asen 5050 spaceflight dynamics orbit transfers prof. jeffrey s. parker university of colorado –...
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ASEN 5050SPACEFLIGHT 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 106th time. 1400 km altitude, 5.6 km/s Vp
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 5050SPACEFLIGHT DYNAMICS
Orbital Maneuvers
Prof. Jeffrey S. Parker
University of Colorado - Boulder
Lecture 10: Orbit Transfers 6
Lecture 10: Orbit Transfers7
Orbital ManeuversHohmann Transfer – Walter Hohmann (1880-1945) showed
minimum energy transfer between two orbits used two tangential burns.
Lecture 10: Orbit Transfers8
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 6563.136 km• GEO: altitude 35,786 km, radius 42,164 km
• VLEO = 7.7932 km/s VGEO = 3.0747 km/s• Vp
T = 10.2521 km/s VaT = 1.5958 km/s
• ΔV1 = 2.4590 km/s ΔV2 = 1.4788 km/s
• Total ΔV = 3.9378 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 Transfers14
Orbital ManeuversBi-elliptic Transfer – Uses two Hohmann transfers. Can save v
in some cases. rb must be greater than rfinal, 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 Transfers16
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 = 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
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 = 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
Lecture 10: Orbit Transfers 21
Moon Radius
Hohmann
Lecture 10: Orbit Transfers22
Hohmann vs Bi-elliptic
Lecture 10: Orbit Transfers23
One-Tangent Burns
Lecture 10: Orbit Transfers24
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
V0
Vf
Changing Inclination
• Let’s start with circular orbits
Lecture 10: Orbit Transfers 28
V0
Vf
Changing Inclination
• Let’s start with circular orbits
Lecture 10: Orbit Transfers 29
V0
Vf
Δi
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 V0 and Vf 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 uinitial and ufinal 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 Maneuvers37
Circular Rendezvous (coplanar)
Target spacecraft; interceptor spacecraft
Lecture 8: Orbital Maneuvers38
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 ΔVLecture 8: Orbital Maneuvers 39
How do we build these?
• Compute the ΔV
Lecture 8: Orbital Maneuvers 40
Lecture 8: Orbital Maneuvers41
Example 6-8
This should be +20°
Lecture 8: Orbital Maneuvers42
Example 6-8
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)
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 Maneuvers44
Circular Coplanar Rendezvous (Different Orbits)
Lecture 8: Orbital Maneuvers45
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 – αL
Lecture 8: Orbital Maneuvers46
Example 6-9
Lecture 8: Orbital Maneuvers47
Example 6-9
I think this should be pi – alpha, not alpha – pi (see Fig 6-17)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
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