On the elliptic restricted three-body problem

The main goal of this paper is to show that the elliptic restricted three-body problem has ejection-collision orbits when the mass parameter µ is small enough. We make use of the blow up techniques. Moreover, we describe the global flow of the elliptic problem when µ = 0 taking into account the singularities due to collision and to infinity.     

Bounded orbits in the elliptic restricted three-body problem

Special solutions of the planar rectilinear elliptic restricted 3-body problem are investigated for the limiting case e=1. Numerical integration is performed for primaries of equal masses. Starting values which define circular orbit solutions lead to bounded solutions if the initial radius a₀ is larger than 3.74 in units of the primaries' semimajor axis a. A comparison with the Eulerian two-fixedcentre problem is presented in order to understand qualitatively the characteristic features of bounded orbits and the transition to escape orbits.     

From the circular to the spatial elliptic restricted three-body problem

We deal with the study of the spatial restricted three-body problem in the case where the small particle is far from the primaries, that is, the so-called comet case. We consider the circular problem, apply double averaging and compute the relative equilibria of the reduced system. It appears that, in the circular problem, we find not only part of the equilibria existing in the elliptic case, but also new ones. These critical points are in correspondence with periodic and quasiperiodic orbits and invariant tori of the non-averaged Hamiltonian. We explain carefully the transition between the circular and the elliptic problems. Moreover, from the relative equilibria of elliptic type, we obtain invariant 3-tori of the original system.     

Integrals of motion in the elliptic three-dimensional restricted three-body problem

Three integrals of motion have been found in the three-dimensional elliptic restricted three-body problem for small eccentricitye of the relative orbit of the primaries and small distancer and eccentricitye of the orbit of the third body around a primary. The integrals are given in the form of formal series in the mass-ratio , the eccentricitiese, e and the coordinates and velocities. These integrals depend periodically on the time.     

Second-species solutions in the circular and elliptic restricted three-body problem

An analytical proof of the existence of some kinds of periodic orbits of second species of Poincaré, both in the Circular and Elliptic Restricted three-body problem, is given for small values of the mass parameter. The proof uses the asymptotic approximations for the solutions and the matching theory developed by Breakwell and Perko. In the paper their results are extended to the Elliptic problem and applied to prove the existence of second-species solutions generated by rectilinear ellipses in the Circular problem and nearly-rectilinear ones in the Elliptic case.     

Integration of the elliptic restricted three-body problem with Lie series

The method of Lie series is used to construct a solution for the elliptic restricted three body problem. In a synodic pulsating coordinate system, the Lie operator for the motion of the third infinitesimal body is derived as function of coordinates, velocities and true anomaly of the primaries. The terms of the Lie series for the solution are then calculated with recurrence formulae which enable a rapid successive calculation of any desired number of terms. This procedure gives a very useful analytical form for the series and allows a quick calculation of the orbit.The project is supported by the Austrian Fonds zur Förderung der wissénschaftlichen Forschung under Project No. 4471.     

Lagrangian coherent structures in the planar elliptic restricted three-body problem

This study investigates Lagrangian coherent structures (LCS) in the planar elliptic restricted three-body problem (ER3BP), a generalization of the circular restricted three-body problem (CR3BP) that asks for the motion of a test particle in the presence of two elliptically orbiting point masses. Previous studies demonstrate that an understanding of transport phenomena in the CR3BP, an autonomous dynamical system (when viewed in a rotating frame), can be obtained through analysis of the stable and unstable manifolds of certain periodic solutions to the CR3BP equations of motion. These invariant manifolds form cylindrical tubes within surfaces of constant energy that act as separatrices between orbits with qualitatively different behaviors. The computation of LCS, a technique typically applied to fluid flows to identify transport barriers in the domains of time-dependent velocity fields, provides a convenient means of determining the time-dependent analogues of these invariant manifolds for the ER3BP, whose equations of motion contain an explicit dependency on the independent variable. As a direct application, this study uncovers the contribution of the planet Mercury to the Interplanetary Transport Network, a network of tubes through the solar system that can be exploited for the construction of low-fuel spacecraft mission trajectories. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Families of symmetric-periodic orbits in the elliptic three-dimensional restricted three-body problem

With an orbit of the three-dimensional circular problem as a starting point, we have calculated families of symmetric-periodic orbits in the three-dimensional elliptic problem with a variation of the mass ratio and the eccentricitye. Afterwards, we have studied their evolution and stability.     

Infinitesimal orbits around lagrange points in the elliptic,restricted three-body problem

This study presents a method of obtaining asymptotic approximations for motions near a Lagrange point in the planar, elliptic, restricted three-body problem by using a von Zeipel-type method. The calculations are carried out for a second-order escape solution in the proximity of the equilateral Lagrange point, L ₄, where the primaries' orbital eccentricity is taken as the small parameter .     

Out-of-plane equilibrium points in the elliptic restricted three-body problem under albedo effect

The aim of this research is to show the significant effects of albedo on the existence of out-of-plane equilibria in the elliptic restricted three-body problem under an oblate primary model. We computed out-of-plane equilibria numerically and graphically for different values of the parameters μ, α, e, k and σ where μ, α, e, k and σ are mass parameter, albedo factor, eccentricity, ratio of the luminosity of smaller primary to luminosity of bigger primary considered as constant and oblateness factor due to smaller primary, respectively. Further, we examined the stability of out-of-plane equilibria and found that these equilibria are unstable in linear sense for all parameters μ, α, e, k and σ. Finally, the three-dimensional periodic orbits are analyzed for different values of albedo factor α and oblateness factor σ.     

A note on independent variables for restricted three-body problems

In studies of the elliptic restricted three-body problem, the true anomaly of the motion of the primaries is often used as the independent variable. The equations of motion then show invariancy in form from the circular case. It is of interest whether other independent variables exist, such that the invariant form of the equations is maintained. It is found that true anomaly is the only such variable.     

Connection between Hill stability and weak stability in the elliptic restricted three-body problem

This paper provides a study on the connection between Hill stability and weak stability in the framework of the spatial elliptic restricted three-body problem. We determine a necessary condition for weak stability by giving an upper and a lower bound of qualitative measure of the Hill stability. The sufficient condition for weak stability and the symmetry of weak stable regions around the planets of the Solar System is also investigated.     

Artificial equilibrium points for a generalized sail in the elliptic restricted three-body problem

Different types of propulsion systems with continuous and purely radial thrust, whose modulus depends on the distance from a massive body, may be conveniently described within a single mathematical model by means of the concept of generalized sail. This paper discusses the existence and stability of artificial equilibrium points maintained by a generalized sail within an elliptic restricted three-body problem. Similar to the classical case in the absence of thrust, a generalized sail guarantees the existence of equilibrium points belonging only to the orbital plane of the two primaries. The geometrical loci of existing artificial equilibrium points are shown to coincide with those obtained for the circular three body problem when a non-uniformly rotating and pulsating coordinate system is chosen to describe the spacecraft motion. However, the generalized sail has to provide a periodically variable acceleration to maintain a given artificial equilibrium point. A linear stability analysis of the artificial equilibrium points is provided by means of the Floquet theory.     

Remarks on the photogravitational restricted three-body problem

The effects of the radiation pressure in the restricted three-body problem are considered and the existence of the out-of-plane equilibrium points is analyzed. It is found that within the framework of the stellar stability, the five Lagrangian points are the only equilibrium points, at least as far as the force of the radiation pressure is taken into account.     

A parametric study of stability and resonances around L 4 in the elliptic restricted three-body problem

The size of the stable region around the Lagrangian point L ₄ in the elliptic restricted three-body problem is determined by numerical integration as a function of the mass parameter and eccentricity of the primaries. The size distribution of the stable regions in the mass parameter-eccentricity plane shows minima at certain places that are identified with resonances between the librational frequencies of motions around L ₄. These are computed from an approximate analytical equation of Rabe relating the frequency, mass parameter and eccentricity. Solutions of this equation are determined numerically and the global behaviour of the frequencies depending on the mass parameter and eccentricity is shown and discussed. The minimum sizes of the stable regions around L ₄ change along the resonances and the relative strength of the resonances is analysed. Applications to possible Trojan exoplanets are indicated. Escape from L ₄ is also investigated.