# Mean anomaly

In celestial mechanics, the **mean anomaly** is an angle used in calculating the position of a body in an elliptical orbit 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.^{[1]}^{[2]}

## Definition[edit]

Define *T* as the time required for a particular body to complete one orbit. In time *T*, the radius vector sweeps out 2π radians or 360°. The average rate of sweep, *n*, is then

which is called the *mean angular motion* of the body, with dimensions of radians per unit time or degrees per unit time.

Define *τ* as the time at which the body is at the pericenter. From the above definitions, a new quantity, *M*, the *mean anomaly* can be defined

which gives an angular distance from the pericenter at arbitrary time *t*,^{[3]} with dimensions of radians or degrees.

Because the rate of increase, *n*, is a constant average, the mean anomaly increases uniformly (linearly) from 0 to 2π radians or 0° to 360° during each orbit. It is equal to 0 when the body is at the pericenter, π radians (180°) at the apocenter, and 2π radians (360°) after one complete revolution.^{[4]} If the mean anomaly is known at any given instant, it can be calculated at any later (or prior) instant by simply adding (or subtracting) *n δt* where *δt* represents the time difference.

Mean anomaly does not measure an angle between any physical objects. It is simply a convenient uniform measure of how far around its orbit a body has progressed since pericenter. The mean anomaly is one of three angular parameters (known historically as "anomalies") that define a position along an orbit, the other two being the eccentric anomaly and the true anomaly.

## Formulae[edit]

The mean anomaly *M* can be computed from the eccentric anomaly *E* and the eccentricity *e* with Kepler's Equation:

Mean anomaly is also frequently seen as

where *M*_{0} is the *mean anomaly at epoch* and *t*_{0} is the *epoch*, a reference time to which the orbital elements are referred, which may or may not coincide with *τ*, the time of pericenter passage. The classical method of finding the position of an object in an elliptical orbit from a set of orbital elements is to calculate the mean anomaly by this equation, and then to solve Kepler's equation for the eccentric anomaly.

Define *ϖ* as the *longitude of the pericenter*, the angular distance of the pericenter from a reference direction. Define l as the *mean longitude*, the angular distance of the body from the same reference direction, assuming it moves with uniform angular motion as with the mean anomaly. Thus mean anomaly is also^{[5]}

Mean angular motion can also be expressed,

where *μ* is a gravitational parameter which varies with the masses of the objects, and *a* is the semi-major axis of the orbit. Mean anomaly can then be expanded,

and here mean anomaly represents uniform angular motion on a circle of radius *a*.^{[6]}

Mean anomaly can be expressed as a series expansion of the eccentricity *e* and the true anomaly ν,^{[7]}

A similar formula gives the true anomaly directly in terms of the mean anomaly:^{[8]}

## See also[edit]

## References[edit]

**^**Montenbruck, Oliver (1989).*Practical Ephemeris Calculations*. Springer-Verlag. p. 44. ISBN 0-387-50704-3.**^**Meeus, Jean (1991).*Astronomical Algorithms*. Willmann-Bell, Inc., Richmond, VA. p. 182. ISBN 0-943396-35-2.**^**Smart, W. M. (1977).*Textbook on Spherical Astronomy*(sixth ed.). Cambridge University Press, Cambridge. p. 113. ISBN 0-521-29180-1.**^**Meeus (1991), p. 183**^**Smart (1977), p. 122**^**Vallado, David A. (2001).*Fundamentals of Astrodynamics and Applications*(second ed.). El Segundo, CA: Microcosm Press. pp. 53–54. ISBN 1-881883-12-4.**^**Smart, W. M. (1953).*Celestial Mechanics*. Longmans, Green and Co., London. p. 38.**^**Roy, A.E. (1988).*Orbital Motion*(1 ed.). Bristol, UK; Philadelphia, PA: A. Hilger. ISBN 0852743602.

## External links[edit]

- Glossary entry
*anomaly, mean*at the US Naval Observatory's*Astronomical Almanac Online*