true anomaly

Other Definitions
true anomaly (dict)

In astronomy, the true anomaly (

T\,\!

, also written

v\

) 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

T\,\!

can be calculated from orbital state vectors as:

T = \arccos { {\mathbf{e} \cdot \mathbf{r}} \over { \mathbf{\left |e \right |} \mathbf{\left |r \right |} }}

(if

\mathbf{r} \cdot \mathbf{v} < 0

then replace T by 2π − T)

where:

$\mathbf{v}\,$ is orbital velocity vector of the orbiting body,
$\mathbf{e}\,$ is eccentricity vector,
$\mathbf{r}\,$ is orbital position vector (segment sp) of the orbiting body.

For circular orbits this can be simplified to:

T = \arccos { {\mathbf{n} \cdot \mathbf{r}} \over { \mathbf{\left |n \right |} \mathbf{\left |r \right |} }}

(if

\mathbf{n} \cdot \mathbf{v} >0

then replace T by 2π − T)

where:

$\mathbf{n}$ is vector pointing towards the ascending node (i.e. the z-component of $\mathbf{n}$ is zero).

For circular orbits with the inclination of zero this can be simplified further to:

T = \arccos { r_x  \over { \mathbf{\left |r \right |}}}

(if

v_x\ > 0

then replace T by 2π − T)

where:

$r_x \,$ is x-component of orbital position vector $\mathbf{r}$ ,
$v_x \,$ is x-component of orbital velocity vector $\mathbf{v}$ .

Other relations

The relation between T and E, the eccentric anomaly, is:

\cos{T} = },\,

or equivalently

\tan{T \over 2} = \sqrt} \tan{E \over 2}.

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

r = a \left ( 1 - e \cdot \cos{E} \right )\,\!

and

r = a{(1 - e^2) \over (1 + e \cdot \cos{T})}\,\!

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

True Anomaly

Calculation from state vectors

Other relations

See also