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 lettersν or θ, or the Latin letterf, and is usually restricted to the range 0–360° (0–2π rad).
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.
Formulas
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From state vectors
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For elliptic orbits, the true anomalyν can be calculated from orbital state vectors as:
For circular orbits the true anomaly is undefined, because circular orbits do not have a uniquely determined periapsis. Instead the argument of latitudeu is used:
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:
Alternatively, a form of this equation was derived by ^{[2]} that avoids numerical issues when the arguments are near $\pm \pi$, as the two tangents become infinite. Additionally, since ${\frac {E}{2}}$ and ${\frac {\nu }{2}}$ are always in the same quadrant, there will not be any sign problems.
^Fundamentals of Astrodynamics and Applications by David A. Vallado
^Broucke, R.; Cefola, P. (1973). "A Note on the Relations between True and Eccentric Anomalies in the Two-Body Problem". Celestial Mechanics. 7 (3): 388–389. Bibcode:1973CeMec...7..388B. doi:10.1007/BF01227859. ISSN 0008-8714. S2CID 122878026.
^ ^{a}^{b}Battin, R.H. (1999). An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series. American Institute of Aeronautics & Astronautics. p. 212 (Eq. (5.32)). ISBN 978-1-60086-026-3. Retrieved 2022-08-02.
^Smart, W. M. (1977). Textbook on Spherical Astronomy(PDF). p. 120 (Eq. (87)). Bibcode:1977tsa..book.....S.
^Roy, A.E. (2005). Orbital Motion (4 ed.). Bristol, UK; Philadelphia, PA: Institute of Physics (IoP). p. 78 (Eq. (4.65)). Bibcode:2005ormo.book.....R. ISBN 0750310154. Archived from the original on 2021-05-15. Retrieved 2020-08-29.
Further reading
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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.)
External links
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Federal Aviation Administration - Describing Orbits