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Laboratory Mission 7: Ballistic Missiles
Mission Objective
- Gain a better understanding of ballistic missile trajectories
- Exercise your ability to apply relevant ballistic trajectory equations
- Create an STK scenario to evaluate your answers
- Manually adjust ballistic trajectory initial conditions in order to hit a target location
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 using the computer lab)
Read Appendix E of Understanding Space
Mission Planning
UNDERSTANDING OF BALLISTIC MISSILE TRAJECTORIES
1.
GEOMETERY OF A BALLISTIC TRAJECTORY: Having read Appendix E of your text,
Understanding Space, you should be acquainted with the six initial conditions of a ballistic
trajectory (RBurnout, VBurnout, ΦBurnout, βBurnout, LBurnout, l Burnout). It is not clear from the text that these
conditions exist at the point where the missile rocket motor shut off—after a period of thrusting and
maneuvering through the atmosphere. In fact, the text concerns itself only with the second phase
of an ICBM “flight.” The three phases are the boost, mid-course, and terminal or re-entry phase
(Figure 1 illustrates all three). We’ve already studied launch velocities and sub phases of the
boost phase in Chapter 9.3 of Understanding Space. Because your text makes the assumption
that the boost and the terminal phases are negligible to the flight path or trajectory, you are not
given this full picture. So the equations you are dealing with assume that     0 and    .
At this level of study, this is appropriate.
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Figure 1: Ballistic Trajectory Geometry
2.
BALLISTIC MISSILE DEFENSE: The US Missile Defense Agency (http://www.mda.mil) is
responsible for developing the US Ballistic Missile Defense System. The first ballistic missile
interceptor was installed in Fort Greely, Alaska, in July 2004. As of September, 2005, there are
nine interceptors deployed in Alaska and Vandenberg AFB, California, providing mid-course
intercept coverage for all 50 states.
This country has struggled with the development of a missile defense for a quarter of a century—
beginning with the Space Defense Initiative (SDI) in the 80s, through the Ballistic Missile Defense
Organization (BMDO) in the 90s, and now with the Missile Defense Agency (MDA). While a
ballistic missile (short range, medium range, or intercontinental) is easiest to detect in the boost
phase because of the high temperatures generated by the motors, the boost phase is so short that
detection leading to interception is problematic. Likewise, in the terminal phase, the re-entry
vehicle is shrouded in high temperature plasma and is easily observable. This phase is even
shorter than the boost phase and aerodynamic forces complicate interception. Conversely, in the
mid-course phase, the vehicle is very cold and harder to detect or observe without prior knowledge
of the phase’s initial conditions. With a reliable detection in the boost phase, we can use the
burnout parameters to model the ballistic missile’s trajectory--a predictable orbit. Once we’ve
calculated where the missile will be, we can launch an interceptor before re-entry and neutralize
the threat in space.
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APPLY RELEVANT BALLISTIC TRAJECTORY EQUATIONS
3.
CALCULATE THE RANGE ANGLE () FOR THE FOLLOWING SCENARIO: Imagine you
are the Chief of Safety for a test launch of an Intercontinental Ballistic Missile (ICBM). The launch
site will be Vandenberg Air Force Base (AFB) just north of Los Angeles (Lo=35°N, lo=121°W). The
target is an island in the South Pacific Ocean (Figure 2)—Yap Island (Lt=10°N, lt=140°E).
Figure 2: Yap Island Targeted
a) The contractor planning the test assures you that the test vehicle, which can achieve a
burnout velocity (VBurnout) of 7200 m/s, will hit the target based on their calculated initial
conditions without error. However, as the Chief of Safety, you must verify their numbers.
Determine what the Range Angle (), Range in km, and Burnout Azimuth (Burnout). Recall
from Appendix E the equation is:
Cos  SinL0 SinLt  CosL0CosLt Cosl
  _____________ or ____________
b) Remember these two answers correspond to the short and the long way around the Earth.
Using the shortest path, find the Range and possible launch azimuths.
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Range  
10,000(km)
900
Range  ___________________
Cos 
km
SinLt  SinL0Cos
CosL0 Sin
  _____________ or ____________
c) One of these azimuths is correct for the short path, the other for the long path around the
Earth.
4.
DRAW THE TRAJECTORY’S GROUND TRACK: Using the Polar Map given in Figure 3,
draw the ground track for the short path from VAFB to Yap Island. Assume the boost phase
places the missile directly above the launch site and the re-entry occurs directly above the target.
Annotate the both the Range Angle (  , centered at the pole) and launch azimuth (  , centered
on your launch site) on your drawing.
Based on your figure, do you expect the launch azimuth to be;
a) 0<  <90
b) 90<  <180
c) 180<  <270
d) 270<  <360?
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Figure 3: Polar Map
5.
DETERMINE THE LAUNCH ELEVATION AND MAXIMUM RANGE: The test vehicle
should obtain a burnout velocity of 7200 m/s at 180 km altitude (RBurnout = 6558 km). What would
the flight path angle (Burnout) be? Use the method provided in Appendix E. Watch your units!
QBurnout 
2
VBurnout
RBurnout


QBurnout 
NOTES
__________________
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 2  QBurnout 
1
  
 sin    
Burnout_ LOW   sin 1 


2
 QBurnout 
2
2
 Burnout_ LOW  __________________
 2  QBurnout 
1
  
 sin    
Burnout_ HIGH  180  sin 1 
2 
 QBurnout 
2
2 
Burnout_ HIGH  __________________
Which Burnout would you choose and why?
Launch elevation is the angle you point the rocket at launch as measured from the local
horizontal to the velocity vector. Assuming the angle you launch the rocket is the same
on the pad as it is at burnout, what would your launch elevation angle be?
What would the maximum range be for this ICBM and the elevation angle needed to
achieve max range?
 QBurnout 
 
 Maximum  2 Sin 1 
2

Q
Burnout  

 Maximum  _______________
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 
( Burnout ) Max _ Range  45 0   max  
 4 
(Burnout) Max _ Range  _______________
6.
DETERMINE THE MISSILE TIME OF FLIGHT: To determine the time of flight for our
ballistic trajectory, the period of a circular orbit (assuming Q=1) with a radius equal to Rburnout must
first be calculated. Then the TOF/Pcircular versus Range Angle plot (Figure 4 below or Figure E-6 in
Appendix E of Understanding Space) can be used to determine the flight path angles.
Pcircular  2
3
Rburnout


Using Figure 4 and the values of Q and range angle,  that you calculated in
Mission Planning, find
TOF
 ____________ or ____________.
Pcircular
 TOF 
 Pcircular  _______________(Low trajectory)
TOF  
P
circular


or ________________(High Trajectory)
From Figure 4, estimate burnout_Low and
NOTES
burnout_High.
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Figure 4: TOF/Pcircular versus Range Angle
7.
COLLECT OUR DATA FOR THE LAUNCH: Complete the Table 1 with the data you
calculated.
Table 1: Ballistic Trajectory Parameters
PARAMETER
LBurnout
lBurnout
LT (Target Latitude)
lT (Target Longitude)
RBurnout
VBurnout
Burnout_ LOW
DATA
STK ENTRY
Launch Latitude
Launch Longitude
----Launch Radius
--Launch Elevation
 Burnout_ HIGH
Launch Elevation
βBurnout
Launch Azimuth
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VELOCITY AT BURNOUT VECTOR: From your skills review, you found that a vector
of magnitude R can be decomposed into its components using the following equations. Find

VBurnout for the low trajectory and the high trajectory.

VBurnout  VBurnout
Vburnout,S  VBurnoutcos  cos 
Vburnout,E  VBurnoutcos  sin 
Vburnout,Z  VBurnoutsin 

VBurnout  VBurnoutcos  cos(  )Ŝ  VBurnoutcos  sin(  ) Eˆ  VBurnoutsin  Ẑ

VBurnout_ LowŜ  VBurnoutcos bo,low cos(  )Ŝ 

VBurnout_ Low Eˆ VBurnoutcos bo,low sin(  ) Eˆ 

VBurnout_ Low Ẑ  VBurnoutsin bo,low Ẑ 

VBurnout_ Low  __________________ Ŝ +____________________ Ê +_____________________ Ẑ

VBurnout_ HighŜ  VBurnoutcos bo,high cos(  )Ŝ 
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
VBurnout_ HighEˆ VBurnoutcos bo,high sin(  ) Eˆ 

VBurnout_ HighẐ  VBurnoutsin bo,high Ẑ 

VBurnout_ High  __________________ Ŝ +____________________ Ê +_____________________ Ẑ
9.
From the K: drive, (K:\Campus\DF\DFAS\A310\STK Data Files) copy the Mission 7 folder
to your hard drive. This folder contains the scenarios you will need in class for Mission Execution.
Mission Execution
During this laboratory mission, you will create a launch facility, a target zone, and the ICBM to match
the planning scenario. After configuring the scenario, evaluate your calculated answers from mission
planning.
CREATE AN STK SCENARIO TO EVALUATE YOUR ANSWERS
10.
Open the saved scenario file Ballistic_Targeting.sc from the Mission 7 folder that you
saved in mission planning. This will have two 2-D windows. One is an Equidistant Cylindrical
projection and the other is a polar plot.
11.
Create a facility object named “VAFB” located at Lo=35°N, lo=121° W. NOTE: negative values
are necessary for west longitudes.
a) Check the 2-D map to ensure it was placed where you wanted it to.
12.
Create a Target object named “Yap_Island” using the Object Browser (
locate it at Lt=10°N, lt=140°E.
a) Check the 2-D map to ensure the Target was placed where you wanted it.
NOTES
) and
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13.
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Create a Missile object that takes the high trajectory to the target. Remember that we
assume the burnout location is directly above the launch site.
a) Open the missile’s Basic/Trajectory property page (Figure 5). Enter the missile’s burnout
latitude (Launch Latitude), burnout longitude (Launch Longitude) and burnout altitude
(Rburnout) (Figure 5). NOTE: negative values are necessary for west longitudes.
b) Change the Launch Latitude – Geodetic box to read Launch Latitude – Geocentric
c) Change the Impact Latitude - Geodetic box to read Launch Elevation (Figure 5). This will
generate an error window. Ignore the error and click OK, then set the box to Launch
Elevation again.
d) Enter your VBurnout_ High in Fixed Delta V field
35°
Change to Geocentric
-121°
Change to Launch
Elevation
RBurnout km
Figure 5: Missile Trajectory
e) Enter your
f)
14.
 Burnout_ High
as your launch elevation and βBurnout as your launch azimuth in
the appropriate fields
Click OK.
Evaluate your scenario
a) Observe the animated scenario in both of the 2-D windows. You may have to zoom
into the area around Yap Island in the cylindrical projection using
.
b) Observe the scenario in a 3-D window. You’ll have to open a new one.
Did you hit your target? Explain.
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c) Remember that the Earth is spinning eastward. From Chapter 9.3 we know that the

km ˆ
velocity of the launch site is VLaunchSite  .4651cos( L0 )
E . Calculate the launch site
sec

velocity and subtract that value from VBurnout_ High to overcome the spin of the Earth to get
the vector Delta V.

VBurnout_ High  _______________ Ŝ +_______________ Ê +_______________ Ẑ

VLaunchSite  _0_ Ŝ +_______________ Ê +_0_ Ẑ

V  _______________ Ŝ +_______________ Ê +_______________ Ẑ
d) Now calculate the magnitude of the Delta V required to get to the given VBurnout
V 
V   V   V 
2
Sˆ
2
Eˆ
2
Zˆ

e) Put this value into the Fixed Delta V
Did you hit your target this time?
15.

Manually adjust your launch azimuth (βBurnout) and Fixed Delta V ( VBurnout ) until you do hit the
target and record your new initial conditions that caused a direct hit on Yap Island in Table 2.
Table 2: Initial Conditions to Hit Yap Island High Trajectory
INITIAL CONDITION
RBurnout
VBurnout
HIGH TRAJECTORY
Burnout
βBurnout
LBurnout
l Burnout
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16.
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If you have time, try repeating steps 13 through 15 using the low trajectory to the target
Table 3: Initial Conditions to Hit Yap Island Low Trajectory
INITIAL CONDITION
RBurnout
VBurnout
LOW TRAJECTORY
Burnout
βBurnout
LBurnout
l Burnout
17.
Close the Ballistic_Targeting.sc scenario.
NOTES
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