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Concept
Kepler's equation
Summary
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.
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 .
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.
Official source
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