## ABSTRACT

In the first problem, a modification of the restricted three-body problem has been considered where the primary (more massive body) is a triaxial rigid body and the secondary (less massive body) is an oblate spheroid and study periodic motions around the collinear equilibrium points. The locations of these points are firstly determined for ten combinations of the parameters of the problem. In all ten cases, the collinear equilibrium points are found to be unstable, as in the classical problem, and the Lyapunov periodic orbits around them have been computed accurately by applying known corrector-predictor algorithms. An extensive study on the families of three-dimensional periodic orbits emanating from these points has also been done. To find suitable starting points, for all the computed families, semi-analytical solutions have been obtained, for both two- and three-dimensional cases, around the collinear equilibrium points using the Lindstedt-Poincaré method. Finally, the stability of all computed periodic orbits has been studied. The second problem presents a third order analytic approximation solution of Lyapunov orbits around the collinear equilibrium points in the planar restricted three-body problem by utilizing the Lindstedt Poincaré method. The primaries are oblate bodies and sources of radiation pressure. The theory has been applied to the binary -Centuari system in six cases. Also, the positions of the collinear equilibrium points have been determined numerically and the effects of the parameters concerned with these equilibrium points shown graphically.

In the third problem, an investigation of three-dimensional periodic orbits and their stability emanating from the collinear equilibrium points of the restricted three-body problem with oblate and radiating primaries is presented. A numerical simulation is done by using five binary systems: Sirius, Procyon, Luhman 16, -Centuari and Luyten 726-8. Firstly, based on the topological degree theory, the total number of the collinear equilibrium points for the five binary systems were obtained and then, their positions were determined numerically. The

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linear stability of these equilibrium points was also examined and found to be unstable in the Lyapunov sense. An analytical approximation of three-dimensional periodic solutions around them was established via the Lindstedt-Poincaré local analysis. Finally, using the analytical solution to obtain starting orbits, the families of three-dimensional periodic orbits emanating from these equilibria have been continued numerically. Additionally, the collinear equilibrium points and periodic motion around them are studied in the framework of the restricted three-body problem where the two primaries are triaxial rigid bodies which emit radiation in the fourth problem. Firstly, the positions and stability of the collinear equilibria are studied for HD 191408, Kruger 60 and HD 155876 binary systems. Then, the planar and three-dimensional periodic motion about these points is considered. The study includes both semi-analytical and numerical determination of these motions. It is found that all families of planar periodic orbits emanating from these points terminate with asymptotic periodic orbits at the triangular equilibrium points while the corresponding families of three-dimensional periodic orbits terminate with planar periodic orbits.

## TABLE OF CONTENTS

Cover Page ………………………………………………………………………………………………………………… i

Title Page …………………………………………………………………………………………………………………. ii

Declaration ………………………………………………………………………………………………………………. iii

Certification …………………………………………………………………………………………………………….. iv

Dedication ………………………………………………………………………………………………………………….v

Acknowledgement ……………………………………………………………………………………………………. vi

Abstract …………………………………………………………………………………………………………………. viii

Table of Contents ………………………………………………………………………………………………………..x

List of Figures …………………………………………………………………………………………………………..xv

List of Tables …………………………………………………………………………………………………………. xxi

List of Symbols …………………………………………………………………………………………………….. xxiii

CHAPTER ONE ……………………………………………………………………………………………………….1

INTRODUCTION……………………………………………………………………………………………………..1

1.1 General Introduction ……………………………………………………………………………………………1

1.2 Statement of the Problem …………………………………………………………………………………….5

1.3 Justification of the Study ……………………………………………………………………………………..6

1.4 Aim and Objectives of the Study…………………………………………………………………………..6

1.5 Thesis Organization …………………………………………………………………………………………….7

CHAPTER TWO ………………………………………………………………………………………………………9

LITERATURE REVIEW ………………………………………………………………………………………….9

2.1 Introduction ………………………………………………………………………………………………………..9

2.2 Periodic Orbits around the Triangular Points ………………………………………………………….9

2.3 Periodic Orbits around the Collinear Equilibrium Points………………………………………..11

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2.4 Periodic Orbits and Variations of the Mass Parameter …………………………………………..13

2.5 Periodic Orbits with Radiation Pressure……………………………………………………………….15

2.6 Periodic Orbits and the Shapes of the Bodies ……………………………………………………….16

2.7 Periodic Orbits with Radiation Pressure and the Shapes of the Bodies …………………….17

2.8 Periodic Horseshoe Orbits ………………………………………………………………………………….19

2.9 Finding Periodic Orbits by using Other Contemporary Methods …………………………….19

2.10 Theoretical Framework ……………………………………………………………………………………..21

2.10.1 Formulation of the Classical CR3BP in a Synodic Reference Frame ……………………..21

2.10.2 Stability of the Equilibrium Points …………………………………………………………………….25

2.10.3 Computing Periodic Orbits ……………………………………………………………………………….26

2.10.4 Fourth order Runge-Kutta method ……………………………………………………………………..31

2.10.5 Radiation Pressure …………………………………………………………………………………………..31

2.10.6 Ellipsoid …………………………………………………………………………………………………………32

2.10.7 Coriolis and Centrifugal Forces …………………………………………………………………………34

2.10.8 Binary Star Systems ………………………………………………………………………………………..35

2.10.9 Romberg Integration ……………………………………………………………………………………….35

CHAPTER THREE …………………………………………………………………………………………………37

PERIODIC MOTIONS AROUND THE COLLINEAR EQUILIBRIUM POINTS OF THE PERTURBED R3BP WHEN THE PRIMARY IS A TRIAXIAL RIGID BODY AND THE SECONDARY IS AN OBLATE SPHEROID …………………………………………..37

3.1 Introduction ……………………………………………………………………………………………………..37

3.2 Equations of Motion and Variation and the Stability Parameters …………………………….37

3.3 Determination of the Collinear Equilibrium Points for the model with triaxial and oblate primaries ………………………………………………………………………………………………..43

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3.4 Motion around the Collinear Equilibrium Points for model with triaxial and oblate primaries ………………………………………………………………………………………………………….43

3.4.1 Second order Taylor Series Expansions of the Equations of Motion for the Model with Triaxial and Oblate Primaries ……………………………………………………………………………..44

3.5 Spatial Periodic Orbits: Second Order Analysis for the Model with Triaxial and Oblate Primaries ………………………………………………………………………………………………………….53

3.5.1 Second Order Expansions for the 3D equations of Motion for the Model with Triaxial and Oblate Primaries …………………………………………………………………………………………..54

3.5.2 Semi-analytical Approximation of the 3D Periodic Orbits for the Model with Triaxial and Oblate Primaries …………………………………………………………………………………………..71

CHAPTER FOUR ……………………………………………………………………………………………………74

PERIODIC MOTIONS AROUND THE COLLINEAR EQUILIBRIUM POINTS OF THE PHOTOGRAVITATIONAL R3BP WITH TRIAXIALITY, OBLATENESS AND SMALL PERTURBATIONS IN THE CORIOLIS AND CENTRIFUGAL FORCES…74

4.1 Introduction ……………………………………………………………………………………………………….74

4.2 Third Order Analytic Approximation Solutions of Lyapunov Orbits Around the Collinear Equilibrium Points: The Planar Case……………………………………………………..75

4.2.1 Two-dimensional Equations of Motion for the Radiating Oblate Primaries ………………75

4.2.2 Determination of the Collinear Equilibrium Points for the Model with Radiating Oblate Primaries in the Plane of Motion …………………………………………………………………………77

4.2.3 Motion around the Collinear Equilibrium points for the Model with Radiation Oblate Primaries in the Plane of Motion …………………………………………………………………………77

4.3 Analytic Approximation of Lyapunov Orbits Around the Collinear Equilibrium Points: The Spatial Case ……………………………………………………………………………………………….89

4.3.1 Three-dimensional Equations of Motion for the Model with Radiating Oblate Primaries of the Spatial Case ………………………………………………………………………………………………89

4.3.2 Equations of Motion and Variation for the Model with Radiating Oblate Primaries …….. ……………………………………………………………………………………………………………………….91

4.3.3 Determination of the Collinear Equilibrium Points for the Model with Radiating Oblate Primares of the Spatial Case ……………………………………………………………………………….93

4.3.4 The Dynamics around the Collinear Equilibrium Points for the Model with Radiating Oblate Primaries of the Spatial Case ……………………………………………………………………..94

4.3.5 Second Order Taylor Series Expansions of the Equations of Motion for the Model with Radiating Oblate Primaries ………………………………………………………………………………….96

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4.3.6 Semi-analytic Approximation of Periodic Solutions for the the Model with Radiating Oblate Primaries of the Spatial Case ……………………………………………………………………..97

4.3.7 Numerical Approximation of Periodic Solutions …………………………………………………..98

4.4 Periodic Solutions around the Collinear Equilibrium Points in the Perturbed Restricted Three-Body Problem with Triaxial and Radiating Primaries …………………………………100

4.4.1 Equations of Motion for the Model with Triaxial and Radiating Primaries ……………..100

4.4.2 Variational Equations for the Model with Triaxial and Radiating Primaries ……………101

4.4.3 Determination of the Collinear Equilibrium points for the Model with Triaxial and Radiating Primaries …………………………………………………………………………………………..105

4.4.4 Motion around the Collinear Equilibrium points for the Model with Triaxial and Radiating Primaries …………………………………………………………………………………………..105

4.4.5 Expansions of the 2D Equations of Motion up to Second Order Terms …………………106

4.4.6 Semi-analytical Approximation of 2D Periodic Orbits for the Model with Triaxial and Radiating Primaries …………………………………………………………………………………………114

4.4.7 Expansions of the 3D Equations of Motion up to Second Order Terms ………………….116

4.4.8 Semi-analytical Approximation of 3D Periodic Orbits for the Model with Triaxial and Radiating Primaries …………………………………………………………………………………………136

CHAPTER FIVE …………………………………………………………………………………………………..139

RESULTS AND DISCUSSION ………………………………………………………………………………139

5.1 Introduction ……………………………………………………………………………………………………139

5.2 Numerical Considerations for Periodic Solutions Around the Collinear Equilibrium Points in the Perturbed R3BP when the Bigger Primary is a Triaxial Rigid Body and the Secondary an Oblate Spheroid ………………………………………………………………………….139

5.2.1 Numerical Results for the Coplanar Periodic Orbits of the model with triaxial and Oblate Primaries ……………………………………………………………………………………………….143

5.2.2 Numerical Results for 3D Periodic orbits having Triaxial and Oblate Primaries ……..163

5.3 Numerical Considerations for the Analytic Approximation of Lyapunov Orbits Around the Collinear Equilibrium Points in the Two-Dimensional R3BP when the Primaries are Radiating and Oblate Spheroidal Bodies. ……………………………………………………………173

5.3.1 Numerical Applications for the 2D case for the Model with Radiating Oblate Primaries . ……………………………………………………………………………………………………………………..173

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5.4 Numerical Considerations for the Periodic Solutions Around the Collinear Equilibrium Points in the Three-Dimensional R3BP when the Primaries are Radiating and Oblate Spheroidal Bodies. …………………………………………………………………………………………..183

5.4.1 Numerical applications for the 3D case for the Model with Radiating Oblate Primaries .. ……………………………………………………………………………………………………………………..184

5.4.2 Numerical approximation results ……………………………………………………………………….200

5.5 Numerical Considerations for Periodic Solutions Around the Collinear Equilibrium Points in the Perturbed R3BP when Both Primaries are Radiating and Triaxial Spheroids …215

5.5.1 Numerical Applications when the Primaries are both Triaxial and Radiating Bodies .215

5.5.2 Numerical Results for the 2D Periodic Orbits for the Model with Triaxial and Radiating Primaries ………………………………………………………………………………………………………….228

5.5.2 Numerical Results for the Spatial Periodic Orbits of the Model with Triaxial and Radiating Primaries …………………………………………………………………………………………..238

CHAPTER SIX ……………………………………………………………………………………………………..245

SUMMARY, CONCLUSION AND RECOMMENDATION ……………………………………245

6.1 Introduction ……………………………………………………………………………………………………245

6.2 Summary………………………………………………………………………………………………………..245

6.3 Conclusion ……………………………………………………………………………………………………..252

6.3 Recommendation …………………………………………………………………………………………….255

6.4 Contribution to Knowledge ………………………………………………………………………………..256

REFERENCES ……………………………………………………………………………………………………….257

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## CHAPTER ONE

INTRODUCTION

1.1 General Introduction

New developments in the mathematical theory of differential equations and dynamical systems have been aided by studies being carried out in the three-body problem (3BP). The 3BP refers to the movement of three bodies under their mutual gravitational influence. Eighteen integrals of motion are needed for the complete solution to the 3BP. Euler (Barrow-Green, 1997) was able to establish that only ten (10) integrals of motion exist and as such, the problem when considered analytically does not have a closed form solution. Six of the scalar integrals are derived from the conservation of linear momentum, and then from the conservation of total angular momentum and the conservation of energy are three and one scalar integrals of motion respectively.

The restricted three-body problem (R3BP) is a reduced form of the 3BP. The problem is termed „restricted‟ on the basis of some assumptions that differentiate it from the general 3BP. One of such is the consideration of three bodies, two of which have finite masses and are often referred to as primaries and the third body whose mass is much smaller and negligible is often called the infinitesimal body. Another assumption which is on the basis of these mass ratios is that the motion of the primaries is not influenced by the gravitational attraction of the infinitesimal body whereas their gravitational attraction completely determines the motion of the infinitesimal body. Hence, the R3BP is to determine the motion of the infinitesimal body under the gravitational influence of the two other bodies or primaries of finite masses. In the instance whereby the primaries move in circular orbits about their common centre of mass the aforementioned problem is often referred to as the

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circular restricted three-body problem (CR3BP) and when the motion of the infinitesimal

body is restricted to the orbital plane of the primaries the problem is further referred to as the

planar circular restricted three-body problem (PCR3BP).

The R3BP in whichever way we consider it (either circular or planar circular) is still not

solvable in closed form. However, particular solutions do exist in the 3BP in order to obtain

an insight into the problem. The five equilibrium solutions or libration points are particular

solutions of this problem.

Joseph Louis Lagrange in 1772 showed that five equilibrium solutions or libration points

exist as earlier mentioned. These consist of three collinear points and two triangular points

denoted by 1 2 3 L , L , L and 4 5 L , L , respectively, as shown in Figure 1.1. In the exact positions

of these equilibrium points the velocity and acceleration of the infinitesimal body when

placed at any of these points is equal to zero and it would remain at rest for all time.

Figure 1. 1: Location of the equilibrium points.

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In addition to the equilibrium points, periodic orbits are also particular solutions in the CR3BP. Periodic orbit is simply an orbit that closes upon itself in the phase space after each revolution. Theoretically and by application, it has been established that periodic orbits exist around the collinear equilibrium points. As for the stable triangular equilibrium points, no special treatment is needed once there. Specific orbital parameters cannot be used to describe these orbits as is common for typical orbits, but families of orbits exist. Goudas in 1963 was able to compute 19 families of three-dimensional periodic orbits in the CR3BP (Howell, 1998). Each of the collinear equilibrium points can be associated with a family of Lyapunov orbits. Also, each Lyapunov orbit within a family is characterized by a particular Jacobi Constant. In Figure 1.2 is shown the Lyapunov orbits around the collinear equilibrium points for the Earth-Moon system.

Figure 1. 2: Lyapunov orbits around the collinear equilibrium points for the Earth-Moon system (Davis (2011)).

In the R3BP, periodic orbits around the collinear equilibrium points that lie in the plane of motion of the primaries are called planar Lyapunov orbits and only one planar periodic

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motion exists around each collinear equilibrium point. In the three-dimensional (3D) aspect

of the R3BP, the corresponding 3D periodic orbits emanating from the collinear points are

called vertical Lyapunov orbits. The Lyapunov orbits around the collinear libration points can

be grouped into the families A, B, C based on Strӧ mgren‟s notation for classifying periodic

orbits.

On the other hand, two classes of periodic orbits also exist around the triangular libration

points in the plane of motion of the primaries; the long-period orbits and the short-period

orbits. The period of the long-period orbits depends on the mass ratio of the primaries while

the short-period orbits have periods on the order of the periods of the primaries about their

common centre of mass. (Goodrich, 1965)

The constant of integration associated with the differential equations was identified by Jacobi

and was named after him as the Jacobian integral or Jacobian constant and is often denoted

by the symbolC . The Jacobian integral may be used to obtain the Zero-velocity surface plots

by assuming that all the velocity components are equal to zero. This surface divides the space

into two regions. One region is known as the region of possible motion while the other is

called the forbidden region. These regions describe the area where the infinitesimal body is

allowed and not allowed to move. Figure 1.3 as shown below illustrates certain projections of

the zero-velocity surfaces as examples of Zero-velocity curves for the Jacobi Constant values

computed at the first and second collinear libration points. The white portions in the figure

indicate the regions of the allowed motion while the shaded portions correspond to the

forbidden regions for the motion of the infinitesimal body.

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Figure 1. 3: Zero velocity curves and forbidden regions. From left to right: C < CL2, CL2 <

C < CL1 and C > CL1 (Vila, 2007).

The knowledge of these forbidden regions provides some insight into the dynamics of the

problem. Furthermore, the Jacobian Constant often serves as a method used in checking the

accuracy of the calculations, especially the accuracy of the numerical integration of the

differential equations. Also, in the case of the planar R3BP which is a dynamical system of

two degrees of freedom the Jacobian constant reduces the initial four-dimensional phase

space into a three-dimensional subspace and by considering the appropriate intersections of

the 3D trajectories, e.g. with the straight line y 0, for certain flow, we obtain the Poincaré

surface of sections which, among others, is a powerful tool for the determination of the

stability of a periodic orbit as well as of chaos detection.

1.2 Statement of the Problem

Jain et al. (2006, 2009) carried out some studies on periodic orbits around the collinear

libration points in the R3BP when:

i. the smaller body is a triaxial rigid body; and

ii. when the bigger primary is a source of radiation pressure and the secondary a

triaxial rigid body,

their investigation was done for periodic orbits with respect to perturbations in the plane of

motion of the primaries, which did not include the perturbations perpendicular to the plane of

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motion of the primaries and lacked the numerical continuation to families of both planar and spatial periodic orbits. In this research work, the continuation of infinitesimal orbits in the vicinity of the collinear equilibrium points to orbits of finite dimensions in the plane and perpendicular to the plane of motion of the primaries in the framework of the generalized CR3BP will be our focus. The problem is generalized in the sense that the primaries are modelled as aspherical (for example, Triaxiallity and oblateness) bodies and the involvement of other perturbing forces like the influence of the Coriolis and centrifugal forces, and the radiation pressure.

1.3 Justification of the Study

The study would play a very important role of separating the classes of orbits when the classification of the totality of orbits is being considered. The study will serve as an avenue to obtain information regarding Lyapunov orbits for nonintegrable dynamical systems when the orbit is periodic. Since the R3BP is still not solvable in closed form, the study will serve as an opening through which the problem can be penetrated. Once a particular solution is given, a periodic solution can be found such that the difference between the two solutions is as small as it is intended for any given length of time.

1.4 Aim and Objectives of the Study

The aim of this research is to investigate periodic orbits using the Lindstedt-Poincaré local analysis and a differential correction scheme as the case may be around the collinear equilibrium points in the generalized CR3BP.

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In order to achieve this aim, the researcher has set the following objectives: to

i. compute and examine the horizontal and vertical stability of the Lyapunov

periodic orbits in the vicinity of the collinear equilibrium points when the primary

is a triaxial rigid body and the secondary an oblate spheroidal body together with

perturbations in the Coriolis and centrifugal forces;

ii. numerically determine the families C, A, and B (in Strӧ mgren‟s notation) around

1 L , 2 L and 3 L respectively for (i).

iii. compute horizontally Lyapunov periodic orbits around the collinear equilibrium

points and make an application to a binary system when the primaries are both

radiating and oblate spheroidal bodies;

iv. compute vertically Lyapunov periodic orbits and make applications to five binary

systems when the primaries are both radiating and oblate spheroidal bodies and

thereafter numerically continue to the corresponding families of periodic orbits;

v. compute and examine the horizontal and vertical stability of the Lyapunov

periodic orbits in the vicinity of the collinear equilibrium points when the

primaries are both radiating and triaxial spheroids together with small

perturbations in the Coriolis and centrifugal forces;

vi. numerically determine the families C, A, and B (in Strӧ mgren‟s notation) around

1 L , 2 L and 3 L respectively for (v).

1.5 Thesis Organization

The organization of this thesis has been done in six chapters: Chapter ONE gives the general

introduction, statement of the problem, justification of the study, aim and objectives of the

study and the theoretical framework. In chapter TWO, the review of related literature based

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on investigations carried out over the years on periodic orbits in the R3BP is given. The periodic motion around the collinear equilibrium points of the R3BP when the primary is a triaxial rigid body and the secondary is an oblate spheroid together with perturbations in the Coriolis and centrifugal forces was investigated in chapter THREE.

In chapter FOUR, Periodic motions around the collinear equilibrium points of the photogravitational R3BP with triaxiality, oblateness and small perturbations in the Coriolis and centrifugal forces was examined. In chapter FIVE, the results and discussions for all the problems investigated in this research work are given.

The summary, conclusion, and recommendation are given in chapter SIX. In the next chapter, we give the literature review as well the theoretical framework as related to this research.

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