Dynamics and Mission Design Near Libration Points

by Gomez, Gerard; Simo, Carles; Llibre, J.; Martinez, R.; Gomez, Gerard

• ISBN13: 9789810242749
• ISBN10: 9810242743
• Format: Hardcover
• Copyright: 2001-04-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

It is well known that the restricted three-body problem has triangular equilibrium points. These points are linearly stable for values of the mass parameter, f, below Routh's critical value, f1. It is also known that in the spatial case they are nonlinearly stable, not for all the initial conditions in a neighborhood of the equilibrium points L4, L5 but for a set of relatively large measures. This follows from the celebrated Kolmogorov-Arnold-Moser theorem. In fact there are neighborhoods of computable size for which one obtains "practical stability" in the sense that the massless particle remains close to the equilibrium point for a big time interval (some millions of years, for example).According to the literature, what has been done in the problem follows to approaches: (a) numerical simulations of more or less accurate models of the real solar system; (b) study of periodic or quasi-periodic orbits of some much simpler problem.The concrete questions that are studied in this volume are: (a) Is there some orbit of the real solar system which looks like the periodic orbits of the second approach? (That is, are there orbits performing revolutions around L4 covering eventually a thick strip? Furthermore, it would be good if those orbits turn out to be quasi-periodic. However, there is no guarantee that such orbits exist or will be quasi-periodic). (b) If the orbit of (a) exists and two particles (spacecraft) are put close to it, how do the mutual distance and orientation change with time?As final conclusion of the work, there is evidence that orbits moving in a somewhat big annulus around L4 and L5 exist, that these orbits have small components out of the plane of the Earth-Moonsystem, and that they are at most mildly unstable.

Table of Contents

Preface	v
Bibliographical Survey	1	(14)
Equations. The Triangular Equilibrium Points and their Stability	1	(1)
Numerical Results for the Motion Around L4 and L5	2	(4)
Analytical Results for the Motion Around L4 and L5	6	(6)
The Models Used	6	(6)
Miscellaneous Results	12	(3)
Station Keeping at the Triangular Equilibrium Points	12	(1)
Some Other Results	12	(3)
Periodic Orbits of the Bicircular Problem and Their Stability	15	(18)
Introduction	15	(1)
The Equations of the Bicircular Problem	16	(3)
Periodic Orbits with the Period of the Sun	19	(2)
The Tools: Numerical Continuation of Periodic Orbits and Analysis of Bifurcations	21	(7)
Numerical Continuation of Periodic Orbits for Nonautonomous and Autonomous Equations	21	(3)
Bifurcations of Periodic Orbits: From the Autonomous to the Nonautonomous Periodic System	24	(2)
Bifurcation for Eigenvalues Equal to One	26	(2)
The Periodic Orbits Obtained by Triplication	28	(5)
Numerical Simulations of the Motion in an Extended Neighborhood of the Triangular Libration Points in the Earth-Moon System	33	(14)
Introduction	34	(1)
Simulations of Motion Starting at the Instantaneous Triangular Points at a Given Epoch	35	(1)
Simulations of Motion Starting Near the Planar Periodic Orbit of Kolenkiewicz and Carpenter	35	(12)
The Equations of Motion	47	(24)
Reference Systems	47	(1)
The Lagrangian	48	(3)
The Hamiltonian and the Related Expansions	51	(1)
Some Useful Expansions	52	(2)
Fourier Analysis: The Relevant Frequencies and the Related Coefficients	54	(8)
Concrete Expansions of the Hamiltonian and the Functions	62	(3)
Simplified Normalized Equations. Tests	65	(6)
Tests of the Simplified Normalized Equations	66	(5)
Periodic Orbits of Some Intermediate Equations	71	(16)
Equations of Motion for the Computation of Intermediate Periodic Orbits	71	(2)
Obtaining the Periodic Orbits Around the Triangular Libration Points for the Intermediate Equations	73	(1)
Results and Comments	74	(13)
Quasi-periodic Solution of the Global Equations: Semianalytic Approach	87	(20)
The Objective	87	(1)
The Algorithm	88	(2)
The Adequate Set of Relevant Frequencies	90	(4)
Avoiding Secular Terms	94	(1)
The Coefficients Related to the Different Frequencies	94	(1)
Determination of the Coefficients of Quasi-periodic Functions Using FFT	95	(8)
Results and Conclusions	103	(4)
Numerical Determination of Suitable Orbits of the Simplified System	107	(14)
The Objective	107	(1)
Description of Two Families of Algorithms. Reduction of the Linearized Equations	108	(4)
Description of the Methods. Comments	112	(4)
Results and Discussion	116	(5)
Relative Motion of Two Nearby Spacecrafts	121	(16)
The Selection of Orbits for the Two Spacecrafts	121	(1)
Variations of the Relative Distance and Orientation. Results	122	(13)
Comments on the Applicability of the Results	135	(2)
Summary	137	(6)
Objectives of the Work	137	(1)
Contribution to the Solution of the Problem	138	(2)
Conclusions	140	(1)
Outlook	141	(2)
Bibliography	143