True anomaly

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The true anomaly is the angle z-s-p

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 $ν$ or $θ$ , 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.

[edit] Formulas

[edit] 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:

$\mathbf{v}\,$ is orbital velocity vector of the orbiting body,
$\mathbf{e}\,$ is eccentricity vector,
$\mathbf{r}\,$ is orbital position vector (segment sp) of the orbiting body.

[edit] Circular orbit

For circular orbits this can be simplified to:

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

where:

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

[edit] Circular orbit with zero inclination

For circular orbits with the inclination of zero this can be simplified further to:

$\nu = \arccos { r_x \over { \mathbf{\left |r \right |}}}$ (if $v_x > 0\$ then replace $\nu\$ by $2\pi-\nu\$ )

where:

$r_x \,$ is x-component of orbital position vector $\mathbf{r}$ ,
$v_x \,$ is x-component of orbital velocity vector $\mathbf{v}$ .

[edit] From the eccentric anomaly

The relation between the true anomaly ν 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)).

[edit] 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

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

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

[edit] See also

[edit] References

Murray, C. D. & Dermott, S. F. 1999, Solar System Dynamics, Cambridge University Press, Cambridge.
Plummer, H.C., 1960, An Introductory treatise on Dynamical Astronomy, Dover Publications, New York. (Reprint of the 1918 Cambridge University Press edition.)