# Longitude of the ascending node

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] 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:

• 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 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 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.

The longitude of the ascending node.

## 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:

{\displaystyle {\begin{aligned}\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}\geq 0;\\2\pi -\arccos {{n_{x}} \over {\mathbf {\left|n\right|} }},&n_{y}<0.\end{cases}}\end{aligned}}}

Here, n=<nx, ny, nz> 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

1. ^ Parameters Describing Elliptical Orbits, web page, accessed May 17, 2007.
2. ^ a b Orbital Elements and Astronomical Terms Archived 2007-04-03 at the Wayback Machine, Robert A. Egler, Dept. of Physics, North Carolina State University. Web page, accessed May 17, 2007.
3. ^ Keplerian Elements Tutorial, amsat.org, accessed May 17, 2007.
4. ^ a b The Binary Stars, R. G. Aitken, New York: Semi-Centennial Publications of the University of California, 1918.
5. ^ a b Celestial Mechanics, Jeremy B. Tatum, on line, accessed May 17, 2007.
Argument of periapsis

The argument of periapsis (also called argument of perifocus or argument of pericenter), symbolized as ω, is one of the orbital elements of an orbiting body. Parametrically, ω is the angle from the body's ascending node to its periapsis, measured in the direction of motion.

For specific types of orbits, words such as perihelion (for heliocentric orbits), perigee (for geocentric orbits), periastron (for orbits around stars), and so on may replace the word periapsis. (See apsis for more information.)

An argument of periapsis of 0° means that the orbiting body will be at its closest approach to the central body at the same moment that it crosses the plane of reference from South to North. An argument of periapsis of 90° means that the orbiting body will reach periapsis at its northmost distance from the plane of reference.

Adding the argument of periapsis to the longitude of the ascending node gives the longitude of the periapsis. However, especially in discussions of binary stars and exoplanets, the terms "longitude of periapsis" or "longitude of periastron" are often used synonymously with "argument of periapsis".

C/2014 Q1 (PANSTARRS)

Comet PANSTARRS is a non-periodic/long period comet discovered in August 2014 by The Panoramic Survey Telescope and Rapid Response System (Pan-STARRS)

Orbital Properties Info

Comet PANSTARRS (C/2014 Q1)

Discovery Date: August 2014

Last Perihelion: July 6, 2015 & 12:15:01 UTC

Next Perihelion: Unknown

Orbital Period: 40,343.2700 years

Semi Major Axis: 175,968,123,726 km

Eccentricity: 0.99973254

Inclination: 43.10640632

Argument of Perihelion: 120.046973107°

Longitude of the ascending node: 8.76268729392°

Earth's orbit

Earth orbits the Sun at an average distance of 149.60 million km (92.96 million mi), and one complete orbit takes 365.256 days (1 sidereal year), during which time Earth has traveled 940 million km (584 million mi). Earth's orbit has an eccentricity of 0.0167. Since the Sun constitutes 99.76% of the mass of the Sun–Earth system, the center of the orbit is extremely close to the center of the Sun.

As seen from Earth, the planet's orbital prograde motion makes the Sun appear to move with respect to other stars at a rate of about 1° (or a Sun or Moon diameter every 12 hours) eastward per solar day. Earth's orbital speed averages about 30 km/s (109044 km/h; 67756 mph), which is fast enough to cover the planet's diameter in 7 minutes and the distance to the Moon in 4 hours.From a vantage point above the north pole of either the Sun or Earth, Earth would appear to revolve in a counterclockwise direction around the Sun. From the same vantage point, both the Earth and the Sun would appear to rotate also in a counterclockwise direction about their respective axes.

Ground track

A ground track or ground trace is the path on the surface of a planet directly below an aircraft or satellite. In the case of a satellite, it is the projection of the satellite's orbit onto the surface of the Earth (or whatever body the satellite is orbiting).

A satellite ground track may be thought of as a path along the Earth's surface which traces the movement of an imaginary line between the satellite and the center of the Earth. In other words, the ground track is the set of points at which the satellite will pass directly overhead, or cross the zenith, in the frame of reference of a ground observer.

High Earth orbit

A high Earth orbit is a geocentric orbit with an altitude entirely above that of a geosynchronous orbit (35,786 kilometres (22,236 mi)). The orbital periods of such orbits are greater than 24 hours, therefore satellites in such orbits have an apparent retrograde motion – that is, even if they are in a prograde orbit (90° > inclination ≥ 0°), their orbital velocity is lower than Earth's rotational speed, causing their ground track to move westward on Earth's surface.

Inclined orbit

A satellite is said to occupy an inclined orbit around Earth if the orbit exhibits an angle other than 0° to the equatorial plane. This angle is called the orbit's inclination. A planet is said to have an inclined orbit around the Sun if it has an angle other than 0° to the ecliptic plane.

Lense–Thirring precession

In general relativity, Lense–Thirring precession or the Lense–Thirring effect (named after Josef Lense and Hans Thirring) is a relativistic correction to the precession of a gyroscope near a large rotating mass such as the Earth. It is a gravitomagnetic frame-dragging effect. It is a prediction of general relativity consisting of secular precessions of the longitude of the ascending node and the argument of pericenter of a test particle freely orbiting a central spinning mass endowed with angular momentum ${\displaystyle S}$.

The difference between de Sitter precession and the Lense–Thirring effect is that the de Sitter effect is due simply to the presence of a central mass, whereas the Lense–Thirring effect is due to the rotation of the central mass. The total precession is calculated by combining the de Sitter precession with the Lense–Thirring precession.

According to a recent historical analysis by Pfister, the effect should be renamed as Einstein–Thirring–Lense effect.

Longitude of the periapsis

In celestial mechanics, the longitude of the periapsis, also called longitude of the pericenter, of an orbiting body is the longitude (measured from the point of the vernal equinox) at which the periapsis (closest approach to the central body) would occur if the body's orbit inclination were zero. It is usually denoted ϖ.

For the motion of a planet around the Sun, this position is called longitude of perihelion ϖ, which is the sum of the longitude of the ascending node Ω, and the argument of perihelion ω.The longitude of periapsis is a compound angle, with part of it being measured in the plane of reference and the rest being measured in the plane of the orbit. Likewise, any angle derived from the longitude of periapsis (e.g., mean longitude and true longitude) will also be compound.

Sometimes, the term longitude of periapsis is used to refer to ω, the angle between the ascending node and the periapsis. That usage of the term is especially common in discussions of binary stars and exoplanets. However, the angle ω is less ambiguously known as the argument of periapsis.

Mean longitude

Mean longitude is the ecliptic longitude at which an orbiting body could be found if its orbit were circular and free of perturbations. While nominally a simple longitude, in practice the mean longitude does not correspond to any one physical angle.

Nodal precession

Nodal precession is the precession of the orbital plane of a satellite around the rotational axis of an astronomical body such as Earth. This precession is due to the non-spherical nature of a rotating body, which creates a non-uniform gravitational field. The following discussion relates to low Earth orbit of artificial satellites, which have no measurable effect on the motion of Earth. The nodal precession of more massive, natural satellites like the Moon is more complex.

Around a spherical body, an orbital plane would remain fixed in space around the gravitational primary body. However, most bodies rotate, which causes an equatorial bulge. This bulge creates a gravitational effect that causes orbits to precess around the rotational axis of the primary body.

The direction of precession is opposite the direction of revolution. For a typical prograde orbit around Earth (that is, in the direction of primary body's rotation), the longitude of the ascending node decreases, that is the node precesses westward. If the orbit is retrograde, this increases the longitude of the ascending node, that is the node precesses eastward. This nodal progression enables heliosynchronous orbits to maintain a nearly constant angle relative to the Sun.

Node

In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex).

Non-inclined orbit

A non-inclined orbit is an orbit coplanar with a plane of reference. The orbital inclination is 0° for prograde orbits, and π (180°) for retrograde ones. If the plane of reference is a massive spheroid body's equatorial plane, these orbits are called equatorial; if the plane of reference is the ecliptic plane, they are called ecliptic.

As these orbits lack nodes, the ascending node is usually taken to lie in the reference direction (usually the vernal equinox), and thus the longitude of the ascending node is taken to be zero. Also, the argument of periapsis is undefined.

Geostationary orbit is a geosynchronous example of an equatorial orbit.

Orbital elements

Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are generally considered in classical two-body systems, where a Kepler orbit is used. There are many different ways to mathematically describe the same orbit, but certain schemes, each consisting of a set of six parameters, are commonly used in astronomy and orbital mechanics.

A real orbit (and its elements) changes over time due to gravitational perturbations by other objects and the effects of relativity. A Keplerian orbit is merely an idealized, mathematical approximation at a particular time.

Orbital inclination change

Orbital inclination change is an orbital maneuver aimed at changing the inclination of an orbiting body's orbit. This maneuver is also known as an orbital plane change as the plane of the orbit is tipped. This maneuver requires a change in the orbital velocity vector (delta v) at the orbital nodes (i.e. the point where the initial and desired orbits intersect, the line of orbital nodes is defined by the intersection of the two orbital planes).

In general, inclination changes can take a very large amount of delta v to perform, and most mission planners try to avoid them whenever possible to conserve fuel. This is typically achieved by launching a spacecraft directly into the desired inclination, or as close to it as possible so as to minimize any inclination change required over the duration of the spacecraft life. Planetary flybys are the most efficient way to achieve large inclination changes, but they are only effective for interplanetary missions.

Orbital node

An orbital node is either of the two points where an orbit intersects a plane of reference to which it is inclined. A non-inclined orbit, which is contained in the reference plane, has no nodes.

Orbital plane (astronomy)

The orbital plane of a revolving body is the geometric plane on which its orbit lies. A common example would be the centers of a massive body, of an orbiting body, and of the orbiting object at another time.

The orbital plane is defined in relation to a reference plane by two parameters: inclination (i) and longitude of the ascending node (Ω). Three non-collinear points in space suffice to determine the orbital plane.

By definition, the reference plane for the Solar System is usually considered to be Earth's orbital plane. This defines the ecliptic, the circular path on the celestial sphere that the Sun appears to follow over the course of a year.

In other cases, for instance a moon or artificial satellite orbiting another planet, it is convenient to define the inclination of the Moon's orbit as the angle between its orbital plane and the planet's equatorial plane.

Plane of reference

In celestial mechanics, the plane of reference (or reference plane) is the plane used to define orbital elements (positions). The two main orbital elements that are measured with respect to the plane of reference are the inclination and the longitude of the ascending node.

Depending on the type of body being described, there are four different kinds of reference planes that are typically used:

The ecliptic or invariable plane for planets, asteroids, comets, etc. within the Solar System, as these bodies generally have orbits that lie close to the ecliptic.

The equatorial plane of the orbited body for satellites with small semi-major axes

The local Laplace plane for satellites with intermediate-to-large semi-major axes

The plane tangent to celestial sphere for extrasolar objectsOn the plane of reference, a zero-point must be defined from which the angles of longitude are measured. This is usually defined as the point on the celestial sphere where the plane crosses the prime hour circle (the hour circle occupied by the First Point of Aries), also known as the equinox.

Semi-synchronous orbit

A semi-synchronous orbit is an orbit with a period equal to half the average rotational period of the body being orbited, and in the same direction as that body's rotation.

For Earth, a semi-synchronous orbit is considered a medium Earth orbit, with a period of just under 12 hours. For circular Earth orbits, the altitude is approximately 20,200 kilometres (12,600 mi).Semi-synchronous orbits are typical for GPS satellites.

Subsynchronous orbit

A subsynchronous orbit is an orbit of a satellite that is nearer the planet than it would be if it were in synchronous orbit, i.e. the orbital period is less than the sidereal day of the planet.

Gravitational orbits
Types
Parameters
Maneuvers
Orbital mechanics

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