Celestial mechanics and astrodynamics theory and practice

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Celestial mechanics and astrodynamics theory and practice

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Astrophysics and Space Science Library 436 Pini Gurfil P Kenneth Seidelmann Celestial Mechanics and Astrodynamics: Theory and Practice Celestial Mechanics and Astrodynamics: Theory and Practice Astrophysics and Space Science Library EDITORIAL BOARD Chairman W B BURTON, National Radio Astronomy Observatory, Charlottesville, Virginia, U.S.A (bburton@nrao.edu); University of Leiden, The Netherlands (burton@strw.leidenuniv.nl) F BERTOLA, University of Padua, Italy C J CESARSKY, Commission for Atomic Energy, Saclay, France P EHRENFREUND, Leiden University, The Netherlands O ENGVOLD, University of Oslo, Norway A HECK, Strasbourg Astronomical Observatory, France E P J VAN DEN HEUVEL, University of Amsterdam, The Netherlands V M KASPI, McGill University, Montreal, Canada J M E KUIJPERS, University of Nijmegen, The Netherlands H VAN DER LAAN, University of Utrecht, The Netherlands P G MURDIN, Institute of Astronomy, Cambridge, UK B V SOMOV, Astronomical Institute, Moscow State University, Russia R A SUNYAEV, Space Research Institute, Moscow, Russia More information about this series at http://www.springer.com/series/5664 Pini Gurfil • P Kenneth Seidelmann Celestial Mechanics and Astrodynamics: Theory and Practice 123 Pini Gurfil Faculty of Aerospace Engineering Technion-Israel Institute of Technology Haifa, Israel P Kenneth Seidelmann Department of Astronomy The University of Virginia Charlottesville, USA ISSN 0067-0057 ISSN 2214-7985 (electronic) Astrophysics and Space Science Library ISBN 978-3-662-50368-3 ISBN 978-3-662-50370-6 (eBook) DOI 10.1007/978-3-662-50370-6 Library of Congress Control Number: 2016943837 © Springer-Verlag Berlin Heidelberg 2016 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Cover illustration: Technion’s Space Autonomous Mission for Swarming and Geo-locating Nanosatellites (SAMSON) Credit: Asher Space Research Institute, Technion Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Berlin Heidelberg I dedicate this book to my children, Eytam, Oshri, and Ohav, and to my parents, Arie and Sara Pini Gurfil This book is dedicated to Bobbie Seidelmann and our family, Holly, Kent, Jutta, Alan, Karen, and Sarah P Kenneth Seidelmann Also we dedicate the book to the scientists that preceded us and taught, mentored, and inspired us Foreword The early contributions to artificial satellite orbit theory were mostly made by the celestial mechanicians, e.g., Brouwer, Garfinkel, Vinti and Kozai Then, as aerospace engineering curricula emerged, their astrodynamics graduates began to make contributions Most of the recent astrodynamics books have been written by engineering graduates This book, co-authored by a celestial mechanician, Ken Seidelmann, and an astrodynamicist, Pini Gurfil, is a welcome addition to the aerospace community as it merges the two backgrounds Chapter begins with a short history of celestial mechanics and then transitions to introductions to some of the key topics covered in the book Topics included that are not usually seen in astrodynamics books are stability, chaos, Poincaré sections, KAM (Kolmogorov-Arnold-Moser) theory, and observation systems Chapter covers the basic math and physics concepts needed for the subjects in the book Chapter provides an excellent discussion of coordinate systems and introduces relativity, a subject not usually included in astrodynamics books but certainly present in celestial mechanics, e.g., the precession of Mercury’s perihelion Chapters and provide a thorough discussion of the central force and two-body problems Included is a section on Einstein’s modification of the orbit equation The focus of Chap is initial orbit determination Chapter provides a thorough discussion of the N-body problem and the integrals associated with this problem Chapter then addresses the special case of the circular restricted 3-body problem (CR3BP) The coverage of the CR3BP is more comprehensive than found in most astrodynamics books and includes a discussion of families of periodic orbits Chapter is an introduction to numerical procedures used in astrodynamics and celestial mechanics This chapter is not a comprehensive coverage and comparison of numerical integration methods but an introduction to the methods needed to understand numerical methods and error computation Chapter 10 begins a group of five chapters that this writer considers very important for astrodynamics and celestial mechanics but is often not found in astrodynamics books I believe that the motion under the influence of conservative perturbations, those derivable from a potential, is best addressed and understood vii viii Foreword using Hamiltonian mechanics and perturbation methods such as Lie series Chapter 10 discusses the basics of Hamiltonian mechanics, canonical transformations, generating functions, and Jacobi’s theorem and applies these to the two-body problem The focus of Chap 11 is perturbation methods, and it begins with an excellent discussion of the variation of parameters (VOP), which leads to Lagrange’s planetary equations Then, with the perturbations expressed as specific disturbing accelerations instead of the accelerations obtained from a potential, Gauss’ variational equations are derived for the accelerations in the radial, transverse, and orbit normal directions and the tangential, normal, and orbit normal directions Included is a discussion of Lagrange brackets, which are needed for the VOP Also in this chapter is the presentation of the Kustaanheimo-Stiefel variables Using the foundations developed in Chap 10, Chap 11 addresses the solution for the 3rd body perturbations, atmospheric drag, and gravitational potential Then Chap 12 focuses on the solution for motion about an oblate planet There are many such solutions beginning with Brouwer’s 1959 paper, and presenting even a few solutions would be prohibitive The solution presented here is the Cid-Lahulla radial intermediary Special perturbation (numerical integration) methods are the most accurate and the general perturbation analytical methods, e.g, Brouwer’s solution, are the most efficient Chapter 13 presents the semianalytical approach, which is more efficient than numerical integration and more accurate than the analytical solution The method is then applied to the four problems, a LEO satellite perturbed by drag, frozen orbits, sun-synchronous and repeat ground track orbits, and the motion of a geosynchronous satellite Chapters 10–13 address the problem of the motion of a space object under the influence of forces derivable from a potential except for the section on the effects of atmospheric drag Chapters 14 and 15 consider the problem of the control of a space object using both continuous and impulsive control Chapter 14 considers the control of specific types of orbits such as sun-synchronous orbits, frozen orbits, and geosynchronous orbits, as well as gravity assists Both impulsive and continuous thrust control are addressed Chapter 15 provides a very thorough coverage of the well-known problem of optimal impulsive orbit transfers Chapter 16 addresses the problem of orbit data processing and presents batch least squares and recursive filtering Also discussed is the use of polynomials for the compression/representation of ephemerides Chapter 17 provides a summary of the problem of space debris including probability of collision and collision avoidance maneuvers The book concludes with another discussion of main contributors to celestial mechanics and the early pioneers of astrodynamics Entire books have been written on the subjects presented in many of the chapters in this book Thus, when writing a book on astrodynamics, there has to be a balance between the amount of material presented and the necessary balance of mathematical rigor and its application to the problem at hand I believe this book has achieved such a balance There is a breadth of topics and each one is presented with the necessary depth needed for the reader to understand the topic The book can Foreword ix be used for a senior/1st-year graduate class in astrodynamics and also for a 2nd-year graduate class in astrodynamics It is a pleasure for me to write this Foreword and recommend this book to the astrodynamics community Texas A&M University, College Station, TX, USA Kyle T Alfriend ... laws and Newton’s Principia Celestial mechanics has evolved into a myriad of approaches, methods, and results, some of which are the bases for astrodynamics Indeed, celestial mechanics and astrodynamics. . .Celestial Mechanics and Astrodynamics: Theory and Practice Astrophysics and Space Science Library EDITORIAL BOARD Chairman W B BURTON,... methods and applications common to celestial mechanics and astrodynamics The book includes classical and emerging topics, manifesting the state of the art and beyond The book contains homogenous and

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  • Foreword

  • Preface

  • Acknowledgments

  • Contents

  • List of Figures

  • List of Tables

  • Notation and Acronyms

  • 1 Introduction

    • 1.1 Definitions

    • 1.2 History

    • 1.3 Properties of Conics

      • 1.3.1 The Ellipse, 0 < e < 1

      • 1.3.2 The Parabola, e = 1

      • 1.3.3 The Hyperbola, e > 1

    • 1.4 Astronomical Background

    • 1.5 Stability and Chaos

      • 1.5.1 Three-Body Problem

      • 1.5.2 Solar System

      • 1.5.3 Resonances, Singularities and Regularization

    • 1.6 Stability Determination

      • 1.6.1 Poincaré Surface of Section

      • 1.6.2 Hill Stability

      • 1.6.3 Lyapunov

      • 1.6.4 Kolmogorov-Arnold-Moser Theorem

      • 1.6.5 Spacecraft Orbit Stability

    • 1.7 Chaos Determination

    • 1.8 Observational Data

      • 1.8.1 Transit Circle

      • 1.8.2 Photographic

      • 1.8.3 Radar Observations

      • 1.8.4 Laser Ranging

      • 1.8.5 VLBI

      • 1.8.6 CCDs

      • 1.8.7 Optical Interferometry

      • 1.8.8 Surveys

      • 1.8.9 GNSS

      • 1.8.10 Satellite Observations

    • References

  • 2 Vectors

    • 2.1 Introduction

    • 2.2 Scalar Product

    • 2.3 Vector Product

    • 2.4 Triple Scalar and Vector Products

    • 2.5 Velocity of Vector

    • 2.6 Rotation of Axes

    • 2.7 Angular Velocity

    • 2.8 Rotating Axes

    • 2.9 Gradient of a Scalar

    • 2.10 Momentum and Energy

      • 2.10.1 Simple Harmonic Motion

      • 2.10.2 Linear Motion in an Inverse Square Field

      • 2.10.3 Foucoult's Pendulum

    • References

  • 3 Reference Systems and Relativity

    • 3.1 Reference Systems

    • 3.2 Relativistic Coordinate Systems

      • 3.2.1 Newtonian Coordinates

      • 3.2.2 Relativistic Coordinates

      • 3.2.3 ICRS, BCRS, GCRS

      • 3.2.4 Geodesic Precession and Nutation

    • 3.3 Reference Frames

      • 3.3.1 Celestial Reference Frames

      • 3.3.2 CIP and CIO

      • 3.3.3 Equation of Equinoxes

      • 3.3.4 Equation of Origins

      • 3.3.5 Terrestrial Reference Frames

      • 3.3.6 Terrestrial Intermediate Origin

      • 3.3.7 ECEF, ECI, ECR

      • 3.3.8 Satellite Geodesy

      • 3.3.9 GNSS Reference Systems

    • 3.4 Time Scales

    • 3.5 Coordinate Systems

      • 3.5.1 Origins and Planes

      • 3.5.2 Horizon Reference Frame

      • 3.5.3 Geocentric Coordinates

      • 3.5.4 Geodetic Coordinates

      • 3.5.5 Geographic Coordinates

      • 3.5.6 Astronomical Coordinates

    • 3.6 Kinematics of the Earth

      • 3.6.1 Earth Orientation

      • 3.6.2 Precession

      • 3.6.3 Nutation

      • 3.6.4 Polar Motion

    • 3.7 Observation Effects

      • 3.7.1 Aberration

      • 3.7.2 Proper Motion

      • 3.7.3 Radial Velocities

      • 3.7.4 Parallax

      • 3.7.5 Refraction

      • 3.7.6 Relativistic Light Deflection

      • 3.7.7 Space Motion

      • 3.7.8 Tidal Effects

    • 3.8 Earth Satellite Equations of Motion in GCRS

    • References

  • 4 Central Force Motion

    • 4.1 Introduction

    • 4.2 Law of Areas

    • 4.3 Linear and Angular Velocities

    • 4.4 Integrals of Angular Momentum and Energy

    • 4.5 Equation of the Orbit

    • 4.6 Inverse Square Law

      • 4.6.1 Eccentricity Vector

      • 4.6.2 From Orbit to Force Law

    • 4.7 Einstein's Modification of the Orbit Equation

    • 4.8 Universality of Newton's Law

    • References

  • 5 The Two-Body Problem

    • 5.1 Introduction

    • 5.2 Classical Orbital Elements

      • 5.2.1 Osculating Orbital Elements

      • 5.2.2 Nonsingular Orbital Elements

    • 5.3 Motion of the Center of Mass

    • 5.4 Relative Motion

    • 5.5 The Integral of Areas

    • 5.6 Elements of the Orbit from Position and Velocity

    • 5.7 Properties of Motion

    • 5.8 The Constant of Gravitation

    • 5.9 Kepler's Equation

      • 5.9.1 Series Expansion

      • 5.9.2 Differential Method

    • 5.10 Position in the Elliptic Orbit

    • 5.11 Position in the Parabolic Orbit

    • 5.12 Position in a Hyperbolic Orbit

    • 5.13 Position on the Celestial Sphere

      • 5.13.1 Heliocentric Coordinates

      • 5.13.2 Geocentric Coordinates

    • References

  • 6 Orbit Determination

    • 6.1 Introduction

    • 6.2 Known Radius Vectors

    • 6.3 Laplace's Method

    • 6.4 Gauss's Method

    • 6.5 Lambert's Theorem

    • 6.6 Parabolic Orbits, Olber's Method

    • 6.7 Circular Orbits

    • References

  • 7 The n-Body Problem

    • 7.1 Introduction

    • 7.2 Equations of Motion

    • 7.3 Angular Momentum, or Areal Velocity, Integral

    • 7.4 Integral of Energy

    • 7.5 Stationary Solutions of the Three-Body Problem

    • 7.6 Generalization to n Bodies

    • 7.7 Equations of Relative Motion

    • 7.8 Energy Integral and Force Function

    • References

  • 8 The Restricted Three-Body Problem

    • 8.1 Introduction

    • 8.2 Equations of Motion

    • 8.3 The Jacobi Constant

    • 8.4 Zero Velocity Curves

    • 8.5 The Lagrangian Points

    • 8.6 Stability of Motion Near the Lagrangian Points

    • 8.7 Hill's Restricted Three-Body Problem

      • 8.7.1 Equations of Motion

      • 8.7.2 Hill's Equations of Motion

      • 8.7.3 Families of Periodic Orbits

    • References

  • 9 Numerical Procedures

    • 9.1 Differences and Sums

    • 9.2 Interpolation

    • 9.3 Lagrangian Methods

    • 9.4 Differentiation

    • 9.5 Integration

    • 9.6 Differential Equations

    • 9.7 Errors

    • 9.8 Numerical Integration

    • 9.9 Numerical Integration by Runge-Kutta Methods

    • 9.10 Accumulation of Errors in Numerical Integration

    • 9.11 Numerical Integration of Orbits

      • 9.11.1 Equations for Cowell's Method

      • 9.11.2 Equations for Encke's Method

      • 9.11.3 Comparison of Cowell's and Encke's Methods

    • 9.12 Equations with Origin at the Center of Mass

    • 9.13 Integration with Augmented Mass of the Sun

    • References

  • 10 Canonical Equations

    • 10.1 Introduction

    • 10.2 Canonical Form of the Equations

    • 10.3 Eliminating the Time Dependency

    • 10.4 Integral of a System of Canonical Equations

    • 10.5 Canonical Transformation of Variables

      • 10.5.1 Necessary Condition

      • 10.5.2 Sufficient Condition

    • 10.6 Examples of Canonical Transformations

      • 10.6.1 Change of Variables by Means of a Generating Function

      • 10.6.2 Conjugate Variables to Qj

    • 10.7 Jacobi's Theorem

    • 10.8 Canonical Equations for the Two-Body Problem

    • 10.9 Application of Jacobi's Theorem to the Two-body Problem

      • 10.9.1 Meaning of the Constants a

      • 10.9.2 Variables Conjugate to Qi

      • 10.9.3 Application to the General Problem

    • 10.10 The Delaunay Variables

    • 10.11 The Lagrange Equations

    • 10.12 Small Eccentricity and Small Inclination

      • 10.12.1 Small Eccentricity

      • 10.12.2 Small Inclination

      • 10.12.3 Universal Variables

    • References

  • 11 General Perturbations Theory

    • 11.1 Introduction

    • 11.2 Variation of Parameters

    • 11.3 Properties of the Lagrange Brackets

    • 11.4 Evaluation of the Lagrange Brackets

    • 11.5 Solution of the Perturbation Equations

    • 11.6 Case I: Radial, Transverse, and Orthogonal Components

    • 11.7 Case II: Tangential, Normal, and Orthogonal Components

    • 11.8 Expansion of the Third-Body Potential

      • 11.8.1 The Factor ( r/r' ) 2

      • 11.8.2 The Factor P2(cosϕ)

    • 11.9 The Earth-Moon System

    • 11.10 Expansion of the Gravitational Potential

    • 11.11 Atmospheric Drag

    • 11.12 Regularization of Perturbed Motion

    • References

  • 12 Motion Around Oblate Planets

    • 12.1 Introduction

    • 12.2 Axially-Symmetric Gravitational Field

    • 12.3 Equatorial Motion

      • 12.3.1 The Orbital Angle and Radial Period

      • 12.3.2 New Orbital Elements

      • 12.3.3 Open Orbits and the Escape Velocity

      • 12.3.4 Circular Orbits

    • 12.4 The Cid-Lahulla Approach

      • 12.4.1 Polar-Nodal Coordinates

      • 12.4.2 The Cid-Lahulla Radial Intermediary

      • 12.4.3 Comparison with Brouwer's Approximation

    • 12.5 Solution for Motion in a Cid-Lahulla Potential

      • 12.5.1 Main Steps Towards a Solution

      • 12.5.2 New Independent Variable

    • References

  • 13 Semianalytical Orbit Theory

    • 13.1 Introduction

    • 13.2 Preliminaries

    • 13.3 Semianalytical Models

      • 13.3.1 The Zonal Part of the Geopotential

      • 13.3.2 Second-Order Effects

      • 13.3.3 The Tesseral-Sectorial Part of the Geopotential

      • 13.3.4 Atmospheric Drag

    • 13.4 Frozen Orbits

    • 13.5 Sun-synchronous and Repeat Ground-track Orbits

    • 13.6 Geostationary Orbits

      • 13.6.1 In-Plane Motion

      • 13.6.2 Out-of-Plane Motion

      • 13.6.3 Averaged Solution

      • 13.6.4 The Perturbed Problem

    • References

  • 14 Satellite Orbit Control

    • 14.1 Introduction

    • 14.2 Stability and Control of Dynamical Systems

    • 14.3 Impulsive and Continuous Maneuvers

    • 14.4 Gravity Assist Maneuvers

      • 14.4.1 Multiple Gravity Assists

      • 14.4.2 Concatenation Rules

    • 14.5 Optimization of Orbits

      • 14.5.1 Static Optimization

      • 14.5.2 Dynamic Optimization

    • 14.6 Linear Orbit Control

    • 14.7 Low Earth Orbit Control

      • 14.7.1 Altitude Correction

      • 14.7.2 Frozen Orbit Control

      • 14.7.3 Sun-synchronous Orbit Control

      • 14.7.4 Repeat Ground-track Orbit Control

    • 14.8 Geostationary Orbit Control

      • 14.8.1 North-South Stationkeeping

      • 14.8.2 East-West Stationkeeping

      • 14.8.3 Eccentricity Correction

    • 14.9 Nonlinear Feedback Control of Orbits

    • 14.10 Fixed-Magnitude Continuous-Thrust Orbit Control

    • 14.11 Comparison of Continuous-Thrust Controllers

    • References

  • 15 Optimal Impulsive Orbit Transfers

    • 15.1 Introduction

    • 15.2 Modified Hohmann Transfer

    • 15.3 Modified Bi-Elliptic and Bi-Parabolic Transfers

      • 15.3.1 Definitions

      • 15.3.2 Modified Bi-Elliptic Transfer

        • 15.3.2.1 Calculating Ycrit

        • 15.3.2.2 Evaluating the effect of X on maneuvers where Y=Ycrit

        • 15.3.2.3 Extending the evaluation to include all Ycrit<Y

      • 15.3.3 Modified Bi-Parabolic Transfer

    • 15.4 Comparison Between the Modified Bi-Parabolic and the Modified Hohmann Transfers

      • 15.4.1 Bi-Elliptic Transfer

      • 15.4.2 Bi-Parabolic Transfer

    • References

  • 16 Orbit Data Processing

    • 16.1 Introduction

    • 16.2 Principle of Least Squares

    • 16.3 Least Squares Approximation

    • 16.4 Orthogonal Polynomials

    • 16.5 Chebyshev Series

      • 16.5.1 Chebyshev Approximation

      • 16.5.2 Other Polynomial Approximations

    • 16.6 Fourier Approximation: Continuous Range

    • 16.7 Fourier Approximation: Discrete Range

    • 16.8 Optimum Polynomial Interpolation

    • 16.9 Chebyshev Interpolation

    • 16.10 Economization of Power Series

    • 16.11 Recursive Filtering

    • 16.12 Mean Elements Estimator

      • 16.12.1 Initial Conditions and Parameter Values

      • 16.12.2 Uncontrolled Orbits, Single Run

      • 16.12.3 Orbits with No Control Inputs, Monte-CarloRuns

      • 16.12.4 Impulsive Maneuvers

      • 16.12.5 Continuous Thrust

    • References

  • 17 Space Debris

    • 17.1 Introduction

    • 17.2 SGP4 Propagator and TLE

    • 17.3 Sizing the Debris

    • 17.4 Time of Closest Approach

    • 17.5 Probability of Collision

    • 17.6 Calculating the Required v

    • References

  • 18 People, Progress, Prospects

    • 18.1 People and Progress

    • 18.2 Future Prospects: Exoplanets

      • 18.2.1 History

      • 18.2.2 Observations

        • 18.2.2.1 Direct Imaging

        • 18.2.2.2 Radial Velocities

        • 18.2.2.3 Transits

        • 18.2.2.4 Astrometry

        • 18.2.2.5 Microlensing

      • 18.2.3 Types of Exoplanets

      • 18.2.4 Orbit Determinations

      • 18.2.5 Planetary Systems

      • 18.2.6 Habitable Zone

      • 18.2.7 Observing Program

    • References

  • Bibliography

  • Index

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