Appendix A: Useful Data Earth gravitational parameter (GM) = 398 600.5 km3/s2 Earth mass (M) = 5.9733 x 1024 kg Earth gravitational constant = 6.673 X 10-20 km3/kgs2 Earth equatorial radius = 6378.14km Earth polar radius = 6356.785km Earth eccentricity = 0.08182 Velocity of light = 299 792.458 km/s Average radius of geostationary orbit = 42164.57km Velocity of geostationary satellite = 3.074689km/s Angular velocity of geostationary satellites = 72.92115 X 10-6 rad/s Geostationary satellite orbital period = 86164.09 s (23 hours, 56 minutes, 4.09 seconds) Boltzmann constant = 1.3803 X 10-23 W/KHz or - 228.6 dB W/K Maximum range of geostationary satellite (0° elevation) = 41680km Minimum range of geostationary satellite (90° elevation) = 35786km Half-angle subtended at the satellite by Earth= 8.69° Coverage limit on Earth (0° elevation) = 81.3° One nautical mile = 1.852km 429 Appendix B: Useful Orbit-related Formulas (1) Doppler effect The equation set included here is general enough to provide Doppler shifts in non-geostationary orbits. The Doppler shift /lfct observed at a given point on the Earth at a frequency ft is given by vr -F ilfct=±-Jt (B.l) c where vr = relative radial velocity between the observer and the satellite transmitter c = velocity of light ft = transmission frequency. The sign of the Doppler shift is positive when the satellite is approaching the observer. The relative velocity can be approximated as (B.2) where p1(t1) and p2(t 2) are satellite ranges at times t1 and t2 respectively; (t2 - t1,) is arbitrarily small. p(t) at any instant t can be obtained from the orbital parameters by using the technique given in a following section ('(9) Satellite position from orbital pa­ rameters'). Range rate can then be obtained by using equation (B.2), at two successive instants. The following equation set may be used for approximate estimation of the range rate of a geostationary satellite. We note that range rate is a function of orbital eccentricity, inclination and satellite drift rate. The range rate for each of these components is given as (Morgan and Gordon, 1989): (a) Eccentricity (B.3) Pm 430 Appendix B: Useful Orbit-related Formulas 431 where Pe = range rate due to eccentricity e = eccentricity a = semi-major axis w• = angular velocity 2'7T where To = orbital period To Pm = mean range from observation point tp = time from perigee. (b) Inclination iaRw . (. ) Pi = ---smOcos wti (B.4) Pm where Pi = range rate due to inclination i = inclination R = Earth radius 0 = latitude of earth station ti = time from ascending node. (c) Drift DaR . A.-I,. Pct = --cosOsm~'+' (B.S) Pm where D = drift rate in radians/s Pct = range rate due to satellite drift Ll¢ = difference in longitude between satellite and earth station. The total range rate at any given time is the sum of range rates due to each of the above components. CCIR Report 214 gives the following approximate relationship for estimating the maximum Doppler shift: -6 Llfctm = ± 3.0(10) fts (B.6) where ft = operating frequency s = number of revolutions/24 hours of the satellite with respect to a fixed point on the Earth. For a more precise treatment of the subject the reader is referred to the literature (e.g. Slabinski, 1974). (2) Near geostationary satellites On various occasions, communication satellites are in near geostationary orbits. Examples are: (a) when orbit inclination is intentionally left uncorrected to 432 Appendix B: Useful Orbit-related Fonnulas conserve on-board fuel and thereby prolong the satellite's useful lifetime and (b) when a satellite is being relocated to another position or a newly launched satellite is being moved to the operational location (such a drifting satellite is sometimes used for communication provided the transmissions do not interfere with other systems). When the satellite orbit is lower than the geostationary orbit altitude, the angular velocity of the satellite is greater than the angular velocity of the Earth. Consequently the satellite drifts in an eastward direction with respect to an earth station. When the satellite altitude is higher than the geostationary height, the satellite drifts westward. The following relationships apply (Morgan and Gordon, 1989): AP Aw (B.7) p w where AP = change in orbital period P = orbital period Aw = change in angular velocity w = angular velocity and ~r = -(~)A: (B.8) where r = orbital radius Ar = change in orbital radius. For example, a change in radius of + 1 km from the nominal causes a west­ ward drift of 0.0128°/day. The required change in satellite velocity Ave to correct the drift is given by 1 Aw Ave= -v- (B.9a) 3 w or 1 -aAw (B.9b) 3 where a = semi-major axis. Effect of inclination The main effect of inclination i on a geostationary satellite is to cause north­ south oscillation of the sub-satellite point, with an amplitude of i and period of Appendix B: Useful Orbit-related Formulas 433 a day. When the inclination is small (the condition is, tan (i) = i in radians), the motion can be approximated as a sinusoid in a right ascension-declination coordinate system. An associated relatively minor effect is an east-west oscillation with a period of half a day. This is caused by the change in rate of variation of the right ascension relative to the average rate. The satellite appears to drift west for the first 3 hours and then east for the next half quarter. The satellite continues to move eastward during the next half quarter and then westward, completing the cycle in half a day. The maximum amplitude of such east-west oscillation for a circular orbit is given by (B.10a) 1 ·2 = -l (B.10b) 229 where i is in degrees. Usually the east-west oscillation is very small (e.g. fori = 2.5°, LlliW; = 0.027°). The net effect of these two motions is the often-quoted figure-of-eight mo­ tion of the sub-satellite point. Effect of eccentricity The effect of eccentricity in a geostationary orbit is to cause east-west oscilla­ tion with a period of a day. The satellite is to the east of its nominal position between perigee and apogee and to the west between apogee and perigee. The amplitude of the oscillation is given by L1EW., = 2e radians (B.ll) For example, an eccentricity of 0.001 produces an east-west oscillation of ±0.1145° about the satellite's nominal position. (3) Coverage contours It is often necessary to plot the coverage contours of geostationary satellites on the surface of the Earth. The satellite antenna boresight (the centre of coverage area) and a specified antenna power beamwidth (usually, half-power beamwidth) are known. In the case of an elliptical antenna beam shape, the sizes of the major and minor axes together with the orientation of the major axis are known. The coverage contour on the Earth is obtained by calculating the latitude/longitude of n points on the periphery of the coverage (Siocos, 1973). 434 Appendix B: Useful Orbit-related Formulas Let us first define the following angles: 'YB, 'Yn = tilt angles of antenna boresight and the nth point on the coverage contour, respectively En = angular antenna beamwidth of the specified power (e.g. half-power) in the direction of the nth point. For a circular beam, En is a constant. To specify the nth coverage point we further define 1/Jn as the angle of rotation, the rotation being referenced to the plane containing the sub-satellite and boresight points (see figure B.l). The following steps are used to specify the nth coverage point Tn. Obtain 'YB using the following equation set {3 = arccos( cos 8B cos cf>sB) (B.l2a) 'YB = arctan[ sinf3/ (6.6235 - cosf3)] (B.l2b) where 8B = latitude of boresight cf>sB = longitude of boresight with respect to sub-satellite point, taken positive when to the west of the sub-satellite point. Then (B.13a) (B.13b) gn = arctan(sincf>sB/tan8B) + c/>n (B.13c) Coverage contour Earth South Figure B.l Coverage contours geometry. S = sub-satellite point, B = boresight point on Earth, Tn = nth point on the coverage contour. Appendix B: Useful Orbit-related Formulas 435 f3n = arcsin(6.6235sinrn)- 'Yn (B.13d) (B.13e) (B.13f) where <Psn = longitude of nth point relative to sub-satellite point (Jn = latitude of nth point. When the beam is elliptical, En depends on 1/Jn as follows: (B.14) where a = rotation of t:1 away from the direction of the azimuth of the bore sight t:1 and t:2 are the semi-major and semi-minor axes. 1/Jn can be varied from 0° to 360° to obtain as many points on the coverage contour as desired. For a multiple beam satellite the above steps are repeated for each beam. (4) Sun transit time Around the equinox periods (March and September), the Sun is directly behind the geostationary orbit and therefore appears within earth stations' antenna beam. Sun transit through an earth station's antenna causes disruption to com­ munication services because of a large increase in system noise temperature caused by the Sun. The transit time of the Sun through an antenna is predict­ able, giving the earth station operator the option to make alternative communi­ cation arrangements or at least not be taken by surprise when communication is disrupted. The position of astronomical bodies such as the Sun is published in a readily available annual publication called the Nautical Almanac (US Government Printing Office). The position is given in the right ascension-declination coordi­ nate system.