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Orbits and Launching Methods
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2.1. State Kepler s three laws of planetary motion. Illustrate in each case their relevance to artificial satellites orbiting the earth. 2.2. Using the results of App. B, show that for any point P, the sum of the focal distances to S and S is equal to 2a. 2.3. Show that for the ellipse the differential element of area dA r2d /2, where d is the differential of the true anomaly. Using Kepler s second law, show that the ratio of the speeds at apoapsis and periapsis (or apogee and perigee for an earth-orbiting satellite) is equal to
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2.4. A satellite orbit has an eccentricity of 0.2 and a semimajor axis of 10,000 km. Find the values of (a) the latus rectum; (b) the minor axis; (c) the distance between foci. 2.5. For the satellite in Prob. 2.4, find the length of the position vector when the true anomaly is 130 . 2.6. The orbit for an earth-orbiting satellite has an eccentricity of 0.15 and a semimajor axis of 9000 km. Determine (a) its periodic time; (b) the apogee height; (c) the perigee height. Assume a mean value of 6371 km for the earth s radius. 2.7. For the satellite in Prob. 2.6, at a given observation time during a south to north transit, the height above ground is measured as 2000 km. Find the corresponding true anomaly. 2.8. The semimajor axis for the orbit of an earth-orbiting satellite is found to be 9500 km. Determine the mean anomaly 10 min after passage of perigee. 2.9. The exact conversion factor between feet and meters is 1 ft 0.3048 m. A satellite travels in an unperturbed circular orbit of semimajor axis a 27,000 km. Determine its tangential speed in (a) km/s, (b) ft/s, and (c) mi/h. 2.10. Explain what is meant by apogee height and perigee height. The Cosmos 1675 satellite has an apogee height of 39,342 km and a perigee height of 613 km. Determine the semimajor axis and the eccentricity of its orbit. Assume a mean earth radius of 6371 km. 2.11. The Aussat 1 satellite in geostationary orbit has an apogee height of 35,795 km and a perigee height of 35,779 km. Assuming a value of 6378 km for the earth s equatorial radius, determine the semimajor axis and the eccentricity of the satellite s orbit.
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2.12. Explain what is meant by the ascending and descending nodes. In what units would these be measured, and in general, would you expect them to change with time 2.13. Explain what is meant by (a) line of apsides and (b) line of nodes. Is it possible for these two lines to be coincident 2.14. With the aid of a neat sketch, explain what is meant by each of the angles: inclination; argument of perigee; right ascension of the ascending node. Which of these angles would you expect, in general, to change with time 2.15. The inclination of an orbit is 67 . What is the greatest latitude, north and south, reached by the subsatellite point Is this orbit retrograde or prograde 2.16. Describe briefly the main effects of the earth s equatorial bulge on a satellite orbit. Given that a satellite is in a circular equatorial orbit for which the semimajor axis is equal to 42,165 km, calculate (a) the mean motion, (b) the rate of regression of the nodes, and (c) the rate of rotation of argument of perigee. 2.17. A satellite in polar orbit has a perigee height of 600 km and an apogee height of 1200 km. Calculate (a) the mean motion, (b) the rate of regression of the nodes, and (c) the rate of rotation of the line of apsides. The mean radius of the earth may be assumed equal to 6371 km. 2.18. What is the fundamental unit of universal coordinated time Express the following times in (a) days and (b) degrees: 0 h, 5 min, 24 s; 6 h, 35 min, 20 s; your present time. 2.19. Determine the Julian days for the following dates and times: midnight March 10, 1999; noon, February 23, 2000; 16:30 h, March 1, 2003; 3 P.M., July 4, 2010. 2.20. Find, for the times and dates given in Prob. 2.19, (a) T in Julian centuries and (b) the corresponding GST in degrees. 2.21. Find the month, day, and UT for the following epochs: (a) day 3.00, year 1999; (b) day 186.125, year 2000; (c) day 300.12157650, year 2001; (d) day 3.29441845, year 2004; (e) day 31.1015, year 2010. 2.22. Find the GST corresponding to the epochs given in Prob. 2.21.
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2.23. The Molnya 3-(25) satellite has the following parameters specified: perigee height 462 km; apogee height 40,850 km; period 736 min; inclination 62.8 . 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 n0. (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.
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