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by J.M.A. Danby, hardbound, 466 pages, $29.95.

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Optional Source Code and demonstration programs in Borland BASIC (Details here) $15.95
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Optional Source Code and demonstration programs in Borland Pascal (Details here)$15.95
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Fundamentals of Celestial Mechanics is an introductory text that should be accessible to a reader having a background in calculus and elementary differential equations. The original edition (published in 1962) has been radically revised, and emphasis is placed on computation. The numerical analysis needed for the computations is derived, and sample programs (run on a IBM-PC) are included. There are introductory chapters on the astronomical background and on vectorial mechanics. Sections dealing with the problem of two bodies include the use of universal variables, several methods (including that of Laguerre) for solving Kepler's equation, and three methods for solving the two point boundary value problem. The chapter on the determination of orbits includes two versions of Gauss' method, the application of least squares and an introduction to recursive methods. The chapter on numerical methods has been expanded, and includes three methods for the numerical integration of differential equations, one of which has full stepsize control. There are also chapters on perturbations, the three- and n-body problems, the motion of the Moon and the rotations of the Earth and Moon. The appendix includes numerical tables and derivations of properties of conic sections that are used in the text. The text includes several hundred problems, and suggested computer projects. The computer listings found in the book are available separately, along with a selection of other programs, on CDROM.

The text is accessible to undergraduates or graduate students in mathematics, physics, engineering or astronomy. As minimum requirements, a user is assumed to have taken introductory courses in calculus, differential equations and mechanics. Other topics, such as numerical analysis, are developed in the text as needed. There is plenty of material for two semesters; so a one semester course can be designed in many ways (even with the exclusion of all computing).

From the Reviewers:

…The select group of scientists and engineers who dedicate their career to celestial mechanics consider the author’s book of fundamental importance. The original edition of 1962 appeared when the author worked at Yale University with several of the giants in our field (Brouwer, Clemence, Eckert, Hagihara, Herget, etc.). The considerable influence of the original edition on our field is surpassed by the present second, revised and significantly enlarged edition. The author often referred to his book as "Fun in celestial mechanics", and while this was certainly true for the original edition, it is even more applicable to the second edition. The significant additions appearing in this edition emphasize computations and numerical analysis essential in celestial mechanics. The author’s inclusion of the listing of some programs is most welcome and will help students and will save considerable time. His use of BASIC as the language for program listing might be a controversial selection but it is considered one of the excellent languages for scientific computations.

The author’s competence as a teacher offering a panoramic view of his subject is best represented in the chapter dedicated to numerical procedures. His discussion of the concepts of "respectable errors" (round-off and truncation errors) and "blunders" shows how to lead the reader along the beautiful but hard path of numerical celestial mechanics. The book contains many well-selected examples and problems, offers a solid astronomical background, gives details of physical and orbital elements of planets and of comets, and lists basic references.

Since the first artificial satellite was placed in orbit, initiating the space age, many introductory textbooks have appeared in the literature. The author’s book is distinguished from those of his rivals for several reasons which make it one of the outstanding texts: clear, readable and understandable style, emphasis on numerical approaches, offering astronomical background and use of vector analysis. This excellent introductory textbook is strongly recommended for undergraduate courses in celestial mechanics, orbit dynamics and astrodynamics.

                                                                                                 Mathematical Reviews — American Mathematical Society

 A new edition of Fundamentals of Celestial Mechanics is a welcome sight to those of us who teach this subject to advanced undergraduates or beginning graduate students. . . One of the best features of the original was its large variety of problems and exercises, and the new edition has even more...The expanded sections and new material have broadened the text's applications. . . this second edition of Danby's Fundamentals of Celestial Mechanics will also become a classic.

                                                                                                      Sky & Telescope magazine

This second edition has been substantially revised and enlarged and demonstrates in a dramatic manner how the use of computers of all sizes has allowed teachers and students alike to gain a much deeper understanding of a subject requiring substantial numerical computation....(it) is most certainly a valuable addition to any teacher's personal library. It is also cheap enough for students, who are interested in the finer details of numerical computation of orbits, to purchase their own copy.
                                                                                                                   The Observatory


Table of Contents

1. The Astronomical Background 1
        1.1. Introduction 1
        1.2. Some Definitions 2
        1.3. Orbital Definitions 3
        1.4. Kepler’s Laws 4
        1.5. The Astronomical Unit 5
        1.6. Bode’s Law 5
        1.7. Astronomical Observations 6
        1.8. The Celestial Sphere 6
        1.9. Precession, Nutation, and Variation of Latitude 9
        1.10. The True and Apparent Places of a Celestial Object 10
        1.11. The Measurement of Time 11
2. Introduction to Vectors 13
        2.1. Scalars and Vectors 13
        2.2. The Law of Addition 15
        2.3. The Scalar Product 19
        2.4. The Vector Product 22
        2.5. The Velocity of a Vector 27
        2.6. Angular Velocity 29
        2.7. Rotating Axes 31
        2.8. The Gradient of a Scalar 36
        2.9. Spherical Trigonometry 37
3. Introduction To Vectorial Mechanics 41
        3.1. Forces as Vectors 41
        3.2. Basic Definitions 41
        3.3. Newton’s Laws of Motion 44
        3.4. The Laws of Energy and Momentum 45
        3.5. Simple Harmonic Motion 47
        3.6. Motion in a Uniform Field, Subject to Resistance Proportional
                        to the Velocity 48
        3.7. Linear Motion in an Inverse Square Field 49
        3.8. Foucault’s Pendulum 50
        3.9. The Equation of Motion of a Rocket, Subject to its Own Propulsion 52
        3.10. Problems 53
4. Central Orbits 57
        4.1. General Properties 57
        4.2. The Stability of Circular Orbits 59
        4.3. Further Basic Formulas 62
        4.4. Newtonian Attraction 63
        4.5. Einstein’s Modification of the Equation of the Orbit 67
        4.6. The Case f(r) = n
2r 68
        4.7. The Case f(r) =  m/ r3: Cotes’ Spirals 69
        4.8. To Find the Law of Force, Given the Orbit 71
        4.9. The “Universality” of Newton’s Law 74
        4.10. Worked Examples 78
        4.11. Problems 82
5 Some Properties of Solid Bodies 89
        5.1. Center of Mass and Center of Gravity 89
        5.2. The Moments and Products of Inertia: The Inertia Tensor 90
        5.3. The Potential of a Sphere 96
        5.4. The Potential of a Distant Body: MacCullagh’s Formula 100
        5.5. The Field of a Homogeneous Ellipsoid 102
        5.6. Laplace’s Equation, Legendre Polynomials, Potential of the Earth 112
        5.7. The Tidal Distortion of a Liquid Sphere Under the Action
                         of a Distant Point Mass 117
        5.8. Ellipsoidal Figures of Rotating Fluid Masses 120
6. Chapter 6 The Two-Body Problem 125
        6.1. The Motion of the Center of Mass 125
        6.2. The Relative Motion 127
        6.3. The Orbit in Time 129
        6.4. Some Properties of the Motion 138
        6.5. The Choice of Units 146
        6.6. The Solution of Kepler’s Equation 149
        6.7. The f and g Functions 162
        6.8. The Initial Value Problem I 165
        6.9. Universal Variables 168
        6.10. The Initial Value Problem II 178
        6.11. The Two-Point Boundary Value Problem I—Application of
                        Lambert’s Theorem 180
        6.12. The Two-Point Boundary Value Problem II—Gauss’ Method 191
        6.13. The Two-Point Boundary Value Problem III—The Method Herrick
                         and Liu 195
        6.14. Some Expansions in Elliptic Motion 198
        6.15. The Orbit in Space 201
        6.16. The Geocentric Coordinates 206
        6.17. The Effects of Planetary Aberration and Parallax 207
        6.18. Projects 209
7.  The Determination of Orbits 213
        7.1. Introduction 213
        7.2. Laplace’s Method 217
        7.3. Gauss’ Method 226
        7.4. Herget’s Method for a Preliminary Orbit Using More Than
                         Three Observations 235
        7.5. The Differential Correction of Orbits 238
                7.5.1. Projects 244
        7.6. Using a Previous Estimate: Recursive Methods 246
8.  The Three-Body Problem 253
        8.1. The Restricted Three-Body Problem: Jacobi’s Integral 253
        8.2. Tisserand’s Criterion for the Identification of Comets 254
        8.3. The Surfaces of Zero Relative Velocity 255
        8.4. The Positions of Equilibrium 260
        8.5. The Stability of the Points of Equilibrium 262
        8.6. The Lagrangian Solutions for the Motion of Three Finite Bodies 266
        8.7. Problems 270
9. The n-Body Problem 273
        9.1. The Center of Mass and the Invariable Plane 273
        9.2. The Energy Integral and the Force Function 274
        9.3. The Virial Theorem 276
        9.4. Transfer of the Origin: the Perturbing Forces 276
        9.5. Application to the Solar System 278
        9.6. Problems 280
10.  Numerical Procedures 283
        10.1. Differences and Sums 283
        10.2. Interpolation 285
        10.3. Differentiation 290
        10.4. Integration 291
        10.5. Errors 293
        10.6. The Numerical Integration of Differential Equations—Runge-Kutta
                         Methods 296
        10.7. The Numerical Integration of Differential Equations—A Multistep
                         Method for First-Order Sys-tems 302
        10.8. The Numerical Integration of Differential Equations—Systems of
                         Second-Order Equations 306
11.  Perturbations 315
        11.1. Introduction 315
        11.2. Cowell’s Method 319
        11.3. Encke’s Method 320
        11.4. The Osculating Orbit 322
        11.5. The Effects of Small Impulses on the Elements 323
        11.6. The Equation for
e 327
        11.7. Modifications When Components Are Tangential and
                         Normal-Drag-Perturbed Orbits 328
        11.8. Hansen’s Method 331
        11.9. The Equations in Terms of  
R , a etc. 331
        11.10. Substitutions for Small e or i 337
        11.11. The General Approach to the Solution of Lagrange’s Planetary
                         Equations 337
        11.12. The Disturbing Function 339
        11.13. General Discussion of the First-Order Solution of the Planetary
                         Equations 341
        11.14. Secular Perturbations 343
        11.15. The Motion of a Satellite in the Field of an Oblate Planet 345
        11.16. The Computation of the Variations of the Elements 349
        11.17. The Activity Sphere 352
        11.18. General Methods 353
        11.19. Problems 359
12.  The Motion of the Moon 371
        12.1. Introduction 371
        12.2. The Perturbing Forces 372
        12.3. The Perturbation of the Nodes 374
        12.4. The Perturbation of the Inclination 376
        12.5. The Perturbations of ? and e 377
        12.6. The Variation 379
        12.7. The Perturbation of the Period and the Annual Equation 380
        12.8. The Parallactic Inequality 381
        12.9. The Secular Acceleration of the Moon 382
        12.10. Theories of the Motion of the Moon 384
        12.11. Problems 385
13. The Earth and its Rotation 389
        13.1. The Eulerian Motion of the Earth 389
        13.2. The Couple Exerted on the Earth by a Distant Body 391
        13.3. The Couples Exerted on the Earth by the Sun and Moon 392
        13.4. The Lunisolar Precession 395
        13.5. Nutation 397
        13.6. Problems 399
14. Chapter 14 The Moon and its Rotation 401
        14.1. Cassini’s Laws 401
        14.2. The Eulerian Equations 401
        14.3. The Libration in Longitude 403
        14.4. Other Oscillations 405
        14.5. Problems 411
Appendix A. Properties of Conics 413
        A.1. General Properties 413
        A.2. The Ellipse 416
        A.3. The Parabola 418
        A.4. The Hyperbola 420
        A.5. Pole and Polar 422
Appendix B. The Rotation of Axes 425
Appendix C. Numerical Values 427

        C.1. Orbital Elements of Planets 427
        C.2. Satellites: Orbital and Physical Data 430
        C.3. Physical Elements of Planets 432
        C.4. The Earth 433
        C.5. The Moon 434
        C.6. The Sun 435
        C.7. Physical Constants 435
        C.8. Miscellaneous Data 436
Appendix D. Miscellaneous Expansions in Series 437
        D.1. f and g Series 437
        D.2. Elliptic Motion 437
Appendix E. The Solution of Linear Systems 439
Appendix F. The Generation on the Computer of Gaussian Deviates 443
Appendix G. Some Orbits of Comets and Minor Planets 445
Appendix H. The Greek Alphabet 449
Appendix I. Random Variables, and Least Squares 451
Appendix J. Notes on Hamiltonian Mechanics 457

        J.1. Elements of Lagrangian Mechanics 457
        J.2. Hamilton’s Equations 458
        J.3. Canonical Transformations 460
        J.4. Canonical Transformations Defined by Functions 462
        J.5. The Hamilton-Jacobi Equation 463
        J.6. The Problem of Two Bodies 464
        J.7. Perturbed Motion 465
References and Bibliography 469
Index 477


These CDROMs contains the same listings of the IBM-PC BASIC programs found in Fundamentals of Celestial Mechanics and they will run without a compiler. Also included on the CDROM is either a set of programs written in Turbo Pascal (version 3) or Turbo BASIC. Source code is provided—you will need the appropriate Borland compiler to run these programs if you want to manipulate the code without modifying it for another compiler. The programs fall into three categories.

  • Procedures that are used, or could be used in other programs.
  • Programs that are concerned with computations related to orbital motion.
  • Programs that have been designed for use as demonstrations.

The second category is closely related to the text, Fundamentals of Celestial Mechanics, second edition. In that text, listings of several programs are given in IBM BASIC. All of these programs appear here, in Pascal, and several additional programs have been added.

Many programs have procedures in common, such as those for generating Stumpff functions, or for solving Kepler's equation in universal variables. These procedures appear on the CDROM as separate files, and are added to the relevant programs by "include" statements: for instance, $I Kepler would cause the file Kepler to be added to the program. The CDROMs include the 18 program listings form Professor Danby's Fundamentals of Celestial Mechanics and over 80 demonstrations or procedures.