# Motion in the Generalized Restricted Three-Body Problem

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**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 *L**n*1 , *L**n*2 due to the potential from the belt, and in the presence of all these perturbations, the equilibrium points *L*1 , *L*3 ,*L*4 ,*L*5 come nearer to the primaries; while *L*2 , *L**n*2 move towards the bigger primary and *L**n*1 moves away from it.

The collinear equilibrium points remain unstable, while the triangular points are stable in 0 < * *< ***c *and unstable in * *£ * *£ 1 , where *c *2 ***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**

- INTRODUCTION………………….. 1
- STATEMENT OF THE PROBLEM…………………… 2
- JUSTIFICATION/SIGNIFICANCE OF THE STUDY……….. 2
- OBJECTIVES OF THE STUDY……………… 3
- THEORETICAL FRAME WORK
- Circular Restricted Three-body Problem…. 3
- Radiation…………….. 8
- Oblateness Coefficients….. 10
- Stability of Equilibrium Points of a System of Differential Equations………… 11

**CHAPTER 2 LITERATURE REVIEW**

**CHAPTER 3: EQUATIONS OF MOTION**

- INTRODUCTION…………………….. 17
- MATHEMATICAL FORMULATIONS OF THE PROBLEM………………. 17
- NON-DIMENSIONAL UNITS OF MEASUREMENT………………… 24
- EQUATIONS OF MOTION IN THE NON-DIMENSIONAL UNITS…………… 25

3.4. THE JACOBIAN INTEGRAL……………. 26

- DISCUSSION…………………….. 26
- CONCLUSION……. 27

**CHAPTER 4: LOCATIONS OF EQUILIBRIUM POINTS**

- INTRODUCTION……………………………… 28
- LOCATIONS OF THE TRIANGULAR POINTS……….. 29
- Numerical Investigation of Triangular Points…………………. 32

- LOCATIONS OF COLLINEAR POINTS………………….. 33
- Numerical Investigations of Collinear Points……………………. 39

- DISCUSSION…………………………….. 41
- CONCLUSION…………….. 42

**CHAPTER 5: STABILITY OF EQUILIBRIUM POINT**S

- INTRODUCTION………………………………… 43
- VARIATIONAL EQUATIONS…………………. 43
- CHARACTERISTIC EQUATION…………………… 47
- STABILITY OF TRIANGULAR POINTS….. 49
- The Critical Mass Parameter
***c……………….*55

- The Critical Mass Parameter
- STABILITY OF COLLINEAR POINTS……… 57
- DISCUSSION………………………. 63
- CONCLUSION……….. 65

**CHAPTER 6: SUMMARY AND CONCLUSION**

- SUMMARY………….. 66
- CONCLUSION……… 67

REFERENCES…. 68

** 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 belong 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 extra solar planetary systems, which are regarded as young analogues of the kuiper belt in our solar system.

REFERENCES

Abdul Raheem, A. and Singh, J. (2006). Combined effects of perturbations, radiation, and oblateness on the stability of equilibrium points in the restricted three-body problem. The Astronomical Journal, 131: 1880–1885.

Allen, C. W. (1973). Astrophysical Quantites. London; Athlone press.

Aumann H.H.,Beichman,C.A.,Gillett, F.C.,de Jong, T., Houck,J.R., Low,F.J., Neugebauer,G.,Walker,R.G. and Wesselius,P.R. (1984). Discovery of a shell around Alpha Lyrae. Astrophysical Journal, 278: L23-L27.

Bhatnagar, K.B.and Chawla, J.M. (1979). A study of the Lagrangian points in the photogravitational restricted three-body problem. Indian J. Pure Appl. Math.10(11): 1443–1451.

Corben, H.C.and Stehle, P. (1977). Classical Mechanics, 2nd ed. Dover Publishers, New York.

Euler, L. (1772). Theoria Motuum Lunae, Typis Academiae Imperialis Scientiarum Petropoli. In: Opera Omnia, Series 2, ed. L. Courvoisier, 22, Lausanne:Orell Fussli Turici, 1958.

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., And Walker, H. J. (1998). A Dust Ring around Eridani: Analog to the young Solar System. The Astrophysical Journal, 506:L133 L137.

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