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 Lecture 10: Orbit Transfers 4 Today: Cassini is flying by Titan for the 106 th time km altitude, 5.6 km/s V p

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 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

7 Orbital Maneuvers Hohmann Transfer – Walter Hohmann ( ) showed minimum energy transfer between two orbits used two tangential burns.

Lecture 10: Orbit Transfers 8 Hohmann Transfer Can also be done using elliptical orbits, but must start at apogee or perigee to be a minimum energy transfer. (Algorithm 36, Example 6-1)

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 Lecture 10: Orbit Transfers 9 V is highest at perigee, thus energy-changing maneuvers are the most efficient at perigee!

Energy Changes Lecture 10: Orbit Transfers 10

Hohmann Transfer Example: LEO to GEO: LEO: altitude 185 km, radius km GEO: altitude 35,786 km, radius 42,164 km V LEO = km/sV GEO = km/s V p T = km/sV a T = km/s ΔV 1 = km/sΔV 2 = km/s Total ΔV = km/s Lecture 10: Orbit Transfers 11

Hohmann Transfer Lecture 10: Orbit Transfers 12 GEO Moon Radius

Hohmann Transfer Lecture 10: Orbit Transfers 13 GEO Moon Radius General radii transfers

Lecture 10: Orbit Transfers 14 Orbital Maneuvers Bi-elliptic Transfer – Uses two Hohmann transfers. Can save  v in some cases. r b must be greater than r final, but can otherwise be optimized.

Bi-elliptic Transfer Equations you need: Lecture 10: Orbit Transfers 15 SIMPLE, because all maneuvers are tangential, co-planar.

Lecture 10: Orbit Transfers 16 Bi-elliptic Transfer Much longer flight times for bi-elliptic transfer, but sometimes less energy. (Algorithm 37, Example 6-2)

Bi-elliptic Transfer LEO – GEO via 100,000 km altitude ΔV ΔV1 = km/s ΔV2 = km/s ΔV3 = km/s Total ΔV: km/s –More than Hohmann! Lecture 10: Orbit Transfers 17

Bi-elliptic LEO-GEO Lecture 10: Orbit Transfers 18 Moon Radius

Bi-elliptic LEO-GEO Lecture 10: Orbit Transfers 19 Moon Radius Hohmann

Bi-elliptic Transfer LEO – 250,000 km via 2.4 million km altitude ΔV ΔV1 = km/s ΔV2 = km/s ΔV3 = km/s Total ΔV: km/s –More than Hohmann (4.058 km/s)! Lecture 10: Orbit Transfers 20

Bi-elliptic 185 km – 250,000 km Lecture 10: Orbit Transfers 21 Moon Radius Hohmann

Lecture 10: Orbit Transfers 22 Hohmann vs Bi-elliptic

Lecture 10: Orbit Transfers 23 One-Tangent Burns

Lecture 10: Orbit Transfers 24 Orbit Transfer Comparison

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 Lecture 10: Orbit Transfers 27 V0V0 VfVf

Changing Inclination Let’s start with circular orbits Lecture 10: Orbit Transfers 28 V0V0 VfVf

Changing Inclination Let’s start with circular orbits Lecture 10: Orbit Transfers 29 V0V0 VfVf ΔiΔi Are these vectors the same length? What’s the ΔV? Is this more expensive in a low orbit or a high orbit?

Changing Inclination More general inclination-only maneuvers Lecture 10: Orbit Transfers 30 Line of Nodes Where do you perform the maneuver? How do V 0 and V f compare? What about the FPA?

Changing Inclination More general inclination-only maneuvers Lecture 10: Orbit Transfers 31

Changing The Node Lecture 10: Orbit Transfers 32

Changing The Node Lecture 10: Orbit Transfers 33 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 u initial and u final in the book for circular orbits. Lambert’s Problem gives easier solutions.

Changing Argument of Perigee Lecture 10: Orbit Transfers 34

Changing Argument of Perigee Lecture 10: Orbit Transfers 35

Changing Argument of Perigee Lecture 10: Orbit Transfers 36 Which ΔV is cheaper?

Lecture 8: Orbital Maneuvers 37 Circular Rendezvous (coplanar) Target spacecraft; interceptor spacecraft

Lecture 8: Orbital Maneuvers 38 Circular Rendezvous (coplanar)

How do we build these? Determine your phase angle, φ Determine how long you want to spend performing the transfer –How many revolutions? Build the transfer Compute the ΔV Lecture 8: Orbital Maneuvers 39

How do we build these? Compute the ΔV Lecture 8: Orbital Maneuvers 40

Lecture 8: Orbital Maneuvers 41 Example 6-8 This should be +20°

Lecture 8: Orbital Maneuvers 42 Example 6-8 This should really be an absolute value (one maneuver is in-track, one is anti-velocity) Should be positive

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

Lecture 8: Orbital Maneuvers 44 Circular Coplanar Rendezvous (Different Orbits)

Lecture 8: Orbital Maneuvers 45 Circular Coplanar Rendezvous (Different Orbits) Use Hohmann Transfer The “wait time”, or time until the interceptor and target are in the correct positions: Synodic Period: π – α L

Lecture 8: Orbital Maneuvers 46 Example 6-9

Lecture 8: Orbital Maneuvers 47 Example 6-9 I think this should be pi – alpha, not alpha – pi (see Fig 6-17)

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