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

Notes

II. 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

Notes

III. 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. The Perturbation Equations of Celestial Mechanics

Survey

Poisson's Equation

Perturbation of the Major Axis

Perturbations of Eccentricity

Inclination and Nodal Line

The First Integral for the Direction of the Perihelion

Perturbation of the Direction of the Perihelion

Perturbation of the Perihelion Time

Summary


5. Perturbations in the Motions of the Planets


6. Perturbations in the Motion of the Moon

The Disturbing Force as a Function of Position

The Disturbing Force as a Function of the Time

Perturbation of the Semi-major Axis

The Perturbations of the Nodal Line, Inclination and Eccentricity

The Perturbation of the Apsidal Line

The Perturbation of the Solar Motion


7. The Perturbation of the Perihelion of Mercury

The General Formula

An Hypothesis about the Disturbing Force


8. Lagrange's Solution of the Three Body Problem

The Triangular Solution

The Rectilinear Solutions


9. The Problème Restreint

The Equations of Motion

Jacobi's Integral

Lagrange's Solutions


10. Summary

Exercises

Notes

IV. The Planets and the Moon as Rigid Bodies


1. Perturbations of the Orbits of Satellites caused by Flattening of the Planets

The Disturbing Force

Assumption of Rigid Body

The Displacement of the Periastron

The Rotation of the Nodal Line


2. The Rotation of the Earth

An Approximate Theory

The Exact Equation

Geometry of Rotations

Euler's Dynamical Equations

Euler's Geometrical Equations

The Free Motion of the Earth's Axis

The Equations for its Forced Motion

Precession and Nutation

Numerical Data


3. The Rotation of the Moon

Qualitative Considerations

Outline of a Quantitative Treatment


4. Summary and Conclusion

Exercises

Notes

Index