Analytic estimates and topological properties of the weak stability boundary

The weak stability boundary (WSB) is the transition region of the phase space where the change from gravitational escape to ballistic capture occurs. Studies on this complicated region of chaotic motion aim to investigate its unique, fuel saving properties to enlarge the frontiers of low energy transfers. This “fuzzy stability” region is characterized by highly sensitive motion, and any analysis of it has been carried out almost exclusively using numerical methods. On the contrary this paper presents, for the planar circular restricted 3 body problem (PCR3BP), 1) an analytic definition of the WSB which is coherent with the known algorithmic definitions; 2) a precise description of the topology of the WSB; 3) analytic estimates on the “stable region” (nearby the smaller primary) whose boundary is, by definition, the WSB

A Dynamical Systems Approach to Schwarzschild Null Geodesics

The null geodesics of a Schwarzschild black hole are studied from a dynamical systems perspective. Written in terms of Kerr-Schild coordinates, the null geodesic equation takes on the simple form of a particle moving under the influence of a Newtonian central force with an inverse-cubic potential. We apply a McGehee transformation to these equations, which clearly elucidates the full phase space of solutions. All the null geodesics belong to one of four families of invariant manifolds and their limiting cases, further characterized by the angular momentum L of the orbit: for |L|>|L_c|, (1) the set that flow outward from the white hole, turn around, then fall into the black hole, (2) the set that fall inward from past null infinity, turn around outside the black hole to continue to future null infinity, and for |L|<|L_c|, (3) the set that flow outward from the white hole and continue to future null infinity, (4) the set that flow inward from past null infinity and into the black hole. The critical angular momentum Lc corresponds to the unstable circular orbit at r=3M, and the homoclinic orbits associated with it. There are two additional critical points of the flow at the singularity at r=0. Though the solutions of geodesic motion and Hamiltonian flow we describe here are well known, what we believe is a novel aspect of this work is the mapping between the two equivalent descriptions, and the different insights each approach can give to the problem. For example, the McGehee picture points to a particularly interesting limiting case of the class (1) that move from the white to black hole: in the limit as L goes to infinity, as described in Schwarzschild coordinates, these geodesics begin at r=0, flow along t=constant lines, turn around at r=2M, then continue to r=0. During this motion they circle in azimuth exactly once, and complete the journey in zero affine time.Comment: 14 pages, 3 Figure

Geometry of Weak Stability Boundaries

The notion of a weak stability boundary has been successfully used to design low energy trajectories from the Earth to the Moon. The structure of this boundary has been investigated in a number of studies, where partial results have been obtained. We propose a generalization of the weak stability boundary. We prove analytically that, in the context of the planar circular restricted three-body problem, under certain conditions on the mass ratio of the primaries and on the energy, the weak stability boundary about the heavier primary coincides with a branch of the global stable manifold of the Lyapunov orbit about one of the Lagrange points

Two Dynamical Classes of Centaurs

The Centaurs are a transient population of small bodies in the outer solar system whose orbits are strongly chaotic. These objects typically suffer significant changes of orbital parameters on timescales of a few thousand years, and their orbital evolution exhibits two types of behaviors described qualitatively as random-walk and resonance-sticking. We have analyzed the chaotic behavior of the known Centaurs. Our analysis has revealed that the two types of chaotic evolution are quantitatively distinguishable: (1) the random walk-type behavior is well described by so-called generalized diffusion in which the rms deviation of the semimajor axis grows with time t as ~t^H, with Hurst exponent H in the range 0.22--0.95, however (2) orbital evolution dominated by intermittent resonance sticking, with sudden jumps from one mean motion resonance to another, has poorly defined H. We further find that these two types of behavior are correlated with Centaur dynamical lifetime: most Centaurs whose dynamical lifetime is less than ~22 Myr exhibit generalized diffusion, whereas most Centaurs of longer dynamical lifetimes exhibit intermittent resonance sticking. We also find that Centaurs in the diffusing class are likely to evolve into Jupiter-family comets during their dynamical lifetimes, while those in the resonance-hopping class do not.Comment: 31 pages, including 12 figures and 2 tables. Accepted for publication in Icaru

On Optimal Two-Impulse Earth-Moon Transfers in a Four-Body Model

In this paper two-impulse Earth-Moon transfers are treated in the restricted four-body problem with the Sun, the Earth, and the Moon as primaries. The problem is formulated with mathematical means and solved through direct transcription and multiple shooting strategy. Thousands of solutions are found, which make it possible to frame known cases as special points of a more general picture. Families of solutions are defined and characterized, and their features are discussed. The methodology described in this paper is useful to perform trade-off analyses, where many solutions have to be produced and assessed

Cantor Set Structure of the Weak Stability Boundary for Infinitely Many Cycles in the Restricted Three-Body Problem

The geometry of the weak stability boundary region for the planar restricted three-body problem about the secondary mass point has been an open problem. Previous studies have conjectured that it may have a fractal structure. In this paper, this region is studied for infinitely many cycles about the secondary mass point, instead of a finite number studied previously. It is shown that in this case the boundary consists of a family of infinitely many Cantor sets and is thus fractal in nature. It is also shown that on two-dimensional surfaces of section, it is the boundary of a region only having bounded cycling motion for infinitely many cycles, while the complement of this region generally has unbounded motion. It is shown that that this shares many properties of a Mandelbrot set. Its relationship to the non-existence of KAM tori is described, among many other properties. Applications are discussed.Comment: 30 pages, 5 figure

Relativistic effects in the chaotic Sitnikov problem

We investigate the phase space structure of the relativistic Sitnikov problem in the first post-Newtonian approximation. The phase space portraits show a strong dependence on the gravitational radius which describes the strength of the relativistic pericentre advance. Bifurcations appearing at increasing the gravitational radius are presented. Transient chaotic behavior related to escapes from the primaries are also studied. Finally, the numerically determined chaotic saddle is investigated in the context of hyperbolic and non-hyperbolic dynamics as a function of the gravitational radius.Comment: 8 pages, 11 figure

Easily retrievable objects among the NEO population

Asteroids and comets are of strategic importance for science in an effort to understand the formation, evolution and composition of the Solar System. Near-Earth Objects (NEOs) are of particular interest because of their accessibility from Earth, but also because of their speculated wealth of material resources. The exploitation of these resources has long been discussed as a means to lower the cost of future space endeavours. In this paper, we consider the currently known NEO population and define a family of so-called Easily Retrievable Objects (EROs), objects that can be transported from accessible heliocentric orbits into the Earth’s neighbourhood at affordable costs. The asteroid retrieval transfers are sought from the continuum of low energy transfers enabled by the dynamics of invariant manifolds; specifically, the retrieval transfers target planar, vertical Lyapunov and halo orbit families associated with the collinear equilibrium points of the Sun-Earth Circular Restricted Three Body problem. The judicious use of these dynamical features provides the best opportunity to find extremely low energy Earth transfers for asteroid material. A catalogue of asteroid retrieval candidates is then presented. Despite the highly incomplete census of very small asteroids, the ERO catalogue can already be populated with 12 different objects retrievable with less than 500 m/s of Δv. Moreover, the approach proposed represents a robust search and ranking methodology for future retrieval candidates that can be automatically applied to the growing survey of NEOs