Section 2.4: Applications of the Vis-Viva Equation: The...
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MAE 5540 - Propulsion Systems Section 2.4: Applications of the Vis-Viva Equation: The Hohmann Transfer Feasible Transfer Orbit Not-Feasible Transfer Orbit Rc 1 Rc 2 1
Transcript of Section 2.4: Applications of the Vis-Viva Equation: The...
MAE 5540 - Propulsion Systems!
Section 2.4: Applications of the Vis-Viva Equation: The Hohmann Transfer!
FeasibleTransfer Orbit
Not-FeasibleTransfer Orbit
Rc1
Rc2
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Orbital Energy Review!
= −µ2a33!
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Total Specific Energy (concluded)
• gathering up the terms
rpraVperigeeVapogee
εT = constant
εT = kineticenergy
V2
2
potential energy
- µ r perigee
= kineticenergy
V2
2
potential energy
- µ r anywhere
totalenergy
= - µ2 a
(6)"
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• !
Total …!
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Vis-Viva Equation for All the Conic-Sections!
See Appendix 2.3.2 for Hyperbolic Trajectory Proof!37!
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Excess-energy transfer orbit"
“Wasted”! ΔV!
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Optimal-energy transfer Orbit"Two curves!Are tangential!In space!
Target!Orbit!
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… The Hohmann Transfer!
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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!
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Hohmann Transfer Steps
• 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
• 0 : Calculate transfer orbit semi-major axis & eccentricity
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Hohmann Transfer: Calculating "the ΔV’s!
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Hohmann Transfer: Calculating "the ΔV’s!
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Hohmann Transfer: Calculating "the ΔV’s!
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Hohmann Transfer: Calculating "the ΔV’s!
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Hohmann Transfer:"Total ΔV!
-1!
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Hohmann Transfer Problem … Solved!!
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Example:"ΔV required for Hohmann Transfer from LEO to GEO!
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Example:"ΔV required for Hohmann Transfer from LEO to
GEO (cont’d)!
• Compute Orbit ratio!
• Compute Normalized ΔV!
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Example:"ΔV required for Hohmann Transfer from LEO to
GEO (cont’d)!
• Compute Initial Orbit Velocity!
• Compute Required ΔV!
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Example:"ΔV required for Hohmann Transfer from LEO to GEO
(cont’d)!
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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!
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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!
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Simple Plane Change!
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Simple Plane Change (cont’d)!
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Simple Plane Change (cont’d)!
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Simple Plane Change (cont’d)!
• ΔV can be expressed in polar form also!
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Simple Plane Change (cont’d)!
• Evaluate magnitude of ΔV!
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Simple Plane Change (cont’d)!
Why no Vr component?!
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Simple Plane Change (cont’d)!• Define “Flight path angle”!
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Simple Plane Change (cont’d)!• Define “Flight path angle”!
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Simple Plane Change (cont’d)!
• In-Plane Velocity Vector!
perifocal!
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Simple Plane Change (cont’d)!
Circular orbit!
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Example Geo-synchronous "Transfer"
iii) Hohmann transfer!To geo-synchronous!orbit!
Calculate required ΔV!
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Orbital Speed!
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Gravitational Specific Energy!• Equivalent specific energy required to lift unit !mass to Orbital altitude !
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Vboost Due to Earth’s Rotation!
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Plane Change ΔV!
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Total Delta V required to Reach Equatorial Leo Orbit from KSC!
than what is required just to obtain orbit!!
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How About Transfer to Geo-Stationary Orbit!
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Hohmann transfer!
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• 28.5° North Latitude LaunchMinimum Total Delta V to GEO
ΔVtotal = ΔVlaunch + ΔV Plane
Change + ΔV1
hohmann + ΔV2
hohmann =
7455.6 + 3846.1 + 2467.0 + 1481.4 m
sec = 15250.0 msec
• THAT'S almost twice orbital velocity ... How do we ever get there?
KSC Launch example: Option 1!
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OK, lets try another approach!
“where V is smaller”!42!
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Option 2!
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Vinitial
VfinalΔI ΔVplane change
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Plane Change at Apogee!
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KSC Launch Option 2!
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KSC Launch Option 2!
• Better …. But can we still do better than this?!
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Combined Plane Change!
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Combined Plane Change (cont’d)!
Angle between vectors!50!
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Combined Plane Change (cont’d)!
Vν2
V1V2
Vν1
Vr
Δi
γ1
γ2
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Combined Plane Change (cont’d)!
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Combined Plane Change (cont’d)!
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Option 3!
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Option 3 (concluded)!
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Option 4!
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Cost to get to GEO from Equatorial Launch!
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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!
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