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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
REVIEW OF CONIC SECTIONS
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
SUMMARY COMMENTS
Hyperbolic Parabolic Elliptic
Circle
Period
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
WHAT IS ACTUAL SCALE OF ORBITS?
NOT EVEN CLOSE TO SCALE
WHAT IS ACTUAL SCALE OF ORBITS?
EARTH
LEO, 300 km
GEO
WHAT IS ACTUAL SCALE OF ORBITS?
LEO
GEO
EARTH
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
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
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
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
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
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
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
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
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
BOEING DELTA IV COMPONENTS
http://www.boeing.com/defense-space/space/delta/delta4/d4h_demo/book04.html
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
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
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
INTRODUCTION TO THE PROBLEM• Analytical and computational thermal modeling of cryogenic rocket propellants
• Examine effects parametrically
LOX Tank
LH2 Tank
LEO TO GEO USING LOW THRUST
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
