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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.

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Semi-major and semi-minor axes

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section.

Mean anomaly

In celestial mechanics, the mean anomaly is the fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periapsis, expressed as an angle which can be used in calculating the position of that body in the classical two-body problem. It is the angular distance from the pericenter which a fictitious body would have if it moved in a circular orbit, with constant speed, in the same orbital period as the actual body in its elliptical orbit. Define T as the time required for a particular body to complete one orbit.

Orbital mechanics

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control. Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of gravity, including both spacecraft and natural astronomical bodies such as star systems, planets, moons, and comets.

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