Part of a series on
Astrodynamics
Orbital mechanics
Orbital elements Apsis Argument of periapsis Eccentricity Inclination Mean anomaly Orbital nodes Semi-major axis True anomaly
Types oftwo-body orbits by eccentricity Circular orbit Elliptic orbit Transfer orbit (Hohmann transfer orbit Bi-elliptic transfer orbit) Parabolic orbit Hyperbolic orbit Radial orbit Decaying orbit
Equations Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation
Celestial mechanics
Gravitational influences Barycenter Hill sphere Perturbations Sphere of influence
N-body orbits Lagrangian points (Halo orbits) Lissajous orbits Lyapunov orbits
Engineering and efficiency
Preflight engineering Mass ratio Payload fraction Propellant mass fraction Tsiolkovsky rocket equation
Efficiency measures Gravity assist Oberth effect
Propulsive maneuvers Orbital maneuver Orbit insertion
v t e

Incelestial mechanics,true anomaly is an angularparameter that defines the position of a body moving along aKeplerian orbit. It is the angle between the direction ofperiapsis and the current position of the body, as seen from the main focus of theellipse (the point around which the object orbits).

The true anomaly is usually denoted by theGreek lettersν orθ, or theLatin letterf, and is usually restricted to the range 0–360° (0–2π rad).

The true anomalyf is one of three angular parameters (anomalies) that defines a position along an orbit, the other two being theeccentric anomaly and themean anomaly.

For elliptic orbits, thetrue anomalyν can be calculated fromorbital state vectors as:

${\displaystyle \nu =\arccos \frac{\mathbf{e}\cdot \mathbf{r}}{\left|\mathbf{e}\right|\left|\mathbf{r}\right|}}$

(ifr ⋅v < 0 then replaceν by2π −ν)

where:

• v is theorbital velocity vector of the orbiting body,
• e is theeccentricity vector,
• r is theorbital position vector (segmentFP in the figure) of the orbiting body.

Forcircular orbits the true anomaly is undefined, because circular orbits do not have a uniquely determined periapsis. Instead theargument of latitudeu is used:

${\displaystyle u=\arccos \frac{\mathbf{n}\cdot \mathbf{r}}{\left|\mathbf{n}\right|\left|\mathbf{r}\right|}}$

(ifr_z < 0 then replaceu by 2π −u)

where:

• n is a vector pointing towards the ascending node (i.e. thez-component ofn is zero).
• r_z is thez-component of theorbital position vectorr

Forcircular orbits with zero inclination the argument of latitude is also undefined, because there is no uniquely determined line of nodes. One uses thetrue longitude instead:

${\displaystyle l=\arccos \frac{r_x}{\left|\mathbf{r}\right|}}$

(ifv_x > 0 then replacel by2π −l)

where:

• r_x is thex-component of theorbital position vectorr
• v_x is thex-component of theorbital velocity vectorv.

The relation between the true anomalyν and theeccentric anomaly${\displaystyle E}$ is:

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

or using thesine^[1] andtangent:

or equivalently:

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

so

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

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

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

The true anomaly can be calculated directly from themean anomaly${\displaystyle M}$ via aFourier expansion:^[3]

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

withBessel 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${\displaystyle \mathcal{O}\left(e^4\right)}$), it can be written as^[3][4][5]

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

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 theequation 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

${\displaystyle r(t)=a\frac{1-e^2}{1+e\cos \nu (t)}}$

wherea is the orbit'ssemi-major axis.

Incelestial mechanics,Projective anomaly is an angularparameter that defines the position of a body moving along aKeplerian orbit. It is the angle between the direction ofperiapsis and the current position of the body in the projective space.

The projective anomaly is usually denoted by the${\displaystyle \theta }$ and is usually restricted to the range 0 - 360 degree (0 - 2${\displaystyle \pi }$ radian).

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

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

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

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

• elliptic orbit

• parabolic orbit${\displaystyle \alpha \beta =1}$

• hyperbolic orbit${\displaystyle \alpha \beta >1}$

• linear orbit${\displaystyle \alpha =\beta }$

• imaginary orbit

where

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

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

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

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

where${\displaystyle \alpha }$ issemi major axis,${\displaystyle e}$ iseccentricity,${\displaystyle q}$ isperihelion distance、${\displaystyle Q}$ isaphelion distance.

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

${\displaystyle x=\frac{-\beta +\alpha \cos \theta }{1+\alpha \beta \cos \theta }}$

${\displaystyle y=\frac{\sqrt{{\alpha }^2-{\beta }^2}\sin \theta }{1+\alpha \beta \cos \theta }}$

${\displaystyle r=\frac{\alpha -\beta \cos \theta }{1+\alpha \beta \cos \theta }}$

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

• Case :

${\displaystyle \tan \frac{\theta }{2}=\sqrt{\frac{1+\alpha \beta }{1-\alpha \beta }}\tan \frac{u}{2}}$

${\displaystyle u-e\sin u=M={\left(\frac{1-{\alpha }^2{\beta }^2}{\alpha (1+{\beta }^2)}\right)}^{3/2}k(t-T_0)}$

• case :${\displaystyle \alpha \beta =1}$

${\displaystyle \frac{s^3}{3}+\frac{{\alpha }^2-1}{{\alpha }^2+1}s=\frac{2k(t-T_0)}{\sqrt{\alpha ({\alpha }^2+1{)}^3}}}$

${\displaystyle s=\tan \frac{\theta }{2}}$

• case :${\displaystyle \alpha \beta >1}$

${\displaystyle \tan \frac{\theta }{2}=\sqrt{\frac{\alpha \beta +1}{\alpha \beta -1}}\tanh \frac{u}{2}}$

${\displaystyle e\sinh u-u=M={\left(\frac{{\alpha }^2{\beta }^2-1}{\alpha (1+{\beta }^2)}\right)}^{3/2}k(t-T_0)}$

The above equations are calledKepler's equation.

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

${\displaystyle \tan \frac{\mathrm{\Theta }}{2}=\lambda \tan \frac{u}{2}}$

The eccentric anomaly, the true anomaly, and the projective anomaly are the cases of${\displaystyle \lambda =1}$,${\displaystyle \lambda =\sqrt{\frac{1+e}{1-e}}}$,${\displaystyle \lambda =\sqrt{\frac{1+\alpha \beta }{1-\alpha \beta }}}$, respectively.

• Sato, I., "A New Anomaly of Keplerian Motion", Astronomical Journal Vol.116, pp.2038-3039, (1997)

• Two body problem
• Mean anomaly
• Eccentric anomaly
• Kepler's equation
• projective geometry
• Kepler's laws of planetary motion
• Projective anomaly
• Ellipse
• Hyperbola

• 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.)

• Federal Aviation Administration - Describing Orbits

• Orbits
• Angle
• Equations of astronomy

• Articles with short description
• Short description is different from Wikidata