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## Über dieses Buch

This book presents recent advances in space and celestial mechanics, with a focus on the N-body problem and astrodynamics, and explores the development and application of computational techniques in both areas. It highlights the design of space transfers with various modes of propulsion, like solar sailing and low-thrust transfers between libration point orbits, as well as a broad range of targets and applications, like rendezvous with near Earth objects. Additionally, it includes contributions on the non-integrability properties of the collinear three- and four-body problem, and on general conditions for the existence of stable, minimum energy configurations in the full N-body problem.
A valuable resource for physicists and mathematicians with research interests in celestial mechanics, astrodynamics and optimal control as applied to space transfers, as well as for professionals and companies in the industry.

## Inhaltsverzeichnis

### Integrability and Non Integrability of Some n Body Problems

Abstract
We prove the non integrability of the colinear 3 and 4 body problem, for any positive masses. To deal with resistant cases, we present strong integrability criterions for 3 dimensional homogeneous potentials of degree $$-1$$, and prove that such cases cannot appear in the 4 body problem. Following the same strategy, we present a simple proof of non integrability for the planar n body problem. Eventually, we present some integrable cases of the n body problem restricted to some invariant vector spaces.
Thierry Combot

### Relative Equilibria in the Full N-Body Problem with Applications to the Equal Mass Problem

Abstract
General conditions for the existence of stable, minimum energy configurations in the full N-body problem are derived and investigated. Then the minimum energy and stable configurations for the spherical, equal mass full body problem are investigated for $$N = 2, 3, 4$$. This problem is defined as the dynamics of finite density spheres which interact gravitationally and through surface forces. This is a variation of the gravitational N-body problem in which the bodies are not allowed to come arbitrarily close to each other (due to their finite density), enabling the existence of resting configurations in addition to orbital motion. Previous work on this problem has outlined an efficient and simple way in which the stability of configurations in this problem can be defined. This methodology is reviewed and derived in a new approach and then applied to multiple body problems. In addition to this, new results on the Hill stability of these configurations are examined and derived. The study of these configurations is important for understanding the mechanics and morphological properties of small rubble pile asteroids. These results can also be generalized to other configurations of bodies that interact via field potentials and surface contact forces.
Daniel J. Scheeres

### Station Keeping Strategies for a Solar Sail in the Solar System

Abstract
In this paper we focus on the station keeping around an equilibrium point for a solar sail in the Earth-Sun system. The strategies that we present use the information on the dynamics of the system to derive the required changes on the sail orientation to remain close to an equilibrium point for a long time. We start by describing the main ideas when we consider the RTBP with the effect of the SRP as a model. Then we will see how to extend these ideas when we consider a more complex dynamical model which includes the gravitational attraction of the main bodies in the solar system. One of the goals of the paper is to check the robustness of the algorithms in a more realistic setting and study the effect of errors both in the position determination of the probe and in the orientation of the sail.

### Minimum Fuel Round Trip from a $$L_2$$ L 2 Earth-Moon Halo Orbit to Asteroid 2006 RH $$_{120}$$ 120

Abstract
The goal of this paper is to design a spacecraft round trip transfer from a parking orbit to asteroid 2006 RH$$_{120}$$ during its geocentric capture while maximizing the final spacecraft mass or, equivalently, minimizing the delta-v. The spacecraft begins in a halo “parking” orbit around the Earth-Moon $$L_2$$ libration point. The round-trip transfer is composed of three portions: the approach transfer from the parking orbit to 2006 RH$$_{120}$$, the rendezvous “lock-in” portion with the spacecraft in proximity to and following the asteroid orbit, and finally the return transfer to $$L_2$$. An indirect method based on the maximum principle is used for our numerical calculations. To partially address the issue of local minima we restrict the control strategy to reflect an actuation corresponding to up to three thrust arcs during each portion of the transfer. Our model is formulated in the circular restricted four-body problem (CR4BP) with the Sun considered as a perturbation of the Earth-Moon circular restricted three body problem. A shooting method is applied to numerically optimize the round trip transfer, and the 2006 RH$$_{120}$$ rendezvous and departure points are optimized using a time discretization of the 2006 RH$$_{120}$$ trajectory.
Monique Chyba, Thomas Haberkorn, Robert Jedicke

### Low-Thrust Transfers Between Libration Point Orbits Without Explicit Use of Manifolds

Abstract
In this paper, we investigate the numerical computation of minimum-energy low-thrust transfers between Libration point orbits in the Circular Restricted Three-Body Problem. We develop a three-step methodology based on optimal control theory, indirect shooting methods and variational equations without using information from invariant manifolds. Numerical results are provided in the case of transfers between Lyapunov orbits around $$L_{1}$$ and $$L_{2}$$ in the Earth-Moon system demonstrating the efficiency of the developed approach for different values of the transfer duration leading to trajectories with one or two revolutions around the Moon.
Richard Epenoy

### Time-Minimum Control of the Restricted Elliptic Three-Body Problem Applied to Space Transfer

Abstract
In this chapter, we investigate time minimal transfers in the elliptic restricted 3-body problem. We study the controllability of the problem and show that it is small-time locally controllable at the equilibrium points. We present results about the structure of the extremal trajectories, based on a previous study of the time minimum control of the circular restricted 3-body problem. We use indirect numerical methods in optimal control to simulate time-minimizing space transfers using the elliptic model from the geostationary orbit to the equilibrium points $$L_1$$ and $$L_2$$ in the Earth-Moon system, as well as a rendezvous mission with a near-Earth asteroid.
Monique Chyba, Geoff Patterson, Gautier Picot

### On Local Optima in Minimum Time Control of the Restricted Three-Body Problem

Abstract
The structure of local minima for time minimization in the controlled three-body problem is studied. Several homotopies are systematically used to unfold the structure of these local minimizers, and the resulting singularity of the path associated with the value function is analyzed numerically.