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The perifocal system is convenient for describing the motion of the satellite in the orbital plane. If the earth were uniformly spherical, the perifocal coordinates would be fixed in space, that is, inertial. However, the equatorial bulge causes rotations of the perifocal coordinate system, as described in Sec. 2.8.1. These rotations are taken into account when the satellite position is transferred from perifocal coordinates to geocentric-equatorial coordinates, described in the next section.
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Example 2.15 Using the values r 7257.5 km and 204.81 obtained in the previous example, express r in vector form in the perifocal coordinate system.
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cos 204.81
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6587.7 km rQ 7257.5 sin 204.81
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3045.3 km Hence r 6587.7P 3045.3Q km
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2.9.6 The geocentric-equatorial coordinate system
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The geocentric-equatorial coordinate system is an inertial system of axes, the reference line being fixed by the fixed stars. The reference line is the line of Aries described in Sec. 2.5. (The phenomenon known as the precession of the equinoxes is ignored here. This is a very slow rotation of this reference frame, amounting to approximately 1.396971 per Julian century, where a Julian century consists of 36,525 mean solar days.) The fundamental plane is the earth s equatorial plane. Figure 2.9 shows the part of the ellipse above the equatorial plane and the orbital angles , w, and i. It should be kept in mind that and w may be slowly varying with time, as shown by Eqs. (2.12) and (2.13). The unit vectors in this system are labeled I, J, and K, and the coordinate system is referred to as the IJK frame, with positive I pointing along the line of Aries. The transformation of vector r from the PQW frame to the IJK frame is most easily expressed by matrix multiplication. If A is an m n matrix and B is an n p matrix, the product AB is an m p matrix (details of matrix multiplication will be found in most good
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Orbits and Launching Methods
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Geocentric-equatorial coordinate system (IJK frame).
texts on engineering mathematics. Many programmable calculators and computer programs have a built-in matrix multiplication function). The transformation matrix in this case is 3 2, given by Eq. (2.33b) and the PQW components form a 2 1 matrix. The components of r in the IJK frame appear as a 3 1 matrix given by rI rJ rK
~ r R c Pd rQ
(2.33a)
~ where the transformation matrix R is given by (cos ~ 5 (sin R cos sin sin cos i) ( cos cos cos sin cos i) ( sin (sin sin i) sin sin cos cos i) sin cos cos cos i) (cos sin i) (2.33b) It should be noted that the angles and w take into account the rotations resulting from the earth s equatorial bulge, as described in Sec. 2.8.1. The matrix multiplication is most easily carried out by computer.
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Example 2.16 Calculate the magnitude of the position vector in the PQW frame
for the orbit specified below. Calculate also the position vector in the IJK frame and its magnitude. Confirm that this remains unchanged from the value obtained in the PQW frame.
Solution
The given orbital elements are 300 ; w 60 ; i 65 ; rP 6500 km; rQ 4000 km
The magnitude of the r vector is r 5 2( 6500)2 7632.2 km Substituting the angle values in Eq. (2.33b) gives 0.567 ~ R 0.25 0.785 The vector components in the IJK frame are rI rJ rK 0.567 0.25 0.785 0.25 0.856 c 6500 d 4000 0.453 0.25 0.856 0.453 (4000)2
4685.3 5047.7 km 3289.1 The magnitude of the r vector in the IJK frame is r 5 2( 4685.3) 7632.2 km This is seen to be the same as that obtained from the P and Q components.
(5047.7)
( 3289.1)
2.9.7 Earth station referred to the IJK frame
The earth station s position is given by the geographic coordinates of latitude lE and longitude fE. (Unfortunately, there does not seem to be any standardization of the symbols used for latitude and longitude. In some texts, as here, the Greek lambda is used for latitude and the Greek phi for longitude. In other texts, the reverse of this happens. One minor advantage of the former is that latitude and lambda both begin with the same la which makes the relationship easy to remember.)
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