# True anomaly

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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).

## Contents

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° (0–2πc).

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

### From state vectors

For elliptic orbits, the true anomalyν can be calculated from orbital state vectors as:

${\displaystyle \nu =\arccos {{\mathbf {e} \cdot \mathbf {r} } \over {\mathbf {\left|e\right|} \mathbf {\left|r\right|} }}}$
(if rv < 0 then replace ν by 2πν)

where:

#### 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:

${\displaystyle u=\arccos {{\mathbf {n} \cdot \mathbf {r} } \over {\mathbf {\left|n\right|} \mathbf {\left|r\right|} }}}$
(if rz < 0 then replace u by 2πu)

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:

${\displaystyle l=\arccos {r_{x} \over {\mathbf {\left|r\right|} }}}$
(if vx > 0 then replace l by 2πl)

where:

### From the eccentric anomaly

The relation between the true anomaly ν and the eccentric anomaly ${\displaystyle E}$ is:

${\displaystyle \cos {\nu }={{\cos {E}-e} \over {1-e\cos {E}}}}$

or using the sine [1] and tangent:

{\displaystyle {\begin{aligned}\sin {\nu }&={{{\sqrt {1-e^{2}\,}}\sin {E}} \over {1-e\cos {E}}}\\[4pt]\tan {\nu }={{\sin {\nu }} \over {\cos {\nu }}}&={{{\sqrt {1-e^{2}\,}}\sin {E}} \over {\cos {E}-e}}\end{aligned}}}

or equivalently:

${\displaystyle \tan {\nu \over 2}={\sqrt {{1+e\,} \over {1-e\,}}}\tan {E \over 2}}$

so

${\displaystyle \nu =2\,\operatorname {arctan} \left(\,{\sqrt {{1+e\,} \over {1-e\,}}}\tan {E \over 2}\,\right)}$

Alternatively, a form of this equation was derived by [2] that avoids numerical issues when the arguments are near ${\displaystyle \pm \pi }$, as the two tangents become infinite. Additionally, since ${\displaystyle {\frac {E}{2}}}$ and ${\displaystyle {\frac {\nu }{2}}}$ are always in the same quadrant, there will not be any sign problems.

${\displaystyle \tan {{\frac {1}{2}}(\nu -E)}={\frac {\beta \sin {E}}{1-\beta \cos {E}}}}$ where ${\displaystyle \beta ={\frac {e}{1+{\sqrt {1-e^{2}}}}}}$

so

${\displaystyle \nu =E+2\operatorname {arctan} \left(\,{\frac {\beta \sin {E}}{1-\beta \cos {E}}}\,\right)}$

### From the mean anomaly

The true anomaly can be calculated directly from the mean anomaly ${\displaystyle M}$ via a Fourier expansion: [3]

${\displaystyle \nu =M+2\sum _{k=1}^{\infty }{\frac {1}{k}}\left[\sum _{n=-\infty }^{\infty }J_{n}(-ke)\beta ^{|k+n|}\right]\sin {kM}}$

with Bessel functions ${\displaystyle J_{n}}$ and parameter ${\displaystyle \beta ={\frac {1-{\sqrt {1-e^{2}}}}{e}}}$.

Omitting all terms of order ${\displaystyle e^{4}}$ or higher (indicated by ${\displaystyle \operatorname {\mathcal {O}} \left(e^{4}\right)}$), it can be written as [3] [4] [5]

${\displaystyle \nu =M+\left(2e-{\frac {1}{4}}e^{3}\right)\sin {M}+{\frac {5}{4}}e^{2}\sin {2M}+{\frac {13}{12}}e^{3}\sin {3M}+\operatorname {\mathcal {O}} \left(e^{4}\right).}$

Note that for reasons of accuracy this approximation is usually limited to orbits where the eccentricity ${\displaystyle e}$ is small.

The expression ${\displaystyle \nu -M}$ is known as the equation of the center, where more details about the expansion are given.

The radius (distance between the focus of attraction and the orbiting body) is related to the true anomaly by the formula

${\displaystyle r=a\,{1-e^{2} \over 1+e\cos \nu }\,\!}$

where a is the orbit's semi-major axis.

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## References

1. Fundamentals of Astrodynamics and Applications by David A. Vallado
2. 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.
3. 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.
4. Smart, W. M. (1977). Textbook on Spherical Astronomy (PDF). p. 120 (Eq. (87)). Bibcode:1977tsa..book.....S.
5. 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.
• 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.)