True anomaly

In astronomy, the true anomaly (, also written ) 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 T by 2π − T)

where:

• is orbital velocity vector of the orbiting body,
• is eccentricity vector,
• is orbital position vector (segment sp) of the orbiting body.

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For circular orbits this can be simplified to:

(if then replace T by 2π − T)

where:

• is vector pointing towards the ascending node (i.e. the z-component of is zero).

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For circular orbits with the inclination of zero this can be simplified further to:

(if then replace T by 2π − T)

where:

• is x-component of orbital position vector ,
• is x-component of orbital velocity vector .

Other relations

The relation between T 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).

See also

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