Theory of Orbital Motion

The following sections are included:

• Preface

• Contents

The following sections are included:

• Background

• Kepler's Laws of Planetary Motion

• Keplerian Ellipse

• Polar Equation of a Conic

• The Slope of a Tangent to an Ellipse

• General Formulation of Particle Dynamics

• Kinematics in a Two-Dimensional Plane

• Auxiliary Circular Reference Orbit

• Speeds at Various Points of an Orbital Ellipse

• General Theorems on Speeds of a Planet

• Velocity Components of a Planet

• Planetary Motion as the Sum of Circular and Linear Motions

• Orbit of the Earth and its Eccentricity

• Bode's Law

• Exercises

The following sections are included:

• Gravity and Gravitation

• Newton's Law of Gravitation

• Centripetal Force and Radius of Curvature

• Kepler's Laws and Newton's Law for Circular Orbits

• Newton's Law of Gravitation from Elliptical Orbits

• The Nature of Gravitational Force

• General Dynamics of a Particle

• Kepler's Third Law as an Extension of the First Two

• The Equation of Energy

• Energy of the Orbit

• The Virial Theorem of Clausius

• Exercises

The following sections are included:

• Concerning the Distance in Kepler's Third Law

• Average Values of the Radial Distance

• True and Eccentric Anomalies

• Kepler's Equation, Mean Anomaly and Mean Motion

• Average Values of Radial Distance using Eccentric Anomaly

• Locations of Extrema of Dynamical Variables

• Centripetal Acceleration in Planetary Motion

• The Jerk Vector in Planetary Motion

• Polar Coordinates in Reference to the Empty Focus

• Estimating the Eccentricity of Earth's Orbit

• Estimating the Distance of a Heavenly Body

• Exercises

The following sections are included:

• Central Forces

• General Dynamics of Angular Motion

• The Planetary Problem in Polar Coordinates

• Alternative Derivation of the Equation of Orbits

• Lagrangian and Hamiltonian Formulations of Classical Mechanics

• The Planetary Problem in Lagrangian and Hamiltonian Formulations

• Variational Principles in Classical Mechanics

• The Planetary Problem from the Variational Principle

• The Effective Potential Energy

• Determination of the Force Law from the Equation of Orbit

• A General Orbit Equation and the Force Law

• Stability of Circular Orbits under Central Forces

• The Precessing Ellipse as Superposition of Two Power Laws

• The Precession of Mercury's Perihelion

• The Runge–Lenz Vector

• Exercises

The following sections are included:

• The Hodograph

• Vector Hodographs in Uniform Circular Motion

• Orientation of the Orbital Ellipse

• Polar Coordinates of Special Points on the Orbital Ellipse

• Hodograph of the Position Vector in Planetary Motion

• Velocities at Special Points of an Elliptical Orbit

• Velocity Hodographs in Planetary Motion

• Acceleration Hodographs in Planetary Motion

• Hodographs of the Jerk Vector in Planetary Motion

• Hodographs of Rotational Quantities in Planetary Motion

• Exercises

The following sections are included:

• Cartesian Coordinates versus Polar Coordinates

• Equations of an Ellipse in Cartesian Coordinates

• Cartesian Coordinates of the Special Points on the Orbital Ellipse

• Kepler's Law of Areas in Cartesian Coordinates

• Inverse-Square Law in Cartesian Coordinates

• Velocity of a Planet in Cartesian Coordinates

• Velocities at Special Points of an Elliptical Orbit

• The Radius of Curvature in Cartesian Coordinates

• The Equation of Energy in Cartesian Coordinates

• The Equation of an Orbit in Cartesian Coordinates

• Acceleration of a Planet in Cartesian Coordinates

• The Jerk Vector in Cartesian Coordinates

• The Runge–Lenz Vector in Cartesian Coordinates

• Lagrange's Equations in Cartesian Coordinates

• The Two-Body Problem

• The Three-Body Problem

• Exercises

The following sections are included:

• Complex Numbers

• Graphical Representation of Complex Numbers

• Polar Coordinate Representation of Complex Numbers

• Complex Numbers as Coplanar Vectors

• Polar Coordinates Representation of Coplanar Vectors

• Rotation of a Vector in Complex Coordinates

• Central Force Motion in Complex Coordinates

• Conservation of Energy in Complex Coordinates

• Conservation of Angular Momentum in Complex Coordinates

• Conservation of the Runge–Lenz Vector in Complex Coordinates

• Exercises

The following sections are included:

• Keplerian Motion in the Solar System

• Attraction of a Thin Uniform Spherical Shell on a Point Mass

• Attraction of a Spherically-Symmetric Body on a Point Mass

• Physical Properties of the Earth and the Moon

• Velocities of Artificial Earth Satellites

• Periods of Artificial Earth Satellites

• Velocity of Escape from the Surface of the Earth

• Time of Travel to the Moon

• Times of Travel to Mars and Venus

• The Mass of the Sun from the Period of the Earth

• The Two-Body Problem in the Solar System

• Estimating the Mass of the Moon

• Determining the Masses of Venus and Mercury

• The Three-Body Problem in the Solar System

• Exercises

The following sections are included:

• Keplerian Motion in Three-Dimensional Space

• Newton's Law of Gravitation in Three Dimensions

• Conservation of Angular Momentum in Three Dimensions

• Conservation of Total Energy in Three Dimensions

• Constancy of the Runge–Lenz Vector in Three Dimensions

• The Equation of Orbit in Three Dimensions

• The Two-Body Problem in Three Dimensions

• The Planetary Problem in Rectangular Coordinates

• The Planetary Problem in Spherical Coordinates

• Exercises

The following sections are included:

• Artificial Earth Satellites

• Frames of Reference

• The Celestial Sphere, Right Ascension and Declination

• The Orbital Elements of a Satellite

• Three Fundamental Vectors

• Determination of Orbital Elements

• The Topocentric-Horizontal Coordinate System

• Approximate Solution of Kepler's Equation

• Ground Track of a Satellite

• Exercises

The following sections are included:

• Perturbations of Satellite Orbits

• Perturbative Forces on a Satellite in Orbit

• Effect of a Perturbing Force on the Semi-Major Axis

• Effect of a Perturbing Force on Eccentricity

• Effect of a Perturbing Force on Inclination

• Effect of a Perturbing Force on the Longitude of an Ascending Node

• Effect of a Perturbing Force on True Anomaly

• Effect of a Perturbing Force on the Argument of Perigee

• Effects of a Pertubing Force on the Period and Mean Motion

• Effects of Finite Impulse on the Orbital Elements

• Effects of Atmospheric Drag on Orbital Elements

• Effects of Earth's Gravitational Field on Orbital Elements

• Satellite Fragmentation and Orbital Debris

• Velocity Perturbations from Orbital Element Changes

• Exercises

The following sections are included:

• Appendix A The Ellipse and its Properties
  • The Ellipse in Everyday Life
  • The Definition of an Ellipse
  • Special Properties of the Ellipse
  • An Ellipse as Sections of Cones and Cylinders
  • Constructions of the Ellipse
  • General Equation of a Conic in Cartesian Coordinates
  • Equations of an Ellipse in Cartesian Coordinates
  • Equation of an Ellipse in Cartesian Coordinates with the Right Focus at the Origin
  • Parametric Equations of the Ellipse
  • The Polar Equation of a Conic
  • The Pedal Equation of the Ellipse
  • The Equation of an Ellipse in Eccentric Polar Coordinates
  • The Equation of an Ellipse in Complex Coordinates

• References

• Index