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Orbital Mechanics Orbital Mechanics •• Planetary launcch and entry ovverview •• Energy and vvelocity in orbit •• Elliliptical orbit parameters •• Orrbiittaal elleemeenttss •• Coplanar orbital transfers •• Nonccoplanar ttransfferss •• Tiime iin orrbiitt •• Interplanetary trajectoriess •• Relative orbital motion (“proximity operations”)) © 2009 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu U N I V E R S I T Y O F Orbital Mechanics MARYLAND Space Launch - The Physics •• Minimum orbital altitude is ~200 km Potential Energy µµ µµ 66 J = − + = 11..99 × 10 kg in orbit rrorbit rrE kg •• Circular orbital velocity there is 7784 m/sec 11 Kinetic Energy µµ 66 J = = 30 × 10 22 22 kg in orbit rrorbit kg •• Total energy per kg in orbit Total Energy 66 J = KE + P E = 32 × 10 kg in orbit kg U N I V E R S I T Y O F Orbital Mechanics MARYLAND Theoretical Cost to Orbit •• Convvert to usual energy units T ootta Eneergyy 66 J W rs = 32 × 10 = 88..888 kg in orbit kg kkgg •• Domestic energy costs are ~$0.05/kWhrr ! Theoretical cost to orbit $0.44/kg U N I V E R S I T Y O F Orbital Mechanics MARYLAND Actual Cost to Orbit •• Deelltta IIV Heeaavvyy – 23,000 kg to LEO – $250 M per flight •• $$10,,990000/kg of paayylloaad •• Faactor of 25,000x higherr than theoretical energyy costs! U N I V E R S I T Y O F Orbital Mechanics MARYLAND What About Airplanes? •• For an aircraft in levvel flight, Weight Lift mg L = , or = Thrust Drag T D •• Energy = force x distancee, so Total Energy thrust × distance T dd ggdd = = = kg mass m L/D •• For an airliner (L/D=25) to equal orbital energy,, d=81,000 km (2 roundtrips NY-Sydney) U N I V E R S I T Y O F Orbital Mechanics MARYLAND Equivalent Airline Costs? • Average economy ticket NY-Sydney round- round-trip (Travelocity 9/3/09) ~$1300 • Average passenger (+ luggage) ~100 kg • Two round trips = $26/kg – Factor of 60x more than electrical energy costs – Factor of 420x less than current launch costs • But… you get to refuel at each stop! U N I V E R S I T Y O F Orbital Mechanics MARYLAND Equivalence to Air Transport • 81,000 km ~ twice around the world • Voyager - one of only two aircraft to ever circle the world non-stop, non-refueled - once! U N I V E R S I T Y O F Orbital Mechanics MARYLAND Orbital Entry - The Physics • 32 MJ/kg dissipated by friction with atmosphere over ~8 min = 66kW/kg • Pure graphite (carbon) high-temperature material: cp=709 J/kg°K • Orbital energy would cause temperature gain of 45,000°K! • (If you’re interesting in how this works out, you can take ENAE 791 Launch and Entry Vehicle Design next term...) U N I V E R S I T Y O F Orbital Mechanics MARYLAND Differences in Inclination Line of Nodes U N I V E R S I T Y O F Orbital Mechanics MARYLAND Choosing the Wrong Line of Apsides U N I V E R S I T Y O F Orbital Mechanics MARYLAND Simple Plane Change v 1 ! v 2 v apogee v v perigee 2 U N I V E R S I T Y O F Orbital Mechanics MARYLAND Patched Conic - Lunar Orbit Insertion • The spacecraft is now in a hyperbolic orbit of the moon. The velocity it will have at the perilune point tangent to the desired 100 km low lunar orbit is 2µm 2(4667.9) km v = v2 + = 1.0182 + = 2.451 pm h rLLO 1878 sec • The required delta-V to slow down into low lunar orbit is 4667.9 km ∆v = v − v = 2.451 − = 0.874 pm cm 1878 sec U N I V E R S I T Y O F Orbital Mechanics MARYLAND ! V Requirements for Lunar Missions To: Low Earth Lunar Low Lunar Lunar Lunar Orbit Transfer Orbit Descent Landing From: Orbit Orbit Low Earth 3.107 Orbit km/sec Lunar 3.107 0.837 3.140 Transfer km/sec km/sec km/sec Orbit Low Lunar 0.837 0.022 Orbit km/sec km/sec Lunar 0.022 2.684 Descent km/sec km/sec Orbit Lunar 2.890 2.312 Landing km/sec km/sec U N I V E R S I T Y O F Orbital Mechanics MARYLAND LOI ! V Based on Landing Site U N I V E R S I T Y O F Orbital Mechanics MARYLAND Hill’s Equations (Proximity Operations) 2 ˙˙ = 3 + 2 ˙ + x n x n y adx ˙˙ = "2 ˙ + y n x ady 2 ˙˙ = " + z n z ad z Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993 U N I V E R S I T Y O F Orbital Mechanics MARYLAND Clohessy-Wiltshire (“CW”) Equations sin(nt ) 2 = " + ˙ + " ˙ x(t ) [ 4 3cos(nt )] xo x o [1 cos(nt )] yo n n 2 4sin(nt ) " 3nt = " + " " ˙ + ˙ y(t ) 6[s in(nt ) nt ] xo yo [1 cos(nt )] xo yo n n z ˙o z t = z nt + nt ( ) o cos( ) sin( ) n z˙ t = " z n nt + z˙ nt ( ) o sin( ) o sin( ) U N I V E R S I T Y O F Orbital Mechanics MARYLAND “V-Bar” Approach Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On-Orbit Satellite Servicing and Resupply , 15th USU Small Satellite Conference, 2001 U N I V E R S I T Y O F Orbital Mechanics MARYLAND “R-Bar” Approach • Approach from along the radius vector (“R- bar”) • Gravity gradients decelerate spacecraft approach velocity - low contamination approach • Used for Mir, ISS docking approaches Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On-Orbit Satellite Servicing and Resupply , 15th USU Small Satellite Conference, 2001 U N I V E R S I T Y O F Orbital Mechanics MARYLAND References for This Lecture • Wernher von Braun, The Mars Project University of Illinois Press, 1962 • William Tyrrell Thomson, Introduction to Space Dynamics Dover Publications, 1986 • Francis J. Hale, Introduction to Space Flight Prentice- Hall, 1994 • William E. Wiesel, Spaceflight Dynamics MacGraw- Hill, 1997 • J. E. Prussing and B. A. Conway, Orbital Mechanics Oxford University Press, 1993 U N I V E R S I T Y O F Orbital Mechanics MARYLAND.
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