FRENCH, DAVID B. Deviation of Earth Threatening Asteroids Using Tether and Ballast. (Under the direction of Dr. Andre P. Mazzoleni).
Raleigh, North Carolina
2009
APPROVED BY:
Dr. Pierre A. Gremaud Dr. Larry M. Silverberg
DEDICATION
ACKNOWLEDGMENTS
First, I would like to thank my wonderful wife for her love and support in this process. She is my perfect wife. Thanks also to my son, who, in spite of consuming curiosity about my research-bearing flash drive, never once threw it into the toilet. Thanks also to my mother-in-law and father-mother-in-law for helping so much with him and everything else during their welcomed visits. Thanks to my dad, who has always been an encouraging and valued advisor in all of my efforts, and to my mother for her constant support and occasional over-the-phone recaps of the news that I missed while at the library. I owe a great debt to my advisor, Dr. Andre Mazzoleni, for his guidance and patience–especially in tolerating my impatience at times–and for always improving my submissions with his thoughtful edits. Thanks to my committee for your guidance and advice. Also, thanks to Heather McNamara Koehler of NASA’s Micrometeoroid Environment Office for her assistance with the MEMR1c software. Thanks must be extended as well to my dog Bishop, for not never getting lost (permanently, anyways) after escaping his confines, though he had far too many opportunities. And thanks to all of my Whitehall Manor neighbors for returning him to us so many times with so little reproach. We will miss you...and not only when the dog gets out. Thanks to the rest of my Apex friends as well, especially my life group buddies. You guys are awesome. Uh, Tim, it’s your turn...
1.2 Model . . . 8
1.2.1 Derivation of Equations of Motion . . . 9
1.2.2 Constraints . . . 17
1.2.3 Verification . . . 17
1.2.4 Trajectory Alteration . . . 18
1.3 Numerical Simulation . . . 19
1.3.1 Baseline parameters . . . 20
1.4 Results . . . 20
1.4.1 Effect of Duration . . . 21
1.4.2 Effect of mass ratio . . . 22
1.4.3 Effect of Tether Length . . . 24
1.4.4 Effect of semimajor axis . . . 25
1.4.5 Effect of eccentricity . . . 27
1.4.6 Effect of initial true anomaly . . . 27
1.4.7 Effect of initial tether angle . . . 31
1.4.8 Effect of initial tether angle rotation rate . . . 33
1.4.9 Summary of parametric results . . . 34
1.4.10 Case Study . . . 37
1.5 Conclusions . . . 38
2 Massless Inelastic Model Parametric Study for Earth Miss Distances . . . 40
2.1 Introduction . . . 41
2.2 Model . . . 42
2.3 Results . . . 46
2.4 Conclusions . . . 51
3 Massless Inelastic Model Survey of Required Tether Lengths Required for Known Threatening Asteroids . . . 52
3.1 Introduction . . . 53
3.2 Model . . . 54
3.3 Results . . . 58
4 Comparison of New Models with Massless Inelastic Model: Massive
In-elastic Model, Massless Elastic Model and Massive Elastic Model . . . 64
4.1 Introduction . . . 65
4.2 Models . . . 65
4.2.1 Inelastic, massive tether model . . . 66
4.2.2 Elastic, bendable, massive tether model . . . 70
4.2.3 Elastic, straight, massless tether model . . . 75
4.3 Methods and assumptions . . . 75
4.4 Results . . . 76
4.4.1 Model Comparisons . . . 77
4.4.2 Slack Tether and High Tension . . . 91
4.4.3 Convergence . . . 95
4.4.4 Numerical Cost . . . 99
4.5 Case Study 1 . . . 100
4.6 Case Study 2 . . . 102
4.7 Conclusions . . . 103
5 Assessment of Micrometeoroid Threat to Very Long Tethers in Interplan-etary Space . . . 105
5.1 Introduction . . . 106
5.2 Methods and Assumptions . . . 106
5.3 Results . . . 108
5.4 Tether thickness variation . . . 113
5.5 Multi-Stranded Tether . . . 114
5.6 Conclusions . . . 115
Bibliography . . . 119
Appendices . . . 125
LIST OF FIGURES
Figure 1.1 Diagram of NEO-tether-ballast system . . . 9
Figure 1.2 Diagram of system forces . . . 11
Figure 1.3 Diagram of an orbit and reference orbit for use with Encke’s Method . . . 15
Figure 1.4 Scalar difference in position over 5 years between NEOs simulated using Cowell’s Method (direct integration) and Encke’s Method. Both NEOs shared the common parameters ˜m=0.1, L=100,000 km,a=2 AU,e= 0.8 . . . 18
Figure 1.5 Final positions of tether after 5 years, simulated using Cowell’s Method (direct integration) and Encke’s Method. NEOs shared common parameters ˜m=0.1,
L=100,000 km,a=2 AU,e= 0.8. . . 19
Figure 1.6 Definition of separation distance metric, ∆ . . . 20
Figure 1.7 Effect of tether mitigation over time shown vs. (a.) true anomaly (angular) scale and (b.) time scale. All unspecified parameters correspond to the baseline case. . . 22
Figure 1.8 Effect of duration over time will be cyclical . . . 23
Figure 1.9 Maximum separation between tethered and untethered NEOs . . . 23
Figure 1.10 Effect of varying mA : mB vs. (a.) true anomaly (angular) scale and (b.) time scale. All unspecified parameters correspond to the baseline case. . . 24
Figure 1.11 Effect of varying tether length vs. (a.) true anomaly (angular) scale and (b.) time scale. All unspecified parameters correspond to the baseline case.. . . 25
Figure 1.12 Effect of various orbit semimajor axes vs. (a.) true anomaly (angular) scale and (b.) time scale . . . 26
Figure 1.13 Effect of changing semimajor axis on separation rate of change for (a) true anomaly at which NEO orbit intersects Earth orbit approaching perihelion and (b) true anomaly at perihelion. All unspecified parameters correspond to the baseline case. . . 26
correspond to the baseline case. . . 29
Figure 1.18 Gain in specific mechanical energy, 2-1, varying (a.) semimajor axis, a,
and (b.) eccentricity,e. . . 31
Figure 1.19 Effect of varying initial tether angle vs. (a.) true anomaly (angular) scale and (b.) time scale. All unspecified parameters correspond to the baseline case. . . 32
Figure 1.20 Parametric view of separation rates vs. initial tether angle for orbits of various (a.) semimajor axes and (b.) eccentricities. Unspecified parameters are baseline values. . . 32
Figure 1.21 Effect of varying initial tether angle rotation rate vs. (a.) true anomaly (angular) scale and (b.) time scale. All unspecified parameters correspond to the baseline case. . . 33
Figure 1.22 Parametric view of separation rates vs. tether rotation rate for orbits of various (a.) semimajor axes and (b.) eccentricities. Unspecified parameters are baseline. . . 34
Figure 1.23 Plot of sixteen cases showing approximate equivalence of equal λcases vs. (a.) true anomaly (angular) scale and (b.) time scale . . . 35
Figure 1.24 Parametric plot of separation rates vs. λfor a. e = 0.3, b. e= 0.5 c. e = 0.7 and d. e= 0.9 . . . 35
Figure 1.25 Parametric plot of separation rates vs. λ for a. a= 0.8, b. a = 1.2 c. a= 1.6 and d. a= 2.0 . . . 36
Figure 1.26 Summary of results using slope of separation curve vs. semimajor axis and eccentricity of NEO orbits. . . 37
Figure 1.28 relative trajectory of a NEO tethered to a ballast mass with respect to that
same NEO’s position if untethered . . . 38
Figure 2.1 Configuration of NEO-tether-ballast system . . . 42
Figure 2.2 Diagram of an orbit and reference orbit for use with Encke’s Method . . . 46
Figure 2.3 Time history of distance between Earth and untethered NEO . . . 47
Figure 2.4 Time history of distance between Earth and tethered NEO . . . 47
Figure 2.5 Time history of distance between untethered and tethered NEO . . . 47
Figure 2.6 Results for all parameter values over 10 years . . . 48
Figure 2.7 Results for all parameter values over 20 years . . . 49
Figure 2.8 Results for all parameter values over 50 years . . . 50
Figure 3.1 Semimajor axis and eccentricity of known potentially hazardous asteroids (PHAs) . . . 54
Figure 3.2 Diagram of NEO-tether-ballast system . . . 55
Figure 3.3 Diagram of an orbit and reference orbit for use with Encke’s Method . . . 58
Figure 3.4 Example of scalar distance between an untethered PHA (2002-UK11) and Earth as a function of time . . . 59
Figure 3.5 Example of scalar distance between tethered PHA (2002-UK11) and Earth as a function of time . . . 59
Figure 3.6 Example of scalar distance between tethered and untethered PHA (2002-UK11) as a function of time . . . 60
Figure 3.7 Tether lengths required to divert the trajectory of an Earth-threatening PHA to miss by approximately one Earth radius, plotted vs. PHA mass. Dotted lines indicate lengths of 1000, 10,000 and 100,000 km. . . 60
Figure 3.8 Tether lengths required to divert the trajectory of an Earth-threatening PHA to miss by approximately one Earth radius, plotted vs. semimajor axis. Dotted lines indicate lengths of 1000, 10,000 and 100,000 km. . . 61
Figure 4.3 Diagram of asteroid-tether-ballast system with three nodes in the bead model tether . . . 71
Figure 4.4 Density of potentially hazardous asteroids (PHAs) in the parametric space (brightness indicates higher density), as well as convergence using MLI model.. . . . 77
Figure 4.5 Massive Inelastic Model vs. Massless Inelastic Model: Plot comparing mas-sive inelastic (set to 0 nodes, so without mass in this case) and massless inelastic tether models. From top to bottom, the plots show the following: 1. the separation distance between the tethered and untethered asteroid, 2. the difference between the predicted position of the asteroid by the model indicated, 3. the rotation state of the tether, 4. the rotation rates of the tethered bodies, and 5. mean tether tension. . . 78
Figure 4.6 Massive Inelastic Model: Plot comparing performance of tethers modeled with 1, 2, 3, 4 and 5 mass nodes with the massive inelastic model. From top to bottom, the plots show the following: 1. the separation distance between the tethered and untethered asteroid, 2. the difference between the predicted position of the asteroid by the model indicated, 3. the rotation state of the tether, 4. the rotation rates of the tethered bodies and 5. mean tether tension. . . 80
Figure 4.7 Massive Inelastic Model: Parametric plot of maximum separation distances between various specified cases. . . 81
Figure 4.8 Massive Inelastic Model: Parametric plot of maximum separation distances between various specified cases, divided into groups by specified semimajor axis (cross-sectional “slices” of Figure 4.7). . . 82
Figure 4.9 Massive Inelastic Model: Parametric plot of maximum tension for various specified cases . . . 83
“slices” of Figure 4.9). . . 84
Figure 4.11 Massless Elastic vs. Massless Inelastic Model: Plot comparing massless elastic and massless inelastic tether models. From top to bottom, the plots show the following: 1. the separation distance between the tethered and untethered asteroid, 2. the difference between the predicted position of the asteroid by the model indicated, 3. the rotation state of the tether (approximated by the rotation state of the first link for multi-link cases), 4. the rotation rates of the tethered bodies (approximated by the rotation of the first link for multi-link cases), and 5. mean tether tension. . . 85
Figure 4.12 Massive Elastic Tether vs. Massless Elastic Tether Model: Plot comparing massive elastic and massless elastic tether models. From top to bottom, the plots show the following: 1. the separation distance between the tethered and untethered asteroid, 2. the difference between the predicted position of the asteroid by the model indicated, 3. the rotation state of the tether (approximated by the rotation state of the first link for multi-link cases), 4. the rotation rates of the tethered bodies (approximated by the rotation of the first link for multi-link cases), and 5. mean tether tension. . . 86
Figure 4.13 Massive Elastic Tether Model: Plot comparing performance as the number of mass nodes is increased. From top to bottom, the plots show the following: 1. the separation distance between the tethered and untethered asteroid, 2. the difference between the predicted position of the asteroid by the model indicated, 3. the rotation state of the tether (approximated by the rotation state of the first link for multi-link cases), 4. the rotation rates of the tethered bodies (approximated by the rotation of the first link for multi-link cases), and 5. mean tether tension. . . 87
Figure 4.14 Massive Elastic Model: Parametric plot of maximum separation distances between various specified cases. . . 88
Figure 4.15 Massive Elastic Model: Parametric plot of maximum separation distances between various specified cases, divided into groups by specified semimajor axis (cross-sectional “slices” of Figure 4.14). . . 89
Figure 4.16 Massive Elastic Model: Parametric plot of maximum tension for various specified cases . . . 91
Figure 4.17 Massive Elastic Model: Parametric plot of maximum tension for various specified cases, divided into groups by specified semimajor axis (cross-sectional “slices” of Figure 4.16). . . 92
marker), the mass nodes (dot markers), and the tether links, either in tension )(solid line) or slack (dotted line). Each plot also includes, in the lower left, an overview plot showing the Sun, the orbit of the Earth (dotted line) the orbit of the asteroid (solid line) and the asteroid position (circle marker). . . 94
Figure 4.21 Massive Elastic Tether Model: Four snapshots of tether final positions after 10 year simulation. Each snapshot shows a range of cases, differentiated by the number of nodes used in the simulation. . . 96
Figure 4.22 Massive Elastic Tether Model: Rotational convergence plot, showing the approximate final angle of the tether vs. number of nodes used for the simulation. Labels correspond to those used in Figure 4.21: a. a=1.2 AU,e=0.3; b. a=1.2 AU,
e=0.5; c. a=1.2 AU, e=0.7; d. a=1.8 AU, e=0.5; e. a=1.8 AU, e=0.7; f. a=2.4 AU,e=0.7. . . 97
Figure 4.23 Massive Elastic Tether Model: Spacial convergence plot, showing the sepa-ration in the predications made by the model usingn nodes vs. predictions using
n-1 nodes, in terms of scalar separation distance. Labels correspond to those used in Figure 4.21: a. a=1.2 AU,e=0.3; b. a=1.2 AU, e=0.5; c. a=1.2 AU,e=0.7; d.
a=1.8 AU, e=0.5; e. a=1.8 AU,e=0.7; f. a=2.4 AU, e=0.7. . . 98
Figure 4.24 Time required to generate data for the massless inelastic model, the massless elastic model, the massive inelastic model, and the massive elastic model. . . 99
Figure 4.25 Case study scenario: 8 node tether going slack. . . 100
Figure 4.26 Massive Elastic Model: Case Study showing baseline case, for which tether goes slack, resulting in order of magnitude increase in maximum tether tension. . . 101
Figure 4.27 Massive Elastic Model: Case Study showing results of deceasing tether length to 45,000 km. . . 101
Figure 4.29 Massive Elastic Model: Case Study showing the baseline tension profile for a tether designed to carry a load of 5000 Newtons. . . 102
Figure 4.30 Massive Elastic Model: Case Study showing the tension profile for a tether designed to carry a load of 2000 Newtons. . . 102
Figure 4.31 Massive Elastic Model: Case Study showing the tension profile for a tether designed to carry a load of 1200 Newtons. . . 103
Figure 5.1 MEMR1c input screen. . . 107
Figure 5.2 Flux velocity charts for points along an orbit with semimajor axis of 1.2 AU and eccentricity of 0.3. . . 109
Figure 5.3 Micrometoroid flux vs. true anomaly . . . 110
Figure 5.4 Running total of number of penetrations vs. orbital true anomaly. . . 111
Figure 5.5 Flux velocity charts for points along an orbit with semimajor axis of 1.2 AU and eccentricity of 0.5. . . 112
Figure 5.6 Micrometoroid flux vs. true anomaly . . . 113
Figure 5.7 Running total of number of penetrations vs. orbital true anomaly. . . 114
Figure 5.8 Flux velocity charts for points along an orbit with semimajor axis of 1.2 AU and eccentricity of 0.7. . . 115
Figure 5.9 Micrometoroid flux vs. true anomaly . . . 116
Figure 5.10 Running total of number of penetrations vs. orbital true anomaly. . . 117
Figure 5.11 Number of predicted tether failures over a single orbit versus tether diameter for semimajor axis of 1.2 and eccentricity of 0.7. . . 117
Figure 5.12 Structure of multi-stranded tether (Hoytether). . . 118
Figure 5.13 Lifetime comparison of equal weight single line and failsafe multiline tethers for low-load mission . . . 118
Figure .14 Difference in rate of convergence for Cowell and Encke Methods . . . 128
δR = scalar difference between distance to actual point and reference point, km
∆ = scalar distance between unattached NEO and attached NEO, RE
δ = angle between the directions of tension and gravity force on point B, rad
∆ = change in specific mechanical energy, m2/s2
δν = difference between true anomalies of actual point and reference point, rad
= specific mechanical energy, m2/s2
1 = specific mechanical energy of unattached system, m2/s2
2 = specific mechanical energy of attached system, m2/s2
γ = angle between the directions of tension and gravity force on point A, rad
λ = mass center offset parameter, product of ˜m and L, km
µSun = gravitational parameter of the Sun, 1.327×1011km3/s2
ν = true anomaly, rad
νref = true anomaly of reference point on reference orbit, rad
a1 = first unit vector in A frame
a2 = second unit vector in A frame
a3 = third unit vector in A frame
aB/O = acceleration of point B with with respect to point O, km/s2
b1 = first unit vector in body frame
b2 = second unit vector in body frame
b3 = third unit vector in body frame
e1 = first unit vector in inertial frame
e2 = second unit vector in inertial frame
e3 = third unit vector in inertial frame
RA/O = vector from point O to point A, km
RB/A = vector from point A to point B, km
vA/O = velocity of point A with with respect to point O, km/s
vB/O = velocity of point B with with respect to point O, km/s
θ = tether angle, rad
θ0 = initial value ofθ, rad
θi = tether offset angle from local vertical (θ1) or previous link axis, rad
˜
m = ballast to NEO mass ratio
Fa
mi/O = acceleration of the ith mass with respect to the Sun in the inertial frame, km s
−2 a = orbit semimajor axis, AU
ai = MATLAB ODE suite absolute error for ith component of state vector
e = orbit eccentricity
Ei = MATLAB ODE suite allowable estimated error
fB1 = arbitrary force on B in theb1 direction, kN
mB = ballast mass, kg
mi = tether node mass, kg
R = scalar distance between Sun and NEO, km
r = MATLAB ODE suite relative error
RB = scalar distance from Sun to point B, km
RE = mean Earth radius, 6378.137 km
Rref = scalar distance from Sun to reference point on reference orbit, km
T = tether tension, kN
Ti = tether link tension, kN
tend = simulation duration, years
The views expressed in this document are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the
Over the course of history, the Earth has experienced many impacts of asteroids and comets, collectively referred to in this dissertation as “Near Earth Objects” (NEOs). The evidence of these impacts exists in the geological evidence of craters, such as the 1.2 km diameter Arizona impact crater, which was formed approximately 50,000 years ago [27], the 180 km diameter Chicxulub crater on the north coast of the Yucatan Peninsula, which is hypothesized to have resulted from a meteor impact about 65 million years ago [4], and the even larger 300 km diameter Vredefort crater in South Africa, which is 2 billion years old and the largest known impact crater on the Earth [28]. On a smaller scale and more recently, a NEO atmospheric entry was witnessed in the Tunguska region of Siberia, Russia on June 20, 1908. The object was observed from 500 km away and the explosion was heard at a distance of 1270 km [6]. It is estimated that the explosion was approximately the equivalent of 15 megatons of TNT [37].
Recently, the threat of future NEO impacts on the Earth has gained increased attention. On December 27, 2004, the Minor Planet Center issued a warning that the asteroid 2004 MN4 (later renamed “99942 Apophis”–after the serpent god in Egyptian
Mythology that brings darkness to the Earth) had a 1 in 38 chance of striking the Earth on April 13, 2029 [50]. This assessment was soon disproved, and there is no chance of an impact during the 2029 encounter. However, there remains a 1 in 40,000 chance [46] that Apophis will impact the Earth in a resonant encounter on April 13, 2036. The publicity from this event, combined with the potential impact risk of a resonant encounter in 2036, served to awaken the public to the risk posed by NEOs. In fact, the Congress of the United States ordered NASA, in 2005, via the George E. Brown, Jr. Near Earth Object Survey Act (HR-1022) [1], to locate threatening asteroids and determine how they should be mitigated: “The Congress declares that the general welfare and security of the United States require that the unique competence of the National Aeronautics and Space Administration in science and engineering systems be directed to detecting, tracking, cataloguing, and characterizing near-Earth asteroids and comets in order to provide warning and mitigation of the potential hazard of such near-Earth objects impacting the Earth.”
re-the Earth. In fact, it is only a matter of time.
The question of how to alter the trajectory of such a threat has been the subject of much research. Several mitigation techniques have been proposed, including detonating nuclear or conventional explosives in, on or near the NEO [3], guiding a retrograde NEO to impact the Earth-threatening NEO [41], taking advantage of the Yarkovsky effect [52], or using a tug of some type, whether connected to the NEO [35] or using gravity to pull the NEO [36].
This dissertation looks at a different approach–rendezvous with and attachment of a long tether and ballast mass to a NEO. This is not to be confused with the method proposed by Chobotov [13], a diversion approach using a relatively short tether. The tether-ballast system attachment proposed in this paper would affect the trajectory of the NEO in two ways. First, the connection of a tether and ballast mass would instantaneously change the center of mass of the system and therefore the orbit. Second, the tether tension would add a perturbing force which also would change the NEO’s trajectory.
inelastic and unbending. The next comparison model included elasticity (without mass). Next, the massive elastic tether model was introduced. Comparisons between the new models and the original massless inelastic model were generally in terms of overall deviation performance, i.e. the scalar distance between positions predicted by various models over time.
Massless Inelastic Model
1.1
Introduction
Over the course of history, the Earth has experienced many impacts of asteroids and comets, collectively referred to in this paper as “Near Earth Objects” (NEOs). The evidence of these impacts exists in the geological evidence of craters, such as the 1.2 km diameter Arizona impact crater, which was formed approximately 50,000 years ago [27], the 180 km diameter Chicxulub crater on the north coast of the Yucatan Peninsula, which is hypothesized to have resulted from a meteor impact about 65 million years ago [4], and the even larger 300 km diameter Vredefort crater in South Africa, which is 2 billion years old and the largest known impact crater on the Earth [28].
Recently, the threat of future NEO impacts on the Earth has gained increased attention. On December 27, 2004, the asteroid Apophis was projected to have a 1 in 38 chance of striking the Earth in 2029 [50]. This impact was ruled out a day later, but the publicity from this event, combined with the potential impact risk of a resonant encounter in 2036, served to awaken the public to the risk posed by NEOs, leading Congress to call on NASA to investigate mitigation alternatives [45].
The question of how to alter the trajectory of such a threat has been the subject of much research. Several mitigation techniques have been proposed, including detonating nuclear or conventional explosives in, on or near the NEO [3], guiding a retrograde NEO to impact the Earth-threatening NEO [41], taking advantage of the Yarkovsky effect [52], or using a tug of some type, whether connected to the NEO [35] or using gravity to pull the NEO [36].
B
ballast and mA is the mass of the NEO.
˜
m= mB
mA
(1.1)
The goal of this study was to determine the extent to which a tether and ballast mass system alone would change the trajectory of threatening NEOs–specifically, NEOs of the Aten and Apollo NEO groups [46] (both Earth orbit crossing). To this end, a simplified model of the system was constructed. The only gravitational force allowed to act on the NEO and ballast mass was that of the Sun. The only additional force considered was the tether tension. The attachment of the fully extended tether and ballast mass was assumed to occur instantaneously, and this attachment was assumed neither to change the position of the NEO nor to impart any change in momentum to the NEO at the time of attachment; therefore, the center of mass of the unattached NEO differed from the center of mass of the NEO-tether-ballast system. This change in the center of mass of the system was the dominant factor in changing the orbit of the NEO, with the acceleration due to the tether tension adding a secondary trajectory deviation (it should be noted that further perturbation effects, such as vibrational and librational motion [43] might further add to the trajectory alteration, but are not included in this study). The tether itself was assumed to be massless, inextensible, and of constant length. Additionally, all masses were assumed to be point masses, and all motion was constrained to the plane of the Earth’s orbit about the Sun.
The system can be seen in Figure 1.1. In this figure, there are three reference frames established. The three unit vectors {e1, e2, e3} comprise the inertial frame. The
Figure 1.1: Diagram of NEO-tether-ballast system
with e3. Finally, the {b1, b2, b3} frame is the “body frame” of the NEO-tether-mass
system, and rotates such thatb1 is always aligned withRB/A and b3 remains aligned with
e3. There are a number of variables that appear in the figure as well. R is the scalar
distance from the Sun to the NEO, ν is the true anomaly, which is the angle between the periapsis direction (same as inertial unit vector e1) andRA/O,Lis the scalar length of the tether, and θis the angle between the directions ofRA/O and RB/A.
1.2.1 Derivation of Equations of Motion
derivation are with respect to the inertial frame unless otherwise noted.
vA/O= ˙R a1+Rν˙ a2 (1.4)
Similarly, the differentiation of equation 1.3 in the inertial frame yields the velocity of the ballast (vB/O), expressed here in the B frame.
vB/O =
h
˙
Rcosθ+Rν˙sinθ+ ˙L
i
b1+
h
−R˙sinθ+Rν˙cosθ+L
˙
ν+ ˙θ
i
b2 (1.5)
Next, equations 1.4 and 1.5 are differentiated again with respect to time to deter-mine the accelerations of the NEO (aA/O) and ballast mass (aB/O).
aA/O=hR¨−Rν˙2ia1+
h
2 ˙Rν˙+Rν¨ia2 (1.6)
aB/O =
¨
Rcosθ+ 2 ˙Rν˙sinθ+Rν¨sinθ+ ¨L−Rν˙2cosθ−L
˙ ν+ ˙θ 2 b1 + h
−R¨sinθ+ 2 ˙Rν˙cosθ+Rν¨cosθ+ 2 ˙L
˙ ν+ ˙θ +L ¨ ν+ ¨θ
+Rν˙2sinθ
i
b2
(1.7)
To form the complete equations of motion using Newton’s Second Law of Motion, the forces acting on the end bodies must be determined.
The forces in Figure 1.2 include T, the tether tension force, which acts in equal magnitude on the NEO and the ballast. FGA is the force of the Sun’s gravity acting on the NEO, and FGB is the gravity force acting on the ballast. The forces fB1 and fB2
Figure 1.2: Diagram of system forces
zero). The angles, δ and γ are used to help orient the force terms, and will be related to other variables later in this derivation, and therefore do not appear in the final equations of motion. R and RB are the scalar distances from the Sun to the NEO and ballast mass, respectively. The relationship between these forces and the accelerations derived in equations 1.6 and 1.7 are expressed here.
(Tcosθ−FGA)a1+Tsinθa2 =mAaA/O (1.8)
(fB1−T−FGBcosγ)b1+ (fB2+FGBsinγ)b2=mBaB/O (1.9)
It is possible to express the sine and cosine of the angle γ in terms of defined variables using the Law of Sines and the Law of Cosines. These expressions are given in equations 1.10 and 1.11.
sinγ= √ Rsinθ
R2+L2+ 2RLcosθ (1.10)
cosγ = √ L+Rcosθ
R2+L2+ 2RLcosθ (1.11)
The Sun’s gravity force can also be expressed in terms of known constants and our primary variables. The expressions for FGA and FGB are given in equation 1.12 and 1.13.
FGA =
mAµSun
fB1−T−mB
µSun(L+Rcosθ) (R2+L2+ 2RLcosθ)3/2
=mB
2 ˙Rν˙+Rν¨
sinθ+
¨
R−Rν˙2
cosθ+ ¨L−L
˙
ν+ ˙θ
2
(1.16)
fB2+mB
µSunRsinθ (R2+L2+ 2RLcosθ)3/2
=mB
h
−R¨+Rν˙2
sinθ+
2 ˙Rν˙+Rν¨
cosθ+ 2 ˙L
˙ ν+ ˙θ +L ¨ ν+ ¨θ i (1.17)
To compress equations 1.16 and 1.17, it is convenient to introduce a new variable substitution, K, which is defined as follows:
K= µSun
(R2+L2+ 2RLcosθ)3/2 (1.18)
Using this substitution and dividing equations 1.14 and 1.15 bymA, and equations 1.16 and 1.17 by mB, the four equations can be written as follows:
− T
mA
cosθ+ µSun
R2 + ¨R−Rν˙
2= 0 (1.19)
− T
mA
sinθ+ 2 ˙Rν˙+Rν¨= 0 (1.20)
−fB1+T
mB
+K(L+Rcosθ) +2 ˙Rν˙+Rν¨sinθ+R¨−Rν˙2cosθ+ ¨L−Lν˙+ ˙θ2 = 0 (1.21)
−fB2
mB
−KRsinθ+
−R¨+Rν˙2
sinθ+
2 ˙Rν˙+Rν¨
cosθ+ 2 ˙L
Equations 1.19 and 1.20 are nearly in their final form. Solving equation 1.19 for ¨
R and equation 1.20 for ¨ν produces the following equations.
¨
R=Rν˙2−µSun
R2 + T mA
cosθ (1.23)
¨
ν= T
mAR
sinθ−2 ˙R
R ν˙ (1.24)
Some additional manipulation of equations 1.21 and 1.22 will be required to get them in their final forms. First, equation 1.22 is multiplied by sinθ and subtracted from equation 1.21 multiplied by cosθ.
−fB1+T
mB
cosθ+fB2
mB
sinθ+KLcosθ+KR+ ¨R−Rν˙2+ ¨Lcosθ
−2 ˙Lν˙+ ˙θsinθ−Lν˙+ ˙θ2cosθ−Lν¨+ ¨θsinθ= 0
(1.25)
Next, equation 1.22 is multiplied by sinθ and thenadded to equation 1.21 multi-plied by cosθ.
−fB1+T
mB
sinθ−fB2
mB
cosθ+KLsinθ+ 2 ˙Rν˙+Rν¨+ ¨Lsinθ
+ 2 ˙Lν˙+ ˙θcosθ−Lν˙+ ˙θ2sinθ+Lν¨+ ¨θcosθ= 0
(1.26)
By subtracting equation 1.19 from equation 1.25, equation 1.27 is derived.
−fB1
mB +T 1 mA + 1 mB
+KL+ ¨L−Lν˙+ ˙θ2
cosθ
+
−fB2
mB
−2 ˙Lν˙+ ˙θ−Lν¨+ ¨θ
sinθ+KR− µSun
R2 = 0
(1.27)
Subtracting 1.20 from 1.26 results in equation 1.28.
−fB1
mB +T 1 mA + 1 mB
+KL+ ¨L−Lν˙+ ˙θ2
sinθ
+
−fB2
mB
+ 2 ˙Lν˙+ ˙θ+Lν¨+ ¨θ
cosθ= 0
(1.28)
mul-The four boxed equations (1.23, 1.24, 1.29 and 1.30) comprise the full system of equations necessary to numerically simulate the NEO-tether-ballast system. However, direct numerical integration (Cowell’s Method) of these equations can be problematic, due to large order of magnitude variances between the variables and the limitations of machine precision (for more details, see Appendix). Therefore, it is desirable to find another method of simulating the system (other than direct integration), as will be discussed below.
Encke’s Method
While Cowell’s Method involves numerically integrating all system accelerations, with Encke’s Method only the perturbing accelerations are integrated [5]. These results are then added to a Keplerian (ie. untethered) reference orbit. This sum provides all necessary information to determine the actual position of the NEO. Figure 1.3 illustrates how this scheme works. The reference orbit is elliptical, and is determined by the initial conditions of the NEO’s position and velocity vectors. At the beginning of the simulation (arbitrarily shown in the figure at perihelion), there is no variation between the Keplerian orbit and the actual orbit. As time goes on, the difference between these orbits will increase. Over time, if the variation, ∆, between the actual and reference position increases to an unacceptable degree, a new reference orbit can be established based on the current position and velocity of the NEO at that time. Classically, Encke’s Method has been employed using a Cartesian coordinate system. It is convenient for this study, however, to employ a polar coordinate system.
Figure 1.3: Diagram of an orbit and reference orbit for use with Encke’s Method
νref, is the angle from perihelion to the reference body andδν is the variation between this and the actual true anomaly. Finally,θis the angle between the local vertical (at the NEO) and the tether.
Mathematically, the variation of the actual NEO position from the reference posi-tion is determined by subtracting Keplerian orbital equaposi-tions from the equaposi-tions of moposi-tion forR(equation 1.23) andν(equation 1.24) derived previously, and integrating the resulting equations over time.
The variations between the actual and reference positions can be described as follows:
R=Rref +δR (1.31)
ν =νref +δν (1.32)
Differentiating these equations twice with respect to time yields the following.
¨
¨ Rref +
µ
R3Rref = 0 (1.37)
Expressing these equations in polar form leads to the following expressions.
¨
Rref =Rrefνref˙ 2−
µSun
R2
ref
(1.38)
¨
νref =− 2 ˙Rref
Rref ˙
νref (1.39)
By combining equation 1.35 with equations 1.23 and 1.38, the first Encke equation of motion is derived.
δR¨= (Rref +δR) 2 ˙νrefδν˙+δν˙2
+δRν˙ref2 +µSun
"
2RrefδR+δR2
R2ref(Rref+δR)2
#
+ T
mA cosθ
(1.40) Similarly, by combining equation 1.36 with equations 1.24 and 1.39, the second Encke equation of motion is derived.
δν¨=
−2R˙ref +δR˙
Rref+δR
( ˙νref +δν˙) + 2 ˙Rref
Rref ˙
νref+
T
(Rref +δR)mA
sinθ (1.41)
1.2.2 Constraints
The tether in this study was assumed to be of constant length. Given this con-straint, ˙L and ¨L must both be equal to zero. Additionally, it is assumed that no external forces are acting on the ballast mass, therefore, fB1 and fB2 are also equal to zero. These
assumptions change equation 1.29 as follows.
¨
θ= 1
L
KR−µSun
R2
sinθ−ν¨ (1.42)
Equation 1.30 changes to equation 1.43 when the assumptions are applied.
¨
L= 0 =−T
1 mA + 1 mB
−KL+Lν˙+ ˙θ2+µSun
R2 −KR
cosθ (1.43)
This expression then leads to the expression for determining the tether tension which can be seen in equation 1.44.
T mA
= m˜ 1 + ˜m
−KL+L
˙ ν+ ˙θ 2 + µSun
R2 −KR
cosθ
(1.44)
Equation 1.42 can either be paired with equations 1.23 and 1.24 and integrated with Cowell’s Method or paired with equations 1.40 and 1.41 and integrated using Encke’s Method. The tether tension over time is required by both methods, and equation 1.44 provides that information.
1.2.3 Verification
Figure 1.4: Scalar difference in position over 5 years between NEOs simulated using Cow-ell’s Method (direct integration) and Encke’s Method. Both NEOs shared the common parameters ˜m=0.1, L=100,000 km,a=2 AU,e= 0.8
1.2.4 Trajectory Alteration
Figure 1.5: Final positions of tether after 5 years, simulated using Cowell’s Method (direct integration) and Encke’s Method. NEOs shared common parameters ˜m=0.1, L=100,000 km,a=2 AU,e= 0.8.
1.3
Numerical Simulation
In order to determine the separation distance between the tethered and untethered NEOs, both cases had to be simulated and then compared. For the untethered NEO, the results could be determined by orbit projection based on its known elliptical path, assuming it followed the restricted two-body equations of motion [5]. For the NEO-tether-ballast system, the results were generated using numerical simulation via a special perturbation method, specifically a modified form of Encke’s Method [5], as described earlier. As stated in the derivation of equations of motion, this perturbation method was necessary due to convergence issues that arose with attempts at using direct integration (for more discussion on convergence, see the Appendix).
Figure 1.6: Definition of separation distance metric, ∆
1.3.1 Baseline parameters
In studying the effect of changing each parameter, it is helpful to specify a baseline set of parameters that will be used in these comparisons. First, the baseline ratio between the ballast mass and the NEO mass will be 1:1000. This value is based on the approximate mass of known potentially hazardous asteroids (PHAs) [46]. Specifically, this would allow a ballast roughly equal in mass to a fueled Saturn V rocket to be used for the smallest known PHA and allow relatively small multiples of this mass to be effective for a large number of PHAs. The baseline tether length will be 10,000 km. This length was deemed reasonable based on the ongoing studies on the space elevator concept, which would require a tether roughly four to ten times as long. The baseline orbital parameters were a semimajor axis of 1.2 astronomical units (AU) and an eccentricity of 0.8. These values were chosen because they are central to the parametric space in which this method obtains favorable results. The baseline true anomaly for tether-ballast attachment was perihelion. The baseline tether angle and rate of this angle were both set to zero. See Table 1.1 for a summary of these parameters.
1.4
Results
Table 1.1: Baseline parameters for this study
parameter symbol value
mass ratio m˜ 0.001
tether length L 10,000 km
semimajor axis a 1.2 AU
eccentricity e 0.8
initial true anomaly νi 0 initial tether angle θ0 0
initial tether angle rate θ˙i 0
reader should note in examining these distances that the NASA Analysis of Alternatives presented to Congress in 2007 indicates (on p.22) that one Earth radius is a sufficient NEO deflection [45]. Specifically, this study will initially involve examining the effect of changing each study parameter on separation distance. The focus will then shift to the rate of change of this separation distance, which will enable a summary of the results that can be applied over any time frame.
1.4.1 Effect of Duration
Figure 1.7 shows the scalar distance between the attached and unattached NEO over time for a particular case. Note that this distance oscillates as the two systems travel through the orbit. The distance is at a maximum at perihelion and at a minimum at aphelion. However, if a particular true anomaly is chosen (chosen true anomaly represented by circles on figure), the effect is shown to be generally linear over time with respect to the chosen value of true anomaly. That is, the displacement, ∆, of the tethered NEO position from its untethered position increases approximately linearly with respect to this specified true anomaly.
a. b.
Figure 1.7: Effect of tether mitigation over time shown vs. (a.) true anomaly (angular) scale and (b.) time scale. All unspecified parameters correspond to the baseline case.
move further and further apart, the curvature of the orbit begins to affect the separation distance more and more. This effect will be seen sooner for cases where the mass ratio is large, or the tether is very long, or both. Figure 1.8 shows an extreme case using a 100,000 km tether and mass ratio of 0.1 (ballast is one tenth the NEO mass) shown over 100 years. The figure shows the cyclical nature of the separation distance. This cycling of the separation distance will occur regardless of the parameters chosen.
Ultimately, the maximum separation distance is approximately the average length of the major axes (twice the semimajor axis) of the two systems, occurring when the tethered and untethered bodies are at opposition with respect to the sun, precisely when one is at periapsis and the other is at apoapsis (Figure 1.9). Most cases will take a very long time to reach this maximum separation.
1.4.2 Effect of mass ratio
By attaching a larger ballast mass to a tether of specified length, the system center of mass is moved in the direction of the ballast mass (this can also be accomplished by increasing tether length).
Figure 1.8: Effect of duration over time will be cyclical
Figure 1.9: Maximum separation between tethered and untethered NEOs
a. b.
Figure 1.10: Effect of varyingmA:mB vs. (a.) true anomaly (angular) scale and (b.) time scale. All unspecified parameters correspond to the baseline case.
the ordinate).
1.4.3 Effect of Tether Length
As stated before, changing the tether length is another way to control the degree to which the system center of mass is shifted. Figure 1.11 shows that, as was the case for the mass ratio, an order of magnitude increase results in an order of magnitude increase in miss distance.
a. b.
Figure 1.11: Effect of varying tether length vs. (a.) true anomaly (angular) scale and (b.) time scale. All unspecified parameters correspond to the baseline case.
Given the near equivalence, the two parameters, the mass ratio, ˜m, and tether length,L, will be combined (in a later section) into a single parameter, λ, to help compress a summary of all results. Specifically, it will be shown in Figure 1.23 that any equivalent product of the mass ratio and the tether length produces a similar result in terms of separa-tion distance. This compression of two parameters will then help to succinctly summarize the results.
1.4.4 Effect of semimajor axis
Up to this point, the parameters changed have been matters of design choice. The semimajor axis, on the other hand, is a property of the chosen NEO’s orbit.
It was found that for an orbit with a larger semimajor axis, a, the effectiveness of the tether-ballast mitigation technique was reduced. As Figure 1.12b shows, a larger semimajor axis results in a lower rate of change of separation distance, ˙∆.
a. b.
Figure 1.12: Effect of various orbit semimajor axes vs. (a.) true anomaly (angular) scale and (b.) time scale
a. b.
Figure 1.13: Effect of changing semimajor axis on separation rate of change for (a) true anomaly at which NEO orbit intersects Earth orbit approaching perihelion and (b) true anomaly at perihelion. All unspecified parameters correspond to the baseline case.
monotonically witha.
1.4.5 Effect of eccentricity
The eccentricity of the NEO orbit was another parameter studied. Without excep-tion, a more eccentric orbit is more susceptible to trajectory alteration than a less eccentric orbit, assuming all other parameters remain the same.
a. b.
Figure 1.14: Effect of various orbit eccentricities vs. (a.) true anomaly (angular) scale and (b.) time scale. All unspecified parameters correspond to the baseline case.
This can be seen in Figure 1.14. In this figure, plotted on a log scale for the y-axis, the slope of the separation distance increases as eccentricity increases.
Figure 1.15 shows that the separation rate increases monotonically with increasing eccentricity, regardless of semimajor axis. Additionally, the two plots, one measuring the effect at an Earth intersecting true anomaly, the other at perihelion, suggest that the relationship is monotonic regardless of the true anomaly at which the rate is measured.
1.4.6 Effect of initial true anomaly
a. b.
Figure 1.15: Effect of changing eccentricity on separation rate of change for (a) true anomaly at which NEO orbit intersects Earth orbit approaching perihelion and (b) true anomaly at perihelion All unspecified parameters correspond to the baseline case.
effectiveness of the tether-ballast system in altering the position of the NEO over time is maximized. It also shows that the rate is minimized by attaching at aphelion.
a. b.
Figure 1.16: Effect of varying initial orbit true anomaly vs. (a.) true anomaly (angular) scale and (b.) time scale. All unspecified parameters correspond to the baseline case.
a typical tether-ballast attachment. Starting at 0 degrees true anomaly (perihelion), the separation rates are at a maximum. The rate decreases to a minimum at 180 degrees (apoapsis) and then returns to a maximum as it approaches perihelion again. These plots demonstrate that the best true anomaly for attachment is periapsis.
a. b.
Figure 1.17: Parametric view of separation rates vs. initial true anomaly plotted for various (a.) semimajor axes and (b.) eccentricities All unspecified parameters correspond to the baseline case.
An explanation of this can be derived from examining the specific mechanical energy at the time of tether-ballast attachment. Determining the total energy before at-tachment is straightforward.
E1 = 1 2mA
˙
R2+R2ν˙2
−mAµSun
R (1.45)
Assuming the attachment of the tether and ballast mass is always performed with the tether pointing radially away from Sun (θ = 0, this assumption will be proven to be beneficial later on) and that the tether is rotating at exactly the rate of change of the true anomaly, ˙ν, at that time ( ˙θ = 0, more effective rotation rates exist, but this is assumed for simplicity), the position vector to the center of mass of the system after tether attachment can be defined as follows.
R2=
R+ mL˜ 1 + ˜m
a1 (1.46)
Combining the kinetic and potential energies yields the total energy.
E2=
1
2(mA+mB)
"
˙
R2+
R+ mL˜ 1 + ˜m
2
˙
ν2
#
−(mA+mB)µSun
R+1+ ˜mL˜m (1.50)
Dividing equations 1.45 and 1.50 by the total mass of each system (mAfor equation 1.45 andmA+mB for equation 1.50) results in equations describing the specific mechanical energies for both the unattached (1) and attached (2) systems.
1 = 1 2
˙
R2+R2ν˙2
− µSun
R (1.51)
2 = 1 2
"
˙
R2+
R+ mL˜ 1 + ˜m
2
˙
ν2
#
− µSun
R+1+ ˜mL˜m (1.52)
Subtracting equation 1.51 from equation 1.52, reveals the difference in the specific mechanical energy between the two systems.
∆=2−1 =
˜
mLν˙2 1 + ˜m
R+ mL˜ 2 + 2 ˜m
+µSun
1
R −
1 + ˜m R+Rm˜ + ˜mL
(1.53)
Figure 1.18 illustrates this difference vs. the true anomaly at which the attachment is made. For Figure 1.18a., several values of semimajor axis are shown, and for 1.18b., various eccentricities are shown.
Figure 1.18 is similar to Figure 1.17 and confirms that more energy is gained for an attachment at periapsis than at any other point in the orbit, regardless of orbit size and shape. It should be noted that, while they are not shown here, various values of L and
a. b.
Figure 1.18: Gain in specific mechanical energy, 2-1, varying (a.) semimajor axis, a, and
(b.) eccentricity, e
approximately proportional to the changes in length or mass ratio, but periapsis attachment still resulted in the largest energy gain.
1.4.7 Effect of initial tether angle
In Figure 1.19, the effect of varying the initial tether angle is shown. The figure appears not to include the θ= 0 degree case, but it is almost indistinguishable from the θ
= 180 degree case, so the plots overlap. These two cases clearly outperform theθ = 90 and 270 degree cases.
The reason for this phenomenon is that if the initial tether angle is at 90 or 270 degrees, the new center of mass is in the same orbit as the original NEO. Attaching at 0 or 180 degrees maximizes the instantaneous change in the perihelion distance of the system. It is also interesting to note the oscillation seen in theθ0 = 90 and 270 degree cases. This
shows the small scale effects of the rotation of the NEO-tether-ballast system about its center of mass.
a. b.
Figure 1.19: Effect of varying initial tether angle vs. (a.) true anomaly (angular) scale and (b.) time scale. All unspecified parameters correspond to the baseline case.
a. b.
Figure 1.20: Parametric view of separation rates vs. initial tether angle for orbits of various (a.) semimajor axes and (b.) eccentricities. Unspecified parameters are baseline values.
1.4.8 Effect of initial tether angle rotation rate
There are logistical reasons that an initial tether rotation rate of 0 is preferable, but variations from this are interesting. By increasing this rate in the direction of increasing true anomaly, the system center of mass is accelerated, increasing the system center of mass velocity at perihelion, resulting in a higher energy orbit and greater separation.
a. b.
Figure 1.21: Effect of varying initial tether angle rotation rate vs. (a.) true anomaly (angular) scale and (b.) time scale. All unspecified parameters correspond to the baseline case.
On the other hand, a slight decrease from a zero rate has a detrimental effect on performance, as Figure 1.21 shows. However, Figure 1.22 shows this is only true for small negative rates. For larger negative rotation rates, separation performance is similar to large positive rates.
Figure 1.22 shows the effect of the tether rotation rate on the separation rate. This chart shows that either a positive or a negative rotation can improve separation rates. For each case, though, there is a minimum performance region near, but less than, 0.
a. b.
Figure 1.22: Parametric view of separation rates vs. tether rotation rate for orbits of various (a.) semimajor axes and (b.) eccentricities. Unspecified parameters are baseline.
1.4.9 Summary of parametric results
As stated earlier, any equivalent product of mass ratio, ˜m, and tether length L, will produce approximately the same separation over time. We therefore define a new parameter, λ.
λ= ˜mL (1.54)
Any two cases in which the value ofλis the same will have very similar results. For example, the case of a mass ratio of 0.01 and a tether length of 10,000 km will produce the same result as a mass ratio of 0.1 and a tether length of 1000 km. Figure 1.23 demonstrates this for sixteen cases that fall into seven approximately equivalent plots.
To create the plots, all possible combinations of the mass ratio vector of [0.1 0.01 0.001 0.0001] and the tether length vector (in kilometers) of [100 1000 10,000 100,000] were plotted. Specifically, the λ = 10,000 km case is a single case combining ˜m of 0.1 and L of 100,000 km. Theλ= 1000 km case is comprised of two cases, ˜mof 0.1 withLof 10,000 km and ˜mof 0.01 withLof 100,000 km. In a similar manner, theλ= 100 km case is comprised of three combinations, λ= 10 km is comprised of four, λ = 1 km is comprised of three,λ
= 0.1 km is comprised of two and λ= 0.01 km is comprised of one combination.
a. b.
Figure 1.23: Plot of sixteen cases showing approximate equivalence of equalλcases vs. (a.) true anomaly (angular) scale and (b.) time scale
a. b.
c. d.
c. d.
Figure 1.25: Parametric plot of separation rates vs. λfor a. a= 0.8, b. a= 1.2 c. a= 1.6 and d. a= 2.0
combinations of mass ratio and tether length were plotted with repeated values ofλ. Thus, these plots reinforce that for a particular value ofλ, the separation rates overlap (all identi-cal markers for each value ofλare approximately equal). Again, the purpose of these plots is to convince the reader of the validity of compressing ˜m and L into a single variable, λ, for the purpose of summary. Moreover, they help support the claim that this is true over the entirea-eparametric space.
Figure 1.26 gives a summary of the separation distance rates, ˙∆ for different values of a, e andλ. The eccentricity is represented on the x-axis, the semimajor axis is represented on the y-axis, and the separation distance slopes are plotted on the z-axis. The extent of each mesh (a-e parametric space spanned by by each value ofλ) reflects the values ofaand
Figure 1.26: Summary of results using slope of separation curve vs. semimajor axis and eccentricity of NEO orbits.
whereλ= ˜mL, results in a proportional increase in separation distance rate of change over time, ˙∆.
1.4.10 Case Study
Figure 1.27: time history of separation between NEO and that same NEO tethered to a ballast mass
Figure 1.28: relative trajectory of a NEO tethered to a ballast mass with respect to that same NEO’s position if untethered
1.5
Conclusions
Massless Inelastic Model
2.1
Introduction
NASA’s “Near Earth Object Program” is responsible for coordinating “NASA-sponsored efforts to detect, track and characterize potentially hazardous asteroids and comets that could approach the Earth.” To date, the program has discovered over 5500 objects with a perihelion distance of less than 1.3 astronomical units (AUs)–this is the re-quirement for the objects to be qualified as “near Earth objects” (NEOs). Of these, nearly 1000 are classified as “potentially hazardous objects” (PHOs) or objects that pass within 0.05 AU of Earth’s orbit. And these objects are quite likely to be only a small fraction of the total number, as discovery rates are increasing each year. In 1995, there were 335 identified near Earth Asteroids (NEAs), the classification that makes up the vast majority of NEOs. In the past three years alone, six times this number were added to the database: 627 in 2005, 641 in 2006, and 648 in 2007. With so many objects passing so near the Earth, it is likely that eventually one of these objects will be predicted to be on a course to impact the Earth.
In fact, this has already occurred. On December 27, 2004, the Minor Planet Center issued a warning that the asteroid 2004 MN4 (later renamed “99942 Apophis”) had a 1 in
38 chance of striking the Earth on April 13, 2029 [50]. This assessment was soon disproved, and there is no chance of an impact during the 2029 encounter. However, there remains a 1 in 40,000 chance [46] that Apophis will impact the Earth in a resonant encounter on April 13, 2036.
completed for this conference [21].
2.2
Model
Figure 2.1: Configuration of NEO-tether-ballast system
Figure 2.1 illustrates the layout of the system. The figure shows three points: the Sun (point O), the asteroid (point A) and the ballast (point B). It also specifies three reference frames. The three unit vectors{e1,e2,e3} comprise the inertial frame. The{a1,
a2,a3}frame rotates such thata1 is always aligned withRA/Oanda3 remains aligned with
e3. Finally, the {b1, b2, b3} frame is the “body frame” of the NEO-tether-mass system,
and rotates such thatb1 is always aligned withRB/A and b3 remains aligned withe3. The
(perihelion) direction. Finally, θ is the angle between RB/A (or b1) and the local vertical
(a1).
In this model, there are several assumptions and constraints. First, all NEOs simulated in this study are assumed to be coplanar with the Earth for the purpose of simplicity. Also, the orbit of the Earth is assumed to be circular, again for simplicity. The Sun, the Earth, the NEO and the ballast are all treated as point masses. The tether is assumed to be inextensible and massless.
This model led to the following set of equations of motion [24]:
¨
R=Rν˙2−µSun
R2 + T mA
cosθ (2.1)
¨
ν= T
mAR
sinθ−2 ˙R
R ν˙ (2.2)
¨
θ= fB2
LmB −2 ˙ L L ˙ ν+ ˙θ +
KR− µSun
R2
sinθ
L −¨ν
(2.3)
¨
L=fB1
mB −T 1 mA + 1 mB
−KL+
L ˙ ν+ ˙θ 2 + µSun
R2 −KR
cosθ
(2.4)
New variables appear in these equations and require definition. First, T is the tether tension,mAis the mass of the NEO,mB is the mass of the ballast (the ratiomB:mA is also expressed as ˜m),fB1 andfB2are arbitrary forces on the ballast (these could be used
for modeling controls or disturbances, but they are set equal to zero for this study) and
µSun is the gravitational parameter of the Sun. Finally, K is a function used to compress the equations of motion and is defined as follows:
K= µSun
(R2+L2+ 2RLcosθ)3/2 (2.5)
The tether for this study was assumed to be massless and inextensible; therefore, ˙
orbit, a minimum distance between the Earth and the PHA could be found. This was accomplished for a range of orbital parameters, specifically semimajor axis and eccentricity. System parameters, including tether length and ballast to NEO ratio, were also varied. Finally, results of these varied cases were determined for durations of 10, 20 and 50 years. A list of the parameters used in the study can be found in Table 2.1.
Table 2.1: Parameter values used in simulations
tend, yrs mB:mA L, km a, AU e
10 1×10−1 100 0.8 0.6 20 1×10−2 1000 1.2 0.7 50 1×10−3 10,000 1.6 0.8 1×10−4 100,000 2.0 0.9 1×10−5
1×10−6
In this table, tend represents the number of years simulated,mB:mA(also denoted as ˜m) is the ratio of ballast to NEO mass,Lis the tether length, andaandeare the orbital semimajor axis and eccentricity, respectively. The time periods chosen represent a range of times that might be available for a mitigation attempt. The ratios represent a span of feasible ratios (depending on NEO size), the lengths range from the conservative 100 km to a very aggressive length of 100,000 km, the same length considered in some space elevator proposals [9]. The orbital parameters represent the parametric space in which the tether and ballast technique were found to be the most effective.
equations. Therefore, a special perturbation method was utilized to improve convergence– Encke’s Method [5]. However, the method was altered somewhat since Encke’s method is generally used in Cartesian coordinates and this problem lends itself to polar coordinates.
Thus the method was revised [24]. Figure 2.2 shows a diagram describing the altered form of Encke’s method. In order to reduce stiffness, the equations of motion (Equations 2.1 and 2.2) reflect only the deviation from a purely elliptical reference orbit. This is determined by subtracting Keplerian orbital equations from the equations of motion forR (equation 2.1) and ν (equation 2.2) for the tethered system.
δR¨ = (Rref +δR) 2 ˙νrefδν˙+δν˙2
+δRν˙ref2 +
µSun
"
2RrefδR+δR2
R2ref(Rref+δR)
#
+ Tcosθ
mA
(2.6)
δν¨=
−2R˙ref +δR˙
Rref +δR
( ˙νref +δν˙) +
2 ˙Rref
Rref ˙
νref+
Tsinθ
(Rref +δR)mA
(2.7)
In these equations,Rref is the scalar distance from the Sun to the reference body, andδRis the variation betweenRref and the scalar distance to the NEO. The reference true anomaly,
νref, is the angle from perihelion to the reference body andδν is the variation between this and the actual true anomaly. All other variables are as previously defined.
Equations 2.6 and 2.7, along with equations 2.3 and 2.4 make up the set of equa-tions of motion required for determining the variation from the reference orbit, which is elliptical. Convergence for the Encke Method approach was much better than for direct integration, generally by several orders of magnitude [24].
Finally, the initial conditions for the numerical simulation should be discussed. There are eight to be defined: Rref,0, νref,0, δR0, δR˙0, δν0,δν˙0, θ0, and ˙θ0. The reference
Figure 2.2: Diagram of an orbit and reference orbit for use with Encke’s Method
depends on the size and shape of the orbit. In using Encke’s Method to simulate the path of the tethered NEO, it is assumed that the NEO is on the reference orbit initially. Therefore,
δR0,δR˙0,δν0,δν˙0 are all set equal to zero.
2.3
Results
Before discussing overall results, it is useful to discuss the result of a single case. The parameters for this single case are as follows: tend of 10 years, ˜m of 0.001,L of 10,000 km, semimajor axis, a, of 0.8 AUs, and eccentricity, e, of 0.8.
Figure 2.3 shows the impact case. The miss distance, in terms of Earth radii, is shows on the ordinate. The time, in years, is shown on the abscissa. In this case, the strike occurs within 10 km of the Earth’s center of mass.
Figure 2.4 shows the distance between thetethered NEO and the Earth over time. At approximately the time at which the untethered NEO would have impacted the Earth, the closest pass of the tethered NEO occurs. In this case, the tethered NEO misses by over 21 Earth radii.
unteth-Figure 2.3: Time history of distance between Earth and untethered NEO
Figure 2.4: Time history of distance between Earth and tethered NEO
Figure 2.5: Time history of distance between untethered and tethered NEO
Figure 2.6: Results for all parameter values over 10 years
For the overall results, this minimum distance is plotted in the context of the orbital and system parameters that produced it. Figure 2.6 shows the overall results for the 10 year case. This figure requires some explanation. First, note that there are six charts in the figure, ranging from left to right. For each chart, the ordinate shows the miss distance and the abscissa represents the eccentricity. Starting on the left, the first chart shows results for the smallest ballast to mass ratio. This mass ratio increases by one order of magnitude for each successive chart, as viewed from left to right, and results in a one order of magnitude increase in miss distance for each successive chart. For each chart, there are four groups of lines, each group with a different line style. Each of these line styles represents a tether length, as shown in the legend in the lower right of the figure. In general, a longer tether results in a larger miss distance, again proportionally–an order of magnitude increase in length results in an order of magnitude increase in miss distance. Each line represents a particular semimajor axis, each semimajor axis having a unique marker as noted in the legend in the upper left of the figure.
eccentricity of the NEO orbit increases and the semimajor axis of the NEO orbit decreases (we see this same dependence on NEO eccentricity and semimajor axis for the 20 year and 50 year cases discussed below).
Also note that for larger mass ratios and longer tethers, there are cases that do not follow the general increasing trend that is evident from left to right in the charts. These outliers are the result of a tethered NEO making its closest pass to the Earth at a time prior to the final NEO-Earth orbit intersection.
Figure 2.7: Results for all parameter values over 20 years
Figure 2.7 displays results from the 20 year case. In general, these cases show an increase in the miss distances over the 10 year case, but also include an increased number of outliers–cases with longer tether length and larger mass ratios that come closest to the Earth before the final NEO-Earth intersection.
Figure 2.8 displays results from the longest duration case–50 years. Note that in general, miss distances increase over the 10 and 20 year cases. However, there are also more outliers than for the 10 and 20 year cases.
Figure 2.8: Results for all parameter values over 50 years
if a lighter ballast had been used, although adding the tether-ballast system still yields an improvement over the non-tethered case.
2.4
Conclusions
3.1
Introduction
On Friday, April 13th, 2029, a 320 meter asteroid will pass close enough to the Earth to be viewed with the naked eye. The asteroid will make a second pass at Earth in 2036, and it is estimated that the asteroid has a 1 in 45,000 chance of impacting during that encounter [26]. The asteroid is Apophis, ominously named after the serpent god in Egyptian Mythology that brings darkness to the Earth. Before the discovery of the threat Apophis posed (it was originally projected to have a significant chance of impacting the Earth during the 2029 encounter [50]), the threat of an asteroid or comet, collectively referred to as near-Earth objects (NEOs), colliding with Earth was largely ignored. Since Apophis, NEO impact mitigation has been widely researched. Several mitigation techniques have been proposed, including detonating nuclear or conventional explosives in, on or near the NEO [3], guiding a retrograde NEO to impact the Earth-threatening NEO [41], using the Yarkovsky effect [52], or using a tug of some type, whether connected to the NEO [35] or using gravity to pull the NEO [36]. This paper investigates the idea of using a tether and ballast mass to divert a PHA on an Earth-intersecting trajectory, and examines the effectiveness of the idea through a study of how many known PHAs could be successfully diverted by such a system.
Goal of this study
NASA has identified nearly 1000 objects that it classifies as “potentially hazardous asteroids” (PHAs), a subset of the NEO population described by the NASA NEO website as “NEAs whose Minimum Orbit Intersection Distance (MOID) with the Earth is 0.05 AU or less and whose absolute magnitude (H) is 22.0 or brighter.” [46]. This study evaluates the effectiveness of a tether and ballast mass for diverting these asteroids. It should be noted here that not all of the PHAs were simulated. This is because not all of the PHAs have Earth-crossing orbits. Aten and Apollo class asteroids cross and hence were included, but Amors, which do not cross Earth’s orbit, were excluded.
Figure 3.1: Semimajor axis and eccentricity of known potentially hazardous asteroids (PHAs)
was held constant for all cases studied. The mass of a fueled Saturn V rocket, 3 million kg, was chosen as the nominal ballast mass. While the choice was somewhat arbitrary, it was desirable to settle upon a mass that was feasible but aggressive (the greater the mass, the shorter the required tether). The results, based on this choice of ballast serve as an initial feasibility study for such a diversion. Since tether lengths of up to 100,000 km have been proposed in the space tether literature (e.g. for the space elevator [9]), we consider tether lengths less than 100,000 km to be feasible in this paper.
3.2
Model
The geometry of the NEO-tether-ballast system is illustrated in Figure 3.2. In this figure, the variables that describe the system geometry can be seen. First, R is the scalar distance from the Sun to the asteroid in question. The variableν is the true anomaly of the asteroid and therefore describes the angle between the vector that points from the sun in the perihelion direction and the vector from the Sun to the asteroid. The scalar variableL is the length of the tether andθis the angle between the vector extending from the sun to the asteroid and the vector that extends from the asteroid to the ballast mass. The figure also indicates the three reference frames (all of which are right handed) used in deriving the equations of motion. The three unit vectors{e1,e2,e3}comprise the inertial
