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Formulas

From state vectors

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

ν = arccos ⁡ e ⋅ r | e | | r | {\displaystyle \nu =\arccos {{\mathbf {e} \cdot \mathbf {r} } \over {\mathbf {\left|e\right|} \mathbf {\left|r\right|} }}} (if r ⋅ v < 0 then replace ν by 2π − ν)

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:

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

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:

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

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 ν and the eccentric anomaly E {\displaystyle E} is:

cos ⁡ ν = cos ⁡ E − e 1 − e cos ⁡ E {\displaystyle \cos {\nu }={{\cos {E}-e} \over {1-e\cos {E}}}}

or using the sine¹ and tangent:

sin ⁡ ν = 1 − e 2 sin ⁡ E 1 − e cos ⁡ E tan ⁡ ν = sin ⁡ ν cos ⁡ ν = 1 − e 2 sin ⁡ E cos ⁡ E − e {\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:

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

so

ν = 2 arctan ⁡ ( 1 + e 1 − e tan ⁡ E 2 ) {\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 ± π {\displaystyle \pm \pi } , as the two tangents become infinite. Additionally, since E 2 {\displaystyle {\frac {E}{2}}} and ν 2 {\displaystyle {\frac {\nu }{2}}} are always in the same quadrant, there will not be any sign problems.

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

so

ν = E + 2 arctan ⁡ ( β sin ⁡ E 1 − β cos ⁡ E ) {\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 M {\displaystyle M} via a Fourier expansion:³

ν = M + 2 ∑ k = 1 ∞ 1 k [ ∑ n = − ∞ ∞ J n ( − k e ) β | k + n | ] sin ⁡ k M {\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 J n {\displaystyle J_{n}} and parameter β = 1 − 1 − e 2 e {\displaystyle \beta ={\frac {1-{\sqrt {1-e^{2}}}}{e}}} .

Omitting all terms of order e 4 {\displaystyle e^{4}} or higher (indicated by O ⁡ ( e 4 ) {\displaystyle \operatorname {\mathcal {O}} \left(e^{4}\right)} ), it can be written as⁴⁵⁶

ν = M + ( 2 e − 1 4 e 3 ) sin ⁡ M + 5 4 e 2 sin ⁡ 2 M + 13 12 e 3 sin ⁡ 3 M + O ⁡ ( e 4 ) . {\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 e {\displaystyle e} is small.

The expression ν − M {\displaystyle \nu -M} 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

r ( t ) = a 1 − e 2 1 + e cos ⁡ ν ( t ) {\displaystyle r(t)=a\,{1-e^{2} \over 1+e\cos \nu (t)}\,\!}

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 the θ {\displaystyle \theta } and is usually restricted to the range 0 - 360 degree (0 - 2 π {\displaystyle \pi } radian).

The projective anomaly θ {\displaystyle \theta } is one of four angular parameters (anomalies) that defines a position along an orbit, the other two being the eccentric anomaly, true anomaly and the mean anomaly.

In the projective geometry, circle, ellipse, parabolla, hyperbolla are treated as a same kind of quadratic curves.

projective parameters and projective anomaly

An orbit type is classified by two project parameters α {\displaystyle \alpha } and β {\displaystyle \beta } as follows,

• circular orbit β = 0 {\displaystyle \beta =0}

• elliptic orbit α β < 1 {\displaystyle \alpha \beta <1}

• parabolic orbit α β = 1 {\displaystyle \alpha \beta =1}

• hyperbolic orbit α β > 1 {\displaystyle \alpha \beta >1}

• linear orbit α = β {\displaystyle \alpha =\beta }

• imaginary orbit α < β {\displaystyle \alpha <\beta }

where

α = ( 1 + e ) ( q − p ) + ( 1 + e ) 2 ( q + p ) 2 + 4 e 2 2 {\displaystyle \alpha ={\frac {(1+e)(q-p)+{\sqrt {(1+e)^{2}(q+p)^{2}+4e^{2}}}}{2}}}

β = 2 e ( 1 + e ) ( q + p ) + ( 1 + e ) 2 ( q + p ) 2 + 4 e 2 {\displaystyle \beta ={\frac {2e}{(1+e)(q+p)+{\sqrt {(1+e)^{2}(q+p)^{2}+4e^{2}}}}}}

q = ( 1 − e ) a {\displaystyle q=(1-e)a}

p = 1 Q = 1 ( 1 + e ) a {\displaystyle p={\frac {1}{Q}}={\frac {1}{(1+e)a}}}

where α {\displaystyle \alpha } is semi major axis, e {\displaystyle e} is eccentricity, q {\displaystyle q} is perihelion distance、 Q {\displaystyle Q} is aphelion distance.

Position and heliocentric distance of the planet x {\displaystyle x} , y {\displaystyle y} and r {\displaystyle r} can be calculated as functions of the projective anomaly θ {\displaystyle \theta }  :

x = − β + α cos ⁡ θ 1 + α β cos ⁡ θ {\displaystyle x={\frac {-\beta +\alpha \cos \theta }{1+\alpha \beta \cos \theta }}}

y = α 2 − β 2 sin ⁡ θ 1 + α β cos ⁡ θ {\displaystyle y={\frac {{\sqrt {\alpha ^{2}-\beta ^{2}}}\sin \theta }{1+\alpha \beta \cos \theta }}}

r = α − β cos ⁡ θ 1 + α β cos ⁡ θ {\displaystyle r={\frac {\alpha -\beta \cos \theta }{1+\alpha \beta \cos \theta }}}

Kepler's equation

The projective anomaly θ {\displaystyle \theta } can be calculated from the eccentric anomaly u {\displaystyle u} as follows,

• Case : α β < 1 {\displaystyle \alpha \beta <1}

tan ⁡ θ 2 = 1 + α β 1 − α β tan ⁡ u 2 {\displaystyle \tan {\frac {\theta }{2}}={\sqrt {\frac {1+\alpha \beta }{1-\alpha \beta }}}\tan {\frac {u}{2}}}

u − e sin ⁡ u = M = ( 1 − α 2 β 2 α ( 1 + β 2 ) ) 3 / 2 k ( t − T 0 ) {\displaystyle u-e\sin u=M=\left({\frac {1-\alpha ^{2}\beta ^{2}}{\alpha (1+\beta ^{2})}}\right)^{3/2}k(t-T_{0})}

• case : α β = 1 {\displaystyle \alpha \beta =1}

s 3 3 + α 2 − 1 α 2 + 1 s = 2 k ( t − T 0 ) α ( α 2 + 1 ) 3 {\displaystyle {\frac {s^{3}}{3}}+{\frac {\alpha ^{2}-1}{\alpha ^{2}+1}}s={\frac {2k(t-T_{0})}{\sqrt {\alpha (\alpha ^{2}+1)^{3}}}}}

s = tan ⁡ θ 2 {\displaystyle s=\tan {\frac {\theta }{2}}}

• case : α β > 1 {\displaystyle \alpha \beta >1}

tan ⁡ θ 2 = α β + 1 α β − 1 tanh ⁡ u 2 {\displaystyle \tan {\frac {\theta }{2}}={\sqrt {\frac {\alpha \beta +1}{\alpha \beta -1}}}\tanh {\frac {u}{2}}}

e sinh ⁡ u − u = M = ( α 2 β 2 − 1 α ( 1 + β 2 ) ) 3 / 2 k ( t − T 0 ) {\displaystyle e\sinh u-u=M=\left({\frac {\alpha ^{2}\beta ^{2}-1}{\alpha (1+\beta ^{2})}}\right)^{3/2}k(t-T_{0})}

The above equations are called Kepler's equation.

Generalized anomaly

For arbitrary constant λ {\displaystyle \lambda } , the generalized anomaly Θ {\displaystyle \Theta } is related as

tan ⁡ Θ 2 = λ tan ⁡ u 2 {\displaystyle \tan {\frac {\Theta }{2}}=\lambda \tan {\frac {u}{2}}}

The eccentric anomaly, the true anomaly, and the projective anomaly are the cases of λ = 1 {\displaystyle \lambda =1} , λ = 1 + e 1 − e {\displaystyle \lambda ={\sqrt {\frac {1+e}{1-e}}}} , λ = 1 + α β 1 − α β {\displaystyle \lambda ={\sqrt {\frac {1+\alpha \beta }{1-\alpha \beta }}}} , 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
• Projective anomaly
• Ellipse
• Hyperbola

Further reading

• 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

• Federal Aviation Administration - Describing Orbits

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. https://ui.adsabs.harvard.edu/abs/1973CeMec...7..388B/abstract ↩

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. 978-1-60086-026-3 ↩

4. 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. 978-1-60086-026-3 ↩

5. Smart, W. M. (1977). Textbook on Spherical Astronomy (PDF). p. 120 (Eq. (87)). Bibcode:1977tsa..book.....S. https://wangsajaya.files.wordpress.com/2015/02/textbook-on-spherical-astronomy-smart-6ed-1977.pdf ↩

6. 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. 0750310154 ↩