Study of the Ballistic Capabilities of a Tethered Satellite System 
 
by 
 
Jennifer Marie Grant 
 
 
 
 
A thesis submitted to the Graduate Faculty of 
Auburn University 
in partial fulfillment of the 
requirements for the Degree of 
Master of Science 
 
Auburn, Alabama 
August 4, 2012 
 
 
 
 
Copyright 2012 by Jennifer Marie Grant 
 
 
Approved by 
 
David Cicci, Chair, Professor of Aerospace Engineering 
Andrew Sinclair, Associate Professor of Aerospace Engineering 
George Flowers, Professor of Mechanical Engineering and Dean of the Graduate School  
 
 
 
 
 
 
 
 
 
 
ii 
 
 
 
 
 
 
 
Abstract  
 
 
This thesis is an investigation into the ballistic capabilities of a Tethered Satellite System 
(TSS) when the sub-satellite is released from the system. This topic is of particular interest 
because TSS could be potentially used as threats, or weapons.  There is a need to determine the 
velocity change or angular velocity that is required to cause a sub-satellite to enter an impact 
trajectory.  Once the sub-satellite enters an impact trajectory, the ground range covered by the 
new trajectory and the time to impact are determined.  A simple dumbbell model is used to 
represent the TSS in a dynamical simulation.  Changes to the velocity of the system were 
introduced at release point in the orbit in order to cause the sub-satellite to enter an impact 
trajectory toward the Earth after release from the TSS.  The parameters of the TSS that affect the 
impact trajectories are the altitude, tether length, and release point.  A comparison is then done 
for changes in these parameters in order to determine the maximum and minimum ground range 
and time to impact for the various cases studied.  An analytical solution is also developed to 
determine the maximum and minimum ranges when given a range of changes in velocity and to 
find the angular velocity and velocity change necessary for a given a set of initial conditions and 
desired impact trajectories. 
  
 
 
 
 
 
iii 
 
 
 
 
 
 
Acknowledgments 
 
 
 I would like to thank my advisor, Dr. Cicci, for helping me to get through graduate 
school and complete this thesis.  I would also like to thank Dr. Cicci and Dr. Gross for their 
support during the more stressful times of my college education.  Without their support I might 
have given up due to all the ups and downs in life. 
 I would also like to thank my father, Richard K. Grant, for his continued support and his 
encouragement for me to pursue an engineering degree.  I will always remember those late night 
calls where he is trying to help me solve homework problems, telling me that I can complete my 
thesis, and telling me to remember to relax and say a little prayer.  I also appreciate the support 
my step-mother, Patty Caldwell Grant, gave me after she married my father.  
 Finally, I would like to dedicate this thesis to my mother, Maria T. Grant.  She never got 
to see her daughter graduate with a Bachelor?s degree in Aerospace Engineering and go after a 
Master?s Degree in Aerospace Engineering; however, without her I would never have considered 
going for a Masters.  Thanks to her I have followed my dreams and now they have come true. 
 
 
 
 
 
 
 
 
 
 
 
 
iv 
 
 
 
 
 
 
Table of Contents 
 
 
Abstract   ....................................................................................................................................... ii 
Acknowledgments ....................................................................................................................... iii 
List of Tables  ............................................................................................................................. vii 
List of Figures  .............................................................................................................................. x 
Nomenclature  ............................................................................................................................. xv 
1. Introduction   ............................................................................................................................. 1 
 1.1 Historical Background   .............................................................................................. 1 
 1.2 TSS Applications   ...................................................................................................... 5 
 1.3 Previous Research   ..................................................................................................... 7 
 1.4 Problem Description   ............................................................................................... 13 
2. Relevant Theory Used in TSS Analysis  ................................................................................ 16 
   2.1 Tethered Satellite System Dynamics   ...................................................................... 16 
  2.1.1 Equations of Motion   ................................................................................ 21 
 2.2 Attitude Dynamics Using Euler Parameters   ........................................................... 23 
 2.3 Rigid Body Dynamics   ............................................................................................. 26 
 2.4 Orbital Elements of a Trajectory   ............................................................................. 29 
 2.5 Numerical Integration Using a 4th Order Runge Kutta   ........................................... 35 
 2.6 Analytical Approach   ............................................................................................... 37 
3: Model and Problem Formulation   .......................................................................................... 39 
v 
 
 3.1 Problem Set-up of the TSS   ..................................................................................... 40 
 3.2 Changes Applied to the System to Create Impact of Sub-Satellite   ........................ 50 
 3.3 Variations to the Problem Set-up  ............................................................................. 51 
 3.4 Analytical Solutions   ................................................................................................ 53 
             3.4.1 Equations to Find the Range and Time to Impact  .................................... 55 
             3.4.2 Algorithm to Find the Range and Time to Impact   ................................... 59 
             3.4.3 Equations to Find the Velocity Change and Angular Velocity.................. 64 
             3.4.4 Algorithm to Find the Velocity Change and Angular Velocity   ............... 66 
4: Results for Analytical Solutions    .......................................................................................... 68 
 4.1 Analytical Results for the Range and Time to Impact .............................................. 68 
  4.1.1 Case A Results ........................................................................................... 69 
  4.1.2 Case B Results ........................................................................................... 72 
  4.1.3 Case C Results ........................................................................................... 75 
  4.1.4 Case D Results ........................................................................................... 78 
 4.2 Analytical Results for the Impulsive Velocity Change and the Angular Velocity   . 80 
  4.2.1 Ground Range of 1500 km and           ............................................ 81 
  4.2.2 Ground Range of 3000 km and           ............................................ 87 
5: Results for Numerical Integration Simulation    ..................................................................... 95 
 5.1 Results for First Configuration or Case 1  ................................................................ 96 
  5.1.1 Impulsive Velocity Change of 1.0 km/s in the Positive Y - Direction ...... 96 
  5.1.2 Impulsive Velocity Change of 1.0 km/s in the Positive Z - Direction ..... 104 
  5.1.3 Impulsive Velocity Change of 3.0 km/s in the Negative Z - Direction ... 113 
  5.1.4 Comparison between Velocity Changes for First Configuration ............. 121 
vi 
 
 5.2 Results for Second Configuration or Case 2 ........................................................... 122 
  5.2.1 Impulsive Velocity Change of 1.0 km/s in the Positive Z- Direction ...... 123 
  5.2.2 Impulsive Velocity Change of 1.5 km/s in the Negative Z - Direction ... 132 
  5.2.3 Impulsive Velocity Change of 1.0 km/s in the Positive Y - Direction .... 140 
  5.2.4 Comparison between Velocity Changes for Second Configuration ........ 149 
 5.3 Comparison between the Analytical Results and the Simulation Results   ............ 150 
6: Conclusions and Future Work    ........................................................................................... 155 
References   ............................................................................................................................... 157 
Appendix A: Sample Results from Numerical Integration   ..................................................... 161 
Appendix B: Sample Results from Analytical Solutions   ........................................................ 164 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
vii 
 
 
 
 
 
 
List of Tables 
 
 
Table 3.1: Sub-satellite Position Components for Cases A - D .................................................. 54 
Table 4.1: Minimum Impulsive Velocity Change Needed for Case A ....................................... 70 
Table 4.2: Maximum Ground Range and Time to Impact for Case A ....................................... 71 
Table 4.3: Minimum Ground Range and Time to Impact for Case A ........................................ 71 
Table 4.4: Minimum Impulsive Velocity Change Needed for Case B ....................................... 74 
Table 4.5: Maximum Ground Range and Time to Impact for Case B ........................................ 74 
Table 4.6: Minimum Ground Range and Time to Impact for Case B ........................................ 75 
Table 4.7: Minimum Impulsive Velocity Change Needed for Case C ....................................... 77 
Table 4.8: Maximum Ground Range and Time to Impact for Case C ........................................ 77 
Table 4.9: Minimum Ground Range and Time to Impact for Case C ........................................ 77 
Table 4.10: Minimum Impulsive Velocity Change Needed for Case D ..................................... 79 
Table 4.11: Maximum Ground Range and Time to Impact for Case D ..................................... 80 
Table 4.12: Minimum Ground Range and Time to Impact for Case D ...................................... 80 
Table 4.13: Impulsive Velocity Change and Angular Velocity for Case A and B at Three  
                    Altitudes When R = 1500 km and ?imp = 270? ........................................................ 83 
 
Table 4.14: Impulsive Velocity Change and Angular Velocity for Case A and B at Three Tether  
                    Lengths When R = 1500 km and ?imp = 270? ......................................................... 85 
 
Table 4.15: Impulsive Velocity Change and Angular Velocity for Case C and D at Three  
                    Altitudes When R = 1500 km and ?imp = 270? ........................................................ 86 
 
Table 4.16: Impulsive Velocity Change and Angular Velocity for Case C and D at Three Tether  
                    Lengths When R = 1500 km and ?imp = 270? ......................................................... 87 
viii 
 
Table 4.17: Impulsive Velocity Change and Angular Velocity for Case A and B at Three  
                    Altitudes When R = 3000 km and ?imp = 270? ........................................................ 89 
 
Table 4.18: Impulsive Velocity Change and Angular Velocity for Case A and B at Three Tether  
                    Lengths When R = 3000 km and ?imp = 270? ......................................................... 91 
 
Table 4.19: Impulsive Velocity Change and Angular Velocity for Case C and D at Three  
                    Altitudes When R = 3000 km and ?imp = 270? ........................................................ 92 
 
Table 4.20: ImpulsiveVelocity Change and Angular Velocity for Case C and D at Three Tether  
                    Lengths When R = 3000 km and ?imp = 270? ......................................................... 94 
 
Table 5.1: Impulsive Velocity Changes for Case 1 .................................................................... 96 
Table 5.2: Range and Impact Time for          km/s at Three Altitudes for Case 1 ............ 97 
Table 5.3: Range and Impact Time for          km/s at Three Tether Lengths  
                 for Case 1 ................................................................................................................. 101 
 
Table 5.4: Maximum and Minimum Values as       Increases for Case 1 ........................... 104 
Table 5.5: Range and Impact Time for          km/s at Three Altitudes for Case 1 .......... 106 
Table 5.6: Range and Impact for          km/s at Three Tether Lengths for Case 1 .......... 109 
Table 5.7: Maximum and Minimum Values as       Increases for Case 1 ........................... 113 
Table 5.8: Range and Impact Time for           km/s at Three Altitudes Case 1 ............. 114 
Table 5.9: Range and Impact Time for           km/s at Three Tether Lengths  
                   for Case 1 ............................................................................................................... 118 
 
Table 5.10: Maximum and Minimum Values as       Increases for Case 1 ......................... 121 
Table 5.11: Impulsive Velocity Changes for Case 2 ................................................................ 123 
Table 5.12: Range and Impact Time for          km/s at Three Altitudes for Case 2 ........ 125 
Table 5.13: Range and Impact Time for          km/s at Three Tether Lengths  
                   for Case 2 ............................................................................................................... 128 
 
Table 5.14: Maximum and Minimum Values as       Increases for Case 2 ......................... 132 
Table 5.15: Range and Impact Time for           km/s at Three Altitudes for Case 2 ..... 133 
 
ix 
 
Table 5.16: Range and Impact Time for           km/s at Three Tether Lengths  
                   for Case 2 ............................................................................................................... 137 
 
Table 5.17: Maximum and Minimum Values as       Increases for Case 2 ......................... 140 
Table 5.18: Range and Impact Time for          km/s at Three Altitudes for Case 2 ........ 141 
Table 5.19: Range and Impact Time for          km/s at Three Tether Lengths  
                   for Case 2 ............................................................................................................... 145 
 
Table 5.20: Maximum and Minimum Values as       Increases for Case 2 ......................... 148 
Table 5.21: Results for Configuration 1 and Case B with          km/s ............................ 150 
Table 5.22: Comparison of Results for Configuration 1 and Case B with          km/s ... 151 
Table 5.23: Results for Configuration 1 and Case B with          km/s ............................ 151 
Table 5.24: Comparison of Results for Configuration 1 and Case B with          km/s ... 152 
Table 5.25: Results for Configuration 2 and Case A with          km/s ............................ 152 
Table 5.26: Comparison of Results for Configuration 2 and Case A with          km/s ... 152 
Table 5.27: Results for Configuration 2 and Case A with          km/s ............................ 153 
Table 5.28: Comparison of Results for Configuration 2 and Case A with          km/s ... 153 
Table A.1: Simulation Results as Altitude Increases for Case 1 at          km/s .............. 161 
Table A.2: Simulation Results as Tether Length Increases for Case 1 at          km/s ..... 162 
Table A.3: Simulation Results as       Increases for Case 1................................................. 163 
Table B.1: Analytical Results as Altitude Increases for Case A When R = 1500 km and 
                            ........................................................................................................... 164 
 
Table B.2: Analytical Results as Tether Length Increases for Case A When R = 1500 km and 
                            ........................................................................................................... 165 
 
Table B.3: Analytical Results as      Increases for Case A When H = 500 km  
                  and    = 4 km .......................................................................................................... 166 
                   
x 
 
 
 
 
 
List of Figures 
 
 
Figure 2.1: Tethered Satellite System Model  ............................................................................ 17 
Figure 2.2: Geocentric Equatorial Coordinate System  .............................................................. 19 
Figure 2.3: Body Fixed Coordinate System for a Negative ?  ................................................... 21 
Figure 2.4: Orbital Elements of a Trajectory  ............................................................................. 30 
Figure 3.1: Initial Set-up of the TSS for a Negative ?  ............................................................... 40 
Figure 3.2: Analytical Cases  ...................................................................................................... 54 
Figure 4.1: Ground Range vs.       for Case A ....................................................................... 69 
Figure 4.2: Time to Impact vs.       for Case A...................................................................... 70 
Figure 4.3: Ground Range vs.       for Case B ....................................................................... 73 
Figure 4.4: Time to Impact vs.       for Case B ...................................................................... 73 
Figure 4.5: Ground Range vs.       for Case C ....................................................................... 76 
Figure 4.6: Time to Impact vs.       for Case C ...................................................................... 76 
Figure 4.7: Ground Range vs.       for Case D ....................................................................... 78 
Figure 4.8: Time to Impact vs.       for Case D...................................................................... 79 
Figure 4.9: Velocity Change vs. Altitude for Case A and B; R = 1500 km and  
                     ?imp = 240? .............................................................................................................. 82 
 
Figure 4.10: Angular Velocity vs. Altitude for Case A and B; R = 1500 km and 
                     ?imp = 240? .............................................................................................................. 82 
 
Figure 4.11: Velocity Change vs. Tether Length for Case A and B; R = 1500 km and  
                     ?imp = 240? .............................................................................................................. 84 
 
xi 
 
Figure 4.12: Angular Velocity vs. Tether Length for Case A and B; R = 1500 km and 
                     ?imp = 240? .............................................................................................................. 84 
 
Figure 4.13: Velocity Change vs. Altitude for Case C and D; R = 1500 km and 
                     ?imp = 240? .............................................................................................................. 86 
 
Figure 4.14: Velocity Change vs. Tether Length for Case C and D; R = 1500 km and 
                     ?imp = 240? .............................................................................................................. 87 
 
Figure 4.15: Velocity Change vs. Altitude for Case A and B; R = 3000 km and  
                     ?imp = 240? .............................................................................................................. 88 
 
Figure 4.16: Angular Velocity vs. Altitude for Case A and B; R = 3000 km and 
                     ?imp = 240? .............................................................................................................. 88 
 
Figure 4.17: Velocity Change vs. Tether Length for Case A and B; R = 3000 km and  
                     ?imp = 240? .............................................................................................................. 90 
 
Figure 4.18: Angular Velocity vs. Tether Length for Case A and B; R = 3000 km and 
                     ?imp = 240? .............................................................................................................. 90 
 
Figure 4.19: Velocity Change vs. Altitude for Case C and D; R = 3000 km and 
                     ?imp = 240? .............................................................................................................. 92 
 
Figure 4.20: Velocity Change vs. Tether Length for Case C and D; R = 3000 km and 
                     ?imp = 240? .............................................................................................................. 93 
 
Figure 5.1: Ground Range vs. Altitude for          km/s in Case 1 ..................................... 96 
Figure 5.2: Time to Impact vs. Altitude for          km/s in Case 1 .................................... 97 
Figure 5.3: Trajectory at 500 km Altitude for          km/s in Case 1 ................................. 98 
Figure 5.4: Trajectory at 1000 km Altitude for          km/s in Case 1 .............................. 99 
Figure 5.5: Trajectory at 1500 km Altitude for          km/s in Case 1 ............................... 99 
Figure 5.6: Ground Range vs. Tether Length for          km/s in Case 1 .......................... 100 
Figure 5.7: Time to Impact vs. Tether Length for          km/s in Case 1 ........................ 100 
Figure 5.8: Trajectory at 55 km Tether Length for          km/s in Case 1 ....................... 102 
Figure 5.9: Trajectory at 100 km Tether Length for          km/s in Case 1 ..................... 102 
xii 
 
Figure 5.10: Ground Range vs.       for Case 1 .................................................................... 103 
Figure 5.11: Time to Impact vs.       for Case 1 .................................................................. 103 
Figure 5.12: Ground Range vs. Altitude for          km/s in Case 1 ................................. 105 
Figure 5.13: Time to Impact vs. Altitude for          km/s in Case 1 ................................ 105 
Figure 5.14: Trajectory at 500 km Altitude for          km/s in Case 1 ............................. 106 
Figure 5.15: Trajectory at 650 km Altitude for          km/s in Case 1 ............................. 107 
Figure 5.16: Trajectory at 850 km Altitude for          km/s in Case 1 ............................. 107 
Figure 5.17: Ground Range vs. Tether Length for          km/s in Case 1 ........................ 108 
Figure 5.18: Time to Impact vs. Tether Length for          km/s in Case 1 ....................... 109 
Figure 5.19: Trajectory at 55 km Tether Length for          km/s in Case 1 ..................... 110 
Figure 5.20: Trajectory at 100 km Tether Length for          km/s in Case 1 ................... 110 
Figure 5.21: Ground Range vs.       for Case 1 .................................................................... 111 
Figure 5.22: Time to Impact vs.       for Case 1 ................................................................... 111 
Figure 5.23: Lofted Trajectory at          km/s for Case 1 ................................................ 112 
Figure 5.24: Ground Range vs. Altitude for           km/s in Case 1 .............................. 114 
Figure 5.25: Time to Impact vs. Altitude for           km/s in Case 1 ............................. 114 
Figure 5.26: Trajectory at 500 km Altitude for           km/s in Case 1 .......................... 115 
Figure 5.27: Trajectory at 1000 km Altitude for           km/s in Case 1 ........................ 116 
Figure 5.28: Trajectory at 1500 km Altitude for           km/s in Case 1 ........................ 116 
Figure 5.29: Ground Range vs. Tether Length for           km/s in Case 1 ..................... 117 
Figure 5.30: Time to Impact vs. Tether Length for           km/s in Case 1 .................... 117 
Figure 5.31: Trajectory at 55 km Tether Length for           km/s in Case 1 .................. 118 
Figure 5.32: Trajectory at 100 km Tether Length for           km/s in Case 1 ................ 119 
xiii 
 
Figure 5.33: Ground Range vs.       for Case 1 .................................................................... 120 
Figure 5.34: Time to Impact vs.       for Case 1 ................................................................... 120 
Figure 5.35: Ground Range vs. Altitude for          km/s in Case 2 ................................. 124 
Figure 5.36: Time to Impact vs. Altitude for          km/s in Case 2 ................................ 124 
Figure 5.37: Trajectory at 500 km Altitude for          km/s in Case 2 ............................. 127 
Figure 5.38: Trajectory at 650 km Altitude for          km/s in Case 2 ............................. 126 
Figure 5.39: Trajectory at 850 km Altitude for          km/s in Case 2 ............................. 126 
Figure 5.40: Ground Range vs. Tether Length for          km/s in Case 2 ........................ 127 
Figure 5.41: Time to Impact vs. Tether Length for          km/s in Case 2 ....................... 128 
Figure 5.42: Trajectory at 55 km Tether Length for          km/s in Case 2 ..................... 129 
Figure 5.43: Trajectory at 100 km Tether Length for          km/s in Case 2 ................... 129 
Figure 5.44: Ground Range vs.       for Case 2 .................................................................... 130 
Figure 5.45: Time to Impact vs.       for Case 2 ................................................................... 130 
Figure 5.46: Lofted Trajectory at          km/s for Case 2 ................................................ 131 
Figure 5.47: Ground Range vs. Altitude for           km/s in Case 2 .............................. 132 
Figure 5.48: Time to Impact vs. Altitude for           km/s in Case 2 ............................. 133 
Figure 5.49: Trajectory at 500 km Altitude for           km/s in Case 2 .......................... 134 
Figure 5.50: Trajectory at 1000 km Altitude for           km/s in Case 2 ........................ 134 
Figure 5.51: Trajectory at 1300 km Altitude for           km/s in Case 2 ........................ 135 
Figure 5.52: Ground Range vs. Tether Length for           km/s in Case 2 ..................... 136 
Figure 5.53: Time to Impact vs. Tether Length for           km/s in Case 2 .................... 136 
Figure 5.54: Trajectory at 55 km Tether Length for           km/s in Case 2 .................. 137 
Figure 5.55: Trajectory at 100 km Tether Length for           km/s in Case 2 ................ 138 
xiv 
 
Figure 5.56: Ground Range vs.       for Case 2 .................................................................... 139 
Figure 5.57: Time to Impact vs.       for Case 2 ................................................................... 139 
Figure 5.58: Ground Range vs. Altitude for          km/s in Case 2 ................................. 140 
Figure 5.59: Time to Impact vs. Altitude for          km/s in Case 2 ................................ 141 
Figure 5.60: Trajectory at 500 km Altitude for          km/s in Case 2 ............................ 142 
Figure 5.61: Trajectory at 1000 km Altitude for          km/s in Case 2 .......................... 142 
Figure 5.62: Trajectory at 1500 km Altitude for          km/s in Case 2 .......................... 143 
Figure 5.63: Ground Range vs. Tether Length for          km/s in Case 2 ....................... 144 
Figure 5.64: Time to Impact vs. Tether Length for          km/s in Case 2 ...................... 144 
Figure 5.65: Trajectory at 55 km Tether Length for          km/s in Case 2 ..................... 146 
Figure 5.66: Trajectory at 100 km Tether Length for          km/s in Case 2 ................... 146 
Figure 5.67: Ground Range vs.       for Case 2 .................................................................... 147 
Figure 5.68: Time to Impact vs.       for Case 2 .................................................................. 148 
 
 
 
 
 
 
 
 
 
 
 
xv 
 
 
 
 
 
 
Nomenclature 
 
 
ATEx Advanced Tether Experiment    
a Semi-major axis 
   First value in the principal axis vector 
   Second value in the principal axis vector 
   Third value in the principal axis vector 
 ? Acceleration vector, Principal axis 
 ?  Acceleration vector of the sub-satellite 
 ?   Acceleration vector of the center of mass 
C Direction cosine matrix 
    Direction cosine matrix element from the ith row and jth column 
E Energy, Eccentric anomaly 
e Eccentricity 
ex X-component of the eccentricity vector 
ey Y-component of the eccentricity vector 
ez Z-component of the eccentricity vector 
 ? Eccentricity vector 
 ? Force Vector 
  ? Gravitational Force 
  Function for the fourth order Runge Kutta 
xvi 
 
G Universal gravitational constant 
H Altitude  
 ? Angular momentum vector 
h Magnitude of the angular momentum vector 
hi X-component of the angular momentum vector 
hj Y-component of the angular momentum vector 
hk Z-component of the angular momentum vector 
 ? Unit vector for the X axis in the geocentric equatorial coordinate system 
 ?  Unit vector for the X2 axis in the body fixed coordinate system 
i  Inclination 
 ? Unit vector for the Y axis in the geocentric equatorial coordinate system 
 ?  Unit vector for the Y2 axis in the body fixed coordinate system 
 ? Unit vector for the Z axis in the geocentric equatorial coordinate system 
 ?  Unit vector for the Z2 axis in the body fixed coordinate system 
   First step in the fourth order Runge Kutta 
   Second step in the fourth order Runge Kutta 
   Third step in the fourth order Runge Kutta 
   Fourth step in the fourth order Runge Kutta 
   Magnitude of the tether length vector 
 ?  Tether length vector, position of    relative to    
    Magnitude of the position vector of   relative to the center of mass of the system 
 ?   Position vector of   relative to the center of mass of the system 
    Magnitude of the position vector of   relative to the center of mass of the system 
xvii 
 
 ?   Position vector of   relative to the center of mass of the system 
M total mass of a TSS; Mean anomaly 
ME Mass of the Earth 
M2 Secondary mass, usually of a planet in a two-body problem 
   Sub-satellite mass 
   Main satellite mass  
NORAD North American Aerospace Defense Command 
N Magnitude of the node vector 
Nj Y-component of the node vector 
 ? Node vector 
n Mean motion 
    Mean motion of the center of mass 
PMG Plasma Motor Generator    
ProSEDS Propulsive Small Expendable Deployer System   
R Ground range covered by the sub-satellite after release 
   Radius of the Earth 
    Position Magnitude of the center of mass 
     X-component of the position of the center of mass of the TSS 
     Y-component of the position of the center of mass of the TSS 
     Z-component of the position of the center of mass of the TSS 
 ?   Position vector of the center of mass 
 ??   First derivative of the position vector of the center of mass 
 ??   Second derivative of the position vector of the center of mass 
xviii 
 
r Magnitude of the position vector 
rp Position magnitude at perigee 
rx X component of the position vector 
ry Y component of the position vector 
rz Z component of the position vector 
 ? Position vector 
   Position magnitude of the sub-satellite 
    X-component of the position of the sub-satellite 
    Y-component of the position of the sub-satellite 
    Z-component of the position of the sub-satellite 
 ?  Position vector of the sub-satellite 
   Position magnitude of the main satellite 
 ?  Position vector of the main satellite 
SEDS Small Expendable Deployment System    
TiPS Tether Physics and Survivability Experiment 
TSS Tethered Satellite System 
     Time at sub-satellite impact 
        Time to impact from release 
   Final time 
   Initial time 
     Time at sub-satellite release 
V Magnitude of the velocity vector 
 ? Velocity vector 
xix 
 
    Velocity magnitude of the center of mass 
Vx X component of the velocity vector 
Vy Y component of the velocity vector 
Vz Z component of the velocity vector 
 ?   Velocity vector of the center of mass 
     Circular velocity of the center of mass 
     Velocity of the center of mass in the X-direction 
     Velocity of the center of mass in the Y-direction 
     Velocity of the center of mass in the Z-direction 
   Velocity magnitude of the sub-satellite 
 ?  Velocity vector of the sub-satellite 
 ?     Velocity vector of the sub-satellite at release 
       Velocity magnitude of the sub-satellite in the X-direction 
       Velocity magnitude of the sub-satellite in the Y-direction 
       Velocity magnitude of the sub-satellite in the Z-direction 
 ?   ?  Relative velocity of    with respect to the center of mass 
    Circular velocity of the sub-satellite 
   Velocity magnitude of the main satellite 
 ?  Velocity vector of the main satellite 
 ?   ?  Relative velocity of    with respect to the center of mass 
    Circular velocity of the main satellite 
 ? State Matrix 
 ?? First derivative of the state matrix 
xx 
 
X, Y, Z Axes for the geocentric equatorial coordinate system 
X2, Y2, Z2 Axes for the body fixed coordinate system. 
   X-position of the sub-satellite 
 ?  X-acceleration of the sub-satellite 
   Y-position of the sub-satellite 
 ?  Y-acceleration of the sub-satellite 
   Z-position of the sub-satellite 
 ?  Z-acceleration of the sub-satellite 
    X-position of the center of mass 
 ?   X-acceleration of the center of mass 
    Y-position of the center of mass 
 ?   Y-acceleration of the center of mass 
    Z-position of the center of mass 
 ?   Z-acceleration of the center of mass 
   Time step  
      Velocity change of the center of mass in the X-direction 
      Velocity change of the center of mass in the Y-direction 
      Velocity change of the center of mass in the Z-direction 
 ? Angular acceleration vector 
   Fourth Euler parameter 
   First Euler parameter 
   Second Euler parameter 
   Third Euler parameter 
xxi 
 
  ?  First derivative of the fourth Euler parameter 
  ?  First derivative of the first Euler parameter 
  ?  First derivative of the second Euler parameter 
  ?  First derivative of the third Euler parameter 
? Angle between the position and velocity vector 
? True anomaly 
     True anomaly at sub-satellite impact 
     True anomaly at sub-satellite release 
? Gravitational parameter 
? True anomaly 
? Rotation angle, Principal angle 
? Longitude of the ascending node 
? Argument of periapsis 
 ? Angular velocity vector 
   X-component of the Angular Velocity 
   Y-component of the Angular Velocity 
   Z-component of the Angular Velocity 
1 
 
 
 
 
Chapter 1: Introduction 
 This research is a conceptual study to evaluate the possible ballistic capabilities of a 
satellite that is launched from a space based platform.  While any space based platform could be 
used, the platform chosen for this study is a Tethered Satellite System (TSS).  A TSS is used as 
the platform because different release point configurations can be created having different 
velocities.  The sub-satellite of the TSS will be the object that is launched or released from the 
system.  The sub-satellite will be released from the TSS after a velocity change is placed on the 
sub-satellite.  This velocity change could be the result of an impulsive velocity change maneuver 
done by the sub-satellite or it could be the result of an increase in the angular velocity of the 
TSS.  The ballistic capabilities that will be studied are the ground range and time to impact after 
the sub-satellite is released from the system.  Different release point configurations and changes 
in the TSS parameters will be analyzed to determine the options in the available ground ranges 
and times to impact.  This chapter will discuss the history and applications of a TSS, the previous 
work done on TSS, and a more detailed problem description for this study. 
1.1 Historical Background 
 A Tethered Satellite System (TSS) is made up of two satellites connected by a cord or 
tether.  The two end bodies are called the main satellite and the sub-satellite.  The sub-satellite is 
in a lower orbit than the main satellite and is deployed from the main satellite using the tether.  
The main satellite can be a larger satellite than the sub-satellite or even a shuttle-type orbiter.  
The idea of a TSS was first introduced in 1895 by a Russian scientist by the name of Tsiolkoskii, 
2 
 
who presented the concept of a ?space tower? or a space elevator.  Tsiolkoskii?s idea of a space 
tower was to have a system that would be attached to the surface of the Earth with the other end 
being in a geosynchronous orbit about the Earth.  The objective of the space tower was to allow 
an object to travel up the tower and be released once it reached the top.  At the top of the tower 
the object would be released into a geosynchronous orbit.  The idea of a space tether is attributed 
to Tsiolkovskii because of his proposal for the space tower.7   
In 1965 the idea of a TSS was put to the test by NASA during the Gemini XI and Gemini 
XII missions.  In Gemini XI, the sub-satellite was connected to the upper stage and both the sub-
satellite and the upper stage swung around the center of mass of the system and generated an 
artificial gravity.7  The Gemini XII mission was designed to test the gravity-gradient stabilization 
of a TSS.17  Both the Gemini XI and Gemini XII missions verified the assumptions and 
calculations done by scientists thereby, proving that tethered satellites could be useful.  Issac 
Artsutanov, was the first person to propose a nonsynchronous sky hook.  This sky hook was a 
rotation tethered system that would approach Earth?s surface with a zero relative velocity at the 
perigee of the tethered system?s orbit.3   
Despite the missions and the ideas presented in the 1960s the ?Era of TSS? did not begin 
until the 1970s when Mario Grossi and Giuseppe Colombo proposed their idea for a skyhook to 
NASA and the Italian Space Agency.7  Grossi was from the Smithsonian Astrophysical 
Observatory and Colombo was working at the University of Padua in Italy.  The skyhook 
proposal involved a small satellite that could be deployed from a shuttle orbiter along a twenty to 
one hundred kilometer tether.  The small satellite could then help with exploration and data 
collection of the Earth?s upper atmosphere.18  This proposal by Grossi and Colombo generated 
interest in TSS for NASA and the Italian Space Agency.   
3 
 
The interest generated by the proposal by Grossi and Colombo resulted in NASA and the 
Italian Space Agency formally agreeing to collaborate on TSS projects.  While several short 
tethered experiments were conducted on the shuttle during the 1980s, it was not until 1992 that 
the first major joint project between the two agencies was launched.18  The satellite was called 
the TSS-1 and was launched from the Atlantis orbiter.  The TSS-1 was the first test of a TSS in 
space.7  The purpose of the TSS-1 mission was to study the electro-dynamic properties of the 
Earth?s ionosphere.  This required that the tether be made of an electrically conducting 
material.18  However, during the mission the reeling mechanism for the tether and sub-satellite 
malfunctioned.  This caused the tether to stop its deployment.  In other words the tether did not 
deploy from the shuttle to its full length.7  Despite this malfunction the TSS-1 mission proved 
that a satellite could be deployed and controlled.  The mission also showed that a TSS could be 
easily controlled and showed that the system was more stable than predicted.5 
The Small Expendable Deployment System 1 (SEDS-1) and the Plasma Motor Generator 
(PMG) missions were conducted during 1993.  The SEDS-1 successfully demonstrated the 
downward free-reeling deployment of the sub-satellite in a TSS.  The PMG?s mission was to 
investigate the possibility of using an electrically conducting tether to generate power that could 
be used to create thrust for the TSS.  This was the first example of propulsion for a system that 
did not require any propellant. A year later the SEDS-2 was deployed and remained in orbit for 
five days until space debris or a micro-meteorite severed the tether.7  The purpose for deploying 
the SEDS-2 was to study long term tether dynamics.18  Even though the tether was severed, the 
SEDS-2 mission collected useful data. 
In 1995 and 1996 three more missions were launched to further study the dynamics and 
applications of TSS.  These missions were the OEDIPUS-C, the TSS-1R, and the Tether Physics 
4 
 
and Survivability Experiment (TiPS).  The OEDIPUS-C system was made up of two rocket 
payloads and was launched in 1995.  After the system reached the apogee of the TSS orbit, the 
tether was intentionally cut in order to study a post-apogee trajectory.18  The TSS-1R and TiPS 
systems were launched in 1996.  The TSS-1R is named because it is the re-launch of the TSS-1 
mission.  During this launch the tether was not severed and the mission was able to prove that a 
tether could be used to create a high voltage charge as it passes through the Earth?s ionosphere.7  
The TiPS satellite was launched by the Naval Research Laboratory.18  The system consisted of 
two small satellites rather than a shuttle orbiter and satellite combination.  The TiPS satellite 
remained in space for several years and provided enough data to further study tether satellite 
dynamics and orbital motion.7 
The last TSS mission launched in the 1990s was the Advanced Tether Experiment 
(ATEx).  The Naval Research Laboratory launched the ATEx system with a goal to demonstrate 
TSS stability, control, and attitude determination of the two end bodies.  The plan behind the 
mission was to deploy the sub-satellite and tether out to 6 km over the course of three and a half 
days; however, after only eighteen minutes of deployment the ATEx system was jettisoned from 
the main satellite.  The reason behind the jettison of the ATEx was because the tether angle 
sensor detected that an out-of-limits condition had been reached and automatically jettisoned the 
system.  It is believed that the condition was detected due to a thermal expansion of the tether.7,18 
Very few tether missions have been launched in the 21st century.  Some missions were 
proposed but later cancelled.  One successful launch was the PICOSat in 2001.  The PICOSat 
consisted of a small TSS that had a mass less than a kilogram.  In 2002 the Propulsive Small 
Expendable Deployer System (ProSEDS) was proposed.  This system was a propulsive 
experiment that was designed to test the ability of a TSS to draw energy from the magnetic field 
5 
 
of the Earth and then use that energy as an electric thruster to raise or lower the TSS?s orbit.7  
Work was done on the ProSEDS until NASA cancelled the project due to the fact that key 
parameters in the performance of the TSS could not be met.27 
1.2 TSS Applications 
TSS applications can be broken into three categories:  space transportation, electrically 
conducting tethers, and atmospheric exploration. The first application that will be discussed is 
the space transportation category, which focuses on the movement of the satellite system.  A TSS 
can be used to put a satellite into a higher or lower orbit than the original orbit without the use of 
boosters.  The system can also be used to return a discarded payload to the Earth and help during 
the docking process of a shuttle to a space station.  These applications can be done because 
energy and momentum exchange occurs between the main satellite and the sub-satellite during 
the deployment of the sub-satellite.  This same momentum and energy exchange can also help 
with controlling the attitude of the main satellite.18  A TSS could be used to perform aero-
braking when approaching a planet.  Aero-breaking occurs when the sub-satellite is deployed 
into the planet?s atmosphere where atmospheric drag would affect the satellite.  When 
aerodynamic drag is exerted on the sub-satellite, the system will theoretically decrease to a 
capture speed from the approach speed before deployment.  This decrease in the satellite speed is 
caused by the tension in the tether instead of using a propulsive maneuver to slow down the 
satellite.7   
 Since the space transportation application for TSS deals with movement of all or part of 
the system, fuel savings and artificial gravity could be a result.18  The fuel savings are beneficial 
for obvious reasons.  The benefit from artificial gravity may not be as obvious.  Artificial gravity 
is created by the spin rate of a spacecraft rotating about its center of mass.  However, in order for 
6 
 
a shuttle orbiter to generate enough artificial gravity the required spin rate would be too high for 
the crew of the shuttle to withstand.  When a TSS is used to create an artificial gravity, the spin 
rate of the system is at an acceptable level for the crew.7   
The second category of TSS applications is electrically conducting tethers, which allows 
for the transportation of energy.  In this case the energy is in the form of electricity that is 
generated from the Earth?s magnetic field.  The electricity will produce a current along the tether 
which will create a force that can move the TSS into a higher or lower orbit.  In other words the 
force generated by the electric current can be used instead of propellant in a propulsive maneuver 
to change the trajectory of the TSS.  If the tether can conduct electricity, it can also be used as an 
antenna for the system, which can transmit electromagnetic waves in the direction that the tether 
is pointed.18  There are two main benefits to using a conducting tether.  The first one is reduced 
cost.  If the conducting tether can provide a force to move the satellite system, the amount of fuel 
needed for the mission is reduced and the fuel cost will be reduced as well.  The second benefit is 
that there is a reduced risk of a chemical propellant contaminating any external part of the TSS.7  
While fuel will always be needed on the TSS, a conducting tether can reduce the amount the 
system carries; thereby, decreasing the risk and cost of the mission. 
The final category of tether applications is atmospheric exploration.  In atmospheric 
exploration an instrument is deployed into the Earth?s atmosphere from a large spacecraft, like 
the shuttle orbiter.  A TSS is a great system for atmospheric exploration because the tether can be 
extended into areas of the upper atmosphere that are too ?rarified? for the aircraft and too dense 
for shuttle orbiters or other large spacecraft.  One use of tethers for atmospheric exploration is 
the gathering of physical, meteorological, and environmental data over a long course of time.  
The tethers could also be used for surveillance and measurements of the Earth?s magnetic and 
7 
 
electric fields.  Perhaps the most useful application for tethers in atmospheric exploration is the 
lowering of a model into the atmosphere for high-speed and high altitude aircraft.  Wind tunnels 
are normally used to determine the characteristics of an aircraft design; however, high altitudes 
and high speeds are harder to replicate inside a wind tunnel.  Overall, the tether can be used to 
collect data for a variety of experiments in atmospheric exploration.18 
Atmospheric exploration is an area of great interest, especially when it comes to how the 
sun interacts with the Earth?s atmosphere.  The peak area for this interaction occurs anywhere at 
or below a four hundred kilometer altitude above the Earth.  This region contains the maximum 
gradient in electron content and temperature in the Earth?s atmosphere.  The higher areas of this 
segment are hard to reach with an aircraft and difficult to maintain for long periods of time for a 
large spacecraft.  By extending a tether into this region long term data can be collected.  It?s 
important to note that once the scientific community has an understanding on how to use TSS to 
study the Earth?s atmosphere this application can be extended to other planets.7 
1.3 Previous Research 
TSS research has mainly focused on the orbit determination and the identification of TSS 
throughout the years.  Some researchers have studied the dynamics and attitude of tethered 
satellites and there has been some research done that looks at the impact trajectory of the sub-
satellite and the new orbital elements of the trajectory for the sub-satellite once it has been 
released from the system; however, there has not been an effort, to this writer?s knowledge, to 
determine what can cause an impact trajectory and what will lead to a maximum range with the 
best time to impact for the sub-satellite.  That being said this section will still discuss some 
previous research that has helped to formulate the problem and model development for this 
research as well as for research studying impacting trajectories. 
8 
 
 In 1983 Hoots, Roehrich, and Szebehely published a study that focused on TSS analysis 
and the effect of the tether on both satellites.  The authors discussed the effect of the tether on the 
speed of the satellites in the system.  The tether causes the sub-satellite to have a lower velocity 
than its expected circular velocity without the tether and the main satellite actually has a higher 
velocity than its circular velocity without the tether.  Hoots, Roehrich, and Szebehely also 
determined the conditions for impact after the sub-satellite is released from the TSS and define 
how to modify the gravitational constant in order to make the Keplerian acceleration valid while 
the tether remains intact.  The authors also pointed out the problem of detecting a TSS system 
and what it does to detection programs, such as the one used by North American Aerospace 
Defense Command, NORAD.13 
 Several papers have also been published concerning the modeling, orbit determination, 
and quick identification of TSS.  Qualls and Cicci24 published a paper in 2007 that studied the 
modifications of classic preliminary orbit determination methods to distinguish between a 
tethered satellite and an untethered satellite.  It was determined that the Herrick ? Gibbs Method 
and the true anomaly iteration technique provided the most accurate results in determining if the 
object in question was tethered or not.24  Rossi, Cicci and Cochran Jr.25 published a paper in 
2004 that discussed an analytical study of the dynamics of a TSS to determine the conditions for 
periodic motions about the equilibrium state of the system.  It was determined that when only the 
gravitational and oblateness forces were placed on the system that the periodic motion of the 
TSS depended on the physical characteristics of the tether; however, if the aerodynamic drag was 
placed on the system, the drag must be bounded in order to create a periodic motion.25  A paper 
was published by Lovell, Cho, Cochran Jr., and Cicci19 in 2003, that examined the factors that 
caused the bodies of a TSS to behave differently and how each of the factors creates a 
9 
 
discrepancy in the determination of a satellite re-entry.  The authors determined that the tether 
force and the librational motion of the system were key factors that caused a tethered satellite to 
appear to be on a re ? entry course.19 In 2002, Cicci, Cochran Jr, Qualls, and Lovell9 published a 
paper that provided a methodology to perform an identification, orbit determination, and orbit 
prediction of a TSS.  For the identification of a TSS preliminary orbit determination methods 
were used and ridge ? type estimation methods were used for the orbit determination of the TSS.  
The authors used a TSS dynamic model for the long term orbit prediction of the TSS.9  Another 
paper would be published by Cicci, Qualls, and Lovell in 2001.8  This paper discussed the use of 
ridge ? type estimation methods to identify a TSS when a ?small arc? of observational data was 
available.8  The final paper to be discussed by this group of authors was published in 1998 by 
Cho, Cochran, Jr., and Cicci.6  In this paper a perturbed two ? body model was used for the TSS 
during orbit determination.  The orbit determination was done using only the observations from 
one of the satellites in the system.  The characteristics of the perturbed motion on the TSS was 
studied using this data.6 
 Another key author in TSS studies is A. K. Misra.  Misra21 published a paper in 2008 
titled ?Dynamics and Control of Tethered Satellite Systems.?  In this paper Misra studies the 
nonlinear roll and pitch motions on a TSS.  Misra found that the aerodynamic drag produced 
librational motion on the TSS and that electrodynamic forces would affect the pitch and roll of 
the TSS; therefore, the drag and electrodyanmic forces would change the stability of the TSS.21  
Modi, Gilardi, and Misra22 published a paper in 1998 that studied the attitude control of a TSS.  
The authors used an Nth order lagrangian formulation to study the attitude control of space 
platform based TSS and used a Liapunov method for the control of the system.  The Liapunov 
control was found to be effective in stabilizing the TSS.22 
10 
 
 Curtis Hilton Stanley wrote his Master?s thesis at the University of Colorado at Colorado 
Springs about apparent impacting trajectories, identification, and orbit determination of TSS in 
2010.  Stanley defines an apparent trajectory as being the ?trajectory that an end mass would 
follow if not tethered.?  If the apparent trajectory would intersect the Earth, then it becomes an 
apparent impact trajectory. He looked at circular and elliptical orbits with and without librational 
motion to determine the apparent impacting trajectory and used batch filters to determine the 
trajectory of the satellite and identify the sub-satellite as being a part of a TSS.  Stanley?s 
research focused on determining the minimum tether length needed to create an apparent 
impacting trajectory and simulating a ballistic missile trajectory along with the apparent 
impacting trajectory to form a comparison between the two.  Once the comparison was 
determined he could use that knowledge to have a program identify an object as being a ballistic 
missile or a satellite attached to a TSS.  Stanley never looked into an actual impacting trajectory 
and the range that the end mass could obtain once released from a TSS.28 
 In 1994 Naigang, Dun, Yuhua, and Naiming23 discussed the calculation of the orbital 
elements for the sub-satellite after it is released from the TSS.  The purpose behind this 
discussion is the momentum transfer between the two satellites that could place a payload in a 
higher orbit about the Earth.  The orbital elements are calculated from position and velocity 
vectors, which are determined using the pitch angle, roll angle, magnitude of the position, the 
true anomaly, and their derivatives.  The authors then compare the orbital elements before 
release to those after the release of the satellite.  They found that the change in the right 
ascension and the inclination angle of the sub-satellite after release is small because of the small 
roll rate present in their examples.  The authors also note that the biggest change is found in the 
semi-major axis.23 
11 
 
 Nammi Jo Choe and Thomas Alan Lovell did research into the orbit determination and 
detection of TSS and state estimation of TSS, respectively.  Choe looked at a two satellite TSS 
and a three satellite TSS in 2003.  She also developed a basic model for the TSS by assuming a 
massless tether and representing the end satellites as point masses.  An algorithm was developed 
by Choe to detect the TSS systems.  Once the TSS was detected, Choe used an estimation 
process to determine the orbit of the system over the orbital period of the systems.  The process 
developed by Choe allowed for the detection, state estimation, and calculation of tether 
parameters for a two and three body TSS.7  Lovell used a ?batch-type differential corrections 
filtering scheme? or a gradient based algorithm and a genetic algorithm to determine the state 
estimation of a TSS.  He then compared the two techniques in order to determine which one had 
better accuracy, speed, and robustness of the two algorithms.  His research showed that the 
genetic algorithm could give an accurate solution over short periods of data.  Since long term 
filters such as the gradient based method depend on the initial approximation, Lovell suggested 
that a hybrid approach be used.  This hybrid approach would use the genetic algorithm to 
determine the initial approximation and the gradient based method would be used to solve for the 
estimation of the state.18 
 Research addressing TSS dynamics includes the study of perturbed motion, constrained 
dynamics, attitude and control, librational motion, and stability of the TSS.  Sungki Cho5 
analyzed the perturbed motion of TSS and used filters to identify the perturbed motion of the 
satellite system in 1999.  The perturbed motion of one detected satellite could be used to identify 
the satellite as part of a TSS.  Cho used a least squares batch filter in order to identify the satellite 
as a member of a TSS and to determine the motion of the system.5  In 2001 Peter Beda used 
Lagrange?s equations to define the equations of motion for the TSS.  He then put a constraint on 
12 
 
the system to make sure that the tether length remained constant.  The equations of motion and 
the control constraint were then used in a numerical simulation to determine the attitude of a 
TSS.2  Aaron Schutte and Brian Dooley26 developed a control law for the constrained motion of 
a TSS.  This control law or constraint equations could then be used to simplify the modeling and 
simulation of a TSS.26   
An extended rigid body model of a TSS was used by Insu Chang, Sang-Young Park, and 
Kyu-Hong Choi4 in 2010 to study the attitude and control of a TSS.  In order to simulate this, the 
authors used a state-dependent Riccati equation to model the nonlinear attitude control of a TSS.  
A numerical integration was used to determine the stability of the system using the Riccati 
equation.  The TSS was found to be asymptotically stable when using the Riccati equation to 
model the nonlinearities in the system.4  Joshua Ellis and Christopher Hall11 also studied the 
stability of a TSS by looking at the system?s out of plane librations.  Equations of motion were 
developed by Ellis and Hall to model the in-plane and out-of-plane librations of the TSS.  The 
eigenvalues for the librational motions were then used to determine the stable and unstable areas 
of motion for a TSS.11   
In 2008 Kosei Ishimura and Ken Higuchi15 wrote a paper that described the coupling 
between the pitch, axial vibration, and orbital motion of TSS.  The authors looked at the effects 
of these motions on the eigenvalues and eigenvectors.  Ishimura and Higuchi also looked into the 
influence of the mass ratio and natural frequency on the characteristics of the system, specifically 
the eigenvalues.  The natural frequency was found to have the greatest influence on the coupling 
between the pitch, axial vibration, and orbital motion while the mass ratio was found to influence 
the pitch motion and found to affect the value for the eigenvalues and eigenvectors.15  Also in 
2008, Hao Wen, Dongping P. Jin, and Haiyan Y. Hu29 discussed the relative advances in the 
13 
 
dynamics and control of TSS and provided a comprehensive look at all the advances in TSS 
dynamics and control.  Some of the topics discussed are controls for tether deployment and 
retrieval of the sub-satellite, further studies into producing artificial gravity with a tethered 
system, re-entry and rendezvous missions assisted by a tether, and stability of a TSS.  Some of 
the studies discussed by Wen, Jin, and Hu focused on a two satellite tethered system, while 
others looked at three or more satellites tethered together.29   
The majority of the research conducted concerning TSS focused on constraints and 
attitude and control and dynamics of tethered systems.  Only a few studies have discussed impact 
trajectories.  Stanley?s28 focus was on the apparent impacting trajectory of a sub-satellite without 
being cut from the TSS.  He compared this apparent impacting trajectory with a typical ballistic 
missile trajectory.  This information was then used to identify a tethered satellite, when only one 
end of the tethered satellite was detected.  After the sub-satellite was identified as part of a TSS, 
Stanley focused on the orbit determination of a TSS.  Naigang, Dun, Yuhua, and Naiming23 
focused on the changes in the orbital elements after the sub-satellite was released from a TSS 
with librational motion.  All the information found in the research stated above was used to help 
formulate the best model and equations of motion for this study. 
1.4 Problem Description 
The purpose of this study is to determine the possible ballistic capabilities of a satellite 
launched from a space based platform.  For the purposes of this study a TSS was chosen for the 
launching platform because it can be used to release the satellite with different release 
configurations.  The sub-satellite of the TSS will be released from the system after a velocity 
change is placed on the sub-satellite.  There are two methods to create such a velocity change on 
the sub-satellite:  an impulsive velocity change maneuver and an increase in the angular velocity 
14 
 
of the system.  The impulsive velocity change is achieved when the sub-satellite produces a 
thrust in a certain direction.  The resulting velocity change from the thrust maneuver will affect 
the velocity of the sub-satellite and may result in an impact trajectory after the sub-satellite is 
released from the TSS.  For the increase in the angular velocity of the system, the sub-satellite 
and the main satellite will produce a thrust that would cause the TSS to rotate faster.  This 
increase in the rotation of the system will affect the velocity of both satellites and may cause the 
sub-satellite to impact the Earth after it is released from the system.  If an impact does occur, the 
range covered by the sub-satellite?s impacting trajectory and the time it takes for the sub-satellite 
to impact the Earth after launch will be calculated.  Range is used to describe the ground range 
covered by the sub-satellite after it is released from the TSS.  The range and time to impact are 
the ballistic capabilities that will be analyzed in this study. 
Changes in the parameters of the space based platform, or the TSS, are also analyzed to 
determine how they might affect the capabilities of the sub-satellite after release.  The 
parameters that are investigated are the altitude of the main satellite and the tether length because 
both of these parameters will affect the launching position of the sub-satellite relative to the 
Earth.  For each combination of parameters an impulsive velocity change maneuver is placed on 
the sub-satellite and it is released from the system.  If the sub-satellite impacts the Earth, the 
range and time to impact of the impacting trajectory are calculated in order to determine the 
impact of the changing on the ballistic capabilities of the sub-satellite. 
            This research takes a threat analysis point of view for a satellite launched from a space 
based platform.  Using the TSS as the platform, the sub-satellite will be placed in different 
release configurations with various altitudes, tether lengths, and velocity changes placed on the 
sub-satellite before it is released.  For each velocity change evaluated, a corresponding angular 
15 
 
velocity of the TSS is calculated which would produce such a velocity change.  The ballistic 
capabilities for different cases are then compared in order to determine a range of ballistic 
capabilities for each configuration. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
16 
 
 
 
 
Chapter 2: Relevant Theory Used in TSS Analysis 
There are many ways to model a TSS with different coordinate systems and assumptions.  
Basic effects caused by the tether on the end bodies will be discussed before developing a 
detailed model for the TSS.  After the detailed model is defined, the equations of motion and 
perturbation effects will be introduced.  Since rotational motion is included in the model, attitude 
control parameters will be used to assist with the transformations between the body coordinate 
system and the inertial coordinate system.  Orbital elements will be described since the elements 
will be used to determine if the trajectory of the sub-satellite after release will impact the Earth.  
Finally, discussions of rigid body dynamics, the integration method used to model the TSS, and 
orbital mechanics will be presented.  In the discussion that follows, it will be assumed that the 
center of mass of the TSS will move in a circular orbit. 
2.1 Tethered Satellite System Dynamics  
 The tether in the system exerts a force on both the main satellite and the sub-satellite.  
This force leads to a change in the velocities of both of the satellites.  Depending on the position 
of the mass, the tether will either increase or decrease the velocity of each satellite to a point 
where it is greater than or less than the satellite?s untethered circular velocity.  For this 
discussion the set-up for the TSS is shown in Figure 2.1.  The main satellite generally has a mass 
equal to or larger than the sub-satellite.  Hoots, Roehrich, and Szebehely13 discussed the effects 
of the tether on the masses and developed equations for the velocity of both masses and the 
center of mass in their study.  The first step was to determine the location of the center of mass.  
17 
 
Once the position of the center of mass was determined, the equation for circular velocity was 
then used to find the circular velocity for the center of mass.13 
 
Figure 2.1: Tethered Satellite System Model 
 
For a circular orbit, the position vector of the center of mass relative to the center of the Earth 
and the circular velocity of the center of mass are calculated using equation (1) and equation (2), 
respectively. 13 
 ?     ?      ?    
     
                                                                              
      ?   
  
                                                                                     
18 
 
The variables    and    are the mass of the sub-satellite and the main satellite, respectively. 
The magnitude of the position vectors of the sub-satellite and the main satellite are   and   ,  ? is 
the gravitational parameter, and the magnitude of the position vector of the center of mass is    . 
 In order to find the velocities of the end masses the mean motion of the center of mass 
must first be calculated.  Equation (3) defines the mean motion as a function of the gravitational 
constant and the position of the center of mass.13  
    ?   
   
                                                                                
Once the mean motion is calculated, the velocities of the end masses can be found as a function 
of the mean motion and the position of each mass.  Equation (4) and Equation (5) define the 
equations used to obtain the velocity of the main satellite and the sub-satellite, respectively.13 
                                                                                         
                                                                                         
Equation (3) can then be substituted into equations (4) and (5) to give equations (6) and (7) for 
the velocities of the main satellite and the sub-satellite, repectively.13 
       
  
?   
  
                                                                           
       
  
?   
  
                                                                           
 When the satellites are not connected by a tether, both the main satellite and sub-satellite 
have a circular velocity that is defined similar to the circular velocity of the center of mass given 
19 
 
by equation (2).  The actual velocities of the satellites when connected with the tether and the 
unconnected circular velocity of the satellites can then be compared.  For the situation shown in 
Figure 2.1, Hoots, Roehrich, and Szebehely13 determined that the actual velocity of the main 
satellite when tethered is larger than the circular velocity of the main satellite when it is not 
tethered.  The sub-satellite?s actual velocity when tethered was found to be smaller than the 
circular velocity the sub-satellite has when not tethered.  This means that when the two masses 
are put together in a TSS formation the main satellite?s mass will speed up, while the mass of the 
sub-satellite will be slowed down relative to their untethered circular velocities.13  
 In order to model a TSS a set of coordinate systems needs to be defined before the 
equations of motion can be determined.  An inertial coordinate system will be used to describe 
the motion of the system, and will be selected to be the geocentric equatorial coordinate system.  
The geocentric - equatorial coordinate, Earth ? centered Inertial, or ECI system is depicted in 
Figure 2.2. 
 
Figure 2.2: Geocentric Equatorial Coordinate System 
20 
 
 
The origin of the geocentric equatorial system is fixed at the center of the Earth.  The positive X-
direction points in the vernal equinox direction.  The positive Z axis goes through the North Pole 
and the positive Y axis is ninety degrees from the positive X and Z axes, such that the XY plane 
lies in the Earth?s equatorial plane.   ?,  ?, and  ? are the unit vectors for the X, Y, and Z axes, 
respectively.1  The TSS is also depicted in Figure 2.2.  The end masses are considered to be point 
masses and the tether is assumed to be a massless, rigid tether.  The position vector for the main 
satellite and the sub-satellite are  ?  and  ? , respectively, while the position vector,  ?  , represents 
the position of the center of mass of the TSS.  The fourth vector,  ? , in Figure 2.2 is the tether 
length vector or the position of    relative to   .  
 Since the end masses are naturally rotating about the center of mass in a counter - 
clockwise rotation about the positive X ? axis due to the gravity gradient affect, a body 
coordinate system will be needed in order to determine the position of the end masses relative to 
the center of mass of the system.  This position will then be transformed into the inertial 
coordinate system.  Figure 2.3 shows the body coordinate system for the TSS.  The fixed 
coordinate system now has an origin located at the center of mass of the TSS.  The X2, Y2, and 
Z2 coordinate system is the body coordinate system after a rotation about the X axis.  The TSS 
rotates about the center of mass; therefore, the body coordinate system will rotate about the X 
axis.  This means that the X2 axis will line up with the X axis and an angle will be created 
between the Y and Y2 axes and the Z and Z2 axes.  The unit vectors for the body fixed coordinate 
system are  ? ,  ? , and  ? , which correspond to the X2, Y2, and Z2 axes, respectively.  The rotation 
angle, ?, represents the angle between the fixed coordinate system and the body coordinate 
system caused by the angular velocity.  The two vectors in Figure 2.3 represent the distances 
21 
 
from the center of mass to the end bodies.  The position vector of   relative to the center of 
mass of the system is  ?  .  The vector,  ?  , is the position vector of   relative to the center of 
mass.  Both vectors are expressed in body components.  The rotation angle can be used to 
express the vectors in the geocentric equatorial system by creating a direction cosine matrix and 
using attitude dynamics.  This methodology will be discussed after the introduction of the 
equations of motion. 
 
Figure 2.3: Body Fixed Coordinate System for a Negative ? 
 
2.1.1 Equations of Motion  
 As stated earlier the end satellites will be modeled as two point masses and the tether will 
be modeled as a massless, rigid tether.  It was also mentioned earlier that due to the tether force 
only the center of mass of the system can be modeled using Keplarian motion.  Therefore, the 
22 
 
TSS will be considered as a rigid body in this study.  In this section the equations of motion for 
the center of mass of the system will be determined using two-body forces.  The position and 
velocity of the two end masses will be determined using rigid body dynamics. 
The equations of motion of a spacecraft relative to the Earth are well-known and can be 
written as 
   
   
 ?     ??                                                                            
where 
                                                                               
Equation (8) is also known as the Keplarian acceleration of the system.  Since equation (8) is a 
vector, it has  ?,  ?, and  ? components.  This means that there are a total of three scalar equations 
that can be used to describe the motion of the system.  These final equations of motion are 
expressed in equations (10) through (12). 
   
   
      ?                                                                          
   
   
      ?                                                                          
   
   
      ?                                                                          
These equations can be used to find the velocity and position components of the center of mass 
of the TSS using numerical integration; however, the position and velocity of the end masses 
have yet to be determined.  In order to find the position of the end masses, the attitude dynamics 
of the TSS must be considered. 
23 
 
2.2 Attitude Dynamics Using Euler Parameters 
 According to Hughes14, spacecraft attitude dynamics is an applied science that aims to 
understand and predict spacecraft orientation and how the orientation evolves over time.  The 
part of attitude dynamics that this research utilizes is the determination of equations to describe 
rotational motion and the differential equations that govern the motion equations.  There are 
many ways to represent the attitude dynamics of a spacecraft.  In this text, the Euler parameters 
are chosen because they have no singularities at any orientation; however, the disadvantage of 
the Euler parameters is that they lack uniqueness. In other words, a positive Euler parameter can 
describe the same attitude orientation as a negative Euler parameter.  The Euler parameters are 
then placed into a direction cosine matrix, which describes the rotation of the system and allows 
for a transformation between two coordinate systems.14  In this case the direction cosine matrix 
will allow for the transformation of the body coordinate system into the fixed geocentric 
equatorial coordinate system. 
 Euler?s theorem states that any rigid body orientation can be achieved by a single rotation 
about the principal axis,  ?, through a principal angle, ?.  The principal axis and angle can be 
determined by examining the eigenvalues and eigenvectors of the direction cosine matrix.  The 
principal axis is a vector made up of three components.  If the principal axis and angle are 
known, the Euler parameters can be found.  Equations (13) through (16) define the four Euler 
parameters in terms of the principal axis and angle.14 
      (  )                                                                            
         (  )                                                                        
24 
 
         (  )                                                                        
         (  )                                                                        
The fourth Euler parameter is labeled using the symbol   .14 
 The direction cosine matrix, C, is a 3x3 matrix and can be expressed in terms of the four 
Euler parameters. 
   [
                                       
                                       
                                       
]                     
If the Euler parameters are not known but a direction cosine matrix is known, the Euler 
parameters can be found using the elements from the direction cosine matrix.  This must be done 
in a series of steps.  The first step is to calculate the squared values of each of the Euler 
parameters. 
               [ ]                                                                  
                    [ ]                                                        
                    [ ]                                                        
                    [ ]                                                        
Where, 
     [ ]                                                                       
25 
 
The second step is to determine the largest squared value from equation (18) through (22).  Once 
that is determined, take the square root of the largest value in order to find one of the Euler 
parameters.  Finally, the remaining three Euler parameters can be determined using the three of 
the following equations.14 
                                                                                    
                                                                                    
                                                                                    
                                                                                    
                                                                                    
                                                                                    
For example, the value for    was found in step two.  In order to find the other three Euler 
parameters equation (21) can be used to find   , equation (28) can be sued to find   , and 
equation (27) can be used to find   .14 
 The Euler parameters will change with time and the introduction of an angular velocity.  
In order to do this kinematics are introduced into the rotational motion equations.  The motion of 
the Euler parameters can be described using the first derivative of the parameters.  The first 
derivative of the Euler parameters is a function of the angular velocity and the original Euler 
parameters.  The first order derivatives are expressed in matrix format in equation (29).14 
26 
 
[
 
  
   
  ?
?
 ? 
 ? ]
 
  
 
    [
          
        
  
  
   
  
 
   
  
 
][
  
  
  
  
]                                               
The   ,   , and   variables are the X, Y, and Z-components of the angular velocity.14  Now 
that the rotational motion equations have been defined, the position and velocity of the end 
masses in the TSS can be determined using rigid body dynamics. 
2.3 Rigid Body Dynamics 
 Since the TSS will be modeled as a rigid body, the center of mass is the only point on the 
TSS that can be modeled using Keplarian motion.  This means that the position and velocity of 
the main satellite and the sub-satellite must be calculated using rigid body dynamics.  According 
to Ginsberg12 the motion of a rigid body is a ?superposition of a translation and a pure rotation.?  
For the translational motion all points on the rigid body follow the movement of an arbitrary 
point on the body.12  The arbitrary point for the TSS is the center of mass of the system.  In 
Figure 2.3, the X, Y, Z coordinate system is placed at the center of mass of the TSS because 
when there is no angular velocity on the system the body coordinate system will line up with the 
geocentric equatorial system.   
Another important note is that the position vectors between the center of mass and the 
two end masses will have constant components relative to the moving reference frame.12  The 
moving reference frame in this case is the body fixed coordinate system; however, the final result 
needs to be expressed in terms of the geocentric equatorial coordinate system.  The position of 
the center of mass is already expressed in terms of the geocentric equatorial coordinate system; 
therefore, all that is left to do is to convert the position vectors between the center of mass and 
27 
 
the two end masses into geocentric equatorial components.  This is done by multiplying the 
direction cosine matrix by the vector of the distance between the center of mass and the end 
masses. 
 ?        [ ] ?                                                                              
 ?        [ ] ?                                                                              
Equations (30) and (31) give the position vector from the center of mass to the sub-satellite and 
to the main satellite, respectively, in geocentric equatorial coordinates.  The next two equations 
express the position of the sub-satellite and the main satellite in geocentric equatorial coordinates 
using the relative position equation.20 
 ?    ?     ?                                                                       
 ?    ?     ?                                                                       
At this point, the disadvantage to using Euler parameters needs to be considered.  Recall that the 
Euler parameters have a lack of uniqueness and the negative Euler parameters could describe the 
same position as positive Euler parameters.  The sign on the position vectors from the center of 
mass to the end masses must be checked to make sure that the addition and subtraction of the 
distances matches up with the model. 
 Now that the position has been found for the end masses, the velocity of the sub-satellite 
and the main satellite needs to be found using rigid body dynamics.  Ginsberg12 and Meriam20 
define the velocity as the velocity of an arbitrary point plus the relative velocity between the 
point where the velocity is to be determined and the arbitrary point.12,20  The arbitrary point 
28 
 
remains the center of mass of the TSS.  Equation (34) and (35) defines the velocity of the sub-
satellite and the main satellite, respectively.12,20 
 ?    ?     ?   ?                                                                    
 ?    ?     ?   ?                                                                    
The second term on the right hand side of both equations is the relative velocity between the two 
end masses and the center of mass.  The relative velocity is defined as the angular velocity vector 
crossed with the position vector between both points.12,20   
 ?   ?    ?    ?                                                                   
 ?   ?    ?    ?                                                                   
The angular velocity is described in geocentric equatorial coordinates and the position vectors 
that will be used are those defined in equations (30) and (31).  Equation (36) can be substituted 
into equation (34) to get the velocity of the sub-satellite.  The velocity of the main satellite can 
be found by substituting equation (37) into equation (35). 
 ?    ?     ?    ?                                                                
 ?    ?     ?    ?                                                                
Equations (32), (33), (38), and (39) are the rigid body equations used to find the position and 
velocity of the end masses of the TSS. 
 The movement of the center of mass of the TSS and the end masses differ slightly.  The 
center of mass will move in a circular orbit.  Rossi25 states that if the TSS is modeled as a solid 
body it will rotate with a constant angular velocity in a counter ? clockwise direction.  If the TSS 
29 
 
moves only in an orbital plane, the system will have a radial configuration or it will form a 
straight line that is tangent to the circular orbit of the center of mass.  The points of the TSS, the 
center of mass and the two end masses, must be collinear because the system is being treated as a 
rigid solid body.25  If there is no librational motion, the center of mass and both the end mass of 
the TSS will move in concentric circles with the lower mass always being the sub-satellite.19  
Kaplan16 states that the lower mass of the TSS will always remain in the sub-satellite position 
because of the gravity gradient on the system.  The gravity gradient will create a spin 
stabilization that will keep the system at a local vertical; therefore, the sub-satellite will always 
be below the main.16  The natural motion of the TSS is a rotation of the system in a counter ? 
clockwise direction with the lower satellite always remaining the sub-satellite because of the 
gravity gradient. 
2.4 Orbital Elements of a Trajectory 
 At the time of the sub-satellite?s release, the orbital elements of the sub-satellite?s impact 
trajectory will be calculated.  The six orbital elements that describe an orbit are the semi-major 
axis (a), the eccentricity (e), the inclination (i), the longitude of the ascending node (?), the 
argument of periapsis (?), and the true anomaly (?).  Figure 2.4 labels some of the orbital 
elements of an orbit. 
30 
 
 
Figure 2.4:  Orbital Elements of a Trajectory10 
The semi-major axis defines the size of the orbit and the eccentricity defines the shape of a conic 
orbit.1  The other four orbital elements are the inclination, the longitude of the ascending node, 
the argument of perigee or periapsis, and the true anomaly.  The inclination is the angle between 
the  ? geocentric equatorial unit vector and the angular momentum vector,  ?.  The longitude of 
the ascending node is measured counter clockwise about the Z-axis from the X-axis to the 
ascending node on the orbit.  The argument of periapsis is the angle between the ascending node 
and the periapsis or perigee point of the orbit.  The argument of periapsis is measured in the 
plane of the satellite?s orbit in the direction of travel of the satellite.  The true anomaly, ?, is 
measure from the perigee point to the position vector of the satellite.  The true anomaly is 
measured in the direction of travel of the satellite.1  Each of these four parameters are shown in 
the orbit depicted in Figure 2.4.10 
31 
 
The orbital elements can be determined when the position and the velocity vector are 
known for a satellite.  The eccentricity can be found by calculating the eccentricity vector. 
 ?  (  
 
   
 
  ) ?   
 
    ?    ?   ?                                                        
The magnitude of the eccentricity vector can then be found to get the eccentricity.  Equation (41) 
is used to calculate the magnitude of the eccentricity vector.10 
   ?                                                                                 
The second orbital element that can be found is the semi-major axis.  In order to find the semi-
major axis the angular momentum must be calculated. 
 ?    ?    ?                                                                          
The magnitude of the angular momentum vector can then be calculated using the same formula 
that was used to find the eccentricity vector.   
  
         
                                                                     
Equation (43) can then be rearranged to find the semi-major axis in terms of the eccentricity, the 
angular momentum, and the gravitational parameter.10 
   
  
 
                                                                               
32 
 
The inclination can then be found by using the vector properties for an angle between two 
vectors.  The two vectors that will be used are the angular momentum vector and the unit vector 
for the Z axis.10 
 ?   ?   | ?|                                                                          
The magnitude of a unit vector is always one and the angular momentum vector dotted with the 
unit vector will lead to the Z-component of the angular momentum vector. 
                                                                                     
Equation (46) can then be arranged to find the inclination angle.10 
        (   )                                                                      
The inclination will always be less than or equal to one hundred and eighty degrees.   
The longitude of the ascending node can be found by forming the node vector,  ?.  The 
node vector is a constant vector that points in the direction of the ascending node.  Equation (48) 
can be used to find the node vector.10 
 ?    ?    ?      ?    ?                                                                  
The node vector consists of the X and Y components of the angular momentum.  The magnitude 
of the node vector can be found in the same way that the magnitude of the eccentricity vector 
was found.  The longitude of the ascending node is the angle between the X unit vector and the 
node vector.10 
 ?    ?   | ?|                                                                         
33 
 
                                                                                 
                                                                                 
Equation (51) can be used find the longitude of the ascending node; however, the quadrant of the 
longitude of the ascending node must be determined.  In order to find the quadrant for the 
longitude of the ascending node the Y-component for the node vector must be looked at. The test 
to determine the quadrant of the longitude of the ascending node is described in equation (52).10 
                
                                                                              
For the case when the Y-component of the node vector is less than zero, the longitude of the 
ascending node must be recalculated.  The new longitude of the ascending node can be found in 
equation (53).10 
                                                                                 
The argument of periapsis is the angle between the node vector and the eccentricity 
vector.  The same process to find the longitude of the ascending node and the inclination are used 
to find the argument of periapsis.10 
        ( 
?    ?
  )                                                                    
The quadrant for the argument of periapsis must also be determined.  This time the variable that 
will be looked at to determine the quadrant is the Z-component of the eccentricity vector.  The 
test is described in Equation (55).10 
34 
 
                
                                                                             
For the case when the Z-component of the eccentricity vector is negative, the argument of 
periapsis must be subtracted from three hundred and sixty degrees.  This will result in an angle 
greater than one hundred and eighty degrees.10 
                                                                                
 The final orbital element to be found is the true anomaly.  The true anomaly is the angle 
between the eccentricity vector and the position vector.  This means that the true anomaly can be 
found using the same method that was used for the previous three orbital elements.  Equation 
(57) gives the true anomaly as a function of the eccentricity vector and position vector, as well as 
their magnitudes.10 
        ( ?    ?  )                                                                   
The quadrant for the true anomaly must be determined as well.  The test involves the result from 
taking the dot product of the position vector with the velocity vector.10 
    ?    ?            
    ?    ?                                                                    
If the dot product is found to be negative, the true anomaly must be recalculated in order to be 
greater than one hundred and eighty degrees.  The new true anomaly for a negative dot product is 
given in equation (59).10 
                                                                                 
35 
 
 The six orbital elements found in this section can be used to describe the trajectory of the 
sub-satellite once it is released from the TSS.  Some of the elements can be used to determine if 
the sub-satellite will impact the Earth without having to propagate the trajectory of the sub-
satellite forward in time.  The elements that are used to determine impact are the semi-major axis 
and the eccentricity vector.  These two orbital elements can be used to calculate the distance at 
the perigee point of the trajectory.  The perigee point on a trajectory is the lowest point of the 
satellite?s orbit.10 
                                                                                    
If the perigee point is found to be less than the radius of the Earth, then the satellite will impact 
the Earth.  Being able to determine if the sub-satellite will impact before propagating the motion 
forward in time will save computational time.  The goal of this study is to determine the impact 
capabilities of the impact trajectories after release and how different changes affect the time to 
impact and range; therefore, if the satellite does not impact the Earth, the propagation of the 
motion of the satellite does not need to be done. 
2.5 Numerical Integration Using a 4th Order Runge Kutta 
 Numerical integration is used to solve ordinary differential equations.  This feature 
allows a state matrix consisting of position, velocity, and constants to be propagated forward in 
time.  Numerical integration methods use a Taylor series expansion to solve the ordinary 
differential equations.  The Euler method approximates the solution to the ordinary differential 
equations using the first-order term from the Taylor series.  The first and second terms of the 
Taylor series are used for the improved Euler method.  The Runge Kutta method used for this 
research approximates the higher derivatives using finite-difference expressions.  This means 
36 
 
that the higher derivatives do not have to be calculated from the original ordinary differential 
equation that is given.17  
 The approximation for the Runge Kutta is done by calculating data through a series of 
time steps from an initial time, to, to a final time, tf.  The order of the Runge Kutta method is 
determined by the number of steps used to estimate the state vector after a time step.  For this 
research a fourth order Runge Kutta is used because it is the most common version of the Runge 
Kutta method.  Equations (61) through (64) list the four steps used for the fourth order Runge 
Kutta method.17 
    [ ?       ]                                                                     
    [ ?                   ]                                                      
    [ ?                    ]                                                     
    [ ?                 ]                                                      
The function,  , is used to find the first derivative of the state vector using the state vector,  ?, at 
the initial time.  After the first step, step two and three are calculated after half of the time step, 
?t, has passed.  The fourth step is calculated at the initial time plus the time step.  After each 
time step a new state vector is calculated.17 
 ?           ?                                                               
The four step process is continued until the final time is reached and a final state vector is 
calculated.  A Runge Kutta method can be implemented in MATLAB using a WHILE loop or a 
37 
 
for loop to get from the initial time to the final time using any time step.  Typical values for the 
final time are the orbital period of the satellite and the time for a single day.  The set-up of the 
state vector and the terms of the first derivative of the state vector are discussed in chapter three. 
2.6 Analytical Approach 
 In order to establish an analytical approach to the TSS problem more orbital mechanics 
equations must be introduced.  The energy of an orbit remains constant and can be calculated by 
using the semi-major axis or the position and velocity magnitudes of the satellite.  Both ways to 
calculate the energy of an orbit are shown in equation (66).10 
    
 
   
 
   
  
                                                                        
The trajectory equation is used to find the position of a satellite when the semi-major axis, the 
eccentricity, and the true anomaly of the satellite are known as given in equation (67). 
         
  
                                                                                 
The energy equation and the trajectory equation will be used later in chapter 3 to find an 
analytical solution to the range and time to impact problem for the sub-satellite. 
 The next equation set will be used to find the time to impact for the analytical solutions.  
Kepler?s Equation gives the mean anomaly in terms of the eccentric anomaly, E, and the 
eccentricity.10 
                                                                            
The mean anomaly can be expressed in terms of the mean motion, n, and a difference in time.10 
38 
 
                                                                              
The difference in time stands for the time at the eccentric anomaly minus the time at perigee.  
Equations (68) and (69) can be defined at two different points, a point one and a point two, and 
combined into the differenced Kepler?s equation.10 
 (                 )                                                      
 (           )                                                               
Where, 
   ?                                                                                 
        [(      )
  ?
   (  )]                                                      
The difference in time expressed in equation (71) is the time from release to impact.  Point one is 
the place in the orbit where the sub-satellite is released from the TSS and point two is the place 
in the impact trajectory of the sub-satellite where the sub-satellite impacts the Earth.  Equations 
(71), (72), and (73) can be used to find the time to impact.  Equation (73) is the same for points 
one and two.  The only value that changes is the true anomaly.  The first true anomaly 
corresponds to the true anomaly at release and the second true anomaly corresponds to the true 
anomaly at impact. 
 
 
39 
 
 
 
 
Chapter 3: Model and Problem Formulation 
In order to investigate this problem, a simplified model is developed for the TSS and the 
initial velocity for the center of mass of the TSS is determined.  A numerical integration method 
is used to propagate the motion of the center of mass of the system using Keplerian motion.  The 
position and velocity of the end masses is then determined relative to the center of mass.  Once 
the system successfully travels in its baseline orbit, changes into the system can be introduced.  
There are three main changes that are applied to the system:  a velocity change on the sub-
satellite, a change in the altitude of the TSS, a change in tether length of the TSS, and a change 
in the release point location.  The change in velocity is needed in order to cause the sub-satellite 
to enter an impact trajectory, while the tether length and altitude are changed in order to 
determine their effect on the range and time to impact.  As stated before, the velocity change 
placed on the sub-satellite will be a result of an impulsive velocity change maneuver or a change 
to the angular velocity of the system.  The tether is cut after the velocity change is implemented 
and the trajectory of the released mass is calculated to determine if the trajectory of the sub-
satellite impacts the Earth after release.  If the sub-satellite impacts the Earth, then the time to 
impact and the range from release to impact are calculated.  This information is then used to 
determine how the changes in the system affect the ballistic capabilities of the sub-satellite after 
release.  An analytical solution is also developed to examine situations where the sub-satellite is 
directly above, below, to the right, and to the left of the main satellite.  At each of these 
orientations the analytical solution can be used to determine the maximum and minimum ranges 
and times to impact when given a range of velocity changes.  A second set of analytical solutions 
40 
 
are also developed in order to determine an equivalent angular velocity that can be done in place 
of an impulsive velocity change.  The set-up and algorithms for the simulation and analytical 
solutions are discussed in this chapter. 
3.1 Problem Set-up of the TSS 
 Before numerical integration of the equations of motion can be performed and before an 
impact of the sub-satellite can be evaluated an initial position and velocity must be determined.  
The initial set-up of the TSS needs to be performed in such a way that the position and velocity 
of the center of mass and the end masses can be easily determined.  The initial set-up for the TSS 
is shown in Figure 3.1. 
 
Figure 3.1: Initial Set-up of the TSS for a Negative ? 
41 
 
The center of mass of the TSS is initially placed along the Z axis so that the position of each of 
the end masses can be determined from the altitude, TSS orientation, and tether length.  The 
direction of travel of the TSS is counter-clockwise about the origin of the geocentric equatorial 
coordinate system and about the positive X - axis.  The orbit for the center of mass is assumed to 
be circular in shape. 
 In order to determine the positions of the end masses and the center of mass the altitude, 
tether length, and the size of the masses must be chosen.  It?s also important to keep in mind that 
the tether itself is modeled as a massless, rigid tether.  A 10:1 ratio between the main satellite and 
the sub-satellite was selected; therefore, the mass of the main satellite was assigned a value of 
1000 kg and the sub-satellite was assigned a mass of 100 kg.  The altitude measures the height of 
the main satellite above the Earth.  The sub-satellite is then ?lowered down? from the main 
satellite until the tether is completely extended.  For the baseline case the altitude was 500 km 
above the Earth?s surface with a tether length of 4 km.  The angular velocity of the system is set 
to zero.  Changes to the baseline values will be made in order to determine the effects of the 
altitude and tether length on the range and time to impact for an impact trajectory of the sub-
satellite.  The masses of the two satellites will always remain constant.   
 With the baseline values determined, the position of the end masses and the position and 
velocity of the center of mass can be determined.  In order to determine the positions of the TSS, 
the system is assumed to be completely aligned with the Z axis. In other words the rotation angle 
is zero.  Since the TSS is aligned with the Z axis of the geocentric equatorial system, the 
positions of the end masses can be determined by adding and subtracting values without having 
to worry about the other two axes directions.  The main satellite position can be calculated by 
adding the radius of the Earth to the altitude.  The tether length can then be subtracted from the 
42 
 
position of the main satellite to get the position of the sub-satellite.  The calculations for the 
position of the main satellite and for the sub-satellite are shown in equations (74) and (75), 
respectively. 
                                                                                   
                                                                                 
Both of the positions are initially in the  ? direction.  The position and velocity of the center of 
mass are then calculated using equations (1) and (2).  The position for the center of mass is along 
the  ? direction.  The velocity of the center of mass that is calculated is the circular velocity and 
is tangent to the circular orbit.  Since the TSS moves in a counter-clockwise direction, the initial 
velocity of the center of mass is in the negative  ? direction. 
 With the position of the center of mass determined, the length between the center of mass 
and the two end masses can be calculated. For this initial case the direction of the lengths 
between the center of mass and the two end masses is in the  ? direction.   
                                                                                  
                                                                                  
The next step after determining the distances between the end masses and the center of mass is to 
find the four Euler parameters with the rotation angle in place.  In order to find the four Euler 
parameters a direction cosine matrix must be generated.  The direction cosine matrix for this 
instant in time and with a rotation angle can be calculated using the rotation transformation about 
an X axis.12  
43 
 
   [
   
         
          
]                                                           9 
The process described in chapter 2 to find the Euler parameters from a given direction cosine 
matrix using equations (18) through (28) are used to find the Euler parameters for the initial set-
up of the TSS.  As the direction cosine matrix changes the direction of the distance between the 
end masses and the center of mass will change; however, the magnitude of the distances will 
remain constant.  In order to determine the new directions the direction cosine matrix must be 
pre-multiplied by the magnitude of the two lengths.  This process is shown in equations (30) and 
(31).  The vector components are then added or subtracted from the position of the center of 
mass to determine the new position of the end masses.  This process is shown in equations (32) 
and (33).   
 Before the numerical integration is started the velocity for the main satellite and sub-
satellite must be calculated.  Equations (38) and (39) are used to find the velocity vector for the 
sub-satellite and the main satellite, respectively.  The determination of the velocity vector and the 
determination of the position vector for the end masses will be done at each time step until the 
sub-satellite is released.  This is done in order to keep track of the position and velocity vector 
for the sub-satellite at release.  Once the position and the velocity vectors have been calculated 
the numerical integration can be started. 
 In order to do a numerical integration, the initial and final time and an initial state vector 
are needed.  The initial time for the system is set to zero and a final time is set to one day.  
Giving a day for the motion of the TSS, the release of the sub-satellite, and the motion of the 
sub-satellite allows for the determination of the time to impact and the range of an impact 
44 
 
trajectory.  The time step used for the numerical integration is five seconds; however, the time 
step is decreased later when the tether length is increased.  Before the release of the sub-satellite, 
the state vector will contain 15 terms.  The terms include the position and velocity components 
of the center of mass, the true anomaly, the four Euler parameters, the angular velocity, and the 
rotation angle. 
 ?   
[
 
 
 
 
 
 
 
 
  ?   ?
  
 
  
  
  
  
 ?
 ]
 
 
 
 
 
 
 
 
 
                                                                                  
 The function in the numerical integration calculates the first derivative of the state vector.  
The derivative of the state vector is calculated in four steps as described in equations (61) 
through (64).  In order to get the derivative of the state vector the derivatives of the individual 
elements must be taken.  The first derivative of the state vector contains the velocity of the center 
of mass, the acceleration of the center of mass, the first derivative of the true anomaly, the first 
derivative of the four Euler parameters, the angular acceleration, and the derivative of the 
rotation angle.  Equation (80) gives the first derivative of the state vector. 
 ??   
[
 
 
 
 
 
 
 
 
  ?   ?
  
 ?
  ?
  ?
  ?
  ?
 ?
 ? ]
 
 
 
 
 
 
 
 
 
                                                                                 
45 
 
The velocity of the center of mass is taken from the original state vector.  The first derivative of 
the velocity of the center of mass is equal to the acceleration of the center of mass.  Since the 
acceleration of the center of mass was not in the original state vector, the acceleration must be 
calculated using equations (10) through (12).  The derivative of the true anomaly must now be 
calculated. 
Lovell?s procedure18 is used to determine the first derivative of the true anomaly using 
the position and velocity of the center of mass.  The first step is to calculate the magnitude of the 
angular momentum in geocentric coordinates and polar coordinates.  Equation (81) gives the 
magnitude of the angular momentum for geocentric coordinates.18 
   | ?   ?|   ?                                                            
Lovell then finds the magnitude of the angular momentum in terms of the body fixed coordinate 
system expressed in polar coordinates.  Equation (82) gives the magnitude for the angular 
momentum using the body fixed coordinates assigned earlier in chapter two.18 
   | ?   ?|   |  ?     ? ?    ? ?  |     ?                                             
The angular momentum of an orbit remains constant and the magnitude will always be the same 
value no matter what coordinate system is used to express the components.  The next step is to 
set equation (81) and (82) equal to each other and solve for the first derivative of the true 
anomaly.  The solution is given below. 
 ?   ?           
                           
                                     
46 
 
The X, Y, and Z components from the position vector and the velocity vector can be taken from 
any position or velocity vector.  For this case the components will come from the position and 
velocity vector of the center of mass, so that the true anomaly of the system can be tracked. 
 The first derivatives of the Euler parameters, the angular acceleration, and the first 
derivative of the rotation angle now have to be determined.  The derivatives of the four Euler 
parameters are calculated using equation (29).  The angular acceleration and angular velocity for 
all cases is kept at zero. The derivative of the rotation angle is equal to the angular velocity.  The 
equivalent angular velocity needed to replace the impulsive velocity change for impact is in the 
negative or positive X2 ? direction of the body coordinate system, which is parallel to the X ? 
direction of the ECI coordinate system.  Since the study is focused on the in-plane motion of the 
sub-satellite, rotation of the TSS about the Y2 and Z2 axes is zero.  This means that the Y2 and Z2 
components of the equivalent angular velocity will be zero. 
 This process is continued within a loop until the point that the sub-satellite is released.  
Prior to release an impulsive velocity change is done by the sub-satellite.  After the velocity 
change, the sub-satellite is released from the system at the desired configuration and the release 
point position and velocity of the sub-satellite are calculated as described below. 
 ?       ?     ?                                                                    
The term on the left hand side of equation (84) stands for the velocity of the sub-satellite at 
release.  An equivalent angular velocity that could cause impact can then be calculated by setting 
equation (84) equal to equation (38). 
 ?      ?     ?    ?                                                                    
47 
 
The angular velocity in the X - direction can then be solved for using equation (85).  There are 
two ways to calculate the angular velocity in the X - Direction.  If the velocity of the sub-satellite 
changes in the Y-direction, equation (86) is used; and equation (87) is used to find the angular 
velocity when the velocity changes in the Z-direction. 
                  
        
                                                                
               
       
                                                                 
Equations (86) and (87) can then be solved for the velocity change of the sub-satellite.  The 
velocity change is expressed in vector form in equation (88). 
  ?   ?   ?                                                                        
From equation (88), it can be seen that the velocity change of the sub-satellite is related to the 
angular velocity of the system.  This is why an impulsive velocity change or a change in the 
angular velocity can lead to the same ballistic capabilities of the sub-satellite. 
Before a new numerical integration is done for the sub-satellite only, the orbital elements 
of the sub-satellite?s impact trajectory are calculated in order to determine if the sub-satellite will 
impact the earth.  Since the position and velocity vectors of the sub-satellite have been calculated 
the orbital elements can be calculated using the process discussed in section 2.4.  Equation (60) 
is then used to calculate the perigee position of the sub-satellite along its impact trajectory.  If the 
perigee position of the sub-satellite is greater than the radius of the Earth, the sub-satellite will 
not impact the earth and the program breaks out of the loop to begin a new test from the 
beginning.  This is done in order to save computational time.  When the perigee position is found 
48 
 
to be less than the radius of the Earth, a new state vector is created in order to model the motion 
of the sub-satellite after release from the system. 
 The state vector for the sub-satellite after release is smaller than the state vector needed 
for the TSS.  The Euler parameters are no longer needed to model the sub-satellite, since the sub-
satellite is considered to be a point mass.  There is also no rotation angle for the sub-satellite 
because all vectors can now be expressed in terms of the geocentric coordinates only.  The new 
state vector that will be used in a 4th order Runge Kutta is expressed in equation (89). 
 ?   [ ?  ?
 
]                                                                                 
The first derivative of the state vector is then calculated for the numerical integration. 
 ??   [ ?  ?
 
]                                                                                
The velocity vector is taken from the original state vector.  The acceleration of the sub-satellite 
can be calculated using the same equations as before by replacing the position and acceleration 
center of mass values with the position and acceleration of the sub-satellite.  
   
  
     ?                                                                             
   
  
     ?                                                                             
   
  
     ?                                                                             
 The state vector for the sub-satellite is then propagated forward in time using the 
numerical integration.  This is done until impact occurs or a full day goes by.  After a full day 
49 
 
goes by, the simulation stops because a day to reach an impact point is too long of a travel time 
for a threat.  When the magnitude of the position vector is equal to or less than the radius of the 
Earth, impact of the sub-satellite will occur.  When this happens, the program breaks out of the 
loop and completes the numerical integration.   
Before the range and time to impact can be calculated, the true anomaly at impact and the 
true anomaly at release must be calculated using the eccentricity and the semi-major axis from 
the impact trajectory.  The first step is to take the trajectory equation in equation (67) and solve 
for the true anomaly. 
           (     
     
   )                                                        
The magnitude for the position of the sub-satellite in equation (94) is dependent upon the true 
anomaly that is trying to be found.  If equation (94) is used to find the true anomaly at impact 
than the position of the sub-satellite is equal to the radius of the Earth. 
           (     
     
   )                                                     
Equation (95) may give a true anomaly at impact that is less than the true anomaly at release.  
Since impact occurs after the true anomaly at release, the true anomaly at impact must be greater 
than the true anomaly at release.  This change can be done by following the test in equation (96). 
             
                                                                                     
With the true anomaly at impact in the correct quadrant, the range and time to impact can be 
calculated using the following equations. 
50 
 
     (|          |)                                                                    
                                                                                         
The angle for the arc length in equation (97) is the absolute value of the true anomaly at impact 
minus the true anomaly at release.  The time to impact subtracts the time at release from the final 
time when impact occurs.  The impact of the sub-satellite can take longer than a day depending 
on the type of changes that are placed on the system.  Usually an impact that takes longer than a 
day means that the impact trajectory of the sub-satellite is hyperbolic.  If the impact takes longer 
than a day, the program continues to run until the end of the day but will not calculate the range 
or time to impact because the sub-satellite has not impacted yet.  For the range and time to 
impact to be calculated the total time for the numerical integration to run must be increased.  
With this test complete the process can be continued with new altitude and tether length values.  
The changes placed on the system to create an impact trajectory will be discussed in the next 
section. 
3.2 Changes Applied to the System to Create Impact of Sub-Satellite 
 For this research two configurations are chosen for the sub-satellite upon release during 
the numerical simulation.  These configurations were chosen to model possible real world 
scenarios for releasing the sub-satellite from the TSS.  The first configuration of the sub-satellite 
at release will have the sub-satellite located below the main satellite with a rotation angle of -0.5 
radians.  The second configuration will place the sub-satellite above the main satellite using a 
rotation angle of about -3.6416 radians.  The rotation angle is depicted in figure 3.1 and 
measures the position of the sub-satellite from the vertical Z ? axis.  The rotation angle also 
51 
 
keeps the sub-satellite in a leading position.  More points can be chosen and the program can be 
easily changed to reflect the new configuration or configurations of the sub-satellite at release. 
 Simply releasing the sub-satellite from the TSS by cutting the tether does not always 
create an impact trajectory.  In order to create impact trajectories an impulsive velocity change of 
the sub-satellite or a change in the angular velocity of the system must be introduced prior to the 
time of release.  The change in velocity of the sub-satellite is increased or decreased in the Z 
direction or the Y direction.  All changes in velocity are tested within the program because a 
velocity change in the positive Y direction may cause the impact of the sub-satellite to occur at a 
small velocity change, while a large change in velocity in the negative Z direction may be 
needed in order to cause an impact.  In order to get the change in the sub-satellite velocity to 
occur, a thrust force is generated by the sub-satellite.  The thrust force is not modeled in the 
program because the research is not concerned with how the velocity change occurs, but on what 
is the minimum velocity change needed to cause impact and how the velocity change will affect 
the range and time to impact for the sub-satellite system.  It is also important to note that a 
change in velocity still may not cause the sub-satellite to impact.  The test for impact is done 
because of the fact that an impact still may not occur even with a large velocity change.  If a 
change in velocity is not done on the sub-satellite, a change in the angular velocity can be placed 
on the system prior to the time of release in order to create an impact.  This angular velocity is 
produced by the main satellite and sub-satellite generating a thrust that would cause the TSS to 
rotate about the center of mass.  Once again, the thrust changes are not modeled. 
 The velocity of the sub-satellite can also be changed by introducing a velocity change in 
the X direction; however, this case is not investigated because the motion of the TSS is restricted 
to the Y-Z plane.  Introducing a change in the X direction will cause out of plane librational 
52 
 
motion that will need to be accounted for.  While the out of plane librational motion can be 
introduced into the program, it will severely complicate the program and the set-up or model of 
the TSS.  One of the goals of this research is to investigate the impact of the changes in altitude, 
tether length, and angular velocity using a simplified model for the TSS.  For this reason velocity 
changes in the X direction on the sub-satellite are not considered in this research.   
3.3 Variations to the Problem Set-up   
 The changes in the velocity of the sub-satellite discussed in the previous section will help 
to lead to an impact and will affect the range and time to impact for the sub-satellite.  The main 
properties of the TSS that will be changed are the altitude and the tether length of the system.  
When a change in the velocity of the sub-satellite leads to an impact, changes in the altitude and 
tether length will be analyzed in order to determine if these changes lead to an increase or a 
decrease in the range and time to impact. 
 The altitude that is being changed is the altitude for the main satellite.  In this case the 
tether length of the TSS is kept constant at four kilometers.  The range of the altitude goes from 
five hundred kilometers to fifteen hundred kilometers.  The minimum value of five hundred 
kilometers was chosen because the drag force due to the Earth?s atmosphere does not have to be 
taken into account at this point.  As the altitude of a satellite decreases and gets closer to the 
Earth, the atmospheric drag must be taken into account.  The altitude is increased in increments 
of fifty kilometers.  At each altitude the program is run as discussed in section 3.1 in order to 
determine the range and time to impact for the sub-satellite after release.  The increase in altitude 
should lead to an increase in the range and time to impact because the sub-satellite is located at a 
higher position at release.  
53 
 
 The range of values used for the tether length is one kilometer to one hundred kilometers.  
Three kilometers are used as the increment change from one kilometer to one hundred 
kilometers.  For this case the altitude is held constant at a value of five hundred kilometers. The 
tether length can either increase or decrease the range and time to impact of the sub-satellite 
depending on the location of the sub-satellite relative to the main satellite at the point of release.  
For example, if the sub-satellite is located above the main satellite, an increase in the tether 
length should lead to an increase in the range and time to impact.  On the other hand, if the sub-
satellite is located below the main satellite, an increase in the tether length should decrease the 
range and time to impact of the sub-satellite.    
3.4 Analytical Solutions 
 This section discusses the formulation of equations and algorithms for the analytical 
solution to the impact problem of a sub-satellite after it is released from a TSS.  The first 
analytical solution that is determined can be used to find the range and time to impact given 
initial conditions for the TSS.  The second analytical solution solves the problem backwards.  In 
other words, given the range, true anomaly at impact, and some other initial conditions find the 
change in the angular velocity of the TSS or the impulsive velocity change needed in order to 
achieve the desired range at impact.  The reason behind the analytical approach is to force the 
sub-satellite to be directly below, above, to the right, or to the left of the main satellite.  The set 
up for the four cases are depicted in figure 3.2, which is located before the next section.   
 The position of the center of mass, the main satellite, and the sub-satellite must be 
determined for each case.  Each position component can be described in terms of the radius of 
the Earth, the altitude of the main satellite, and the tether length.  The position of the main 
satellite will always be in the positive Z ? direction and equal to the radius of the Earth plus the 
54 
 
altitude. Table 3.1 gives the position of the sub-satellite for each of the four cases. Case A, B, C, 
and D correspond to when the sub-satellite is located above, below, to the right, and to the left of 
the main satellite, respectively. In table 3.1 the components of the sub-satellite position are given 
for each case.  The X-component of the sub-satellite position vector is always equal to zero 
because the orbital motion is kept in the Y-Z plane.   
Table 3.1 Sub-satellite Position Components for Cases A - D 
Case X-Component (km) Y-Component (km) Z-Component (km) 
A 0 0 Re + H + Lt 
B 0 0 Re + H - Lt 
C 0  Lt Re + H 
D 0  - Lt Re + H 
 
 
Figure 3.2 Analytical Cases 
55 
 
3.4.1 Equations to Find the Range and Time to Impact 
 For this set of analytical equations the goal is to find expressions for the range and time 
to impact.  It is assumed that the altitude of the main satellite, the length of the tether, the angular 
velocity of the TSS, and the impulsive velocity change of the sub-satellite are known.  The 
angular velocity can have three components in the X2, Y2, and Z2 ? directions; however, the Y2 
and Z2 ? components of the angular velocity are equal to zero because the trajectory of the TSS 
remains in the Y-Z plane.  For all cases an impulsive velocity change is placed on the sub-
satellite; therefore, the angular velocity in the X2 or X direction will also be zero .  Since the only 
known values are the altitude, tether length, angular velocity, and the velocity change of the sub-
satellite, the range and time to impact must be functions of these four known values for the given 
mass ratio. 
 In order to find the range the true anomaly at impact and the true anomaly at release, 
equation (97) must be expressed in terms of the four known variables.  Since the radius of the 
Earth is a constant and the magnitude of the position vector is a function of the altitude, the 
tether length, and the radius of the Earth, the semi-major axis and the eccentricity of the 
impacting trajectory must now be expressed in terms of the four known values.  The energy 
equation, or equation (66), can be used to find the semi-major axis. 
     
           
 
                                                                       
Equation (43) can then be used to solve for the eccentricity. 
   ?    
 
                                                                      
56 
 
Equation (99) can then be substituted into equation (100). 
   ?    
            
                                                              
Equation (102) expresses the angular momentum of the sub-satellite?s impact trajectory. 
    ?   ?   | ?|| ?|                                                            
The point of release for the sub-satellite is located at the point near the apogee of the impact 
trajectory, where the flight path angle is zero.  The angular momentum can now be expressed in 
terms of the magnitudes of the position and velocity vectors of the sub-satellite after release. 
                                                                                      
The velocity of the sub-satellite can be written as 
 ?    ?     ?    ?    ?                                                          
The velocity of the center of mass can be found by using equation (2).  The angular velocity only 
has a single vector component. 
 ?      ?                                                                         
The velocity vector for the sub-satellite can be written as 
 ?  [                  ]  ?  [         (        )]  ?                     
The final step is to calculate the magnitude of the sub-satellite velocity at release.  A change in 
velocity can then be added to the velocity of the sub-satellite to get the velocity at release, as 
57 
 
shown in equation (84).  The magnitude of the sub-satellite velocity can then be found by taking 
the square root of the components squared. 
 Now that all the terms have been found for the true anomaly the range equation can be 
defined.  The true anomaly at release is assumed to be one hundred and eighty degrees.  
Equations (99), (101), and (102) are substituted into equation (95) to get the true anomaly at the 
impact point.  The true anomaly at the impact point is then substituted into equation (97).  The 
range formula given in equation (107) is now a function of the altitude, tether length, angular 
velocity, and velocity change of the sub-satellite.   
    (|     (     
     
   )   |)                                            
Where, 
    
           
 
                                                                    
   ?           
 
                                                                  
 In order to get the time to impact for the impact trajectory, the time at release and the 
time at impact must be calculated.  The time at release can then be found by taking half of the 
orbital period, given by 
           ? 
 
                                                                     
58 
 
Since the true anomaly at impact is known, it can be substituted into equation (73) to get the 
eccentric anomaly at impact.  The time at impact can then be found by using equations (68) and 
(69), giving 
                      
?    
                                                       
The initial time in equation (111) is equal to zero because it is the time at perigee. 
                  
?    
                                                           
This same method can be used to find the time at release if the true anomaly at release is not 
equal to one hundred and eighty degrees.  The true anomaly at release would be used in equation 
(73) to find the eccentric anomaly at release. 
The time to impact can now be found by subtracting the time at impact from the time at 
release, as 
                                                                                  
 Another method can be used to find the position of the sub-satellite if the center of mass 
position and velocity vector, and the rotational angle are known in addition to the other values.  
The position vector between the center of mass and the sub-satellite is defined in equation (114). 
 ?             ?            ?                                               
Substituting into equation (32) gives 
59 
 
 ?  (            )  ?                  ?                                   
which has a magnitude of 
   ?                                                                    
For the case when the angular momentum of the sub-satellite is 
 ?  (    ?     ?)  (       ?        ?)                                              
Equation (117) can then be simplified to 
 ?  (                   ) ?                                                            
Having the magnitude of  
  ?                                                                                  
Equation (116) and (119) can then be used in the second method to find the range and time to 
impact, when the center of mass position and velocity vector, and the rotational angle are known.  
3.4.2 Algorithm to Find the Range and Time to Impact 
 Since all the equations are defined for the analytical solution to find the range and time to 
impact, an algorithm can be developed for use in a computer program, such as MATLAB, or for 
use in a hand calculation of the results.  Remember that the known values for this algorithm are 
the altitude of the main satellite, the tether length, the angular velocity, and the impulsive 
velocity change of the sub-satellite.  The algorithm is listed in step format beginning on the next 
page. 
60 
 
Step 1: Calculate the position of the sub-satellite at release, position of the main satellite, and the 
position of the center of mass.  The position of the sub-satellite is calculated using table 3.1 and 
the position of the center of mass is calculated using 
 ?     ?      ?    
     
                                                                              
Step 2: Calculate the velocity of the center of mass using 
      ?   
  
                                                                                     
Step 3: Calculate the velocity of the sub-satellite at release using  
 ?  [                  ]  ?  [         (        )]  ?                      
 ?       ?     ?                                                                        
Step 4:  Calculate the semi-major axis using  
     
           
 
                                                                         
Step 5: Calculate the angular momentum using 
                                                                                      
Step 6: Calculate the eccentricity using  
   ?    
 
                                                                             
61 
 
Step 7: Calculate the perigee position using  
                                                                                   
and test for impact using equation (119). 
                            
                                                                          
Step 8: Calculate true anomaly at impact using 
           (     
     
   )                                                     
And assign the true anomaly at release value using equation (121).  
                                                                                     
The true anomaly at impact must then be placed in the correct quadrant using  
             
                                                                                   
Step 9: Calculate the range using  
     (|          |)                                                                 
Note: Step eight and nine can be combined into one step by using  
    (|     (     
     
   )   |)                                            
Where, 
62 
 
    
           
 
                                                                    
   ?           
 
                                                                  
Step 10: Calculate the time at release using 
           ? 
 
                                                                     
Step 11: Calculating the eccentric anomaly at impact using 
        [(      )
  ?
   (  )]                                                      
Step 12: Calculating the time at impact using  
                  
?    
                                                           
If the true anomaly at release is much less than or much greater than one hundred and eight 
degrees use equation (122) to find the time at impact. 
                                                                         
Step 13: Calculate the time to impact. If the true anomaly at impact was found to already be 
greater than the true anomaly at release use 
                                                                                  
63 
 
If the true anomaly at impact was changed in step 8, use equation (123) to calculate the time to 
impact. This eliminates the issue caused by a negative eccentric anomaly.  
                                                                               
If the true anomaly at release is much less than or much greater than one hundred and eight 
degrees use equations (122) and (113) to find the time to impact. 
 If the center of mass position and velocity vectors and the rotational angle are known in 
addition to the other values, some steps in the algorithm will change. The changes to the 
algorithm are listed below. 
Step 1: Calculate the length between the center of mass and the sub-satellite using 
                                                                                ) 
Then calculate the position of the sub-satellite at release using  
 ?  (            )  ?                  ?                                   
Step two can then be omitted.  
Step 5: Calculate the angular momentum using equation (119). 
  ?                                                                                  
Step 8: Calculate the true anomaly at release using  
           (     
     
   )                                                        
64 
 
The final steps remain the same for this set of given values.   
3.4.3 Equations to Find the Velocity Change and Angular Velocity 
 In this section equations are developed to find the impulsive velocity change of the sub-
satellite and the equivalent angular velocity of the TSS needed to cause an impact if an impulsive 
velocity change is not to be placed on the system given a set of known values.  Five known 
values are needed for each solution.  Four of the known values remain the same for each 
solution.  These are the range, the true anomaly at impact, the altitude, and the tether length.  The 
fifth known value depends upon what is being solved for.  If the velocity change of the sub-
satellite is to be determined, the angular velocity of the TSS must be known.  When the change 
in the angular velocity of the TSS is being solved for, the velocity change of the sub-satellite 
caused by the angular velocity must be known.  Since the known values have changed the 
equations found in section 3.4.1 need to be rearranged to find the velocity change and the change 
angular velocity. 
 With this analytical solution the assumption that the point of release is located at the 
apogee of the impact trajectory no longer holds.  This means that the true anomaly at release 
must be determined.  This is done by first getting rid of the absolute value in equation (97) by 
multiply the left hand side by a negative one and then solving for the true anomaly at the release 
point for the sub-satellite. 
             
 
                                                                 
       
 
                                                                       
65 
 
The next step is to use the true anomaly at impact to find the semi-major axis of the impacting 
trajectory in terms of the eccentricity of the impacting trajectory.  This is done by solving for the 
semi-major axis in equation (99). 
                                                                                 
Equation (126) is then substituted into equation (94). 
           (  (          )       
 
)                                           
Equation (127) is then used to solve for the eccentricity of the impact trajectory. 
        
                  
                                                        
 Two different equations must be developed for finding the impulsive velocity change of 
the sub-satellite and for finding the change in angular velocity of the TSS.  In order to find the 
velocity change of the sub-satellite when given an angular velocity, equation (38) must be 
substituted into equation (84). 
 ?      ?     ?   ?     ?    ?      ?                                        
Since the angular velocity of the TSS is assumed to be zero when an impulsive velocity change 
is applied, the cross product in equation (129) is dropped and the velocity change can be solved 
for by using equations (130) and (131). 
                                                                             
                                                                               
66 
 
The equivalent angular velocity for a velocity change of the sub-satellite can be found using 
equations (86) and (87) depending upon the direction of the velocity change.  
3.4.4 Algorithm to Find the Velocity Change and Angular Velocity 
 The algorithm to find the impulsive velocity change of the sub-satellite and the algorithm 
to find the change in the angular velocity of the TSS to cause an impact of the sub-satellite given 
a set of known parameters vary slightly.  The first algorithm that will be looked at is the 
algorithm to solve for the impulsive velocity change of the sub-satellite.   
Step 1 ? 6: Same as the steps discussed in section 3.4.2. 
Step 7: Calculate the velocity change using equation (130) or equation (131). 
                                                                             
                                                                               
If the velocity change calculated is reasonable, then the algorithm can be stopped.  If the velocity 
change is not reasonable, the altitude or tether length must be increases for decreased to reach a 
reasonable value. 
 The second algorithm that will be looked at is the algorithm to solve for the equivalent 
angular velocity change of the sub-satellite.   
Step 1 ? 6: Same as the steps discussed in section 3.4.2. 
Step 7: Calculate the equivalent angular velocity using equation (86) or equation (87). 
                  
        
                                                                
67 
 
               
       
                                                                 
If the angular velocity change calculated is reasonable, then the algorithm can be stopped.  If the 
angular velocity change is not reasonable, the altitude or tether length must be increases for 
decreased to reach a reasonable value. The effects of the change in altitude and tether length on 
the angular velocity and on the velocity change of the sub-satellite are shown in chapter four. 
 
 
 
 
 
 
 
 
 
 
 
 
 
68 
 
 
 
 
Chapter 4: Results for Analytical Solutions 
The cases that will be considered for this chapter are cases A through D, which were 
described in chapter three, section four.  Case A and B are when the sub-satellite is located 
directly above and directly below the main satellite, while Case C and D are when the sub-
satellite is located immediately to the right, in the direction opposite the velocity vector, and 
immediately to the left, in the same direction as the velocity vector, of the main satellite.  The 
first section of this chapter will discuss the ground range and the time to impact found by using 
the algorithm discussed in chapter three, section 3.2.  The second section discusses the results 
when the algorithm discussed in chapter three, section 3.4 is used to find the impulsive velocity 
change and the change in the angular velocity of the TSS. 
4.1 Analytical Results for the Range and Time to Impact 
 For this case the altitude, tether length and the impulsive velocity change are known and 
are used to calculate the ground range and time to impact.  For the results presented here, the 
altitude and tether length are kept constant, while the impulsive velocity change is increased 
from zero kilometers per second to six kilometers per second.  Three different set-ups for the 
altitude and the tether length are evaluated.  The three set-ups are a five hundred kilometer 
altitude and a four kilometer tether length, an eight hundred kilometer altitude and a four 
kilometer tether length, and a five hundred kilometer tether length and a forty kilometer tether.  
These three set-ups are used in order to investigate the results obtained by increasing the altitude 
and increasing the tether length. 
69 
 
4.1.1 Case A Results 
 The position of the TSS at the point of release was shown in figure 3.2.  In this situation 
the velocity of the sub-satellite is directed solely in the negative Y ? direction and it is assumed 
any impulsive velocity change that takes place must be done in the positive Y ? direction to 
create impact.  The trends for the ground range and the time to impact as the magnitude of the 
impulsive velocity change increases hold for all three set-ups. 
 
Figure 4.1: Ground Range vs. +?V1y for Case A 
0
2000
4000
6000
8000
10000
12000
14000
16000
0 1 2 3 4 5 6 7
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Change in Velocity in the Y - Direction, Delta Vy (km/s) 
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
70 
 
 
Figure 4.2: Time to Impact vs. +?V1y Increases for Case A 
A minimum impulsive velocity change in the positive Y ? direction is required in order to 
create an impact trajectory for each of the three set-ups.  The minimum impulsive velocity 
needed to create an impact point changes when the altitude is increased to eight hundred 
kilometers; however, a change in the tether length does not result in a change to the minimum 
impulsive velocity change needed for an impact trajectory.  The minimum impulsive velocity 
change values needed for the three set-ups are listed in table 4.1. 
Table 4.1: Minimum Impulsive Velocity Change Needed for Case A 
Altitude (km) Tether Length (km) 
Minimum Velocity Change needed in 
+ Y - direction (km/s) 
500 4 0.2 
800 4 0.3 
500 40 0.2 
 
 The maximum ground range and time to impact occur at the minimum impulsive velocity 
change, while the minimum ground range and time to impact occur at the maximum impulsive 
0
500
1000
1500
2000
2500
0 1 2 3 4 5 6 7
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
 
Change in Velocity in the Y - Direction, Delta Vy (km/s) 
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
71 
 
velocity change.  Depending upon the altitude and the tether length the value for the ground 
range and time to impact could increase or decrease.  For these three cases an increase in the 
tether length to a value of forty kilometers, results in the largest increase in the ground range and 
the time to impact.  The increase in the altitude to eight hundred kilometers results in a small 
decrease in the ground range and time to impact.  These effects are shown in figures 4.1 and 4.2.  
The next two tables list the actual maximum and minimum values for the ground range and time 
to impact for each of the set-ups.   The maximum values occur at the minimum impulsive 
velocity change placed on the sub-satellite for the three cases listed. 
Table 4.2 Maximum Ground Range and Time to Impact for Case A 
Altitude 
(km) 
Tether Length 
(km) 
Ground Range 
(km) 
Time to Impact 
(s) 
500 4 13065.3 1793.5 
800 4 13220.3 1899.3 
500 40 15165.6 2077.0 
 
The maximum value for the ground range and the time to impact occur when the tether length 
increases to forty kilometers and the velocity change is 0.2 kilometers per second in the positive 
Y ? direction.  The increase in altitude leads to an increase in the ground range and the time to 
impact, but an increased velocity change is needed in order to create an impact point at the 
higher altitude.  This results in the smaller increase to the ground range and time to impact. 
Table 4.3: Minimum Ground Range and Time to Impact for Case A 
Altitude 
(km) 
Tether Length 
(km) 
Ground Range 
(km) 
Time to Impact 
(s) 
500 4 549.9 349.8 
800 4 636.5 456.4 
500 40 570.1 363.8 
 
72 
 
The minimum values for the ground range and time to impact occur when the maximum 
impulsive velocity change is placed on the sub-satellite.  For Cases A through D this maximum 
velocity change is six kilometers per second in the positive Y ? direction.  As the velocity change 
increases the ground range and the time to impact are decreased.  The true minimum value 
comes from the first set-up, where the altitude is five hundred kilometers and the tether length is 
four kilometers.  This time the increase in altitude results in a higher minimum value for the 
ground range and time to impact.  The third set-up serves as a midpoint for the minimum values 
that can be obtained. 
4.1.2 Case B Results 
 For case B, the sub-satellite is located below the main satellite.  The center of mass, main 
satellite, and sub-satellite have a velocity solely in the negative Y ? direction.  In order to cause 
the sub-satellite to enter an impact trajectory, the impulsive velocity change of the sub-satellite 
must be in the positive Y ? direction.  The ground range and time to impact decrease as the 
impulsive velocity change in the positive Y ? direction is increase.  The decrease in the ground 
range and the time to impact as the impulsive velocity change increases are shown in figures 4.3 
and 4.4 on the next page. 
73 
 
 
 
Figure 4.3: Ground Range vs. +?V1y for Case B 
 
 
Figure 4.4: Time to Impact vs. +?V1y for Case B 
The minimum impulsive velocity change in the positive Y ? direction needed for the sub-satellite 
to have an impact trajectory for each of the three set-ups is listed in table 4.4. 
 
0
2000
4000
6000
8000
10000
12000
14000
0 1 2 3 4 5 6 7
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Change in Velocity in the Y - Direction, Delta Vy (km/s) 
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 1 2 3 4 5 6 7
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
 
Change in Velocity in the Y - Direction, Delta Vy (km/s) 
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
74 
 
Table 4.4: Minimum Impulsive Velocity Change Needed for Case B 
Altitude (km) Tether Length (km) 
Minimum Velocity Change needed in 
+ Y - direction (km/s) 
500 4 0.2 
800 4 0.3 
500 40 0.2 
 
The maximum and minimum ground range and time to impact occur at the minimum impulsive 
velocity change and the maximum impulsive velocity change of six kilometers per second, 
respectively.  An increase in the altitude leads to a larger increase in the ground range and time 
to impact, than an increase in the tether length. 
Table 4.5: Maximum Ground Range and Time to Impact for Case B 
Altitude 
(km) 
Tether Length 
(km) 
Ground Range 
(km) 
Time to Impact 
(s) 
500 4 12696.5 1743.2 
800 4 12980.0 1865.2 
500 40 11279.4 1548.9 
 
The maximum ground range and time to impact occur when the altitude is increased because as 
the altitude increases the sub-satellite moves further away from the surface of the Earth.  The 
lowest maximum value occurs when the tether length is increased, because the sub-satellite gets 
closer to the surface of the Earth as the tether increases.  If the altitude is increased further, the 
ground range and time to impact should also increase; however, if the tether length increases 
instead, the ground range and time to impact will decrease.   
 
 
75 
 
Table 4.6: Minimum Ground Range and Time to Impact for Case B 
Altitude 
(km) 
Tether Length 
(km) 
Ground Range 
(km) 
Time to Impact 
(s) 
500 4 545.3 346.7 
800 4 633.1 453.7 
500 40 524.3 332.4 
 
The minimum values for the ground range and the time to impact are affected by the increase in 
the altitude and tether length in the same way that the maximum values are affected.  The largest 
ground range and time to impact happens when the altitude increases, while the true minimum 
occurs when the tether length is increased.  The ground range and time to impact calculated for 
case B in all set-ups is less than the ground range and time to impact calculated in case A. 
4.1.3 Case C Results 
 The sub-satellite is located immediately to the right of the main satellite.  In this position 
the sub-satellite?s velocity is only in the negative Y ? direction; therefore, an impulsive velocity 
change in the positive Y ? direction is needed in order to create an impact for the sub-satellite.  
As the impulsive velocity change in the positive Y ? direction is increased, the ground range and 
the time to impact decrease.  The trends mentioned above can be seen in figures 4.5 and 4.6 on 
the next page. 
76 
 
 
Figure 4.5: Ground Range vs. +?V1y for Case C 
  
Figure 4.6: Time to Impact vs. +?V1y for Case C 
The minimum impulsive velocity change needed to create the first impact trajectory for each set-
up is listed on the next page in table 4.7. 
 
 
 
0
2000
4000
6000
8000
10000
12000
14000
16000
0 1 2 3 4 5 6 7
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Change in Velocity in the Y - Direction, Delta Vy (km/s) 
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 1 2 3 4 5 6 7
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
 
Change in Velocity in the Y - Direction, Delta Vy (km/s) 
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
77 
 
Table 4.7: Minimum Impulsive Velocity Change Needed for Case C 
Altitude (km) Tether Length (km) 
Minimum Velocity Change needed 
in + Y - direction (km/s) 
500 4 0.2 
800 4 0.3 
500 40 0.2 
 
The maximum ground range and time to impact will occur at the minimum impulsive velocity 
changes listed in table 4.7 for each set-up.  When the impulsive velocity change reaches the 
maximum value of six kilometers per second, the ground range and time to impact will be at 
their minimum values.  The maximum and minimum values are listed for the ground range and 
time to impact below in table 4.8 and table 4.9. 
Table 4.8: Maximum Ground Range and Time to Impact for Case C 
Altitude 
(km) 
Tether Length 
(km) 
Ground Range 
(km) 
Time to Impact 
(s) 
500 4 12945.5 1758.1 
800 4 13140.0 1875.6 
500 40 13535.9 1667.9 
 
Table 4.9: Minimum Ground Range and Time to Impact for Case C 
Altitude 
(km) 
Tether Length 
(km) 
Ground Range 
(km) 
Time to Impact 
(s) 
500 4 547.8 348.2 
800 4 635.0 455.0 
500 40 549.4 347.2 
 
The largest maximum value that is obtained for case C occurs when the tether is increased, 
because an increase in the tether length places the sub-satellite further away from the impact 
78 
 
point. The largest minimum value for case C occurs when the altitude is increased because the 
altitude places the sub-satellite in a higher orbit.  
4.1.4 Case D Results 
 The position of the sub-satellite for case D is directly to the left main satellite.  In this set-
up the velocity of the main satellite, the center of mass, and the sub-satellite is only in the 
negative Y ? direction; therefore, the impulsive velocity change needs to be in the positive Y ? 
direction in order to cause the sub-satellite to enter an impact trajectory.  The ground range and 
time to impact for all three set-ups decreases as the impulsive velocity change in the positive Y ? 
direction increases.  The change in ground range is shown below, while the change in the time to 
impact is shown in figure 4.8 on the next page. 
 
Figure 4.7: Ground Range vs. +?V1y for Case D 
0
2000
4000
6000
8000
10000
12000
14000
16000
0 1 2 3 4 5 6 7
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Change in Velocity in the Z - Direction, Delta Vz (km/s) 
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
79 
 
 
 
Figure 4.8: Time to Impact vs. +?V1y for Case D 
The minimum impulsive velocity change needed to send the sub-satellite on an impact trajectory 
follows the same trends that are in the previous cases.   
Table 4.10 Minimum Impulsive Velocity Change Needed for Case D 
Altitude (km) Tether Length (km) 
Minimum Velocity Change needed in 
+ Y - direction (km/s) 
500 4 0.2 
800 4 0.3 
500 40 0.2 
 
The maximum and minimum values for the ground range and the time to impact are listed in 
tables 4.11 and 4.12, respectively.  The maximum ground range and time to impact occur when 
the minimum impulsive velocity changes listed in table 4.10 are placed on the sub-satellite.  The 
minimum ground range and time to impact occur when the impulsive velocity change is equal to 
six kilometers per second. 
 
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 1 2 3 4 5 6 7
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
 
Change in Velocity in the Z - Direction, Delta Vz (km/s) 
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
80 
 
Table 4.11: Maximum Ground Range and Time to Impact for Case D 
Altitude 
(km) 
Tether Length 
(km) 
Ground Range 
(km) 
Time to Impact 
(s) 
500 4 12945.5 1758.1 
800 4 13140.2 1875.6 
500 40 13535.1 1667.9 
 
Table 4.12: Minimum Ground Range and Time to Impact for Case D 
Altitude 
(km) 
Tether Length 
(km) 
Ground Range 
(km) 
Time to Impact 
(s) 
500 4 547.8 348.2 
800 4 635.0 455.0 
500 40 549.4 347.2 
 
The values for the maximum and minimum ground range and time to impact are the same as the 
values in case C.  The only difference is the position vector of the sub-satellite in both cases; 
therefore, the trajectories are only shifted over because of the sub-satellite?s orientation with the 
main satellite.   
4.2 Analytical Results for the Impulsive Velocity Change and the Angular Velocity 
 In this section it is assumed that the altitude, tether length, ground range, and true 
anomaly at impact are known.  The algorithm discussed in chapter three, section 3.4 is used to 
get the results in this section.  Two scenarios are looked at to use the analytical results to find the 
impulsive velocity change and change in angular velocity of the TSS at different altitudes and 
tether lengths.  The first scenario sets the ground range at fifteen hundred kilometers with a true 
anomaly at impact of two hundred and forty degrees or 4.1888 radians.  The second scenario sets 
the ground range at three thousand kilometers with a true anomaly at impact of two hundred and 
forty degrees.  As each scenario is looked at, case A and B are placed on the same plot because 
81 
 
in both situations the velocity of the sub-satellite and the impulsive velocity change are along the 
Y ? axis, while case C and D are placed on the same plot.  For all cases the velocity of the sub-
satellite is in the negative Y ? direction and the impulsive velocity change is in the positive Y ? 
direction.   
4.2.1 Ground Range of 1500 km and           
 The impulsive velocity change and the equivalent change in angular velocity of the TSS 
were first found for case A and case B.  The altitude starts at five hundred kilometers and is then 
increased to fifteen hundred kilometers in increments of one hundred kilometers.  As the altitude 
increases the impulsive velocity change and the change in angular velocity of the TSS needed to 
reach a ground range of fifteen hundred kilometers decreases in magnitude.  The impulsive 
velocity change for case A and B are slightly different as the altitude increases and are in the 
same direction.  For case A the change in angular velocity must be in the negative X ? direction 
to cause a clockwise rotation of the TSS to create the same velocity change.  The change in 
angular velocity must be in the positive direction for case B to cause a counter-clockwise 
rotation.  The plots of the impulsive velocity change and the change in angular velocity as a 
function of changing altitude are shown on the next page and are followed by a table that gives 
specific values for the velocity change and the angular velocity at three different altitudes. 
82 
 
 
 
Figure 4.9 Velocity Change vs. Altitude for Case A and B; R = 1500 km and ?imp = 240? 
 
 
Figure 4.10 Angular Velocity vs. Altitude for Case A and B; R = 1500 km and ?imp = 240? 
 
 
 
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 500 1000 1500 2000
Ch
an
ge
 in
 Ve
loc
ity 
in t
he
 Y 
- D
ire
ction
, 
De
lta 
Vy
 (km
/s)
 
Altitude, H (km) 
Case A
Case B
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 500 1000 1500 2000
An
gu
lar
 Ve
loc
ity,
 Om
eg
a_
x (r
ad
s/s)
 
Altitude, H (km) 
Case A
Case B
83 
 
Table 4.13 Impulsive Velocity Change and Angular Velocity at Three Altitudes for Case A and 
B When R = 1500 km and ?imp = 240? 
 
Case A Case B 
Altitude 
(km) 
Velocity Change 
(km/s) 
Angular 
Velocity (rads/s) 
Velocity Change 
(km/s) 
Angular 
Velocity (rads/s) 
500 0.6048 -0.1663 0.5981 0.1645 
1000 0.5012 -0.1378 0.5025 0.1382 
1500 0.0681 -0.0187 0.0731 0.0201 
 
From table 4.13 it can be seen that the values for the impulsive velocity change and the 
magnitude of the change in angular velocity differ slightly, because of the placement of the sub-
satellite with respect to the main satellite. For both cases as the altitude increases the required 
impulsive velocity change and change in angular velocity of the system is decreased.  
 The next change that is placed on case A and B is an increase in the tether length.  The 
tether length starts at four kilometers and is increased to one hundred and fourteen kilometers in 
increments of ten kilometers.  As the tether length increases the impulsive velocity change of the 
sub-satellite is decreased for case B, while the impulsive velocity change increases for case A.  
The magnitude of the equivalent angular velocity needed to reach the same ground range and 
true anomaly at impact decreases as the tether length increases.  These trends are shown in 
figures 4.11 and 4.12.  The plots are then followed by a table that lists specific impulsive velocity 
changes and angular velocities that are needed to achieve the desired ground range and impact 
point. 
84 
 
 
 
Figure 4.11 Velocity Change vs. Tether Length for Case A and B; R = 1500 km and ?imp = 270? 
  
Figure 4.12 Angular Velocity vs. Tether Length for Case A and B; R = 1500 km and ?imp = 240? 
 
 
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 20 40 60 80 100 120
Ch
ange i
n Vel
oc
ity
 in
 th
e Y
 - D
irec
tion
, 
De
lta 
Vy
 (km
/s)
 
Tether Length, Lt (km) 
Case A
Case B
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 20 40 60 80 100 120
An
gu
lar
 Ve
loc
ity,
 Om
eg
a_
x (r
ad
s/s)
 
Tether Length, Lt (km) 
Case A
Case B
85 
 
Table 4.14: Impulsive Velocity Change and Angular velocity at Three Tether Lengths for Case A 
and B When R = 1500 km and ?imp = 240? 
 
Case A Case B 
Tether 
Length 
(km) 
Velocity Change 
(km/s) 
Angular 
Velocity (rads/s) 
Velocity Change 
(km/s) 
Angular 
Velocity (rads/s) 
4 0.6048 -0.1663 0.5981 0.1645 
54 0.6431 -0.0131 0.5515 0.0112 
114 0.6804 -0.0066 0.4856 0.0047 
 
The impulsive velocity changes increase for case A as the tether length increases because the 
sub-satellite is located above the main satellite.  The impulsive velocity change decreases for 
case B because the sub-satellite is located below the main satellite.  The change in angular 
velocity of the TSS needed to create the same ground range and impact point in case A and B 
decrease because the increase in tether length is a larger value than the velocity change.   
 Now the effect of increasing altitude and tether length will be investigated for cases C 
and D. The change in angular velocity of the TSS for both cases was found to be zero because an 
impulsive change in velocity in the positive Y ? direction on the sub-satellite will not lead to a 
rotation of the TSS.  A change in velocity in the positive or negative Z ? direction will lead to a 
rotation.  For a ground range of fifteen hundred kilometers and a true anomaly of two hundred 
and forty degrees, the increase in altitude will results in a decrease of the magnitude of the 
velocity change. 
86 
 
 
Figure 4.13 Velocity Change vs. Altitude for Case C and D; R = 1500 km and ?imp = 240? 
Table 4.15: Impulsive Velocity Change and Angular Velocity at Three Altitudes for Case C and 
D When R = 1500 km and ?imp = 270? 
 
Case C Case D 
Altitude 
(km) 
Velocity Change 
(km/s) 
Angular 
Velocity (rads/s) 
Velocity Change 
(km/s) 
Angular 
Velocity (rads/s) 
500 0.6015 N/A 0.6015 N/A 
1000 0.5019 N/A 0.5019 N/A 
1500 0.0706 N/A 0.0706 N/A 
 
At each altitude the magnitude of the impulsive velocity change is the same for each case.  This 
similarity occurs because the sub-satellite in case C and D has the same altitude as the main 
satellite. 
 The increasing tether length for case C and D leads to an increase in the magnitude of the 
impulsive velocity change.  The change in angular velocity is still equal to zero because of the 
reasons explained for the changing altitude.  Three values for impulsive velocity change and 
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 500 1000 1500 2000
Ch
an
ge
 in
 Ve
loc
ity 
in t
he
 Z-
Dir
ec
tion
, 
De
lta 
Vz (
km
/s)
 
Altitude, H (km) 
Case C
Case D
87 
 
change in angular velocity are shown in table 4.16 for both cases at an altitude of five hundred 
kilometers, one thousand kilometers, and fifteen hundred kilometers. 
 
Figure 4.14 Velocity Change vs. Tether Length for Case C and D; R = 1500 km and ?imp = 240?  
Table 4.16: Impulsive Velocity Change and Angular Velocity at Three Tether Lengths for Case 
C and D When R = 1500 km and ?imp = 240? 
 
Case C Case D 
Tether 
Length 
(km) 
Velocity Change 
(km/s) 
Angular 
Velocity (rads/s) 
Velocity Change 
(km/s) 
Angular 
Velocity (rads/s) 
4 0.6015 N/A 0.6015 N/A 
54 0.6017 N/A 0.6017 N/A 
114 0.6023 N/A 0.6023 N/A 
 
4.2.2 Ground Range of 3000 km and           
 The same increase in altitude and tether length are placed on the four cases; however, the 
desired ground range has been doubled in order to see if there are any changes to the impulsive 
velocity change and change in angular velocity trends by increasing the altitude and tether 
0.6014
0.6015
0.6016
0.6017
0.6018
0.6019
0.602
0.6021
0.6022
0.6023
0.6024
0 20 40 60 80 100 120
Ch
an
ge
 in
 Ve
loc
ity 
in t
he
 Y-
Dir
ec
tion
, 
De
lta 
Vy
 (km
/s)
 
Tether Length, Lt (km) 
Case C
Case D
88 
 
length.  As in the previous section, the first two cases that will be looked at are case A and B and 
the first change that will be investigated is the increase in the altitude of the main satellite.  The 
increase in altitude leads to an increase in the magnitude of the impulsive velocity change and 
change in angular velocity for both cases.  The impulsive velocity change and change in angular 
velocity of the TSS as a function of altitude are shown in figures 4.15 and 4.16. 
 
Figure 4.15 Velocity Change vs. Altitude for Case A and B; R = 3000 km and ?imp = 240? 
 
 
Figure 4.16 Angular Velocity vs. Altitude for Case A and B; R = 3000 km and ?imp = 240? 
0
0.2
0.4
0.6
0.8
1
1.2
0 500 1000 1500 2000
Ch
an
ge
 in
 Ve
loc
ity 
in t
he
 Y 
- 
Dir
ec
tion
, D
elta 
Vy
 (k
m/
s) 
Altitude, H (km) 
Case A
Case B
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 500 1000 1500 2000
An
gu
lar
 Vel
oc
ity
, Ome
ga_
x 
(rad
s/s)
 
Altitude, H (km) 
Case A
Case B
89 
 
Table 4.17: Impulsive Velocity Change and Angular Velocity at Three Altitudes for Case A and 
B When R = 3000 km and ?imp = 240? 
 
Case A Case B 
Altitude 
(km) 
Velocity Change 
(km/s) 
Angular 
Velocity (rads/s) 
Velocity Change 
(km/s) 
Angular 
Velocity (rads/s) 
500 0.5968 -0.1641 0.5857 0.1611 
1000 0.9200 -0.2530 0.9128 0.2510 
1500 1.0687 -0.2939 1.0641 0.2926 
 
With this ground range the increase in altitude leads to an increase in the impulsive velocity 
change because a greater velocity change is needed in order to create an impact at a higher 
altitude and to achieve a greater ground range.  As the velocity change increases, the change in 
angular velocity increases as well because a larger equivalent angular velocity is needed to create 
the same impact point.  The angular velocity for case A is negative because the angular velocity 
of the system needs to be in the negative X ? direction in order to create a clockwise rotation.  
The angular velocity is positive in order to create a counter-clockwise rotation for case B. 
 The increase in tether length will also affect the impulsive velocity change and the 
change in angular velocity for case A and B.  The impulsive velocity change placed on the sub-
satellite at release decreases with the increasing tether length for case B; however, the impulsive 
velocity change increases with increase tether length in case A.  The increase in tether length 
also causes the change in angular velocity of the TSS to decrease in magnitude.  This decrease in 
angular velocity is a result of the increase in the distance between the center of mass and the sub-
satellite.  As the distance increases the amount of rotation or angular velocity that is required to 
place the TSS in the proper orientation to cause the sub-satellite to enter an impact trajectory 
decreases.  The impulsive velocity change and change in angular velocity as a function of 
increasing tether length are shown in figures 4.17 and 4.18. 
90 
 
 
 
Figure 4.17 Velocity Change vs. Tether Length Case A and B; R = 3000 km and ?imp = 240? 
 
 
 
Figure 4.18 Angular Velocity vs. Tether Length for Case A and B; R = 3000 km and ?imp = 240? 
 
 
 
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 20 40 60 80 100 120
Ch
ange i
n Vel
oc
ity
 in
 th
e Y
 - D
irec
tion
, 
De
lta 
Vy
 (km
/s)
 
Tether Length, Lt (km) 
Case A
Case B
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 20 40 60 80 100 120
An
gu
lar
 Ve
loc
ity,
 Om
eg
a_
x (r
ad
s/s)
 
Tether Length, Lt (km) 
Case A
Case B
91 
 
Table 4.18: Impulsive Velocity Change and Angular Velocity at Three Tether Lengths for Case 
A and B When R = 3000 km and ?imp = 240? 
 
Case A Case B 
Tether 
Length 
(km) 
Velocity Change 
(km/s) 
Angular Velocity 
(rads/s) 
Velocity Change 
(km/s) 
Angular Velocity 
(rads/s) 
4 0.5968 -0.1641 0.5857 0.1611 
54 0.6642 -0.0135 0.5149 0.0105 
114 0.7414 -0.0072 0.4260 0.0041 
 
The impulsive velocity change for case A increases with increasing tether length because the 
sub-satellite is placed in a higher orbit away from the Earth.  The impulsive velocity change for 
case B decreases with increasing tether length because the sub-satellite is lowered into an orbit 
closer to the Earth.  The change in angular velocity of the TSS needed for impact decreases with 
increasing tether length.  The magnitudes of the angular velocities differ between the two cases 
because of the different impulsive velocity changes.  The angular velocity in case A is in the 
negative X ? direction because a clockwise rotation is needed for impact. 
 The increase in altitude and increase in tether length will now be applied to case C and D.  
From the previous case in section 4.2.1, it is expected that the results for case C and D for each 
change should be equal in magnitude and direction.  The first parameter that will be increased on 
case C and D is the altitude.  As the altitude increases the impulsive velocity change required for 
the sub-satellite to reach a ground range of three thousand kilometers at a true anomaly at impact 
of two hundred and forty degrees increase as well.  The change in angular velocity remains at 
zero because the impulsive velocity change direction in the positive Y ? direction does not create 
a rotation on the system.  Figure 4.19 shows the impulsive velocity change as a function of the 
92 
 
increase in altitude, and is followed by a table that list specific impulsive velocity change and 
change in angular velocity values for three altitudes. 
 
Figure 4.19 Velocity Change vs. Altitude for Case C and D; R = 3000 km and ?imp = 240? 
Table 4.19: Impulsive Velocity Change and Angular Velocity at Three Altitudes for Case C and 
D When R = 3000 km and ?imp = 240? 
 
Case C Case D 
Altitude 
(km) 
Velocity Change 
(km/s) 
Angular 
Velocity (rads/s) 
Velocity Change 
(km/s) 
Angular 
Velocity (rads/s) 
500 0.5912 N/A 0.5912 N/A 
1000 0.9164 N/A 0.9164 N/A 
1500 1.0664 N/A 1.0664 N/A 
 
As expected the magnitudes and directions of the impulsive velocity change are equal between 
the two cases.  This is a result of the placement of the sub-satellite in the same orbit height as the 
main satellite at release.  The larger ground range causes the impulsive velocity change to 
increase in order to reach the desired impact. 
0
0.2
0.4
0.6
0.8
1
1.2
0 500 1000 1500 2000
Ch
an
ge
 in
 Ve
loc
ity 
in t
he
 Y-
Dir
ec
tion
, 
De
lta 
Vy
 (km
/s)
 
Altitude, H (km) 
Case C
Case D
93 
 
 The last change that will be looked at is the increase in the tether length for case C and D.  
As the tether length increases the impulsive velocity change increases.  The change in angular 
velocity of the TSS is once again zero because of the direction of the velocity change.  The 
increase in the impulsive velocity change as a function of the tether length is shown below.  
Table 4.20 lists the values for the impulsive velocity change and the change in angular velocity 
at a tether length of four kilometers, fifty-four kilometers, and one hundred and fourteen 
kilometers.  The table can be found on the next page. 
 
 
Figure 4.20 Velocity Change vs. Tether Length for Case C and D; R = 3000 km and ?imp = 240? 
 
 
 
 
 
 
 
 
0.591
0.5912
0.5914
0.5916
0.5918
0.592
0.5922
0.5924
0.5926
0.5928
0 20 40 60 80 100 120
Ch
an
ge
 in
 Ve
loc
ity 
in t
he
Y-D
ire
ction
, 
De
lta 
Vy
 (km
/s)
 
Tether Length, Lt (km) 
Case C
Case D
94 
 
Table 4.20: Impulsive Velocity Change and Angular Velocity at Three Tether Lengths for case C 
and D When R = 3000 km and ?imp = 240? 
 
Case C Case D 
Tether 
Length 
(km) 
Velocity Change 
(km/s) 
Angular 
Velocity (rads/s) 
Velocity Change 
(km/s) 
Angular 
Velocity (rads/s) 
4 0.5912 N/A 0.5912 N/A 
54 0.5916 N/A 0.5916 N/A 
114 0.5926 N/A 0.5926 N/A 
 
The impulsive velocity change for both case C and D have the same magnitudes and directions 
because the sub-satellite for both cases is located at the same orbit height as the main satellite.   
 
 
 
 
 
 
 
 
 
 
95 
 
 
 
 
Chapter 5: Results for Numerical Integration Simulation 
The process discussed in section 3.1 is now implemented with the changes discussed in 
sections two and three in chapter three.  Three different impulsive velocity changes were chosen 
for the two different release configurations.  For each configuration and impulsive velocity 
change the effect of changes in altitude and tether length were investigated.  The first 
configuration corresponds to when the sub-satellite is below the main satellite.  The second 
configuration corresponds to when the sub-satellite is above the main satellite.  For each change 
in altitude and tether length three values are chosen to discuss in detail after the general trend is 
introduced.  For the change in altitude the tether length is kept constant at four kilometers.  The 
altitude is kept at five hundred kilometers when the tether length is changed.  A range of 
impulsive velocity changes are then looked at to determine the minimum impulsive velocity 
change needed to impact and the change in angular velocity of the TSS needed to impact.  For 
the range of impulsive velocity changes the altitude is kept at five hundred kilometers and the 
tether length is kept at four kilometers.  For all cases and release points the rotation angle is -0.5 
radians because it is a good mid-range oscillation that will keep the sub-satellite leading the 
system.  Some sample results for the numerical integration simulation can be found in Appendix 
A. 
 
 
96 
 
5.1 Results for First Configuration or Case 1 
 Three values were chosen for the impulsive velocity change of the sub-satellite as shown 
in table 5.1.  An impulsive velocity change in the negative Y-direction was not chosen because 
the velocity change would only increase the sub-satellite velocity and an impact trajectory would 
not be created. 
Table 5.1: Impulsive Velocity Changes for Case 1 
  Velocity Change (km/s) 
     1.0 
     N/A 
     1.0 
     -3.0 
 
5.1.1 Impulsive Velocity Change of 1.0 km/s in the Positive Y-Direction 
 The first change that will be looked at is the change in altitude.  Remember that the 
altitude defines the height above the Earth for the main satellite and the sub-satellite is a distance 
away from the main satellite equal to the tether length. 
 
Figure 5.1 Ground Range vs. Altitude for ?V1y = 1.0 km/s in Case 1 
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 500 1000 1500 2000
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Altitude, H (km) 
97 
 
For the first configuration an increase in the altitude of the main satellite leads to an increase in 
the range.  This makes sense because as the altitude is increased the sub-satellite is in a higher 
orbit and it travels for a longer time before impact is reached.  The time increase can be seen in 
Figure 5.2.  As the orbit of the sub-satellite gets higher the ground range covered by the sub-
satellite in its impact trajectory will increase as well.   
 
Figure 5.2 Time to Impact vs. Altitude for ?V1y = 1.0 km/s in Case 1 
An altitude of five hundred kilometers, one thousand kilometers, and one thousand five hundred 
kilometers are looked at in more detail.  Table 5.2 lists the ground range covered by the sub-
satellite after release and the time to impact. 
Table 5.2 Range and Impact Time for          km/s at three Altitudes for Configuration 1 
Altitude (km) Ground Range (km) Time to Impact (s) 
500 4505.7 700 
1000 6377.7 1055 
1500 7822.9 1370 
 
0
200
400
600
800
1000
1200
1400
1600
0 200 400 600 800 1000 1200 1400 1600
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
 
Altitude, H (km) 
98 
 
The lowest ground range and time to impact happen at an altitude of five hundred kilometers, 
while the maximum for each happens at an altitude of one thousand five hundred kilometers.  To 
achieve maximum ground range covered by the sub-satellite with a constant tether length and 
impulsive velocity change the altitude should be increased.  If a minimum ground range is 
needed the altitude should be decreased.  The next three figures show the trajectory for the sub-
satellite after release at the three altitude values and illustrate how the range and time to impact 
increase as the altitude increases.   
 
Figure 5.3 Trajectory at 500 km Altitude for ?V1y = 1.0 km/s in Case 1 
99 
 
 
 
Figure 5.4 Trajectory at 1000 km Altitude for ?V1y = 1.0 km/s in Case 1 
 
 
 
Figure 5.5 Trajectory at 1500 km Altitude for ?V1y = 1.0 km/s in Case 1 
100 
 
Increasing the tether length also affects the time to impact and the ground range covered 
by the sub-satellite after release.  As the tether length increases the ground range and the time to 
impact decrease as well.  This is due to the fact that the sub-satellite is located below the main 
satellite. 
 
Figure 5.6 Ground Range vs. Tether Length for ?V1y = 1.0 km/s in Case 1 
 
 
Figure 5.7 Time to Impact vs. Tether Length for ?V1y = 1.0 km/s in Case 1 
4050
4100
4150
4200
4250
4300
4350
4400
4450
4500
4550
0 20 40 60 80 100 120
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Tether Length, Lt (km) 
630
640
650
660
670
680
690
700
710
0 20 40 60 80 100 120
Tim
e t
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pac
t, t_i
mp
ac
t (s)
 
Tether Length, Lt (km) 
101 
 
 
A tether length of four kilometers, fifty-five kilometers, and one hundred kilometers are 
examined in more detail.  The ground range and time to impact for these three cases can be 
found in table 5.3. 
Table 5.3: Range and Impact Time for          km/s at Three Tether Lengths for Case 1 
Tether Length 
(km) Ground Range (km) Time to Impact (s) 
4 4505.7 699 
55 4297.5 665 
100 4110.8 635 
 
The maximum range and time to impact occur when the tether length is shorter.  If a 
shorter ground range or time to impact is desired, an increase in tether length can achieve the 
desired results instead of decreasing the altitude.  Figure 5.3 shows the impact trajectory of the 
sub-satellite at an altitude of five hundred kilometers and a tether length of four kilometers.  The 
next two figures show the impact trajectories of the sub-satellite for the other two tether lengths.  
The changes in the impact trajectory of the sub-satellite after release are small.  The sub-satellite 
travels along a similar impact trajectory path for each tether length; however, as the tether 
increases the point of impact moves closer to the point of release.  This movement of the impact 
point results in a decrease of the ground range and time to impact of the sub-satellite. 
102 
 
 
Figure 5.8 Trajectory at 55 km Tether Length for ?V1y = 1.0 km/s for Case 1 
 
 
 
Figure 5.9 Trajectory at 100 km Tether Length for ?V1y = 1.0 km/s for Case 1 
103 
 
 Finally, the impulsive velocity change of the sub-satellite is increased while the altitude 
and tether length are kept constant at five hundred kilometers and four kilometers, respectively.  
The effect of the impulsive velocity change increase on the ground range and time to impact can 
be determined by inspecting figures 5.10 and 5.11. 
 
Figure 5.10 Ground Range vs. +?V1y for Case 1 
 
Figure 5.11 Time to Impact vs. +?V1y for Case 1 
0
2000
4000
6000
8000
10000
12000
14000
0 1 2 3 4 5 6 7
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Change in Velocity in the Y-Direction, Delta Vy (km/s) 
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 1 2 3 4 5 6 7
Tim
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Change in Velocity in the Y-Direction, Delta Vy (km/s) 
104 
 
As the impulsive velocity change is increased the ground range and the time to impact 
decreases because the larger the impulsive velocity change is the smaller the impact orbit will be.  
In other words the point of impact will move closer to the release point.  The minimum 
impulsive velocity change needed to cause an impact was 0.2 km/s in the positive Y-direction.  
The maximum and minimum values for the ground range and time to impact are shown in table 
5.4. 
Table 5.4 Maximum and Minimum Values as       Increases for Case 1 
Velocity 
Change 
(km/s) 
Ground Range 
(km) 
Time to Impact 
(s) 
0.2 12746.8 1755 
3 1924.3 430 
6 545.6 350 
 
 
While an impulsive velocity change of six kilometers achieves the shortest ground range and 
time to impact, the fuel needed to achieve this impulsive velocity change and impact values will 
be greater than the fuel needed to achieve impact at the lowest impulsive velocity change.  It is 
important to remember that there are trade-offs when selecting an impulsive velocity change.  
5.1.2 Impulsive Velocity Change of 1.0 km/s in the Positive Z ? Direction  
 For this impulsive velocity change, an increase in the altitude will also lead to an increase 
in the ground range covered by the sub-satellite at release and the time to impact.  This matches 
with the results found in the previous section.  The increase of the ground range and time to 
impact can be seen in figures 5.12 and 5.13.  The maximum and minimum values for the ground 
range and time to impact are listed in table 5.5. 
105 
 
 
Figure 5.12 Ground Range vs. Altitude for ?V1z = 1.0 km/s in Case 1 
 
Figure 5.13 Time to impact vs. Altitude for ?V1z = 1.0 km/s in Case 1 
The impulsive velocity change placed on the sub-satellite does not result in an impact at higher 
altitudes; however, if the impulsive velocity change for the sub-satellite is increased an impact 
can occur.  For this velocity change an altitude of five hundred kilometers, six hundred and fifty 
kilometers, and eight hundred and fifty kilometers will be looked at in more detail. 
 
23000
24000
25000
26000
27000
28000
29000
0 200 400 600 800 1000
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Altitude, H (km) 
0
1000
2000
3000
4000
5000
6000
0 200 400 600 800 1000
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
 
Altitude, H (km) 
106 
 
Table 5.5 Range and Impact Time for          km/s at Three Altitudes for Case 1 
 
Altitude (km) 
Ground Range 
(km) Time to Impact (s) 
500 24015.7 3920 
650 25498.6 4235 
850 28753.9 4825 
 
The ground range and time to impact are larger than the values in the previous section in table 
4.2 because the sub-satellite is lofted up higher in its impact trajectory.  Another way to phrase it 
is that the angle of the sub-satellite?s impact trajectory at release is greater than the angle for the 
impact trajectory at release when the velocity change in the positive Y ? Direction was done.  As 
the altitude is increased the impact trajectory of the sub-satellite becomes longer and the point of 
impact moves further around the Earth.  The impact trajectories for the sub-satellite at an altitude 
of five hundred kilometers, six hundred and fifty kilometers, and eight hundred and fifty 
kilometers can be seen in figures 5.14, 5.15, and 5.16. 
 
Figure 5.14 Trajectory at 500 km Altitude for ?V1z = 1.0 km/s in Case 1 
107 
 
 
 
Figure 5.15 Trajectory at 650 km Altitude for ?V1z = 1.0 km/s in Case 1 
 
5.16 Trajectory at 850 km Altitude for ?V1z = 1.0 km/s in Case 1 
 An increasing tether length for an impulsive velocity change in the positive Z ? direction 
has the same impact on the ground range and time to impact as it did when the impulsive 
108 
 
velocity change was in the positive Y ? direction.  Both values decrease as the tether length 
increases; therefore, the maximum ground range and time to impact will occur at the smallest 
tether length and the minimum ground and time to impact will occur at the largest tether length 
value.  The decrease in ground range and time to impact can be seen in figures 5.17 and 5.18.  A 
list of ground ranges and times to impact are given in table 5.6. 
 
Figure 5.17 Ground Range vs. Tether Length for ?V1z = 1.0 km/s in Case 1 
21500
22000
22500
23000
23500
24000
24500
0 20 40 60 80 100 120
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Tether Length, Lt (km) 
109 
 
 
 
Figure 5.18 Time to Impact vs. Tether Length for ?V1z = 1.0 km/s in Case 1 
 
Table 5.6 Range and Impact Time for          km/s at Three Tether Lengths for Case 1 
 
Tether Length (km) 
Ground Range 
(km) Time to Impact (s) 
4 24015.7 3920 
55 22804.3 3700 
100 21821.4 3520 
 
The ground range and time to impact for an impulsive velocity change in the positive Z ? 
direction is greater than the ground range and time to impact for an impulsive velocity change in 
the positive Y ? direction for an increasing tether length because of the lofted trajectory.  Figure 
5.14 shows the impact trajectory for an altitude of 500 km and a tether length of 4 km.  The other 
two impact trajectories are shown in figures 5.19 and 5.20. 
3450
3500
3550
3600
3650
3700
3750
3800
3850
3900
3950
4000
0 20 40 60 80 100 120
Tim
e t
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pac
t, t_i
mp
ac
t (s)
 
Tether Length, Lt (km) 
110 
 
 
Figure 5.19 Trajectory at 55 km Tether Length for ?V1z = 1.0 km/s in Case 1 
 
Figure 5.20 Trajectory at 100 km Tether Length for ?V1z = 1.0 km/s in Case 1 
 Figures 5.21 to 5.22 show the decrease of the ground range and the increase of the time to 
impact as the impulsive velocity change is increased. 
111 
 
 
Figure 5.21 Ground Range vs. +?V1z for Case 1 
 
 
 
Figure 5.22 Time to Impact vs. +?V1z for Case 1 
The increase of the impulsive velocity change in the positive Z ? direction also leads to a 
decrease in the ground range of the sub-satellite; however, the time to impact starts to increase 
after a three kilometer per second impulsive velocity change.  This happens because the sub-
satellite is getting lofted higher but is impacting at a closer point to the release point.  Figure 5.23 
0
5000
10000
15000
20000
25000
30000
35000
0 1 2 3 4 5 6 7
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Change in Velocity in the Z - Direction, Delta Vz (km/s) 
0
5000
10000
15000
20000
25000
0 1 2 3 4 5 6 7
Tim
e t
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pac
t, t_i
mp
ac
t (s)
 
Change in Velocity in the Z - Direction, Delta Vz (km/s) 
112 
 
shows the impact trajectory for an impulsive velocity change of three kilometers per second.  
The sub-satellite is lofted higher after it is released from the TSS with an impulsive velocity 
change of three kilometers per second in the positive Z ? direction than it is when the impulsive 
velocity change is one kilometer per second.    
 
Figure 5.23 Lofted Trajectory at ?V1z = 3.0 km/s for Case 1 
  The minimum impulsive velocity change needed to cause impact was 0.6 km/s in the 
positive Y-direction.  The maximum and minimum values for the ground range and time to 
impact are shown in table 5.7 on the next page. 
 
 
 
113 
 
Table 5.7 Maximum and Minimum Values as      Increases for Case 1 
Velocity 
Change 
(km/s) 
Ground Range 
(km) 
Time to Impact 
(s) 
0.6 28675.1 4270 
3 21282.8 5600 
6 20654.0 22965 
 
For an impulsive velocity change in the positive Z ? direction a smaller ground range can be 
achieved by increasing the impulsive velocity change; however, the time to impact will increase 
as the impulsive velocity change becomes greater than three kilometers per second. 
5.1.3 Impulsive Velocity Change of 3.0 km/s in the Negative Z ? Direction 
 The third impulsive velocity change investigated in this section is higher than the 
previous two sections because a larger impulsive velocity change in the negative Z ? Direction 
was needed in order to create impacts at higher altitudes.  During this altitude range the ground 
range covered by the sub-satellite during its impact trajectory and the time to impact increases 
because of reasons explained in the previous two sections.  Figures 5.24 and 5.25 show the 
increasing trend of the ground range and time to impact as the altitude of the sub-satellite 
increases. The values for the ground range and time to impact for three altitudes are given in 
table 5.8. 
114 
 
 
Figure 5.24 Ground Range vs. Altitude for ?V1z = -3.0 km/s in Case 1 
 
 
Figure 5.25 Time to Impact vs. Altitude for ?V1z = -3.0 km/s in Case 1 
Table 5.8 Range and Impact Time for           km/s at Three Altitudes for Case 1 
Altitude (km) 
Ground Range 
(km) Time to Impact (s) 
500 21297.8 170 
1000 22536.7 345 
1500 23795.4 530 
 
21000
21500
22000
22500
23000
23500
24000
0 200 400 600 800 1000 1200 1400 1600
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Altitude, H (km) 
0
100
200
300
400
500
600
0 200 400 600 800 1000 1200 1400 1600
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
 
Altitude, H (km) 
115 
 
The ground range and time to impact are lower than the ground range and time to impact that 
were calculated in the previous section.  The time to impact is significantly smaller because of 
the steep descent of the impact trajectory caused by the large impulsive velocity change in the 
negative Z - direction.  The large impulsive change in velocity in the negative Z ? direction, 
causes the sub-satellite to enter a very small orbit that will send the sub-satellite into the Earth 
quicker than the other impulsive velocity changes. The impact trajectories can be seen in the next 
three figures. 
 
Figure 5.26 Trajectory at 500 km Altitude for ?V1z = -3.0 km/s in Case 1 
116 
 
 
 
Figure 5.27 Trajectory at 1000 km Altitude for ?V1z = -3.0 km/s in Case 1 
  
Figure 5.28 Trajectory at 1500 km Altitude for ?V1z = -3.0 km/s in Case 1 
 An impact is found for all values of the tether length from one kilometer to one hundred 
kilometers.  As the tether length is increased for this impulsive change in velocity the ground 
117 
 
range and the time to impact decrease.  These two trends can be seen in Figures 5.29 and 5.30.  
Table 5.9 gives specific values for three tether lengths. 
 
Figure 5.29 Ground Range vs. Tether Length for ?V1z = -3.0 km/s in Case 1 
 
 
Figure 5.30 Time to Impact vs. Tether Length for ?V1z = -3.0 km/s in Case 1 
 
 
 
20850
20900
20950
21000
21050
21100
21150
21200
21250
21300
21350
0 20 40 60 80 100 120
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Tether Length, Lt (km) 
135
140
145
150
155
160
165
170
0 20 40 60 80 100 120
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
 
Tether Length, Lt (km) 
118 
 
Table 5.9 Range and Impact Time for           km/s at Three Tether Lengths for Case 1 
Tether Length (km) 
Ground Range 
(km) Time to Impact (s) 
4 21297.8 167 
55 21080.4 152.5 
100 20884.8 140 
 
The decrease in ground range and time to impact are caused by the impact trajectory of the sub-
satellite after release.  The impact point of the sub-satellite slowly moves closer to the release 
point; however, the impact trajectory becomes steeper as the tether length increases.  This 
increase in the steepness of the impact trajectory leads to the decrease in time that is seen in 
figure 5.30.  The impact trajectory of the sub-satellite with a four kilometer tether length is 
shown in figure 5.26.  The impact trajectory for the sub-satellite with a fifty ? five kilometer 
tether length and a one hundred kilometer tether length are shown in figure 5.31 and 5.32. 
 
Figure 5.31 Trajectory at 55 km Tether Length for ?V1z = -3.0 km/s in Case 1 
119 
 
 
 
Figure 5.32 Trajectory at 100 km Tether Length for ?V1z = -3.0 km/s in Case 1 
 When the impulsive velocity change placed on the sub-satellite is increased in the 
negative Z ? direction, the ground range and the time to impact are decreased.  This decrease 
occurs because the impulsive velocity change increase results in a steeper descent toward the 
impact point and the movement of the impact point is toward the point of release.  Figures 5.33 
and 5.34 show ground range and time to impact as a function of the impulsive velocity change 
increase and are located on the next page.   
120 
 
 
Figure 5.33 Ground Range vs. ??V1z for Case 1 
  
Figure 5.34 Time to Impact vs. ??V1z for Case 1 
The minimum impulsive velocity change required for an impact trajectory at a five hundred 
kilometer altitude and a tether length of four kilometers is 0.6 kilometers per second in the 
negative Z ? direction.  This impulsive velocity change is increased until it reaches 6.0 
0
5000
10000
15000
20000
25000
30000
35000
-7 -6 -5 -4 -3 -2 -1 0
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Change in Velocity in Z - Direction, Delta Vz (km/s) 
0
200
400
600
800
1000
1200
1400
-7 -6 -5 -4 -3 -2 -1 0
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
 
Change in Velocity in Z - Direction, Delta Vz (km/s) 
121 
 
kilometers per second.  The maximum and minimum values for the ground range and time to 
impact are shown in table 5.10. 
Table 5.10: Maximum and Minimum Values as ??V1z Increases for Case 1 
Velocity 
Change 
(km/s) 
Ground Range 
(km) 
Time to 
Impact (s) 
-0.6 28929.6 1160 
-3 21297.8 170 
-6 20667.6 85 
 
5.1.4 Comparison between Velocity Changes for First Configuration 
 For the first configuration point a small impulsive velocity change in the positive Y and Z 
? directions leads to an impact trajectory; however, a minimum value of 0.6 kilometers per 
second in the negative Z ? direction is needed to create the first impact point when the altitude is 
at five hundred kilometers and the tether length is at four kilometers.  If the altitude is increased, 
the minimum impulsive velocity change needed in the positive Z ? direction has to increase as 
well in order to achieve an impact at an altitude greater than eight hundred and fifty kilometers.  
The other two impulsive velocity changes do not require a further increase in order to achieve an 
impact trajectory for the altitudes tested in the simulation.   
 The impulsive velocity changes in the Y and Z ? directions responded to changes in 
altitude and tether length in the same way.  The range and time to impact increased for an 
increasing altitude and decreased for an increasing tether length.  The tether length decreased the 
range in both cases by less than two hundred kilometers.  An increase in the altitude for an 
impulsive velocity change in the positive Z ? direction lead to a greater increase in the range 
when compared to the increase found in the range for an impulsive velocity change in the 
122 
 
positive Y ? direction; therefore, if the objective is to have the longest range increasing the 
altitude and having an impulsive velocity change in the positive Z ? direction will lead to the 
maximum range that can be achieved for this configuration.  The minimum range can be 
achieved by increasing the tether length and implementing an impulsive velocity change in the 
positive Y ? direction. 
 Another way to decrease the range in all three examples is to increase the impulsive 
velocity change.  As the impulsive velocity change gets higher the range and time to impact 
decrease for a change in the positive Y ? direction and for a change in the negative Z ? direction.  
For the case when the impulsive velocity change is increased in the positive Z ? direction, the 
range decreases but the time to impact starts to increase again after an impulsive velocity change 
of three kilometers per second.  This occurs because the impact trajectory of the sub-satellite 
becomes lofted.  As the sub-satellite is lofted into a higher orbit, the time it takes to impact the 
Earth will increase but the range may remain the same or slightly decrease.  For this attempt in 
decreasing range, it is better to increase the impulsive velocity change in the positive Y ? 
direction or in the negative Z ? direction; however, having an impulsive velocity change of six 
kilometers in either direction will require a lot more fuel than changing the tether length or 
altitude of the TSS. 
5.2 Results for Second Configuration or Case 2 
 For the second configuration, the sub-satellite is located above the main satellite.  This is 
similar to case A for the analytical solutions.  The three impulsive velocity changes that were 
applied to the sub-satellite in the second configuration are shown in table 5.11 on the next page. 
 
123 
 
Table 5.11: Impulsive Velocity Changes for Case 2 
  Velocity Change (km/s) 
     1.0 
     -1.5 
     N/A 
     1.0 
 
An impulsive velocity change in the positive Y ? direction is not investigated because the 
velocity of the sub-satellite is in the positive Y ? direction; therefore, an increase in the velocity 
in the positive Y ? direction will only increase the size of the sub-satellite trajectory after release 
and its perigee point, which will not result in an impact of the sub-satellite. 
5.2.1 Impulsive Velocity Change of 1.0 km/s in the Positive Z ? Direction 
 For an increase in the altitude, the ground range covered by the impact trajectory and the 
time to impact increases; however, at altitudes greater than eight hundred and fifty kilometers the 
sub-satellite does not impact the Earth.  This means that a greater impulsive velocity change in 
the positive Z ? direction is needed in order to create an impact trajectory. The ground range and 
time to impact as a function of altitude are shown in figures 5.35 and 5.36.  The figures are then 
followed by a table that lists the ground range and time to impact for an altitude of five hundred 
kilometers, six hundred and fifty kilometers, and eight hundred and fifty kilometers. 
124 
 
 
Figure 4.35 Ground Range vs. Altitude for ?V1z = 1.0 km/s in Case 2 
 
 
 
Figure 4.36 Time to Impact vs. Altitude for ?V1z = 1.0 km/s in Case 2 
 
 
 
 
 
 
 
0
5000
10000
15000
20000
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30000
35000
0 200 400 600 800 1000 1200 1400 1600
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125 
 
Table 5.12 Ground Range and Impact Time for ?V1z = 1.0 km/s at Three Altitudes for Case 2 
Altitude (km) 
Ground Range 
(km) Time to Impact (s) 
500 24210.6 3955 
650 25730.4 4275 
850 29736.1 4955 
  
The impact trajectory of the sub-satellite is lofted until a maximum apogee is reached and then it 
begins its return toward the Earth.  The increase in the ground range is caused by the movement 
of the impact point further away from the release point as the altitude increases.  The increase in 
the time to impact is caused by the fact that as the altitude increases the area of the impact 
trajectory that is lofted increases as well.  The impact trajectories for a five hundred kilometer 
altitude, a six hundred and fifty kilometer altitude, and an eight hundred and fifty kilometer 
altitude are shown below. 
 
Figure 5.37 Trajectory at 500 km Altitude for ?V1z = 1.0 km/s in Case 2 
126 
 
 
 
Figure 5.38 Trajectory at 650 km Altitude for ?V1z = 1.0 km/s in Case 2 
  
Figure 5.39 Trajectory at 850 km Altitude for ?V1z = 1.0 km/s in Case 2 
 
127 
 
 The next change that is looked at for this impulsive velocity change is an increase in the 
tether length.  The time to impact and the ground range increase as the tether length becomes 
longer.  This means that the minimum range that can be achieved will occur at the shortest tether 
length, while the maximum range will occur at the longest length that the tether can reach.  The 
increasing trends for the range and time to impact are shown in figures 5.40 and 5.41. 
 
Figure 5.40 Ground Range vs. Tether Length for ?V1z = 1.0 km/s in Case 2 
 
24000
24500
25000
25500
26000
26500
27000
0 20 40 60 80 100 120
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Tether Length, Lt (km) 
128 
 
 
Figure 5.41 Time to Impact vs. Tether Length for ?V1z = 1.0 km/s in Case 2 
A list of ground ranges and times to impact for a tether length of four kilometers, fifty-five 
kilometers, and one hundred kilometers is given in table 5.29. 
Table 5.13 Range and Impact Time for ?V1z = 1.0 km/s at Three Tether Lengths for Case 3 
Tether Length (km) 
Ground Range 
(km) Time to Impact (s) 
4 24210.6 3955 
55 25495.2 4180 
100 26800.6 4400 
 
The increase in tether length causes the ground range to increase because the point of impact 
moves further away from the point of release and the height of the sub-satellite is increasing.  
This same change in the impact point also causes an increase in the time to impact because if the 
impact point moves further away from the release point the time to get to an impact will increase.  
The impact trajectory at a four kilometer tether length is shown in figure 5.37.  The other two 
impact trajectories are shown in the figures below. 
3900
4000
4100
4200
4300
4400
4500
0 20 40 60 80 100 120
Tim
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Tether Length, Lt (km) 
129 
 
 
Figure 5.42 Trajectory at 55 km Tether Length for ?V1z = 1.0 km/s in Case 2 
 
 
Figure 5.43 Trajectory at 100 km Tether Length for ?V1z = 1.0 km/s in Case 2 
 The last change that is looked at in order to cause an impact trajectory is the increase of 
the magnitude for the impulsive velocity change while keeping the altitude and tether length 
130 
 
constant at five hundred kilometers and four kilometers.  The effect of the increase in the 
impulsive velocity change on the ground range and time to impact are shown in figures 5.44 to 
5.45. 
 
Figure 5.44 Ground Range vs. +?V1z for Case 2 
  
Figure 5.45 Time to Impact vs. +?V1z for Case 2 
0
5000
10000
15000
20000
25000
30000
0 2 4 6 8
Gr
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Change in Velocity in the Z - direction, Delta Vz (km/s) 
0
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Change in Velocity in the Z - direction, Delta Vz (km/s) 
131 
 
The increase in the time to impact is caused by the lofted trajectory after the sub-satellite release.  
Figure 5.46 shows the lofted trajectory of the sub-satellite when an impulsive velocity change of 
three kilometers per second in the positive Z ? direction is placed on the system. 
 
Figure 5.46 Lofted Trajectory at ?V1z = 1.0 km/s for Case 2 
 
The ground range decreases because the impact point moves closer to the release point.    The 
minimum impulsive velocity change in the positive Z ? direction that was needed to cause an 
impact at a five hundred kilometer altitude and a tether length of four kilometers was 0.7 
kilometers per second.  The ground range and time to impact achieved during the minimum 
impulsive velocity change, a velocity change of 3.0 kilometers per second, and a velocity change 
of 6.0 kilometers per second are listed in table 5.14 
 
 
132 
 
Table 5.14 Maximum and Minimum Values as +?V1z Increases for Case 3 
Velocity 
Change 
(km/s) 
Ground Range 
(km) 
Time to Impact 
(s) 
0.7 26743.1 4090 
3 21344.7 5625 
6 20690.8 23140 
 
5.2.2 Impulsive Velocity Change of 1.5 km/s in the Negative Z ? Direction 
 For an impulsive velocity change of 1.5 kilometers per second in the negative Z ? 
direction an increase in the altitude leads to an increase in the range and time to impact; however, 
no impact trajectory is created at altitudes greater than thirteen hundred kilometers.  A minimum 
impulsive velocity change of three kilometers per second in the negative Z ? direction is required 
to have an impact at all altitudes that are tested.  The increase in the range and time to impact as 
the altitude is increased for a velocity change of 1.5 kilometers per second our shown in figures 
5.47 and 5.48. 
 
Figure 5.47 Ground Range vs. Altitude for ?V1z = -1.5 km/s in Case 2 
0
5000
10000
15000
20000
25000
30000
35000
0 200 400 600 800 1000 1200 1400
Gr
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 Ran
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, R
 (km
) 
Altitude, H (km) 
133 
 
 
  
Figure 5.48 Time to Impact vs. Altitude for ?V1z = -1.5 km/s in Case 2 
Since an impact trajectory does not occur at altitudes greater than thirteen hundred 
kilometers, the results for an impact trajectory at five hundred kilometers, one thousand 
kilometers, and thirteen hundred kilometers will be examined in more detail.  A list of the ground 
range and time to impact achieved at these three altitude values are given in table 5.15. 
Table 5.15 Range and Impact Time for ?V1z = -1.5 km/s at Three Altitudes for Case 2 
Altitude (km) 
Ground Range 
(km) Time to Impact (s) 
500 22683.8 345 
650 25665.0 760 
800 28841.2 1180 
 
The small time to impact is caused by the steep impact trajectory of the sub-satellite after release.  
The impact trajectories of the sub-satellite at the three different altitudes are shown in figures 
5.49 to 5.51.   
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400
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800
1000
1200
1400
0 200 400 600 800 1000 1200 1400
Tim
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Altitude, H (km) 
134 
 
 
Figure 5.49 Trajectory at 500 km Altitude for ?V1z = -1.5 km/s in Case 2 
  
Figure 5.50 Trajectory at 1000 km Altitude for ?V1z = -1.5 km/s in Case 2 
 
135 
 
 
 
Figure 5.51 Trajectory at 1300 km Altitude for ?V1z = -1.5 km/s in Case 2 
After inspecting the impact trajectory figures, it can be deduced that as the altitude increases the 
point of impact of the sub-satellite moves further along the surface of the Earth.  This movement 
of the impact point results in an increase in the ground range.  The increase of the time to impact 
is a direct result of the increase of the altitude of the TSS. 
 Even though an impact trajectory cannot be obtained for all altitudes, an impact trajectory 
can still be found for all the tested tether lengths.  The trends for an impulsive velocity change of 
1.5 kilometers per second in the negative Z ? direction are shown in figures 5.52 and 5.53. 
136 
 
 
Figure 5.52 Ground Range vs. Tether Length for ?V1z = -1.5 km/s in Case 2 
 
Figure 5.53 Time to Impact vs. Tether Length for ?V1z = -1.5 km/s in Case 2 
Since the ground range and time to impact increase as the tether length increases, the maximum 
range and time to impact will result at the maximum tether length.  The ground range, time to 
impact, and impact trajectories will be looked at in more detail for the same tether length values 
used in previous sections. 
22600
22700
22800
22900
23000
23100
23200
23300
23400
23500
23600
0 20 40 60 80 100 120
Gr
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 (km
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Tether Length, Lt (km) 
340
345
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385
390
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Tether Length, Lt (km) 
137 
 
Table 5.16 Range and Impact Time for ?V1z = -1.5 km/s at Three Tether Lengths for Case 2 
Tether Length (km) 
Ground Range 
(km) Time to Impact (s) 
4 22683.8 344.5 
55 23144.0 366.5 
100 23540.6 385.5 
 
The cause behind the increase in ground range and time to impact as the tether length increases 
can be found by examining the impact trajectories at each tether length.  The impact trajectory 
for the four kilometer tether length and five hundred kilometer case is shown in figure 5.49.  The 
other two impact trajectories also have an altitude of five hundred kilometers. The impact 
trajectories for these other two tether length values are shown below. 
 
Figure 5.54 Trajectory at 55 km Tether Length for ?V1z = -1.5 km/s in Case 2 
138 
 
 
 
Figure 5.55 Trajectory at 100 km Tether Length for ?V1z = -1.5 km/s in Case 2 
The increase in range is caused by the movement of the impact point further along the Earth?s 
surface, while the time to impact is increased because the orbit is at a higher position.  As the 
position or height of the orbit increases so will the time to impact. 
 The final change that is looked at in this section is an increase in the impulsive velocity 
change.  The altitude and tether length are kept constant at five hundred kilometers and four 
kilometers, respectively.  As the magnitude of the impulsive velocity change is increased in the 
negative Z ? direction, the ground range and time to impact decrease.  The decrease in the 
ground range and the time to impact can be seen in figures 5.56 through 5.57. 
139 
 
 
Figure 5.56 Ground Range vs. ??V1z for Case 2 
 
 
 
Figure 5.57 Time to Impact vs. ??V1z for Case 2 
The decrease in the ground range and the time to impact follows the trends displayed in 
the previous sections for different release points and impulsive velocity changes. In order to 
create an impact trajectory at this configuration a minimum impulsive velocity change value of 
0
5000
10000
15000
20000
25000
30000
-7 -6 -5 -4 -3 -2 -1 0
Gr
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, R
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Change in Velocity in the Z - Direction, Delta Vz (km/s) 
0
100
200
300
400
500
600
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900
1000
-7 -6 -5 -4 -3 -2 -1 0
Tim
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Change in Velocity in the Z - Direction, Delta Vz (km/s) 
140 
 
0.7 kilometers per second in the negative Z ? direction is needed.  Table 5.17 list values for the 
ground range and time to impact for three impulsive velocity changes. 
Table 5.17 Maximum and Minimum Values as ??V1z Increases for Case 2 
Velocity 
Change 
(km/s) 
Ground Range 
(km) 
Time to Impact 
(s) 
-0.7 26663.4 860 
-3 21329.7 170 
-6 20677.2 85 
 
5.2.3 Impulsive Velocity Change of 1.0 km/s in the Positive Y ? Direction 
 The ground range and time to impact increase as the altitude is increased and the tether 
length is kept constant at four kilometers with an impulsive velocity change of one kilometer per 
second in the positive Y ? direction.  The results for the ground range and time to impact as the 
altitude increases are shown below. 
 
Figure 5.58 Ground Range vs. Altitude for ?V1y = 1.0 km/s in Case 2 
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 500 1000 1500 2000
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Altitude, H (km) 
141 
 
 
Figure 5.59 Time to Impact vs. Altitude for ?V1y = 1.0 km/s in Case 2 
Altitudes of five hundred kilometers, one thousand kilometers, and fifteen hundred kilometers 
are examined in more detail.  The values for the ground range and time to impact at these 
altitudes are listed in table 5.18. 
Table 5.18: Range and Impact Time for ?V1y = 1.0 km/s at Three Altitudes for Case 2 
Altitude (km) 
Ground Range 
(km) Time to Impact (s) 
500 4544.8 705 
1000 6412.0 1060 
1500 7856.0 1375 
 
The range and time to impact achieved with this impulsive velocity change are less than the 
ground range and time to impact calculated for different altitudes for the other two impulsive 
velocity changes.  This is a result of the type of impact trajectory that is created after the velocity 
change is put into place and the sub-satellite is released from the TSS.  The impact trajectory of 
the sub-satellite at each altitude is shown on the next two pages. 
0
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1600
0 200 400 600 800 1000 1200 1400 1600
Tim
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Altitude, H (km) 
142 
 
 
Figure 5.60 Trajectory at 500 km Altitude for ?V1y = 1.0 km/s in Case 2 
  
Figure 5.61 Trajectory at 1000 km Altitude for ?V1y = 1.0 km/s in Case 2 
143 
 
 
 
Figure 5.62 Trajectory at 1500 km Altitude for ?V1y = 1.0 km/s in Case 2 
From the impact trajectories depicted in figure 5.60 through 5.62, it can be concluded that the 
ground range increases because as the altitude increases the release point occurs earlier in the 
TSS orbit and the impact point occurs further on the surface of the Earth.  The increase in the 
time to impact happens because the impact trajectory is placed on a higher orbit that requires 
more time to travel around in order to get to the impact point. 
 An increase in the tether length of the TSS affects the ground range and time to impact of 
the impact trajectory after the sub-satellite is released from the system.  For a constant impulsive 
velocity change of one kilometer per second in the positive Y ? direction, an increase in tether 
length leads to an increase in the ground range and time to impact. 
144 
 
 
Figure 5.63 Ground Range vs. Tether Length for ?V1y = 1.0 km/s in Case 2 
  
Figure 5.64 Time to Impact vs. Tether Length for ?V1y = 1.0 km/s in Case 2 
The list of the ground range and the time to impact at three tether lengths are listed in 
table 5.19 on the next page. 
 
4400
4500
4600
4700
4800
4900
5000
5100
5200
0 20 40 60 80 100 120
Gr
ou
nd
 Ran
ge
, R
 (km
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Tether Length, Lt (km) 
690
700
710
720
730
740
750
760
0 20 40 60 80 100 120
Tim
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Tether Length, Lt (km) 
145 
 
Table 5.19 Range and Impact Time for ?V1y = 1.0 km/s at Three Tether Lengths for Case 2 
Tether Length (km) 
Ground Range 
(km) Time to Impact (s) 
4 4544.8 703 
55 4839.0 729 
100 5103.0 752 
 
 
The ground range and time to impact are increased because an increase in the tether length 
causes the sub-satellite to be in a slightly higher orbit, which in turn causes the impact trajectory 
to start from a higher point and move the impact point further along the Earth?s surface.  In order 
to see if this is true the impact trajectories must be examined.  The impact trajectory of the sub-
satellite at a four kilometer tether length is shown in figure 5.60.  The impact trajectories for the 
other two tether lengths are shown below.  As the tether length increases the height in the impact 
trajectories are increased.  This increase is what results in the increase in the time to impact 
shown in table 5.37.   The plot of the impact trajectories also show that the impact point of the 
sub-satellite travels forward along the surface of the Earth.  The movement of the impact point is 
not significantly large; therefore, a small increase in the ground range results from the increase of 
the tether length. 
146 
 
 
Figure 5.65 Trajectory at 55 km Tether Length for ?V1y = 1.0 km/s in Case 2 
  
Figure 5.66 Trajectory at 100 km Tether Length for ?V1y = 1.0 km/s in Case 2 
  
147 
 
 An increase of the magnitude of the impulsive velocity change in the positive Y ? 
direction will also lead to a change in the ground range and time to impact.  The increase in the 
impulsive velocity change results in a decrease of the ground range and the time to impact.  This 
happens because the impact trajectory of the sub-satellite after release decreases in size as the 
impulsive velocity change increases; therefore, the sub-satellite will impact the Earth sooner at 
higher impulsive velocity changes.  The effect of the impulsive velocity change on the ground 
range and time to impact values are shown in figures 5.67 through figures 5.68. 
 
Figure 5.67 Ground Range vs. +?V1y for Case 2 
 
0
2000
4000
6000
8000
10000
12000
14000
0 1 2 3 4 5 6 7
Gr
ou
nd
 Ran
ge
, R
 (km
) 
Change in Velocity in the Y - Direction, Delta Vy (km/s) 
148 
 
 
 
Figure 5.68 Time to Impact vs. +?V1y for Case 2 
The minimum impulsive velocity change in the positive Y ? direction that is needed to create an 
impact trajectory for the second configuration is 0.2 kilometers per second.  Note that this is the 
minimum impulsive velocity change needed when the altitude of the main satellite is five 
hundred kilometers and the tether length is kept at four kilometers.  A maximum impulsive 
velocity change of six kilometers per second in the positive Y ? direction is place on the system; 
therefore, the minimum time to impact and the minimum ground range will occur at this 
maximum impulsive velocity change.  Values of the ground range and time to impact are listed 
in the table below. 
Table 5.20: Maximum and Minimum Values at +?V1y Increases for Case 2 
Velocity 
Change 
(km/s) 
Ground Range 
(km) 
Time to Impact 
(s) 
0.2 13058.2 1785 
3 1938.1 435 
6 549.4 350 
 
0
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1000
1200
1400
1600
1800
2000
0 1 2 3 4 5 6 7
Tim
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Change in Velocity in the Y - Direction, Delta Vy (km/s) 
149 
 
5.2.4 Comparison between Velocity Changes For Second Configuration 
 The impulsive velocity changes that are placed on the sub-satellite result in different 
values for the ground range and the time to impact.  For the second configuration an impulsive 
velocity change placed in the positive Z ? direction will result in the maximum ground range and 
time to impact for the sub-satellite.  These values for ground range and time to impact can be 
further increased by increasing the altitude and the tether length; however, a larger impulsive 
velocity change in the positive Z ? direction will be needed to cause an impact trajectory at 
altitudes above eight hundred and fifty kilometers.  For the case of an impulsive velocity change 
in the positive Z ? direction, the tether length can be increased in order to increase the ground 
range without needing an increase in the magnitude of the impulsive velocity change.  In order to 
save on fuel, the increase in tether length is the better way to achieve a longer ground range for 
this impulsive velocity change direction. 
 An impulsive velocity change in the positive Y ? direction results in the minimum ground 
range and time to impact for the second configuration.  The ground ranges achieved with this 
impulsive velocity change can be increased by placing the TSS in a higher altitude or by 
increases the tether length.  The increase in altitude leads to a larger increase for the ground 
range, while the increase in tether length has a smaller effect on the ground range and time to 
impact.  In this case, the method used to increase the ground range depends upon how big of an 
increase is needed in order to reach the desired impact point. 
 As at any release point, the magnitude of the impulsive velocity change can be increased 
in order to achieve a lower ground range.  The time to impact will also decrease with the ground 
range when the impulsive velocity change is in the negative Z ? direction or the positive Y ? 
direction.  When an impulsive velocity change is placed in the positive Z ? direction, the time to 
150 
 
impact will increase because the sub-satellite is lofted into a higher orbit upon release.  In other 
words, one has to consider the tradeoff of an increase in the time to impact in order to achieve a 
shorter ground range.   Overall, impact trajectories will result at these impulsive velocity changes 
for the second configuration; however, which impulsive velocity change to be used depends 
upon if a maximum or minimum ground range is desired.   
5.3 Comparison between the Analytical Results and the Simulation Results 
 In order to compare the analytical approach and the simulation results the algorithm to 
find the ground range and time to impact for the analytical calculation is used.  The first 
configuration from the simulation is compared to case B from the analytical solution because 
both have the sub-satellite below the main satellite.  Case A from the analytical solution is then 
compared to the second configuration from the simulation because both have the sub-satellite 
located above the main satellite.  For each comparison an impulsive velocity change of one 
kilometer per second and six kilometers per second in the positive Y ? direction will be placed 
on the sub-satellite. 
 The first comparison will be done using the first release configuration.  The impulsive 
velocity change that is used for the simulation and the analytical calculations is one kilometer per 
second in the positive Y ? direction.  The ground range and time to impact calculated by the 
simulation and the analytical approach are listed in table 5.21. 
Table 5.21: Results for Configuration 1 and Case B with ?V1y = 1.0 km/s 
 
Ground Range (km) Time to Impact (s) 
Simulation 4.5057232E+03 700 
Analytical 4.4980330E+03 696.98 
 
151 
 
The percent error and the difference between the simulation value and the analytical value for the 
ground range and time to impact are calculated and listed below. 
Table 5.22 Comparison of Results for Configuration 1 and Case B with ?V1y = 1.0 km/s 
 
Ground Range (km) Time to Impact (s) 
% Error 0.17068 0.43108 
Difference 7.69028 3.01755 
 
The percent error between the two ground ranges is less than one percent and the actual 
difference is about 7.7 kilometers.  The percent error for the time to impact is also less than one 
percent but the actual difference between the two times is about 3.02 seconds with the time to 
impact from the simulation being the greater value.   
 The next comparison will be done at the second configuration with an impulsive velocity 
change of 6.0 kilometers per second in the positive Y ? direction. The calculated values for the 
ground range and time to impact for both methods are listed below. 
Table 5.23: Results for Configuration 1 and Case B with ?V1y = 6.0 km/s 
 
Ground Range (km) Time to Impact (s) 
Simulation 5.4562965E+02 350 
Analytical 5.4529027E+02 346.72 
 
The percent error and the direct difference between the ground ranges and times to impact are 
given in table 5.24 on the next page. 
 
 
152 
 
Table 5.24 Comparison of Results for Configuration 1 and Case B with ?V1y = 6.0 km/s 
 
Ground Range (km) Time to Impact (s) 
% Error 0.062201 0.936190 
Difference 0.339384 3.276665 
 
The percent error for the ground range and the time to impact are less than one.  The difference 
between the ground ranges is 0.34 kilometers and the difference between the times is about 3.28 
seconds.  So far the analytical solution is giving similar results to those from the simulation. 
 The third comparison is done with the second configuration and an impulsive velocity 
change placed on the sub-satellite of one kilometer per second in the positive Y ? direction.  The 
simulation results and the analytical approach results are listed below in table 5.25. 
Table 5.25: Results for Configuration 2 and Case A with ?V1y = 1.0 km/s 
 
Ground Range (km) Time to Impact (s) 
Simulation 4.5448026E+03 705 
Analytical 4.5458193E+03 704.69 
 
The analytical result for the ground range is greater than the ground range obtained from the 
simulation results.  The time to impact calculated during the simulation is greater than the time to 
impact calculated by the analytical approach.  The actual difference between the simulation 
values and the analytical values and the percent error are listed below. 
Table 5.26: Comparison of Results for Configuration 2 and Case A with ?V1y = 1.0 km/s 
 
Ground Range (km) Time to Impact (s) 
% Error 0.02237 0.04446 
Difference 1.01666 0.31343 
 
153 
 
The percent errors between the ground ranges and the times to impact are less than one percent.  
This is the same trend that has been seen in the past two comparisons.  The difference between 
the ground range determined by the simulation and the ground range calculated using the 
analytical approach is about one kilometer. The difference between the times to impact is about 
0.313 seconds.   
 The final comparison will be done with the second configuration and an impulsive 
velocity change of 6.0 kilometers per second in the positive Y ? direction. 
Table 5.27: Results for Configuration 2 and Case A with ?V1y = 6.0 km/s 
 
Ground Range (km) Time to Impact (s) 
Simulation 5.4939447E+02 350 
Analytical 5.4986409E+02 349.86 
 
The ground range calculated by the analytical approach is slightly larger than the ground range 
calculated by the simulation.  The percent error and the actual difference between the two values 
are presented in table 5.28. 
Table 5.28: Comparison of Results for Configuration 2 and Case A with ?V1y = 6.0 km/s 
 
Ground Range (km) Time to Impact (s) 
% Error 0.08548 0.04084 
Difference 0.46964 0.14295 
 
The percent error between the two ground ranges and the percent error between the two times to 
impact is less than one percent.  The difference between the two ground ranges was about 0.45 
kilometers.  The time to impact calculated by the simulation is greater than the time to impact 
that was calculated using the simulation approach by almost 0.14 seconds. 
154 
 
 More comparisons are not shown in this write up because the results have a percent error 
of less than one percent.  The maximum difference between the times was found to be about 3.3 
seconds, while the maximum difference for the ground ranges was 7.7 kilometers.  The errors 
between the simulation and the analytical approach are caused by the rotation angle that is 
introduced into the simulation in order to keep the sub-satellite in a leading position.  Overall the 
analytical approach provides a good estimate of what the time to impact and ground range will 
be after the sub-satellite release.  A simulation should still be run in order to determine the 
impact trajectory after the sub-satellite is released and to place the impulsive velocity change in 
different directions on the sub-satellite. 
 
 
 
 
 
 
 
 
 
 
155 
 
 
 
 
Chapter 6: Conclusions and Future Work 
The altitude, tether length, and the impulsive velocity change have been varied to 
determine the type of ground range and time to impact that can be achieved.  The effects of the 
altitude, tether length, and impulsive velocity change on the ground range and time to impact 
were investigated using a numerical simulation approach and an analytical approach.  The 
increase in altitude caused the ground range and time to impact to increase for all configurations 
and impulsive velocity changes.  If the sub-satellite is located below the main satellite, an 
increase in tether length will cause the ground range and time to impact to decrease; however, if 
the sub-satellite is located above the main satellite, an increase in the tether length will result in 
an increase in the ground range and time to impact.  An increase in the impulsive velocity change 
on the sub-satellite will always cause the ground range and the time to impact to decrease; 
however, there are a few occasions where the time to impact will increase because of the lofted 
impact trajectory.  The changes in altitude and tether length are the better methods to increase or 
decrease the ground range and time to impact because an increase in an impulsive velocity 
change will result in a greater fuel usage.  The analytical solutions were found to follow these 
exact same trends for a sub-satellite located above or below the main satellite.  For the cases 
when a sub-satellite had the same altitude as the main satellite, the increase in altitude resulted in 
an increase in the ground range and time to impact, while an increase in the tether length resulted 
in an increase to the ground range and a decrease to the time to impact. 
156 
 
 The final part of this research was to compare the analytical solution to the numerical 
simulation by comparing the ground range and the time to impact.  The percent error between the 
ground ranges and times to impact for the analytical solution and the numerical solution was 
found to always be less than one percent.  The percent errors and differences between the two 
approaches are caused by the fact that a rotation angle was introduced into the simulation in 
order to model the fact that the sub-satellite is leading the main satellite.  The analytical approach 
can be used to provide an initial estimate of the ground range and time to impact; however, the 
numerical simulation should be used in order to track the trajectory of the sub-satellite before 
and after its release from the TSS. 
 In order to improve the ground range and time to impact calculated in the numerical 
simulation a more detailed model of the TSS should be put into place for future research.  The 
effect of the Earth?s oblateness and the aerodynamic drag on the satellites can be added to the 
simulation.  The aerodynamic drag will impact the movement of the sub-satellite as it gets closer 
to the Earth?s surface.  The inclusion of the gravitational gradient in the model will keep the TSS 
in a natural orbital motion where the sub-satellite will remain in a lower orbit than the main 
satellite at all times.  By adding the above three conditions to the model that is being used in the 
numerical simulation, the movement of the TSS will be model more accurately and the 
simulation will provide a more accurate result for the position of the sub-satellite at release.  If 
the effects of the Earth?s oblateness and the effects of the aerodynamic drag are placed on the 
sub-satellite after release, a more accurate model for the movement of the sub-satellite after 
release will be in place and more precise ground ranges and times to impact can be calculated. 
 
157 
 
 
 
 
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158 
 
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161 
 
 
 
 
Appendix A: Sample Results from Numerical Integration 
Table A.1: Simulation Results as Altitude Increases for Case 1 at ?V1y = 1.0 km/s 
Altitude 
(km) Time to Impact (s) Ground Range (km)  
500 700 4505.723 
550 740 4726.313 
600 775 4937.022 
650 815 5139.085 
700 850 5333.504 
750 885 5521.106 
800 920 5702.580 
850 955 5878.512 
900 990 6049.402 
950 1020 6215.682 
1000 1055 6377.728 
1050 1085 6535.871 
1100 1120 6690.403 
1150 1150 6841.584 
1200 1180 6989.646 
1250 1215 7134.797 
1300 1245 7277.226 
1350 1275 7417.105 
1400 1305 7554.589 
1450 1340 7689.820 
1500 1370 7822.930 
 
 
 
 
 
 
 
 
 
 
162 
 
Table A.2: Simulation Results as Tether Length Increases for Case 1 at ?V1y = 1.0 km/s 
Tether 
Length 
(km) Time to Impact (s) Ground Range (km)  
1 701 4517.880 
4 699 4505.723 
7 697 4493.558 
10 695 4481.383 
13 693 4469.199 
16 691 4457.006 
19 689 4444.803 
22 687 4432.590 
25 685 4420.367 
28 683 4408.133 
31 681 4395.889 
34 679 4383.634 
37 677 4371.369 
40 675 4359.092 
43 673 4346.803 
46 671 4334.502 
49 669 4322.190 
52 667 4309.865 
55 665 4297.528 
58 663 4285.178 
61 661 4272.815 
64 659 4260.439 
67 657 4248.049 
70 655 4235.645 
73 653 4223.227 
76 651 4210.795 
79 649 4198.348 
82 647 4185.886 
85 645 4173.409 
88 643 4160.917 
91 641 4148.408 
94 639 4135.884 
97 637 4123.343 
100 635 4110.786 
  
163 
 
Table A.3: Simulation Results as +?V1y Increases for Case 1 
Velocity 
Change 
(km/s) Time to Impact (s) Ground Range (km)  
0     
0.1     
0.2 1755 12746.853 
0.3 1335 9553.004 
0.4 1130 7953.634 
0.5 1000 6929.881 
0.6 905 6197.195 
0.7 840 5636.828 
0.8 785 5188.739 
0.9 740 4818.745 
1 700 4505.723 
1.1 670 4235.805 
1.2 640 3999.452 
1.3 620 3789.851 
1.4 595 3601.989 
1.5 580 3432.081 
1.6 560 3277.210 
1.7 545 3135.081 
1.8 530 3003.864 
1.9 520 2882.078 
2 510 2768.509 
2.1 500 2662.151 
2.2 490 2562.160 
2.3 480 2467.828 
2.4 470 2378.548 
2.5 465 2293.802 
2.6 455 2213.141 
2.7 450 2136.178 
2.8 440 2062.572 
2.9 435 1992.024 
3 430 1924.271 
 
 
164 
 
 
 
 
Appendix B: Sample Results from Analytical Solutions 
Table B.1: Analytical Results as Altitude Increases for Case A When R = 1500 km and ?imp = 
240? 
Altitude (km) 
Velocity Change in the Y 
(km/s) Angular Velocity (rads/s) 
500 0.60484 -0.16633 
600 0.62511 -0.17191 
700 0.62185 -0.17101 
800 0.59824 -0.16452 
900 0.55718 -0.15322 
1000 0.50120 -0.13783 
1100 0.43258 -0.11896 
1200 0.35332 -0.09716 
1300 0.26517 -0.07292 
1400 0.16966 -0.04666 
1500 0.06812 -0.01873 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
165 
 
Table B.2: Analytical Results as Tether Length Increases for Case A When R = 1500 km and 
?imp = 240? 
Tether Length (km) 
Velocity Change in the Y 
(km/s) Angular Velocity (rads/s) 
4 0.60484 -0.16633 
14 0.61305 -0.04817 
24 0.62098 -0.02846 
34 0.62864 -0.02034 
44 0.63602 -0.01590 
54 0.64314 -0.01310 
64 0.64999 -0.01117 
74 0.65658 -0.00976 
84 0.66292 -0.00868 
94 0.66900 -0.00783 
104 0.67483 -0.00714 
114 0.68041 -0.00657 
 
 
 
 
 
 
 
 
 
 
 
 
 
166 
 
Table B.3: Analytical Results as ?V1y Increases for Case A When H = 500 km and Lt = 4 km 
Velocity 
Change 
(km/s) Ground Range (km) Time to Impact (s) 
0     
0.1     
0.2 13065.345 1793.531 
0.3 9700.880 1355.315 
0.4 8053.841 1142.390 
0.5 7007.363 1008.651 
0.6 6261.278 914.513 
0.7 5692.005 843.645 
0.8 5237.520 787.848 
0.9 4862.674 742.480 
1 4545.819 704.687 
1.1 4272.780 672.600 
1.2 4033.819 644.941 
1.3 3821.997 620.798 
1.4 3632.213 599.504 
1.5 3460.618 580.556 
1.6 3304.248 563.568 
1.7 3160.775 548.237 
1.8 3028.342 534.322 
1.9 2905.447 521.628 
2 2790.860 509.997 
2.1 2683.561 499.298 
2.2 2582.698 489.421 
2.3 2487.551 480.274 
2.4 2397.508 471.778 
2.5 2312.045 463.867 
2.6 2230.707 456.484 
2.7 2153.102 449.579 
2.8 2078.887 443.108 
2.9 2007.758 437.034 
3 1939.451 431.323