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9789812709127
Preface vii
Kepler's Laws of Planetary Motion 1
Background 1
Kepler's Laws of Planetary Motion 2
Keplerian Ellipse 3
Polar Equation of a Conic 5
The Slope of a Tangent to an Ellipse 6
General Formulation of Particle Dynamics 8
Kinematics in a Two-Dimensional Plane 8
Auxiliary Circular Reference Orbit 11
Speeds at Various Points of an Orbital Ellipse 13
General Theorems on Speeds of a Planet 15
Velocity Components of a Planet 18
Planetary Motion as the Sum of Circular and Linear Motions 21
Orbit of the Earth and its Eccentricity 22
Bode's Law 24
Newton's Law of Gravitation 26
Gravity and Gravitation 26
Newton's Law of Gravitation 27
Centripetal Force and Radius of Curvature 27
Kepler's Laws and Newton's Law for Circular Orbits 30
Newton's Law of Gravitation from Elliptical Orbits 31
The Nature of Gravitational Force 35
General Dynamics of a Particle 37
Kepler's Third Law as an Extension of the First Two 39
The Equation of Energy 41
Energy of the Orbit 43
The Virial Theorem of Clausius 45
Average and Extremum Values of Variables 47
Concerning the Distance in Kepler's Third Law 47
Average Values of the Radial Distance 49
True and Eccentric Anomalies 52
Kepler's Equation, Mean Anomaly and Mean Motion 54
Average Values of Radial Distance using Eccentric Anomaly 55
Locations of Extrema of Dynamical Variables 59
Centripetal Acceleration in Planetary Motion 61
The Jerk Vector in Planetary Motion 64
Polar Coordinates in Reference to the Empty Focus 69
Estimating the Eccentricity of Earth's Orbit 72
Estimating the Distance of a Heavenly Body 76
The Central Force Problem 78
Central Forces 78
General Dynamics of Angular Motion 79
The Planetary Problem in Polar Coordinates 80
Alternative Derivation of the Equation of Orbits 83
Lagrangian and Hamiltonian Formulations of Classical Mechanics 86
The Planetary Problem in Lagrangian and Hamiltonian Formulations 88
Variational Principles in Classical Mechanics 89
The Planetary Problem from the Variational Principle 90
The Effective Potential Energy 92
Determination of the Force Law from the Equation of Orbit 94
A General Orbit Equation and the Force Law 97
Stability of Circular Orbits under Central Forces 100
The Precessing Ellipse as Superposition of Two Power Laws 102
The Precession of Mercury's Perihelion 104
The Runge-Lenz Vector 106
Vector Hodographs in Planetary Motion 112
The Hodograph 112
Vector Hodographs in Uniform Circular Motion 113
Orientation of the Orbital Ellipse 115
Polar Coordinates of Special Points on the Orbital Ellipse 116
Hodograph of the Position Vector in Planetary Motion 116
Velocities at Special Points of an Elliptical Orbit 118
Velocity Hodographs in Planetary Motion 120
Acceleration Hodographs in Planetary Motion 122
Hodographs of the Jerk Vector in Planetary Motion 123
Hodographs of Rotational Quantities in Planetary Motion 124
Planetary Motion in Cartesian Coordinates 127
Cartesian Coordinates versus Polar Coordinates 127
Equations of an Ellipse in Cartesian Coordinates 128
Cartesian Coordinates of the Special Points on an Orbital Ellipse 131
Kepler's Law of Areas in Cartesian Coordinates 132
Inverse-Square Law in Cartesian Coordinates 134
Velocity of a Planet in Cartesian Coordinates 135
Velocities at Special Points of an Elliptical Orbit 136
The Radius of Curvature in Cartesian Coordinates 138
The Equation of Energy in Cartesian Coordinates 142
The Equation of an Orbit in Cartesian Coordinates 142
Acceleration of a Planet in Cartesian Coordinates 143
The Jerk Vector in Cartesian Coordinates 144
The Runge-Lenz Vector in Cartesian Coordinates 144
Lagrange's Equations in Cartesian Coordinates 145
The Two-Body Problem 146
The Three-Body Problem 151
Planetary Problem in Complex Coordinates 158
Complex Numbers 158
Graphical Representation of Complex Numbers 159
Polar Coordinate Representation of Complex Numbers 160
Complex Numbers as Coplanar Vectors 160
Polar Coordinates Representation of Coplanar Vectors 162
Rotation of a Vector in Complex Coordinates 163
Central Force Motion in Complex Coordinates 164
Conservation of Energy in Complex Coordinates 165
Conservation of Angular Momentum in Complex Coordinates 166
Conservation of the Runge-Lenz Vector in Complex Coordinates 166
Keplerian Motion in the Solar System 169
Keplerian Motion in the Solar System 169
Attraction of a Thin Uniform Spherical Shell on a Point Mass 170
Attraction of a Spherically-Symmetric Body on a Point Mass 171
Physical Properties of the Earth and the Moon 172
Velocities of Artificial Earth Satellites 174
Periods of Artificial Earth Satellites 175
Velocity of Escape from the Surface of the Earth 176
Time of Travel to the Moon 177
Times of Travel to Mars and Venus 178
The Mass of the Sun from the Period of the Earth 180
The Two-Body Problem in the Solar System 181
Estimating the Mass of the Moon 182
Determining the Masses of Venus and Mercury 183
The Three-Body Problem in the Solar System 184
Planetary Motion in Three-Dimensional Space 189
Keplerian Motion in Three-Dimensional Space 189
Newton's Law of Gravitation in Three Dimensions 189
Conservation of Angular Momentum in Three Dimensions 191
Conservation of Total Energy in Three Dimensions 192
Conservation of the Runge-Lenz Vector in Three Dimensions 193
The Equation of Orbit in Three Dimensions 194
The Two-Body Problem in Three Dimensions 195
The Planetary Problem in Rectangular Coordinates 196
The Planetary Problem in Spherical Coordinates 199
Motion of Artificial Earth Satellites 205
Artificial Earth Satellites 205
Frames of Reference 206
The Celestial Sphere, Right Ascension and Declination 207
Orbital Elements of a Satellite 208
Three Fundamental Vectors 210
Determination of Orbital Elements 211
The Topocentric-Horizontal Coordinate System 215
Approximate Solution of Kepler's Equation 217
Ground Track of a Satellite 218
Perturbations of Satellite Orbits 221
Perturbations of Satellite Orbits 221
Perturbative Forces on a Satellite in Orbit 222
Effect of a Perturbing Force on the Semi Major Axis 224
Effect of a Perturbing Force on Eccentricity 227
Effect of a Perturbing Force on Inclination 228
Effect of a Perturbing Force on the Longitude of an Ascending Node 229
Effect of a Perturbing Force on True Anomaly 232
Effect of a Perturbing Force on the Argument of Perigee 233
Effects of a Pertubing Force on the Period and Mean Motion 235
Effects of Finite Impulse on the Orbital Elements 236
Effects of Atmospheric Drag on Orbital Elements 237
Effects of Earth's Gravitational Field on Orbital Elements 240
Satellite Fragmentation and Orbital Debris 242
Velocity Perturbations from Orbital Element Changes 243
The Ellipse and its Properties 251
The Ellipse in Everyday Life 251
The Definition of an Ellipse 253
Special Properties of the Ellipse 255
An Ellipse as Sections of Cones and Cylinders 257
Constructions of the Ellipse 258
General Equation of a Conic in Cartesian Coordinates 263
Equations of an Ellipse in Cartesian Coordinates 264
Equation of an Ellipse in Cartesian Coordinates with the Right Focus at the Origin 266
Parametric Equations of the Ellipse 267
The Polar Equation of a Conic 268
The Pedal Equation of the Ellipse 269
The Equation of an Ellipse in Eccentric Polar Coordinates 270
The Equation of an Ellipse in Comple Coordinates 270
References 272
Index 276
- ISBN-10:
- 9812709126
- ISBN-13:
- 9789812709127
- Pub. Date:
- 01/07/2008
- Publisher:
- World Scientific Publishing Company, Incorporated
- ISBN-10:
- 9812709126
- ISBN-13:
- 9789812709127
- Pub. Date:
- 01/07/2008
- Publisher:
- World Scientific Publishing Company, Incorporated
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Overview
Orbital motion is a vital subject which has engaged the greatest minds in mathematics and physics from Kepler to Einstein. It has gained in importance in the space age and touches every scientist in any field of space science. Still, there is almost a total dearth of books in this important field at the elementary and intermediate levels — at best a chapter in an undergraduate or graduate mechanics course.This book addresses that need, beginning with Kepler's laws of planetary motion followed by Newton's law of gravitation. Average and extremum values of dynamical variables are treated and the central force problem is formally discussed. The planetary problem in Cartesian and complex coordinates is tackled and examples of Keplerian motion in the solar system are also considered. The final part of the book is devoted to the motion of artificial Earth satellites and the modifications of their orbits by perturbing forces of various kinds.
Product Details
| ISBN-13: | 9789812709127 |
|---|---|
| Publisher: | World Scientific Publishing Company, Incorporated |
| Publication date: | 01/07/2008 |
| Pages: | 282 |
| Product dimensions: | 6.29(w) x 8.96(h) x 0.60(d) |
Table of Contents
Preface vii
Kepler's Laws of Planetary Motion 1
Background 1
Kepler's Laws of Planetary Motion 2
Keplerian Ellipse 3
Polar Equation of a Conic 5
The Slope of a Tangent to an Ellipse 6
General Formulation of Particle Dynamics 8
Kinematics in a Two-Dimensional Plane 8
Auxiliary Circular Reference Orbit 11
Speeds at Various Points of an Orbital Ellipse 13
General Theorems on Speeds of a Planet 15
Velocity Components of a Planet 18
Planetary Motion as the Sum of Circular and Linear Motions 21
Orbit of the Earth and its Eccentricity 22
Bode's Law 24
Newton's Law of Gravitation 26
Gravity and Gravitation 26
Newton's Law of Gravitation 27
Centripetal Force and Radius of Curvature 27
Kepler's Laws and Newton's Law for Circular Orbits 30
Newton's Law of Gravitation from Elliptical Orbits 31
The Nature of Gravitational Force 35
General Dynamics of a Particle 37
Kepler's Third Law as an Extension of the First Two 39
The Equation of Energy 41
Energy of the Orbit 43
The Virial Theorem of Clausius 45
Average and Extremum Values of Variables 47
Concerning the Distance in Kepler's Third Law 47
Average Values of the Radial Distance 49
True and Eccentric Anomalies 52
Kepler's Equation, Mean Anomaly and Mean Motion 54
Average Values of Radial Distance using Eccentric Anomaly 55
Locations of Extrema of Dynamical Variables 59
Centripetal Acceleration in Planetary Motion 61
The Jerk Vector in Planetary Motion 64
Polar Coordinates in Reference to the Empty Focus 69
Estimating the Eccentricity of Earth's Orbit 72
Estimating the Distance of a Heavenly Body 76
The Central Force Problem 78
Central Forces 78
General Dynamics of Angular Motion 79
The Planetary Problem in Polar Coordinates 80
Alternative Derivation of the Equation of Orbits 83
Lagrangian and Hamiltonian Formulations of Classical Mechanics 86
The Planetary Problem in Lagrangian and Hamiltonian Formulations 88
Variational Principles in Classical Mechanics 89
The Planetary Problem from the Variational Principle 90
The Effective Potential Energy 92
Determination of the Force Law from the Equation of Orbit 94
A General Orbit Equation and the Force Law 97
Stability of Circular Orbits under Central Forces 100
The Precessing Ellipse as Superposition of Two Power Laws 102
The Precession of Mercury's Perihelion 104
The Runge-Lenz Vector 106
Vector Hodographs in Planetary Motion 112
The Hodograph 112
Vector Hodographs in Uniform Circular Motion 113
Orientation of the Orbital Ellipse 115
Polar Coordinates of Special Points on the Orbital Ellipse 116
Hodograph of the Position Vector in Planetary Motion 116
Velocities at Special Points of an Elliptical Orbit 118
Velocity Hodographs in Planetary Motion 120
Acceleration Hodographs in Planetary Motion 122
Hodographs of the Jerk Vector in Planetary Motion 123
Hodographs of Rotational Quantities in Planetary Motion 124
Planetary Motion in Cartesian Coordinates 127
Cartesian Coordinates versus Polar Coordinates 127
Equations of an Ellipse in Cartesian Coordinates 128
Cartesian Coordinates of the Special Points on an Orbital Ellipse 131
Kepler's Law of Areas in Cartesian Coordinates 132
Inverse-Square Law in Cartesian Coordinates 134
Velocity of a Planet in Cartesian Coordinates 135
Velocities at Special Points of an Elliptical Orbit 136
The Radius of Curvature in Cartesian Coordinates 138
The Equation of Energy in Cartesian Coordinates 142
The Equation of an Orbit in Cartesian Coordinates 142
Acceleration of a Planet in Cartesian Coordinates 143
The Jerk Vector in Cartesian Coordinates 144
The Runge-Lenz Vector in Cartesian Coordinates 144
Lagrange's Equations in Cartesian Coordinates 145
The Two-Body Problem 146
The Three-Body Problem 151
Planetary Problem in Complex Coordinates 158
Complex Numbers 158
Graphical Representation of Complex Numbers 159
Polar Coordinate Representation of Complex Numbers 160
Complex Numbers as Coplanar Vectors 160
Polar Coordinates Representation of Coplanar Vectors 162
Rotation of a Vector in Complex Coordinates 163
Central Force Motion in Complex Coordinates 164
Conservation of Energy in Complex Coordinates 165
Conservation of Angular Momentum in Complex Coordinates 166
Conservation of the Runge-Lenz Vector in Complex Coordinates 166
Keplerian Motion in the Solar System 169
Keplerian Motion in the Solar System 169
Attraction of a Thin Uniform Spherical Shell on a Point Mass 170
Attraction of a Spherically-Symmetric Body on a Point Mass 171
Physical Properties of the Earth and the Moon 172
Velocities of Artificial Earth Satellites 174
Periods of Artificial Earth Satellites 175
Velocity of Escape from the Surface of the Earth 176
Time of Travel to the Moon 177
Times of Travel to Mars and Venus 178
The Mass of the Sun from the Period of the Earth 180
The Two-Body Problem in the Solar System 181
Estimating the Mass of the Moon 182
Determining the Masses of Venus and Mercury 183
The Three-Body Problem in the Solar System 184
Planetary Motion in Three-Dimensional Space 189
Keplerian Motion in Three-Dimensional Space 189
Newton's Law of Gravitation in Three Dimensions 189
Conservation of Angular Momentum in Three Dimensions 191
Conservation of Total Energy in Three Dimensions 192
Conservation of the Runge-Lenz Vector in Three Dimensions 193
The Equation of Orbit in Three Dimensions 194
The Two-Body Problem in Three Dimensions 195
The Planetary Problem in Rectangular Coordinates 196
The Planetary Problem in Spherical Coordinates 199
Motion of Artificial Earth Satellites 205
Artificial Earth Satellites 205
Frames of Reference 206
The Celestial Sphere, Right Ascension and Declination 207
Orbital Elements of a Satellite 208
Three Fundamental Vectors 210
Determination of Orbital Elements 211
The Topocentric-Horizontal Coordinate System 215
Approximate Solution of Kepler's Equation 217
Ground Track of a Satellite 218
Perturbations of Satellite Orbits 221
Perturbations of Satellite Orbits 221
Perturbative Forces on a Satellite in Orbit 222
Effect of a Perturbing Force on the Semi Major Axis 224
Effect of a Perturbing Force on Eccentricity 227
Effect of a Perturbing Force on Inclination 228
Effect of a Perturbing Force on the Longitude of an Ascending Node 229
Effect of a Perturbing Force on True Anomaly 232
Effect of a Perturbing Force on the Argument of Perigee 233
Effects of a Pertubing Force on the Period and Mean Motion 235
Effects of Finite Impulse on the Orbital Elements 236
Effects of Atmospheric Drag on Orbital Elements 237
Effects of Earth's Gravitational Field on Orbital Elements 240
Satellite Fragmentation and Orbital Debris 242
Velocity Perturbations from Orbital Element Changes 243
The Ellipse and its Properties 251
The Ellipse in Everyday Life 251
The Definition of an Ellipse 253
Special Properties of the Ellipse 255
An Ellipse as Sections of Cones and Cylinders 257
Constructions of the Ellipse 258
General Equation of a Conic in Cartesian Coordinates 263
Equations of an Ellipse in Cartesian Coordinates 264
Equation of an Ellipse in Cartesian Coordinates with the Right Focus at the Origin 266
Parametric Equations of the Ellipse 267
The Polar Equation of a Conic 268
The Pedal Equation of the Ellipse 269
The Equation of an Ellipse in Eccentric Polar Coordinates 270
The Equation of an Ellipse in Comple Coordinates 270
References 272
Index 276
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