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Laboratory Mission 3: Orbital Maneuvers
Mission Objective
- Reinforce your understanding of Hohmann Transfers
- Reinforce your understanding of Plane Changes
- Reinforce your understanding of Combined Plane Changes
Resources/Requirements
For this laboratory mission, you must have:
Successfully installed STK and borrowed a license for a period covering the class in which this
mission is to be executed (unless working in the computer lab)
Read Chapter 6 of the Understanding Space textbook
Mission Planning
1.
HOHMANN TRANSFER: The Air Force is studying the effects of charged particles on the
Global Positioning System (GPS) satellites. The Joint Program Office (JPO) has asked the 1st
Space Operations Squadron (1SOPS) at Schriever AFB Colorado to change the orbit of an
obsolete NAVSTAR GPS satellite so the space shuttle can rendezvous with it and bring it back to
Earth for analysis. The COEs for this NAVSTAR GPS satellite are given in Table 1.
Table 1: Classical Orbital Elements for the NAVSTAR GPS Satellite
Semi-major axis, a
26562 km
Eccentricity, e
0
Inclination, i
28.5°
RAAN, 
0°
The Space Shuttle can achieve a height of 450 km (a=6828 km) at 28.5 degrees inclination. Orbital
Analysts in the 1SOPS will use a Hohmann transfer to maneuver the GPS spacecraft down to the
shuttle.
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Case 0: Calculate the initial and final Vs for this Hohmann transfer to 5 decimal places.
Keep in mind that the GPS satellite in MEO (at an orbital radius of R1) is transferring
to the smaller Shuttle orbit in LEO (at an orbital radius of R2).
t  
V1 

2at

R1


( R1  R2 )
V2 


R2



Vt1  2   t  
 R1



Vt 2  2   t  
 R2

VHT1  Vt1  V1 
VHT 2  V2  Vt 2 
VHT1 = _____________


km
sec
VHT 2 = _____________
km
sec
Record these values in the appropriate locations in Table 2 on page 3-8.
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Figure 1: STK scenario of the Hohmann transfer between the GPS orbit
(a=26,562 km) and the Shuttle orbit (a=6,828 km).
2.
SIMPLE PLANE CHANGE: After further consideration, the 1SOPS informs the NAVSTAR
SPO that the space vehicle they want to maneuver doesn’t have enough propellant on-board to
perform the desired operation. However, another satellite does. Its COEs are the same as the
first NAVSTAR GPS except for its inclination which is 55 degrees. Orbital Analysts are
prepared to perform three Delta Vs (a simple plane change maneuver and a Hohmann transfer) in
order to rendezvous with the Space Shuttle. There are two ways to do this. In the first case, the
plane change is performed before the Hohmann. In the second case, the Hohmann is performed
before the plane change (reference page 206 in Understanding Space).

a) Case 1 (Completed for you) Calculate the Delta V Simple: VS  2Vi Sin ( ) .
2

The initial velocity, Vi, is the velocity of the GPS satellite in its original orbit (a=26,562 km).
Therefore,
 55  28.5 

3.986 x105 km3 / s 2
 
 
 

VS1  2Vi Sin   2V1 sin    2
sin    2
sin 
R1
26562km
2
2
2
2


VS1  1.77576 km/sec

To be useful in STK, it is necessary to calculate the vector components of VS that you
calculated above. Figure 2 illustrates the body mounted coordinate system used by STK
to describe the direction of thrust. Calculate the vector components of the Delta V Simple
using the following equation

km
VS1  (V f cos   Vi ) xˆ  (V f sin  ) yˆ 
sec
Note: you will not be responsible for remembering the above equation.
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Figure 2: Body Mounted Coordinate System

km
VS1  ((3.8738 cos 26.5)  3.8738) xˆ  (3.8738 sin  26.5) yˆ  0 zˆ 
sec
Note: Since you are calculating the actual vector for the VS burn, the angle for the
plane change is important. At the ascending node, this scenario has a negative plane change
angle since the orbit inclination is decreasing. If the plane change occurs at the descending
node, the plane change angle is positive.
VS1  (-0.40700 x̂ -1.72849 ŷ + 0 ẑ )

km
sec
Figure 3 shows the simple plane change maneuver.
Figure 3: Making a Simple Plane Change
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b) Case 2 (Do on your own) Calculate the Delta V required to get the NAVSTAR satellite into the
proper inclination after the Hohmann Transfer (you calculated these Delta Vs in step 1).
 
VS 2  2Vi Sin  =
2
VS 2  _________________

km
sec
Calculate the vector components of the Delta V Simple using the following equation

km
VS 2  (V f cos  Vi ) xˆ  (V f sin  ) yˆ 
sec
Note: In the STK scenario, this burn is designed to occur at the ascending node, so the
plane change angle is negative for this case.
VS 2  (_____________ x̂ +_______________ ŷ + 0 ẑ )

3.
km
sec
Record these values in the appropriate locations in Table 2 on page 3-8.
COMBINED PLANE CHANGE: The SPO instructs the 1SOPS team to reduce the orbital
maneuver propellant requirement by performing a combined plane change in conjunction with a
Hohmann Transfer.
a) Case 3 (Completed for you): In this case, the first Delta V burn is used to place the NAVSTAR
GPS satellite into an elliptical transfer without changing its inclination (The first part of the
Hohmann transfer calculated in step 1). The second Delta V burn occurs at perigee and will
circularize the orbit plus change its inclination. The Delta V Combined equation is
VC 2  Vi  V f  2ViV f Cos( )
2

2
The initial velocity for this maneuver, Vi, is the spacecraft velocity in the transfer orbit just
before the ΔVC2 maneuver or Vt2. The final velocity, Vf, is the spacecraft velocity just after
the ΔVC2 maneuver or V2.
VC 2  Vi  V f  2ViV f Cos( )
2
NOTES
2
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 

km
Vi  Vt 2  2   t  =__9.63738__
sec
 R2

V f  V2 

R2
 __7.64050__
km
sec
VC 2  (9.63738) 2  (7.64050) 2  2(9.63738)(7.63738) cos( 26.5  )
km
VC 2 = 4.41140
sec

This is the magnitude of the Delta V. In order to use this in STK, it must be broken into x̂ ,
ŷ , and ẑ components.
Note: In the STK scenario, this burn is designed to occur at the descending node, so
the plane change angle is positive for this case.

km
km
VC 2  (V f cos   Vi ) xˆ  (V f sin  ) yˆ 
= (-2.79963 x̂ + 3.40910 ŷ + 0 ẑ )
sec
sec
b) Case 4 (Do on your own): In this case, determine the first Delta V burn to change the orbit’s
inclination and put the vehicle into a transfer orbit (ΔVC1 ). Design the second Delta V burn to
occur at perigee of the transfer orbit. Once the second burn is complete, the NAVSTAR GPS
satellite should be in the Shuttle’s orbit.
VC1  Vi  V f  2ViV f Cos( ) =
2
2
Note: For this case the values of Vi and Vf are different from those in Case 3.
Vi 
Vi  ____________________
km
sec
Vf 
V f  ____________________
NOTES
km
sec
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VC1  Vi  V f  2ViV f Cos( )
2
2
VC1 ___________

km
sec
Now calculate the x̂ , ŷ , and ẑ components to use in the STK scenario.

VC1  ((V f cos   Vi ) xˆ  (V f sin  ) yˆ )
Note: In the STK scenario, this burn is designed to occur at the ascending node, so the
plane change angle is negative for this case.
VC1 (_____________________ x̂ +_____________________ ŷ + 0 ẑ )
4.

km
sec

Record these values in the appropriate locations in Table 2 on page 3-8.

Figure 4 Shows the orbit after the combined plane change burn, but before the second
Hohmann transfer burn.
From the K: drive, copy the Mission 3 folder to your hard drive
This folder contains the scenarios you will need in class for Mission Execution
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Figure 4: After a Combined Plane Change Burn
Table 2: Maneuver Options Summarized
Hohmann Transfer
Case 0:
VHT1 =
__________
km
sec
Simple Plane Change & Hohmann
Case 1:
Case 2

VS1 
VHT1 =
- 0.40700 x̂
__________
Combined Plane Change & ½ Hohmann
Case 3
Case 4

VC1 
VHT1 =
km
sec
__________
km
sec
- 1.72848 ŷ
+ 0 ẑ
VHT 2 =
__________
+ __________ ŷ
km
sec
+ 0 ẑ
VHT1 =
km
sec
___________
___________ x̂

VC 2 
VHT 2 =
km
sec
__________
km
sec
km
sec
VHT 2 =
- 2.79963 x̂
__________
km
sec
+ 3.40910 ŷ
+ 0 ẑ
Note: These are
magnitudes. You’ll
need to add direction
to make them
vectors in order to
use them in STK.
NOTES

VS 2 
VHT 2 =
___________
km
sec
km
sec
___________ x̂
+ __________ ŷ
+ 0 ẑ
km
sec
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Mission Execution
Having calculated the Delta Vs required to maneuver the NAVSTAR GPS satellite down to the
Shuttle’s orbit, you must verify that your calculations are correct using STK.
REINFORCE YOUR UNDERSTANDING OF HOHMANN TRANSFERS
(CASE 1)
Using the scenarios you downloaded from the K: drive and your hand calculations, analyze the
Hohmann Transfer.
1.
Open the saved scenario file Hohmann_Transfer.sc from your Mission 3/Hohmann
Transfer folder that you saved in mission planning
2.
Double-Click on the NAVSTAR_GPS Satellite icon to open the Properties Browser and go
to the Basic/Orbit page.
a) STK uses an associated program called “Astrogator” to propagate an orbit through time.
Luckily you are not required to master Astrogator in this mission. The scenario has already
been created with Astrogator enabled. In this Mission, you simply have to adjust the inputs to
the Astrogator engine to get your satellite to the desired orbit.
Figure 3: NAVSTAR_GPS Orbit
b) Select the Initial Delta V segment of the maneuver
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c) Insert the value for ΔVHT1 found in Case 0 of Table 2 into the X Velocity field (Figure 4).
d) Click Apply and look at the 3-D window.
Figure 4:Hohmann Transfer Initial Delta V
e) Select the Final Delta V segment of the maneuver.
f) Insert the value for ΔVHT2 found in Case 0 of Table 2 into the X Velocity field.
g) Click Apply and look at the 3-D window.
3.
Run the scenario and note the ground track
Is your final orbit smaller or larger than your initial orbit?
o A Delta V in the direction of velocity (positive) will increase your semimajor axis. A
negative Delta V will decrease your orbit’s semimajor axis.
Is your final orbit 450 km in height?
Is the inclination the same for both orbits?
What is the total Delta V required to perform this maneuver?
VT  VHT1  VHT 2
4.
Switch to the 2-D window, animate the scenario and watch the ground track
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Does the satellite in the final ground track more slowly or more quickly than the satellite in
its initial orbit ground track?
5.
Close the Hohmann_Transfer scenario
REINFORCE YOUR UNDERSTANDING OF SIMPLE PLANE CHANGES
(CASES 1 & 2)
Use STK to verify your simple orbital plane change calculations are correct.
6.
Open the Simple_Plane_Change.sc file from your Mission 3/Simple Plane Change
folder
7.
o
8.
Open the NAVSTAR_GPS Basic/Orbit property page.
Note there are three Delta Vs in this scenario. One for the plane change and two for the
Hohmann Transfer (Figure 5).
Perform a simple plane change followed by the Hohmann Transfer (Case 1)
a) Change the X Velocity and Y Velocity fields of the Initial Delta V using ΔVS from Case 1
of Table 2.
b) Click Apply and look at the 3-D window.
c) Change the X Velocity field of the Intermediate Delta V using ΔVHT1 from Case 1 of
Table 2.
d) Click Apply and look at the 3-D window.
e) Change the X Velocity field of the Final Delta V using ΔVHT2 from Case 1 of Table 2.
f) Click Apply and look at the 3-D window.
Figure 5: Simple Plane Change Segments
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9.
Switch to the 3-D window and animate the scenario
Is your second orbit (red track) correctly inclined?
o A positive Delta V in the ŷ direction will increase your inclination at the ascending node
and decrease the inclination at the descending node.
o If your inclination increased instead of decreasing, change the sign of your ŷ component
of the Delta V.
Does the second burn take place at apogee or perigee of the transfer orbit? Which node?
o
Does your transfer orbit decrease or increase the semimajor axis?
Remember, thrust in the direction of velocity will increase the orbit size
What is the value of your semimajor axis in the final orbit?
What is the total Delta V required to perform this maneuver?
VT  VS  VHT1  VHT 2
10.
Switch to the 2-D window, animate the scenario and watch the ground track
Does the satellite in the final ground track more slowly or more quickly than the satellite in
its initial orbit ground track?
11.
Perform the Hohmann Transfer followed by a Simple Plane Change (CASE 2)
a) Change the X Velocity field of the Initial Delta V using ΔVHT1 from Case 2 of Table 2.
o Remember to zero out the Y Velocity component.
b) Click Apply and look at the 3-D window.
c) Change the X Velocity field of the Intermediate Delta V using ΔVHT2 from Case 2 of
Table 2.
d) Click Apply and look at the 3-D window.
e) Change the X Velocity and Y Velocity fields of the Final Delta V using ΔVS2 from Case 2
of Table 2.
f) Click Apply and look at the 3-D window.
12.
Switch to the 3-D window and animate the scenario
Is your transfer orbit (red track) correctly inclined?
Does the second burn take place at apogee or perigee of the transfer orbit? Which node?
NOTES
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Is your final orbit correctly inclined and have the right semimajor axis?
What is the value of your semimajor axis in the final orbit?
What is the total Delta V required to perform this maneuver?
VT  VHT1  VHT 2  VS 2
Where is the best place to do the simple plane change when going from an outer to an inner
orbit? Before or after the Hohmann? At apogee or perigee? Is the answer the same if you
are going from an inner to outer orbit?
13.
Switch to the 2-D window, animate the scenario and watch the ground track
Does the final ground track repeat more slowly or more quickly than the initial orbit ground
track?
14.
Close the Simple_Plane_Change scenario.
REINFORCE YOUR UNDERSTANDING OF COMBINED PLANE
CHANGES (CASES 3 & 4)
Use STK to verify your combined orbital plane change calculations are correct.
15.
16.
17.
Open the Combined_Plane_Change.sc file from your Mission 3/Combined Plane
Change folder
Open the NAVSTAR_GPS Basic/Orbit property page
o Note there are only two Delta Vs in this scenario. One for the combined plane
change and the other for half of the Hohmann Transfer
Perform a Combined Plane Change followed by the second Hohmann Transfer maneuver
(Case 3)
a) Change the X Velocity and Y Velocity fields of the Initial Delta V using ΔVHT1 from Case 3 of
Table 2.
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b) Click Apply and look at the 3-D window.
c) Change the X Velocity field of the Final Delta V using ΔVC2 from Case 3 of Table 2.
d) Click Apply and look at the 3-D window.
18.
Switch to the 3-D window and animate the scenario
Is your transfer orbit (red track) correctly inclined?
Does the second burn take place at apogee or perigee of the transfer orbit? Which node?
Is your final orbit correctly inclined and have the right semimajor axis?
What is the value of your Semimajor axis in the final orbit?
What is the total Delta V required to perform this maneuver?
VT  VHT1  VC 2
19.
Switch to the 2-D window, animate the scenario and watch the ground track
Does the satellite in the final ground track more slowly or more quickly than the satellite in
its initial orbit ground track?
20. Perform a Combined Plane Change followed by the last Hohmann Transfer maneuver (Case 4)
a) Change the X Velocity and Y Velocity fields of the Initial Delta V using ΔVC1 from Case 4 of
Table 2.
b) Click Apply and look at the 3-D window.
c) Change the X Velocity field of the Final Delta V using ΔVHT2 from Case 4 of Table 2.
 Remember to zero out the Y Velocity component.
d) Click Apply and look at the 3-D window.
21.
Switch to the 3-D window and animate the scenario
Is your transfer orbit (red track) correctly inclined?
Does the second burn take place at apogee or perigee of the transfer orbit? Which node?
NOTES
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Is your final orbit correctly inclined and have the right semimajor axis?
What is the value of your semimajor axis in the final orbit?
What is the total Delta V required to perform this maneuver?
VT  VC1  VHT 2
Where is the best place to do the combined plane change when going from an outer to an
inner orbit? At apogee or perigee? Is the answer the same if you are going from an inner to
outer orbit?
22.
Switch to the 2-D window, animate the scenario and watch the ground track
Does the final ground track repeat more slowly or more quickly than the initial orbit ground
track?
23.
Close the Combined_Plane_Change scenario
NOTES
Last Major Revision by
Capt Sobers on 5 May 2006