True anomaly

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In astronomy, the true anomaly \nu\,\! (Greek nu, also written \theta\ 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.

Image:anomalies.svg

[edit] Calculation from state vectors

For elliptic orbits true anomaly \nu\,\! can be calculated from orbital state vectors as:

\nu = \arccos { {\mathbf{e} \cdot \mathbf{r}} \over { \mathbf{\left |e \right |} \mathbf{\left |r \right |} }}   (if \mathbf{r} \cdot \mathbf{v} < 0 then replace \nu\ by 2\pi-\nu\ )

where:

For circular orbits this can be simplified to:

\nu = \arccos { {\mathbf{n} \cdot \mathbf{r}} \over { \mathbf{\left |n \right |} \mathbf{\left |r \right |} }}   (if \mathbf{n} \cdot \mathbf{v} >0 then replace \nu\ by 2\pi-\nu\ )

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:

\nu = \arccos { r_x \over { \mathbf{\left |r \right |}}}   (if v_x > 0\ then replace \nu\ by 2\pi-\nu\ )

where:

[edit] Other relations

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

\cos{\nu} = {{\cos{E} - e} \over {1 - e \cdot \cos{E}}},\,

or equivalently

\tan{\nu \over 2} = \sqrt{{{1+e} \over {1-e}}} \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{\nu})}\,\!

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.

[edit] See also