True anomaly

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In astronomy, the true anomaly ([T\,\!], also written [ v\ ]) 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 [T\,\!] can be calculated from orbital state vectors as:

[ T = \arccos \cdot \mathbf} \over \mathbf }}] (if [\mathbf \cdot \mathbf < 0] then replace T by 2π − T)

where:

[ \mathbf\,] is orbital velocity vector of the orbiting body,
[ \mathbf\,] is eccentricity vector,
[ \mathbf\,] is orbital position vector (segment sp) of the orbiting body.

For circular orbits this can be simplified to:

[ T = \arccos \cdot \mathbf} \over \mathbf }}] (if [\mathbf \cdot \mathbf >0] then replace T by 2π − T)

where:

[ \mathbf ] is vector pointing towards the ascending node (i.e. the z-component of [ \mathbf ] is zero).

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

[ T = \arccos }}] (if [ v_x\ > 0] then replace T by 2π − T)

where:

[r_x \,] is x-component of orbital position vector [\mathbf],
[v_x \,] is x-component of orbital velocity vector [\mathbf].

Other relations

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

[\cos = },\,]

or equivalently

[\tan = \sqrt \tan.]

The relations between the radius (position vector magnitude) and the anomalies are:

[r = a \left ( 1 - e \cdot \cos \right )\,\!]

and

[r = a)}\,\!]

where a is the orbit's semi-major axis (segment cz).

True anomaly

Calculation from state vectors

Other relations

See also