True anomaly
In astronomy, the true anomaly (
, also written 
) is the angle between the direction z-s of periapsis and the current position p of an object on its orbit, measured at the focus s of the ellipse (the point around which the object orbits). In the diagram below, true anomaly is the angle z-s-p.
Missing image
Kepler's-equation-scheme.png
Image:Kepler's-equation-scheme.png
Calculation from state vectors
For elliptic orbits true anomaly can be calculated from orbital state vectors as:
(if 
then replace T by 2π − T)
where:
-
is orbital velocity vector of the orbiting body,
-
is eccentricity vector,
-
is orbital position vector (segment sp) of the orbiting body.
For circular orbits this can be simplified to:
(if 
then replace T by 2π − T)
where:
-
is vector pointing towards the ascending node (i.e. the z-component of is zero).
For circular orbits with the inclination of zero this can be simplified further to:
(if 
then replace T by 2π − T)
where:
-
is x-component of orbital position vector 
,
-
is x-component of orbital velocity vector 
.
Other relations
The relation between T and E, the eccentric anomaly, is:
or equivalently
The relations between the radius (position vector magnitude) and the anomalies are:
and
where a is the orbit's semi-major axis (segment cz).