Z

C. Roll, Pitch, and Yaw (RPY) Coordinates

D. Celestial Coordinates

Fig. 5-1. Coordinate Systems In Common Use. See Table 5-1 for characteristics.

C. Roll, Pitch, and Yaw (RPY) Coordinates

D. Celestial Coordinates

Fig. 5-1. Coordinate Systems In Common Use. See Table 5-1 for characteristics.

of 0.014 deg/yr. Because of this slow drift, celestial coordinates require a corresponding date to accurately define the position of the vernal equinox. The most commonly used systems are 1950 coordinates, 2000 coordinates, and true of date, or TOD. The latter coordinates use the same epoch as the orbit parameters and are traditionally used for spacecraft orbit analysis. The small corrections required to maintain TOD coordinates are conveniently done by standard computer subroutines. They are important for precise numerical work, but are not critical for most problems in mission analysis.

Once we have defined a coordinate system, we can specify a direction in space by a unit vector, or vector of unit magnitude, in that direction. While a unit vector will have three components, only two will be independent because the magnitude of the vector must be one. We can also define a unit vector by defining the two coordinates of its position on the surface of a sphere of unit radius, called the celestial sphere, centered on the origin of the coordinate system. Clearly, every unit vector corresponds to one and only one point on the celestial sphere, and every point on the surface of the sphere corresponds to a unique unit -vector, as illustrated in Fig. 5-2. Because either representation is mathematically correct, we can shift back and forth between them as the problem demands. Unit vector analysis is typically the most convenient form for computer computations, while the celestial sphere approach provides the geometrical and physical insight so important to mission analysis. In Fig. 5-2 it is difficult to estimate the X, Y, and Z components of the unit vector on the left, whereas we can easily determine the two coordinates corresponding to a point on the celestial sphere from the figure on the right. Perhaps more important, the celestial sphere allows us easily to represent a large collection of points or the trace of a moving vector by simply drawing a line on the sphere. We will use the celestial sphere throughout most of this chapter, because it gives us more physical insight and more ability to convey precise information in an illustration.

Fig. 5-2. Alternative Representations of Unit Vectors. In (B) it is clear that the small circle is of 10 deg radius centered at (15°, 30°) and that the single vector is at (60°, 40°). In (A) even the quadrant Is difficult to estimate. Also note that the body of the unit vectors from the center of the sphere can be omitted since any point on the sphere implies a corresponding unit vector. Thus, the 3-dlmenslonal pointing geometry is reduced to a 2-dlmenslonal representation. See Fig. 5-5 for the definition of notation.

0 0

Post a comment