ASEN 5050 SPACEFLIGHT DYNAMICS Orbit Transfers Prof. Jeffrey S. Parker University of Colorado – Boulder Lecture 10: Orbit Transfers 1 Announcements • Homework #4 is due Friday 9/26 at 9:00 am – You’ll have to turn in your code for this one. – Again, write this code yourself, but you can use other code to validate it. • Concept Quiz #8 is active after this lecture; due before Wednesday’s lecture. • Mid-term Exam will be handed out Friday, 10/17 and will be due Wed 10/22. (CAETE 10/29) – Take-home. Open book, open notes. – Once you start the exam you have to be finished within 24 hours. – It should take 2-3 hours. • Today’s office hours are at 2:00. • Reading: Chapter 6 (SIX, we jumped a few) Lecture 10: Orbit Transfers 2 Space News • Sunday: MAVEN arrived at Mars! Lecture 10: Orbit Transfers 3 Space News Today: Cassini is flying by Titan for the 106th time. 1400 km altitude, 5.6 km/s Vp Lecture 10: Orbit Transfers 4 Space News • Then Tuesday: MOM arrives at Mars! • MOI: Tuesday at 20:00 Mountain – It will enter occultation at 20:04 – MOI will end at 20:24 – We’ll know if it’s successful around 20:30 – Notice that I write “Tuesday” here. It’ll be Wednesday in India and that keeps throwing me off Aw, time conversions! – Not sure if there will be media coverage. Try http://www.spaceflightnow.com/ or NASA TV. Lecture 10: Orbit Transfers 5 ASEN 5050 SPACEFLIGHT DYNAMICS Orbital Maneuvers Prof. Jeffrey S. Parker University of Colorado - Boulder Lecture 10: Orbit Transfers 6 Orbital Maneuvers Hohmann Transfer – Walter Hohmann (1880-1945) showed minimum energy transfer between two orbits used two tangential burns. Lecture 10: Orbit Transfers 7 Hohmann Transfer atrans rinitial + rfinal = 2 v initial = v final = m rinitial m rfinal Dv a = v transA - v initial Dv total = Dv a + Dv b t trans 3 Ptrans atrans = =p 2 m v transA = v transB = 2m rinitial 2m rfinal - m atrans m atrans Dv b = v final - v transB (Algorithm 36, Example 6-1) Can also be done using elliptical orbits, but must start at apogee or perigee to be a minimum energy transfer. Lecture 10: Orbit Transfers 8 Hohmann Transfer • We just argued that the Hohmann Transfer is (usually) the most energy-efficient orbital transfer. • Why? – Consider Elliptical—Elliptical transfer – Tangential Burns – Energy efficiency considerations 2 V m E= 2 R dE =V Þ dV Lecture 10: Orbit Transfers DE » V × DV V is highest at perigee, thus energy-changing maneuvers are the most efficient at perigee! 9 Energy Changes Lecture 10: Orbit Transfers 10 Hohmann Transfer • Example: LEO to GEO: • LEO: altitude 185 km, radius 6563.136 km • GEO: altitude 35,786 km, radius 42,164 km • VLEO = 7.7932 km/s • VpT = 10.2521 km/s • ΔV1 = 2.4590 km/s VGEO = 3.0747 km/s VaT = 1.5958 km/s ΔV2 = 1.4788 km/s • Total ΔV = 3.9378 km/s Lecture 10: Orbit Transfers 11 Hohmann Transfer GEO Moon Radius Lecture 10: Orbit Transfers 12 Hohmann Transfer GEO Moon Radius Lecture 10: Orbit Transfers General radii transfers 13 Orbital Maneuvers Bi-elliptic Transfer – Uses two Hohmann transfers. Can save Dv in some cases. rb must be greater than rfinal, but can otherwise be optimized. Lecture 10: Orbit Transfers 14 Bi-elliptic Transfer • Equations you need: SIMPLE, because all maneuvers are tangential, co-planar. Lecture 10: Orbit Transfers 15 Bi-elliptic Transfer atrans 1 rinitial + rb = 2 v initial = v trans 1 B = v trans 2 C = atrans 2 m v trans 1 A = rinitial 2m rb - 2m rfinal m v trans 2 B = atrans 1 - m v final = atrans 2 Dv a = v trans 1 A - v initial Dv c = v final - v trans 2 C t trans = p 3 atrans 1 m rb + rfinal = 2 +p 2m rinitial 2m rb - - m atrans 1 m atrans 2 m rfinal Dv b = v trans 2 B - v trans 1 B Dv total = Dv a + Dv b + Dv c 3 atrans 2 (Algorithm 37, Example 6-2) m Much longer flight times for bi-elliptic transfer, but sometimes less energy. Lecture 10: Orbit Transfers 16 Bi-elliptic Transfer • LEO – GEO via 100,000 km altitude ΔV • • • • ΔV1 = 2.903 km/s ΔV2 = 0.799 km/s ΔV3 = 0.605 km/s Total ΔV: 4.307 km/s – More than Hohmann! Lecture 10: Orbit Transfers 17 Bi-elliptic LEO-GEO Moon Radius Lecture 10: Orbit Transfers 18 Bi-elliptic LEO-GEO Hohmann Moon Radius Lecture 10: Orbit Transfers 19 Bi-elliptic Transfer • LEO – 250,000 km via 2.4 million km altitude ΔV • • • • ΔV1 = 3.192 km/s ΔV2 = 0.329 km/s ΔV3 = 0.327 km/s Total ΔV: 3.849 km/s – More than Hohmann (4.058 km/s)! Lecture 10: Orbit Transfers 20 Bi-elliptic 185 km – 250,000 km Hohmann Moon Radius Lecture 10: Orbit Transfers 21 Hohmann vs Bi-elliptic Lecture 10: Orbit Transfers 22 One-Tangent Burns Lecture 10: Orbit Transfers 23 Orbit Transfer Comparison Lecture 10: Orbit Transfers 24 Changing Orbital Elements • • • • • • Δa Hohmann Transfer Δe Hohmann Transfer Δi Plane Change ΔΩ Plane Change Δω Coplanar Transfer Δν Phasing/Rendezvous Lecture 10: Orbit Transfers 25 Changing Inclination • Δi Plane Change • Inclination-Only Change vs. Free Inclination Change Lecture 10: Orbit Transfers 26 Changing Inclination • Let’s start with circular orbits Vf V0 Lecture 10: Orbit Transfers 27 Changing Inclination • Let’s start with circular orbits Vf V0 Lecture 10: Orbit Transfers 28 Changing Inclination • Let’s start with circular orbits Are these vectors the same length? Vf Δi V0 What’s the ΔV? Is this more expensive in a low orbit or a high orbit? Lecture 10: Orbit Transfers 29 Changing Inclination • More general inclination-only maneuvers Where do you perform the maneuver? How do V0 and Vf compare? What about the FPA? Line of Nodes Lecture 10: Orbit Transfers 30 Changing Inclination • More general inclination-only maneuvers Lecture 10: Orbit Transfers 31 Changing The Node Lecture 10: Orbit Transfers 32 Changing The Node Where is the maneuver located? Neither the max latitude nor at any normal feature of the orbit! There are somewhat long expressions for how to find uinitial and ufinal in the book for circular orbits. Lambert’s Problem gives easier solutions. Lecture 10: Orbit Transfers 33 Changing Argument of Perigee Lecture 10: Orbit Transfers 34 Changing Argument of Perigee Lecture 10: Orbit Transfers 35 Changing Argument of Perigee Which ΔV is cheaper? Lecture 10: Orbit Transfers 36 Circular Rendezvous (coplanar) Target spacecraft; interceptor spacecraft Lecture 8: Orbital Maneuvers 37 Circular Rendezvous (coplanar) Lecture 8: Orbital Maneuvers 38 How do we build these? • Determine your phase angle, φ • Determine how long you want to spend performing the transfer – How many revolutions? wtgt = m 3 atgt =n t phase = Dt = t1 - t0 = • Build the transfer • Compute the ΔV Lecture 8: Orbital Maneuvers K tgt ( 2p ) + J 2 ö1/3 æ æ t phase ö ÷ ç ÷ a phase = m çç ç è K int ( 2p ) ÷ø ÷ è ø rp = 2a phase - ra wtgt Ktgt = 1, 2, 3,... K int = 1, 2, 3,... (must be > rÅ ) 39 How do we build these? • Compute the ΔV wtgt = m 3 atgt =n t phase = Dt = t1 - t0 = K tgt ( 2p ) + J wtgt 2 ö1/3 æ æ t phase ö ÷ ç ÷ a phase = m çç ç è K int ( 2p ) ÷ø ÷ è ø rp = 2a phase - ra Ktgt = 1, 2, 3,... K int = 1, 2, 3,... (must be > rÅ ) First Dv = v phase - v int Total Dv = 2 v phase - v int = 2 Lecture 8: Orbital Maneuvers 2m m m atgt a phase atgt 40 Example 6-8 This should be +20° Lecture 8: Orbital Maneuvers 41 Example 6-8 Should be positive This should really be an absolute value (one maneuver is in-track, one is anti-velocity) This should really be an absolute value (one maneuver is in-track, one is anti-velocity) Lecture 8: Orbital Maneuvers 42 Conclusions • Better to use as many revolutions as possible to save fuel. • Trade-off is transfer duration • If you perform the transfer quickly, be sure to check your periapse altitude. Lecture 8: Orbital Maneuvers 43 Circular Coplanar Rendezvous (Different Orbits) Lecture 8: Orbital Maneuvers 44 Circular Coplanar Rendezvous (Different Orbits) Use Hohmann Transfer t phase = p 3 atrans a L = wtgtt trans m phase angle q = a L - p π – αL The “wait time”, or time until the interceptor and target are in the correct positions: Synodic Period: Lecture 8: Orbital Maneuvers 45 Example 6-9 Lecture 8: Orbital Maneuvers 46 Example 6-9 I think this should be pi – alpha, not alpha – pi (see Fig 6-17) Lecture 8: Orbital Maneuvers 47 Announcements • Homework #4 is due Friday 9/26 at 9:00 am – You’ll have to turn in your code for this one. – Again, write this code yourself, but you can use other code to validate it. • Concept Quiz #8 is active after this lecture; due before Wednesday’s lecture. • Mid-term Exam will be handed out Friday, 10/17 and will be due Wed 10/22. (CAETE 10/29) – Take-home. Open book, open notes. – Once you start the exam you have to be finished within 24 hours. – It should take 2-3 hours. • Today’s office hours are at 2:00. • Reading: Chapter 6 (SIX, we jumped a few) Lecture 10: Orbit Transfers 48
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