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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 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

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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) Typical Delta 4 Medium launch sequence to geosynchronous transfer orbit from Cape

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2 nd STAGE OVERVIEW LOX Tank LH 2 Tank

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OVERVIEW: WHAT CAN HAPPEN INSIDE TANKS? 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.), 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 n.htmlhttp://www.shef.ac.uk/physics/people/vdhillon/teaching/phy105/phy105_derivatio n.html References on Discount Airfare

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