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 second primary 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 buoyancy force. The relevant equations of motion are established, and a special case where the density of the fluid and that of the infinitesimal mass are equal (D = 0) is discussed. It is observed in the first case that there are two axial libration points on the line joining the centers 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, while the second one is conditionally stable, and the circular points are unstable. The location of the libration point and its stability when the infinitesimal mass is denser than the medium (D > 0), in the second case, are also studied and it is found that the origin (0, 0, 0) of the system is the only libration point, and this point is stable.

TABLE OF CONTENTS

Cover page…………………………………………………………………………………………………………… i
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Title page ……………………………………………………………….. Error! Bookmark not defined.
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
1.1 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 ……………………………………………………. 16
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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 ……………………………………………. 23
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 …………………………………………………………………………………………………………… 32
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
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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 . 95
5.1 Introduction …………………………………………………………………………………………….. 95
5.2 Summary ………………………………………………………………………………………………… 95
5.3 Conclusions …………………………………………………………………………………………….. 96
5.4 Recommendations ……………………………………………………………………………………… 97
REFERENCES ………………………………………………………………………………………………….. 98
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CHAPTER ONE

GENERAL INTRODUCTION
1.1 Introduction
The most celebrated problem of space dynamics is the problem of three bodies, known as the three-body problem (3BP). The problem is defined in terms of three bodies with arbitrary 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, when they 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 in the Earth-Moon space. In the general 3BP, 18 first order, coupled, nonlinear differential equations govern the motion. However, only ten integrals of the motion are known to exist; they are derived from the conservation of linear momentum, angular momentum and energy. Thus, the equations 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 the bodies is assumed to posses’ infinitesimal mass. The R3BP describes the motion of an infinitesimal mass moving under the gravitational effects of two finite masses, called primaries, which move in circular orbits around the center of mass on the account of their mutual attraction and the infinitesimal mass not influencing the motion of the primaries.
The difference between the general 3BP and the R3BP is first of all that in the latter the masses of only two particles are arbitrary; the third mass is much smaller than the other two. The general problem allows any sets of initial conditions for the three particles involved; the restricted problem requires circular orbits for the primaries and requires that
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the motion of the infinitesimal body takes place in the plane defined by the motion of the primaries. The R3BP is one of the most widely studied areas in space dynamics as well as celestial mechanics; and very significant results have been produced by well known mathematicians and scientists in an attempt to understand and predict the motion of natural bodies. The application of R3BP spans solar system dynamics, lunar theory, motion of space craft and stellar dynamics. A typical example of the R3BP is seen in a system made up of the Sun and Jupiter as primaries, and then a Trojan asteroid assuming the role of the infinitesimal mass in the Sun-Jupiter system. The Robe’s R3BP, referred to as Robe’s problem, is a peculiar case of R3BP, where the infinitesimal mass is embedded in the first primary which is filled with homogenous incomparable fluid of known density, the second primary is a point mass located outside an orbiting shell. The infinitesimal mass is under the gravitational attraction of the primaries as well as buoyancy force due to the fluid. The study of the oscillations of the Earth’s inner core taking into cognizance the attraction of the Moon; and the motion of an artificial Earth satellite located inside another satellite, constitute models for the Robe’s problem. This Robe’s problem has been varied by the introduction of perturbing forces such as oblatenes and triaxiality.
1.2 Statement of The Problem
The present problem describes the motion of a third body of an infinitesimal mass in the gravitational field of two massive bodies which move in a circular orbits around their center of mass on account of their mutual attraction. In the first problem, the first primary is a shell filled with an incompressible homogenous fluid of density such that, the hydrostatic equilibrium figure of the first primary is an oblate spheroid; the second primary
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is also considered as an oblate spheroid with oblateness coefficients up to the second zonal harmonic, located outside the shell and describes a circular orbit around the first one and the infinitesimal mass is a solid sphere of density , located inside the shell. In the second problem, we assume the first primary is a fluid of density in the shape of a Roche ellipsoid, the second primary be a triaxial rigid body. Our effort in this research work is to study the combined effect of the full buoyancy force of the fluid of the first primary and (i) the oblateness of the primaries, (ii) the triaxiality of the second primary; on their locations and linear stability of the libration points.
1.3 Aim and Objectives of Study
The aim of this study is to investigate 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 second primary 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. The objectives of the study are to
i) Establish the equations of motion of the infinitesimal mass of the stated problems
ii) Examine possible libration points,
iii) Investigate the effect of zonal harmonic and triaxiality of the primaries on the possible libration points and their stability.
1.4 Significant / Justification of the Study
In the classical restricted three-body problem, the participating bodies are assumed to be spherical in shape, but we found that in reality, several bodies are sufficiently oblate or triaxial. Earth, Jupiter, Saturn and Ragulus are oblate while Moon, Pluto and Charon are
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triaxial. The meteoroids and the minor planets have irregular shapes. The lack of spherisity oblateness, or triaxiality of the celestial bodies causes large perturbations from the two-body orbit. Examples of this are the motions of artificial Earth satellite. Many researchers like Subbarao and Sharma (1975), Singh and Ishwar (1999), Singh and Begha (2011), Singh and Leke (2012), Singh and Sandah (2012), Singh and Mohammed (2013) and Singh and Omale (2014) have included obaleteness and triaxiality in their study of R3BP. It is reasonable to modify the model by considering first, the hydrostatic equilibrium figure of the first primary as an oblate spheroid, the shape of the second one as oblate spheroid too with oblateness coefficients up to the second zonal harmonic. The shape of the first primary in the second case is considered as Roche-ellipsoid and the second primary as a triaxial rigid body; and the full buoyancy of the fluid is taken into account as well as other forces acting on the infinitesimal mass when the density parameter is not zero . This Robe’s model under consideration will have many applications in various astronomical problems as it may provide insight into the problem of small oscillations of the earth’s core in the gravitational field of Earth-Moon system, and the stability of the Earth centre as an libration point of the Robe’s problem and the motion of the artificial satellites in the Earth-Moon vicinity. It can also be applicable in space mission design.
1.5 Research Methodology
The equations of motion of the infinitesimal body with respect to the primaries are established by adopting a synodic coordinate system.
1.6 Preliminary Ideas
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1.6.1 Circular restricted three-body problem
In general, R3BP involves three particles of arbitrary masses moving freely under their mutual gravitational attraction. The particle with infinitesimal mass, sometimes called a massless particle, does not perturb the motions of the two massive bodies. In the circular problem, the two finite masses are fixed in a coordinate system rotating at the orbital angular velocity, with the origin (axis of rotation) at the centre of mass of the two bodies. Lagrange showed that in this rotating frame there were five libration points at which the infinitesimal body would remain fixed if placed there. Three of these points called collinear libration points were found by L. Euler (1767), there lie on the straight line connecting the two finite masses: one between the masses and one outside each of the masses. While the other two called the triangular libration points found by J. L Lagrange (1772), are located equidistant from the two finite masses at a distance equal to the finite mass separation. The two masses and the triangular libration points are thus located at the vertices of equilateral triangles in the plane of the circular orbit. These are illustrated in the figure below, by the use of Sun-Earth-Moon system. The velocity and acceleration are zero at these points.
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Fig. 1.1 The Five Lagrangian points in CR3BP (Sun-Earth-Moon system)
1.6.2 Equations of motion of two-body problem
Let the masses of the two primaries be and their respective vectors from the fixed point 0 in the space of the motion to the respective masses be 1 2, and be the radius vector between 4the bodies.
By Newton law of gravitation, the force of attraction on the particle of mass due to another particle of mass and the force acting on the particle of mass due to the particle of mass are
where is the gravitational constant and is the distance between the primaries. By Newton’s second law, we have Equating the equations in system (1.1) and (1.2), we have
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Equations of system (1.3) are the vectorial equations of motion of the bodies of arbitrary masses .
1.6.3 Equations of relative motion
Let the velocity of the bodies be 1 2 respectively, and R be the velocity vector of the body
Then R2 1 (1.4) Now the vectorial equations of system (1.3) recast to the form Subtracting the equations in system (1.5), we have Using Eq. (1.4), we have
Let , we have
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Equation (1.6) is refers to as the relative motion of the body of mass with respect to the body of mass , where r is the distance between the bodies.
1.7 Theoretical Framework
The theoretical framework upon which the derivations and finding of this research work is built on is given in this section
1.7.1 Relative motion of Robe’s restricted three-body
The volume of the fluid of density embedded in the heavier primary is given by
so the mass of the fluid with density can be written as
Therefore, the mass of the shell which is denoted by is such that From Newton second law of motion,
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Using the Newton second law of motion in system (1.9), we have
Where and are the gravitational constant and the distance between the primaries respectively.
The vectorial equations of motion of the bodies of masses are the equations of system (1.10). Subtracting the equations of system (1.10), we have
Therefore, the relative motion of the mass about the spherical shell of mass of density .is given by equation (1.11)
1.7.2 The Robe’s restricted three-body problem
Robe (1977) formulated a new kind of R3BP in which one of the primaries of mass is a rigid spherical shell, filled with homogenous, incompressible fluid of density; the second one is a point mass outside the shell, and the third body of mass is a small solid sphere of density moving inside the shell, with assumption that the mass and radius of are infinitesimal; as illustrated in figure 1.3. He showed the centre of the first primary as a libration point and examined its linear stability in two cases. In the first case,
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the orbit of around is circular and in the second case, the orbit is elliptic, but the shell is empty, or the densities and are equal. In evaluating buoyancy force, he assumed that the pressure field of the fluid has spherical symmetry around the centre of the shell, which are due to the own gravitational field of the fluid.
Figure 1.2 The Robe’s restricted three-body problem
The equations of motion of the infinitesimal mass of Robe’s circular R3BP are given in this section, since in our work; we shall consider a modified Robe’s model. It can be written as where
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1.7.3 The buoyancy force
The force exerted on an object that is wholly or partially immersed in a fluid is known as buoyancy force. It acts upward, generally opposite to the direction of the frame of reference acceleration and its magnitude is equal to the weight of the fluid displaced by the submerged object. This force is caused by the difference between the pressure at the top of the object, which pushes downward and the pressure at the bottom of the object which pushes upward; every submerged object feels upward buoyancy force, because the pressure at the bottom of the object is always greater than the pressure at the top.
Suppose the orbital plane of around (the shell with fluid ) be taken as , and also let the origin of the coordinate system be at the center of mass, 0, of the two finite masses. Then the forces acting on the third body are;
1) The attraction of
2) The gravitation force exerted by the fluid of density is given by
By the use of equation (1.7), we have
Since the body of mass maintains a spherical symmetry about (center of ) due to the pressure of the fluid inside , in the case, the radius R becomes Therefore, the gravitational force equation (1.13) is recast to the form
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3) The buoyancy force obtained from equation (1.14) is given as
1.8 Definition of Terms
1.8.1 An ellipsoid
An ellipsoid is a closed quadric surface that is a three-dimensional analogue of an ellipse
Figure 1.3 shape of an ellipsoid
1.8.2 Oblatenesss
In the above figure, if we have an oblate spheroid. In other words, an ellipsoid having a polar axis shorter than the diameter of the equatorial circle whose plane bisects it is known as an oblate spheroid, thus, its two moments of inertial are equal out of the three. It is the approximated shape of many planets and celestial bodies, including Saturn, Jupiter and to a lesser extent the Earth. It is therefore the most used geometric figure for defining reference ellipsoid, upon which cartographic and geodetic system are based.
1.8.3 Triaxiality
If we have a triaxial rigid body; that is, triaxiality is a situation that involves three axes. A rigid body that has three mutual perpendicular symmetric axes is called a
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triaxial rigid body, for example, an ellipsoid. A triaxial body has all its moments of inertial to be distinct.
1.8.4 Roche ellipsoid
Roche ellipsoid is a general triaxial ellipsoid with all axes having different length.
1.8.5 Zonal hamomic
A harmonic function is a function f that satisfies the Laplace’s equation the spherical harmonic is an infinite set of harmonic functions defined on a sphere. Zonal harmonic is a spherical harmonic projection of functions that have rotational symmetry around the axis.
1.8.6 Moment of Inertia
The moment of inertial of a system or particle with respect to a plane, point or line is defined as the sum of the products of the masses and the square of their respective perpendicular distances from the plane, point or line. If are the masses of the particles with perpendicular distances, then the moment of inertia is given as
For a body, we have
where is the density of the body and subscript M shows that the integration is done over the whole mass of the body.
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The potential of the body with the origin at the center of the mass is given as
where the moment of inertia about the origin while I is the moment of inertia about a line fixed from the center of the body to a point outside the body with distance . Thus the potential between the infinitesimal mass and the primaries can be written as;
Where are the principal moments of inertia of the oblate spheroid with and are the principal moment of inertia of the triaxial rigid body with . From Kepler’s third law, the mean motion is expressed as In the first case where both primaries are oblate spheroid with oblateness coefficients up to the second zonal harmonic and the full buoyancy of the fluid is taken into account, the mean motion of the primaries is given (McCuskey 1963) by while the second case where the primaries form a Roche ellipsoid-triaxial system, the mean motion of the primaries given also by McCuskey 1963 is
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1.9 Linear Stability of the Solutions of Dynamical Systems
The stability of linear systems of ordinary differential equations is determined completely by the eigen values of the coefficient matrix. Due to perturbations, the position of the infinitesimal body would be displaced a little from the libration point. If the resultant motion of the infinitesimal body is a rapid departure from the vicinity of the point, we call such a position of libration point an “unstable one” if however the body merely oscillates about the libration point, it is said to be a “stable position” (in the sense of Lyapunov). To investigate the stability of the orbit in the surrounding area of the libration points, the small displacement method is applied by shifting the coordinates or the origin of the infinitesimal mass, and then linearizing the equations of motion around the coordinates of the libration points to derive the variational equations of motion corresponding to the dynamical system. These equations are transformed to a matrix form through the trial solutions and a characteristic equation of the dynamical system is obtained. The roots of the characteristic equation are obtained and the stability of the solutions depends on the nature of the characteristic roots; for complex roots, the libration point is asymptotically stable when all roots have negative real parts, and unstable when some or all roots have positive real part. For pure imaginary roots, the motion is oscillatory and the libration point is stable, though not asymptotically stable. If there are multiple roots, the solution contains mixed terms (periodic and secular terms) the libration point is unstable. If all the roots are both real and negative then the libration point is stable, but if any of the roots is positive, the libration point will be unstable.

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