In astronomy, the true anomaly (<math>T\,\!</math>, also written <math> \nu\ </math>) 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. 
Calculation from state vectors
For elliptic orbits true anomaly <math>T\,\!</math> can be calculated from orbital state vectors as:
- <math> T = \arccos { {\mathbf{e} \cdot \mathbf{r}} \over { \mathbf{\left |e \right |} \mathbf{\left |r \right |} }}</math> (if <math>\mathbf{r} \cdot \mathbf{v} < 0</math> then replace T by 2π − T)
where:
- <math> \mathbf{v}\,</math> is orbital velocity vector of the orbiting body,
- <math> \mathbf{e}\,</math> is eccentricity vector,
- <math> \mathbf{r}\,</math> is orbital position vector (segment sp) of the orbiting body.
For circular orbits this can be simplified to:
- <math> T = \arccos { {\mathbf{n} \cdot \mathbf{r}} \over { \mathbf{\left |n \right |} \mathbf{\left |r \right |} }}</math> (if <math>\mathbf{n} \cdot \mathbf{v} >0</math> then replace T by 2π − T)
where:
- <math> \mathbf{n} </math> is vector pointing towards the ascending node (i.e. the z-component of <math> \mathbf{n} </math> is zero).
For circular orbits with the inclination of zero this can be simplified further to:
- <math> T = \arccos { r_x \over { \mathbf{\left |r \right |}}}</math> (if <math> v_x\ > 0</math> then replace T by 2π − T)
where:
- <math>r_x \,</math> is x-component of orbital position vector <math>\mathbf{r}</math>,
- <math>v_x \,</math> is x-component of orbital velocity vector <math>\mathbf{v}</math>.
Other relations
The relation between T and E, the eccentric anomaly, is:
- <math>\cos{T} = {{\cos{E} - e} \over {1 - e \cdot \cos{E}}},\,</math>
or equivalently
- <math>\tan{T \over 2} = \sqrt{{{1+e} \over {1-e}}} \tan{E \over 2}.</math>
The relations between the radius (position vector magnitude) and the anomalies are:
- <math>r = a \left ( 1 - e \cdot \cos{E} \right )\,\!</math>
and
- <math>r = a{(1 - e^2) \over (1 + e \cdot \cos{T})}\,\!</math>
where a is the orbit's semi-major axis (segment cz). Note that z is the periapsis (closest approach to the focus or object being orbited) and also one of two points where the semi-major axis (furthest distance from the centre of the ellipse) can be measured, the other point being the apoapsis (furthest distance from the focus being orbited and 180 degrees around from the periapsis).
