THE CIRCULAR RESTRICTED THREE-BODY PROBLEM
RICHARD FRNKA
ABSTRACT. The restricted three body problem considers two large finite masses
orbiting circularly around their common center of mass. A significantly smaller
third body is introduced, and the objective is to find the motion of this body
under the gravitational influence of the two larger masses. This paper undertakes
a thorough investigation of the equations of motion that define the behavior of the
third body. These equations demonstrate chaotic behavior so it is necessary to
understand Lagrange points, stability of those points, and zero velocity curves to
comprehend how the smaller mass is affected. Using these ideas allows for greater
understanding of the unpredictable nature of the motion of the body and provides
astronomers with useful tools as they look at the motion of asteroids and satellites
entering two-body systems.
CONTENTS
List of Figures 2
1. Introduction 3
2. Initializing the Problem 4
3. Equations of Motion 5
4. Lagrange Points 10
4.1. Stability of Lagrange Points 12
4.2. Quasi-periodic orbits around L4 and L5 points 17
5. Zero Velocity Curves 18
1
,2 RICHARD FRNKA
6. Practical Applications 25
7. Conclusion 26
8. Acknowledgements 27
References 27
LIST OF FIGURES
1 3-Body Visualization 5
2 Common µ values 9
3 Common Systems and Lagrange Points 11
4 Location of Lagrange Points 12
5 Quasi-Periodic Orbit Around L4 17
6 Poincare Section of Periodic Orbit 18
7 Zero Velocity Curves 20
8 L1 Contour 21
9 Restricted Orbit from Left of L1 Point 22
10 Restricted Orbit from Right of L1 Point 22
11 Restricted Orbit from Left of L2 Point 23
12 Restricted Orbit from Right of L3 Point 24
, THE CIRCULAR RESTRICTED THREE-BODY PROBLEM 3
1. INTRODUCTION
The three-body problem has been an interesting topic for mathematicians and
physicists for centuries. The statement of the problem is simple: ”Three particles
move in space under their mutual gravitational attraction; given their initial condi-
tions, determine their subsequent motion.”[1] Newton first explored the idea in 1687,
trying to determine the differential equations that would solve for the motion of the
bodies. Although many other scientists have studied the problem, it was relatively
unsolved until Poincare made groundbreaking work in the area in 1890.
The typical three-body problem involves 18 first order differential equations. Through
use of conservation equations and calculus, it can be reduced to 6. It has still not
been solved because there are not enough conservation quantities to allow for further
simplification. We can, however, look at a more restricted case to get a general idea
of what will happen for limited examples.
The restricted three-body problem has two large masses that orbit their common
center of mass in a circular fashion. A third body (that is significantly smaller than
the other two) is introduced into the system. The goal of the problem is then to
deduce the motion of this object as it experiences the gravitational forces of both
larger bodies. Though the limitations that have been introduced seem to restrict
the problem to the point of impracticality, there are many situations in the solar
system to which the circular restricted three-body problem applies. Examples such
as a comet entering the Sun-Earth system or a satellite traveling in the Earth-Moon
system are both applications in which physicists and astronomers are extremely
RICHARD FRNKA
ABSTRACT. The restricted three body problem considers two large finite masses
orbiting circularly around their common center of mass. A significantly smaller
third body is introduced, and the objective is to find the motion of this body
under the gravitational influence of the two larger masses. This paper undertakes
a thorough investigation of the equations of motion that define the behavior of the
third body. These equations demonstrate chaotic behavior so it is necessary to
understand Lagrange points, stability of those points, and zero velocity curves to
comprehend how the smaller mass is affected. Using these ideas allows for greater
understanding of the unpredictable nature of the motion of the body and provides
astronomers with useful tools as they look at the motion of asteroids and satellites
entering two-body systems.
CONTENTS
List of Figures 2
1. Introduction 3
2. Initializing the Problem 4
3. Equations of Motion 5
4. Lagrange Points 10
4.1. Stability of Lagrange Points 12
4.2. Quasi-periodic orbits around L4 and L5 points 17
5. Zero Velocity Curves 18
1
,2 RICHARD FRNKA
6. Practical Applications 25
7. Conclusion 26
8. Acknowledgements 27
References 27
LIST OF FIGURES
1 3-Body Visualization 5
2 Common µ values 9
3 Common Systems and Lagrange Points 11
4 Location of Lagrange Points 12
5 Quasi-Periodic Orbit Around L4 17
6 Poincare Section of Periodic Orbit 18
7 Zero Velocity Curves 20
8 L1 Contour 21
9 Restricted Orbit from Left of L1 Point 22
10 Restricted Orbit from Right of L1 Point 22
11 Restricted Orbit from Left of L2 Point 23
12 Restricted Orbit from Right of L3 Point 24
, THE CIRCULAR RESTRICTED THREE-BODY PROBLEM 3
1. INTRODUCTION
The three-body problem has been an interesting topic for mathematicians and
physicists for centuries. The statement of the problem is simple: ”Three particles
move in space under their mutual gravitational attraction; given their initial condi-
tions, determine their subsequent motion.”[1] Newton first explored the idea in 1687,
trying to determine the differential equations that would solve for the motion of the
bodies. Although many other scientists have studied the problem, it was relatively
unsolved until Poincare made groundbreaking work in the area in 1890.
The typical three-body problem involves 18 first order differential equations. Through
use of conservation equations and calculus, it can be reduced to 6. It has still not
been solved because there are not enough conservation quantities to allow for further
simplification. We can, however, look at a more restricted case to get a general idea
of what will happen for limited examples.
The restricted three-body problem has two large masses that orbit their common
center of mass in a circular fashion. A third body (that is significantly smaller than
the other two) is introduced into the system. The goal of the problem is then to
deduce the motion of this object as it experiences the gravitational forces of both
larger bodies. Though the limitations that have been introduced seem to restrict
the problem to the point of impracticality, there are many situations in the solar
system to which the circular restricted three-body problem applies. Examples such
as a comet entering the Sun-Earth system or a satellite traveling in the Earth-Moon
system are both applications in which physicists and astronomers are extremely
