3.2.2.2 Earth coordinate system

The Earth coordinate system is the pre­ferred coordinate system of the Earth’s frame triad e, ei, and «з (see Fig. 3.6). Its Iе axis pierces through the unit sphere at the intersection of the equator with the Greenwich meridian. The 3E axis overlays the Earth’s spin axis, and the 2E axis completes the right-handed coordinate system (see Fig. 3.14).

To relate the Earth coordinates to the inertial coordinates, we have to heed the Earth’s rotation. Every 24 h the Earth presents the same face to the sun. This is called the solar day. However, a full rotation of the Earth relative to the stars, the so-called sidereal day, is actually shorter by about 4 min. The lengthening of the solar day is caused by the progression of the Earth on the ecliptic during 24 h. To present the same face to the sun, the Earth has to rotate further. We are interested in the sidereal time that has elapsed since the Iе axis coincided with the 11 axis. Rather than using time, however, we use an angular measurement, equating 360° to one sidereal day. The angle between Iе and Iе is called the hour angle E and establishes the Greenwich meridian relative to the meridian of the vernal equinox.

The transformation matrix [TJE/ of the Earth coordinates wrt the inertial coor­dinates is obtained by inspection:

The rules, just given, should enable you to do the same. Figure 3.14 displays also the frugal orange peel schematic, which in its simplicity conveys all of the important information of the picture on the left.