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Orbits and Launching Methods
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Repeat the calculations in Prob. 2.23 for an inclination of 63.435 .
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2.25. Determine the orbital condition necessary for the argument of perigee to remain stationary in the orbital plane. The orbit for a satellite under this condition has an eccentricity of 0.001 and a semimajor axis of 27,000 km. At a given epoch the perigee is exactly on the line of Aries. Determine the satellite position relative to this line after a period of 30 days from epoch. 2.26. For a given orbit, K as defined by Eq. (2.11) is equal to 0.112 rev/day. Determine the value of inclination required to make the orbit sun synchronous. 2.27. A satellite has an inclination of 90 and an eccentricity of 0.1. At epoch, which corresponds to time of perigee passage, the perigee height is 2643.24 km directly over the north pole. Determine (a) the satellite mean motion. For 1 day after epoch determine (b) the true anomaly, (c) the magnitude of the radius vector to the satellite, and (d) the latitude of the subsatellite point. 2.28. The following elements apply to a satellite in inclined orbit: 0 0 ; w0 90 ; M0 309 ; i 63 ; e 0.01; a 7130 km. An earth station is situated at 45 N, 80 W, and at zero height above sea level. Assuming a perfectly spherical earth of uniform mass and radius 6371 km, and given that epoch corresponds to a GST of 116 , determine at epoch the orbital radius vector in the (a) PQW frame; (b) IJK frame; (c) the position vector of the earth station in the IJK frame; (d) the range vector in the IJK frame; (e) the range vector in the SEZ frame; and (f ) the earth station look angles. 2.29. A satellite moves in an inclined elliptical orbit, the inclination being 63.45 . State with explanation the maximum northern and southern latitudes reached by the subsatellite point. The nominal mean motion of the satellite is 14 rev/day, and at epoch the subsatellite point is on the ascending node at 100 W. Calculate the longitude of the subsatellite point 1 day after epoch. The eccentricity is 0.01. 2.30. A no name satellite has the following parameters specified: perigee height 197 km; apogee height 340 km; period 88.2 min; inclination 64.6 . Using an average value of 6371 km for the earth s radius, calculate (a) the semimajor axis and (b) the eccentricity. (c) Calculate the nominal mean motion n 0 . (d) Calculate the mean motion. (e) Using the calculated value for a, calculate the anomalistic period and compare with the specified value. Calculate (f ) the rate of regression of the nodes, and (g) the rate of rotation of the line of apsides. 2.31. Given that 0 250 , w0 85 , and M0 30 for the satellite in Prob. 2.30, calculate, for 65 min after epoch (t0 0) the new values of , w, and M. Find also the true anomaly and radius. 2.32. From the NASA bulletin given in App. C, determine the date and the semimajor axis.
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2.33. Determine, for the satellite listed in the NASA bulletin of App. C, the rate of regression of the nodes, the rate of change of the argument of perigee, and the nominal mean motion n0. 2.34. From the NASA bulletin in App. C, verify that the orbital elements specified are for a nominal S N equator crossing. 2.35. A satellite in exactly polar orbit has a slight eccentricity (just sufficient to establish the idea of a perigee). The anomalistic period is 110 min. Assuming that the mean motion is n n0 calculate the semimajor axis. Given that at epoch the perigee is exactly over the north pole, determine the position of the perigee relative to the north pole after one anomalistic period and the time taken for the satellite to make one complete revolution relative to the north pole. 2.36. A satellite is in an exactly polar orbit with apogee height 7000 km and perigee height 600 km. Assuming a spherical earth of uniform mass and radius 6371 km, calculate (a) the semimajor axis, (b) the eccentricity, and (c) the orbital period. (d) At a certain time the satellite is observed ascending directly overhead from an earth station on latitude 49 N. Give that the argument of perigee is 295 calculate the true anomaly at the time of observation. 2.37. The 2-line elements for satellite NOAA 18 are as follows:
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NOAA 18 1 28654U 05018A 05154.51654998-.00000093 00000-0-28161-4 0 189 2 28654 98.7443 101.8853 0013815 210.8695 149.1647 14.10848892 1982 Determine the approximate values of (a) the semimajor axis, and (b) the latitude of the subsatellite point at epoch. 2.38. Using the 2-line elements given in Prob. 2.37, determine the longitude, of the subsatellite point and the LST at epoch. 2.39. Equation 2.34, gives the GST in degrees as GST 99 .9610 36000 .7689 T 0.0004 T2 UT
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where T is the number of Julian centuries that have elapsed since noon, January 0, 1900. The GST equation is derived from (Wertz 1984) GST 180 UT s where as is the right ascension of the mean sun. Determine the right ascension of the mean sun for noon on June 5, 2005. 2.40. Assuming that the orbits detailed in Table 2.5 are circular, and using Eq. (2.2) to find the semimajor axis, calculate the regression of the nodes for these orbits. 2.41. Determine the standard zone time in the following zones, for 12 noon GMT: (a) 285 E, (b) 255 E, (c) 45 E, (d) 120 E.
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