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

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

Appendix M References Sorted by Categories 159

Index 199