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 specified reference direction, called the * origin of longitude *, to the direction of the ascending node, as measured in a specified 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 geocentric orbits, 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**(RAAN). The angle is measured eastwards (or, as seen from the north, counterclockwise) from the First Point of Aries to the node.^{ [2] }^{ [3] }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 systems). - For heliocentric orbits, 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 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 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 simply labeled **longitude of the node**, Ω, and represents the longitude of whichever node has a longitude between 0 and 180 degrees.^{ [5] }^{, chap. 17;}^{ [4] }^{, p. 72.}

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.

- Equinox
- Kepler orbits
- List of orbits
- Orbital node
- Perturbation of the orbital plane can cause precession of the ascending node.

The **ecliptic** or **ecliptic plane** is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic against the background of stars. The ecliptic is an important reference plane and is the basis of the ecliptic coordinate system.

In mathematics, the **polar coordinate system** is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the *pole*, and the ray from the pole in the reference direction is the *polar axis*. The distance from the pole is called the *radial coordinate*, *radial distance* or simply *radius*, and the angle is called the *angular coordinate*, *polar angle*, or *azimuth*. Angles in polar notation are generally expressed in either degrees or radians.

In mathematics, a **spherical coordinate system** is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the *radial distance* of that point from a fixed origin, its *polar angle* measured from a fixed zenith direction, and the *azimuthal angle* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

An **azimuth** is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.

The **ecliptic coordinate system** is a celestial coordinate system commonly used for representing the apparent positions, orbits, and pole orientations of Solar System objects. Because most planets and many small Solar System bodies have orbits with only slight inclinations to the ecliptic, using it as the fundamental plane is convenient. The system's origin can be the center of either the Sun or Earth, its primary direction is towards the vernal (March) equinox, and it has a right-hand convention. It may be implemented in spherical or rectangular coordinates.

**Kinematics** is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics.

In physics, **angular velocity** or **rotational velocity**, also known as **angular frequency vector**, is a pseudovector representation of how fast the angular position or orientation of an object changes with time. The magnitude of the pseudovector represents the *angular speed*, the rate at which the object rotates or revolves, and its direction is normal to the instantaneous plane of rotation or angular displacement. The orientation of angular velocity is conventionally specified by the right-hand rule.

In physics, **angular acceleration** refers to the time rate of change of angular velocity. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular acceleration, called spin angular acceleration and orbital angular acceleration respectively. Spin angular acceleration refers to the angular acceleration of a rigid body about its centre of rotation, and orbital angular acceleration refers to the angular acceleration of a point particle about a fixed origin.

**Orbital elements** are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. 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.

**Rotation** in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have sign : a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (*n* − 1)-dimensional flat of fixed points in a n-dimensional space.

In celestial mechanics, **true anomaly** is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse.

The **argument of periapsis**, 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.

In celestial mechanics, the **longitude of the periapsis**, also called **longitude of the pericenter**, of an orbiting body is the longitude at which the periapsis would occur if the body's orbit inclination were zero. It is usually denoted *ϖ*.

**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.

In celestial mechanics, the **plane of reference** 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.

**Spacecraft flight dynamics** is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.

**Rotation around a fixed axis** is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will appear.

**Orbit determination** is the estimation of orbits of objects such as moons, planets, and spacecraft. One major application is to allow tracking newly observed asteroids and verify that they have not been previously discovered. The basic methods were discovered in the 17th century and have been continuously refined.

In mathematics, the **Euclidean plane** is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point, which includes affine notions of parallel lines, and also metrical notions of distance, circles, and angle measurement.

In celestial mechanics, a **Kepler orbit** is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

- ↑ Parameters Describing Elliptical Orbits, web page, accessed May 17, 2007.
- 1 2 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.
- ↑ Keplerian Elements Tutorial, amsat.org, accessed May 17, 2007.
- 1 2
*The Binary Stars*, R. G. Aitken, New York: Semi-Centennial Publications of the University of California, 1918. - 1 2
*Celestial Mechanics*, Jeremy B. Tatum, on line, accessed May 17, 2007.

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