Kepler's equation
In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force. It was derived by Johannes Kepler in 1609 in Chapter 60 of his Astronomia nova, and in book V of his Epitome of Copernican Astronomy (1621) Kepler proposed an iterative solution to the equation. This equation and its solution, however, first appeared in 9th century work of Habash al-Hasib al-Marwazi related to problems of parallax. The equation has played an important role in the history of both physics and mathematics, particularly classical celestial mechanics. Equation

Kepler's equation is

where  is the mean anomaly,  is the eccentric anomaly, and  is the eccentricity. The 'eccentric anomaly'  is useful to compute the position of a point moving in a Keplerian orbit. As for instance, if the body passes the periastron at coordinates , , at time , then to find out the position of the body at any time, you first calculate the mean anomaly  from the time and the mean motion  by the formula , then solve the Kepler equation above to get , then get the coordinates from:

where  is the semi-major axis,  the semi-minor axis. Kepler's equation is a transcendental equation because sine is a transcendental function, meaning it cannot be solved for  algebraically. Numerical analysis and series expansions are generally required to evaluate . Alternate forms 

There are several forms of Kepler's equation. Each form is associated with a specific type of orbit. The standard Kepler equation is used for elliptic orbits (). The hyperbolic Kepler equation is used for hyperbolic trajectories (). The radial Kepler equation is used for linear (radial) trajectories (). Barker's equation is used for parabolic trajectories (). When , the orbit is circular. Increasing  causes the circle to become elliptical. When , there are three possibilities:
 a parabolic trajectory,
 a trajectory going in or out along an infinite ray emanating from the centre of attraction,
 or a trajectory that goes back and forth along a line segment from the centre of attraction to a point at some distance away. A slight increase in  above 1 results in a hyperbolic orbit with a turning angle of just under 180 degrees. Further increases reduce the turning angle, and as  goes to infinity, the orbit becomes a straight line of infinite length. Hyperbolic Kepler equation

The Hyperbolic Kepler equation is:

where  is the hyperbolic eccentric anomaly. This equation is derived by redefining M to be the square root of −1 times the right-hand side of the elliptical equation:

(in which  is now imaginary) and then replacing  by . Radial Kepler equation
The Radial Kepler equation is:

where  is proportional to time and  is proportional to the distance from the centre of attraction along the ray. This equation is derived by multiplying Kepler's equation by 1/2 and setting  to 1:

and then making the substitution

Inverse problem 

Calculating  for a given value of  is straightforward. However, solving for  when  is given can be considerably more challenging. There is no closed-form solution. One can write an infinite series expression for the solution to Kepler's equation using Lagrange inversion, but the series does not converge for all combinations of  and  (see below). Confusion over the solvability of Kepler's equation has persisted in the literature for four centuries. Kepler himself expressed doubt at the possibility of finding a general solution:

Fourier series expansion (with respect to ) using Bessel functions is 

With respect to , it is a Kapteyn series. Inverse Kepler equation

The inverse Kepler equation is the solution of Kepler's equation for all real values of :

 

Evaluating this yields:

 

These series can be reproduced in Mathematica with the InverseSeries operation. InverseSeries[Series[M - Sin[M], {M, 0, 10}]]
 InverseSeries[Series[M - e Sin[M], {M, 0, 10}]]

These functions are simple Maclaurin series. Such Taylor series representations of transcendental functions are considered to be definitions of those functions. Therefore, this solution is a formal definition of the inverse Kepler equation. However,  is not an entire function of  at a given non-zero . Indeed, the derivative

goes to zero at an infinite set of complex numbers when  the nearest to zero being at  and at these two points

(where inverse cosh is taken to be positive), and  goes to infinity at these values of . This means that the radius of convergence of the Maclaurin series is  and the series will not converge for values of  larger than this. The series can also be used for the hyperbolic case, in which case the radius of convergence is  The series for when  converges when . While this solution is the simplest in a certain mathematical sense,, other solutions are preferable for most applications. Alternatively, Kepler's equation can be solved numerically. The solution for  was found by Karl Stumpff in 1968, but its significance wasn't recognized. One can also write a Maclaurin series in . This series does not converge when  is larger than the Laplace limit (about 0.66), regardless of the value of  (unless  is a multiple of ), but it converges for all  if  is less than the Laplace limit. The coefficients in the series, other than the first (which is simply ), depend on  in a periodic way with period . Inverse radial Kepler equation

The inverse radial Kepler equation () can also be written as:
 

Evaluating this yields:

To obtain this result using Mathematica:
InverseSeries[Series[ArcSin[Sqrt[t]] - Sqrt[(1 - t) t], {t, 0, 15}]]

Numerical approximation of inverse problem

Newton's method 

For most applications, the inverse problem can be computed numerically by finding the root of the function:

 

This can be done iteratively via Newton's method:

 

Note that  and  are in units of radians in this computation. This iteration is repeated until desired accuracy is obtained (e.g. when  < desired accuracy). For most elliptical orbits an initial value of  is sufficient. For orbits with , a initial value of  can be used. Numerous works developed accurate (but also more complex) start guesses. If  is identically 1, then the derivative of , which is in the denominator of Newton's method, can get close to zero, making derivative-based methods such as Newton-Raphson, secant, or regula falsi numerically unstable. In that case, the bisection method will provide guaranteed convergence, particularly since the solution can be bounded in a small initial interval. On modern computers, it is possible to achieve 4 or 5 digits of accuracy in 17 to 18 iterations. A similar approach can be used for the hyperbolic form of Kepler's equation. In the case of a parabolic trajectory, Barker's equation is used. Fixed-point iteration 

A related method starts by noting that . Repeatedly substituting the expression on the right for the  on the right yields a simple fixed-point iteration algorithm for evaluating . This method is identical to Kepler's 1621 solution. function E(e, M, n)
    E = M
    for k = 1 to n
        E = M + e*sin E
    next k
    return E

The number of iterations, , depends on the value of . The hyperbolic form similarly has . This method is related to the Newton's method solution above in that

 

To first order in the small quantities  and ,

. See also 
Kepler's laws of planetary motion
Kepler problem
Kepler problem in general relativity
Radial trajectory

References

External links 

 
 
 
 
 
 
 
 
 
 
 
 
 

 Kepler's Equation at Wolfram Mathworld

Eponymous equations of physics
Johannes Kepler
Orbits