Classical Restricted Three-body Problem Research Articles

Equilibrium dynamics of the restricted 3-body problem with prolate primaries

In this paper, the equilibrium dynamics of the circular restricted (2+1)-body problem with two elongated main bodies is investigated. More precisely, the influence of the mass parameter and the non-sphericity parameter on the position and stability of the libration points of this characteristic configuration is studied through numerical and semi-analytical methods. In addition, the effect of these two parameters of the system on the fractality of the basins of convergence is also investigated, for this, two quantitative indicators are used: the boundary basin entropy and the uncertainty dimension. It is shown that when the two objects have the same value of the non-sphericity parameter, the appearance of new collinear and non-collinear libration points takes place. Compared to the classical circular restricted three-body problem, our numerical study reveals that when both primaries are prolate, some of the collinear libration points may be stable.

Artificial equilibrium points and their linear stability analysis in the solar sail problem with triaxial second primary

Motion of a solar sail or a spacecraft equipped with a solar sail is one of the interesting problem of celestial mechanics. This paper deals with stationary solution of solar sail problem with radiating first primary and triaxial second primary under the frame of circular restricted three body problem (RTBP). First, equations of motion is derived with triaxiality and sail lightness number and then artificial equilibrium points (AEPs) are determined. It is noticed that impact of triaxiality parameters (σ1andσ2) of second primary is less than that of sail lightness number (β) due to which a considerable deviations in the positions of AEPs are occurred. Further, linear stability test is performed for all AEPs and it is found that similar to equilibrium points in classical RTBP, collinear AEPs are unstable for all mass parameter μ∈(0,1/2] and triangular AEPs are stable in the range 0<μ<μc=0.03758902. Moreover, stability range decreases with the values of β and σ1 but increases with σ2. After this we analyze the periodic orbits of the solar sail in the vicinity of AEPs and it is noticed that nature of motion is unaffected but time period is influenced from the sail number β and triaxiality parameters σ1 and σ2. The findings are useful to describe the more generalized solar sail problem with other perturbations such as oblateness, albedo, drag forces etc.

A new class of intermediate thrust arcs in the restricted three-body problem

The Mayer’s variational problem of determining spacecraft optimal trajectories in the context of a classical restricted three-body problem is considered. Integrability of differential equations of a controllable motion in the non-central gravitational field represents well known and challenging task. It is shown that in the absence of sufficient number of first integrals, an explicit dependability of unknown integrals on certain variables can be used to explore the existence of previously unknown particular integrals and solutions of these equations utilizing Dokshevich’s method of analytical dynamics. A new class of extremal analytical solutions of the problem for intermediate thrust arcs is presented. An illustrative example of utilizing these solutions in minimizing the spacecraft characteristic velocity of a transfer from a specified initial position to some final position in the Earth-Moon system is discussed.

Synodic Resonant Halo Orbits of Solar Sails in Restricted Four-Body Problem

To extend the usability of solar sails in cislunar space, synodic resonant halo orbits around the Earth–moon L1 and L2 points are very promising. The dynamics is dominated by the gravitational forces not only from the Earth and moon but also from the sun, as well as solar radiation pressure (SRP); but, these forces are not considered in the classical circular restricted three-body problem. In this paper, we analyze 2:1, 3:1, and 4:1 synodic resonant halo orbits in two kinds of restricted four-body problems that consider the influence of SRP: the bicircular restricted four-body problem (BCR4BP), and the higher-fidelity ephemeris model. The orbits in the BCR4BP are computed by two different continuation schemes. Because the characteristics of the orbits depend on the magnitude of the SRP, which is determined by the sail area-to-spacecraft mass ratio (), various values of the are comprehensively applied to the computation. Furthermore, on the basis of the orbits designed in the BCR4BP, orbits in the ephemeris model are computed for two typical epochs to investigate the influence of perturbations that are not considered in the BCR4BP. The results provide comprehensive insights into the characteristics of the quasi-halo orbits of solar sails, which are useful for cislunar missions.

Effects of perturbations on the stability of equilibrium points in the CR3BP with luminous and heterogeneous spheroid primaries

This paper studies the position and stability of equilibrium points in the circular restricted three-body problem under the influence of small perturbations in the Coriolis and centrifugal forces when the primaries are radiating and heterogeneous oblate spheroids. It is seen that there exist five libration points as in the classical restricted three-body problem, three collinear L_{i} ,(i = 1,2,3) and two triangular L_{i} ,(i = 4,5). It is also seen that the triangular points are no longer to form equilateral triangles with the primaries rather they form simple triangles with line joining the primaries. It is further observed that despite all perturbations the collinear points remain unstable while the triangular points are stable for 0 < mu < mu _{c} and unstable for mu _{c} le mu le frac{1}{2} , where mu _{c} is the critical mass ratio depending upon aforementioned parameters. It is marked that small perturbation in the Coriolis force, radiation and heterogeneous oblateness of the both primaries have destabilizing tendencies. Their numerical examination is also performed.

Effect of three-body interaction on the topology of basins of convergence linked to the libration points in the R3BP

The modified circular restricted three-body problem is numerically investigated by exploring the effect of three-body interaction on the basins of convergence connected to the in-plane as well as out-of-plane equilibrium points. The evolution of the positions of the libration points and their stability are illustrated as a function of parameter k due to three-body interaction. It is observed that the number of equilibrium points strongly depends on the sign and the magnitude of the three-body interaction parameter and there exist at most seven libration points where four are non-collinear and three are collinear with the primaries. Moreover, in the Copenhagen case we have found seven collinear libration points. Moreover, the non-collinear as well as collinear libration points are stable for various combinations of mass parameter μ and k. The collinear as well as non-collinear libration points are stable for even those values of μ which are higher than the μcrit of the classical restricted three-body problem. The study of the regions of possible motion shows that as the value of the Jacobian constant decreases, the forbidden region decreases significantly. The attracting domains of the basins of convergence, on several types of two-dimensional planes, are unveiled by applying the multivariate Newton-Raphson iterative scheme. In an attempt to analyse the effect of parameter k on the topology of basins of convergence, a systematic and thorough investigations are presented. The degree of fractality is also unveiled by determining the basin entropy of the convergence plane.

Motion in the Restricted Three-Body Problem at the Nanoscale

This paper studies the classical restricted three-body problem of a carbon atom in the vicinity of two carbon 60 fullerenes ( fullerenes) at the nanoscale. The total molecular energy between the two fullerenes is determined analytically by approximating the pairwise potential energies between the carbon atoms on the fullerenes by a continuous approach. Using software MATHEMATICA, we compute the positions of the stationary points and their stability for a carbon atom at the nanosacle and it is observed that for each set of values, there exists at least one complex root with the positive real part and hence in the Lyapunov sense, the stationary points are unstable. Since only attractive Van der Waals forces contribute to the orbiting behavior, no orbiting phenomenon can be observed for , where the Van der Waals forces becomes repulsive. Although the orbital is speculative in nature and also presents exciting possibilities, there are still many practical challenges that would need to be overcome before the orbital might be realized. However, the present theoretical study is a necessary precursor to any of such developments.

К ОГРАНИЧЕННОЙ ЗАДАЧЕ ТРЕХ ТЕЛ

The paper analytically investigates the classical restricted three-body problem in a special non-inertial central coordinate system, with the origin at center of forces. In this coordinate system, an analytical expression of the invariant of the centre of forces is given. The existence of the invariant of the centre of forces admits the correct division of the problem into two problems. The first is a triangular restricted three-body problem. The second is a collinear restricted three-body problem. In this paper the collinear restricted three-body problem is investigated. Using the properties of the invariant of centre of forces of the restricted three-body problem in the special non- inertial central coordinate system, the basic differential equations of motion for the collinear restricted three-body problem are obtained when three bodies lie on the same line during all motion. Differential equations of the collinear restricted three-body problem in the rotating non-inertial central coordinate system in pulsating variables are derived. New differential equations of motion for the collinear restricted three-body problem in three regions of possible location of the massless body with stationary solutions corresponding to the three Euler libration points have been derived. The circular collinear restricted three-body problem is investigated in detail. The corresponding Jacobi integrals are obtained. New exact non-stationary partial analytical solutions of the obtained new differential equations of motion of the collinear restricted three-body problem have been found for the considered case.

Motion in the Restricted Three-Body Problem at the Nanoscale

This paper studies the classical restricted three-body problem of a carbon atom in the vicinity of two carbon 60 fullerenes ( fullerenes) at the nanoscale. The total molecular energy between the two fullerenes is determined analytically by approximating the pairwise potential energies between the carbon atoms on the fullerenes by a continuous approach. Using software MATHEMATICA, we compute the positions of the stationary points and their stability for a carbon atom at the nanosacle and it is observed that for each set of values, there exists at least one complex root with the positive real part and hence in the Lyapunov sense, the stationary points are unstable. Since only attractive Van der Waals forces contribute to the orbiting behavior, no orbiting phenomenon can be observed for , where the Van der Waals forces becomes repulsive. Although the orbital is speculative in nature and also presents exciting possibilities, there are still many practical challenges that would need to be overcome before the orbital might be realized. However, the present theoretical study is a necessary precursor to any of such developments.

Science orbits in the Saturn–Enceladus circular restricted three-body problem with oblate primaries

This contribution investigates the properties of a category of orbits around Enceladus. The motivation is the interest in the in situ exploration of this moon following Cassini’s detection of plumes of water and organic compounds close to its south pole. In a previous investigation, a set of heteroclinic connections were designed between halo orbits around the equilibrium points L1 and L2 of the circular restricted three-body problem with Saturn and Enceladus as primaries. The kinematical and geometrical characteristics of those trajectories makes them good candidates as science orbits for the extended observation of the surface of Enceladus: they are highly inclined, they approach the moon and they are maneuver-free. However, the low heights above the surface and the strong perturbing effect of Saturn impose a more careful look at their dynamics, in particular regarding the influence of the polar flattening of the primaries. Therefore, those solutions are here reconsidered by employing a dynamical model that includes the effect of the oblateness of Saturn and Enceladus, individually and in combination. The dynamical equivalents of the halo orbits around the equilibrium points L1 and L2 and their stable and unstable hyperbolic invariant manifolds are obtained in the perturbed models, and maneuver-free heteroclinic connections are identified in the new framework. A systematic comparison with the corresponding solutions of the unperturbed problem shows that qualitative and quantitative features are not significantly altered when the oblateness of the primaries is taken into account. Furthermore, it is found that the J2 coefficient of Saturn plays a larger role than that of Enceladus. From a mission perspective, the results confirm the scientific value of the solutions obtained in the classical circular restricted three-body problem and suggests that this simpler model can be used in a preliminary feasibility analysis.

Effect of variation of charge in the circular restricted three-body problem with variable masses

In the present paper, we are concerned by some investigation on circular restricted three-body problem (CR3BP), where we assume that the primaries have variable masses and variable charges. Among the principal tools used in the present study, we cite the well known Meshcherskii transformation. We have derived the equations of motion and Jacobi integral which differ by variation constant k and charge q from the classical restricted three-body problem. More exactly, in this paper, we have drawn the equilibrium points, the zero-velocity curves, the periodic orbits, the surfaces and the basins of attraction for the different values of charge. We have found one equilibrium point when the charge is q=0.4 and three equilibrium points when its value is q=0.501. We also have drawn the periodic orbits for these two values of charge and found that they are periodic. We have also plotted the zero-velocity surfaces for these two values of charges and found a tremendous variation in these two surfaces. We notice that the Poincaré surfaces of section are shifting away from the origin, when we increase the value of charge. We also got different surfaces for the motion of infinitesimal body, with respect to the variations of charge. The basins of attraction have been drawn for these two values of charge by using Newton-Raphson iterative method. We also noticed that by increasing the values of charge, the basins of attraction are shrinking. For the stability of the equilibrium points that we have studied, we found that, among them, one is stable and three others are unstable.

The restricted three body problem on surfaces of constant curvature

We consider a symmetric restricted three-body problem on surfaces Mκ2 of constant Gaussian curvature κ≠0, which can be reduced to the cases κ=±1. This problem consists in the analysis of the dynamics of an infinitesimal mass particle attracted by two primaries of identical masses describing elliptic relative equilibria of the two body problem on Mκ2, i.e., the primaries move on opposite sides of the same parallel of radius a. The Hamiltonian formulation of this problem is pointed out in intrinsic coordinates. The goal of this paper is to describe analytically, important aspects of the global dynamics in both cases κ=±1 and determine the main differences with the classical Newtonian circular restricted three-body problem. In this sense, we describe the number of equilibria and its linear stability depending on its bifurcation parameter corresponding to the radial parameter a. After that, we prove the existence of families of periodic orbits and KAM 2-tori related to these orbits.

Investigation of the effect of albedo and oblateness on the circular restricted four variable bodies problem

Abstract The present paper investigates the motion of the variable infinitesimal body in circular restricted four variable bodies problem. We have constructed the equations of motion of the infinitesimal variable mass under the effect of source of radiation pressure due to which albedo effects are produced by another two primaries and one primary is considered as an oblate body which is placed at the triangular equilibrium point of the classical restricted three-body problem and also the variation of Jacobi Integral constant has been determined. We have studied numerically the equilibrium points, Poincaré surface of sections and basins of attraction in five cases (i. Third primary is placed at one of the triangular equilibrium points of the classical restricted three-body problem, ii. Variation of masses, iii. Solar radiation pressure, iv. Albedo effect, v. Oblateness effect.) by using Mathematica software. Finally, we have examined the stability of the equilibrium points and found that all the equilibrium points are unstable.

Investigation of the restricted problem of three bodies of variable masses using computer algebra

The classical restricted three-body problem for bodies with variable masses is discussed for the case when the masses of two bodies vary isotropically at different rates and their sum varies according to the joint Meshchersky law. Within the perturbation theory in Hills’s approximation, differential equations determining the secular perturbations of the orbital elements are obtained and it is shown that they are integrable under arbitrary laws governing the mass variations satisfying the joint Meshchersky law. The influence of the variations of the body masses on the evolution of the orbital parameters is comparable with the perturbations created by massive bodies. All symbolic computations are performed in Wolfram Mathematica.

An approximation theorem in classical mechanics

A theorem by K. Meyer and D. Schmidt says that The reduced three-body problem in two or three dimensions with one small mass is approximately the product of the restricted problem and a harmonic oscillator [7]. This theorem was used to prove dynamical continuation results from the classical restricted circular three-body problem to the three-body problem with one small mass. We state and prove a similar theorem applicable to a larger class of mechanical systems. We present applications to spatial $(N+1)$-body systems with one small mass and gravitationally coupled systems formed by a rigid body and a small point mass.

Extension of Earth-Moon libration point orbits with solar sail propulsion

This paper presents families of libration point orbits in the Earth-Moon system that originate from complementing the classical circular restricted three-body problem with a solar sail. Through the use of a differential correction scheme in combination with a continuation on the solar sail induced acceleration, families of Lyapunov, halo, vertical Lyapunov, Earth-centred, and distant retrograde orbits are created. As the solar sail circular restricted three-body problem is non-autonomous, a constraint defined within the differential correction scheme ensures that all orbits are periodic with the Sun’s motion around the Earth-Moon system. The continuation method then starts from a classical libration point orbit with a suitable period and increases the solar sail acceleration magnitude to obtain families of orbits that are parametrised by this acceleration. Furthermore, different solar sail steering laws are considered (both in-plane and out-of-plane, and either fixed in the synodic frame or fixed with respect to the direction of Sunlight), adding to the wealth of families of solar sail enabled libration point orbits presented. Finally, the linear stability properties of the generated orbits are investigated to assess the need for active orbital control. It is shown that the solar sail induced acceleration can have a positive effect on the stability of some orbit families, especially those at the $L_{2}$ point, but that it most often (further) destabilises the orbit. Active control will therefore be needed to ensure long-term survivability of these orbits.

Identification of new orbits to enable future mission opportunities for the human exploration of the Martian moon Phobos

One of the paramount stepping stones towards NASA׳s long-term goal of undertaking human missions to Mars is the exploration of the Martian moons. Since a precursor mission to Phobos would be easier than landing on Mars itself, NASA is targeting this moon for future exploration, and ESA has also announced Phootprint as a candidate Phobos sample-and-return mission. Orbital dynamics around small planetary satellites are particularly complex because many strong perturbations are involved, and the classical circular restricted three-body problem (R3BP) does not provide an accurate approximation to describe the system׳s dynamics. Phobos is a special case, since the combination of a small mass-ratio and length-scale means that the sphere-of-influence of the moon moves very close to its surface. Thus, an accurate nonlinear model of a spacecraft׳s motion in the vicinity of this moon must consider the additional perturbations due to the orbital eccentricity and the complete gravity field of Phobos, which is far from a spherical-shaped body, and it is incorporated into an elliptic R3BP using the gravity harmonics series-expansion (ER3BP-GH). In this paper, a showcase of various classes of non-keplerian orbits is identified and a number of potential mission applications in the Mars–Phobos system are proposed: these results could be exploited in upcoming unmanned missions targeting the exploration of this Martian moon. These applications include: low-thrust hovering and orbits around Phobos for close-range observations; the dynamical substitutes of periodic and quasi-periodic Libration Point Orbits in the ER3BP-GH to enable unique low-cost operations for space missions in the proximity of Phobos; their manifold structure for high-performance landing/take-off maneuvers to and from Phobos’ surface and for transfers from and to Martian orbits; Quasi-Satellite Orbits for long-period station-keeping and maintenance. In particular, these orbits could exploit Phobos’ occulting bulk and shadowing wake as a passive radiation shield during future manned flights to Mars to reduce human exposure to radiation, and the latter orbits can be used as an orbital garage, requiring no orbital maintenance, where a spacecraft could make planned pit-stops during a round-trip mission to Mars.

A Note on the Existence of a Smale Horseshoe in the Planar Circular Restricted Three-Body Problem

It has been proved that, in the classical planar circular restricted three-body problem, the degenerate saddle point processes transverse homoclinic orbits. Since the standard Smale-Birkhoff theorem cannot be directly applied to indicate the chaotic dynamics of the Smale horseshoe type, we in this note alternatively apply the Conley-Moser conditions to analytically prove the existence of a Smale horseshoe in this classical restricted three-body problem.

Equilibrium Points and Related Periodic Motions in the Restricted Three-Body Problem with Angular Velocity and Radiation Effects

The paper deals with a modification of the restricted three-body problem in which the angular velocity variation is considered in the case where the primaries are sources of radiation. In particular, the existence and stability of its equilibrium points in the plane of motion of the primaries are studied. We find that this problem admits the well-known five planar equilibria of the classical problem with the difference that the corresponding collinear points may be stable depending on the parameters of the problem. For all planar equilibria, sufficient parametric conditions for their stability have been established which are used for the numerical determination of the stability regions in various parametric planes. Also, for certain values of the parameters of the problem for which the equilibrium points are stable, the short and long period families have been computed. To do so, semianalytical expressions have been found for the determination of appropriate initial conditions. Special attention has been given to the continuation of the long period family, in the case of the classical restricted three-body problem, where we show numerically that periodic orbits of the short period family, which are bifurcation points with the long period family, are connected through the characteristic curve of the long period family.

Effect of three-body interaction on the number and location of equilibrium points of the restricted three-body problem

In this paper we deal with a modified version of the classical restricted three-body problem in which the additional effect of a three-body interaction is considered. This new force component appears in the potential of the classical problem as a new additional term. Our aim is to study the existence of all equilibrium points of this new model-problem as well as to determine their number and location, in the full range of the parameters of the problem, using analytical techniques. It is found that, depending on the sign and magnitude of the three-body interaction, the number of collinear and triangular equilibrium points may vary from 1 to 7 and 0 to 4, respectively, contrary to the classical case where the number of these equilibria is fixed to 3 and 2, correspondingly. Also, another remarkable result is that out of the orbital plane equilibria may exist.