Thelongitude of the ascending node, also known as theright ascension of the ascending node, is one of theorbital elements used to specify theorbit of an object in space. Denoted with the symbolΩ, it is the angle from a specified reference direction, called theorigin of longitude, to the direction of theascending node (☊), as measured in a specifiedreference plane.[1] The ascending node is the point where the orbit of the object passes through the plane of reference, as seen in the adjacent image.
Commonly used reference planes and origins of longitude include:
Forgeocentric orbits (e.g., artificial satellites around earth),Earth'sequatorial plane as the reference plane, and theFirst Point of Aries (FPA) as the origin of longitude. In this case, the longitude is also called theright ascension of the ascending node (RAAN). The angle is measured eastwards (or, as seen from thenorth,counterclockwise) from the FPA to the node.[2][3] An alternative orbital element to the RAAN is thelocal time of the ascending node (LTAN), defined as thelocal mean time at which the spacecraft crosses the equator traveling northward. Similar definitions exist for satellites around other planets (seeplanetary coordinate systems).
Forheliocentric orbits, theecliptic as the reference plane, and the FPA as the origin of longitude. The angle is measured counterclockwise (as seen from north of the ecliptic) from theFirst Point of Aries to the node.[2]
For orbits outside theSolar System, the plane tangent to thecelestial sphere at the point of interest (called theplane of the sky) as the reference plane, and north (i.e. theperpendicular projection of the direction from the observer to thenorth celestial pole onto the plane of the sky) as the origin of longitude. The angle is measured eastwards (or, as seen by the observer, counterclockwise) from north to the node.[4], pp. 40, 72, 137; [5], chap. 17.
In the case of abinary star known only from visual observations, it is not possible to tell which node is ascending and which is descending. In this case the orbital parameter which is recorded is simply labeledlongitude of the node, ☊, and represents the longitude of whichever node has a longitude between 0 and 180 degrees.[5], chap. 17;[4], p. 72.
Here,n = ⟨nx,ny,nz⟩ is a vector pointing towards theascending node. The reference plane is assumed to be thexy-plane, and the origin of longitude is taken to be the positivex-axis.k is the unit vector (0, 0, 1), which is the normal vector to thexy reference plane.
Fornon-inclined orbits (withinclination equal to zero), ☊ is undefined. For computation it is then, by convention, set equal to zero; that is, the ascending node is placed in the reference direction, which is equivalent to lettingn point towards the positivex-axis.
