MA 4362 Astrodynamics

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MA 4362Astrodynamics
Midterm Exam Open Book, Open Note Show ALL
work Vallado, Chapters 1,2,3, and 6
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First TDRS-IMission Failure
TDRS-I
21 March 2002 TDRS-I Stranded in Useless
Orbit Editor's note Sources tell NASA Watch
that the recently-launched TDRS-I satellite
experienced propulsion (pressurization) problems
during its movement up to GEO. The current
predicament leaves the satellite stranded in a
useless orbit. The satellites current orbit
apogee is 35,800 kilometers (22,236 miles) and
the perigee is approximately 8,000 kilometers
(4,970 miles). The satellite is still in its
launch inclination of 28.5?
RS-68
Delta IV
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First TDRS-IMission Failure (contd)
TDRS-I
Boeing is working diligently on raising the
orbit of the satellite to Geo-stationary
altitude However, the total DV of the
upper-stage apogee-kick-motor is limited to only
2.4 km/sec Preliminary analyses says that the
total DV required To reach an equatorial
geo-synchronous (geo-stationary) Orbit, exceeds
the capacity of the upper-stage motor Is the
Mission doomed? Lets find out
RS-68
Delta IV
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Hohmann Transfer Orbit Parameters
Compute semi-major Axis and Eccentricity of
Initial and Hohmann Transfer orbits Compute
Required DV1, DV2, DVtotal to put satellite Into
Geo-synchronous (28.5?) orbit
m 3.986 x 105 km3/sec2
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Hohmann Transfer Orbit Parameters (contd)
Compute semi-major Axis and Eccentricity of
Initial and Hohmann Transfer orbits
km
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Hohmann Transfer Orbit Parameters (contd)
Compute Required DV1, DV2, DVtotal to put
satellite Into Geo-synchronous (28.5?) orbit
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Plane Change to Geo-Stationary (0?) Orbit
In order for the TDRS-I spacecraft to be
useful, it must be placed into a geo-stationary
(0????????????) orbit As we will derive later
in the class, the required DV for this simple
plane change is Is it cheaper in terms of
fuel to change planes before or after the DV1?,
the DV2 burn? (Consider ONLY simple plane change)
Why? Evaluate the minimum required DV for a
simple plane Change (Di 28.5 ?) can the
satellite be rescued with a Hohmann Transfer?
Ooops! Sorry typo error!
Space craft velocity at time of plane change
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Plane Change to Geo-Stationary (0?) Orbit
Is it cheaper in terms of fuel to change planes
before or after the DV1?, the DV2 burn?
(Consider ONLY simple plane change) Why?
Plane Change after DV1, but before DV2, where
(apogee) velocity of transfer orbit is
slowest Evaluate the minimum required DV for a
simple plane Change (Di 28.5 ?)
DVmax 2.4 km/sec Hohmann Transfer wont
work!
Based on incorrect formula
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Plane Change to Geo-Stationary (0?) Orbit
OK correct answer
DVmax 2.4 km/sec looks like
Hohmann Transfer WILL WORK AFTER ALL
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Bi-Elliptic Lunar-Assist Rescue Mission
Well learn more about This later when we study
Three-body problems
Lets try a different Approach Use
lunar gravity assist to rescue Spacecraft
Perform Plane change In Lunar Vicinity via
gravity assist
Lunar orbit is inclined at -10.3? to earths
equator
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Lunar Assist
Well learn more about This later when we study
Three-body problems
Earth Only
Earth / Moon Combined
Lunar Sphere of Influence
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Bi-Elliptic Orbit Parameters
Do we want to Rendezvous with the moon at
apogee or perigee? Why?
Use lunar perigee, because it requires less
change in orbital energy also gives a bigger
gravity boost
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Bi-Elliptic Orbit Parameters (contd)
Compute aT1 eT1 -gt Lunar trajectory
semi-major
axis , eccentricity
1
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Bi-Elliptic Orbit Parameters (contd)
Compute aT2 eT2 -gtEarth return
trajectory semi-major
axis, eccentricity
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Bi-Elliptic Orbit Parameters (contd)
Compute DV1 -gt Velocity change required to
insert TDRS into Lunar transfer orbit DV2 -gt
Velocity change required by gravity assist to
get onto proper Earth-return trajectory
But DVL2 is free because its via gravity
assist
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Bi-Elliptic Orbit Parameters (contd)
Compute DV3 -gt Velocity change required to
circularize final orbit at GEO altitude
Retrograde!
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Bi-Elliptic Orbit Parameters (contd)
How about Lunar rendezvous at Lunar apogee?
Compute aT1 eT1 -gt Lunar trajectory
semi-major
axis , eccentricity
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Bi-Elliptic Orbit Parameters (contd)
How about Lunar rendezvous at Lunar apogee?
Compute aT2 eT2 -gtEarth return
trajectory semi-major
axis, eccentricity
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Bi-Elliptic Orbit Parameters (contd)
How about Lunar rendezvous at Lunar apogee?
Compute DV1 -gt Velocity change required to
insert TDRS into Lunar transfer orbit
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Bi-Elliptic Orbit Parameters (contd)
How about Lunar rendezvous at Lunar apogee?
Compute DV2 -gt Velocity change required by
gravity assist to get onto proper Earth-return
trajectory
But DVL2 is free because its via gravity
assist
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Bi-Elliptic Orbit Parameters (contd)
How about Lunar rendezvous at Lunar apogee?
Compute DV3 -gt Velocity change required to
circularize final orbit at GEO altitude
Retrograde!
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Lunar Apogee or Perigee Rendezvous?
Lunar Perigee Rendezvous
Perigee wins!
Lunar Apogee Rendezvous
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Plane Change Schematic
Moon
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Plane Change to Geo-Stationary (0?) Orbit
In order for the TDRS-I spacecraft to be
useful, it must be placed into a geo-stationary
(0????????????) orbit What is the minimum DV
required for this plane change? (Consider ONLY
simple plane change)
Ooops! Sorry typo error!
Space craft velocity at time of plane change
DVplane change is free because its via
gravity assist
Lunar orbit is inclined at -10.3? to earths
equator
We can Exploit this -10.3? to give the plane
change but we wont do it here
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Plane Change to Geo-Stationary (0?) Orbit
What is the minimum DV required for this plane
change? (Consider ONLY simple plane change)
Answer 1 assume velocity of transfer orbit 2
apogee behind moon

Based on incorrect formula
km/sec
Correct answer -gt 0.101983 km/sec
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Plane Change to Geo-Stationary (0?) Orbit
What is the minimum DV required for this plane
change? (Consider ONLY simple plane change)
Answer 2 assume velocity of transfer orbit 1
apogee behind moon

Based on incorrect formula
km/sec
Correct answer -gt 0.2245 km/sec
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Plane Change to Geo-Stationary (0?) Orbit
What is the minimum DV required for this plane
change? (Consider ONLY simple plane change)
Answer 3 assume mean velocity behind moon

Based on incorrect formula
km/sec
Correct answer -gt 0.1767 km/sec
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Compare Total DV required for Hohmann Transfer to
DV Required for Bi-EllipticLunar Gravity Assist
Transfer
Assume that DV due to Gravity Assist is
Free .. I.e. dont count it Can the
Satellite be rescued at all?
DVmax 2.4 km/sec
Correct answer
The lunar mission buys some extra DV
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Compare Total DV required for Hohmann Transfer to
DV Required for Bi-EllipticLunar Gravity Assist
Transfer
How long will the rescue take take? (assume
all burns happen impulsively)
Ignore orbit phasing
Period of Trans-Lunar orbit
Period of Earth-return orbit
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Compare Total DV required for Hohmann Transfer to
DV Required for Bi-EllipticLunar Gravity Assist
Transfer
How long will the rescue take take? (assume
all burns happen impulsively)
Total Transfer Time
Acceptable! Hey let go rescue the thing!!
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and this leads us to
Orbits in three Dimensions