Coordinate and Time Systems
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Coordinate and Time Systems
Satellites orbit around the Earth or travel in the planet system of the sun. They are generally observed from the Earth. To describe the orbits of the satellites (positions and velocities), suitable coordinate and time systems have to be defined.
2.1 Geocentric Earth-Fixed Coordinate Systems It is convenient to use the Earth-Centred Earth-Fixed (ECEF) coordinate system to describe the location of a station on the Earth’s surface. The ECEF coordinate system is a right-handed Cartesian system (x, y, z). Its origin and the Earth’s centre of mass coincide, while its z-axis and the mean rotational axis of the Earth coincide; the x-axis points to the mean Greenwich meridian, while the y-axis is directed to complete a right-handed system (Fig. 2.1). In other words, the z-axis points to a mean pole of the Earth’s rotation. Such a mean pole, defined by international convention, is called the Conventional International Origin (CIO). The xy-plane is called the mean equatorial plane, and the xz-plane is called the mean zero-meridian.
Fig. 2.1 Earth-Centred Earth-Fixed coordinates
G. Xu, Orbits, c Springer-Verlag Berlin Heidelberg 2008
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2 Coordinate and Time Systems
The ECEF coordinate system is also known as the Conventional Terrestrial System (CTS). The mean rotational axis and mean zero-meridian used here are necessary. The true rotational axis of the Earth changes its direction all the time with respect to the Earth’s body. If such a pole is used to define a coordinate system, then the coordinates of the station would also change all the time. Because the survey is made in our true world, it is obvious that the polar motion has to be taken into account and will be discussed later. The ECEF coordinate system can, of course, be represented by a spherical coordinate system (r, φ , λ ), where r is the radius of the point (x, y, z), and φ and λ are the geocentric latitude and longitude, respectively (Fig. 2.2). λ is counted eastward from the zero-meridian. The relationship between (x, y, z) and (r, φ , λ ) is obvious: ⎧
⎛ ⎞ ⎛ ⎞ ⎪ r = x2 + y2 + z2 , x r cos φ cos λ ⎨ ⎝ y ⎠ = ⎝ r cos φ sin λ ⎠ or tan λ = y/x, (2.1)
⎪ ⎩ z r sin φ 2 2 tan φ = z/ x + y . An ellipsoidal coordinate system (ϕ , λ , h) may also be defined on the basis of the ECEF coordinates; however, geometrically, two additional parameters are needed to define the shape of the ellipsoid (Fig. 2.3). ϕ , λ and h are geodetic latitude, longitude and height, respectively. The ellipsoidal surface is a rotational ellipse. The ellipsoidal system is also called the geodetic coordinate system. Geocentric longitude and geodetic longitude are identical. The two geometric parameters could be the semi-major radius (denoted by a) and the semi-minor radius (denoted by b) of the rotating ellipse, or the semi-major radius and the flattening (denoted by f ) of the ellipsoid. They are equivalent sets of parameters. The relationship between (x,y,z) and (ϕ , λ , h) is (see, e.g., Torge, 1991): ⎛ ⎞ ⎛ ⎞ (N + h) cos ϕ cos λ x ⎝ y ⎠ = ⎝ (N + h) cos
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