Longitude of the ascending node: Difference between revisions

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{{Short description|Defining the orbit of an object in space}}

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]].

[[Image:Orbits-OrbitalParameters-001Orbit1.PNGsvg|thumbnailthumb|leftright|700px400px|'''LongitudeThe longitude of the [[ascending node''']] and(bright othergreen) as a part of a diagram of [[Orbital elements|orbital parameters]].]]<br style="clear:both;" />

The '''longitude of the ascending node''', also known as the '''right ascension of the ascending node''', is one of the [[orbital elements]] used to specify the [[orbit]] of an object in space. Denoted with the symbol '''Ω''', it is the angle from a specified reference direction, called the ''[[origin of longitude]]'', to the direction of the [[ascending node]] (☊), as measured in a specified [[reference plane]].<ref>[http://www.lns.cornell.edu/~seb/celestia/orbital-parameters.html Parameters Describing Elliptical Orbits], web page, accessed May 17, 2007.</ref> The ascending node is the point where the orbit of the object passes through the plane of reference, as seen in the adjacent image.

==Types==

Commonly used reference planes and origins of longitude include:

* For [[geocentric orbit]]s, [[Earth]]'s [[equator]]ial plane as the reference plane, and the [[First Point of Aries]] (FPA) as the origin of longitude. In this case, the longitude is also called the '''[[right ascension]] of the ascending node''' ('''RAAN'''). The angle is measured eastwards (or, as seen from the [[north]], [[counterclockwise]]) from the FPA to the node.<ref name="egler" /><ref>[http://www.amsat.org/amsat/keps/kepmodel.html Keplerian Elements Tutorial] {{Webarchive|url=https://web.archive.org/web/20021014232553/http://www.amsat.org/amsat/keps/kepmodel.html |date=2002-10-14 }}, amsat.org, accessed May 17, 2007.</ref> An alternative is the '''local time of the ascending node''' ('''LTAN'''), based on the [[local mean time]] at which the spacecraft crosses the equator. Similar definitions exist for satellites around other planets (see [[planetary coordinate system]]s).

* For [[heliocentric orbit]]s, the [[ecliptic]] 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 the [[First Point of Aries]] to the node.<ref name="egler">[http://www.physics.ncsu.edu/courses/astron/orbits.html Orbital Elements and Astronomical Terms] {{webarchive|url=https://web.archive.org/web/20070403095234/http://www.physics.ncsu.edu/courses/astron/orbits.html |date=2007-04-03 }}, Robert A. Egler, Dept. of Physics, [[North Carolina State University]]. Web page, accessed May 17, 2007.</ref>

* For orbits outside the [[Solar System]], the plane tangent to the [[celestial sphere]] at the point of interest (called the ''[[plane of the sky]]'') as the reference plane, and north (i.e. the [[orthographic projection|perpendicular projection]] 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.<ref name="aitken">''The Binary Stars'', R. G. Aitken, New York: Semi-Centennial Publications of the University of California, 1918.</ref>^{, pp.&nbsp;40, 72, 137;&nbsp;}<ref name="tatum">[http://astrowww.phys.uvic.ca/~tatum/celmechs.html ''Celestial Mechanics''], [[Jeremy B. Tatum]], on line, accessed May 17, 2007.</ref>^{, chap. 17.}

In the case of a [[binary 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 labeled '''longitude of the node''', ☊, and represents the longitude of whichever node has a longitude between 0 and 180 degrees.<ref name="tatum" />^{, chap. 17;}<ref name="aitken" />^{, p.&nbsp;72.}

==Calculation from state vectors==

In [[astrodynamics]], forthe [[elliptic orbits]] '''longitude of the ascending node''' <math>can \Omegabe \,</math>calculated isfrom the angle between reference direction (e.g. [[vernalspecific equinox]])relative andangular the [[ascending nodemomentum]] andvector can'''h''' beas calculated from [[orbital state vectors]] asfollows:

:<math>\begin{align}

:<math> \Omega = arccos { {n_x} \over { \mathbf{\left |n \right |}}}</math>

\mathbf{n} &= \mathbf{k} \times \mathbf{h} = (-h_y, h_x, 0) \\

\Omega &= \begin{cases}

\arccos { {n_x} \over { \mathbf{\left |n \right |}}}, &n_y \ge 0; \\

:<math> \Omega = 2\pi-\arccos { {n_x} \over { \mathbf{\left |n \right |}}}, &n_y </math> 0.

\end{cases}

\end{align}</math>

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. '''k''' is the unit vector (0, 0, 1), which is the normal vector to the ''xy'' reference plane.

:(if <math>n_y < 0 \,</math> then <math>\Omega = 2 \pi - \Omega \,</math>)

For equatorial[[non-inclined orbitsorbit]]s (i.e. orbits with [[orbital inclination|inclination]] equal to zero) <math> \Omega\, </math>☊ is undefined. For computationscomputation it is then, by convention, set equal to zero; i.e.that is, the "ascending node" is placed in the reference direction, which is equivalent to settingletting <math> \mathbf{'''n}''' / \mathbf{\left |n \right |} = (1,0,0) </math> for right-handed system with the x-axis pointingpoint towards the vernalpositive equinox (or other reference direction) and the z''x''-axis pointing upwards.

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).

== See also ==

*[[Equinox]]

*[[Kepler orbit|Kepler orbits]]

*[[List of orbits]]

*[[Orbital node]]

*[[Orbital perturbation analysis (spacecraft)#Perturbation of the orbital plane 2|Perturbation of the orbital plane]] can cause [[nodal precession|precession]] of the ascending node.

==References==

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.

{{Reflist}}

{{orbits}}

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