Applications of the Vis-Viva Equation: the Hohmann Transfer!
Section 2.4: Applications of the Vis-Viva Equation: The Hohmann Transfer!
Rc 2
Rc 1 Not-Feasible Transfer Orbit
Feasible Transfer Orbit
1! MAE 5540 - Propulsion Systems! Orbital Energy Review!
µ = − 2a
33! MAE 5540 - Propulsion Systems! Total Specific Energy (concluded)(6)"
εT = constant Vapogee Vperigee ra rp
• gathering up the terms
kinetic potential kinetic potential total energy energy energy energy energy εT = = 2 µ 2 µ µ V - V - = - 2 r perigee 2 r anywhere 2 a
32! MAE 5540 - Propulsion Systems! Total …!
• !
2! MAE 5540 - Propulsion Systems! Vis-Viva Equation for All the Conic-Sections!
See Appendix 2.3.2 for Hyperbolic Trajectory Proof! 37! MAE 5540 - Propulsion Systems! Excess-energy transfer orbit"
“Wasted”! ΔV!
3! MAE 5540 - Propulsion Systems! Optimal-energy transfer Orbit" Two curves! Are tangential! In space!
Target! Orbit!
4! MAE 5540 - Propulsion Systems! … The Hohmann Transfer!
5! MAE 5540 - Propulsion Systems! The Hohmann transfer!
• Most fuel efficient method – All velocity changes are tangential – (change velocity magnitude but not direction)
• Between circular or (aligned elliptical orbits)
• Takes longer than other less efficient transfers
• Tangential elliptical transfer orbit
• (example: Geosynchronous Transfer Orbit GTO)
6! MAE 5540 - Propulsion Systems! Hohmann Transfer Steps • 0 : Calculate transfer orbit semi-major axis & eccentricity • 1: Calculate circular velocity of parking orbit • 2: Calculate perigee velocity of transfer orbit • 3: Determine perigee delta V • 4: Calculate apogee velocity of transfer orbit • 5: Calculate circular velocity of final orbit • 6: Determine apogee delta V • 7: Determine total delta V
7! MAE 5540 - Propulsion Systems! Hohmann Transfer: Calculating " the ΔV’s!
8! MAE 5540 - Propulsion Systems! Hohmann Transfer: Calculating " the ΔV’s!
9! MAE 5540 - Propulsion Systems! Hohmann Transfer: Calculating " the ΔV’s!
10! MAE 5540 - Propulsion Systems! Hohmann Transfer: Calculating " the ΔV’s!
11! MAE 5540 - Propulsion Systems! 12! MAE 5540 - Propulsion Systems! Hohmann Transfer:" Total ΔV!
-1!
13! MAE 5540 - Propulsion Systems! Hohmann Transfer Problem … Solved!!
14! MAE 5540 - Propulsion Systems! 15! MAE 5540 - Propulsion Systems! Example:" ΔV required for Hohmann Transfer from LEO to GEO!
16! MAE 5540 - Propulsion Systems! Example:" ΔV required for Hohmann Transfer from LEO to GEO (cont’d)!
• Compute Orbit ratio!
• Compute Normalized ΔV!
17! MAE 5540 - Propulsion Systems! Example:" ΔV required for Hohmann Transfer from LEO to GEO (cont’d)!
• Compute Initial Orbit Velocity!
• Compute Required ΔV!
18! MAE 5540 - Propulsion Systems! Example:" ΔV required for Hohmann Transfer from LEO to GEO (cont’d)!
19! MAE 5540 - Propulsion Systems! Orbital Plane Changes (1)! • Once Launch Systems burns out … .. And payload is placed ! in orbit … … ignoring the small effects of … drag and ! gravity perturbations … your orbit inclination is fixed unless ! You … add energy to the orbit!
20! MAE 5540 - Propulsion Systems! Orbital Plane Changes (2)! • Launch Puts you into a fixed orbit inclination! • What does it cost (ΔV) to change orbit planes?!
• You can only change planes! When the planes at your ! orbits cross! 21! MAE 5540 - Propulsion Systems! Simple Plane Change!
22! MAE 5540 - Propulsion Systems! Simple Plane Change (cont’d)!
23! MAE 5540 - Propulsion Systems! Simple Plane Change (cont’d)!
24! MAE 5540 - Propulsion Systems! Simple Plane Change (cont’d)!
• ΔV can be expressed in polar form also!
25! MAE 5540 - Propulsion Systems! Simple Plane Change (cont’d)!
• Evaluate magnitude of ΔV!
26! MAE 5540 - Propulsion Systems! Simple Plane Change (cont’d)!
Why no Vr component?!
27! MAE 5540 - Propulsion Systems! Simple Plane Change (cont’d)! • Define “Flight path angle”!
28! MAE 5540 - Propulsion Systems! Simple Plane Change (cont’d)! • Define “Flight path angle”!
29! MAE 5540 - Propulsion Systems! Simple Plane Change (cont’d)!
• In-Plane Velocity Vector!
perifocal! 30! MAE 5540 - Propulsion Systems! Simple Plane Change (cont’d)!
Circular orbit!
31! MAE 5540 - Propulsion Systems! 32! MAE 5540 - Propulsion Systems! Example Geo-synchronous" Transfer"
iii) Hohmann transfer! To geo-synchronous! orbit!
Calculate required ΔV!
33! MAE 5540 - Propulsion Systems! Orbital Speed!
34! MAE 5540 - Propulsion Systems! Gravitational Specific Energy!
• Equivalent specific energy required to lift unit ! mass to Orbital altitude !
35! MAE 5540 - Propulsion Systems! Vboost Due to Earth’s Rotation!
36! MAE 5540 - Propulsion Systems! Plane Change ΔV!
37! MAE 5540 - Propulsion Systems! Total Delta V required to Reach Equatorial Leo Orbit from KSC!
than what is required just to obtain orbit!!
38! MAE 5540 - Propulsion Systems! How About Transfer to Geo-Stationary Orbit!
39! MAE 5540 - Propulsion Systems! Hohmann transfer!
40! MAE 5540 - Propulsion Systems! KSC Launch example: Option 1! • 28.5° North Latitude Launch Minimum Total Delta V to GEO
ΔVtotal = ΔVlaunch + ΔV Plane + ΔV1 + ΔV2 = Change hohmann hohmann
m m 7455.6 + 3846.1 + 2467.0 + 1481.4 sec = 15250.0 sec
• THAT'S almost twice orbital velocity ... How do we ever get there?
41! MAE 5540 - Propulsion Systems! OK, lets try another approach!
“where V is smaller”!
42! MAE 5540 - Propulsion Systems! Option 2!
43! MAE 5540 - Propulsion Systems! Vfinal ΔI ΔVplane change
Vinitial
44! MAE 5540 - Propulsion Systems! 45! MAE 5540 - Propulsion Systems! Plane Change at Apogee!
46! MAE 5540 - Propulsion Systems! KSC Launch Option 2!
47! MAE 5540 - Propulsion Systems! KSC Launch Option 2!
• Better …. But can we still do better than this?!
48! MAE 5540 - Propulsion Systems! Combined Plane Change!
49! MAE 5540 - Propulsion Systems! Combined Plane Change (cont’d)!
Angle between vectors! 50! MAE 5540 - Propulsion Systems! Combined Plane Change (cont’d)!
V1 V 2 Vr
γ2 Vν γ 1 1 Δi V ν2
51! MAE 5540 - Propulsion Systems! Combined Plane Change (cont’d)!
52! MAE 5540 - Propulsion Systems! Combined Plane Change (cont’d)!
53! MAE 5540 - Propulsion Systems! Option 3!
54! MAE 5540 - Propulsion Systems! 55! MAE 5540 - Propulsion Systems! 56! MAE 5540 - Propulsion Systems! Option 3 (concluded)!
57! MAE 5540 - Propulsion Systems! 58! MAE 5540 - Propulsion Systems! Option 4!
59! MAE 5540 - Propulsion Systems! 60! MAE 5540 - Propulsion Systems! 61! MAE 5540 - Propulsion Systems! 62! MAE 5540 - Propulsion Systems! 63! MAE 5540 - Propulsion Systems! 64! MAE 5540 - Propulsion Systems! Cost to get to GEO from Equatorial Launch!
65! MAE 5540 - Propulsion Systems! Cost to get to GEO from Equatorial Launch!
• That’s the bottom Line … get a satellite to GEO-stationary! Orbit from KSC costs you >3% ΔV Compared to! Equatorial LAUNCH!
66! MAE 5540 - Propulsion Systems! 67! MAE 5540 - Propulsion Systems!