True anomaly


In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse.
The true anomaly is usually denoted by the Greek letters or, or the Latin letter, and is usually restricted to the range 0–360°.
As shown in the image, the true anomaly is one of three angular parameters that defines a position along an orbit, the other two being the eccentric anomaly and the mean anomaly.

Formulas

From state vectors

For elliptic orbits, the true anomaly can be calculated from orbital state vectors as:
where:
For circular orbits the true anomaly is undefined, because circular orbits do not have a uniquely determined periapsis. Instead the argument of latitude u is used:
where:
For circular orbits with zero inclination the argument of latitude is also undefined, because there is no uniquely determined line of nodes. One uses the true longitude instead:
where:
The relation between the true anomaly and the eccentric anomaly E is:
or using the sine and tangent:
or equivalently:
so
An equivalent form avoids the singularity as e → 1, however it does not produce the correct value for :
or, with the same problem as e → 1 ,
In both of the above, the function arg is the polar argument of the vector, available in many programming languages as the library function named atan2.

From the mean anomaly

The true anomaly can be calculated directly from the mean anomaly via a Fourier expansion:
where the "big-O" notation means that the omitted terms are all of order e4.
The expression is known as the equation of the center.

Radius from true anomaly

The radius is related to the true anomaly by the formula
where a is the orbit's semi-major axis.