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**Coordinates System**

The __inertial reference frame__ is defined by the position of the vernal equinox ‘X’ at a certain epoch. The ‘Z’ axis is defined by the spin axis of the Earth (North Pole), and the ‘Y’ axis completes the orthogonal set of the right handed inertial reference frame.

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**Keplerian Elements**

The next figure illustrates the geometric properties of the usual set of orbital elements used to describe the motion of a satellite in Earth orbit, well characterized by the Keplerian elements of an elliptical orbit.

* Click to enlarge*

a**: Semi-major axis of orbital ellipse** is the semi-major axis of the ellipse defining the orbit.

e**: Numerical eccentricity of the orbit** is the eccentricity of the orbital ellipse. Eccentricity is a measure of how an orbit deviates from circular. A perfectly circular orbit has an eccentricity of zero; higher numbers indicate more elliptical orbits.

i**: Inclination of orbital plane** is the angle between the orbital plane and the equator.

Ω**: Right ascension of Ascending Node (RAAN)** defines the relative angular phasing between the orbital plane and the **Vernal Equinox**, which is the point of intersection between the Sun’s trajectory and the Earth’s equatorial plane. Due to the Oblateness of the Earth, the RAAN is decreasing about 10 degrees per year.

**NOTE:** *the intersection of equatorial plane and orbital plane is called “Nodal Line”. Its intersection with the unit sphere defines two points: the “Ascending Node”, through which the satellite crosses to the region of the positive Z-axis, and the “Descending Node”. “Right Ascension” is counter-clockwise sense viewed from the positive Z-axis.*

ω**: Argument of perigee** is the angle between the ascending node and perigee directions, measured along the orbital plane. The perigee is the point of closest approach of the satellite to the centre of mass of the earth. The most distant position is the Apogee. Both are in the orbital ellipse semi-major axis direction.

v**: True anomaly** is the geocentric angle between perigee direction and satellite direction. The sum of the True Anomaly and the Argument of Perigee defines the “Argument of Latitude”. Notice that for a circular orbit (e = 0) the Argument of Perigee and the True Anomaly are undefined. The satellite position, however, can be specified by the Argument of Latitude.

u**: Argument of latitude** is the sum of argument of perigee and true anomaly. It is the angle measured from the equator to the satellite at a particular epoch. For the circular orbits the argument of perigee is not well defined, therefore it is more convenient to use the argument of latitude instead. In the table, the argument of perigee has been set to zero, therefore the argument of latitude and the true anomaly are identical.

The **satellite height** is characterized by the orbit’s semimajor axis **‘***a***’**, the variation in the radial distance due to the ellipticity of the orbit (the eccentricity **‘***e***’**), and the angular distance **‘***v***’** (the true anomaly) from the point of closest approach in the orbit (called the Perigee).

Slot**:** The Galileo reference constellation has a total of 30 Medium Earth Orbit (MEO) satellites, including 6 spares, in a so called Walker 24/3/1 constellation. This particular Walker configuration implies that the Galileo constellation consists of 24 satellites homogenously distributed in three different orbital planes (A, B and C) separated in the equatorial plane by 120 degrees.

*Click to enlarge*