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 or , or the Latin letter , and is usually restricted to the range 0–360° (0–2π rad).
The true anomaly 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 can be calculated from orbital state vectors as:
(if then replace by )
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:
(if then replace )
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:
(if then replace by )
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 and the eccentric anomaly is:
or using the sineFundamentals of Astrodynamics and Applications by David A. Vallado and tangent:
or equivalently:
so
Alternatively, a form of this equation was derived by R. Broucke and P. Cefola that avoids numerical issues when the arguments are near , as the two tangents become infinite. Additionally, since and are always in the same quadrant, there will not be any sign problems.
where
so
From the mean anomaly The true anomaly can be calculated directly from the mean anomaly via a Fourier expansion:
with Bessel functions and parameter .
Omitting all terms of order or higher (indicated by ), it can be written as
Note that for reasons of accuracy this approximation is usually limited to orbits where the eccentricity is small.
The expression is known as the equation of the center, where more details about the expansion are given.
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.
In celestial mechanics, Projective 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 in the projective space.
The projective anomaly is usually denoted by and is usually restricted to the range 0 - 360 degrees (0 - 2 radians).
The projective anomaly is one of four angular parameters (anomalies) that defines a position along an orbit, the other three being the eccentric anomaly, the true anomaly, and the mean anomaly.
In the projective geometry, circles, ellipses, parabolae, and hyperbolae are treated as the same kind of quadratic curves.
Projective parameters and projective anomaly
An orbit type is classified by two project parameters and as follows,
circular orbit elliptic orbit parabolic orbit hyperbolic orbit linear orbit imaginary orbit
where
where is semi major axis, is eccentricity, is perihelion distance, and is aphelion distance.
Position and heliocentric distance of the planet , and can be calculated as functions of the projective anomaly :
Kepler's equation
The projective anomaly can be calculated from the eccentric anomaly as follows,
Case :
case :
case :
The above equations are called Kepler's equation.
Generalized anomaly
For arbitrary constant , the generalized anomaly is related as
The eccentric anomaly, the true anomaly, and the projective anomaly are the cases of , , , respectively.
Sato, I., "A New Anomaly of Keplerian Motion", Astronomical Journal Vol.116, pp. 2038-3039, (1997)
See also Two body problem Mean anomaly Eccentric anomaly Kepler's equation Projective geometry Kepler's laws of planetary motion Ellipse Hyperbola
References
Further reading Murray, C. D. & Dermott, S. F., 1999, Solar System Dynamics, Cambridge University Press, Cambridge. Plummer, H. C., 1960, An Introductory Treatise on Dynamical Astronomy, Dover Publications, New York. (Reprint of the 1918 Cambridge University Press edition.)
External links
Federal Aviation Administration - Describing Orbits