True anomaly

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 $f$.

As shown in the image, the true anomaly $f$ is one of three angular parameters ("anomalies") that defines a position along an orbit, the other two being the eccentric anomaly and the mean anomaly. Note that the satellite P orbits around the planet which is at position F.

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:

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

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:

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

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 using sin and tan

$\sin{\nu} = {{\sqrt{1-e^2} \sin{E}} \over {1 - e \cos{E}}}$

$\tan{\nu} = {{\sin{\nu}} \over {\cos{\nu}}} = {{\sqrt{1-e^2} \sin{E}} \over {\cos{E} -e}}$

or equivalently

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

Therefore

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

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

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

• Kepler's laws of planetary motion
• Eccentric anomaly
• Mean anomaly
• Ellipse
• Hyperbola

References

• 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.)