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Kepler's Equation

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KeplersEquation

Kepler's equation gives the relation between the polar coordinates of a celestial body (such as a planet) and the time elapsed from a given initial point. Kepler's equation is of fundamental importance in celestial mechanics, but cannot be directly inverted in terms of simple functions in order to determine where the planet will be at a given time.

Let M be the mean anomaly (a parameterization of time) and E the eccentric anomaly (a parameterization of position on the ellipse) of a body with eccentricity 0<=e<1. Then

M=E-esinE.
(1)

For every real number M, Kepler's equation has a unique real solution E, since its right side is strictly increasing in E. It is a transcendental equation and generally cannot be solved using elementary functions. However, many algorithms have been derived for solving the equation as a result of its importance in celestial mechanics.

Writing E as a power series in e gives

E=M+sum_(n=1)^inftya_ne^n,
(2)

where the coefficients are given by the Lagrange inversion theorem as

a_n=1/(2^(n-1)n!)sum_(k=0)^(|_n/2_|)(-1)^k(n; k)(n-2k)^(n-1)sin[(n-2k)M]
(3)

(Wintner 1941, Moulton 1970, Henrici 1974, Finch 2003), where |_n/2_| uses the floor function. This series converges for every real number M when 0<=e<L, where

L=0.6627434193...
(4)

(OEIS A033259) is the Laplace limit. For e>L, it can diverge, for example at M=pi/2, although at multiples of pi all the coefficients vanish and E=M. The condition e<L is equivalent to rho<1, where

rho=e/(1+sqrt(1+e^2))exp(sqrt(1+e^2))
(5)

(Finch 2003).

There is also a series solution in Bessel functions of the first kind,

E=M+sum_(n=1)^infty2/nJ_n(ne)sin(nM).
(6)

This series converges for 0<=e<1. For 0<e<1, its coefficients have exponential decay with factor

rho=e/(1+sqrt(1-e^2))exp(sqrt(1-e^2)).
(7)

The equation can also be solved by letting psi be the angle between the planet's motion and the direction perpendicular to the radius vector. Then

tanpsi=(esinE)/(sqrt(1-e^2)).
(8)

Alternatively, let r be the distance from the body to the focus, p the semilatus rectum, and v the corresponding true anomaly. Define phi by

e=sinphi
(9)

with 0<=phi<pi/2. Then

sin[1/2(v-E)]=sqrt(r/p)sin(1/2phi)sinv
(10)
sin[1/2(v+E)]=sqrt(r/p)cos(1/2phi)sinv.
(11)

Iterative methods such as the simple

E_(i+1)=M+esinE_i
(12)

converge for 0<=e<1 from any real number initial value E_0, including 0, by the Banach fixed point theorem. Newton's method gives the iteration

E_(i+1)=E_i+(M+esinE_i-E_i)/(1-ecosE_i).
(13)

Convergence of Newton's method depends on the initial value.

See also

Eccentric Anomaly, Kapteyn Series, Laplace Limit

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References

Belur, S. V. "Solution of Kepler's Equation by Newton-Raphson Method." https://www.oocities.org/SiliconValley/2902/kepler.htm.Colwell, P. Solving Kepler's Equation over Three Centuries. Richmond, VA: Willmann-Bell, 1993.Danby, J. M. Fundamentals of Celestial Mechanics, 2nd ed., rev. ed. Richmond, VA: Willmann-Bell, 1988.Dörrie, H. "The Kepler Equation." §81 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 330-334, 1965.Finch, S. R. "Laplace Limit Constant." §4.8 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 266-268, 2003.Goldstein, H. Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, pp. 101-102 and 123-124, 1980.Goursat, E. A Course in Mathematical Analysis, Vol. 2: Functions of a Complex Variable & Differential Equations. New York: Dover, p. 120, 1959.Henrici, P. Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Location of Zeros. New York: Wiley, 1974.Ioakimids, N. I. and Papadakis, K. E. "A New Simple Method for the Analytical Solution of Kepler's Equation." Celest. Mech. 35, 305-316, 1985.Ioakimids, N. I. and Papadakis, K. E. "A New Class of Quite Elementary Closed-Form Integrals Formulae for Roots of Nonlinear Systems." Appl. Math. Comput. 29, 185-196, 1989.Le Lionnais, F. Les nombres remarquables. Paris, France: Hermann, p. 36, 1983.Marion, J. B. and Thornton, S. T. "Kepler's Equations." §7.8 in Classical Dynamics of Particles & Systems, 3rd ed. San Diego, CA: Harcourt Brace Jovanovich, pp. 261-266, 1988.Montenbruck, O. and Pfleger, T. "Mathematical Treatment of Kepler's Equation." §4.3 in Astronomy on the Personal Computer, 4th ed. Berlin, Germany: Springer-Verlag, pp. 62-63 and 65-68, 2000.Moulton, F. R. An Introduction to Celestial Mechanics, 2nd rev. ed. New York: Dover, pp. 159-169, 1970.Plummer, H. An Introductory Treatise of Dynamical Astronomy. New York: Dover, 1960.Siewert, C. E. and Burniston, E. E. "An Exact Analytical Solution of Kepler's Equation." Celest. Mech. 6, 294-304, 1972.Sloane, N. J. A. Sequences A033259 and A085984 in "The On-Line Encyclopedia of Integer Sequences."Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.Wintner, A. The Analytic Foundations of Celestial Mechanics. Princeton, NJ: Princeton University Press, 1941.

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Kepler's Equation

Cite this as:

Weisstein, Eric W. "Kepler's Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KeplersEquation.html

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