Cotes' Spiral


A spiral that gives the solution to the central orbit problem under a radial force law


where mu is a positive constant. There are three solution regimes,

 r={Asec(ktheta+epsilon)   for mu<h^2; Asech(k^'theta+epsilon)   for mu>h^2; A/(theta+epsilon)   for mu=h^2,

where A and epsilon are constants,


and h is the specific angular momentum (Whittaker 1944, p. 83). The case mu>h^2 gives an epispiral, while mu=h^2 leads to a hyperbolic spiral.

See also

Epispiral, Hyperbolic Spiral, Spiral

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Cotes, R. Harmonia Mensurarum. p. 31 and 98, 1722.Danby, J. M. "The Case f(r)=mu/r^3--Cotes' Spiral." §4.7 in Fundamentals of Celestial Mechanics, 2nd ed., rev. ed. Richmond, VA: Willmann-Bell, pp. 69-71, 1988.Symon, K. R. Mechanics, 3rd ed. Reading, MA: Addison-Wesley, p. 154, 1971.Whittaker, E. T. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies. New York: Dover, p. 83, 1944.

Referenced on Wolfram|Alpha

Cotes' Spiral

Cite this as:

Weisstein, Eric W. "Cotes' Spiral." From MathWorld--A Wolfram Web Resource.

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