Longitude of the ascending node

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Longitude of the ascending node

The longitude of the ascending node (? or O) is one of the orbital elements

Orbital elements

In celestial mechanics, the elements of an orbit are the parameters needed to specify that orbit uniquely. Orbital elements are generally considered in classical mechanics two-body systems, where a Kepler orbit is used ....
used to specify the orbit

ORBit

ORBit is a Common Object Request Broker Architecture 2.4 compliant Object Request Broker . It features mature C , C++ and Python bindings, and less developed bindings for Perl, Lisp , Pascal , Ruby , and Tcl....
of an object in space. It is the angle from a reference direction, called the origin of longitude, to the direction of the ascending node, measured in a reference plane. Commonly used reference planes and origins of longitude include:

In the case of a binary star

Binary star

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The longitude of the ascending node (? or O) is one of the orbital elements

Orbital elements

ORBit

For a geocentric orbit
Geocentric orbit

A geocentric orbit is an orbit of any object orbital revolution the Earth, such as the Moon or artificial satellites. Currently there are approximately 2465 artificial satellites orbiting the Earth and 6216 pieces of space debris as tracked by the Goddard Space Flight Center....
, Earth
Earth

Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
's equator
Equator

The equator is the intersection of the Earth's surface with the Plane perpendicular to the Earth's rotation and containing the Earth's center of mass....
ial plane as the reference plane, and the First Point of Aries as the origin of longitude. In this case, the longitude is also called the right ascension of the ascending node, or RAAN. The angle is measured eastwards (or, as seen from the north
North

North is one of the four cardinal directions, specifically the direction that, in Western culture, is treated as the fundamental direction:...
, counterclockwise) from the First Point of Aries to the node.
For a heliocentric orbit
Heliocentric orbit

A heliocentric orbit is an orbit around the Sun. In our Solar System, all planets, comets, and asteroids are in such orbits, as are many artificial Space probe and pieces of Space debris....
, the ecliptic
Ecliptic

The ecliptic is the apparent path that the Sun traces out in the sky during the year. As it appears to move in the sky in relation to the stars, the apparent path aligns with the planets throughout the course of the year....
as the reference plane, and the First Point of Aries as the origin of longitude. The angle is measured counterclockwise (as seen from north of the ecliptic) from the First Point of Aries to the node.
For an orbit outside the Solar System
Solar System

The Solar System consists of the Sun and those Astronomical object bound to it by gravity: the eight planets and five dwarf planets, their 173 known Natural satellite, and billions of Small Solar System body....
, the plane through the primary perpendicular to a line through the observer and the primary (called the plane of the sky) as the reference plane, and north, i.e., the perpendicular projection
Orthographic projection

Orthographic projection is a means of representing a Three-dimensional space object in 2D.It is a form of parallel projection, where the view direction is orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface....
of the direction from the observer to the North 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.^{, pp. 40, 72, 137;}^{, chap. 17.}

In the case of a binary star

Binary star

A binary star is a star system consisting of two stars orbiting around their common center of mass. The brighter star is called the primary and the other is its companion star or secondary....
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 the longitude of the node, Ω, which is the longitude of whichever node has a longitude between 0 and 180 degrees.^{, chap. 17;}^{, p. 72.}

Calculation from state vectors

In astrodynamics

Astrodynamics

Orbital mechanics or astrodynamics is the application of celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft....
, the longitude of the ascending node can be calculated from orbital state vectors

Orbital state vectors

In astrodynamics or celestial dynamics orbital state vectors are vectors of position and velocity that together with their time uniquely determine the state of an orbiting body....
as follows:

Here, n=(n_x, n_y, n_z) is a vector pointing towards the ascending node. The reference plane is assumed to be the xy-plane, and the origin of longitude is taken to be the positive x-axis.

For non-inclined orbits (with inclination equal to zero), Ω 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 letting n point towards the positive x-axis.

Longitude of the ascending node

Calculation from state vectors

See also