True anomaly

In celestial mechanics, the 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 Roman 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.

Contents

Formulas

From state vectors

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

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:

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

where:

From the eccentric anomaly

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%2Be} \over {1-e}}} \tan{E \over 2}.

Therefore

\nu = 2 \, \mathop{\mathrm{arg}}\left(\sqrt{1-e} \, \cos\frac{E}{2} , \sqrt{1%2Be}\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), beware that the arguments are usually in reversed order).

Radius from true anomaly

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 %2B e \cdot \cos\nu}\,\!

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

See also

References