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![Page 1: MAE 4262: ROCKETS AND MISSION ANALYSIS Orbital Mechanics and Hohmann Transfer Orbit Summary Mechanical and Aerospace Engineering Department Florida Institute.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649c7c5503460f94930bad/html5/thumbnails/1.jpg)
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 Section Eccentricity Orbital 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 largerbody 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 andorbital 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 r1 = 6,678 km from center of Earth
– GEO is at 35,786 km above sea level or r2 = 42,164 km from center of Earth
• Step 1: Calculate Vc1 and Vc2 at r1 and r2, respectively
• Step 2: Add some V1 to into elliptical transfer, called GTO
– Perpendicular to r1
– Impulse applied at perigee of ellipse, spacecraft moving fastest
– Spacecraft arrives at apogee moving slowest
• Step 3: Apply some V2 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 r1 and velocity Vc1
• Desired GEO orbit has radius r2 and velocity Vc2
• At LEO (r1), Vc1 = 7,724 m/s
• At GEO (r2), Vc2 = 3,074 m/s
• Could accomplish this in many ways
LEO
GEO
r1
r2
Vc1
Vc2
11 r
MGVc
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HOHMANN TRANSFER SUMMARY
• We want to move spacecraft from LEO → GEO
• Initial LEO orbit has radius r1 and velocity Vc1
• Desired GEO orbit has radius r2 and velocity Vc2
• At LEO (r1), Vc1 = 7,724 m/s
• At GEO (r2), Vc2 = 3,074 m/s
• Could accomplish this in many ways
LEO
GEO
r1
r2
Vc1
Vc2
r
MGVc
1
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HOHMANN TRANSFER SUMMARY
• We want to move spacecraft from LEO → GEO
• Initial LEO orbit has radius r1 and velocity Vc1
• Desired GEO orbit has radius r2 and velocity Vc2
• At LEO (r1), Vc1 = 7,724 m/s
• At GEO (r2), Vc2 = 3,074 m/s
• Could accomplish this in many ways
LEO
GEO
r1
r2
Vc1
Vc2
r
MGVc
1
![Page 12: MAE 4262: ROCKETS AND MISSION ANALYSIS Orbital Mechanics and Hohmann Transfer Orbit Summary Mechanical and Aerospace Engineering Department Florida Institute.](https://reader035.fdocuments.us/reader035/viewer/2022062308/56649c7c5503460f94930bad/html5/thumbnails/12.jpg)
HOHMANN TRANSFER SUMMARY
• We want to move spacecraft from LEO → GEO
• Initial LEO orbit has radius r1 and velocity Vc1
• Desired GEO orbit has radius r2 and velocity Vc2
• At LEO (r1), Vc1 = 7,724 m/s
• At GEO (r2), Vc2 = 3,074 m/s
• Accomplish this using Hohmann Transfer Orbit
– Special illustrative case
LEO
GEO
r1
r2
Vc1
Vc2
r
MGVc
1
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HOHMANN TRANSFER SUMMARY
• Impulsive V1 is applied to get on geostationary transfer orbit (GTO) at perigee:
• Leave LEO (r1) with a total velocity of V1
12111
22
rrrrV
LEO
GEO
r1
r2
Vc1V1
Vc2
GTO
MG
V1
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HOHMANN TRANSFER SUMMARY
• Impulsive V1 is applied to get on geostationary transfer orbit (GTO) at perigee:
• Leave LEO (r1) with a total velocity of V1
• Transfer orbit is elliptical shape
– Perigee located at r1
– Apogee located at r2
12111
22
rrrrV
LEO
GEO
r1
r2
Vc1V1
Vc2
GTO
MG
V1
Perigee
Apogee
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HOHMANN TRANSFER SUMMARY
• Arrive at GEO (apogee) with V2
• When arriving at GEO, which is at apogee or elliptical transfer orbit, must apply some V2 in order to circularize:
• This is exactly the V that should be applied to circularize the orbit at GEO (r2)
– Vc2 = V2 + V2
• If this V is not applied, spacecraft will continue on dashed elliptical trajectory
LEO
GEO
r1
r2
Vc1V1
V1
V2V2
Vc2
GTO
21222
22
rrrrV
MG
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HOHMANN TRANSFER SUMMARY
• Initial LEO orbit has radius r1 and velocity Vc1
• Desired GEO orbit has radius r2 and velocity Vc2
• Impulsive V1 is applied to get on geostationary transfer orbit (GTO) at perigee:
• Coast to apogee and apply impulsive V2:
12111
22
rrrrV
LEO
GEO
r1
r2
Vc1V1
V1
V2V2
Vc2
GTO
21222
22
rrrrV
r
MGVc
1
MG
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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 r1
– 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 Vtotal
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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 togeosynchronous transfer orbit from Cape
http://www.spaceflightnow.com/news/n0201/28delta4mate/delta4medium.html
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2nd STAGE OVERVIEW
http://www.pratt-whitney.com/prod_space_rl10.asp
LOX Tank
LH2 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.htmlXSS-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 (LH2 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
LH2 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_derivation.html
• http://home.cvc.org/science/kepler.htm
• References on Discount Airfare
• http://www.orbitz.com