Multiple Poincaré sections method for finding the quasiperiodic orbits of the restricted three body problem

A new fully numerical method is presented which employs multiple Poincaré sections to find quasiperiodic orbits of the Restricted Three-Body Problem (RTBP). The main advantages of this method are the small overhead cost of programming and very fast execution times, robust behavior near chaotic regions that leads to full convergence for given family of quasiperiodic orbits and the minimal memory required to store these orbits. This method reduces the calculations required for searching two-dimensional invariant tori to a search for closed orbits, which are the intersection of the invariant tori with the Poincaré sections. Truncated Fourier series are employed to represent these closed orbits. The flow of the differential equation on the invariant tori is reduced to maps between the consecutive Poincaré maps. A Newton iteration scheme utilizes the invariance of the circles of the maps on these Poincaré sections in order to find the Fourier coefficients that define the circles to any given accuracy. A continuation procedure that uses the incremental behavior of the Fourier coefficients between close quasiperiodic orbits is utilized to extend the results from a single orbit to a family of orbits. Quasi-halo and Lissajous families of the Sun–Earth RTBP around the L2 libration point are obtained via this method. Results are compared with the existing literature. A numerical method to transform these orbits from the RTBP model to the real ephemeris model of the Solar System is introduced and applied.     

Periodic orbits in the photogravitational restricted problem with the smaller primary an oblate body

In this paper, we have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when more massive body is a source of radiation and the smaller primary is an oblate body. We have determined periodic orbits for fixed values of μ, σ and different values of p and h (μ mass ratio of the two primaries, σ oblate parameter, p radiation parameter and h energy constant). These orbits have been determined by giving displacements along the tangent and normal to the mobile co-ordinates as defined by Karimov and Sokolsky (in Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of radiation pressure on the periodic orbits by taking some fixed values of μ and σ.     

Periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries is an oblate body

We have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries is an oblate body. We have determined the periodic orbits for different values of μ, h and A (h is energy constant, μ is mass ratio of the two primaries and A is an oblateness factor). These orbits have been determined by giving displacements along the tangent and normal to the mobile coordinates as defined by Karimov and Sokolsky (Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of oblateness by taking some fixed values of μ, A and h. As starters for our method, we use some known periodic orbits in the classical restricted three body problem.     

Area-preserving Poincaré mappings of the unit disk

The best way to investigate the long-time behaviour of dynamical systems is to introduce an appropriate Poincaré mapping P and study its iterates.Two cases of physical interest arise: Conservative and dissipative systems. While the latter has been considered by a great many authors, much less is known for the first one (according to Liouville's theorem, here the mapping leaves a certain measure in phase space invariant). In this paper, we concentrate our attention on compact phase spaces (or, rather, surfaces of section). This assumption is mathematically useful and physically reasonable.We consider the simplest possible (2-dimensional) systems whehre the phase space is the compact unit disk D in ². A family of simple area-preserving mappings from D onto itselves will be given and discussed in detail.It is shown that general characteristics of the dynamics are quite similar to those of e.g. the Hénon-Heiles system, while other features, as the structure of invariant curves, are different.     

Periodic orbits of the generalized Friedmann–Robertson–Walker Hamiltonian systems

The averaging theory of first order is applied to study a generalization of the Friedmann–Robertson–Walker Hamiltonian systems with three parameters. We provide sufficient conditions on the three parameters of the generalized system to guarantee the existence of continuous families of periodic orbits parameterized by the energy, and these families are given up to first order in a small parameter.     

The Hori–Deprit method for averaged motion equations of the planetary problem in elements of the second Poincaré system

We consider an algorithm to construct averaged motion equations for four-planetary systems by means of the Hori–Deprit method. We obtain the generating function of the transformation, change-variable functions and right-hand sides of the equations of motion in elements of the second Poincaré system. Analytical computations are implemented by means of the Piranha echeloned Poisson processor. The obtained equations are to be used to investigate the orbital evolution of giant planets of the Solar system and various extrasolar planetary systems.     

Consecutive collision orbits in the limiting case μ=0 of the elliptic restricted problem

A timing condition for consecutive collision orbits in the planar, circular three-body problem has been extended to the elliptic restricted problem for =0. The expression developed relates eccentric anomalies at the time of collision. Some families of solutions are presented.     

Computing periodic orbits of the three-body problem: Effective convergence of Newton’s method on the surface of section

We consider Newton’s method for computing periodic orbits of dynamical systems as fixed points on a surface of section and seek to clarify and evaluate the method’s uncertainty of convergence. Several fixed points of various multiplicities, both stable and unstable are computed in a new version of Hill’s problem. Newton’s method is applied with starting points chosen randomly inside the maximum possible—for any method—circle of convergence. The employment of random starting points is continued until one of them leads to convergence, and the process is repeated a thousand times for each fixed point. The results show that on average convergence occurs with very few starting points and non-converging iterations being wasted.     

Combined effects of perturbations, radiation and oblateness on the periodic orbits in the restricted three-body problem

This paper investigates the periodic orbits around the triangular equilibrium points for 0<μ<μ _c , where μ _c is the critical mass value, under the combined influence of small perturbations in the Coriolis and the centrifugal forces respectively, together with the effects of oblateness and radiation pressures of the primaries. It is found that the perturbing forces affect the period, orientation and the eccentricities of the long and short periodic orbits.     

Oscillatory orbits in the planar three‐body problem with equal masses

In the free‐fall three‐body problem, distributions of escape, binary, and triple collision orbits are obtained. Interpretation of the results leads us to the existence of oscillatory orbits in the planar three‐body problem with equal masses. A scenario to prove their existence is described. This revised version was published online in July 2006 with corrections to the Cover Date.     

Periodic solutions about the ‘out of plane’ equilibrium points in the photogravitational restricted three-body problem

The regions of stability for the out of plane equilibrium points of the photogravitational restricted three-body problem are given. Second order expansions of periodic solutions around these points are constructed and the corresponding families are computed. It is found that two such families exist. One of them originates and terminates on the same equilibrium point while the other terminates by flattening on the orbital plane.     

On the existence of periodic solutions of Poincaré's second sort in the general problem of three bodies moving in plane

In his Méthodes Nouvelles de la Mécanique Cèlèste (Sections 42–47), Poincaré envisaged the existence of a class (the second sort) of periodic solutions of the general problem of three bodies, arising by analytical continuation from unperturbed elliptic motion of two bodies about a primary, in which the two orbits are of commensurable period, and of non-zero eccentricity. Existence proofs have been given for the restricted problem case (in which the mass of the third body remains zero) by Arenstorf and Barrar, and for the lunar theory case by Arenstorf. A proof is offered here for the general problem, making use of the symmetry of the equations of motion (with use of the Mirror Theorem of Roy and Ovenden) in an extension of the approach used by Barrar for the restricted problem, and using a device proposed by Poincaré himself which enables the extension to the general problem to be made.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.     

On the elimination of the critical terms of a first order theory of Jupiter perturbed by Saturn carried out through Hori's method and Poincaré canonical variables

The elimination of the critical terms inside the Hamiltonian of a first order theory of Jupiter perturbed by Saturn is carried out through the Poincaré canonical variables and the Hori's procedure. Powers of the eccentricities and the sines of inclinations which are>3 are neglected. The Poincaré variablesL ₁,H ₁,P ₁, ₁,K ₁,Q ₁ of Jupiter which result from a previous elimination of the short period terms are expressed in terms of the Poincaré canonical variablesL _u ,H _u ,P _u , _u ,Q _u ;u=1, 2; index 1 Jupiter, index 2 Saturn resulting from the elimination of the short period and critical terms. The differential equations inH _u ,P _u ,K _u ,Q _u are solved through the method of Lagrange and the analytical expressions ofL ₁,H ₁,P ₁, ₁,K ₁,Q ₁ as functions of timet are finally obtained.     

The ‘Halo’ family of 3-dimensional periodic orbits in the Earth-Moon restricted 3-body problem

The Halo orbits originating in the vicinities of both,L ₁ andL ₂ grow larger, but shorter in period, as they shift towards the Moon. There is in each case a narrow band of stable orbits roughly half-way to the Moon. Nearer to the Moon, the orbits are fairly well-approximated by an almost rectilinear analysis. TheL ₂ family shrinks in size as it approaches the Moon, becoming stable again shortly before penetrating the lunar surface. TheL ₁-family becomes longer and thinner as it approaches the Moon, with a second narrow band of stable orbits with perilune, however, below the lunar surface.     

Periodic orbits of solar sail equipped with reflectance control device in Earth–Moon system

In this paper, families of Lyapunov and halo orbits are presented with a solar sail equipped with a reflectance control device in the Earth–Moon system. System dynamical model is established considering solar sail acceleration, and four solar sail steering laws and two initial Sun-sail configurations are introduced. The initial natural periodic orbits with suitable periods are firstly identified. Subsequently, families of solar sail Lyapunov and halo orbits around the \(L_{1}\) and \(L_{2}\) points are designed with fixed solar sail characteristic acceleration and varying reflectivity rate and pitching angle by the combination of the modified differential correction method and continuation approach. The linear stabilities of solar sail periodic orbits are investigated, and a nonlinear sliding model controller is designed for station keeping. In addition, orbit transfer between the same family of solar sail orbits is investigated preliminarily to showcase reflectance control device solar sail maneuver capability.     

A study of halo orbits at the Sun–Mars L1 Lagrangian point in the photogravitational restricted three-body problem

Photogravitational Restricted Three-Body Problem (PGRTBP) is considered and halo orbits are generated in the vicinity of the Sun–Mars L₁ Lagrangian point. Deviation of properties such as time period, size and velocity variation in the halo orbits with Sun as a source of radiation are discussed. With increase in solar radiation pressure, the halo orbits are found to elongate and move towards the Sun and the time period of the halo orbits is found to increase. The variation in the behaviour of invariant manifolds with change in radiation pressure is also computed and it is found that as the radiation pressure increases, the transition from Mars-centric path to heliocentric path is delayed. Certain implications of the velocity profile of the invariant manifolds are also discussed.     

New doubly-symmetric families of comet-like periodic orbits in the spatial restricted (N + 1)-body problem

For any positive integer N ≥ 2 we prove the existence of a new family of periodic solutions for the spatial restricted (N +1)-body problem. In these solutions the infinitesimal particle is very far from the primaries. They have large inclinations and some symmetries. In fact we extend results of Howison and Meyer (J. Diff. Equ. 163:174–197, 2000) from N = 2 to any positive integer N ≥ 2.      

Element sets for high-order Poincaré mapping of perturbed Keplerian motion

The propagation and Poincaré mapping of perturbed Keplerian motion is a key topic in Celestial Mechanics and Astrodynamics, e.g., to study the stability of orbits or design bounded relative trajectories. The high-order transfer map (HOTM) method enables efficient mapping of perturbed Keplerian orbits using the high-order Taylor expansion of a Poincaré or stroboscopic map. The HOTM is only accurate close to the expansion point and therefore the number of revolutions for which the map is accurate tends to be limited. The proper selection of coordinates is of key importance for improving the performance of the HOTM method. In this paper, we investigate the use of different element sets for expressing the high-order map in order to find the coordinates that perform best in terms of accuracy. A new set of elements is introduced that enables extremely accurate mapping of the state, even for high eccentricities and higher-order zonal perturbations. Finally, the high-order map is shown to be very useful for the determination and study of fixed points and center manifolds of Poincaré maps.     

Study of the transfer between libration point orbits and lunar orbits in Earth–Moon system

This paper is devoted to the study of the transfer problem from a libration point orbit of the Earth–Moon system to an orbit around the Moon. The transfer procedure analysed has two legs: the first one is an orbit of the unstable manifold of the libration orbit and the second one is a transfer orbit between a certain point on the manifold and the final lunar orbit. There are only two manoeuvres involved in the method and they are applied at the beginning and at the end of the second leg. Although the numerical results given in this paper correspond to transfers between halo orbits around the \(L_1\) point (of several amplitudes) and lunar polar orbits with altitudes varying between 100 and 500 km, the procedure we develop can be applied to any kind of lunar orbits, libration orbits around the \(L_1\) or \(L_2\) points of the Earth–Moon system, or to other similar cases with different values of the mass ratio.