Introduction to the Mechanics of the Solar System

Introduction to the Mechanics of the Solar System

1st Edition - January 1, 1959

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  • Author: Rudolf Kurth
  • eBook ISBN: 9781483147284

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Introduction to the Mechanics of the Solar System introduces the reader to the mechanics of the solar system and covers topics ranging from the periods of the planets to their flattening and its effects on the orbits of satellites. Kepler's three laws of planetary motion are also discussed, along with the law of gravity; the two-body problem; and perturbations in the motions of the moon and the planets. This book is comprised of four chapters and begins with an analysis of the kinematics of a single planet, focusing on the work of Johannes Kepler, particularly his determination of the orbits of the Earth and Mars and his formulation of his three laws of planetary motion. The following chapters explore systems of ordinary differential equations; determination of orbits using Laplace's method and Gauss' method; the equations of motion and their integrals; the perturbation equations of celestial mechanics; and Lagrange's solution of the three-body problem. The notations of the Earth and the moon are also considered. This monograph is intended for astronomers and astronomy students.

Table of Contents

  • 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



    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



    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


    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



    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


    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


    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


    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



    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




Product details

  • No. of pages: 188
  • Language: English
  • Copyright: © Pergamon 1959
  • Published: January 1, 1959
  • Imprint: Pergamon
  • eBook ISBN: 9781483147284

About the Author

Rudolf Kurth

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