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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
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Space News
Today: Cassini is flying by Titan for the 106th time. 1400 km altitude, 5.6 km/s Vp
Lecture 10: Orbit Transfers
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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
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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
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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
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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
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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
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Hohmann Transfer
GEO
Moon
Radius
Lecture 10: Orbit Transfers
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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
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Bi-elliptic Transfer
• Equations you need:
SIMPLE, because all
maneuvers are tangential,
co-planar.
Lecture 10: Orbit Transfers
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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
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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
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Bi-elliptic LEO-GEO
Moon
Radius
Lecture 10: Orbit Transfers
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Bi-elliptic LEO-GEO
Hohmann
Moon
Radius
Lecture 10: Orbit Transfers
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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
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Bi-elliptic 185 km – 250,000 km
Hohmann
Moon
Radius
Lecture 10: Orbit Transfers
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Hohmann vs Bi-elliptic
Lecture 10: Orbit Transfers
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One-Tangent Burns
Lecture 10: Orbit Transfers
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Orbit Transfer Comparison
Lecture 10: Orbit Transfers
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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
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Changing Inclination
• Δi  Plane Change
• Inclination-Only Change vs. Free Inclination Change
Lecture 10: Orbit Transfers
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Changing Inclination
• Let’s start with circular orbits
Vf
V0
Lecture 10: Orbit Transfers
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Changing Inclination
• Let’s start with circular orbits
Vf
V0
Lecture 10: Orbit Transfers
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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
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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
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Changing Inclination
• More general inclination-only maneuvers
Lecture 10: Orbit Transfers
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Changing The Node
Lecture 10: Orbit Transfers
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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
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Changing Argument of Perigee
Lecture 10: Orbit Transfers
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Changing Argument of Perigee
Lecture 10: Orbit Transfers
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Changing Argument of Perigee
Which ΔV is
cheaper?
Lecture 10: Orbit Transfers
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Circular Rendezvous (coplanar)
Target spacecraft; interceptor spacecraft
Lecture 8: Orbital Maneuvers
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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
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Circular Coplanar Rendezvous (Different Orbits)
Lecture 8: Orbital Maneuvers
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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
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Example 6-9
Lecture 8: Orbital Maneuvers
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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