Z Apogee View in Orbit Plane Equatorial Plane hemisphere at the vernal equinox. The z axis points to the north and the y axis is in the equatorial plane and points to the winter solstice. The elements shown are longitude or right ascension of the ascending node Q2 measured in the equatorial plane, the orbit's inclination angle i relative to the equatorial plane; the ellipse semimajor axis length a, the ellipse eccentricity e, the argument(angle)of perigee @, measured in the orbit plane from the ascending node to the satellites closest approach to the earth; and the true anomaly(angle)in the orbit plane from the perigee to the satellite v The mean anomaly M is the angle from perigee that would be traversed by a satellite moving at its mean ngular velocity n Given an initial value Me usually taken as 0 for a particular epoch( time) at perigee, the mean anomaly at time tis M=M+n(t-t), where n= u/a. The eccentric anomaly E may then be found from Keplers transcendental equation M=E-e sinE which must be solved numerically by, for example, guessing an initial value for Eand using a root finding method. For small eccentricities, the series approximation E=M+ esinM+(e/2)sin2M +(e/8)(3sin3M-sinM) yields good accuracy [Morgan and Gordon, 1989, p 806]. Other useful quantities include the orbit radius, r, the period, P, of the orbit, [i.e, for n(t-t )=2T), the velocity, V, and the radial velocity, V: afl= e cos E) (74.2) P=2兀a2/μ (744) e(μa)sinE (745) e cos e .. Figure 74.4 depicts quantities useful for communications links in the plane formed by the satellite,a point the earths surface and the earth's center Shown to approximate scale for comparison are satellites at altitudes representing LEO, MEO, and GEO orbits. For a satellite at altitude h, and for the earths radius at the equator r=6378.14 km, the slant range r,, levation angle to the satellite from the local horizon el, and the satellite's nadir angle 8, are related by simple c 2000 by CRC Press LLC
© 2000 by CRC Press LLC hemisphere at the vernal equinox. The z axis points to the north and the y axis is in the equatorial plane and points to the winter solstice. The elements shown are longitude or right ascension of the ascending node W measured in the equatorial plane, the orbit’s inclination angle i relative to the equatorial plane; the ellipse semimajor axis length a, the ellipse eccentricity e, the argument (angle) of perigee w, measured in the orbit plane from the ascending node to the satellite’s closest approach to the earth; and the true anomaly (angle) in the orbit plane from the perigee to the satellite n. The mean anomaly M is the angle from perigee that would be traversed by a satellite moving at its mean angular velocity n. Given an initial value Mo, usually taken as 0 for a particular epoch (time) at perigee, the mean anomaly at time t is M = Mo + n(t – to ), where n = . The eccentric anomaly E may then be found from Kepler’s transcendental equation M = E – e sinE which must be solved numerically by, for example, guessing an initial value for E and using a root finding method. For small eccentricities, the series approximation E ª M + e sinM + (e2 /2)sin2M + (e3 /8)(3sin3M – sinM) yields good accuracy [Morgan and Gordon, 1989, p. 806]. Other useful quantities include the orbit radius, r, the period, P, of the orbit, [i.e., for n(t – to ) = 2p], the velocity, V, and the radial velocity, Vr : (74.2) (74.3) (74.4) (74.5) Figure 74.4 depicts quantities useful for communications links in the plane formed by the satellite, a point on the earth’s surface and the earth’s center. Shown to approximate scale for comparison are satellites at altitudes representing LEO, MEO, and GEO orbits. For a satellite at altitude h, and for the earth’s radius at the equator re = 6378.14 km, the slant range rs, elevation angle to the satellite from the local horizon el, and the satellite’s nadir angle q, are related by simple FIGURE 74.3 Orbital elements. m a3 § r = = a(1 e cos E) P = 2 a m 3 p V r a 2 2 1 = m - Ê Ë Á ˆ ¯ ˜ V e a E a e E r = (m ) ( - ) 1 2 1 sin cos
MEO c:0 IGURE 74.4 Geometry for a satellite in the plane defined by the satellite, the center of the earth, and a point on the arths surface. The elevation angle, el, is the angle from the local horizon to the satellite. Shown to approximate scale are satellites at LEO, MEO (or ICO), and GEO. trigonometry formulas. Note that 0+el+y=90%, where y is the earths central angle and the ground range IS k=+ (746) () (cos Y-1/k) (74.7) r,=rv1+ k2-2k cosy (748) The earth station azimuth angle to the satellite measured clockwise from north in the horizon plane is given terms of the satellites declination d, the observers latitude, o, and the difference of the east longitudes of observer and satellite. AA. Then: n ( coso tan8- sin cos△M) taking due account of the sign of the denominator to ascertain the quadrant. The fraction of the earths surface area covered by the satellite within a circle for a given elevation angle, el, and the corresponding earth central angle, y is COS (74.10) 74.5 Communications Link Figure 74.5 illustrates the elements of the radio frequency(RF) link between a satellite and earth terminals The overall link performance is determined by computing the link equation for the uplink and downlink separately and then combining the results along with interference and intermodulation effects. For a radio link with only thermal noise, the received carrier-to-noise power ratio is =(p84x7人7人天人4元) (74.1la) The same quantities expressed in dB are e 2000 by CRC Press LLC
© 2000 by CRC Press LLC trigonometry formulas. Note that q + el + g = 90°, where g is the earth’s central angle and the ground range from the subsatellite point is gre. Then, (74.6) (74.7) (74.8) The earth station azimuth angle to the satellite measured clockwise from north in the horizon plane is given in terms of the satellite’s declination d, the observer’s latitude, f, and the difference of the east longitudes of observer and satellite, Dl. Then: (74.9) taking due account of the sign of the denominator to ascertain the quadrant. The fraction of the earth’s surface area covered by the satellite within a circle for a given elevation angle, el, and the corresponding earth central angle, g, is (74.10) 74.5 Communications Link Figure 74.5 illustrates the elements of the radio frequency (RF) link between a satellite and earth terminals. The overall link performance is determined by computing the link equation for the uplink and downlink separately and then combining the results along with interference and intermodulation effects. For a radio link with only thermal noise, the received carrier-to-noise power ratio is (74.11a) The same quantities expressed in dB are FIGURE 74.4 Geometry for a satellite in the plane defined by the satellite, the center of the earth, and a point on the earth’s surface. The elevation angle, el, is the angle from the local horizon to the satellite. Shown to approximate scale are satellites at LEO, MEO (or ICO), and GEO. k r h r el e e = + = cos( ) sin q tan cos sin el k ( ) = ( g - ) g 1 r r k k s e = 1 2 + - 2 cos g tan sin cos tan sin cos A = ( - ) D D l f d f l a a c e = 1 - 2 cos g c n p g r g T k a b t t s r Ê Ë Á ˆ ¯ ˜ = ( ) Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜ Ê Ë Á ˆ ¯ ˜( ) Ê Ë Á ˆ ¯ ˜ 1 4 1 4 1 1 2 2 p l p r