# Theory of Orbits. The Restricted Problem of Three Bodies

**Book**· January 1967

*with*34 Reads

Publisher: New York: Academic Press

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- ... A complete review of the literature for the CRTBP is beyond the scope of this work, and we direct the interested reader to the watershed studies of E. Strömgren [1], of M. Hénon [2], [3], and of R. Broucke [4]. We also refer to the books of Moser [5], of Szebehely [6], of Meyer and Hall [7], and of Belbruno [8]; and also to the lecture notes of Chenciner [9] for much more complete discussions. ...... Section 5 is devoted to results, where we prove existence of periodic orbits in the CRFBP by showing existence of solutions of F = 0 using a rigorous computer-assisted a-posteriori analysis. 6 ...We use validated numerical methods to prove the existence of spatial pe- riodic orbits in the equilateral restricted four body problem. We study each of the vertical Lyapunov families (up to symmetry) in the triple Copenhagen problem, as well as some halo and axial families bifurcating from planar Lyapunov families. We consider the system with both equal and nonequal masses. Our method is constructive and non-perturbative, being based on a-posteriori analysis of a certain nonlinear operator equation in the neighborhood of a suitable approximate solution. The approximation is via piecewise Chebyshev series with coefficients in a Banach space of rapidly decaying sequences. As byproduct of the proof we obtain useful quantitative information about the location and regularity of the solution.
Show abstract - ... 1. If the primaries are spherical bodies (classical prob- lem), see the book [27] (A 1i = A 2i = A 3i = 0, i = 1, 2), thereby one obtains unperturbed mean mo- tion n = 1. 2. If the bigger primary is an oblate spheroid and the smaller is a spherical body [28] (A 11 = A 21 , A 31 = 0 , A 12 = A 22 = A 32 = 0), n 2 = 1+ 3 2 (A 11 − A 31 ), then the perturbed mean motion is faster than the un- perturbed motion (Keplerian motion) for the oblate body with A 11 > A 31 . However, If A 31 > A 11 (when the bigger primary is a prolate body) the perturbed mean motion is slower than the Keplerian motion. ...... The character of the solution of the variational dynami- cal system depends on the character of the solution for ω 2 from the quadratic of Eq. (43). The solution is stable only if the quadratic has two unequal negative roots for ω 2 , see [27] for more details. From Eqs. (26), (27) and Eq. ...ArticleFull-text available
- Mar 2017

In the restricted problem of three bodies when the primaries are triaxial rigid bodies, the necessary and sufficient conditions to find the locations of the three libration collinear points are stated. In addition, the Linear stability of these points is studied for the case of the Euler angles of rotational motion being θi = 0, ψi + φi = π/2, i = 1, 2 accordingly. We underline that the model studied in this paper has special importance in space dynamics when the third body moves in gravitational fields of planetary systems and particularly in a Jupiter model or a problem including an irregular asteroid.Show abstract - ... Problems in celestial navigation and orbit design helped drive the explosion of scientific computing during the space race of the mid Twentieth century. A fascinating account and a much more complete bibliography is found in the book [43]. ...ArticleFull-text available
- Jan 2019

We develop a systematic approach for proving the existence of spatial choreogra-phies in the gravitational n body problem. After changing to rotating coordinates and exploiting symmetries, the equation of a choreographic configuration is reduced to a delay differential equation (DDE) describing the position and velocity of a single body. We study periodic solutions of this DDE in a Banach space of rapidly decaying Fourier coefficients. Imposing appropriate constraint equations lets us isolate choreographies having prescribed symmetries and topological properties. Our argument is constructive and makes extensive use of the digital computer. We provide all the necessary analytic estimates as well as a working implementation for any number of bodies. We illustrate the utility of the approach by proving the existence of some spatial torus knot choreographies for n = 4, 5, 7, and 9 bodies.Show abstract - ... Problems in celestial navigation and orbit design helped drive the explosion of scientific computing during the space race of the mid Twentieth century. A fascinating account and a much more complete bibliography is found in the book [43]. ...Preprint
- Jan 2019

We develop a systematic approach for proving the existence of spatial choreographies in the gravitational $n$ body problem. After changing to rotating coordinates and exploiting symmetries, the equation of a choreographic configuration is reduced to a delay differential equation (DDE) describing the position and velocity of a single body. We study periodic solutions of this DDE in a Banach space of rapidly decaying Fourier coefficients. Imposing appropriate constraint equations lets us isolate choreographies having prescribed symmetries and topological properties. Our argument is constructive and makes extensive use of the digital computer. We provide all the necessary analytic estimates as well as a working implementation for any number of bodies. We illustrate the utility of the approach by proving the existence of some spatial torus knot choreographies for $n=4,5,7$, and $9$ bodies.Show abstract - ... Asteroid flyby trajectories are designed first in the medium-fidelity Circular Restricted Three-Body Problem (CR3BP) [16][17][18], considering solely the gravitational influences of the Sun and the Earth-Moon barycenter (EMB), and then results are verified in a high- fidelity ephemeris model considering the actual positions and gravitational attractions of the major influencing bodies in the Solar System. ...Conference PaperFull-text available
- Oct 2018

In response to the current interest in CubeSats and potential applications for planetary exploration, this work studies the feasibility of using autonomous CubeSats to flyby near-Earth asteroids. Considering the limited performance of current propulsion systems for CubeSats, low-energy (impulsive and low-thrust) trajectories are designed to encounter near-Earth asteroids in the medium-fidelity Circular Restricted Three-Body Problem, and their existence in a high-fidelity ephemeris model is also verified. The use of large ground antennas for deep-space communications might represent a major portion of CubeSat mission budgets, and thus the feasibility of performing optical navigation to autonomously estimate and correct the trajectory of the CubeSat is also evaluated through Monte Carlo simulations. Preliminary results show that approximately 4 asteroids per year could be reached by a 3U CubeSat if deployed around the first or second Sun-Earth Lagrange points. According to the limited performance of current CubeSat components, flyby altitudes of the order of 100-500 kilometers are determined possible using only observations of the Sun and of the target asteroid for autonomous navigation.Show abstract - ... Jules 5 Henri Poincaré (1854-1912 investigated the nature of various trajectories and came up with a visualization technique using maps in 1899 [12], now denoted as Poincaré Maps, to represent different dynamical behavior of the trajectories as noted by saddle, focus, center or nodes. In 1967, Victor Szebehely's Theory of Orbits [13] summarizes the improvements done in the circular restricted three body problem till then and substantiates theories with appropriate numerical results. ...ThesisFull-text available
- Dec 2017

The libration point orbits in the Sun-Earth/Moon system are formed by concurrent gravitational influences by various celestial bodies, originating in a nonlinear dynamical regime. Coupled with the unstable nature of the orbit, the impact of any perturbations are expected to increase rapidly. The feasibility of a flow-based, Cauchy-Green tensor control strategy for station-keeping is examined. An orbit consistent with the mission objectives is selected for examination. The station-keeping process is stochastic, thus Gaussian random errors are introduced for simulation. The evolution of a velocity perturbation over time is monitored, beyond which the attainable state in the accessible region nearest to the target state is employed as a feedback to compute the necessary full, three-component corrective maneuver. The application and appropriateness of single axis control maneuvers for orbit maintenance are also evaluated. The selection procedure for certain parameters such as tolerances and weighting values are developed to incorporate the available dynamical information, yielding a versatile and straightforward strategy. Weighting matrices within the target point approach are effective in influencing the station-keeping costs as well as size and direction of maneuvers. Moreover, selection of appropriate tolerance values in the application of the Cauchy-Green tensor exploits the dominant stretching direction of the perturbation magnitude to inform the maneuver construction process. The work is demonstrated in the context of the upcoming Aditya-1 mission to a Sun-Earth/Moon L1 halo orbit for solar observations and the James Webb Telescope to a Sun-Earth/Moon L2 halo orbit for astronomy.Show abstract - Chapter
- Jul 2017

Basic physical and mathematical formulations of the general three-body problem are presented and used to derive the standard as well as Hamiltonian equations of motion. Then, the derived equations are simplified by using the assumptions underlying the circular and elliptic restricted three body problems. The obtained equations of motion for the circular restricted three-body problem are written in rotating and non-rotating coordinate systems, whose origins are the center of mass of the primaries. The equations of motion for the elliptic restricted three-body system are written in terms of the rotating-pulsating coordinates.Show abstract - Article
- Jan 2018
- GEN RELAT GRAVIT

We illustrate the proposal, nicknamed LAGRANGE, to use spacecraft, located at the Sun–Earth Lagrange points, as a physical reference frame. Performing time of flight measurements of electromagnetic signals traveling on closed paths between the points, we show that it would be possible: (a) to refine gravitational time delay knowledge due both to the Sun and the Earth; (b) to detect the gravito-magnetic frame dragging of the Sun, so deducing information about the interior of the star; (c) to check the possible existence of a galactic gravitomagnetic field, which would imply a revision of the properties of a dark matter halo; (d) to set up a relativistic positioning and navigation system at the scale of the inner solar system. The paper presents estimated values for the relevant quantities and discusses the feasibility of the project analyzing the behavior of the space devices close to the Lagrange points.Show abstract - Article
- Dec 2017
- NONLINEAR DYNAM

We propose a polynomial expansion method to investigate periodic motions around both collinear and triangular libration points in the planar circular restricted three-body problem. The polynomial expansion method focuses on the approximate nonlinear polynomial relations between the two directions in the motion plane, which provide an alternative way to inspect the periodic motions. Based on the nonlinear polynomial relations, the planar two-dimensional system is decoupled. As an example, the \(3\mathrm{rd}\) order analytical solutions for the periodic motion are determined by solving the decoupled system. The efficiency of the polynomial expansion method has been validated by numerical method.Show abstract - The motion of the moonlet Dactyl in the binary system 243 Ida is investigated in this paper. First, periodic orbits in the vicinity of the primary are calculated, including the orbits around the equilibrium points and large-scale orbits. The Floquet multipliers’ topological cases of periodic orbits are calculated to study the orbits’ stabilities. During the continuation of the retrograde near-circular orbits near the equatorial plane, two period-doubling bifurcations and one Neimark–Sacker bifurcation occur one by one, leading to two stable regions and two unstable regions. Bifurcations occur at the boundaries of these regions. Periodic orbits in the stable regions are all stable, but in the unstable regions are all unstable. Moreover, many quasi-periodic orbits exist near the equatorial plane. Long-term integration indicates that a particle in a quasi-periodic orbit runs in a space like a tire. Quasi-periodic orbits in different regions have different styles of motion indicated by the Poincare sections. There is the possibility that moonlet Dactyl is in a quasi-periodic orbit near the stable region I, which is enlightening for the stability of the binary system.
Show abstract - Article
- Oct 2017
- J DYN DIFFER EQU

We consider a restricted three body problem on surfaces of constant curvature. As in the classical Newtonian case the collision singularities occur when the position particle with infinitesimal mass coincides with the position of one of the primaries. We prove that the singularities due to collision can be locally (each one separately) and globally (both as the same time) regularized through the construction of Levi-Civita and Birkhoff type transformations respectively. As an application we study some general properties of the Hill’s regions and we present some ejection–collision orbits for the symmetrical problem.Show abstract - Conference Paper
- Aug 2017

We study the stability of relative equilibrium positions in the planar circular restricted four-body problem formulated on the basis of the Euler collinear solution of the three-body problem. The stability problem is solved in a strict nonlinear formulation in the framework of the KAM theory. We obtained algebraic equations determining the equilibrium positions and showed that there are 18 different equilibrium configurations of the system for any values of the two system parameters \(\mu _1\), \(\mu _2\). Canonical transformation of Birkhoff’s type reducing the Hamiltonian of the system to the normal form is constructed in a general symbolic form. Combining symbolic and numerical calculations, we showed that only 6 equilibrium positions are stable in Lyapunov’s sense if parameters \(\mu _1\) and \(\mu _2\) are sufficiently small, and the corresponding points in the plane \(O\mu _1\mu _2\) belong to the domain bounded by the second order resonant curve. It was shown also that the third order resonance results in instability of the equilibrium positions while in case of the fourth order resonance, either stability or instability can take place depending on the values of parameters \(\mu _1\) and \(\mu _2\). All relevant symbolic and numerical calculations are done with the aid of the computer algebra system Wolfram Mathematica.Show abstract - Article
- Aug 2017
- ASTROPHYS SPACE SCI

Innovated by the nonlinear modes concept in the vibrational dynamics, the vertical periodic orbits around the triangular libration points are revisited for the Circular Restricted Three-body Problem. The \(\zeta \)-component motion is treated as the dominant motion and the \(\xi\) and \(\eta \)-component motions are treated as the slave motions. The slave motions are in nature related to the dominant motion through the approximate nonlinear polynomial expansions with respect to the \(\zeta \)-position and \(\zeta \)-velocity during the one of the periodic orbital motions. By employing the relations among the three directions, the three-dimensional system can be transferred into one-dimensional problem. Then the approximate three-dimensional vertical periodic solution can be analytically obtained by solving the dominant motion only on \(\zeta \)-direction. To demonstrate the effectiveness of the proposed method, an accuracy study was carried out to validate the polynomial expansion (PE) method. As one of the applications, the invariant nonlinear relations in polynomial expansion form are used as constraints to obtain numerical solutions by differential correction. The nonlinear relations among the directions provide an alternative point of view to explore the overall dynamics of periodic orbits around libration points with general rules.Show abstract - Chapter
- Jul 2017

The most important theoretical developments (historical and recent) in the three-body problem are presented and discussed. The first part of the presentation is devoted to periodic solutions to the equations of motion, the second part to non-periodic solutions, and in the third part a detailed description of different stability criteria is given. Each part contains extensive discussions of the general, circular restricted and elliptical restricted three-body problem, as well as the Hill problem.Show abstract - Chapter
- Jul 2017

All attempts to solve the three-body problem described in the previous chapters have greatly enriched celestial and classical mechanics. However, the ubiquity of fast computers with modes of operation that allow for parallel computations, numerical solutions have been the driving force in finding and studying possible solutions to the three-body problem. In this chapter, we provide a brief overview of common numerical schemes in terms of the mathematics of the algorithms used, as well as short examples written in the open-source programming language Python. The topics included in our discussion deal with numerical integration of the three-body problem, Fourier analysis, determination of mean motion resonances, and chaos indicators.Show abstract - Article
- Jan 2019
- CELEST MECH DYN ASTR

This paper globally explores two-impulse, low-energy Earth–Moon transfers in the planar bicircular restricted four-body problem with transfer time of up to 200 days. A grid search combined with a direct transcription and multiple shooting technique reveals numerous families of optimal low-energy solutions, including some that have not been reported yet. We investigate characteristics of solutions in terms of parameters in two- and three-body dynamics, and discuss a trade-off between cost and transfer time based on Pareto-optimal solutions, with and without lunar gravity assists. Analysis of orbital characteristics reveals the role of the Sun, the Earth, and the Moon in the transfer dynamics.Show abstract - Article
- Mar 2017

Periodic orbits in irregular gravitational fields are significant for an understanding of dynamical behaviors around asteroids as well as the engineering aspect for deep space explorations. The rotating mass dipole, referred to as the Chermnykh problem, is a good alternative model to study qualitative dynamical environments near elongated asteroids, like the asteroid 1620 Geographos, 216 Kleopatra or 25143 Itokawa. In this paper a global searching method is adopted to search for periodic orbits around the dipole model based on the concept of Poincaré section of surface. Representative families of periodic orbits are illustrated with respect to all three topological cases of the dipole model. Topological transitions of orbits during iso-energetic continuations are also presented as well as identification of new types of periodic orbits.Show abstract - Article
- Jul 2017
- Astrophys Space Sci

The goal of this research is to study the energy variations in trajectories of a spacecraft that performs a powered aero-gravity-assist maneuver considering the effects of the lift and drag forces around the Earth. It means that the spacecraft makes a passage by the Earth that is close enough to get effects from the atmosphere. Besides those effects, an impulse is applied to the spacecraft when it is passing by the periapsis of the trajectory, which also helps to modify its trajectory. It is varied the ballistic coefficient, lift to drag ratio, magnitude and direction of the impulse and the angle of approach of the maneuver. The mathematical model assumes that a spacecraft comes from an orbit around the Sun, traveling under the restricted three-body problem dynamics, makes a close approach passing inside the atmosphere of the Earth and then goes back to another orbit around the Sun, which is different from the initial orbit. The initial position and velocity of the spacecraft are given according to the usual pure gravity-assisted parameters, using the angle of approach and the perigee altitude and velocity. The energy is measured before and after the passage. The equations of motion are numerically integrated using a Runge–Kutta–Fehlberg 7/8 method. For the numerical examples, the perigee altitude is fixed in 120 km and the velocity at this point is fixed in 0.5 canonical units, which is near 14.89 km/s.Show abstract - Article
- Jan 2018
- ASTROPHYS SPACE SCI

The strongly perturbed environment of a small body, such as an asteroid, can complicate the prediction of orbits used for close proximity operations. Inaccurate predictions may make the spacecraft collide with the asteroid or escape to the deep space. The main forces acting in the dynamics come from the solar radiation pressure and from the body’s weak gravity field. This paper investigates the feasibility of using bi-impulsive maneuvers to avoid the aforementioned non-desired phenomena (collisions and escapes) by connecting orbits around the triple system asteroid 2001SN263, which is the target of a proposed Brazilian space mission. In terms of a mathematical formulation, a recently presented rotating dipole model is considered with oblateness in both primaries. In addition, a “two-point boundary value problem” is solved to find a proper transfer trajectory. The results presented here give support to identifying the best strategy to find orbits for close proximity operations, in terms of long orbital lifetimes and low delta-\(V\) consumptions. Numerical results have also demonstrated the significant influence of the spacecraft orbital elements (semi-major axis and eccentricity), angular position of the Sun and spacecraft area-to-mass ratio, in the performance of the bi-impulsive maneuver.Show abstract - Article
- Jun 2017
- CELEST MECH DYN ASTR

We outline some aspects of the dynamics of an infinitesimal mass under the Newtonian attraction of three point masses in a symmetric collinear relative equilibria configuration when a repulsive Manev potential (−1/r+e/r2), e>0, is applied to the central mass. We investigate the relative equilibria of the infinitesimal mass and their linear stability as a function of the mass parameter β, the ratio of mass of the central body to the mass of one of two remaining bodies, and e. We also prove the nonexistence of binary collisions between the central body and the infinitesimal mass.Show abstract - Article
- May 2017

Utilizing concepts from dynamical systems theory, we demonstrate how the existence of light rings, or fixed points, in a spacetime will give rise to families of periodic orbits and invariant manifolds in phase space. It is shown that these structures define the shape of the black hole shadow as well as a number of salient features of the spacetime lensing. We illustrate this through the analysis of lensing by a hairy black hole.Show abstract - Article
- Mar 2017
- J ASTRONAUT SCI

As we know that the Sun is a source of radiation in our solar system, the other planets or asteroids absorb some of the radiations incident on it and some reflected back into the space, these reflected radiations are called Albedo. The spacecraft is affected by both radiations i.e direct radiations as well as albedo. In this paper this is investigated how albedo perturbed the libration points and its stability in restricted three-body problem when less massive primary is an ellipsoid? It is found that there exist five libration points, three collinear and two non-collinear, the non-collinear libration points are stable for a critical value of mass parameter μ≤μc, where μc= 0.0385208965 …− (0.00891747 + 0.222579k) α– 0.02206859 σ1 – 0.04071097 σ2 but collinear libration points are still unstable. Also, an example of Sun-Earth system is taken in the last as a real application.Show abstract - Article
- Apr 2017
- ASTROPHYS SPACE SCI

Low-thrust propulsion is incorporated into circular restricted three-body problem to balance the gravitational and centrifugal forces, and then artificial equilibrium points can be generated. The linear dynamics indicates that there are stable and unstable artificial equilibrium points. Around the unstable artificial equilibrium points, there are center and hyperbolic invariant manifolds. In this work, invariant manifolds around artificial equilibrium points are expressed as formal series of amplitudes corresponding to hyperbolic and center dynamics, and high-order series solutions are constructed up to an arbitrary order. By taking advantage of the series expansions constructed, the motions around unstable artificial equilibrium points can be parameterized. In order to check the validity, the practical convergence of series solutions truncated at different orders is considered. Finally, series expansions of invariant manifolds are applied to designing transfer trajectories from the primary to periodic orbits around artificial equilibrium points which are located inside \(L_{1}\) and beyond \(L_{2}\) points.Show abstract - Article
- Mar 2017

The positions and stability of the collinear equilibrium points in the photogravitational ER3BP with zonal harmonics of the secondary is investigated. The effects of the perturbing forces: - oblateness, eccentricity and radiation pressure—on the positions and stability of collinear points \((L_{1,2,3})\) of an infinitesimal mass in the framework of the photogravitational ER3BP with zonal harmonics of the secondary are established. These effects on the positions of the binary systems Zeta Cygni, 54 Piscium, Procyon A/B and Regulus A are shown graphically and numerically from the analytic results obtained. It is observed that as the zonal harmonic \(J_{4}\) and eccentricity e increase, the collinear points shift towards the origin, while the reverse is observed with increase in the semi-major axis. The stability behavior however is unaffected by the introduction of these parameters, the collinear points remain linearly unstable.Show abstract - Article
- Dec 2016
- SOLAR SYST RES+

The development of a methodology for designing trajectories of spacecraft intended for the contact and remote studies of Jupiter and its natural satellites is considered. This methodology should take into account a number of specific features. Firstly, in order to maintain the propellant consumption at an acceptable level, the flight profile, ensuring the injection of the spacecraft into orbit around the Jovian moon, should include a large number of gravity assist maneuvers both in the interplanetary phase of the Earth-to-Jupiter flight and during the flight in the system of the giant planet. Secondly, the presence of Jupiter’s powerful radiation belts also imposes fairly strict limitations on the trajectory parameters.Show abstract - Article
- Jun 2017
- CELEST MECH DYN ASTR

This paper globally searches for low-thrust transfers to the Moon in the planar, circular, restricted, three-body problem. Propellant-mass optimal trajectories are computed with an indirect method, which implements the necessary conditions of optimality based on the Pontryagin principle. We present techniques to reduce the dimension of the set over which the required initial costates are searched. We obtain a wide range of Pareto solutions in terms of time of flight and mass consumption. Using the Tisserand–Poincaré graph, a number of solutions are shown to exploit high-altitude lunar flybys to reduce fuel consumption.Show abstract - Article
- Jan 2018
- Astrophys Space Sci

In this paper, we study the invariant manifold and its application in transfer trajectory problem from a low Earth parking orbit to the Sun-Earth L1 and L2 -halo orbits with the inclusion of radiation pressure and oblateness. Invariant manifold of the halo orbit provides a natural entrance to travel the spacecraft in the solar system along some specific paths due to its strong hyperbolic character. In this regard, the halo orbits near both collinear Lagrangian points are computed first. The manifold’s approximation near the nominal halo orbit is computed using the eigenvectors of the monodromy matrix. The obtained local approximation provides globalization of the manifold by applying backward time propagation to the governing equations of motion. The desired transfer trajectory well suited for the transfer is explored by looking at possible intersection between the Earth’s parking orbit of the spacecraft and the manifold.Show abstract - Librational motion in celestial mechanics is generally associated with the existence of stable resonant configurations and signified by the existence of stable periodic solutions and oscillation of critical (resonant) angles. When such an oscillation takes place around a value different than 0 or $\pi$, the libration is called asymmetric. In the context of the planar circular restricted three-body problem (CRTBP), asymmetric librations have been identified for the exterior mean-motion resonances (MMRs) 1:2, 1:3 etc. as well as for co-orbital motion (1:1). In exterior MMRs the massless body is the outer one. In this paper, we study asymmetric librations in the 3-dimensional space. We employ the computational approach of Markellos (1978) and compute families of asymmetric periodic orbits and their stability. Stable, asymmetric periodic orbits are surrounded in phase space by domains of initial conditions which correspond to stable evolution and librating resonant angles. Our computations were focused on the spatial circular restricted three-body model of the Sun-Neptune-TNO system (TNO= trans-Neptunian object). We compare our results with numerical integrations of observed TNOs, which reveal that some of them perform 1:2-resonant, inclined asymmetric librations. For the stable 1:2 TNOs librators, we find that their libration seems to be related with the vertically stable planar asymmetric orbits of our model, rather than the 3-dimensional ones found in the present study.
Show abstract - Article
- Dec 2017
- Comput Appl Math

A study of Earth–Moon bi-impulsive trajectories is presented in this paper. The motion of the space vehicle is described by the classic planar circular restricted three-body problem. The velocity increments are computed through analytical expressions, which are derived from the development of the Jacobi Integral expression. To determine the trajectories, a new two-point boundary value problem (TPBVP) with prescribed value of Jacobi Integral is formulated. Internal and external trajectories are determined through the solution of this new TPBVP for several times of flight. A relation between the Jacobi Integral and the Kepler’s energy at arrival is derived and several kinds of study are performed. Critical values of the Jacobi Integral, for which the Kepler’s energy of the space vehicle on the arrival trajectory becomes negative, are calculated for several configurations of arrival at the low Moon orbit in both directions: clockwise and counterclockwise. Results show that the proposed method allows the estimation of the fuel consumption before solving the TPBVP, and it facilitates the determination of trajectories with large time of flight. However, increasing values of the time of flight are not necessarily related with the increase of the Jacobi Integral value, which means that the obtaining of new trajectories becomes more difficult as the Jacobi Integral increases. Moreover, the proposed method provides results to be used as initial guess for more complex models and for optimization algorithms in order to minimize the total fuel consumption. For this case, this paper presents an example where an internal trajectory with large time of flight is optimized considering the Sun’s attraction.Show abstract - We consider a planetary system consisting of two primaries, namely a star and a giant planet, and a massless secondary, say a terrestrial planet or an asteroid, which moves under their gravitational attraction. We study the dynamics of this system in the framework of the circular and elliptic restricted three-body problem, when the motion of the giant planet describes circular and elliptic orbits, respectively. Originating from the circular family, families of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion resonances are continued in the circular and the elliptic problems. New bifurcation points from the circular to the elliptic problem are found for each of the above resonances and thus, new families, continued from these points are herein presented. Stable segments of periodic orbits were found at high eccentricity values of the already known families considered as whole unstable previously. Moreover, new isolated (not continued from bifurcation points) families are computed in the elliptic restricted problem. The majority of the new families mainly consist of stable periodic orbits at high eccentricities. The families of the 5/1 resonance are investigated for the first time in the restricted three-body problems. We highlight the effect of stable periodic orbits on the formation of stable regions in their vicinity and unveil the boundaries of such domains in phase space by computing maps of dynamical stability. The long-term stable evolution of the terrestrial planets or asteroids is dependent on the existence of regular domains in their dynamical neighbourhood in phase space, which could host them for long time spans. This study, besides other celestial architectures that can be efficiently modelled by the circular and elliptic restricted problems, is particularly appropriate for the discovery of terrestrial companions among the single-giant planet systems discovered so far.
Show abstract - Chapter
- Jan 2019

Spaceflight involving orbital transfers around irregularly shaped bodies or in the gravity field of several large bodies is fundamentally different from the flight in the gravity field of a single spherical body, which was covered in the previous chapters. The primary reason for this difference is that the spacecraft is no longer in a time-invariant gravity field of the two-body problem, but instead encounters a time-dependent field due to the relative motion of the multiple large bodies with respect to one another, or due to the changing position of the spacecraft relative to a rotating, non-spherical body.Show abstract - Article
- Oct 2017

The theorem of mirror trajectories, proven almost six decades ago by Miele, states that for a given path in the restricted problem of three bodies (with primaries in mutual circular orbits) there exists a mirror trajectory (in two dimensions) and three mirror paths (in three dimensions). The theorem at hand regards feasible trajectories and proved extremely useful for investigating the spacecraft natural dynamics in the circular restricted problem of three bodies, by identifying special solutions, such as symmetric periodic orbits and free return paths. This theorem has recently been extended to optimal mirror trajectories, thus substantiating Miele’s conjecture based on numerical evidence. Unlike the theorem of mirror paths, which refers to natural (unpowered) orbital motion, the theorem of optimal mirror trajectories establishes the existence, characteristics, and optimal control time history of the returning path, once the outgoing optimal trajectory has been determined. This theorem applies to (i) finite-thrust trajectories, for which a limiting value of the thrust acceleration exists, (ii) constant-thrust-acceleration paths, (iii) impulsive trajectories, and (iv) artificial periodic orbits (that use very low thrust propulsion or solar sails). This work illustrates the theorem of optimal mirror trajectories applied to two cases of practical interest: (a) continuous, low-thrust orbit transfer, and (b) continuous-thrust lunar descent (with soft touchdown) and ascent (with final orbit injection). In both cases, the theorem allows the immediate and straightforward identification of the optimal control law of the returning path, once the outgoing optimal trajectory has been determined.Show abstract - Article
- Oct 2017

The theorem of mirror trajectories was proven almost six decades ago by Miele, and states that for a given path in the restricted problem of three bodies (with primaries in mutual circular orbits) there exists a mirror trajectory (in two dimensions) and three mirror paths (in three dimensions). This theorem regards feasible trajectories and proved extremely useful for investigating the natural spacecraft dynamics in the circular restricted problem of three bodies. Several trajectories of crucial importance for mission analysis and design can be identified by using the theorem of mirror trajectories, i.e. (a) free return paths, (b) periodic orbits, and (c) homoclinic connections. Free return paths have been designed and flown in the Apollo missions, because they allow safe ballistic return toward the Earth in case of failure of the main propulsive system. These trajectories belong to the class of mirror paths with a single orthogonal crossing of the axis that connects the Earth and the Moon in the synodic reference system. Instead, if two orthogonal crossings of the same axis exist, then the resulting path is a periodic orbit. A variety of such orbits can be found, encircling either both primaries, a single celestial body, or a collinear libration point. In all cases, periodic orbits represent repeating trajectories that can be traveled indefinitely, (ideally) without any propellant expenditure. Homoclinic connections are special paths that belong to both the stable and the unstable manifold associated with a periodic orbit. These trajectories depart asymptotically from a periodic orbit and can encircle both primaries (even with close approaches) before converging asymptotically toward the initial periodic orbit. After almost six decades, the theorem of mirror trajectories, which clarified the fundamental symmetry properties related to the motion of a third body (or, more concretely, a space vehicle), still excerpts a considerable influence in space mission analysis and design.Show abstract - Article
- Mar 2019
- FEW-BODY SYST

The motion of an infinitesimal mass in the restricted four body problem is studied assuming both the orbits of motion of the three primaries to be elliptical. It is assumed that the forces governing the motion of the bodies is the mutual gravitational attractions of the primaries, however the problem is restricted in the sense that the effect of the fourth body of infinitesimal mass is assumed to be negligible. We have derived the equations of motion. The location and stability of the planar equilibrium points is studied for Sun–Earth–Moon system and graphically presented. The pulsating ZVC and the basin of attraction for the system are analyzed.Show abstract - Article
- Nov 2018
- CELEST MECH DYN ASTR

The fragmentation of interstellar molecular clouds has been investigated with great effort by many authors. In this paper, a simple model is given to describe the dynamics of two fragments moving in a special cylindrical potential. Using a modified version of the restricted three-body problem and the corresponding Jacobian integral, some constraints are given for the motion of the fragments.Show abstract - Article
- Aug 2018
- CELEST MECH DYN ASTR

In this work, periodic attitudes and bifurcations of periodic families are investigated for a rigid spacecraft moving on a stationary orbit around a uniformly rotating asteroid. Under the second degree and order gravity field of an asteroid, the dynamical model of attitude motion is formulated by truncating the integrals of inertia of the spacecraft at the second order. In this dynamical system, the equilibrium attitude has zero Euler angles. The linearised equations of attitude motion are utilised to study the stability of equilibrium attitude. It is found that there are three fundamental types of periodic attitude motions around a stable equilibrium attitude point. We explicitly present the linear solutions around a stable equilibrium attitude, which can be used to provide the initial guesses for computing the true periodic attitudes in the complete model. By means of a numerical approach, three fundamental families of periodic attitudes are studied, and their characteristic curves, distribution of eigenvalues, stability curves and stability distributions are determined. Interestingly, along the characteristic curves of the fundamental families, some critical points are found to exist, and these points correspond to tangent and period-doubling bifurcations. By means of a numerical approach, the bifurcated families of periodic attitudes are identified. The natural and bifurcated families constitute networks of periodic attitude families.Show abstract - Article
- Jul 2018
- CELEST MECH DYN ASTR

A multiple grid search strategy is implemented to generate a broad database of axisymmetric three-body periodic orbits for planets and main planetary satellites in the Solar system. The periodic orbit search is performed over 24 pairs of bodies that are well approximated by the circular restricted three-body problem (CR3BP), resulting in approximately 3 million periodic solutions. The periodic orbit generation is implemented in a two-level grid search scheme. First, a global search is applied to each CR3BP system in order to capture the global structure of most existing families, followed by a local grid search, centered around a few fundamental families, where useful, highly sensitive periodic orbits emerge. A robust differential corrector is implemented with a full second-order trust region method in order to efficiently converge the highly sensitive solutions. The periodic orbit database includes solutions that (1) remain in the vicinity of the secondary only; (2) circulate the primary only via inner or outer resonances; and (3) connect both resonance types with orbits bound to the secondary, approximating heteroclinic connections that leads to natural escape/capture mechanisms. The periodic solutions are characterized and presented in detail using a descriptive nomenclature. Initial conditions, stability indices, and other dynamical parameters that allow for the solution characterization are computed and archived. The data and sample scripts are made available online.Show abstract - Research
- May 2018

The present work aims to study the trajectories of a hypothetical spacraft, which starts from an orbit around the Earth to the sphere of influence of the Moon. This study was carried out through the computational simulation of the patched- conic aproximation method, which allowed us to find analytically the arrival conditions in the sphere of lunar influence, from particular cases of initial conditions in the starting geocentric orbit. Having given us as result some orbital parameters, relevants to the lunar missions project. Keywords: Patched conic approximation. Trans-lunar trajectories. Two-body problem.Show abstract - Article
- Jun 2018

This document reflects the effort of constructing a basis for understanding attitude motion within a multi-body problem with application to spacecraft flight dynamics. The circular restricted three-body problem (CR3BP) is employed as a model for the orbital motion. Then, attitude dynamics is discussed within the CR3BP. Conditions for bounded attitude librations and techniques for the identification of such behavior are presented: initially for a spacecraft fixed at an orbital equilibrium point, and later for a vehicle that moves on non-linear periodic orbit. While previous works focus on specific challenges, this analysis serves to create a more general framework for attitude dynamics within the CR3BP. A larger framework enables additional observations. For example, a linkage is noted between regions of bounded motion that may appear on an attitude grid search map and families of periodic attitude solutions. Finally, coupling effects between attitude and orbit dynamics within the CR3BP, ones that enable new options for trajectory design, are considered an important opportunity, and should be included in a general framework. As a proof of that concept, sailcraft trajectories are generated within a coupled orbit-attitude model only using a sequence of constant commands for the attitude actuators.Show abstract - Article
- Jan 2018
- IRAN J SCI TECHNOL A

In the present work, the periodic orbits around the relativistic restricted three-body problem (in brief relativistic RTBP) is treated. The equations of motion of the relativistic restricted three-body problem are presented. The locations of the equilibrium points are computed. The stability of equilibrium points is outlined. Elliptic and hyperbolic orbits in the neighborhood of the collinear points are obtained and then represented graphically for the whole domain of mass ratio as well as for different initial conditions. Degenerate hyperbolic orbits around the collinear points are obtained. Elliptic orbits in the neighborhood of the triangular point L4 are deduced and then represented graphically for the whole domain of mass ratio, for a selected range of frequency, for selected range of time as well as for different initial conditions. All obtained figures are analyzed dynamically.Show abstract - Article
- Apr 2018
- NONLINEAR DYNAM

The non-integrability of the Hill problem makes its global dynamics be necessarily approached numerically. However, the analytical approach is feasible in the computation of relevant solutions. In particular, the nonlinear dynamics of the Hill problem has been thoroughly investigated by perturbation methods in two cases: when the motion is close to the origin, and in the case of motion about the libration points. Out of the Hill sphere, the analytical approach is also feasible, at least in the case of distant retrograde orbits. Previous analytical investigations of this last case succeeded in the qualitative description of the dynamics, but they commonly failed in providing accurate results. This is a consequence of the essential dependance of the dynamics on elliptic functions, a fact that makes progress in the perturbation approach beyond the lower orders of the solution really difficult. An alternative perturbation approach is proposed here that allows to provide a very simple low-order analytical solution in trigonometric functions, on the one hand, and, while still depending on special functions, to compute higher orders of the solution, on the other.Show abstract - Article
- Apr 2018
- CELEST MECH DYN ASTR

The saddle points are locations where the net gravitational accelerations balance. These regions are gathering more attention within the astrophysics community. Regions about the saddle points present clean, close-to-zero background acceleration environments where possible deviations from General Relativity can be tested and quantified. Their location suggests that flying through a saddle point can be accomplished by leveraging highly nonlinear orbits. In this paper, the geometrical and dynamical properties of the Sun-Earth saddle point are characterized. A systematic approach is devised to find ballistic orbits that experience one or multiple passages through this point. A parametric analysis is performed to consider spacecraft initially on L1, L2 Lagrange point orbits. Sun-Earth saddle point ballistic fly-through trajectories are evaluated and classified for potential use. Results indicate an abundance of short-duration, regular solutions with a variety of characteristics.Show abstract - Article
- Mar 2018
- SOLAR SYST RES+

The existence and linear stability of the planar equilibrium points for photogravitational elliptical restricted three body problem is investigated in this paper. Assuming that the primaries, one of which is radiating are rotating in an elliptical orbit around their common center of mass. The effect of the radiation pressure, forces due to stellar wind and Poynting–Robertson drag on the dust particles are considered. The location of the five equilibrium points are found using analytical methods. It is observed that the collinear equilibrium points L1, L2 and L3 do not lie on the line joining the primaries but are shifted along the y-coordinate. The instability of the libration points due to the presence of the drag forces is demonstrated by Lyapunov’s first method of stability.Show abstract - Article
- Apr 2018
- CELEST MECH DYN ASTR

The motion of a massless particle in the gravity of a binary asteroid system, referred as the restricted full three-body problem (RF3BP), is fundamental, not only for the evolution of the binary system, but also for the design of relevant space missions. In this paper, equilibrium points and associated periodic orbit families in the gravity of a binary system are investigated, with the binary (66391) 1999 KW4 as an example. The polyhedron shape model is used to describe irregular shapes and corresponding gravity fields of the primary and secondary of (66391) 1999 KW4, which is more accurate than the ellipsoid shape model in previous studies and provides a high-fidelity representation of the gravitational environment. Both of the synchronous and non-synchronous states of the binary system are considered. For the synchronous binary system, the equilibrium points and their stability are determined, and periodic orbit families emanating from each equilibrium point are generated by using the shooting (multiple shooting) method and the homotopy method, where the homotopy function connects the circular restricted three-body problem and RF3BP. In the non-synchronous binary system, trajectories of equivalent equilibrium points are calculated, and the associated periodic orbits are obtained by using the homotopy method, where the homotopy function connects the synchronous and non-synchronous systems. Although only the binary (66391) 1999 KW4 is considered, our methods will also be well applicable to other binary systems with polyhedron shape data. Our results on equilibrium points and associated periodic orbits provide general insights into the dynamical environment and orbital behaviors in proximity of small binary asteroids and enable the trajectory design and mission operations in future binary system explorations.Show abstract - ArticleFull-text available
- Jun 2018

This paper presents a method for measuring the periodicity of quasi-periodic trajectories by applying discrete Fourier transform (DFT) to the trajectories and analyzing the frequency domain within the concept of entropy. Having introduced the concept of entropy, analytical derivation and numerical results indicate that entropies increase as a logarithmic function of time. Periodic trajectories typically have higher entropies, and trajectories with higher entropies mean the periodicities of the motions are stronger. Theoretical differences between two trajectories expressed as summations of trigonometric functions are also derived analytically. Trajectories in the Henon-Heiles system and the circular restricted three-body problem (CRTBP) are analyzed with the indicator entropy and compared with orthogonal fast Lyapunov indicator (OFLI). The results show that entropy is a better tool for discriminating periodicity in quasiperiodic trajectories than OFLI and can detect periodicity while excluding the spirals that are judged as periodic cases by OFLI. Finally, trajectories in the vicinity of 243 Ida and 6489 Golevka are considered as examples, and the numerical results verify these conclusions. Some trajectories near asteroids look irregular, but their higher entropy values as analyzed by this method serve as evidence of frequency regularity in three directions. Moreover, these results indicate that applying DFT to the trajectories in the vicinity of irregular small bodies and calculating their entropy in the frequency domain provides a useful quantitative analysis method for evaluating orderliness in the periodicity of quasi-periodic trajectories within a given time interval.Show abstract - Article
- Mar 2018
- COSMIC RES+

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.Show abstract - Article
- May 2018
- FEW-BODY SYST

The pulsating surfaces of zero velocity of the elliptic restricted three body problem when both the primaries are luminous oblate spheroids is investigated considering the effect of the oblateness of the infinitesimal. It is observed that if the third participating body in the restricted problem has sufficiently high value of the oblateness factor, the shape of the pulsating zero velocity surfaces changes for certain values of true anomaly. The projection of the zero velocity surfaces on xy- and xz-plane is also studied along with the low-velocity sub-regions in the respective planes. Employing the multivariate Newton–Raphson iterative scheme, the basins of attraction of the equilibrium points on the xy- and xz-plane is determined. Simulation technique is applied to demonstrate the influence of the oblateness of the infinitesimal on the basins of attraction.Show abstract - Article
- Mar 2018
- Nonlinear Dynam

Periodicity of motion around the collinear libration point associated with the Elliptic Restricted Three-Body Problem is studied. A survey of periodic solutions in the Circular Restricted Three-Body Problem is presented considering both Sun--Earth and Earth--Moon systems. Halo, Lyapunov and Vertical families around L1, L2 and L3 points are investigated, and their orbital period ranges through the entire family are reported. Resonant motions within the orbit families in the circular problem are identified and selected as suitable initial guess to find periodic orbits in the elliptic problem, which are targeted using a differential correction algorithm. Periodic solutions found are cataloged depending on the number of revolutions around libration points. Geometry, dynamical behavior and stability properties of single-revolution orbits are shown, as well as double-, triple- and quadruple-revolution solutions.Show abstract - Article
- Apr 2018
- ASTROPHYS SPACE SCI

This work studies a special type of cislunar periodic orbits in the circular restricted three-body problem called resonance transition periodic orbits, which switch between different resonances and revolve about the secondary with multiple loops during one period. In the practical computation, families of multiple periodic orbits are identified first, and then the invariant manifolds emanating from the unstable multiple periodic orbits are taken to generate resonant homoclinic connections, which are used to determine the initial guesses for computing the desired periodic orbits by means of multiple-shooting scheme. The obtained periodic orbits have potential applications for the missions requiring long-term continuous observation of the secondary and tour missions in a multi-body environment.Show abstract - In this paper, a time and space-coordinate transformation, commonly known as Kustaanheimo-Stiefel (KS)-transformation is applied to reduce the order of singularities arising due to the motion of infinitesimal body in the vicinity of smaller primary in the three body system. In this work, the Sun-Earth system is considered assuming the Sun as a radiating body and the Earth as an oblate spheroid. The study covers motion around collinear Lagrangian L1 and L2 points. Numerical computations are performed for both regularized and non-regularized equations of motion and results are compared for non-periodic as well as periodic motion. In the transformed space, time is also computed as a function of solar radiation pressure (q ) and oblateness of the Earth (A2 ). The two parameters (q, A2) have a significant impact on time in the transformed space. It is found that KS-regularization reduces the order of pole from five to three at the point of singularity of the governing equations of motion.
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