ASEN 5050
SPACEFLIGHT DYNAMICS
Orbit Transfers
Prof. Jeffrey S. Parker
University of Colorado – Boulder
Lecture 11: 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 #9 is active after this lecture; due before
Friday’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.
• Reading: Chapter 6
Lecture 11: Orbit Transfers
2
Space News
• Last night: MOM arrived at Mars!
Lecture 11: Orbit Transfers
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Space News
• SpaceX’s Dragon berthed with the ISS
Lecture 11: Orbit Transfers
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Quiz #8
Lecture 8: Orbital Maneuvers
5
Quiz #8
Lecture 8: Orbital Maneuvers
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Quiz #8
Lecture 8: Orbital Maneuvers
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Quiz #8
Lecture 8: Orbital Maneuvers
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ASEN 5050
SPACEFLIGHT DYNAMICS
Orbital Maneuvers
Prof. Jeffrey S. Parker
University of Colorado - Boulder
Lecture 11: Orbit Transfers
9
Changing Orbital Elements
•
•
•
•
•
•
Δa  Hohmann Transfer
Δe  Hohmann Transfer
Δi  Plane Change
ΔΩ  Plane Change
Δω  Coplanar Transfer
Δν  Phasing/Rendezvous
Lecture 11: Orbit Transfers
10
Circular Rendezvous (coplanar)
Target spacecraft; interceptor spacecraft
Lecture 11: Orbit Transfers
11
Circular Rendezvous (coplanar)
Lecture 11: Orbit Transfers
12
How do we build these?
Lecture 11: Orbit Transfers
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How do we build these?
Lecture 11: Orbit Transfers
14
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 11: Orbit Transfers
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 )
15
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 11: Orbit Transfers
2m
m
m
atgt a phase
atgt
16
Example 6-8
Lecture 11: Orbit Transfers
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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 11: Orbit Transfers
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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 11: Orbit Transfers
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Circular Coplanar Rendezvous (Different Orbits)
Lecture 11: Orbit Transfers
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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:
- +
Lecture 11: Orbit Transfers
Synodic Period:
21
Example Circular Coplanar Rendezvous
• Build me a transfer from one circular equatorial orbit
to another.
• Orbit 1: radius = 15,000 km, longitude = 10 deg
• Orbit 2: radius = 30,000 km, longitude = 45 deg
Lecture 11: Orbit Transfers
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Example Circular Coplanar Rendezvous
• Step 1: draw a picture.
?
x
Orbit 1
Lecture 11: Orbit Transfers
Orbit 2
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Example Circular Coplanar Rendezvous
• Step 2: Hohmann.
?
x
Orbit 1
Lecture 11: Orbit Transfers
Orbit 2
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Example Circular Coplanar Rendezvous
• Step 3: Phasing.
How far will
the target move
during the
transfer?
x
Orbit 1
Lecture 11: Orbit Transfers
Orbit 2
25
Example Circular Coplanar Rendezvous
α
• Step 3: Phasing.
x
Orbit 1
Lecture 11: Orbit Transfers
Orbit 2
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Example Circular Coplanar Rendezvous
• Step 3: Phasing.
The target will
advance 116.9
deg during
the transfer.
x
Orbit 1
Lecture 11: Orbit Transfers
Orbit 2
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Example Circular Coplanar Rendezvous
• Step 3: Phasing.
The vehicles
start 35 deg
apart.
ϑ
They need to
be 63.1 deg
apart
(180-116.9 deg)
Lecture 11: Orbit Transfers
x
Orbit 1
Orbit 2
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Example Circular Coplanar Rendezvous
• Step 3: Phasing.
They need to
be 63.1 deg
apart.
x
Orbit 1
Lecture 11: Orbit Transfers
Orbit 2
29
Example Circular Coplanar Rendezvous
• Step 3: Phasing.
They need to
be 63.1 deg
apart.
Orbit 2
Orbit 1
x
Lecture 11: Orbit Transfers
30
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:
- +
Lecture 11: Orbit Transfers
Synodic Period:
31
Example 6-9
Lecture 11: Orbit Transfers
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Example 6-9
I think this should be pi – alpha,
not alpha – pi (see Fig 6-17)
Lecture 11: Orbit Transfers
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Circular Non-Coplanar Phasing
Lecture 8: Orbital Maneuvers
34
Circular Non-Coplanar Phasing
Requires proper nodal alignment as well as proper phasing.
Because of the long wait times, an intermediate phasing orbit
is usually used to set up the proper phasing
3
ainitial + a final
atrans
atrans =
t trans = p
2
m
a L = wtgtt trans
Must determine time to reach node:
Dq int
(360 - n)
wint
(Movement of target
ltrue 1 = ltrue 0 + wtgt Dt node during Dt)
Dt node =
Lecture 8: Orbital Maneuvers
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Circular Non-Coplanar Phasing
Lecture 8: Orbital Maneuvers
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ASEN 5050
SPACEFLIGHT DYNAMICS
Launch
Prof. Jeffrey S. Parker
University of Colorado - Boulder
Lecture 11: Orbit Transfers
37
Launch
Launching a satellite:
For a direct launch, the launch site latitude must be less than or
equal to the desired inclination, otherwise we must change the
inclination of the orbit.
Lecture 8: Orbital Maneuvers
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Right Spherical Triangle
Lecture 8: Orbital Maneuvers
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Launch
We can show this using spherical trigonometry for a
right spherical triangle (eq. C-23):
i = inclination
cos i = cos f gc sin b
Thus, sinb =
cos i
cos f gc
f gc = launch site latitude
b = launch azimuth
Because |sin b | ≤ 1, the launch latitude fgc ≤ i
Cannot direct launch into orbit with inclination < fgc
Another useful relation: sin fgc = sin(i) sin(w+n)
Lecture 8: Orbital Maneuvers
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Launch
The launch site velocity is:
v L = wÅ ´ rsite or v L = wÅ ´ rsite = wÅ rsite cos fgc
Note all the velocity is Eastward in the SEZ system, so launching
from the equator on a 90 azimuth may be best.
The velocity at the equator is vL = 0.465 km/s. Westward
launches must make this up, so difference is 0.93 km/s.
Lecture 8: Orbital Maneuvers
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Launch Azimuths
Launch sites and allowable azimuths
Lecture 8: Orbital Maneuvers
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Launch Sites
Lecture 8: Orbital Maneuvers
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Noncoplanar Transfers
Lecture 8: Orbital Maneuvers
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Noncoplanar Transfers
Launch window – select UT to achieve orbit’s desired
initial nodal location (determine qgst)
First determine launch azimuth b (inverse sine gives
two possible answers: b and 180-b, for ascending (90 < u < 90) and descending (90 < u < 270)
passes.)
cos i
sinb =
cos f gc
cos b
Now, determine the auxiliary angle from: coslu =
sin i
The values lu and 360-lu represent prograde and
retrograde orbits respectively.
Lecture 8: Orbital Maneuvers
lu = q LST - W but, q LST = q GST + l
q GST = W + lu - l
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Noncoplanar Transfers
Tolerance on ascending node (±DW) creates “launch window”,
or range of values of qGST. Once qGST is chosen:
UT =
q GST - q GST 0
wÅ
Substitution of qGST0 for each day (GST at 0 hrs on that day)
gives the launch time on each day.
Dv is more complicated.
Lecture 8: Orbital Maneuvers
46
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 #9 is active after this lecture; due before
Friday’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.
• Reading: Chapter 6
Lecture 11: Orbit Transfers
47