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True anomaly - Definition and Overview |
| Related Words: Aberration, Abnormality, Conceit, Crank, Crankism, Crotchet, Curiosity, Derangement, Deviation, Difference, Divergence, Eccentricity |
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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:
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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).
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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:
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).
See also
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