True anomaly

In astronomy, the true anomaly $u,!$ (Greek nu, also written $heta$ or $f$) 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 $u,!$ can be calculated from orbital state vectors as:

:$u\; =\; arccos\; \{\; \{mathbf\{e\}\; cdot\; mathbf\{r\; over\; \{\; mathbf\{left\; |e\; ight\; mathbf\{left\; |r\; ight$   (if then replace $u$ by $2pi-\; u$)

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:

:$u\; =\; arccos\; \{\; \{mathbf\{n\}\; cdot\; mathbf\{r\; over\; \{\; mathbf\{left\; |n\; ight\; mathbf\{left\; |r\; ight$   (if $mathbf\{n\}\; cdot\; mathbf\{v\}\; >0$ then replace $u$ by $2pi-\; u$)

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:

:$u\; =\; arccos\; \{\; r\_x\; over\; \{\; mathbf\{left\; |r\; ight\; |\}$   (if $v\_x\; >\; 0$ then replace $u$ by $2pi-\; u$)

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 "ν" and "E", the eccentric anomaly, is:

:$cos\{\; u\}\; =$cos{E} - e} over {1 - e cdot cos{E},,

or equivalently

:$an\{\; u\; over\; 2\}\; =\; sqrt${1+e} over {1-e} an{E over 2}.

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

:$r\; =\; a\; left\; (\; 1\; -\; e\; cdot\; cos\{E\}\; ight\; ),!$

and

:$r\; =\; acdot\{1\; -\; e^2\; over\; 1\; +\; e\; cdot\; cos\; u\},!$

where "a" is the orbit's semi-major axis (segment "cz"). Note that z is the closest approach to the focus s or object being orbited but also the furthest point from the center c.

ee also

* Kepler's laws of planetary motion
* Eccentric anomaly
* Mean anomaly