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Solving Kepler's Equation

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Solving Kepler's EquationBy Colwell. Product Information: 6. 00" by 9. 00", 202 pages, hardbound, published 1993. 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

By Colwell.

Product Information: 6.00" by 9.00", 202 pages, hardbound, published 1993.

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.

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West Palm Beach, US
★★★★★ 5
UPDATE TO MY REVIEW / It's NOT indestructible!
Color: Carrot Orange
UPDATE: I stand by my review for the original "Plush Toy". BUT ... the seller contacted me and offered to help, because they don't want their customers to be unhappy with their products. I described the problem, and they said they'd contact their supplier to correct things. AND ... they offered to send another dog toy that actually was indestructible. My dog, a Border Collie - Retriever mix, loved the replacement they sent, an Orange Peanut. (He even took it to bed the first night.) This was much appreciated. But what I most appreciated was their Customer Service and Care. They don't just say they want their customers to be happy, they actually demonstrate it by their actions. This type of care is rare these days, and I commend them for it. Notice I've changed my rating from 1-star to 5-stars, this is in recognition of their Customer Service. To me, this type of care is more important than any product. So I now fully endorse this seller. If you have an issue with a product, just contact them (through Amazon), and they will do their utmost to make things right, and to make sure you're happy. ****************************************** I bought this because it was described as "Dog Plush Toy for Large Aggressive Chewers, Indestructible Dog Squeaky Toy" (from their listing on Amazon). My dog is a Golden Retriever mix, and he's only a "medium size aggressive chewer". And yet, this toy lasted 10 minutes at the most. Very disappointing ... save yourself a "dog toy heartbreak".
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Mistelle Lanko
Bozeman, US
★★★★★ 3
Not indestructible however not terrible.
Color: Shocking Pink
Not indestructible however not terrible. My strong chewer had quickly devoured the plush stomach and eaten the squeaker. But the rope parts and rough face pieces are holding up well. Not a disaster.
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Reviewed in the United States on March 19, 2026
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Chris
Chelsea, US
★★★★★ 4
Good toy but not indestructible
Color: Royal Blue
The toy is nice, but my dog killed it in about 30 minutes. The only reason I didn’t give it five stars is because it says indestructible Toy in the description. Other than that it is a nice chew toy.
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★★★★★ 5
Dog loves it. So 5 starts
Color: Carrot Orange
A great value toy that is very hard to destroy. Our dog usually runs through softer toys and this one has lasted a long time. Its not chewable like a plushy but harder fabric that doesn't rip apart. The dogs are having a blast with it. And so are we. The colors fit the description. The size was idk 2ft ish. It met all expectations. I wouldn't expect to need more than one of them unless your dog is a super destroyer.
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Not indestructible
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It took 20 minutes for my dog to have the squeaker removed!
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Reviewed in the United States on May 28, 2026

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