True anomaly
The true anomaly (<math> v\ <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> v\ <math> can be calculated from orbital state vectors as:
- <math> v = \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 <math>v = 2 \pi - v\ <math>)
where:
- <math> \mathbf{v}\,<math> is orbital velocity vector of the orbiting body's,
- <math> \mathbf{e}\,<math> is eccentricity vector,
- <math> \mathbf{r}\,<math> is orbital position vector of the orbiting body's.
For circular orbits this can be simplified to:
- <math> v = \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 <math>v = 2 \pi - v\ <math>)
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> v = \arccos { r_x \over { \mathbf{\left |r \right |}}}<math> (if <math> v_x\ > 0<math> then <math>v\ = 2 \pi - v<math>)
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>.
See also
fr:Anomalie vraie