Introduction to the Mechanics of the Solar System

I. The Kinematics of a Single Planet 1. The Periods of the Planets Kepler's Problem Specification of Fixed Directions Lower and Upper Bounds for the Period of a Planet Approximate Determination of the Period The Periods of Mercury and Venus Observational Results 2. Kepler's Determination of the Orbits of the Earth and Mars: His First Law Introductory Remarks Kepler's Construction of the Earth's Orbit His Construction of Mars' Orbit Empirical Results 3. Kepler's Second and Third Laws Kepler's Second Law His Third Law Applications Summary 4. The Elements of a Planetary Orbit The System of Co-ordinates The Geometrical Elements Transformations of Co-ordinates The Kinematical Elements 5. The Computation of the Motion of a Planet A direct Method Derivation of Kepler's Equation Its Analytical Solution Successive Approximations Relations between Eccentric and True Anomalies The Vector of Position as a Function of the Elements and the Eccentric Anomaly Its Analytical Properties 6. Orbital Determinations and their Improvement Determination of the Elements from Two known Positions in Space Determination from Known Position and Velocity Vectors Determinations from Three Known Directions Correction of Elements The Influence of Observational Errors The Earth's Motion Aberration Time 7. Summary and Discussion Exercises NotesII. The Dynamics of a Single Planet 1. The Law of Gravity and the Law of Motion Postulates for the Concept of Force Preliminary Formulation of Newton's Laws Critical Remarks Final Form of Newton's Laws Summary The Constant of Gravitation 2. Systems of Ordinary Differential Equations The Principal Problem of Dynamics Successive Approximations Power Series Application to the Two Body Problem 3. The Two Body Problem The Integral of Area The Energy Integral The Relative Orbit The Relative Motion Parabolic Motions Rectilinear Motions The Relations between Period and Semi-major Axis The Masses of the Planets 4. The Determination of Orbits Laplace's Method Gaus's Method 5. Summary Exercises NotesIII. The Dynamics of the Planetary System 1. The Equations of Motion and their Integrals The Equations of Motion Critical Note Conservation of Momentum The Center of Mass Uniform Translations of the System of Co-ordinates Conservation of Moment of Momentum Generalization Conservation of Energy The Use of Known First Integrals Relative Motions 2. Perturbations in the Co-ordinates The Equations of Motion Fourier Series Expansions The Analytical Character of the Integrated Series Gaus's Interpretation of the Secular Term Determination of Neptune's Orbit Perturbations in the Elements 3. Perturbations in the Elements Variation of Parameters (Lagrange's Method) Variation of Parameters (Poisson's Method) Power Series of a Small Parameter Jacobi's Variational Equations Homogeneous Linear Systems Inhomogeneous Linear Systems Constant Coefficients Small Oscillations Periodic Coefficients 4.