Kepler's Equation

Let $M$ be the mean anomaly and $E$ the Eccentric Anomaly of a body orbiting on an Ellipse with Eccentricity $e$, then

$$M = E-e\sin E.$$ (1)

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

Writing a $E$ as a Power Series in $e$ gives

$$E=M+\sum_{n=1}^\infty a_n e^n,$$ (2)

where the coefficients are given by the Lagrange Inversion Theorem as

$$a_n={1\over 2^{n-1}n!} \sum_{k=0}^{\left\lfloor{n/2}\right\rfloor } (-1)^k{n\choose k}(n-2k)^{n-1}\sin[(n-2k)M]$$ (3)

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

$$e>0.6627434193\ldots,$$ (4)

a value known as the Laplace Limit. In fact, $E$ converges as a Geometric Series with ratio

$$r={e\over 1+\sqrt{1+e^2}}\mathop{\rm exp}\nolimits (\sqrt{1+e^2}\,)$$ (5)

(Finch).

There is also a series solution in Bessel Functions of the First Kind,

$$E=M+\sum_{n=1}^\infty {2\over n} J_n(ne)\sin(n M).$$ (6)

This series converges for all $e<1$ like a Geometric Series with ratio

$$r={e\over 1+\sqrt{1-e^2}}\mathop{\rm exp}\nolimits (\sqrt{1-e^2}\,).$$ (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

$$\tan\psi = {e\sin E\over\sqrt{1-e^2}}.$$ (8)

Alternatively, we can define $e$ in terms of an intermediate variable $\phi$

$$e\equiv\sin\phi,$$ (9)

then

$$\sin[{\textstyle{1\over 2}}(v-E)] =\sqrt{r\over p}\,\sin({\textstyle{1\over 2}}\phi)\sin v$$ (10)

$$\sin[{\textstyle{1\over 2}}(v+E)] =\sqrt{r\over p}\,\cos({\textstyle{1\over 2}}\phi)\sin v.$$ (11)

Iterative methods such as the simple

$$E_{i+1}=M+e\sin E_i$$ (12)

with $E_0=0$ work well, as does Newton's Method,

$$E_{i+1}=E_i+{M+e\sin E_i-E_i\over 1-e\cos E_i}.$$ (13)

In solving Kepler's equation, Stieltjes required the solution to

$$e^x(x-1)=e^{-x}(x+1),$$ (14)

which is 1.1996678640257734... (Goursat 1959, Le Lionnais 1983).

See also Eccentric Anomaly

References

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. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/lpc/lpc.html

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. 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.

Siewert, C. E. and Burniston, E. E. ``An Exact Analytical Solution of Kepler's Equation.'' Celest. Mech. 6, 294-304, 1972.

Wintner, A. The Analytic Foundations of Celestial Mechanics. Princeton, NJ: Princeton University Press, 1941.