**Motion in the Generalized Restricted Three-Body Problem**

**ABSTRACT**

This thesis investigates motion in the generalized restricted three-body problem. It is generalized in the sense that both the primaries are radiating oblate bodies, together with the effect of gravitational potential from a belt. It derives the equations of motion, locates the positions of the equilibrium points, and examines their linear stability. It has been found that in addition to the usual five equilibrium points, there appear two new collinear points Ln1, n2 L due to the potential from the belt, and in the presence of all these perturbations, the equilibrium points 1 L, 3 L, 4 L, 5 L come nearer to the primaries; while 2 L, n2 L move towards the bigger primary and n1 L moves away from it. The collinear equilibrium points remain unstable, while the triangular points are stable in 0 c and unstable in 1, c 2 where c is the critical mass ratio influenced by the oblateness and radiation of the primaries and potential from the belt. This model can be applied in the study of binary systems, especially motion near oblate, radiating binary stars.

**TABLE OF CONTENTS**

TITLE PAGE…………………………………………………………………….…………..i

DECLARATION ……………………………………………………..……..………………ii

CERTIFICATION ……………………………………………………..…………………….iii

DEDICATION ……………………………………………………………………………..………iv

ACKNOWLEDGMENT………………………………………………………….……..……v

ABSTRACT …………………………………………………………………………..……vi

TABLE OF CONTENTS……………….……..………………………..…….……………vii

LIST OF FIGURES…………………….……………………………………….…………..ix

LIST OF TABLES…………………………………………………………………………….ix

CHAPTER 1: GENERAL INTRODUCTION

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

1.1 STATEMENT OF THE PROBLEM………………………………………….…………2

1.2 JUSTIFICATION/SIGNIFICANCE OF THE STUDY……………………….…….…..2

1.3 OBJECTIVES OF THE STUDY…………………………………………………………3

1.4 THEORETICAL FRAMEWORK

1.4.1 Circular Restricted Three-body Problem………………………………………3

1.4.2 Radiation …………………………………………………………………..….8

1.4.3 Oblateness Coefficients……………….…………………………..…………..10

1.4.4 Potential of the Belt………………………………………….…….………….10

1.4.5 Stability of Equilibrium Points of a System of Differential Equations……….11

CHAPTER 2

LITERATURE REVIEW………………………………………….…..……………….12

CHAPTER 3: EQUATIONS OF MOTION

3.0 INTRODUCTION ………………………………………………….……..……..……17

3.1 MATHEMATICAL FORMULATIONS OF THE PROBLEM…………………………17

3.2 NON-DIMENSIONAL UNITS OF MEASUREMENT……………………………….24

3.3 EQUATIONS OF MOTION IN THE NON-DIMENSIONAL UNITS………….…….25

3.4. THE JACOBIAN INTEGRAL………………………………………………..…….….26

3.5 DISCUSSION ………………………………………………………………..… ….…26

3.6 CONCLUSION……………………………………………………………..…….……27

CHAPTER 4: LOCATIONS OF EQUILIBRIUM POINTS

4.0 INTRODUCTION…………………………………………………..…………………..28

4.1 LOCATIONS OF THE TRIANGULAR POINTS…………………………….……….29

4.1.1 Numerical Investigation of Triangular Points………………………….……..32

4.2 LOCATIONS OF COLLINEAR POINTS…………………………………..……….33

4.2.1 Numerical Investigations of Collinear Points…………………….………….39

4.3 DISCUSSION ………………………………………………………………..………..41

4.4 CONCLUSION …………………………………………………….…..….…………42

CHAPTER 5: STABILITY OF EQUILIBRIUM POINTS

5.0 INTRODUCTION………………………………………………………..……………43

5.1 VARIATIONAL EQUATIONS ………………………………………………………..43

5.2 CHARACTERISTIC EQUATION………………………………………………..……47

5.3 STABILITY OF TRIANGULAR POINTS……………………………………..……..49

5.3.1 The Critical Mass Parameter. c …………………………………………55

5.4 STABILITY OF COLLINEAR POINTS………………………………………………57

5.5 DISCUSSION ………………………………………………………………..….……63

5.6 CONCLUSION ………………….……………………………………….……………65

CHAPTER 6: SUMMARY AND CONCLUSION

6.1 SUMMARY………………………………………………………….……….….……..66

6.2 CONCLUSION………………………………………………….………………………67

REFERENCES……………………..……………………….…………………………….68

LIST OF FIGURES

Figure:

1.1 Planer view of the rotating frame of reference………………………………………….5

1.2 Positions of the equilibrium points……………………………………..………………7

3.1. The rotating frame of reference………………………………………………………………………….18

5.1 Graph of g(x) against x…………………………………………………………………58

**CHAPTER ONE**

**INTRODUCTION**

The restricted three-body problem is a famous model of classical mechanics. It describes the motion of an infinitesimal mass moving under the gravitational effects of the two finite masses, called primaries, which move in circular orbits around their center of mass on account of their mutual attraction and the infinitesimal mass not influencing the motion of the primaries. The approximate circular motion of the planets around the sun and the small masses of asteroids and the satellites of planets compared to the planet’s masses originally suggested the formulation of the restricted problem. In certain stellar dynamics problems, it is altogether inadequate to consider solely the gravitational interaction force. For example, when a star acts upon a particle in a cloud of gas and dust, the dominant factor is by no means gravity, but the repulsive force of the radiation pressure (Poynting, 1903). Since a large fraction of all-stars belongs to binary systems Allen (1973), the particle motion in the field of a double star offers special interest.

There are disks of dust with various masses in the extrasolar planetary systems, which are regarded as young analogs of the Kuiper belt in our solar system. The gravitational potential due to these belts also has a great influence on the infinitesimal body (Jiang and Yeh, 2004a).

The participating bodies in the classical restricted three-body problem are strictly spherical, but in actual situations, we find that some of the natural and artificial bodies moving in the space do not point to masses or spherical rather they are oblate bodies. For instance, Saturn, Jupiter, Regulus (star) and Peanut binary stars are sufficiently oblate.

Singh and Ishwar (1999), pointed out that, lack of sphericity or oblateness of the primaries affects the motion of an infinitesimal body. The motions of artificial Earth satellites are examples.

In the classical problem, the effects of the gravitational attraction of the infinitesimal body and other perturbations have been ignored. Thus, the classical restricted three-body problem is inadequate to explain the motion of the infinitesimal body in the presence of any perturbing forces such as radiation pressure, oblateness of a body, and gravitational potential from a belt.

Hence, it becomes imperative to modify the classical restricted three-body problem by including some of these perturbing forces.

**1.1 STATEMENT OF THE PROBLEM**

Consider an infinitesimal mass (e.g. dust particle) moving in the orbital plane of oblate binary stars. Then, the problem is to study its motion in the generalized restricted three-body problem. The problem is generalized in the sense that both the primaries are radiating oblate bodies, together with the effect of gravitational potential from the belt. Thus we are to study the combined effect of radiation and oblateness of the primaries and gravitational potential from a belt on the stability of equilibrium points in a restricted three-body problem.

**1.2 JUSTIFICATION/SIGNIFICANCE OF THE STUDY**

The human species stands on the edge of a new frontier, the transition from a planet-bound to a space-faring civilization. Just as the transition from hunter-gatherer to farmer necessitated new approaches to solve new problems, so the expansion into space, in terms of dynamics of an artificial satellite, requires the formulation of new models that include the effects of some of the perturbing forces on the satellite. Motion in the generalized restricted three-body problem is one of such models that considered the effects of radiation and oblateness of the primaries and gravitational potential from a belt on the satellite. Thus, this model will be very helpful in the study of binary stars, especially dynamics near oblate binary stars. We choose the primaries as oblate spheroid: peanut stars, the two stars appear to be nearly identical, each 15 to 20 times the mass of our sun (Jenks, 2008).

**1.3 OBJECTIVES OF THE STUDY**

The objectives of the study include:

1. To derive equations of motion of an infinitesimal body under the influence of radiating oblate primaries and gravitational potential from a belt in the restricted three-body problem.

2. To examine the effects of these perturbations on the locations of the equilibrium

points;

3. To investigate the effects of these perturbations on the linear stability of the equilibrium points in a restricted three-body problem.

**1.4 THEORETICAL FRAMEWORK**

The outline of the theoretical bases on which the problem is built is given here:

**1.4.1 Circular Restricted Three-body Problem**

The three-body problem involves the motion of three celestial bodies under their mutual gravitational attraction. It is an old problem and logically follows from the two-body problem which was solved by Newton in his Principia in 1687. Newton also considered the three-body problem in connection with the motion of the Moon under the influences of the Sun and the Earth, the consequences of which included a headache. Unlike the two-body problem, there is no closed-form analytical solution for the differential equations governing the motion in the three-body problem. However, it is still possible, although not easy, to gain insight into the qualitative nature of the solutions in this system.

This task is more tractable if several simplifying assumptions are introduced. In reducing the general three-body equations, the first assumption is that the mass of one of the bodies is infinitesimal, that is, it does not affect the motion of the other two bodies. Thus, the two massive bodies, or primaries, move-in Keplerian orbits about their common center of mass.

This reduced model is called the restricted three-body problem and was formalized by Euler in the late 18th century (Szebehely, 1967). The problem is further simplified by constraining the primaries to move in circular orbits about their center of mass and are kept fixed on the x-axis, the (x, y), the plane is the plane of motion of the primaries, and the z-axis is orthogonal to the (x, y) plane. These coordinates are sometimes called synodical. The resulting simplified model has usually labeled the circular restricted three-body problem (CR3BP). Although a less complex dynamical model than the general problem (in terms of the number of equations and the number of dependent variables), analysis in the circular problem offers a further understanding of the motion in a regime that is of increasing interest to space science.

In the restricted circular three-body problem, the units are usually chosen in such a way that the properties of the system depend on a single parameter.

**References**

Abdul Raheem, A., Singh, J.: Astron. J. 131, 1880 (2006)

Collins, G.W.: Celest. Mech. 119 (2004)

Das, M.K., Narang, P., Mahajan, S., Yuasa, M.: Astrophys. Space Sci. 314, 261 (2008)

Greaves, J.S., Holland, W.S., Moriarty-Schieven, G., Jenness, T., Dent, W.R.F., Zuckerman, B., Mccarthy, C., Webb, R.A., Butner, H.M., Gear, W.K., Walker, H.J.: Astrophys. J. 506, 133 (1998)

Ishwar, B., Elipe, A.: Astrophys. Space Sci. 277, 67 (2001)

Jiang, I.G., Yeh, L.C.: Int. J. Bifurc. Chaos 13, 534 (2003)

Jiang, I.G., Yeh, L.C.: Astrophys. Space Sci. 305, 341–348 (2006)

Kushvah, B.S.: Astrophys. Space Sci. 318, 41 (2008)

Luk’yanov, L.G.: Sov. Astron. 33, 92 (1989)

Miyamoto, M., Nagai, R.: Publ. Astron. Soc. Jpn. 27, 533 (1975)

Poynting, J.H.: Philos. Trans. R. Soc. Lond. 202, 525 (1903)

Radzievskii, V.V.: Astron. J. 27(5), 250 (1950)

Radzievskii, V.V.: Astron. Zh. 30, 225 (1953)

Sharma, R.K.: Astrophys. Space Sci. 135, 271 (1987)

Sharma, R.K., Subba Rao, P.V.: Celest. Mech. 12, 189 (1975)

Sharma, R.V., Taqvi, Z.A., Bhatnagar, K.B.: Indian J. Pure Appl. Math. 32(2), 255 (2001a)

Sharma, R.V., Taqvi, Z.A., Bhatnagar, K.B.: Indian J. Pure Appl. Math. 32(7), 981 (2001b)

Simmons, J.F.L., McDonald, J.C., Brown, J.C.: Celest. Mech. 35, 145 (1985)

Singh, J., Ishwar, B.: Bull. Astron. Soc. India 27, 415–424 (1999)

Singh, J.: Astron. J. 137, 3286–3292 (2009)

Singh, J., Begha, J.M.: Astrophys. Space Sci. 331, 511–519 (2011)

Szebehely, V.: Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press, New York (1967)

Trilling, D.E., Stansberry’J, A., Stapelfeldt, K.R., Rieke, G.H., Su, K.Y.L., Gray, R.O., Corbally, C.J., Bryden, G., Chen, C.H., Boden, A., Beichman, C.A.: Astrophys. J. 658, 1289 (2007)

Yeh, L.C., Jiang, I.G.: Astrophys. Space Sci. 306, 189 (2006)

*Copyright © 2023 Author(s) retain the copyright of this article.
This article is published under the terms of the Creative Commons Attribution License 4.0*

**If you like this article, see others like it:**

- 11 Simple Pure and Applied Mathematics Project Topics and Ideas
- Dynamic Buckling of Imperfection-Sensitive Elastic Structures Under Slowly-Varying Time Dependent Loading
- Mathematical Modelling And Control Of Blood Glucose/Insulin Concentrations in An Insulin Dependent Diabetic Subject
- Iterative Approximation of Equilibrium Points of Evolution Equations
- Convergence in Norm of Modified Krasnoselskii-Mann Iteration for Fixed Points of Asymptotically Demicontractive Mappings