLEM: INTERNAL CASE, NEW RESULTS

We investigate the spatial restricted circular three-body problem in the nonresonant case. Namely, we apply Gaussian averaging to obtain averaged equations in terms of osculating elements and then investigate them. Keplerian ellipse with a focus in the main body (the Sun) is taken as an unperturbed orbit assuming the semi-major axis of the ellipse to be less than the radius of the orbit of the outer planet (internal problem). Using the Parseval formula we have derived the twice-averaged perturbed force function of the problem in the form of an explicit analytical series with coefficients expressed in terms of the Gauss and Clausen hypergeometric functions. An investigation of averaged force function along it’s curves of non-analyticity showed that the series are asymptotic by Poincaré. For a reduced system with one degree of freedom, phase portraits of oscillations in the plane of the Keplerian elements \(e\), \(\omega \) are constructed in the second and fourth approximations. It is shown the topology of the phase portrait is more complicated in fourth approximation then in first and second approximations provided that the constant \({{c}_{1}}\) of the Lidov–Kozai integral belongs to the interval (0, 0.382).

Quaternion Methods and Regular Models of Celestial and Space Flight Mechanic: Using Euler (Rodrigues–Hamilton) Parameters to Describe Orbital (Trajectory) Motion. II: Perturbed Spatial Restricted Three-Body Problem

Quaternion Methods and Regular Models of Celestial and Space Flight Mechanic: Using Euler (Rodrigues–Hamilton) Parameters to Describe Orbital (Trajectory) Motion. II: Perturbed Spatial Restricted Three-Body Problem

Quaternion Methods and Regular Models of Celestial and Space Flight Mechanic: Using Euler (Rodrigues-Hamilton) Parameters to Describe Orbital (Trajectory) Motion. II: Perturbed Spatial Restricted Three-Body Problem

Quaternion Methods and Regular Models of Celestial and Space Flight Mechanic: Using Euler (Rodrigues-Hamilton) Parameters to Describe Orbital (Trajectory) Motion. II: Perturbed Spatial Restricted Three-Body Problem

Global hypersurfaces of section in the spatial restricted three-body problem

We propose a contact-topological approach to the spatial circular restricted three-body problem, for energies below and slightly above the first critical energy value. We prove the existence of a circle family of global hypersurfaces of section for the regularized dynamics. Below the first critical value, these hypersurfaces are diffeomorphic to the unit disk cotangent bundle of the two-sphere, and they carry symplectic forms on their interior, which are each deformation equivalent to the standard symplectic form. The boundary of the global hypersurface of section is an invariant set for the regularized dynamics that is equal to a level set of the Hamiltonian describing the regularized planar problem. The first return map is Hamiltonian, and restricts to the boundary as the time-1 map of a positive reparameterization of the Reeb flow in the planar problem. This construction holds for any choice of mass ratio, and is therefore non-perturbative. We illustrate the technique in the completely integrable case of the rotating Kepler problem, where the return map can be studied explicitly.

A generalized Poincaré–Birkhoff theorem

We prove a generalization of the classical Poincaré–Birkhoff theorem for Liouville domains, in arbitrary even dimensions. This is inspired by the existence of global hypersurfaces of section for the spatial case of the restricted three-body problem (Moreno and van Koert in Global hypersurfaces of section in the spatial restricted three-body problem, 2020).

Study of the ejection/collision orbits in the spatial RTBP using the McGehee regularization

In this paper we analyse the McGehee regularization of the collision in the spatial Restricted three-body problem (3D RTBP). As a particular application, we study the ejection (collision) orbits. The parameterization method is applied up to high order to obtain suitable initial conditions of ejection (collision) orbits. Moreover, assuming ejection orbits, different methods are discussed to detect which of them are also collision orbits. Finally we explore the so called ejection-collision (EC) orbits, that is, orbits where the particle ejects from one primary, reaches a maximum in the distance with respect to the same primary, and ends at collision with that primary. Some results concerning the existence of spatial EC orbits are described.

Quaternion methods and models of regular celestial mechanics and astrodynamics

This paper is a review, which focuses on our work, while including an analysis of many works of other researchers in the field of quaternionic regularization. The regular quaternion models of celestial mechanics and astrodynamics in the Kustaanheimo-Stiefel (KS) variables and Euler (Rodrigues-Hamilton) parameters are analyzed. These models are derived by the quaternion methods of mechanics and are based on the differential equations of the perturbed spatial two-body problem and the perturbed spatial central motion of a point particle. This paper also covers some applications of these models. Stiefel and Scheifele are known to have doubted that quaternions and quaternion matrices can be used efficiently to regularize the equations of celestial mechanics. However, the author of this paper and other researchers refuted this point of view and showed that the quaternion approach actually leads to efficient solutions for regularizing the equations of celestial mechanics and astrodynamics.This paper presents convenient geometric and kinematic interpretations of the KS transformation and the KS bilinear relation proposed by the present author. More general (compared with the KS equations) quaternion regular equations of the perturbed spatial two-body problem in the KS variables are presented. These equations are derived with the assumption that the KS bilinear relation was not satisfied. The main stages of the quaternion theory of regularizing the vector differential equation of the perturbed central motion of a point particle are presented, together with regular equations in the KS variables and Euler parameters, derived by the aforementioned theory. We also present the derivation of regular quaternion equations of the perturbed spatial two-body problem in the Levi-Civita variables and the Euler parameters, developed by the ideal rectangular Hansen coordinates and the orientation quaternion of the ideal coordinate frame.This paper also gives new results using quaternionic methods in the perturbed spatial restricted three-body problem.

On the Stability of Lagrange Solutions in the Spatial Near-Circular Restricted Three-Body Problem

On the Stability of Lagrange Solutions in the Spatial Near-Circular Restricted Three-Body Problem

Integrability of close encounters in the spatial restricted three-body problem

We extend to the spatial case a technique of integration of the close encounters formulated by Tullio Levi-Civita for the planar restricted three-body problem. We consider the Hamiltonian introduced in the Kustaanheimo–Stiefel regularization and construct a complete integral of the related Hamilton–Jacobi equation by means of a series convergent in a neighborhood of the collisions with the primary or secondary body.

A Perturbative Treatment of the Retrograde Co-orbital Motion

Abstract An important area of research in celestial mechanics is the analysis of dynamics in mean-motion resonances (MMR). In this paper, we consider a retrograde MMR 1:1 corresponding to a co-orbital motion, in which the asteroid and the planet revolve around the Sun in opposite directions. The motivation was provided by the recent discovery of real celestial bodies moving in such resonances with giant planets (the most famous example being 514107 Ka‘epaoka‘awela, a retrograde co-orbital asteroid of Jupiter). Our study is conducted in the context of a spatial restricted circular three-body problem. Applying double numerical averaging, we construct the equations describing the secular evolution of the asteroid's orbit and generate phase portraits in an e − ω space. In some cases, the averaging procedure is complicated by the coexistence of several modes of resonant motion. It leads to different scenarios of the long-term behavior and, consequently, to the splitting of the phase portrait into several sheets.

On the Nonlinear Stability of the Triangular Points in the Circular Spatial Restricted Three-body Problem

The well-known problem of the nonlinear stability of L4 and L5 in the circular spatial restricted three-body problem is revisited. Some new results in the light of the concept of Lie (formal) stability are presented. In particular, we provide stability and asymptotic estimates for three specific values of the mass ratio that remained uncovered. Moreover, in many cases we improve the estimates found in the literature.

Integrability of the spatial restricted three-body problem near collisions (an announcement)

Integrability of the spatial restricted three-body problem near collisions (an announcement)

Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: II

Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: II

Dynamics of the spatial restricted three-body problem stabilized by Hamiltonian structure-preserving control

The local invariant (stable, unstable, and center) manifolds of a saddle $$\times $$ center $$\times $$ center equilibrium point can be used to construct a three-dimensional Hamiltonian structure-preserving (HSP) controller. The linear stability of the controlled Hamiltonian system is verified when one of the proposed criteria is satisfied. The nonlinear dynamics of a linearly stabilized Hamiltonian system is analyzed by normalizing the Hamiltonian high-order perturbation terms using the Lie series method. The analytical solutions of the invariant tori are then obtained in trigonometric series form. A theorem for the Nekhoroshev stability of the controlled Hamiltonian system is provided. When the constructed HSP controller is applied to a photogravitational restricted three-body problem with oblateness, a hyperbolic artificial equilibrium point at which a criterion is satisfied can be stabilized, and bounded Lissajous trajectories near the equilibrium point are obtained. The allocation law demonstrates that the attitude angles and lightness number of the solar sail can serve as control parameters to provide the required acceleration of the HSP controller.

Regularization of the Perturbed Spatial Restricted Three-Body Problem by L-Transformations

Equations of motion for the perturbed circular restricted three-body problem have been regularized in canonical variables in a moving coordinate system. Two different L-matrices of the fourth order are used in the regularization. Conditions for generalized symplecticity of the constructed transform have been checked. In the unperturbed case, the regular equations have a polynomial structure. The regular equations have been numerically integrated using the Runge–Kutta–Fehlberg method. The results of numerical experiments are given for the Earth–Moon system parameters taking into account the perturbation of the Sun for different L-matrices.

Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: I

We develop a quaternion method for regularizing the differential equations of the perturbed spatial restricted three-body problem by using the Kustaanheimo–Stiefel variables, which is methodologically closely related to the quaternion method for regularizing the differential equations of perturbed spatial two-body problem, which was proposed by the author of the present paper. A survey of papers related to the regularization of the differential equations of the two- and threebody problems is given. The original Newtonian equations of perturbed spatial restricted three-body problem are considered, and the problem of their regularization is posed; the energy relations and the differential equations describing the variations in the energies of the system in the perturbed spatial restricted three-body problem are given, as well as the first integrals of the differential equations of the unperturbed spatial restricted circular three-body problem (Jacobi integrals); the equations of perturbed spatial restricted three-body problem written in terms of rotating coordinate systems whose angular motion is described by the rotation quaternions (Euler (Rodrigues–Hamilton) parameters) are considered; and the differential equations for angular momenta in the restricted three-body problem are given. Local regular quaternion differential equations of perturbed spatial restricted three-body problem in the Kustaanheimo–Stiefel variables, i.e., equations regular in a neighborhood of the first and second body of finite mass, are obtained. The equations are systems of nonlinear nonstationary eleventhorder differential equations. These equations employ, as additional dependent variables, the energy characteristics of motion of the body under study (a body of a negligibly small mass) and the time whose derivative with respect to a new independent variable is equal to the distance from the body of negligibly small mass to the first or second body of finite mass. The equations obtained in the paper permit developing regular methods for determining solutions, in analytical or numerical form, of problems difficult for classicalmethods, such as the motion of a body of negligibly small mass in a neighborhood of the other two bodies of finite masses.

Numerical verification of the steepness of three and four degrees of freedom Hamiltonian systems

We describe a new algorithm for the numerical verification of steepness, a necessary property for the application of Nekhoroshev’s theorem, of functions of three and four variables. Specifically, by analyzing the Taylor expansion of order four, the algorithm analyzes the steepness of functions whose Taylor expansion of order three is not steep. In this way, we provide numerical evidence of steepness of the Birkhoff normal form around the Lagrangian equilibrium points L4–L5 of the spatial restricted three-body problem (for the only value of the reduced mass for which the Nekhoroshev stability was still unknown), and of the four-degrees-of-freedom Hamiltonian system obtained from the Fermi-Pasta-Ulam problem by setting the number of particles equal to four.

Quasi-satellite orbits in the general context of dynamics in the 1:1 mean motion resonance: perturbative treatment

Our investigation is motivated by the recent discovery of asteroids orbiting the Sun and simultaneously staying near one of the Solar System planets for a long time. This regime of motion is usually called the quasi-satellite regime, since even at the times of the closest approaches the distance between the asteroid and the planet is significantly larger than the region of space (the Hill's sphere) in which the planet can hold its satellites. We explore the properties of the quasi-satellite regimes in the context of the spatial restricted circular three-body problem "Sun-planet-asteroid". Via double numerical averaging, we construct evolutionary equations which describe the long-term behaviour of the orbital elements of an asteroid. Special attention is paid to possible transitions between the motion in a quasi-satellite orbit and the one in another type of orbits available in the 1:1 resonance. A rough classification of the corresponding evolutionary paths is given for an asteroid's motion with a sufficiently small eccentricity and inclination.