True anomaly

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The true anomaly of point P is the angle f. The center of the ellipse is point C, and the focus is point F.

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 \,\nu or \,\theta, or the Latin letter  f .

The true anomaly is one of three angular parameters ("anomalies") that define a position along an orbit, the other two being the eccentric anomaly and the mean anomaly.

Formulas[edit]

From state vectors[edit]

For elliptic orbits true anomaly \nu\,\! can be calculated from orbital state vectors as:

 \nu = \arccos { {\mathbf{e} \cdot \mathbf{r}} \over { \mathbf{\left |e \right |} \mathbf{\left |r \right |} }}   (if \mathbf{r} \cdot \mathbf{v} < 0 then replace \nu\ by 2\pi-\nu\ )

where:

Circular orbit[edit]

For circular orbits the true anomaly is undefined because circular orbits do not have a uniquely determined periapsis. Instead one uses the argument of latitude u\,\!:

 u = \arccos { {\mathbf{n} \cdot \mathbf{r}} \over { \mathbf{\left |n \right |} \mathbf{\left |r \right |} }}   (if \mathbf{n} \cdot \mathbf{v} >0 then replace u\ by 2\pi-u\ )

where:

  •  \mathbf{n} is vector pointing towards the ascending node (i.e. the z-component of  \mathbf{n} is zero).

Circular orbit with zero inclination[edit]

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  \over { \mathbf{\left |r \right |}}}   (if  v_x > 0\ then replace l\ by 2\pi-l\ )

where:

From the eccentric anomaly[edit]

The relation between the true anomaly \,\nu and the eccentric anomaly E is:

\cos{\nu} = {{\cos{E} - e} \over {1 - e \cdot \cos{E}}}

or equivalently

\tan{\nu \over 2} = \sqrt{{{1+e} \over {1-e}}} \tan{E \over 2}.

Therefore

  \nu   = 2 \, \mathop{\mathrm{arg}}\left(\sqrt{1-e} \, \cos\frac{E}{2} , \sqrt{1+e}\sin\frac{E}{2}\right)

where \operatorname{arg}(x, y) is the polar argument of the vector \left(x, y\right) (available in many programming languages as the library function atan2(y, x) in Fortran and MATLAB, or as ArcTan(x, y) in Wolfram Mathematica).

Radius from true anomaly[edit]

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

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

where a is the orbit's semi-major axis (segment cz).

See also[edit]

References[edit]

  • 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.)