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Milds # 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. 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:

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., chap. 17;, p. 72.

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• ✪ # Keplerian Elements : Orbital Inclination and the Right Ascension of Ascending Node (RAAN)
• ✪ White Dwarf Accretion Model
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#### Transcription

Up next, the orbital inclination. So let’s assume this is our earth, and this is our ECI frame. This is the x axis, this is the y axis and this is the z axis. This will be the equator and this is the equatorial plane. Now imagine the satellite going somewhat like this in the orbit. This is our satellite, it will be going down like this. It will go down and down and from here on it will start ascending and then it will come back to this place and then it will descend again. Now this angle, the angle between the satellites orbital plane and the equatorial plane is the angle of inclination or simply inclination ‘I’. While moving in the orbit, the satellite pierces the equatorial plane twice, at a point when it is going up and at a point when it is going down. The point where the satellite intersects the equatorial plane and goes up is called the ascending node and the point where the satellite intersects the equatorial plane and goes down is called the descending node. Now if we join these two points , we have the nodal line for the satellite. This is called the nodal line for the satellite. Now, the angle between the nodal line and the x axis of the ECI frame is called the right ascention of ascending node or the RAAN. We need to consider the angle from the ascending side of the nodal line so this angle is our RAAN.

## 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:

{\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.