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
be the mean anomaly (a parameterization of time) and the eccentric anomaly
(a parameterization of polar angle) of a body orbiting on an ellipse
with eccentricity , then

(1)

For
not a multiple of ,
Kepler's equation has a unique solution, but is a transcendental
equation and so cannot be inverted and solved directly for given an arbitrary . However, many algorithms have been derived for solving the
equation as a result of its importance in celestial mechanics.

Writing a
as a power series in gives

(2)

where the coefficients are given by the Lagrange
inversion theorem as

(3)

(Wintner 1941, Moulton 1970, Henrici 1974, Finch 2003). Surprisingly, this series diverges for

(4)

(OEIS A033259 ), a value known as the Laplace limit . In fact,
converges as a geometric series with ratio

(5)

(Finch 2003).

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

(6)

This series converges for all like a geometric series
with ratio

(7)

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

(8)

Alternatively, we can define in terms of an intermediate variable

(9)

then

(10)

(11)

Iterative methods such as the simple

(12)

with
work well, as does Newton's method ,

(13)

The point at which Laplace's formula for solving Kepler's equation begins diverging
is known as the Laplace limit .

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." http://www.geocities.com/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: 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. Moulton, F. R. An
Introduction to Celestial Mechanics, 2nd rev. ed. New York: Dover, pp. 159-169,
1970. Montenbruck, O. and Pfleger, T. "Mathematical Treatment of
Kepler's Equation." §4.3 in Astronomy
on the Personal Computer, 4th ed. Berlin: Springer-Verlag, pp. 62-63
and 65-68, 2000. 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. Referenced
on Wolfram|Alpha Kepler's Equation
Cite this as:
Weisstein, Eric W. "Kepler's Equation."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/KeplersEquation.html

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