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MAE 4262: ROCKETS AND MISSION ANALYSIS Orbital Mechanics and Hohmann Transfer Orbit Summary Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

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REVIEW OF CONIC SECTIONS

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ORBITAL MECHANICS: SUMMARY Conic SectionEccentricityOrbital Energy Ellipse < 1 E < 0 Parabola = 1 E = 0 Hyperbola > 1 E > 0 Circle = 0 E = -GmM’/2r Equation for conic sections (polar coordinates) Force balance on orbiting body, m, about larger body M’ under influence of gravity =eccentricity, h=angular momentum (constant) Conservation of orbital energy = constant Orbital energy in terms of semi-major axis Eccentricity in terms of angular momentum and orbital energy

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SUMMARY COMMENTS Hyperbolic Parabolic Elliptic Circle Period

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INTERPLANETARY TRAJECTORY: HOHMANN ORBIT Main idea through example of moving spacecraft from LEO → GEO –Average radius of Earth is about 6,378 km –LEO is at 300 km above sea level or r 1 = 6,678 km from center of Earth –GEO is at 35,786 km above sea level or r 2 = 42,164 km from center of Earth Step 1: Calculate V c1 and V c2 at r 1 and r 2, respectively Step 2: Add some V 1 to into elliptical transfer, called GTO –Perpendicular to r 1 –Impulse applied at perigee of ellipse, spacecraft moving fastest –Spacecraft arrives at apogee moving slowest Step 3: Apply some V 2 to circularize orbit –If this is not done, spacecraft will stay in elliptical orbit

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WHAT IS ACTUAL SCALE OF ORBITS? NOT EVEN CLOSE TO SCALE

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WHAT IS ACTUAL SCALE OF ORBITS? EARTH LEO, 300 km GEO

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WHAT IS ACTUAL SCALE OF ORBITS? LEO GEO EARTH

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HOHMANN TRANSFER SUMMARY We want to move spacecraft from LEO → GEO Initial LEO orbit has radius r 1 and velocity V c1 Desired GEO orbit has radius r 2 and velocity V c2 At LEO (r 1 ), V c1 = 7,724 m/s At GEO (r 2 ), V c2 = 3,074 m/s Could accomplish this in many ways LEO GEO r1r1 r2r2 V c1 V c2

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HOHMANN TRANSFER SUMMARY We want to move spacecraft from LEO → GEO Initial LEO orbit has radius r 1 and velocity V c1 Desired GEO orbit has radius r 2 and velocity V c2 At LEO (r 1 ), V c1 = 7,724 m/s At GEO (r 2 ), V c2 = 3,074 m/s Could accomplish this in many ways LEO GEO r1r1 r2r2 V c1 V c2

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HOHMANN TRANSFER SUMMARY We want to move spacecraft from LEO → GEO Initial LEO orbit has radius r 1 and velocity V c1 Desired GEO orbit has radius r 2 and velocity V c2 At LEO (r 1 ), V c1 = 7,724 m/s At GEO (r 2 ), V c2 = 3,074 m/s Could accomplish this in many ways LEO GEO r1r1 r2r2 V c1 V c2

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HOHMANN TRANSFER SUMMARY We want to move spacecraft from LEO → GEO Initial LEO orbit has radius r 1 and velocity V c1 Desired GEO orbit has radius r 2 and velocity V c2 At LEO (r 1 ), V c1 = 7,724 m/s At GEO (r 2 ), V c2 = 3,074 m/s Accomplish this using Hohmann Transfer Orbit –Special illustrative case LEO GEO r1r1 r2r2 V c1 V c2

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HOHMANN TRANSFER SUMMARY Impulsive V1 is applied to get on geostationary transfer orbit (GTO) at perigee: Leave LEO (r 1 ) with a total velocity of V 1 LEO GEO r1r1 r2r2 V c1 V1V1 V c2 GTO V1V1

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HOHMANN TRANSFER SUMMARY Impulsive V1 is applied to get on geostationary transfer orbit (GTO) at perigee: Leave LEO (r 1 ) with a total velocity of V 1 Transfer orbit is elliptical shape –Perigee located at r 1 –Apogee located at r 2 LEO GEO r1r1 r2r2 V c1 V1V1 V c2 GTO V1V1 Perigee Apogee

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HOHMANN TRANSFER SUMMARY Arrive at GEO (apogee) with V 2 When arriving at GEO, which is at apogee or elliptical transfer orbit, must apply some V 2 in order to circularize: This is exactly the V that should be applied to circularize the orbit at GEO (r 2 ) –V c2 = V 2 + V 2 If this V is not applied, spacecraft will continue on dashed elliptical trajectory LEO GEO r1r1 r2r2 V c1 V1V1 V1V1 V2V2 V2V2 V c2 GTO

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HOHMANN TRANSFER SUMMARY Initial LEO orbit has radius r 1 and velocity V c1 Desired GEO orbit has radius r 2 and velocity V c2 Impulsive V1 is applied to get on geostationary transfer orbit (GTO) at perigee: Coast to apogee and apply impulsive V2 : LEO GEO r1r1 r2r2 V c1 V1V1 V1V1 V2V2 V2V2 V c2 GTO

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SUMMARY Hohmann Transfer Orbit –Minimum energy trajectory –Least fuel consumption (cheapest) –Tends to be longest –Reference Figure 10.16 in textbook Oberth Transfer Orbit –Same basic idea: directly launch into transfer orbit –Larger V at r 1 –Lower overall V –Minimum propulsive requirement to arrive in orbit General Comments –Time does not appear in these expression Depends on orbital characteristics –No Drag, No maneuvering near planet –Faster trajectories require greater V total

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BOEING DELTA IV COMPONENTS http://www.boeing.com/defense-space/space/delta/delta4/d4h_demo/book04.html

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OVERVIEW During LEO → GEO transfer, upper stage coasts for several hours Upper stage must re-start at conclusion of coast phase for insertion Delta-4M+(4,2) (Delta-4240) http://www.skyrocket.de/space/ Typical Delta 4 Medium launch sequence to geosynchronous transfer orbit from Cape http://www.spaceflightnow.com/news/n0201/28delta4mate/delta4medium.html

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2 nd STAGE OVERVIEW http://www.pratt-whitney.com/prod_space_rl10.asp LOX Tank LH 2 Tank http://www.spaceflightnow.com/news/n0201/28delta4mate/delta4upperstage.html

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OVERVIEW: WHAT CAN HAPPEN INSIDE TANKS? http://www.boeing.com/defense-space/space/delta/delta4/d4h_demo/book14.html XSS-10 view of Delta II rocket: An Air Force Research Laboratory XSS-10 micro-satellite uses its onboard camera system to view the second stage of the Boeing Delta II rocket during mission operations Jan. 30. (Photo courtesy of Boeing.), http://www.globalsecurity.org/space/systems/xss.htm Stage exposed to solar heating Propellants (LH 2 and LOX) may thermally stratify Propellants may boil Slosh events during maneuvers

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INTRODUCTION TO THE PROBLEM Analytical and computational thermal modeling of cryogenic rocket propellants Examine effects parametrically LOX Tank LH 2 Tank

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LEO TO GEO USING LOW THRUST

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REFERENCES References on Orbits http://www.shef.ac.uk/physics/people/vdhillon/teaching/phy105/phy105_derivatio n.htmlhttp://www.shef.ac.uk/physics/people/vdhillon/teaching/phy105/phy105_derivatio n.html http://home.cvc.org/science/kepler.htm References on Discount Airfare http://www.orbitz.com

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