MAE 4262: ROCKETS AND MISSION ANALYSIS

1. MAE 4262: ROCKETS AND MISSION ANALYSIS Orbital Mechanics and Hohmann Transfer Orbit Summary October 23, 2012 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

2. REVIEW OF CONIC SECTIONS

3. ORBITAL MECHANICS: SUMMARY Equation for conic sections (polar coordinates) Force balance on orbiting body, m, about larger body M’ under influence of gravity e=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

4. SUMMARY COMMENTS Hyperbolic Elliptic Parabolic Circle Period

5. 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 DV1 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 DV2 to circularize orbit • If this is not done, spacecraft will stay in elliptical orbit

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

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

8. WHAT IS ACTUAL SCALE OF ORBITS? GEO LEO EARTH

9. HOHMANN TRANSFER SUMMARY Vc2 • 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 GEO LEO r1 Vc1 r2

10. HOHMANN TRANSFER SUMMARY Vc2 • 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 GEO LEO r1 Vc1 r2

11. HOHMANN TRANSFER SUMMARY Vc2 • 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 GEO LEO r1 Vc1 r2

12. HOHMANN TRANSFER SUMMARY Vc2 • 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 GEO LEO r1 Vc1 r2

13. HOHMANN TRANSFER SUMMARY Vc2 • Impulsive DV1 is applied to get on geostationary transfer orbit (GTO) at perigee: • Leave LEO (r1) with a total velocity of V1 GEO GTO LEO r1 DV1 Vc1 r2 V1

14. HOHMANN TRANSFER SUMMARY Vc2 Apogee • Impulsive DV1 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 GEO GTO LEO r1 DV1 Vc1 r2 V1 Perigee

15. HOHMANN TRANSFER SUMMARY Vc2 • Arrive at GEO (apogee) with V2 • When arriving at GEO, which is at apogee or elliptical transfer orbit, must apply some DV2 in order to circularize: • This is exactly the DV that should be applied to circularize the orbit at GEO (r2) • Vc2= DV2 + V2 • If this DV is not applied, spacecraft will continue on dashed elliptical trajectory V2 DV2 GEO GTO LEO r1 DV1 Vc1 r2 V1

16. HOHMANN TRANSFER SUMMARY Vc2 • Initial LEO orbit has radius r1 and velocity Vc1 • Desired GEO orbit has radius r2 and velocity Vc2 • Impulsive DV1 is applied to get on geostationary transfer orbit (GTO) at perigee: • Coast to apogee and apply impulsive DV2: V2 DV2 GEO GTO LEO r1 DV1 Vc1 r2 V1

17. 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 DV at r1 • Lower overall DV • 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 DVtotal • See Figures 10.17 – 10.19 for lots of details

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

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

20. 2nd STAGE OVERVIEW LH2 Tank LOX Tank http://www.pratt-whitney.com/prod_space_rl10.asp http://www.spaceflightnow.com/news/n0201/28delta4mate/delta4upperstage.html

21. OVERVIEW: WHAT CAN HAPPEN INSIDE TANKS? • Stage exposed to solar heating • Propellants (LH2 and LOX) may thermally stratify • Propellants may boil • Slosh events during maneuvers 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

22. INTRODUCTION TO THE PROBLEM • Analytical and computational thermal modeling of cryogenic rocket propellants • Examine effects parametrically LH2 Tank LOX Tank

23. LEO TO GEO USING LOW THRUST

24. 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