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Applications of the Vis-Viva Equation: the Hohmann Transfer!

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

Feasible

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 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 )

• 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 • 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 ! 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 !

34! MAE 5540 - Propulsion Systems! Gravitational Specific Energy!

• Equivalent specific energy required to lift unit ! to Orbital altitude !

35! MAE 5540 - Propulsion Systems! Vboost Due to ’s !

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 to GEO-stationary! Orbit from KSC costs you >3% ΔV Compared to! Equatorial LAUNCH!

66! MAE 5540 - Propulsion Systems! 67! MAE 5540 - Propulsion Systems!