True anomaly

In astronomy, the true anomaly (<math>T\,\!<math>, also written <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>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:

or equivalently

The relations between the radius (position vector magnitude) and the anomalies are:

<math>r = a \left ( 1 – e \cdot \cos{E} \right )\,\!<math>

and

where a is the orbit's semi-major axis (segment cz).

True anomaly

Calculation from state vectors

Other relations

See also