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True anomaly
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In astronomy, the true anomaly (Greek nu, also written or ) 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.
elliptic orbits true anomaly can be calculated from orbital state vectors as:
(if then replace by )
where:
-
For circular orbits this can be simplified to:
(if then replace by )
where:
-
For circular orbits with the inclination of zero this can be simplified further to:
(if then replace by )
where:
relation between ν and E, the eccentric anomaly, is:
or equivalently
The relations between the radius (position vector magnitude) and the anomalies are:
and
where a is the orbit's semi-major axis (segment cz).

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Encyclopedia
In astronomy, the true anomaly (Greek nu, also written or ) 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 can be calculated from orbital state vectors as:
(if then replace by )
where:
- is orbital velocity vector of the orbiting body,
- is eccentricity vector,
- is orbital position vector (segment sp) of the orbiting body.
-
For circular orbits this can be simplified to:
(if then replace by )
where:
- is vector pointing towards the ascending node (i.e. the z-component of is zero).
-
For circular orbits with the inclination of zero this can be simplified further to:
(if then replace by )
where:
- is x-component of orbital position vector ,
- is x-component of orbital velocity vector .
Other relations
The relation between ν and E, the eccentric anomaly, is:
or equivalently
The relations between the radius (position vector magnitude) and the anomalies are:
and
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.
See also
Mathematical definitions:
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