Robe’s Circular Restricted Three-Body Problem With Zonal Harmonics And A Roche Ellipsoid-Triaxial System.
ABSTRACT
This research work analyzes the motion of an infinitesimal mass in the framework of Robe’s circular restricted three-body problem in two cases:
(i) when the hydrostatic equilibrium figure of the first primary is an oblate spheroid, the shape of the secondprimary is considered as an oblate spheroid with oblateness coefficients up to the second zonal harmonic, and
(ii) when the primary bodies form a Roche ellipsoid-triaxial system.Without ignoring any component in both cases, a full treatment is given to the buoyancyforce.
The relevant equations of motion are established, and a special case where thedensity of the fluid and that of the infinitesimal mass are equal (D = 0) is discussed.
It isobserved in the first case that there are two axial libration points on the line joining thecenters of the primaries, points on the circle within the first primary are also libration points under certain conditions.
It is further found that the first axial point is stable, whilethe second one is conditionally stable, and the circular points are unstable.
TABLE OF CONTENTS
Declaration ……….iii
Certification……….iv
Acknowledgements………… v
Dedication ………… vi
Abstract …………vii
Table of Contents……..viii
List of Figures ………. xi
List of Notations……….xii
CHAPTER ONE: GENERAL INTRODUCTION
1.1 Introduction …………. 1
1.2 Statement of The Problem…….. 2
1.3 Aim and Objectives of Study ……. 3
1.4 Significant / Justification of the Study ……….. 3
1.5 Research Methodology……… 4
1.6 Preliminary Ideas……… 4
1.6.1 Circular restricted three-body problem ……….. 5
1.9 Linear Stability of the Solutions of Dynamical Systems …. 15
CHAPTER TWO: LITERATURE REVIEW
2.1 Introduction …………………. 16
2.2 Libration Points…………. 16
2.3 Oblateness and Triaxiality…………… 17
2.4 Robe’s Problem………… 19
CHAPTER THREE: EQUATIONS OF MOTION
3.1 Introduction …………………….. 23
3.2 Mathematical Model ………….. 23
3.2 The Case …………… 30
3.3 Conclusion……… 31
CHAPTER FOUR: LOCATIONS AND LINEAR STABILITY OF LIBRATION POINTS
4.1 Introduction ………………. 32
4.2 Stability of Libration Points………….. 33
4.3 Variational and Characteristic Equations…….. 33
4.4 Jacobi Integral ……………… 37
4.5 The Libration Points (Case 1) ……… 37
4.5.1 Axial libration points……………. 38
4.5.2 Locations of circular points……….. 50
4.6 Location of Libration Point When (Case 2)………. 54
4.7 Stability of Axial Libration Points (Case 1)……….. 55
4.7.1 Stability of the axial libration point …………. 63
4.7.2 Stability of the axial libration point ……… 68
4.7.3 Stability of circular points…….. 79
4.8 Stability of Axial Libration Points (Case 2)…….. 91
4.9 Conclusion ……… 94
CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMENDATIONS
5.1 Introduction ……… 95
5.2 Summary …………. 95
5.3 Conclusions……… 96
5.4 Recommendations……… 97
REFERENCES…… 98
GENERAL INTRODUCTION
The most celebrated problem of space dynamics is the problem of three bodies, known asthe three-body problem (3BP).
The problem is defined in terms of three bodies witharbitrary masses attracting one another according to Newtonian law of gravitation, and is free to move in space.
A classical example of 3BP is the Sun-Earth-Moon system, whenthey are considered as point masses; they form the main problem of the lunar theory.
Another approximate example of the 3BP is the Earth, the Moon and the space vehicle inthe Earth-Moon space.In the general 3BP, 18 first order, coupled, nonlinear differential equations govern themotion.
However, only ten integrals of the motion are known to exist; they are derivedfrom the conservation of linear momentum, angular momentum and energy.
Thus, theequations of motion are not solvable analytically. In an attempt to solve the problem,Lanrange reduced the 3BP to the restricted three-body problem (R3BP), where one of thebodies is assumed to posses’ infinitesimal mass.
REFERENCES
AbdulRaheem, A. and Singh J. (2006). Combined effects of perturbations, radiation andoblateness on the stability of libration points in the restricted three-body problem.
Astronomical journal, 131, 1880-1885Bhavneet, K. and Aggarwal R. (2013). Robe’s restricted problem of 2+2 bodies when thebigger primary is a Roche ellipsoid and the smaller primary is an oblate body.
Astrophysics and Space Science, 349.57-69Chandrasekhar, S. (1987). Ellipsoidal figures of libration. Dover publication, New York.(First edition)Euler, L. (1767).
The motion in the rectilinear three-body problem. Nov. Comm. Petrop.(First edition)Giordano, C. M., Plastino, A.R. and Plastino, A. (1996).
Robe’s restricted three-bodyproblem with drag. Celestial Mechanics and Dynamical Astronomy, 66, 229-242.Khanna, M. and Bhatnager, K.B. (1999).
Existence and stability of libration points in therestricted three-body problem when the smaller primary is a triaxial rigid body andthe bigger one an oblate spheroid. Indian Journal of pure and applied Mathematics,30, 721- 733Hallan, P.P. and Rana, N. (2001).
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