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.

Definition

Define as the time required for a particular body to complete one orbit. In time , the radius vector sweeps out 2π radians or 360°. The average rate of sweep, , 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 as the time at which the body is at the pericenter. From the above definitions, a new quantity, , the mean anomaly can be defined

which gives an angular distance from the pericenter at arbitrary time , with dimensions of radians or degrees.

Because the rate of increase, , 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. 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) where represents the time difference.