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The Longitude of the ascending node (<math> \Omega \,<math>) 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.
Calculation from state vectors
In astrodynamics for elliptic orbits longitude of the ascending node <math> \Omega \,<math> is the angle between reference direction (e.g. vernal equinox) and the ascending node and can be calculated from orbital state vectors as:
- <math> \Omega = arccos { {n_x} \over { \mathbf{\left |n \right |}}}<math>
- (if <math>n_y < 0 \,<math> then <math>\Omega = 2 \pi - \Omega \,<math>)
where:
- <math> n_x \,<math> is the x-component of <math> \mathbf{n} <math>,
- <math> \mathbf{n} <math> is cartesian vector pointing towards the ascending node (i.e. the z-component of <math> \mathbf{n} <math> is zero).
For equatorial orbits (i.e. orbits with orbital inclination equal to zero) <math> \Omega\, <math> 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 <math> \mathbf{n} / \mathbf{\left |n \right |} = (1,0,0) <math> for 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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