Mean anomaly

In celestial mechanics, the 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 at the focus in equal intervals of time.

The mean anomaly increases uniformly from 0 to 2π 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 2π 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 δt 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 = M₀ + 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

• Kepler orbit
• Ellipse
• Eccentric anomaly
• True anomaly

References

• Murray, C. D. & Dermott, S. F. 1999, Solar System Dynamics, Cambridge University Press, Cambridge.
• Plummer, H.C., 1960, An Introductory treatise on Dynamical Astronomy, Dover Publications, New York. (Reprint of the 1918 Cambridge University Press edition.)