Mean anomaly

In celestial mechanics, 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 in equal intervals of time by a line joining the focus and the orbiting body (Kepler's second law).

The mean anomaly increases uniformly from 0 to 2\pi radians during each orbit. However, it is not an angle. Due to Kepler's second law, the mean anomaly is proportional to the area swept by the focus-to-body line since the last periapsis.

The mean anomaly is usually denoted by the letter M, and is given by the formula:

M = n \, t = \sqrt{\frac{ G( M_\star \! + \!m ) } {a^3}} \,t

where n is the mean motion, a is the length of the orbit's semi-major axis, M_\star and m are the orbiting masses, and G is the gravitational constant.

The mean anomaly is the time since the last periapsis multiplied by the mean motion, and the mean motion is 2\pi radians divided by the duration of a full orbit.

The mean anomaly is one of three angular parameters ("anomalies") that define a position along an orbit, the other two being the eccentric anomaly and the true anomaly. 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) \sqrt{\frac{ G( M_\star \! + \!m ) } {a^3}} \,\delta t where \delta t represents the time difference. The other anomalies can hence be calculated.

Formulas

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

M = E - e \cdot \sin E

To find the position of the object in an elliptic Kepler orbit at a given time t, the mean anomaly is found by multiplying the time and the mean motion, then it is used to find the eccentric anomaly by solving Kepler's equation.

It is also frequently seen:

M = M_0 + nt,

Again n is the mean motion. However, t, in this instance, is the time since epoch, which is how much time has passed since the measurement of M0 was taken. The value M0 denotes the mean anomaly at epoch, which is the mean anomaly at the time the measurement was taken.

See also

References