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Long established as one of the premier references in the fields of astronomy, planetary science, and physics, the fourth edition of **Orbital Motion **continues to offer comprehensive coverage of the analytical methods of classical celestial mechanics while introducing the recent numerical experiments on the orbital evolution of gravitating masses and the astrodynamics of artificial satellites and interplanetary probes.

Following detailed reviews of earlier editions by distinguished lecturers in the USA and Europe, the author has carefully revised and updated this edition. Each chapter provides a thorough introduction to prepare you for more complex concepts, reflecting a consistent perspective and cohesive organization that is used throughout the book. A noted expert in the field, the author not only discusses fundamental concepts, but also offers analyses of more complex topics, such as modern galactic studies and dynamical parallaxes.

**New to the Fourth Edition:**

• Numerous updates and reorganization of all chapters to encompass new methods

• New results from recent work in areas such as satellite dynamics

• New chapter on the Caledonian symmetrical *n*-body problem

Extending its coverage to meet a growing need for this subject in satellite and aerospace engineering, **Orbital Motion, Fourth Edition** remains a top reference for postgraduate and advanced undergraduate students, professionals such as engineers, and serious amateur astronomers.

**PREFACE TO FOURTH EDITION**

**THE RESTLESS UNIVERSE**

Introduction

The Solar System

Stellar Motions

Clusters of Galaxies

Conclusion

Bibliography

**COORDINATE AND TIME-KEEPING SYSTEMS**

Introduction

Position on the Earth’s Surface

The Horizontal System

The Equatorial System

The Ecliptic System

Elements of the Orbit in Space

Rectangular Coordinate Systems

Orbital Plane Coordinate Systems

Transformation of Systems

Galactic Coordinate System

Time Measurement

Bibliography

**THE REDUCTION OF OBSERVATIONAL DATA**

Introduction

Observational Techniques

Refraction

Precession and Nutation

Aberration

Proper Motion

Stellar Parallax

Geocentric Parallax

Review of Procedures

Bibliography

**THE TWO-BODY PROBLEM**

Introduction

Newton’s Laws of Motion

Newton’s Law of Gravitation

The Solution to the Two-Body Problem

The Elliptic Orbit

The Parabolic Orbit

The Hyperbolic Orbit

The Rectilinear Orbit

Barycentric Orbits

Classification of Orbits with Respect to the Energy Constant

The Orbit in Space

The f and g Series

The Use of Recurrence Relations

Universal Variables

Bibliography

**THE MANY-BODY PROBLEM**

Introduction

The Equations of Motion in the Many-Body Problem

The Ten Known Integrals and Their Meanings

The Force Function

The Virial Theorem

Sundman’s Inequality

The Mirror Theorem

Reassessment of the Many-Body Problem

Lagrange’s Solutions of the Three-Body Problem

General Remarks on the Lagrange Solutions

The Circular Restricted Three Body Problem

The General Three-Body Problem

Jacobian Coordinates for the Many-Body Problem

The Hierarchical Three-Body Stability Criterion

Bibliography

**THE CALEDONIAN SYMMETRIC N-BODY PROBLEM**

Introduction

The Equations of Motions

Sundman’s Inequality

Boundaries of Real and Imaginary Motion

The Caledonian Symmetric Model for n = 1

The Caledonian Symmetric Model for n = 2

The Caledonian Symmetric Model for n = 3

The Caledonian Symmetric N-Body Model for odd N

Bibliography

**GENERAL PERTURBATIONS**

The Nature of the Problem

The Equations of Relative Motion

The Disturbing Function

The Sphere of Influence

The Potential of a Body of Arbitrary Shape

Potential at a Point within a Sphere

The Method of the Variation of Parameters

Lagrange’s Equations of Motion

Hamilton’s Canonic Equations

Derivation of Lagrange’s Planetary Equations from Hamilton’s Canonic Equations

Bibliography

**SPECIAL PERTURBATIONS**

Introduction

Factors in Special Perturbation Problems

Cowell’s Method

Encke’s Method

The Use of Perturbational Equations

Regularization Methods

Numerical Integrations Methods

Bibliography

**THE STABILITY AND EVOLUTION OF THE SOLAR SYSTEM**

Introduction

Chaos and Resonance

Planetary Ephemerides

The Asteroids

Rings, Shepherds, Tadpoles, Horseshoes, and Co-Orbitals

Near-Commensurable Satellite Orbits

Large-Scale Numerical Integrations

Empirical Stability Criteria

Conclusions

Bibliography

**LUNAR THEORY**

Introduction

The Earth–Moon System

The Saros

Measurement of the Moon’s Distance, Mass, and Size

The Moon’s Rotation

Selenographic Coordinates

The Moon’s Figure

The Main Lunar Problem

The Sun’s Orbit in the Main Lunar Problem

The Orbit of the Moon

Lunar Theories

The Secular Acceleration of the Moon

Bibliography

**ARTIFICIAL SATELLITES**

Introduction

The Earth as a Planet

Forces Acting on an Artificial Earth Satellite

The Orbit of a Satellite about an Oblate Planet

The Use of Hamilton–Jacobi Theory in the Artificial Satellite Problem

The Effect of Atmospheric Drag on an Artificial Satellite

Tesseral and Sectorial Harmonics in the Earth’s Gravitational Field

Bibliography

**ROCKET DYNAMICS AND TRANSFER ORBITS**

Introduction

Motion of a Rocket

Transfer between Orbits in a Single Central Force Field

Transfer Orbits in Two or More Force Fields

Bibliography

**INTERPLANETARY AND LUNAR TRAJECTORIES**

Introduction

Trajectories in Earth–Moon Space

Feasibility and Precision Study Methods

The Use of Jacobi’s Integral

The Use of the Lagrangian Solutions

The Use of Two-Body Solutions

Artificial Lunar Satellites

Interplanetary Trajectories

The Solar System as a Central Force Field

Minimum-Energy Interplanetary Transfer Orbits

The Use of Parking Orbits in Interplanetary Missions

The Effect of Errors in Interplanetary Orbits

Bibliography

**ORBIT DETERMINATION AND INTERPLANETARY NAVIGATION**

Introduction

The Theory of Orbit Determination

Laplace’s Method

Gauss’s Method

Olbers’ Method for Parabolic Orbits

Orbit Determination with Additional Observational Data

The Improvement of Orbits

Interplanetary Navigation

Bibliography

**BINARY AND OTHER FEW-BODY SYSTEMS**

Introduction

Visual Binaries

The Mass–Luminosity Relation

Dynamical Parallaxes

Eclipsing Binaries

Spectroscopic Binaries

Combination of Deduced Data

Binary Orbital Elements

The Period of a Binary

Apsidal Motion

Forces Acting on a Binary System

Triple Systems

The Inadequacy of Newton’s Law of Gravitation

The Figures of Stars in Binary Systems

The Roche Limits

Circumstellar Matter

The Origin of Binary Systems

Bibliography

**MANY-BODY STELLAR SYSTEMS**

Introduction

The Sphere of Influence

The Binary Encounter

The Cumulative Effect of Small Encounters

Some Fundamental Concepts

The Fundamental Theorems of Stellar Dynamics

Some Special Cases for a Stellar System in a Steady State

Galactic Rotation

Spherical Stellar Systems

Modern Galactic Studies

Bibliography

**ANSWERS TO PROBLEMS**

**APPENDIX I:**Astronomical and Related Constants

**APPENDIX II:**The Earth’s Gravitational Field

**APPENDIX III:**Mean Elements of the Planetary Orbits; Approximate Elements of the Ten Largest Asteroids

**APPENDIX IV:**Physical Elements of the Planets; Planetary Ring Systems; Satellite Elements and Dimensions

**INDEX**

### Biography

A.E. Roy, Professor Emeritus of Astronomy, Honorary Senior Research Fellow, University of Glasgow.

“Roy updates this fourth edition of an established text to include new research results … While the text is intended for advanced undergraduate and graduate students in disciplines ranging from astronomy and planetary science to aerospace and satellite engineering, its discussion of orbital computation will be of interest to serious amateur astronomers.”

—SciTech Book News, December 2006

“Each chapter is accompanied by exercises (with answers and some hints) that are designed to give the student confidence. This book retains its usefulness as a comprehensive text on introductory celestial mechanics… .”

— James Collett,Physical Science Educational Reviews,Vol. 7 Issue 1, June 2006

Praise for the Third Edition“…a classic text … on the orbits of everything from galactic clusters down to grapefruit-sized Earth satellites.”

—Planet Space Science