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Longitude of the ascending node

The Longitude of the ascending node (☊, also noted Ω) is one of the orbital elements used to specify the orbit of an object in space. For a Sun-orbiting body, it is the angle formed at the Sun from the First Point of Aries to the body's ascending node, measured in the reference plane (the ecliptic) and in the direct sense.

Calculation from state vectors

In astrodynamics, for elliptic orbits the longitude of the ascending node $\Omega \,$ is the angle between the reference direction (e.g. the vernal equinox) and the ascending node. It can be calculated from orbital state vectors as:

$\Omega = arccos { {n_x} \over { \mathbf{\left |n \right |}}}$

(if $n_y < 0 \,$ then $\Omega = 2 \pi - \Omega \,$ )

where:

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

For equatorial orbits (i.e. orbits with orbital inclination equal to zero) $\Omega\,$ is undefined. For computations it is then by convention set to zero i.e. "ascending node" is placed in the reference direction which is equivalent to setting $\mathbf{n} / \mathbf{\left |n \right |} = (1,0,0)$ for a right-handed system with the x-axis pointing towards the vernal equinox (or other reference direction) and the z-axis pointing upwards.

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