Kepler's equation
Encyclopedia
Kepler's equation is
,
where M is the mean anomaly
, E is the eccentric anomaly
, and
is the eccentricity
.
Kepler's equation was first derived by Johannes Kepler
in 1619, and has played an important role in the history of both physics and mathematics.
It is used to solve problems in orbital mechanics; for specific applications of Kepler's equation, see Kepler's laws of planetary motion
.
<1). The hyperbolic Kepler equation is used for hyperbolic orbits (
>1). The radial Kepler equation is used for linear (radial) orbits (
=1). Barker's equation is used for parabolic orbits (
=1). When 
=1, Kepler's equation is not associated with an orbit.
When
is 0, the orbit is circular. Increasing 
causes the circle to flatten into an ellipse. When 
=1, the orbit is completely flat, and it appears to be a either a segment if the orbit is closed, or a ray if the orbit is open. An infinitesimal increase to 
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 
goes to infinity, the orbit becomes a straight line of infinite length.
,
where H is the hyperbolic eccentric anomaly.
This equation is derived by multiplying Kepler's equation by the square root of -1, and replacing E with
.
where t is related to the time, and x is related to the distance.
This equation is derived by multiplying Kepler's equation by 1/2, replacing E with
, and setting 
= 1.
Kepler's equation is transcendental
, which means it cannot be solved for E algebraically
. 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. It is often erroneously claimed that Kepler's equation "cannot be solved analytically". Many authors make the absurd claim that it cannot be solved at all.
The first to make this assertion was Kepler himself:
:
Evaluating this yields:
These series can be reproduced in Mathematica
with the InverseSeries operation.
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
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
was discovered by Karl Stumpff
in 1968, but its significance wasn't recognized.
,where M is the mean anomaly
Mean anomaly
In celestial mechanics, the mean anomaly is a parameter relating position and time for a body moving in a Kepler orbit. It is based on the fact that equal areas are swept at the focus in equal intervals of time....
, E is the eccentric anomaly
Eccentric anomaly
In celestial mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit.For the point P orbiting around an ellipse, the eccentric anomaly is the angle E in the figure...
, and
is the eccentricityEccentricity (mathematics)
In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...
.
Kepler's equation was first derived by Johannes Kepler
Johannes Kepler
Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
in 1619, and has played an important role in the history of both physics and mathematics.
It is used to solve problems in orbital mechanics; for specific applications of Kepler's equation, see Kepler's laws of planetary motion
Kepler's laws of planetary motion
In astronomy, Kepler's laws give a description of the motion of planets around the Sun.Kepler's laws are:#The orbit of every planet is an ellipse with the Sun at one of the two foci....
.
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≤<1). The hyperbolic Kepler equation is used for hyperbolic orbits (>1). The radial Kepler equation is used for linear (radial) orbits (=1). Barker's equation is used for parabolic orbits (=1). When =1, Kepler's equation is not associated with an orbit.When
is 0, the orbit is circular. Increasing causes the circle to flatten into an ellipse. When =1, the orbit is completely flat, and it appears to be a either a segment if the orbit is closed, or a ray if the orbit is open. An infinitesimal increase to 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 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, and replacing E with
.Radial Kepler equation
The Radial Kepler equation is:-
,
where t is related to the time, and x is related to the distance.
This equation is derived by multiplying Kepler's equation by 1/2, replacing E with
, and setting = 1.
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 is transcendental
Transcendental function
A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...
, which means it cannot be solved for E algebraically
Algebraic function
In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...
. Kepler's equation can be solved for E analytically
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
by Lagrange inversion
Lagrange inversion theorem
In mathematical analysis, the Lagrange inversion theorem, also known as the Lagrange-Bürmann formula, gives the Taylor series expansion of the inverse function of an analytic function.-Theorem statement:...
. 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. It is often erroneously claimed that Kepler's equation "cannot be solved analytically". Many authors make the absurd claim that it cannot be solved at all.
The first to make this assertion was Kepler himself:
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
Mathematica
Mathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...
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
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
was discovered by Karl StumpffKarl Stumpff
Karl Johann Nikolaus Stumpff was a German astronomer. The Stumpff functions, used in the universal variable formulation of the two-body problem, are named after him.-External links:*....
in 1968, but its significance wasn't recognized.
Inverse radial Kepler equation
The inverse radial Kepler equation is:-
Evaluating this yields:
To obtain this result using MathematicaMathematicaMathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...
:InverseSeries[Series[ArcSin[Sqrt[t]] - Sqrt[(1 - t) t], {t, 0, 15}]]
See also
- Kepler's laws of planetary motionKepler's laws of planetary motionIn astronomy, Kepler's laws give a description of the motion of planets around the Sun.Kepler's laws are:#The orbit of every planet is an ellipse with the Sun at one of the two foci....
- Kepler problemKepler problemIn classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force F that varies in strength as the inverse square of the distance r between them. The force may be either attractive or repulsive...
- Kepler problem in general relativityKepler problem in general relativityThe two-body problem in general relativity is to determine the motion and gravitational field of two bodies interacting with one another by gravitation, as described by the field equations of general relativity. Solving the Kepler problem is essential to calculate the bending of light by gravity...
- Radial TrajectoryRadial trajectoryIn astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line.- Classification :...
External links