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Colwell, 6.00" by 9.00", 202 pages, hardbound, published 1993, 1 Lb. 10 Ozs. ship wt., $29.95.

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The sole subject of Solving Kepler's Equation work is Kepler's Equation (KE) M = E - e sin E . In its narrowest form, the Kepler problem is to solve KE for E , given M in the interval [0,p]and e in the interval [0,1]. In virtually every decade from 1650 to the present there have appeared papers devoted to the Kepler problem and its solution. We can see from a list of them that the problem has enticed a wide variety of scientists to comment on or involve themselves in its solution.

It is surely not unique in science for a specific problem to be given so much attention over so long a period—particularly if it resists solution, if its partial solutions are inadequate or unsatisfactory, or if it is recharged with new interpretations and new applications. Still, it is curious that the Kepler problem should have continued to be this interesting to so many for so long. Admittedly it is a problem central to celestial mechanics, but it is a technical little problem for which a number of satisfactory solutions are long known. With the advent of calculators and computers, there is no impediment to achieving quick solutions of great accuracy. The problem has neither the broad appeal of an Olbers Paradox, nor the depth and intractability of a many-body problem.

In common with almost any scientific problem which achieves a certain longevity and whose literature exceeds a certain critical mass, the Kepler problem has acquired an undeniable luster and allure for the modern practitioner. Any new technique for the treatment of transcendental equations should be applied to this illustrious test case; any new insight, however slight, lets its conceiver join an eminent list of contributors.

The Kepler problem has been “on the scene” in Western civilization science for over three centuries. To gather its story is to view this science through a narrow-band filter, and our goal is to make the picture at one wavelength instructive and interesting.

The idea to gather all the work on the Kepler problem is not new. Early work was surveyed in Melander,1767, Detmoldt,1798 and Brinkley,1803, and much of the recent work was described in Danby,1983, Burkhardt,1983 and Gooding,1985. In addition, various partial bibliographies have appeared which contain extensive references to the Kepler problem. Perhaps the most cited has been Radau,1900; others are Struve,1860, Houzeau,1882, Herglotz,1910, and Wood,1950.

Our concern will be almost exclusively for the elliptic case of Kepler's problem. What coverage we attempt of the hyperbolic case will appear in Chapter 8 when we look at universal forms of KE, and the parabolic case, e = 1, will be considered in Chapter 5, where treatments for high eccentricity (e near 1) cases are gathered. Related to KE and the Kepler problem is the subject of Lambert's theorem and the Lambert orbital problem. Except for a few bibliography entries, we won't describe this area at all, even though it has an extensive literature and independent interest. The references [Gooding,1988,1990] are to be recommended for a look at the subject.

In the language of the mathematician, there exists a unique solution of the Kepler problem, but there are many methods to describe or approximate it. It isn't feasible to say how many there are: the meaning of "solution" depends too much on the solver's motivations. In our classification of references bearing on KE which appear in the Bibliography, we have made seven primary categories and five secondary categories of methods of solution. These have been adequate to describe almost all our references.

Table of Contents

Preface iii
Introduction ix
Chapter 1 Origins, Antecedents, and Early Developments 1
          The Anomalies and Kepler's Equation 1
          Kepler's Solution 4
          Parallax and Arab Encounters with KE 4
Chapter 2 Nonanalytic Solutions 7
          Solutions Not Ascribing to Kepler's Second Law 7
          The Cassini Solution 12
          The Horrocks Solution 16
          The Horrebow Solution 18
          Solution by Cycloid 20
          Solution by “Curve of Sines” 21
Chapter 3 Infinite Series Solutions 23
          Solution by Lagrange's Theorem 23
          KE and Bessel Functions 27
          Levi-Civita's Solution of KE 38
          A Lie-Series Solution of KE 41
Chapter 4 Solutions of KE by Iteration 45
          Kepler's Solution Revisited 46
          Newton's Method and KE 48
          Ivory's Geometric Iteration 54
Chapter 5 Solutions of KE for High Eccentricity 57
          Barker's Equation and Parabolic Approximations 57
          Gauss' Method 61
Chapter 6 Cauchy and KE 67
          Cauchy's Treatment of Lagrange's Theorem and KE 69
          Following Cauchy from 1849 to 1941 74
Chapter 7 Calculations, Auxiliary Tables, and Analogue Devices 79
          Tables and Approximation Formulas 80
          Analogue Devices 86
Chapter 8 Modern Treatments of KE 93
          The Period 1930–1950 93
          “Universal” Forms for KE 98
          Numerical Experiments with KE 98
          KE and Methods for Transcendental Equations 108
          The Burniston-Siewert Method 109
          The Ioakimidis-Papadakis Method 111
          The Delves-Lyness Method 112
          Newton's Method for Power Series 113
Appendix A Geometric Parallax and KE 119
Appendix B Error in the Horrocks Solution 121
Appendix C Machin's E1 125
Appendix D Coefficients for the Lagrange Solution 127
Appendix E Coefficients of the Levi-Civita Solution 129
Appendix F Autonomous Differential Equations and Lie-Series 135
Appendix G Coefficients of the Lie-Series Solution 137
Appendix H Binary Systems and KE 139
Appendix I Hyperbolas and Battin's Universal KE 141
Appendix J Boltz's Parameters 147
Appendix K Riemann Boundary Value Problems and
        the Burniston-Siewert Method 149

          Riemann Boundary Value Problem 149
          The Burniston-Siewert Method and KE 151
Appendix L Newton's Method for Formal Power Series 155
Appendix M References Sorted by Categories 159
Index 199