HW6 - MAE146 Astronautics Homework#6 Assigned Friday...

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Unformatted text preview: MAE146: Astronautics «- Homework #6 Assigned: Friday, February 20th, 2008 Due date: Friday , February 27th, 20087 (at the beginning of the lecture) Note: “Textbook” refers to Wiesel’s book, “Spaceflight dynamics". Please justify all of your answers. Problem 1 (20 pts): On geostationary orbits Textbook, page 93, #2 1 l. A geosynchronous orbit has a period of 23h 56‘“ 4.09“, or one sidereal day. Calculate the radius of a geosynchronous orbit. Starting from a parking orbit at r, = 6578 km and an inclination of 28 . calculate the two maneuvers required to reach geosynchronous, equatorial orbit. Do the required 28 inclination change at apogee as one burn by combining it with the second Hohmann transfer maneuver. Problem 2 (30 pts): On Apollo Lunar mission Textbook, page 93-94, #2 2. A lunar module (LM) lifts off from the lunar surface and flies a powered trajectory to its burnout point at 30 km altitude as shown in Figure 3. l0. The velocity vector of the LM is parallel to the lunar surface at burnout. It then coasts halfway around the moon, where it must rendezvous with the Apollo command module (CM) in a 250-km circular orbit. The mass of the moon is 0.012l3 times the mass of the earth, and the radius of the moon is l740 km. (a) What is the burnout speed of the LM‘? in kilometers per second? (b) What is the magnitude of the maneuver required to rendezvous with the command module in kilometers per second? (c) What is the coast time for the LM? ((2’) In order to assure a quick rendezvous, it is desirable that the LM and CM arrive at the rendezvous point together (or at least close). Where must the CM be in relation to the LM at burnout? Cite your answer as either a time differential or angle differential of the CM ahead of or behind the LM burnout point. (This sets the “launch window” for the LM takeoff.) Problem 3 (50 pts): On piano changes Testhoek, pages 94$st #5 2 A sateihte leaves a parking orbit at mahoattert z“ amt! executes a fiohmaae traesfer to geosynchronous eqoaa _ , . a _ ' f- - “ are 3 i2 hart of the required inclinatioo change Air is perfoi éflftfig the i * are the remainder A52 = i - at; during the second weaver. if the :33 m the cheater orbits are v61 and veg, respeo tively, and the perigee arrd'apegoe has is the Hohmamt transfer ellipse are up and ad, Show that the total Av for both maneuvers is Av = (v:1 +2): - 2v€1vp cos Axial/2 + [1132 + v: —— 212621)“ cos(i -~ Afar/2 (3.70) Write the condition that minimizes the total required Av as a function of Ail, the amount of inclination change performed during the first maneuver. Argue that Ail = O (as in problem 1) is not optimal. Problem 3 (50 has}: Greased Tracks Textbook, page 9596, #6 6. Whoa the orbit ot a sateiiite ts prJo jectetl back onto the rotating earth the resulting curve is caiied a ground mire. As shots n in Figure 3.13 this often looks neatly sinusoidal for a lower! ititode cirrttiar othit Show that tracks for sequLntial orbits are schtrutLd by a longitude difference of (3.?!) at the equator, where we} is the angular—rotation rate of the earth. Also, show that the satellite can see a swath of angular width R L 9 = 2003" £2 (3.72; a measured from the center of the earth. For polar orbits, 6 and AA can be directly compared, while for lower inclinations the ground—trace separation is less. To get a comparison, assume a polar orbit and compare 6 and AA numerically from a = le to a = 3&3, where R69 is the radius of the earth. For overlapping coverage, we wish that 6 > AA. Are there any altitude bands in this range of a for which this is not possihie? ...
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