# Kepler's equation

For specific applications of Kepler's equation, see Kepler's laws of planetary motion.
Kepler's equation solutions for five different eccentricities between 0 and 1

In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force.

It was first derived by Johannes Kepler in 1609 in Chapter 60 of his Astronomia nova,[1][2] and in book V of his Epitome of Copernican Astronomy (1621) Kepler proposed an iterative solution to the equation.[3][4] 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 M is the mean anomaly, E is the eccentric anomaly, and e is the eccentricity.

The 'eccentric anomaly' E 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 x = a(1 − e), y = 0, at time t = t0, then to find out the position of the body at any time, you first calculate the mean anomaly M from the time and the mean motion n by the formula M = n(tt0), then solve the Kepler equation above to get E, then get the coordinates from:

Kepler's equation is a transcendental equation because sine is a transcendental function, meaning it cannot be solved for E algebraically. Numerical analysis and series expansions are generally required to evaluate E.

## 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 (0 e < 1). The hyperbolic Kepler equation is used for hyperbolic orbits (e ≫ 1). The radial Kepler equation is used for linear (radial) orbits (e = 1). Barker's equation is used for parabolic orbits (e = 1). When e = 1, Kepler's equation is not associated with an orbit.

When e = 0, the orbit is circular. Increasing e causes the circle to flatten into an ellipse. When e = 1, the orbit is completely flat, and it appears to be either a segment if the orbit is closed, or a ray if the orbit is open. An infinitesimal increase to e results in a hyperbolic orbit with a turning angle of 180 degrees, and the orbit appears to be a ray. Further increases reduce the turning angle, and as e goes to infinity, the orbit becomes a straight line of infinite length.

### Hyperbolic Kepler equation

The Hyperbolic Kepler equation is:

where H is the hyperbolic eccentric anomaly. This equation is derived by multiplying Kepler's equation by the square root of −1; i = √−1 for imaginary unit, and replacing

to obtain

where t is time, and x is the distance along an x-axis. This equation is derived by multiplying Kepler's equation by 1/2 making the replacement

and setting e = 1 gives

## Inverse problem

Calculating M for a given value of E is straightforward. However, solving for E when M is given can be considerably more challenging.

Kepler's equation can be solved for E analytically by Lagrange inversion. The solution of Kepler's equation given by two Taylor series below.

Confusion over the solvability of Kepler's equation has persisted in the literature for four centuries.[5] Kepler himself expressed doubt at the possibility of ﬁnding a general solution.

 “ I am sufficiently satisfied that it [Kepler's equation] cannot be solved a priori, on account of the different nature of the arc and the sine. But if I am mistaken, and any one shall point out the way to me, he will be in my eyes the great Apollonius. ” — Johannes Kepler[6]

### Inverse Kepler equation

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

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 Taylor series. 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. While this solution is the simplest in a certain mathematical sense, for values of e near 1 the convergence is very poor, other solutions are preferable for most applications. Alternatively, Kepler's equation can be solved numerically.

The solution for e ≠ 1 was discovered by Karl Stumpff in 1968,[7] but its significance wasn't recognized.[8]

### Algorithm for the inverse Kepler equation

Note that E = M+e·sin(E). Repeatedly substituting the expression on the right for the E on the right yields a simple fixed point iteration algorithm for evaluating E(e,M).

The number of iterations, n, depends on the value of e.

Using Newton's method to find the zero of f = E - e·sin(E) - M one gets,

to first order in the small quantities M - E and e, or,

.

The algorithm actually uses a simplified version of Newton's method to improve on a value for E. Although Newton's method usually converges faster to the correct value for E there are situations where it requires division by 0 and the method above is then preferred.

The inverse radial Kepler equation is:

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

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 E and M are in units of radians in this computation. This iteration is repeated until desired accuracy is obtained (e.g. when f(E) < desired accuracy). For most elliptical orbits an initial value of E0 = M(t) is sufficient. For orbits with e > 0.8, an initial value of E0 = π should be used.[9] 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.