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).
As shown in the image, the true anomaly f is one of three angular parameters ("anomalies") that defines a position along an orbit, the other two being the eccentric anomaly and the mean anomaly. Note that the satellite P orbits around the planet which is at position F.
From state vectors
For elliptic orbits, the true anomaly ν can be calculated from orbital state vectors as:
- (if r ⋅ v < 0 then replace ν by 2π − ν)
- v is the orbital velocity vector of the orbiting body,
- e is the eccentricity vector,
- r is the orbital position vector (segment fp) of the orbiting body.
- (if rz < 0 then replace u by 2π − u)
- n is a vector pointing towards the ascending node (i.e. the z-component of n is zero).
Circular orbit with zero inclination
- (if vx > 0 then replace l by 2π − l)
- 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 is:
where arg(x, y) is the polar argument of the vector (x, y) (available in many programming languages as the library function atan2(y, x) in Fortran and MATLAB, or as ArcTan(x, y) in Wolfram Mathematica).
Radius from true anomaly
The radius (distance from the focus of attraction and the orbiting body) is related to the true anomaly by the formula
where a is the orbit's semi-major axis (segment cz).
- Fundamentals of Astrodynamics and Applications by David A. Vallado