Leaving the Moon by means of invariant manifolds of libration point orbits

The aim of this work is to look for rescue trajectories that leave the surface of the Moon, belonging to the hyperbolic manifolds associated with the central manifold of the Lagrangian points L₁ and L₂ of the Earth–Moon system. The model used for the Earth–Moon system is the Circular Restricted Three-Body Problem. We consider as nominal arrival orbits halo orbits and square Lissajous orbits around L₁ and L₂ and we show, for a given Δv, the regions of the Moon’s surface from which we can reach them. The key point of this work is the geometry of the hyperbolic manifolds associated with libration point orbits. Both periodic/quasi-periodic orbits and their corresponding stable invariant manifold are approximated by means of the Lindstedt–Poincaré semi-analytical approach.     

Periodic Image Trajectories in Earth–Moon Space

The problem of identifying orbits that enclose both the Earth and the Moon in a predictable way has theoretical relevance as well as practical implications. In the context of the restricted three-body problem with primaries in circular orbits, periodic trajectories exist and have the property that a third body (e.g. a spacecraft) can describe them indefinitely. Several approaches have been employed in the past for the purpose of identifying similar orbits. In this work the theorem of image trajectories, proven five decades ago, is employed for determining periodic image trajectories in Earth–Moon space. These trajectories exhibit two fundamental features: (i) counterclockwise departure from a perigee on the far side of the Earth, and (ii) counterclockwise arrival to a periselenum on the far side of the Moon. An extensive, systematic numerical search is performed, with the use of a modified Poincaré map, in conjunction with a numerical refinement process, and leads to a variety of periodic orbits, with various interesting features for possible future lunar missions.     

High-order solutions around triangular libration points in the elliptic restricted three-body problem and applications to low energy transfers

High-order series expansions around triangular libration points in the elliptic restricted three-body problem (ERTBP) are constructed first, and then with the aid of the series solutions, two-impulse and low-thrust low energy transfers to the triangular point orbits of the Earth–Moon system are designed in this paper. The equations of motion of ERTBP in the pulsating synodic reference frame have the same symmetries as the ones in circular restricted three-body problem (CRTBP), and also have five equilibrium points. Considering the stable dynamics of triangular libration points, the analytical solutions of the motion around them in ERTBP are expressed as formal series of four amplitudes: the orbital eccentricity of the primary, the long, short and vertical periodic amplitudes. The series expansions truncated at arbitrary order are constructed by means of Lindstedt–Poincaré method, and then the quasi-periodic orbits around triangular libration points in ERTBP can all be parameterized. In particular, when the eccentricity of the primary is zero, the series expansions constructed can be reduced to describe the motion around triangular libration points in CRTBP. In order to check the validity of the series expansions constructed, the domain of convergence corresponding to different orders is studied by using numerical integration. After obtaining the analytical solutions of the bounded orbits around triangular points, the target orbits in practical missions can be expressed by several related parameters. Thanks to the series expansions constructed, two missions are planned to transfer a spacecraft from the Earth to the short periodic orbits around triangular libration points of Earth–Moon system. To complete the missions with less fuel cost, low energy transfers (two-impulse and low-thrust) are investigated by means of numerical optimization methods (both global and local optimization techniques). Simulation results indicate that (a) the low-thrust, low energy transfers outperform the corresponding two-impulse, low energy transfers in terms of propellant fraction, and (b) compared with the traditional Hohmann-like transfers, both the two-impulse, low energy and low-thrust, low energy transfers perform very efficiently, at the cost of flight time.     

George Darwin's lectures on Hill's lunar theory

In the late 1870s two papers were published in which G W Hill presented a novel method for calculating the orbital motion of the Moon. These much-acclaimed papers were a turning point in the history of the three-body problem and proved inspirational to mathematicians such as Poincaré and G H Darwin in their work on periodic orbits.

Two decades later Darwin gave a series of lectures on Hill's lunar theory at the University of Cambridge. His lectures were an attempt to make Hill's theory more accessible, particularly to students of astronomy, and also mark the beginning of his own research in the field. This article, which expands upon a talk first given at the BSHM's Research in Progress meeting in March 2008, describes both the content and context of Darwin's lectures.     

Efficient invariant-manifold, low-thrust planar trajectories to the Moon

Two-impulse trajectories as well as mixed invariant-manifold, low-thrust efficient transfers to the Moon are discussed. Exterior trajectories executing ballistic lunar capture are formalized through the definition of special attainable sets. The coupled restricted three-body problems approximation is used to design appropriate first guesses for the subsequent optimization. The introduction of the Moon-perturbed Sun-Earth restricted four-body problem allows us to formalize the idea of ballistic escape from the Earth, and to take explicitly advantage of lunar fly-by. Accurate first guess solutions are optimized, through a direct method approach and multiple shooting technique.     

Comparisons of integrators on a diverse collection of restricted three-body test problems

The restricted three-body problem has many important astronomicalapplications. We present a diverse collection of 28 symmetricand asymmetric, stable and unstable periodic orbits for thetwo- and three-dimensional problems and compare the performanceof eight general purpose integrators on the orbits.     

Existence and differential geometric properties of continuous families of periodic three-body motions with non-uniform mass distributions

Using the method of analytic continuation in an equivariant differential geometric setting, we exhibit two interesting families of vanishing angular momentum periodic orbits for the Newtonian three-body problem with non-uniform mass distributions having two equal masses which connect at the celebrated figure-8 orbit, exhibited by A. Chenciner and R. Montgomery (2000) in the case of equal masses, and yield a continuous family of periodic three-body motions in the plane.At one end of the family, when the two equal masses are infinitesimal and the third one reaches the value of +1, we arrive at a solution of a double Kepler problem; at the other end of the family, when the third mass is infinitesimal, we have a special case of periodic solution of a restricted three-body problem.     

Isolating blocks near the collinear relative equilibria of the three-body problem

The collinear relative equilibrium solutions are among the few explicitly known periodic solutions of the Newtonian three-body problem. When the energy and angular momentum constants are varied slightly, these unstable periodic orbits become normally hyperbolic invariant spheres whose stable and unstable manifolds form separatrices in the integral manifolds. The goal of this paper is to construct simple isolating blocks for these invariant spheres analogous to those introduced by Conley in the restricted three-body problem. This allows continuation of the invariant set and the separatrices to energies and angular momenta far from those of the relative equilibrium.

     

On the conditions under which a satellite orbit intersects the surface of a central body of finite radius in the restricted double-averaged three-body problem

This paper is concerned with the applied problem of choosing long-living orbits of artificial Earth satellites whose evolution under the influence of gravitational perturbation from the Moon and the Sun may result in the collision of the satellite with the central body, as was shown by M.L. Lidov for the well-known example of “Vertical Moon.” We use solutions of the completely integrable system of evolution equations obtained by Lidov in 1961 by averaging twice the spatial circular restricted three-body problem in the Hill approximation. In order to apply the integrability of this problem in practice, we study the foliation of the manifold of levels of first integrals and the change of motion under crossing the bifurcation manifolds separating the foliated cells. As a result, we describe the manifold of initial conditions under which the orbit evolution leads to an inevitable collision of the satellite with the central body. We also find a lower bound for the practical applicability of the results, which is determined by the presence of gravitational perturbations caused by a polar flattening of the central body.     

From the restricted to the full three-body problem

The three-body problem with all the classical integrals fixed and all the symmetries removed is called the reduced three-body problem. We use the methods of symplectic scaling and reduction to show that the reduced planar or spatial three-body problem with one small mass is to the first approximation the product of the restricted three-body problem and a harmonic oscillator. This allows us to prove that many of the known results for the restricted problem have generalizations for the reduced three-body problem.

For example, all the non-degenerate periodic solutions, generic bifurcations, Hamiltonian-Hopf bifurcations, bridges and natural centers known to exist in the restricted problem can be continued into the reduced three-body problem. The classic normalization calculations of Deprit and Deprit-Bartholomé show that there are two-dimensional KAM invariant tori near the Lagrange point in the restricted problem. With the above result this proves that there are three-dimensional KAM invariant tori near the Lagrange point in the reduced three-body problem.     

Continuation of relative periodic orbits in a class of triatomic Hamiltonian systems

We study relative periodic orbits (i.e. time-periodic orbits in a frame rotating at constant velocity) in a class of triatomic Euclidean-invariant (planar) Hamiltonian systems. The system consists of two identical heavy atoms and a light one, and the atomic mass ratio is treated as a continuation parameter. Under some nondegeneracy conditions, we show that a given family of relative periodic orbits existing at infinite mass ratio (and parametrized by phase, rotational degree of freedom and period) persists for sufficiently large mass ratio and for nearby angular velocities (this result is valid for small angular velocities). The proof is based on a method initially introduced by Sepulchre and MacKay [J.-A. Sepulchre, R.S. MacKay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity 10 (1997) 679–713] and further developed by Muñoz-Almaraz et al. [F.J. Muñoz-Almaraz, et al., Continuation of periodic orbits in conservative and Hamiltonian systems, Physica D 181 (2003) 1–38] for the continuation of normal periodic orbits in Hamiltonian systems. Our results provide several types of relative periodic orbits, which extend from small amplitude relative normal modes [J.-P. Ortega, Relative normal modes for nonlinear Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 665–704] up to large amplitude solutions which are not restrained to a small neighborhood of a stable relative equilibrium. In particular, we show the existence of large amplitude motions of inversion, where the light atom periodically crosses the segment between heavy atoms. This analysis is completed by numerical results on the stability and bifurcations of some inversion orbits as their angular velocity is varied.     

Existence of either a periodic collisional orbit or infinitely many consecutive collision orbits in the planar circular restricted three-body problem

Mathematische Zeitschrift - In the restricted three-body problem, consecutive collision orbits are those orbits which start and end at collisions with one of the primaries. Interests for such...     

On the conditions under which a satellite orbit intersects the surface of a central body of finite radius in the restricted double-averaged three-body problem

This paper is concerned with the applied problem of choosing long-living orbits of artificial Earth satellites whose evolution under the influence of gravitational perturbation from the Moon and the Sun may result in the collision of the satellite with the central body, as was shown by M.L. Lidov for the well-known example of “Vertical Moon.” We use solutions of the completely integrable system of evolution equations obtained by Lidov in 1961 by averaging twice the spatial circular restricted three-body problem in the Hill approximation. In order to apply the integrability of this problem in practice, we study the foliation of the manifold of levels of first integrals and the change of motion under crossing the bifurcation manifolds separating the foliated cells. As a result, we describe the manifold of initial conditions under which the orbit evolution leads to an inevitable collision of the satellite with the central body. We also find a lower bound for the practical applicability of the results, which is determined by the presence of gravitational perturbations caused by a polar flattening of the central body. Original Russian Text ? V.I. Prokhorenko, 2007, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 156–173.     

Numerical Continuation of Hamiltonian Relative Periodic Orbits

The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years, there has been rapid progress in the development of a bifurcation theory for dynamical systems with structure, such as symmetry or symplecticity. But as yet, there are few results on the numerical computation of those bifurcations. The methods we present in this paper are a first step toward a systematic numerical analysis of generic bifurcations of Hamiltonian symmetric periodic orbits and relative periodic orbits (RPOs). First, we show how to numerically exploit spatio-temporal symmetries of Hamiltonian periodic orbits. Then we describe a general method for the numerical computation of RPOs persisting from periodic orbits in a symmetry breaking bifurcation. Finally, we present an algorithm for the numerical continuation of non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair. Our path following algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a tangential continuation method with implicit reparametrization. We apply our methods to continue the famous figure eight choreography of the three-body system. We find a relative period doubling bifurcation of the planar rotating eight family and compute the rotating choreographies bifurcating from it.      

Billiards and Nonholonomic Distributions

In this note, we consider billiards with full families of periodic orbits. It is shown that the construction of a convex billiard with a “rational” caustic (i.e., carrying only periodic orbits) can be reformulated as a problem of finding a closed curve tangent to an (N - 1)-dimensional distribution on a (2N - 1)-dimensional manifold. We describe the properties of this distribution, as well as some important consequences for billiards with rational caustics. A very particular application of our construction states that an ellipse can be infinitesimally perturbed so that any chosen rational elliptic caustic will persist. Bibliography: 13 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 56–64.     

The -limit sets of a flow and periodic orbits

In this paper we discuss the ω-limit sets of a flow using the Conley theory, chain recurrence and Morse decompositions. Our results generalize and improve the related result in [Schropp J. A reduction principle for ω-limit sets. Z Angew Math Meth 1996;76(6):349–56], and we also show how they can be used as a basis for some new criteria for the existence of periodic orbits.     

Minimal dynamical system with givenD-function and topological entropy

TheD-function is a new topological invariant introduced by the author in [3] to classify the minimal dynamical system and to generalize Sharkovskii's theorem on the coexistence of periodic orbits. We show that theD-function and the topological entropy are independent.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 2, pp. 287–292, February, 1993.     

On Hamiltonian Systems with a Homoclinic Orbit to a Saddle-Center

We consider a real analytic Hamiltonian system with two degrees of freedom having a homoclinic orbit to a saddle-center equilibrium (two nonzero real and two nonzero imaginary eigenvalues). We take a two-parameter unfolding for such a system and show that in the nonresonance case, there are countable sets of multi-round homoclinic orbits to a saddle-center. We also find families of periodic orbits accumulating at homoclinic orbits. Bibliography: 6 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 187–193.     

Quasi‐Periodic Almost‐Collision Orbits in the Spatial Three‐Body Problem

In a system of particles, quasi‐periodic almost‐collision orbits are collisionless orbits along which two bodies become arbitrarily close to each other—the lower limit of their distance is zero but the upper limit is strictly positive—and are quasi‐periodic in a regularized system up to a change of time. Their existence was shown in the restricted planar circular three‐body problem by A.~Chenciner and J. Llibre, and in the planar three‐body problem by J. Féjoz. In the spatial three‐body problem, the existence of a set of positive measure of such orbits was predicted by C. Marchal. In this article, we present a proof of this fact.© 2015 Wiley Periodicals, Inc.     

An analytic estimation for the instability interval near second order resonances in the restricted three-body problem

Summary Periodic orbits of the first kind of the planar circular restricted three-body problem are linearly stable except in the neighbourhood of second order resonances, where an interval of unstable orbits appears when the mass parameter is not zero. Our main result is the estimation of the length of the instability interval.