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AAS 10-137

EARTH-IMPACT MODELING AND ANALYSIS OF A

NEAR-EARTH OBJECT FRAGMENTED AND DISPERSED BY

NUCLEAR SUBSURFACE EXPLOSIONS

Bong Wie* and David Dearborn†

This paper describes the orbital dispersion modeling, analysis, and simulation ofa near-Earth object (NEO) fragmented and dispersed by nuclear subsurface ex-plosions. It is shown that various fundamental approaches of Keplerian orbitaldynamics can be effectively employed for the orbital dispersion analysis of frag-mented NEOs. The nuclear subsurface explosion is the most powerful method formitigating the impact threat of hazardous NEOs although a standoff explosion isoften considered as the preferred approach among the nuclear options. In additionto non-technical concerns for using nuclear explosives in space, a common con-cern for such a powerful nuclear option is the risk that the deflection mission couldresult in fragmentation of the NEO, which could substantially increase the damageupon its Earth impact. However, this paper shows that under certain conditions,proper disruption (i.e., fragmentation and large dispersion) using a nuclear subsur-face explosion even with shallow burial (< 5 m) is a feasible strategy providingconsiderable impact damage reduction if all other approaches failed.

INTRODUCTION

Despite the lack of a known immediate threat from a near-Earth object (NEO) impact, historicalscientific evidence suggests that the potential for a major catastrophe created by an NEO impactingEarth is very real. It is a matter of when, and humankind must be prepared for it.

If an NEO on an Earth-impacting course can be detected with a mission lead time of at least sev-eral years, the challenge becomes mitigating its threat. For a small body impacting in a sufficientlyunpopulated region, mitigation may simply involve evacuation. However, larger bodies, or bodiesimpacting sufficiently developed regions may be subjected to mitigation by either disrupting (i.e.,destroying or fragmenting with large dispersion), or by altering its trajectory so that it will missEarth. When the time to impact exceeds a decade, the velocity perturbation needed to alter the orbitis small (≈ 2 cm/s). A variety of schemes, including nuclear standoff explosions, kinetic-energyimpactors, and slow-pull gravity tractors, have already been extensively investigated for the NEOdeflection problem.1−10 The feasibility of each approach to deflect an incoming hazardous NEOdepends on its size, spin rate, composition, the mission lead time, and many other factors. Whenthe time to impact is short, the necessary velocity change may become very large.

Because nuclear energy densities are nearly a million times higher than those possible with chem-ical bonds, it is the most mass-efficient means for storing energy with today’s technology. Conse-quently, a nuclear standoff explosion, which is often considered as the preferred approach amongthe nuclear options, is much more effective than all other non-nuclear alternatives, especially for

∗Vance Coffman Endowed Chair Professor, Asteroid Deflection Research Center, Department of Aerospace Engineering,Iowa State University, 2271 Howe Hall, Room 2325, Ames, IA 50011-2271, bongwie@iastate.edu.

†Research physicist/astrophysicist, Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94550,dearborn2@llnl.gov.

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larger NEOs with a short mission lead time.1−4 Other nuclear technique involving the subsurfaceuse of nuclear explosives is in fact more efficient than the standoff explosion. The nuclear subsur-face method even with shallow burial (< 5 m) delivers large energy so that there is a likelihood oftotally disrupting the NEO. A common concern for such a powerful nuclear option is the risk thatthe deflection mission could result in fragmentation of the NEO, which could substantially increasethe damage upon its Earth impact as discussed in References 1-3, 7, and 11-13. In fact, if the NEObreaks into a small number of large fragments but with very small dispersion speeds, the multipleimpacts on Earth might cause far more damage than a single, larger impact. Thus, the nuclear dis-ruption approach has not been recommended as a valid technique for mitigation at this time in arecent NEO study report by National Research Council’s Committee to Review Near-Earth ObjectSurveys and Hazard Mitigation Strategies.7 Additional research, including a suite of independentcalculations and laboratory experiments, but particularly including experiments on real comets andasteroids, has been recommended to prove that nuclear disruption can be a valid method.7

However, despite various uncertainties inherent to the nuclear disruption approach, disruption canbecome an effective strategy if most fragments disperse at speeds in excess of the escape velocityso that a very small fraction of fragments impacts the Earth. In this paper, we show that for somerepresentative cases (principally when the warning time is very short), disruption is the only feasiblestrategy, especially if all other deflection approaches were to fail.

A critical element of any mitigation strategy is the capability of available launch vehicles. Apreliminary conceptual design of an interplanetary ballistic missile (IPBM) system carrying nuclearpayloads is presented in Reference 9. The proposed IPBM system basically consists of a launchvehicle (LV) and an integrated space vehicle (ISV). The ISV consists of an orbital transfer vehicleand a terminal maneuvering vehicle carrying nuclear payloads. A Delta IV Heavy launch vehiclecan be chosen as a baseline LV of a primary IPBM system for delivering a 1500-kg (mass) nuclearexplosive for a rendezvous mission with a target NEO. A secondary IPBM system using a Delta IIclass launch vehicle (or a Taurus II) with a smaller ISV carrying a 500-kg nuclear explosive is alsodescribed in Reference 9. Basically, a primary IPBM system studied in Reference 9 can deliver a1.5-MT nuclear payload while a secondary system can deliver a 500-KT nuclear payload. A kilotons(KT) of TNT equals to an energy of 4.18 × 1012 Joules.

This paper presents the fundamentals of orbital dispersion modeling, analysis, and simulation ofan NEO fragmented and dispersed by nuclear subsurface explosions. In this paper, we demonstratethat various fundamental approaches of Keplerian orbital dynamics (e.g., References 14-16) can beeffectively employed for the orbital dispersion analysis of disrupted NEOs.

ORBITAL DISPERSION MODELING

Basic Problem Formulation

All of the orbital analysis and simulation results presented in this paper have been validatedusing the various fundamental approaches of Keplerian orbital dynamics described in this section.The Standard Dynamical Model (SDM) of the solar system for trajectory predictions of NEOsnormally includes the gravity of the Sun, nine planets, Earth’s Moon, and at least the three largestasteroids. However, significant orbit prediction error can result when we use even the SDM due tothe unmodeled or mismodeled non-gravitational perturbation acceleration caused by the Yarkovskyeffect and solar radiation pressure. The Yarkovsky effect is the thermal radiation thrust due to theanisotropic radiation of heat from a rotating body in space. Consequently, in this paper, we consider

569

Y

Z

X

Vernal Equinox� Direcion

�

Line of Nodes

i

�

Perihelion

r

f

Ecliptic Plane

�� = f+

Figure 1. Illustration of an elliptical reference orbit of a target NEO

a fictive NEO on an Earth-impacting course along its nominal Keplerian trajectory and then examinethe orbital dispersion of the NEO fragmented by a nuclear subsurface explosion. The effect of themutual gravitational forces from all fragments (i.e., the self-gravity effect) is not considered in thispaper because the fragments (caused by nuclear subsurface explosions) disperse at speeds muchgreater than the escape velocity. However, a preliminary study result for the self-gravity effect aswell as the computational issue caused by the mutual gravitational forces can be found in Reference13.

Keplerian Two-Body Model

The Keplerian orbital motion of an asteroid (prior to deflection and/or fragmentation) in a helio-centric elliptical orbit is simply described by

�r +μ

r3�r = 0 (1)

where �r = Xc�I + Yc

�J + Zc�K is the position vector of the asteroid from the center of the sun,

r =√

X2c + Y 2

c + Z2c , μ ≈ μ� = 132, 715× 106 km3/s2, and {�I, �J, �K} is a set of basis vectors for

the heliocentric ecliptic reference frame. A heliocentric elliptical orbit is illustrated in Figure 1. Inprinciple, Eq. (1) can be numerically integrated for given initial conditions �r(t0) and �r(t0) to findits solution �r(t). However, the Keplerian orbital motion is uniquely determined by the six classicalorbital elements

(a, e, i,Ω, ω, tp)

where a is the semimajor axis, e is the eccentricity, i is the orbit inclination angle, Ω is the longitudeof the ascending node, ω is the argument of the perihelion, and tp is the perihelion passage time(often replaced by the mean anomaly M0 at epoch).

It is assumed that a subsurface nuclear explosion results in instantaneous fragmentation of anasteroid, as illustrated in Figure 2. The Keplerian orbital motion of each fragment is then described

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Sun

x

y

Asteroid Fragments

Perihelion

Elliptical Reference

Orbit

c

�i

�j

�

f

Line of Nodes

�R

r�

�

LVLH Reference

Frame (x, y, z)

�� = f+

��r = x

�i + y�j + z�k

Figure 2. Illustration of the local-vertical and local-horizontal (LVLH) reference frame

by�R +

μ

R3�R = 0 (2)

where �R = X�I + Y �J + Z �K is the inertial position vector of a fragment and R = |�R| =√X2 + Y 2 + Z2. The relative position vector of a fragment becomes

δ�r = �R − �r = (X − Xc)�I + (Y − Yc) �J + (Z − Zc) �K (3)

A rotating local-vertical and local-horizontal (LVLH) reference frame (x, y, z) is employed to ex-press the relative position vector of a fragment as

δ�r = x�i + y�j + z�k (4)

where (x, y, z) are the radial, transverse, and normal components of the relative position vector inthe LVLH frame. The origin of the LVLH frame is located at the center of mass of the unfragmentedasteroid, and we have

δ�r = x�i + y�j + z�k = (X − Xc)�I + (Y − Yc) �J + (Z − Zc) �K (5)

Consequently, we have the following relationships between (x, y, z, x, y, z) and (X,Y,Z, X, Y , Z):

⎡⎢⎣ x

yz

⎤⎥⎦ = C

⎡⎢⎣ X − Xc

Y − Yc

Z − Zc

⎤⎥⎦ (6)

⎡⎢⎣ x

yz

⎤⎥⎦ = C

⎡⎢⎣ X − Xc

Y − Yc

Z − Zc

⎤⎥⎦ −

⎡⎢⎣ 0 −θ 0

θ 0 00 0 0

⎤⎥⎦

⎡⎢⎣ x

yz

⎤⎥⎦ (7)

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C = C3(θ)C1(i)C3(Ω) =

⎡⎢⎣ cos θ sin θ 0

− sin θ cos θ 00 0 1

⎤⎥⎦

⎡⎢⎣ 1 0 0

0 cos i sin i0 − sin i cos i

⎤⎥⎦

⎡⎢⎣ cosΩ sinΩ 0

− sinΩ cosΩ 00 0 1

⎤⎥⎦

(8)where (θ, i,Ω) are three angles associated with the nominal reference orbit, θ = ω + f is the truelatitude, and f is the true anomaly.

Solution to Kepler’s Problem

Given the initial relative position and velocity components of each fragment, (x, y, z, x, y, z), wecan determine the heliocentric position and velocity components, (X,Y,Z, X, Y , Z), as follows:

⎡⎢⎣ X

YZ

⎤⎥⎦ =

⎡⎢⎣ Xc

Yc

Zc

⎤⎥⎦ + CT

⎡⎢⎣ x

yz

⎤⎥⎦ (9)

⎡⎢⎣ X

Y

Z

⎤⎥⎦ =

⎡⎢⎣ Xc

Yc

Zc

⎤⎥⎦ + CT

⎧⎪⎨⎪⎩

⎡⎢⎣ x

yz

⎤⎥⎦ +

⎡⎢⎣ 0 −θ 0

θ 0 00 0 0

⎤⎥⎦

⎡⎢⎣ x

yz

⎤⎥⎦⎫⎪⎬⎪⎭ (10)

Then, given the initial position and velocity vectors of a fragment (i.e., �R0 and �V0 at t = t0), we candetermine �R(t) and �V (t) by solving Kepler’s problem as follows:14

[�R(t)�V (t)

]=

[F G

F G

] [�R0

�V0

](11)

where F and G are the Lagrangian coefficients expressed as14

F = 1 − a

R0(1 − cosΔE) (12a)

G = (t − t0) −√

a3

μ(ΔE − sinΔE) (12b)

F = −√

μa

RR0sinΔE (12c)

G = 1 − a

R(1 − cosΔE) (12d)

R0 = |�R0|; R = |�R(t)|; ΔE = E − E0 (12e)

and the semimajor axis a and ΔE can be determined from the following set of equations:

V 20

2− μ

R0= − μ

2a(13)

R0 = a(1 − e cosE0) (14)

t − t0 =

√a3

μ[2kπ + (E − e sinE) − (E0 − e sinE0)] (15)

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where k is the number of times the object passes through perihelion between t0 and t.

Equation 11 can also be written as16

[X Y Z X Y Z

]T=

[F I3×3 G I3×3

F I3×3 G I3×3

] [X0 Y0 Z0 X0 Y0 Z0

]T(16)

where I3×3 is a 3 × 3 identity matrix. The 6 × 6 matrix in Eq. (16) is not a state transitionmatrix because the Lagrangian coefficients (F,G) are also functions of the initial state vector(X0, Y0, Z0, X0, Y0, Z0). In fact, we have the following set of six scalar equations.

X = FX0 + GX0

Y = FY0 + GY0

Z = FZ0 + GZ0

X = FX0 + GX0

Y = F Y0 + GY0

Z = FZ0 + GZ0

Elliptical CWH Equations of Relative Motion

The relative orbital motion of a fragment with respect to the LVLH reference frame of an ellipticalreference orbit can also be directly described as16

x =(

2μr3

+ θ2)

x + 2θy + θy (17a)

y =(− μ

r3+ θ2

)y − 2θx − θx (17b)

z = − μ

r3z (17c)

where r =√

X2c + Y 2

c + Z2c , (x, y, z) are the LVLH coordinates of the fragment, r is the radial

distance of the reference orbit from the sun, θ = f + ω is the true latitude, f is the true anomaly,and μ is the gravitational parameter of the sun. It is assumed that

√x2 + y2 + z2 << r. Equations

(17) are called the elliptical Clohessy-Wiltshire-Hill equations of motion in this paper.

The elliptical reference orbit is then described by

r = rθ2 − μ

r2(18)

θ = −2rθr

(19)

Furthermore, we have

r =p

1 + e cos f=

p

1 + e(cosω cos θ + sinω sin θ)(20a)

r =√

μ/p (e sin f) =√

μ/p e(sin θ cosω − cos θ sinω) (20b)

θ =√

μ/p3 (1 + e cos f)2 =√

μ/p3 [1 + e(cosω cos θ + sinω sin θ)]2 (20c)

where p = a(1 − e2).

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State Transition Matrix Solution

The elliptical CWH equations can be numerically integrated for orbital dispersion simulationand analysis of fragmented bodies. However, a state transition matrix (STM) approach can also beemployed to avoid such direct numerical simulation of many fragments.

In this section, we briefly review an STM approach, which has been demonstrated in Reference17 to be computationally effective and accurate.

Let�r = r�i (21)

thenδ�r = δr�i + r δ�i = δr�i + r(�Θ ×�i) (22)

where�Θ = Θx

�i + Θy�j + Θz

�k (23)

Because ⎡⎢⎣ Θx

Θy

Θz

⎤⎥⎦ =

⎡⎢⎣ 0

0δθ

⎤⎥⎦ + C3(θ)

⎡⎢⎣ δi

00

⎤⎥⎦ + C3(θ)C1(i)

⎡⎢⎣ 0

0δΩ

⎤⎥⎦ (24)

we obtain

�Θ = (cos θ δi + sin θ sin i δΩ)�i + (− sin θ δi + cos θ sin i δΩ)�j + (δθ + cos i δΩ)�k (25)

and

δ�r = x�i + y�j + z�k (26a)

δ�r = x�i + y�j + z�k (26b)

where

x = δr (27a)

y = r(δθ + cos i δΩ) (27b)

z = r(sin θ δi − cos θ sin i δΩ) (27c)

x = δr (27d)

y = r δθ + r cos i δΩ + r δθ (27e)

z =d

dt(r sin θ) δi − d

dt(r cos θ) sin i δΩ (27f)

The preceding results can be rewritten as[x y z x y z

]T= D

[δr δθ δr δθ δi δΩ

]T(28)

where

D =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 0 00 r 0 0 0 r cos i0 0 0 0 r sin θ −r sin i cos θ0 0 1 0 0 00 r 0 r 0 r cos i

0 0 0 0 ddt(r sin θ) − d

dt(r cos θ) sin i

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(29)

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A state transition matrix description for the variational parameter vector is expressed as

[δr δθ δr δθ δi δΩ

]T= B(t, t0)

[δr0 δθ0 δr0 δθ0 δi δΩ

]T(30)

where

B(t, t0) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂r∂r0

∂r∂θ0

∂r∂r0

∂r∂θ0

0 0∂θ∂r0

∂θ∂θ0

∂θ∂r0

∂θ∂θ0

0 0∂r∂r0

∂r∂θ0

∂r∂r0

∂r∂θ0

0 0∂θ∂r0

∂θ∂θ0

∂θ∂r0

∂θ∂θ0

0 00 0 0 0 1 00 0 0 0 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(31)

and the initial condition vector is determined as

[δr0 δθ0 δr0 δθ0 δi δΩ

]T= D−1(t0)

[x0 y0 z0 x0 y0 z0

]T(32)

where

D−1(t0) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 0 0 0 0 0

0 1/r cos i(r sin θ+r cos θθ)

r2 sin iθ0 0 − cos i sin θ

r sin iθ0 0 0 1 0 00 −r/r2 0 0 1/r 0

0 0 −(r cos θ−r sin θθ)

r2θ0 0 cos θ

rθ

0 0 −(r sin θ+r cos θθ)

r2 sin iθ0 0 sin θ

r sin iθ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

t=t0

(33)

Finally, we obtain the state transition matrix description as

[x y z x y z

]T= Φ(t, t0)

[x0 y0 z0 x0 y0 z0

]T(34)

whereΦ(t, t0) = D(t)B(t, t0)D−1(t0) (35)

Φ(t0, t0) = D(t0)B(t0, t0)D−1(t0) = I6×6 (36)

r =r40 θ

20/μ

1 + (r30 θ

20/μ − 1) cos(θ − θ0) − (r2

0 r0θ/μ) sin(θ − θ0)(37)

r = r0 cos(θ − θ0) + [r0θ0 − μ/(r20 θ0)] sin(θ − θ0) (38)

θ =μ2

r60 θ

30

[1 + (r30 θ

20/μ − 1) cos(θ − θ0) − (r2

0 r0θ0/μ) sin(θ − θ0)]2 (39)

M − M0 = n(t − t0) = E − E0 +r0r0√

μa[1 − cos(E − E0)] + (r0/a − 1) sin(E − E0) (40)

r = a[1 + (r0/a − 1) cos(E − E0) +r0r0√

μasin(E − E0)] (41)

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where n =√

μ/a3. The semimajor axis a can be determined using the following energy equation:

12(r2

0 + r20 θ

20) −

μ

r0= − μ

2a(42)

Let

ΔE = E − E0

K1 =r0r0√

μa

K2 =r0

a− 1

Then we obtain

n(t − t0) = ΔE + K1(1 − cosΔE) + K2 sinΔE (43)

r

a= 1 + K1 sinΔE + K2 cosΔE (44)

It can be shown that

∂n

∂( )(t − t0) =

r

a

∂ΔE

∂( )+

∂K1

∂( )(1 − cosΔE) +

∂K2

∂( )sinΔE

which is rewritten as

∂ΔE

∂( )=

a

r

[∂n

∂( )(t − t0) − ∂K1

∂( )(1 − cosΔE) − ∂K2

∂( )sinΔE

](45)

By using

cosΔθ = 1 − (r30 θ

20/μ)(a/r)(1 − cosΔE) (46)

sinΔθ = (a/r)[(r20 θ0r0/μ)(1 − cosΔE) + (r0θ0/

√μa) sinΔE] (47)

we obtain the partial derivatives of θ with respect to r0, θ0, r0, and θ0. A detailed description of allpartial derivatives can be found in Reference 17.

Relative Motion of Fragments in the LVLH Frame

Figure 3 shows the pattern of a debris cloud (in an ideal circular orbit) as a function of trueanomaly for an isotropic distribution of initial relative velocity. After the first orbit, the debriscloud will look like a simple stretching along the orbital path. It should be mentioned that thecharacteristics of the orbit are the governing factors behind this phenomenon, and that all initialdistributions with material at a nonzero distance from the center of mass will result in similar cloudshapes (in an ideal circular orbit).

A typical relative motion of a fragment, in the LVLH frame of an elliptical orbit with e = 0.538,with an initial dispersion velocity of 1 m/s along the orbital direction is shown in Figure 4.

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x

y

180 deg

270 deg

360 deg

90 deg

Figure 3. Evolution of the fragment clouds in a circular orbit, with increasing trueanomaly (for an isotropic distribution of initial dispersal velocity)

���� ��� � �� �������

����

����

����

���

�

��

������� ���

������������� ���

Figure 4. Relative motion of a fragment in the LVLH frame of a reference ellipticalorbit for two orbital periods

REFERENCE ELLIPTICAL ORBIT AND HYPERBOLIC APPROACH ORBIT

In this paper we consider a reference elliptical orbit of a fictive NEO as illustrated in Fig. 5. Itssix classical orbital elements are also provided in Table 1.

The impact parameter b of this fictive NEO, illustrated in Fig. 6, can be estimated as14

b = R⊕

√1 +

(Ve

V∞

)2

≈ 1.5R⊕ (48)

where Ve is the Earth escape speed of 11.2 km/s and V∞ is the hyperbolic approach speed of thisNEO (≈ 9.98 km/s).

NUCLEAR SUBSURFACE EXPLOSION MODELS

Gravitational Binding Energy

In astrophysics, the energy required to disassemble a celestial body consisting of loose mate-rial, which is held together by gravity alone, into space debris such as dust and gas is called the

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Table 1. Orbital elements of a reference orbit of a fictive NEO with its impact date of January 3, 2009

Orbital Elements Value

Semimajor axis a, AU 2.10529Eccentricity e 0.53806Inclination i, deg 11.2738Right ascension longitude Ω, deg 282.232Perihelion argument ω, deg 194.383Perihelion passage time tp, days 2453730.11 JD

(25 December 2005)

37.19 km/s

30.28 km/s

12.3 deg

�V = 9.98 km/s�

Earth's Orbit

Asteroid

Impact Date�3 Jan 2009�(2454834.50 JD)

1000 days�before Impact

a = 2.1 AU�e = 0.538�i = 11.27 deg�� = 282 deg�� = 194 deg�tp = 2453730.11 JD��Period = 3.05 yrs� = 1115.7 days

Sun

Vernal Equinox�Direction

Ascending � Node

Figure 5. A fictive NEO in an elliptical orbit with its impact date of January 3, 2009

gravitational binding energy.

The gravitational binding energy of a spherical body of mass M , uniform density ρ, and radiusR is given by

E =3GM2

5R=

3G5R

(4πρR3

3

)2

=π2ρ2G

30D5 (49)

where G = 6.67259 × 10−11 N·m2/kg2 is the universal gravitational constant and D = 2R is thediameter of a spherical body. The escape speed from its surface is given by

Ve =

√2GM

R(50)

For example, for a 200-m (diameter) asteroid with a uniform density of ρ = 2,720 kg/m3 and a massof M = 1.1×1010 kg, its gravitational binding energy is estimated to be 4.8×107 J and the escapespeed from its surface is 12 cm/s. The escape speed of a 1-km asteroid is less than 1 m/s.

In References 1 and 2, the disruption energy per unit asteroid mass is predicted to be 150 J/kg forstrength-dominated asteroids. Also in Reference 2, the energy (per unit asteroid mass) required for

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Hyperbolic Orbit

2 b

Collision

Cross Section

V�

V�

Self Gravity

Figure 6. Illustration of the impact parameter and the collision cross section

Figure 7. Internal composition model of an Apophis-sized (270 m) NEO

both disruption and dispersion of a 1-km asteroid is predicted to be 5000 J/kg.

A ton of TNT is a unit of energy equal to 4.18 × 109 Joules. Thus, a 300 kiloton (KT) nuclearsubsurface explosion has a sufficient energy of 12×1014 J to disrupt and disperse a 200-m asteroid.A 1-MT nuclear subsurface explosion has also sufficient energy of 4.18 × 1015 J to disrupt anddisperse a 1-km asteroid.

Fragmentation Models

Nuclear energy densities are nearly a million times higher than those possible with chemicalbonds, and thus it is the most mass efficient means for storing energy. For example, on the Earth’ssurface (in dry soil), a 1-MT nuclear explosive device creates a crater that is about 150 m in radiusand 75 m deep, ejecting approximately 3.5 × 109 kg of material.13 As discussed in Reference 3, anuclear excavation experiment showed that a burial improves the energy coupling and results in amuch larger crater from a smaller yield device. In this paper, we consider such a nuclear subsurfaceexplosion with shallow burial (< 5 m) for a large diameter (1 km) as well as a small diameter (270m) model of NEOs.

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Figure 8. Computational modeling illustration of nuclear subsurface explosion

Figure 7 illustrates a two-component (inhomogeneous) spherical structure with a high densitycore consistent with granite (density = 2.63 g/cm3), and a lower density (1.91g/cm3) mantle. Thebulk density of the structures was 1.99 g/cm3, close to that measured for asteroid Itokawa (density= 1.95 g/cm3).18

A large 1-km NEO model has a total mass of 1.05×1012 kg or just over a billion tons, placing it inthe class of potentially catastrophic impactors. The smaller, 270-m model was sized to approximateApophis with a total mass of 2.058 × 1010 kg or a bit over 20 million tons.

Sourcing energies corresponding to 900 KT into the 1-km model and 300 KT into the 270-m bodysimulated surface explosions. The source region is cylindrical, and the dimensions in the smallerbody are about 1 m in diameter and 5 m long, as illustrated in Figure 8. The source volume is about4.5 m3 containing a bit over 8 tons of material. In the larger body, the source region has a volumenear 10 m3.

As shown in Figure 9, two-dimensional hydrodynamic modeling of the subsequent explosion ledto expanding clouds of debris. The structures were modeled with different strength approximations,including no material strength, a linear strength model (strength proportional to pressure, limited bya crush strength) often used for shock propagation in rubble, and a model that includes full strengthin the core. The yield strength in the core is set to 14.6 MPa, with a shear modulus of 35 MPa. Thisis somewhat weaker than measured for most granite, and is near the low-end for limestone.

The energy source region expands creating a shock that propagates through the body resulting infragmentation and dispersal. While the material representations used have been tested in a terrestrialenvironment, there are low-density objects, like Mathilde, where crater evidence suggests a veryporous regolith with efficient shock dissipation. Shock propagation may be less efficient in suchporous material, generally reducing the net impulse from a given amount of energy coupled into the

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Figure 9. Computational results from the hydrodynamic code

Figure 10. Speed distribution of the fragmentation model Ap300 (6 seconds after the burst)

surface. More work is needed to understand the limits of very low porosity.

The models were evolved until the bodies had substantially expanded, and the velocity gradientsindicated homologous expansion. After about 20 seconds, the large models showed material withexpansion speeds up to 50 m/s. The mass averaged speed in the no-strength model (M97) was 12m/s, and it was nearly 14 m/s in the linear-strength model (M20e). The smaller model (Ap300) wasrun with a linear-strength model, and after 6 seconds, the mass averaged speed of the fragments wasnear 50 m/s with peak near 30 m/s (Figure 10)

A three-dimensional fragmentation model (Figure 11) was then constructed from the 2D hydro-dynamics models by using the position, speed and mass of each zone, and rotating it to a randomlyassigned azimuth about the axis of symmetry. If more fragments were desired, we interpolated ineach 2D cell to provide more fragments (4, 9, 16, ...).

For a 1-km NEO, two basic models (M97 and M20e) were developed, Both models sourced 900KT into a cubic surface region of the same 1 km diameter object, with an initial mass of 1.047×109

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Figure 11. Fragmentation model Ap300, with 18,220 fragments, for a 270-m NEOdisrupted by 300-KT nuclear subsurface explosion

Figure 12. Fragmentation model M97, with 31,984 fragments, for a 1-km NEO dis-rupted by 900-KT nuclear subsurface explosion

tons. The difference is that M97 was a finely zoned model. Approximately 20 seconds after theenergy deposition, M97 had 31,984 zones of asteroid material for 9.6732 × 108 tons, 92.8% of theinitial mass. The missing 7.2% was ejected from the mesh at high speed prior to the end of thehydrodynamics simulation. In Figure 12, the fragmentation model M97 for a 1-km NEO disruptedby 900-KT nuclear subsurface explosion is shown. This model has 31,984 fragments.

For intercept-to-impact times longer than 300 days, the models are interpolated dividing each 2Dzone into 16 (M97) or 49 (M20e) fragments of equal masses. The position and velocity of thesefragments are found by linearly interpolating from the position and speed of the 4 nodes defining thecorners of the zone. The resulting M97i model, shown in Figure 13, has 511,744 fragments whilethe M20e model has 10,5742 fragments.

These 3D fragment distributions could then be placed and oriented on the asteroids trajectory atdifferent times before impact. For example, the X-axis in Figure 14 can be aligned with the orbitalflight direction or some other direction.

RESULTS AND DISCUSSIONS

A computer code developed by the second author was used for orbital dispersion simulationand analysis. The results have been validated independently by using the various fundamental

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Figure 13. Fragmentation model M97i (interpolated), with 511,744 fragments, for a1-km NEO disrupted by 900-KT nuclear subsurface explosion

Figure 14. Coordinates for initial dispersion position and velocity components

approaches of Keplerian orbital dynamics.

To assess the degree of mitigation, the code includes gravitational focusing effect of the Earth onthose fragments that pass near the earth, and provide a census of those that hit (with a minimumdistance < 1 Earth radius). The code then has two modes of use as described below.

(i) Orbital elements are evaluated for each fragment and the code is used to project the fragmentforward in time. This is used to show the debris cloud evolution over the whole time from interceptto impact time.

(ii) The orbital elements of each fragment are used to define its position and velocity at a timejust prior to the original impact date. Times ranging from 5 days to 6 hours prior to the nominalimpact time can be selected. A subset of the fragments that will pass within about 15 earth radii isselected. These fragments are directly integrated, accounting for the gravity of the Earth and moon.Those fragments that pass within 1 Earth radius are impacts.

After the analytic step places the debris field near the Earth, the relative velocity of each fragmentwith respect to the Earth is used to calculate its closest approach. All fragments that pass within 15earth radii are selected for integration.

This selection process was tested up to 5 days out from nominal impact, and successfully identi-fied essentially all of the fragments that would pass nearer the Earth than the selected limit. Further,and as expected, changing the radius of the selected cylinder from 15 to 5 Earth radii resulted inno change in the number of hits. While direction of fragments that pass within a couple of Earth

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radii are strongly changed, the fragments near the outer portion of the selected group show onlymodest angular deviations. The two basic models M97 and M20e were tested first. Less materialis lost from this simulation because the surrounding area was larger, and the run time was shorter.For 10 to 300 days to impact, these basic models with each zone treated as a fragment were usuallysufficient to obtain enough hits to estimate the amount of mass that remained a threat. As the timeto impact increases, the fragments become more dispersed, and number that hit decreases. Whenfewer than 10 fragments hit the earth, the statistical measure of how much of the asteroid remainsa threat is poor. When only a few fragments hit the earth, the fraction of the asteroid mass thatthreatens (impacts) is merely suggestive.

Figure 15 summarizes the result of intercepting and disrupting a 1-km NEO with 900 KT nuclearsubsurface explosion 1000 days before the impact date of January 3, 2009. For this case, the X-axisof the hydrodynamics fragmentation model has been aligned with the orbital flight direction. Forthis case, there are only 10 impacts with a total mass of 3,000 tons. For an interception 100 daysbefore impact, the M97 model (1/16th fragments of the M97i model) provides sufficient impacts,and the histograms in Figure 16 show more fragments pass near the Earth.

The result for the Ap300 model with 15 days before impact is provided in Figure 17. Basically,only 3% of the initial mass resulted in impacting the Earth even for such a very short time afterinterception. The impact mass can be further reduced to 0.2% if the X-axis of the hydrodynamicfragmentation model is aligned along the inward or outward direction of the orbit, i.e., perpendicularto NEO’s orbital flight direction. Such a sideways push is known to be an optimal when a targetNEO is in the last orbit before the impact. However, a conservative estimation of the impact mass fora worst-case scenario is considered in this paper. Obviously, the impact mass can be further reducedby increasing the intercept-to-impact time or by increasing the energy level of nuclear explosives(i.e., higher yields).

In Figure 18, a performance summary of the nuclear subsurface explosions is presented. Themass that impacts the Earth is converted to energy (MT) using V∞ of 9.98 km/s. The simplest wayof fragmenting the hydrodynamic model involves dividing it up based on the mesh. The results ofthis approach are shown in blue in Figure 18. For longer periods the debris cloud is more dispersed,and there are not enough impacts to be statistically meaningful (i.e., we may get no impacts forthree rolls of the dice, and then three impacts in one). For these longer flight times, more fragmentswere desirable and so the mesh of the hydrodynamic model was interpolated to create smaller (butmore) fragments (results shown in green in Figure 18).

From Figure 18, we notice that a 1-MT nuclear disruption mission for a 1-km NEO requires anintercept-to-impact time of 200 days if we want to reduce the impact mass to that of the Tunguskaevent. A 270-m NEO requires an intercept-to-impact time of 20 days for its 300-KT nuclear disrup-tion mission to reduce the impact mass to that of the Tunguska event. Therefore, it can be concludedthat under certain conditions, disruption (with large dispersal) is the only feasible strategy provid-ing considerable impact threat mitigation for some representative, worst-case scenarios. An optimalinterception can further reduce the impact mass percentage shown in Figure 18. However, a furtherstudy is necessary for assessing the effects of inherent physical modeling uncertainties and missionconstraints.

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Figure 15. Result of an interception 1000 days before impact for the fragmentationmodel M97i (1 km NEO)

Figure 16. Result of an interception 100 days before impact for the fragmentationmodel M97 (1 km NEO)

CONCLUSION

The orbital dispersion modeling, analysis, and simulation of a near-Earth object (NEO) frag-mented and disrupted by nuclear subsurface explosions have been discussed in this paper. For somerepresentative cases for which all other approaches were to fail, this paper has shown that disruptionusing a nuclear subsurface explosion even with shallow burial (< 5 m) is a feasible strategy. How-ever, more research work is needed to assess the effect of physical modeling uncertainties inherentto applying nuclear subsurface explosions to asteroid fragmentation and large dispersion. The fun-damental approaches of Keplerian orbital dynamics can still be effectively used for examining theorbital dispersion problem affected by various physical modeling uncertainties and space missionconstraints (e.g., optimal approach angle, rendezvous vs intercept, impact penetration velocity, etc.).

ACKNOWLEDGMENT

This research work by the first author was supported by a research grant from the Iowa SpaceGrant Consortium (ISGC) awarded to the Asteroid Deflection Research Center at Iowa State Uni-versity. The first author would like to thank Dr. William Byrd (Director, ISGC) for his support ofthis research work. Part of this work by the second author was performed under the auspices of

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Figure 17. Result of an interception 15 days before impact for the fragmentationmodel Ap300 (270 m NEO)

the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. The authors thank Dr. Daero Lee and Dakshesh Patel for validating the orbitaldispersion analysis results of this paper by using the various fundamental approaches of Keplerianorbital dynamics, including the state transition matrix approach of Reference 17.

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