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2.9.9 The subsatellite point
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The point on the earth vertically under the satellite is referred to as the subsatellite point. The latitude and longitude of the subsatellite point and the height of the satellite above the subsatellite point can be determined from knowledge of the radius vector r. Figure 2.13 shows the meridian plane which cuts the subsatellite point. The height of the terrain above the reference ellipsoid at the subsatellite point is denoted by HSS, and the height of the satellite above this, by hSS. Thus the total height of the
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Orbits and Launching Methods
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hSS HSS
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Geometry for determining the subsatellite point.
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satellite above the reference ellipsoid is h HSS hSS (2.50)
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Now the components of the radius vector r in the IJK frame are given by Eq. (2.33). Figure 2.13 is seen to be similar to Fig. 2.11, with the difference that r replaces R, the height to the point of interest is h rather than H, and the subsatellite latitude lSS is used. Thus Eqs. (2.39) through (2.42) may be written for this situation as N rI rJ rK aE 21 (N (N SNQ1 e2 sin 2lSS E h) cos lSS cos LST h) cos lSS sin LST e2 R E hT sin lSS (2.52) (2.53) (2.54) (2.51)
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We now have three equations in three unknowns, LST, lE, and h, which can be solved. In addition, by analogy with the situation shown in Fig. 2.10, the east longitude is obtained from Eq. (2.35) as EL LST GST (2.55)
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where GST is the Greenwich sidereal time.
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Two
Example 2.21 Determine the subsatellite height, latitude, and LST for the satellite in Example 2.16.
Solution From Example 2.16, the known components of the radius vector r in the IJK frame can be substituted in the left-hand side of Eqs.(2.52) through (2.54). The known values of aE and eE can be substituted in the right-hand side to give
a a a
6378.1414 21 0.08182 sin lSS 6378.1414 21 0.081822 sin 2lSS (1
2 2 2
hb cos lSS cos LST
hb cos lSS sin LST
6378.1414 21
0.081822)
0.08182 sin 2lSS
hb cos lSS cos LST
Each equation contains the unknowns LST, lss, and h. Unfortunately, these unknowns cannot be separated out in the form of explicit equations. The following values were obtained by a computer solution. lss > 25.654 h > 1258.012 km LST > 132.868
2.9.10 Predicting satellite position
The basic factors affecting satellite position are outlined in the previous sections. The NASA two-line elements are generated by prediction models contained in Spacetrack report no. 3 (ADC USAF, 1980), which also contains Fortran IV programs for the models. Readers desiring highly accurate prediction methods are referred to this report. Spacetrack report No. 4 (ADC USAF, 1983) gives details of the models used for atmospheric density analysis.
2.10 Local Mean Solar Time and Sun-Synchronous Orbits The celestial sphere is an imaginary sphere of infinite radius, where the points on the surface of the sphere represent stars or other celestial objects. The points represent directions, and distance has no significance for the sphere. The orientation and center of the sphere can be selected to suit the conditions being studied, and in Fig. 2.14 the
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North celestial pole
P Sun 0 ap as Celestial equator
Sun-synchronous orbit.
sphere is centered on the geocentric-equatorial coordinate system (see Sec. 2.9.6). What this means is that the celestial equatorial plane coincides with the earth s equatorial plane, and the direction of the north celestial pole coincides with the earth s polar axis. For clarity the IJK frame is not shown, but from the definition of the line of Aries in Sec. 2.9.6, the point for Aries lies on the celestial equator where this is cut by the x-axis, and the z-axis passes through the north celestial pole. Also shown in Fig. 2.14 is the sun s meridian. The angular distance along the celestial equator, measured eastward from the point of Aries to the sun s meridian is the right ascension of the sun, denoted by as. In general, the right ascension of a point P, is the angle, measured eastward along the celestial equator from the point of Aries to the meridian passing through P. This is shown as aP in Fig. 2.14. The hour angle of a star is the angle measured westward along the celestial equator from the meridian to meridian of the star. Thus for point P the hour angle of the sun is (ap as) measured westward (the hour angle is measured in the opposite direction to the right ascension). Now the apparent solar time of point P is the local hour angle of the sun, expressed in hours, plus 12 h. The 12 h is added because zero hour angle corresponds to midday, when the P meridian coincides with the sun s meridian. Because the earth s path around the sun is elliptical rather than circular, and also because the plane containing the path of the earth s orbit around the sun (the ecliptic plane) is inclined at an angle of approximately 23.44 , the apparent solar time does not measure out
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