In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse (the point around which the object orbits). The true anomaly is usually denoted by the Greek letters ν or θ, or the Latin letter f, and is usually restricted to the range 0–360° (0–2π rad). The true anomaly f is one of three angular parameters (anomalies) that can be used to define a position along an orbit, the other two being the eccentric anomaly and the mean anomaly.
Etymology The term "anomaly" originally referred to the irregular movements of planets. Over time, that phrase evolved to refer to the general angular location of a planet around its orbit. The word true was added to refer to the actual angular location as opposed to other estimations of calculations of angular locations, for example mean anomaly.
Formulas
From state vectors For elliptic orbits, the true anomaly ν can be calculated from orbital state vectors as:
ν = arccos e ⋅ r | e | | r | {\displaystyle \nu =\arccos {{\mathbf {e} \cdot \mathbf {r} } \over {\mathbf {\left|e\right|} \mathbf {\left|r\right|} }}}
(if r ⋅ v < 0 then replace ν by 2π − ν) where:
v is the orbital velocity vector of the orbiting body, e is the eccentricity vector, r is the orbital position vector (segment FP in the figure) of the orbiting body.
Circular orbit For circular orbits the true anomaly is undefined, because circular orbits do not have a uniquely determined periapsis. Instead the argument of latitude u is used:
u = arccos n ⋅ r | n | | r | {\displaystyle u=\arccos {{\mathbf {n} \cdot \mathbf {r} } \over {\mathbf {\left|n\right|} \mathbf {\left|r\right|} }}}
(if rz < 0 then replace u by 2π − u) where:
n is a vector pointing towards the ascending node (i.e. the z-component of n is zero). rz is the z-component of the orbital position vector r
Circular orbit with zero inclination For circular orbits with zero inclination the argument of latitude is also undefined, because there is no uniquely determined line of nodes. One uses the true longitude instead:
l = arccos r x | r | {\displaystyle l=\arccos {r_{x} \over {\mathbf {\left|r\right|} }}}
(if vx > 0 then replace l by 2π − l) where:
rx is the x-component of the orbital position vector r vx is the x-component of the orbital velocity vector v.
From the eccentric anomaly The relation between the true anomaly ν and the eccentric anomaly E {\displaystyle E} is:
cos ν = cos E − e 1 − e cos E {\displaystyle \cos {\nu }={{\cos {E}-e} \over {1-e\cos {E}}}}
or using the sine and tangent:
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