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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,^[1][2] and in book V of his Epitome of Copernican Astronomy (1621) Kepler proposed an iterative solution to the equation.^[3][4] This equation and its solution, however, first appeared in 9th century work of Habash al-Hasib al-Marwazi related to problems of parallax.^[5][6][7][8] 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 ${\displaystyle M}$ is the mean anomaly, ${\displaystyle E}$ is the eccentric anomaly, and ${\displaystyle e}$ is the eccentricity.

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

where ${\displaystyle a}$ is the semi-major axis, ${\displaystyle b}$ the semi-minor axis.

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

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 (${\displaystyle 0\leq e<1}$). The hyperbolic Kepler equation is used for hyperbolic trajectories (${\displaystyle e>1}$). The radial Kepler equation is used for linear (radial) trajectories (${\displaystyle e=1}$). Barker's equation is used for parabolic trajectories (${\displaystyle e=1}$).

When ${\displaystyle e=0}$, the orbit is circular. Increasing ${\displaystyle e}$ causes the circle to become elliptical. When ${\displaystyle e=1}$, 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 ${\displaystyle e}$ above 1 results in a hyperbolic orbit with a turning angle of just under 180 degrees. Further increases reduce the turning angle, and as ${\displaystyle e}$ goes to infinity, the orbit becomes a straight line of infinite length.

Hyperbolic Kepler equation

The Hyperbolic Kepler equation is:

where ${\displaystyle H}$ 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:

${\displaystyle M=i\left(E-e\sin E\right)}$

(in which ${\displaystyle E}$ is now imaginary) and then replacing ${\displaystyle E}$ by ${\displaystyle iH}$.

Radial Kepler equation

The Radial Kepler equation is:

where ${\displaystyle t}$ is proportional to time and ${\displaystyle x}$ 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 ${\displaystyle e}$ to 1:

${\displaystyle t(x)={\frac {1}{2}}\left.}$

and then making the substitution

${\displaystyle E=2\sin ^{-1}({\sqrt {x}}).}$

Inverse problem

Calculating ${\displaystyle M}$ for a given value of ${\displaystyle E}$ is straightforward. However, solving for ${\displaystyle E}$ when ${\displaystyle M}$ 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 ${\displaystyle e}$Zdroj:https://en.wikipedia.org?pojem=Kepler's_equation
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