True anomaly
From Freepedia
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.
Image:Kepler's-equation-scheme.png
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:
- <math>\cos{T} = {{\cos{E} - e} \over {1 - e \cdot \cos{E}}},\,</math>
or equivalently
- <math>\tan{T \over 2} = \sqrt{{{1+e} \over {1-e}}} \tan{E \over 2}.</math>
The relations between the radius (position vector magnitude) and the anomalies are:
- <math>r = a \left ( 1 - e \cdot \cos{E} \right )\,\!</math>
and
- <math>r = a{(1 - e^2) \over (1 + e \cdot \cos{T})}\,\!</math>
where a is the orbit's semi-major axis (segment cz).
