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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).
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 in the figure) 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).
- rz is the z-component of the orbital position vector r
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:
An equivalent form avoids the singularity as e → 1, however it does not produce the correct value for :
or, with the same problem as e → 1 ,
In both of the above, the function arg(x, y) is the polar argument of the vector (x y), available in many programming languages as the library function named atan2(y,x) (note the reversed order of x and y).
From the mean anomaly
where the means that the omitted terms are all of order e4 or higher. Note that for reasons of accuracy this approximation is usually limited to orbits where the eccentricity (e) is small.
The expression is known as the equation of the center.
Radius from true anomaly
The radius (distance between 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.
- Murray, C. D. & Dermott, S. F., 1999, Solar System Dynamics, Cambridge University Press, Cambridge. ISBN 0-521-57597-4
- Plummer, H. C., 1960, An Introductory Treatise on Dynamical Astronomy, Dover Publications, New York. OCLC 1311887 (Reprint of the 1918 Cambridge University Press edition.)