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Orbits and Launching Methods
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Position vector R of the earth relative to the IJK
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Care also must be taken regarding the sign conventions used for latitude and longitude because different systems are sometimes used, depending on the application. In this book, north latitudes will be taken as positive numbers and south latitudes as negative numbers, zero latitude, of course, being the equator. Longitudes east of the Greenwich meridian will be taken as positive numbers, and longitudes west, as negative numbers. The position vector of the earth station relative to the IJK frame is R as shown in Fig. 2.10. The angle between R and the equatorial plane, denoted by yE in Fig. 2.10, is closely related, but not quite equal to, the earth station latitude. More will be said about this angle shortly. R is obviously a function of the rotation of the earth, and so first it is necessary to find the position of the Greenwich meridian relative to the I axis as a function of time. The angular distance from the I axis to the Greenwich meridian is measured directly as Greenwich sidereal time (GST), also known as the Greenwich hour angle, or GHA. Sidereal time is described in Sec. 2.9.4. GST may be found using values tabulated in some almanacs (see Bate et al., 1971), or it may be calculated using formulas given in Wertz (1984). In general, sidereal time may be measured in time units of sidereal days, hours, minutes, seconds, or it may be measured in angular units (degrees, minutes, seconds, or radians). Conversion is easily accomplished, since 2p radians or 360 correspond to 24 sidereal hours. The formula for GST in degrees is GST 99.9610 36000.7689 T 0.0004 T2 UT (2.34)
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Here, UT is universal time expressed in degrees, as given by Eq. (2.19). T is the time in Julian centuries, given by Eq. (2.20). Once GST is known, the local sidereal time (LST) is found by adding the east longitude of the station in degrees. East longitude for the earth station will be denoted as EL. Recall that previously longitude was expressed in positive degrees east and negative degrees west. For east longitudes, EL fE, while for west longitudes, EL 360 fE. For example, for an earth station at east longitude 40 , EL 40 . For an earth station at west longitude 40 , EL 360 ( 40) 320 . Thus the LST in degrees is given by LST GST EL (2.35)
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The procedure is illustrated in the following examples
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Example 2.17
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Find the GST for 13 h UT on 18 December 2000. 1.00963838 . The individual terms of Eq. (2.34)
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From Example 2.11: T
are: X Y UT 5 GST 36000.7689 0.0004 13 24 99.6910 X Y UT 360 T
347.7578 (mod 360 )
0.00041 (mod 360 ) 195
282.4493 (mod 360 )
Example 2.18 Find the LST for Thunder Bay, longitude 89.26 W for 13 h UT on December 18, 2000.
Solution
Expressing the longitude in degrees west: WL 360 GST ( 89.26 ) EL 270.74 270.74
89.26
In degrees east this is EL LST
93.189 (mod 360 )
Knowing the LST enables the position vector R of the earth station to be located with reference to the IJK frame as shown in Fig. 2.10. However, when R is resolved into its rectangular components, account must be taken of the oblateness of the earth. The earth may be modeled as an oblate spheroid, in which the equatorial plane is circular, and any
Orbits and Launching Methods
meridional plane (i.e., any plane containing the earth s polar axis) is elliptical, as illustrated in Fig. 2.11. For one particular model, known as a reference ellipsoid, the semimajor axis of the ellipse is equal to the equatorial radius, the semiminor axis is equal to the polar radius, and the surface of the ellipsoid represents the mean sea level. Denoting the semimajor axis by aE and the semiminor axis by bE and using the known values for the earth s radii gives aE bE 6378.1414 km 6356.755 km (2.36) (2.37)
From these values the eccentricity of the earth is seen to be eE 5 2a2 b2 E E aE 0.08182 (2.38)
In Figs. 2.10 and 2.11, what is known as the geocentric latitude is shown as yE. This differs from what is normally referred to as latitude. An imaginary plumb line dropped from the earth station makes an angle lE with the equatorial plane, as shown in Fig. 2.11. This is known as the geodetic latitude, and for all practical purposes here, this can be taken as the geographic latitude of the earth station.
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