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File:Orbit1.svg
The longitude of the ascending node (☊ or Ω) is
one of the orbital elements used to specify the
orbit 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.^{[1]}
Commonly used reference planes and origins of longitude
include:
- For a geocentric orbit, Earth's equatorial 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, counterclockwise) from the First Point of
Aries to the node.^{[2]}^{[3]}
- For a heliocentric orbit, the ecliptic 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.^{[2]}
- For an orbit outside the Solar System, 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 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.^{[4]}^{
, pp. 40, 72, 137; }^{[5]}^{
, 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 the longitude of the node, Ω, which
is the longitude of whichever node has a longitude between 0 and
180 degrees.^{[5]}^{
, chap. 17;}^{[4]}^{
, p. 72.}
Calculation from state
vectors
In astrodynamics, the longitude of the
ascending node can be calculated from the specific relative
angular momentum vector h 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.
k is the unit vector (0, 0, 1), which is
the normal vector to the xy reference plane.
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.
References
- ^
Parameters Describing
Elliptical Orbits, web page, accessed May 17, 2007.
- ^ ^{a}
^{b}
Orbital Elements and
Astronomical Terms, Robert A. Egler, Dept. of Physics, North Carolina State
University. Web page, accessed May 17, 2007.
- ^
Keplerian Elements
Tutorial, amsat.org, accessed May 17, 2007.
- ^ ^{a}
^{b}
The Binary Stars, R. G. Aitken, New York: Semi-Centennial
Publications of the University of California, 1918.
- ^ ^{a}
^{b}
Celestial
Mechanics, Jeremy B. Tatum, on line, accessed May 17,
2007.
See also