Results from the New Approaches Team of the NASA Frontier
Development Lab: The Deflector Selector
Nicolas Erasmus1,2 , Adam Greenberg1,3 , Erika Nesvold1,4 , Elmarie van Heerden1,5 , J.L.
Galache1,6 , Eric Dahlstrom1,7 , and Franck Marchis1,8
1
2
1
NASA Frontier Development Lab, Mountain View, CA, USA
South African Astronomical Observatory, Cape Town, South Africa
3
University of California Los Angeles, Los Angeles, CA, USA
4
Carnegie Institute of Washington, Washington, DC, USA
5
University of Oxford, Oxford, UK
6
IAU Minor Planet Center, Boston, MA, USA
7
International Space University, Menlo Park, CA, USA
8
SETI Institute, Mountain View, CA, USA
Introduction
Impacts on the Earth by natural Solar System objects (i.e., asteroids and comets) can pose a significant threat
to human lives and infrastructure. Geological evidence of an impact crater near Chicxulub, Mexico indicates
that an asteroid or comet impact 66 million years ago may have been responsible for a mass extinction of
approximately 3/4 of Earth’s plant and animal species and the disruption the global climate [11]. While
the Chicxulub impactor was at least 10 km in diameter, impactors too small to cause mass extinctions can
still pose a regional threat to life and property. Even the ∼ 17 m impactor that struck the atmosphere at a
shallow angle over Chelyabinsk Oblast in 2013 caused $33 million (USD) in infrastructure damage and over
∼1,000 injuries [9].
Several ground- and space-based observing campaigns have been dedicated to detecting and tracking
Near Earth Asteroids (NEAs), and the Minor Planet Center (MPC) maintains a database of asteroid and
comet astrometric observations and orbits [2]. However, research concerning the best methods for deflecting
an impactor once it has been detected is still in the theoretical stages.
Several technologies have been proposed for impactor deflection, including nuclear explosives, kinetic
impactors, and gravity tractors. However, none of these technologies have been developed and fully tested
in space. The kinetic impactor method was tested by the Deep Impact mission, which collided a spacecraft
with the comet Tempel 1, but the subsequent change in velocity of the comet was not measured. While
nuclear explosives are a well-studied technology on the Earth, the testing of nuclear explosives in space
was prohibited by the Outer Space Treaty of 1967 [6]. Developing and testing every proposed deflection
technology is currently prohibitively expensive. However, if humanity waits until a clear impact threat is
detected to select which technologies to use, there may not be time to develop and deploy the chosen deflection
technique between the detection of the hazard and the impact. Determining now which technologies are most
likely to be useful would allow policy and funding decision-makers to effectively prioritize a subset of the
proposed deflection technologies.
Theoretical studies of the various proposed deflection technologies have focused either on modeling the
capabilities of a single technology, or comparing the abilities of the different technologies to address specific
impact scenarios. No comprehensive study comparing effectiveness of the various proposed technologies on
deflecting the likely hazardous object population has been published.
1
Figure 1: Illustration of the impact scenario and a few of the parameters considered by the model.
We have developed a model to map the distribution of parameters of a hypothetical impactor population
to the set of technologies that can deflect these objects. In this work, we describe our model and use it to
address the following questions:
1. Which deflection method has the highest likelihood of deflecting the broadest range of possible impactors?
2. Which impactor characteristics is the choice of deflection method most sensitive to?
3. Which areas of the impactor parameter space are not covered by current deflection technologies?
The model consists of a machine learning algorithm that takes as its input the characteristics of a
hazardous object (e.g., orbital parameters, size, etc.) and outputs the deflection technologies capable of
deflecting the object. To train the algorithm, we produced a set of training data using orbital simulations to
simulate the application of a change in velocity, ∆V , to deflect a hazardous object, and a literature search
of deflection technologies to calculate which technologies could apply that ∆V , given the object’s size. The
general considerations and parameters in the orbital simulations and technology calculations are illustrated
in Figure 1.
Section 2 describes the N -body orbital simulations we used to test the success of the application of various
∆V values to hazardous objects. Section 3 summarizes the results of our calculations of the capabilities of
various deflection technologies. In Section 4 we describe our machine learning algorithm and summarize its
results. We describe our plans for future improvements to the model in Section 5 and discuss our conclusions
in Section 6.
2
Figure 2: Schematic of two types of Earth crossing orbits: Apollos and Aten objects.
2
Orbital Simulations
2.1
2.1.1
Hazardous Object Orbital Parameter Synthesis
Asteroids
The first step in simulating the efficacy of various deflection techniques is to generate a population of
impactors, whose members can then be subsequently deflected. However, the parameters of such a population
are not known, since the rate of Earth-impacts is (thankfully) relatively low. Therefore, we generated a
simulated population of objects by modifying the orbits of all known Apollo and Aten objects (Figure 2),
such that these objects were guaranteed to be Earth-impacting. These modifications were done such that
the shape of the orbit was unchanged, and changes to the orbital orientation were minimized.
1. We first generate Earth-crossing orbit using the following method:
• Pull aa , ea , ia , Ωa from Aten/Apollo distribution to partially define asteroid orbital ellipse (Semimajor axis a,eccentricity e,inclination i, argument of the periapse ω and longitude of the node Ω
fully define an orbit, along with mean anomaly M ).
Earth’s orbital parameters are known to be ae , ee , ie , ωe , Ωe (Ωe ≡ 0 with an ecliptic reference
frame).
• Find θ such that f (θ) is minimized, where
f (θ) = | re − ||~r(Ma = θ, ωa = 0, aa , ea , ia , Ωa )|| |
and re = ||~r(Me = Ωa , ωe , ae , ee , ie , Ωe )||. r is the Sun-object distance.
• Find φ such that g(φ) is minimized, where
g(φ) = | ~r(Ma = θ, ωa = φ, aa , ea , ia , Ωa ) · ~z |
and ~z is Earths orbital normal vector.
3
2. We then choose a (random, with constraints) lead time t` . We set the impactor’s mean anomaly
Ma = θa0 and Earth mean anomaly Me = θe0 such that
√
t` GM∗
θe0 = Ωa −
3
2πae 2
and
θa0
=θ−
√
t` GM∗
3
2πaa 2
,
where M∗ is the mass of the central body.
For purely Keplerian orbits, the preceding steps will generate an object with orbit aa , ea , aa , ωa = φ, Ωa
and mean anomaly Ma = θa0 that will impact an Earth with orbit ae , ee , ie , ωe , Ωe and mean anomaly Me = θe0
after time t` .
However, while purely Keplerian Earth-impacting orbits can be generated using the preceding steps,
the orbits used for this study were not purely Keplerian, as perturbational gravitational effects were taken
into account (see section 2.2). These perturbations will, over time, make an object’s position deviate from
its location predicted from the initial orbital parameters. Therefore, while the steps described above can
guarantee an initial object-Earth intersection, they will not necessarily generate an impact for objects with
a predicted Keplerian impact sufficiently far in the future.
To avoid this complication, we generated a population of impactors using step 1 as described above.
These impactors were then integrated backwards t` in time. Due to the time-symmetric properties of the
leap-frog integrator we employed (see section 2.2), the resulting population was guaranteed to impact after
time t` .
2.1.2
Long-period comets
Relative to near-Earth asteroids, much less is known about the orbital parameter distribution of long-period
comets (LPCs). LPCs are much less likely to impact the Earth due to the low rate at which they enter the
inner Solar System - however, our incomplete knowledge of their orbits, and their high relative velocities,
suggests that humanity would have much less warning time prior to a cometary impact. LPCs can also be
substantially larger than their asteroidal counterparts, further increasing the damage infliced from a potential
Earth impact.
Therefore, while their impact risk is low, the distribution of LPCs arguably present a higher expected
damage risk for humanity than the population of near-Earth asteroids.
As mentioned above, the distribution of orbital parameters for LPCs is not well constrained. To create
a synthetic population of comets, we generated orbital parameters according to the following constraints:
2000AU < a < 100, 000 AU
0o < i < 180o
6R < rp < 1 AU,
where rp is the perihelion distance. After generating a population of LPCs, these objects were made to
be Earth-impacting using the same prescription described in Section 2.1.1.
We should note that while we did generate the population of comets described above, due to time
constraints (and suggestions from experts), we refrained from including these objects in our analyses. We
plan on adding in the comet orbital simulations as the project grows beyond the initial six-week program.
4
Figure 3: Summary of the instantaneous deflection orbital simulations described in Section 2.2. The colors
indicate the percentage of successful deflections for a given lead time and ∆V applied. Black lines indicate
curves fit to the contours for success rates of 50%, 75%, and 100%.
2.2
N -body integrator
We performed orbital simulations using the N -body integrator REBOUND [8], run on the Carnegie Institute
of Washington’s Memex cluser. Our simulations included the gravitational effects of Jupiter, Venus, and
Mars, as well as the Sun and the Earth. The leap-frog integrator used for these simulations used a constant
time-step of dt = 1.6 × 10−6 yr, and a gravitational softening parameter of rs = R⊕ .
As mentioned in Section 2.1, gravitational perturbations make guaranteeing an Earth impact a nontrivial, non-analytic problem. To solve this, we placed each object at it’s Earth-crossing point at time t = 0
(as calculated using the methods described in Section 2.1), and then integrated their orbits backwards to
t = −15 years. Due to the time-reversibility of the leap-frog integrator used, this guaranteed contact between
the impactor and the Earth at t = 0.
To augment these data, each initial impactor was assigned random lead times, {t`,i }, and random ∆V s,
{∆Vi }. The set of lead times were chosen from a linear distribution, constrained by 0 years < t`,i < 15 years.
The set of ∆V s were chosen from a logarithmic distribution constrained by 10−6 m/s < ∆Vi < 1 m/s. For
instantaneous-push deflection technologies (see Section 3), we then applied an change in velocity of ∆Vi at
lead time t = −t`,i for each of copy of the original, unperturbed orbit. For slow-push deflection technologies
(again, see Section 3), we applied a change in velocity of ∆Vi /year for every time step ∆t satisfying the
constraint −t`,i < ∆t < 0. All velocity changes were applied parallel to the object’s velocity vector at the
time of application.
These orbits were tracked forward in time, from the point at which the ∆V was applied until either the
object impacted the Earth, or until 15 years had passed since the original, unperturbed orbit would have
impacted the Earth (i.e. until t = 15 years).
2.3
Orbital Simulation Results
We ran ∼ 200 instantaneous-push simulations and ∼ 100 slow-push simulations for each of 8,000 Apollo and
Aten orbits that had been altered to impact the Earth. Figure 3 summarizes the results of the instantaneouspush simulations. For a given lead time and ∆V , the color of the plot in Figure 3 indicates the percentage of
our simulations in that bin that represented successful deflections. Larger ∆V values increase the proportion
of successful deflections, and the magnitude of the ∆V required to increase this success rate increases sharply
for decreasing lead times.
5
1
2
3
4
5
6
7
Launch vehicle
NASA SLS Block 2
SpaceX Falcon Heavy [3]
Delta IV Heavy
Long March 7
Proton-M
Ariane 5ECA
H-IIB
Country
USA
USA
USA
China
Russia
EU
Japan
Payload to LEO (T)
130.70
54.40
28.79
25.00
23.00
21.00
19.00
Payload to Asteroid (est.) (T)
∼ 32
∼ 13
∼7
∼6
∼6
∼5
∼5
Status
Under Development
Testing
8/9 Succ. Launches
1/1 Succ. Launches
88/98 Succ. Launches
55/56 Succ. Launches
5/5 Succ. Launches
Table 1: Summary of launch vehicles with corresponding payloads to low Earth orbit or to a hazardous
asteroid.
1
2
3
4
5
6
7
Country
USA
USA
USA
China
Russia
EU
Japan
Location
Florida, USA
California, USA
Florida, USA
Hainan, China
Tyuratam, Russia
Kourou, French Guyana
Tanegashima, Japan
Launch Site
Kennedy Space Center LC-39B
Vandenberg SLC-4E
Kennedy Space Center SLC-37B
Wenchang SLC 1
Baikonur Cosmodrome LC-200/39
Guiana Space Centre ELA-3
Tanegashima LA-Y
Launch Vehicle
NASA SLS Block 2
SpaceX Falcon Heavy
Delta IV Heavy
Long March 7
Proton-M
Ariane 5 ECA
H-IIB
Table 2: Summary of possible launch sites corresponding to the launch vehicles listed in Table 2.
3
Deflection Technologies
The orbital simulations described in Section 2 can only reveal which values of ∆V are required to deflect an
incoming hazardous object, given its orbit and a lead time. To map these ∆V s to the proposed deflection
technologies, we conducted a literature search, summarized below, in order to calculate the ∆V values that
each technology can apply, given the object’s size. We considered the three most plausible technologies:
nuclear explosives, kinetic impactors, and gravity tractors.
3.1
Launch Vehicles and Sites
The successful execution of any chosen deflection method is ultimately dependent on getting that appropriate
technology into space and to the asteroid. To reach space, a launch vehicle and a launch site is required.
Arguably, the most critical parameter to consider is the mass of the payload associated with the deflection
method since this will determine the launch vehicle. A launch site that can accommodate that particular
launch vehicle can then be chosen.
The payload each launch vehicle can handle is dependent on the thrust power of the rocket but also on
the insertion orbit. Reaching Low Earth Orbit (LEO) requires less power than for instance Geosynchronous
Equatorial Orbit (GEO) which will again require less effort than a rendezvous with Mars. A summary of
established and promising planned launch vehicles and their associated payload capabilities are given in
Table 1.
A summary of launch sites and the launch vehicles that can be launched from them is given in Table 2.
In this section, we assume that only one technology is used for a given attempted deflection of an asteroid,
and that only a single launch vehicle is used for deploying that technology..
3.2
Nuclear Explosive
Although nuclear explosives are highly controversial and conditional to international treaties, they still
remain an extremely attractive option because the efficiency of ∆V imparted on the asteroid versus the
required launch mass is much higher compared to other techniques. Nuclear explosives also stem from well
6
established and existing technology. However, apart from the obvious safety concerns of a malfunctioning
nuclear cargo launch, there is also still the unpredictable outcome of such a violent deflection mechanism.
Until these issues are addressed this method will remain a divisive topic.
To harness the explosive energy of a nuclear detonation to impart a ∆V on a body there are three possible
approaches: a stand-off, surface-, and a sub-surface detonation. For this study we focus on the stand-off
detonation method because it has lower execution complexity and therefore a more realistic implementation,
but should still be representative of the deflecting power of a nuclear explosion.
Nuclear radiation as a possible deflection method for NEOs has been previously studied [4],[10]. To model
the impulse that a nuclear device can impart onto an object we use the equations derived by [4].
∗
F0,rad
=
1.33 × 105 ηrad Y (M T )∆Ω µrad
(2r)2
v
(1)
∗
Here, F0,rad
is a unitless value associated with the corresponding radiation (x-rays, neutrons or gammarays), Y is the total energy yield of the nuclear explosion in megatons TNT, ηrad is the fraction of the
total yield contributing to x-rays, neutrons or gamma-rays, ∆Ω is the fractional solid angle of radiation
that
√
impedes onto the asteroid, which is dependent on the stand off distance H (optimal at H = ( 2 − 1)r), r
is the radius of the asteroid, µrad is the mass absorption coefficient and v is vaporization energy per unit
mass. The ∆V associated with each radiation is then given by
√ !
hp
io
√ np ∗
2 2
εv
F0 − 1 − tan−1
F0∗ − 1 ,
∆Vrad =
(2)
mµrad
with m = ρr/B the area mass density of the asteroid (B = 3 for a sphere) .
Although in general only a small contribution can be expected by the impact of the payload capsule debris,
for completeness the ∆Vdeb is expressed as
βSsc mdeb Vdeb
,
(3)
M
with β the momentum enhancement factor conservatively set to 2, Ssc = 2/π the scattering angle of the
debris. mdeb and Vdeb are the mass and velocity of the debris respectively and calculated as follows:
∆Vdeb =
mdeb = Fdeb mi
r
2ηdeb E
Vdeb =
,
mi
(4)
(5)
p
p
where Fdeb = 1/2 − H/2 (H + 2r)/(r + H) is the fraction of the debris that hit the asteroid, mi is the
mass of the payload capsule that the debris originates from, ηdeb is the fraction of the total nuclear yield
contributing to the debris acceleration, and E is the total energy yield of the nuclear explosion. The total
∆V expected from a stand-off nuclear detonation is then
∆VT = ∆Vx−ray + ∆Vneutron + ∆Vgamma + ∆Vdeb .
(6)
Plotted in Figure 4 is the expected ∆VT we can expect from a stand-off nuclear detonation as a function
of asteroid mass on the x-axis and the potential nuclear energy on the y-axis. The choice of the asteroid
size range is based on the so called current “blind spot” range, meaning that these are the asteroids that are
large enough that they can cause significant damage when impacting earth but they are small enough that
it is expected that only around 50 − 60% of the expected population these asteroids have been discovered.
The y-axis range is determined by the maximum payload that the available launch vehicles can handle (see
Section 3.1). From the plot we can see that the expected ∆V s from a nuclear detonation for the given
asteroid size range will be in the order of meters to tens of centimeters per second.
7
Nuclear Impactor
NASA SLS ∗
20
15
10
5
10 2
10 1
∆V (m/s)
25
1E+01
Nuclear Energy (Mt)
30
?
1E+00
35
10 0
SpaceX Falcon Heavy ∗ ?
Delta IV Heavy ∗
1E-01
200 400 600 800
Diameter of Asteroid (m)
10 -1
-2
1000 10
Figure 4: Plotted is the ∆V that can be expected from a stand-off nuclear detonation as a function of asteroid
size and potential nuclear energy. The position of the indicated launch vehicles on the plot is determined by
the maximum payload (i.e. number of warheads) each launcher can handle. The question marks highlight the
fact that the SpaceX and NASA launch vehicles are still in the testing and developmental phase respectively
and can’t be relied on for any imminent PHO. ∗ A single B83 warhead weighs 1.1 tons and can deliver 1.2Mt.
8
Kinetic Impactor
45
Vrel = 20.0km/s
β = 2.0
∆V (m/s)
25
10 -2
1E-
20
10 -3
15
10
10 0
10 -1
03
30
1E-02
35
1E-01
Mass of Impactor (tons)
40
10 1
200 400 600 800
Diameter of Asteroid (m)
-4
1000 10
Figure 5: Plotted is the ∆V that can be expected from a kinetic impactor with a relative velocity between
the asteroid and the impactor of 20km/s and a β parameter of 2. The mass on the y-axis is the sum total
of the maximum payload, the dry mass of the payload capsule and the dry mass of the final booster (see
section 3.1 for more detail).
3.3
Kinetic Impactor
A kinetic impactor has the appeal that it is the least complex deflection approach, essentially launching the
largest mass and impacting the asteroid at maximum achievable velocity to accomplish the greatest ∆V .
Due to the lack of actual real-world measurements there are still some ongoing concerns. The β parameter
of the hazardous object, which determines how well the momentum of the kinetic impactor is transferred to
the object, is poorly understood and difficult to predict based on observations. This parameter can also be
highly dependent on the internal structure and composition of the object. Whether the object will remain
intact and not break up into smaller (but still hazardous) fragments is also an area that still needs to be
better understood with models and experimental testing.
The ∆V applied by a kinetic impactor can be estimated as follows:
∆V = β
mi
Vrel ,
(M + mi )
(7)
where β is the previously mentioned β parameter, mi and M are the mass of the kinetic impactor and hazardous object, respectively, and Vrel is the relative velocity between the hazardous object and the impactor.
Plotted in Figure 5 is the expected ∆V achievable from a kinetic impactor. Here we have chosen an
ambitious, but still realistic, relative velocity of 20 km/s and a conservative β value of 2. From the plot we
can see that the ∆V s for the given asteroid size range will be on the order of millimeters to centimeters per
second.
9
physical attachment by using gravity as a towline. The thrusters must be canted outboard to
keep them from blasting the surface (which
Asteroid
φ
r
d
Spacecraft
m, T
ρ, M
⎞r⎞ 3 ⎞ & ⎞
$1.7 ⎠!⎠ !3⎠ !
d ⎠2'10 2
where G is the gravitatio
for definition of other
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tow an asteroid of 200 m
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total thrust T of just ove
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oid per second of hover
Gm
!
(v$ !
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d
Figure 6:Figure
Geometry
and basic geometry
principle of aof
gravity
tractor. Figure taken from [5].
1 | Towing
a gravitational
3.4
tractor. The asteroid (assumed to be spherical) has
Gravity Tractor
radius r, density & and mass M. The spacecraft has
So the velocity change im
in our example in a sin
1.9'10 m s!1. Becau
pendent of the asteroid’s
composition, the effect
In contrast to the previous
a gravity
tractor
a slow-push deflection
technology and requires
masstwo
m, methods,
total thrust
T and
an isexhaust-plume
halfa substantial lead time to be effective. This approach relies on modifying the trajectory of an asteroid by
#. It hovers
at distance
from the
width
using the gravitational
force exerted
by a spacecraft
on andasteroid
as a asteroid’s
towline [5]. The spacecraft hovers
!3
where
its in
netthethrust
its weight.
near the asteroid andcentre,
slowly exerts
a tug
desiredbalances
direction with
thrusters. The major advantage of
this method is that it
is not
at all sensitive
to theoutwards
structure, surface
properties and rotation state of the
The
thrusters
are tilted
to prevent
asteroid.
impinging on the asteroid surface.
The acceleration aexhaust
t imparted on a target of radius r by a spacecraft of mass msc at a distance d = f r
from the target’s surface is
at =
Gmsc
,
(f r)2
(8) Nature Pub
©2005
where G is the gravitational constant. Equation 8 suggests that in the optimal case, a spacecraft would be
positioned as close to the target as possible (i.e. f = 1). However, due to potential interactions between
expelled propellant and the target’s surface, the spacecraft’s thrusters must be pointed away from the
© 2005
target. This in turn means that at least two thrusters must be used. Furthermore, in order
for a Nature
spacecraftPublishing
to maintain a constant separation d from the target, the on-board propulsion system must be capable of
achieving an effective thrust, Teff , greater than the gravitational acceleration imparted on the spacecraft by
the target.
In other words,
G 4 πr3 ρ
Teff
≥ 3 2 ,
(9)
msc
(f r)
where ρ is the target’s bulk density. Here we use Teff to indicate the nominal thrust T projected onto the
radial vector ~r pointing from the spacecraft to the target. For a pair of thrusters, each with plume angle φ,
1
Teff = 2 × T cos(arcsin( ) + φ).
f
(10)
As mentioned above, the magnitude of the effective thrust is limited due to the necessity to avoid propellanttarget interactions.
Therefore, the spacecraft thrust T presents the limiting factor for how close the spacecraft can get to
the target’s surface. Specifically, since the target’s acceleration is maximimized for minimum f , Equations 9
and 10 imply that maximum acceleration can be achieved when f = fmin , where fmin is defined such that
2π
Grρmsc
− T = 0.
2 cos(arcsin( 1 ) + φ)
3 fmin
fmin
10
(11)
Gr
Gravity Tractor with Ttot = 0.8N
∆V (m/s) per year
10
10 -1
10 -2
15
1E-02
Mass of Tractor (tons)
20
10 -3
5
04
1E-
3
1E-0
50
200
350
Diameter of Asteroid (m)
10 -4
-5
500 10
Figure 7: Plotted is the ∆V per year that can be expected from a gravity tractor as a function of tractor
mass on the y-axis and a thrusting power of 0.8N.
Thus, a gravity tractor’s maximum acceleration on a target is
at =
Gmsc
.
(fmin r)2
(12)
Figure 7 shows the ∆V per year achievable for a gravity tractor weighing between 1 and 20 tons and able
to exert a thrust of ∼ 1 N. ∆V s below 1mm/s per year can be expected, so it is clear that a substantial lead
time (&5 years) would be required to have a successful deflection.
3.5
Deflecting Capability Comparison
In Sections 3.2, 3.3 and 3.4, we have estimated ∆Vtech , the ∆V each technology can supply during a deflection
attempt. However, the actual deflecting capability of each technology can only be compared by including
the N -body simulations results seen in Figure 3, which shows curves that we have fit to several success rate
contours (50%, 75% and 100% ). Now, given a certain impact scenario with a associated lead time and
asteroid size, we can do the following:
1. Use the given lead time and contour fit in Figure 3 to determine ∆Vreq , the ∆V required to achieve a
desired deflection success rate associated with the corresponding contour fit.
2. Use the given asteroid size to determine ∆Vmax,tech , the maximum ∆Vtech each technology can impart
on an asteroid of that size.
3. If ∆Vmax,tech ≥ ∆Vreq , then we mark that technology as capable for that specific impact scenario.
Figure 8, we summarize the results of this process for each of the three technologies discussed in this
section. In each plot, for a given object size and lead time, we have predicted the success rate of the
technology, given the orbital simulation results shown in Figure 3. Figure 8 allows us to directly compare
11
Asteroid Diameter (m)
1000
Nuclear Impactor
Kinetic Impactor
Gravity Tractor
800
600
400
200
0
2
4
6
8
10 12
Lead Time (years)
14
0
2
4
6
8
10 12
Lead Time (years)
14
Figure 8: Predicted success rate of each of the three technologies discussed in Section 3, given an object’s
diameter and the lead time between application of the technology and the time of impact.
the proposed technologies’ capability of successfully deflecting a hazardous object. From Figure 8 it is clear
that the nuclear explosive will be successful in most cases, except for large objects with very short (∼ 1
yr) lead times. The kinetic impactor, which imparts generally smaller values of ∆V , is more likely to be
successful only for smaller objects (. 300 m in diameter) with lead times & 1 − 2 yr. The gravity tractor
is effective only for the smallest objects with longer lead times. Because the gravity tractor is a slow-push
technology, its effects are cumulative over time, and the object diameter for which it is effective increases
roughly linearly with time.
4
Machine Learning: Decision Tree
The simulations and calculations discussed in Sections 2 and 3 can produce large quantities of data. The
orbital simulations described in Section 2 can determine, given an object’s orbit, a lead time, and an applied
∆V , whether the deflection will be successful. The calculations described in Section 3 can determine, given
an object’s size and a lead time, the maximum amount of ∆V that can be applied by a given technology. The
questions posed in Section 1 require us to be able to determine, given an object’s orbit, size, and lead time,
which technologies can successfully deflect the object? In Section 3.5 we demonstrated how to combine the
orbital simulations and technology calculations to compare the effectiveness of the deflection technologies as
a function of object size and lead time. However, this example and similar types of analyses can only explore
two or three dimensions of the multi-dimensional parameter space of this problem, which include object size,
lead time, and the orbital parameters of the object. In addition, if we want to add any new data, e.g., a new
object orbit, we must run further orbital simulations, which are computationally expensive.
Machine learning can address these problems by allowing us to produce an algorithm that can predict,
given an object’s characteristics, which technologies can successfully deflect the object, much faster than
orbital simulations can be run. We used the data we produced for our analyses in Sections 2 and 3 as
a training data set for a machine learning algorithm. We chose to train a decision tree algorithm, which
is effective for classification problems such as this one, and produces a resulting decision tree that can be
intuitively understood by the use.
Tree-based methods partition the feature space into a set of rectangles, and then models each region by
a constant outcome. In this section a popular tree-based method called classification and regression tree
(CART) by Breiman et al. [1] is explained.
Assume the outcome y ∈ {1, 2, ..., K}. A classification tree repeatedly partitions the feature space into
a set of rectangles by recursive binary splitting. First, the original feature space is split into two regions
and the response is modeled by the majority vote of y in each region. Subsequently, one or both of these
12
regions are split into two more regions. This process is continued until some stopping criteria is reached. At
each step, the variable and split-point that achieves the best fit have to be determined. The aforementioned
process is illustrated by the following example: four splits x1 = t1 , x2 = t2 , x1 = t3 , x2 = t4 to partition the
features space into five regions Region1 , Region2 , Region3 , Region4 , Region5 shown in Figure 9a.
Region2
X1 ≤ t1
Region5
X2
t4
Region3
t2
X2 ≤ t2
Region1
X1 ≤ t3
Region4
t3
t1
R1
R2
X1
R3
X 2≤ t 4
R4
(a) An example partitioning of a
two-dimensional feature space by
recursive binary splitting.
R5
(b) The tree corresponding to the
two-dimensional partitioning.
Figure 9: Illustration of a feature space partitioned by a single tree.
The full dataset sits at the top node of the tree. Observations satisfying the conditions at each junction
are assigned to the left branch, and the others to the right branch. The terminal nodes of the tree correspond
to the regions Region1 , Region2 , Region3 , Region4 , Region5 (see Fig. 9b).
A key advantage of the recursive binary tree is its interpretability. The feature space partition is fully
described by a single tree as seen in Figure 9b.
4.1
Mathematical description
We define a tree T as a collection of nodes (t) and splits (s). Let |T | denote the number of terminal nodes
in T . In what follows, we denote the N observations as
Set = {(x1 , y1 = f (x1 )) , (x2 , y2 = f (x2 )) , ..., (xN , yN = f (xN ))} ,
(13)
where xi ∈ IRp denotes the p−dim feature vector and yi ∈ {1, 2, ..., K} denotes the class number.
The tree-based algorithm needs to automatically decide on the splitting variables and split points, and
also what topology the tree should have. Three processes are necessary to grow a tree, namely a splitting,
a partitioning, and a pruning process.
The split process involves choosing split variables and split points and then applying the goodness of split
criterion ϑ(s, t) to evaluate any split s of any node t. Each split depends on the values of only one unique
variable xu . For the splitting variable u and split point s, define the pair of half-planes
Region1 (u, s) = {x|xu 6 s}, and
Region2 (u, s) = {x|xu > s}.
(14)
Each split produces two sub-nodes. The tree-based algorithm scans through all the inputs and all the
possible splits to determine the best pair (u, s) yielding the most “pure” nodes, i.e., finding the splitting
variable u and split point s that solve
min[ϑRegion1 + ϑRegion2 ],
u,s
(15)
where ϑRegionm is some purity measure of node Regionm for m = 1, 2. A node is more pure if one class
dominates the node than if multiple classes equally present in the node.
In a node m, representing a region Regionm with Nm observations, let
X
1
p̂mk = Pr(k|m) =
I(yi = k),
(16)
Nm
xi ∈Regionm
13
which is the proportion of class k observations in node m. An observation in node m is assigned to class
k(m) = arg max p̂mk ,
(17)
k
by the majority class in node m. Different measures ϑm (T ) of node impurity include the following:
Misclassification error:
Gini index:
ϑ=
ϑ=
1
Nm
K
X
X
I(yi 6= k(m)), = 1 − p̂mk(m) .
(18)
i∈Regionm
p̂mk (1 − p̂mk ).
(19)
k=1
Cross - entropy:
ϑ=−
K
X
p̂mk log p̂mk .
(20)
k=1
All three impurity measures are similar, but cross-entropy and the Gini index are differentiable, and
hence more amenable to numerical optimization.
The binary partitioning process of the feature space is recursively repeated until the tree is large enough.
A very large tree might overfit the data and a small tree might not capture the important structure. Hence,
the optimal tree size should be adaptively chosen from the data. The preferred approach is to grow a large
tree T0 , stopping the splitting process only when some minimum node size (say 5) is reached.
The weakest link pruning
procedure successively collapse the internal node that produces the smallest
P
per-node increase in
Nm ϑm (T ), and continue until a single node tree is produced. This gives a finite
m
sequence of sub-trees. The cost-complexity for each sub-tree T is measured by
CCα (T ) =
|T |
X
Nm ϑm + α|T |,
(21)
m=1
where m’s run over all the terminal nodes in T , and α governs a trade-off between the tree size |T | and its
goodness of fit to the data. A large α results in smaller trees; a small α results in large trees. Breiman
et al. ([1]) have shown that for each α, there is a unique smallest sub-tree Tα that minimizes CCα (T ).
Furthermore, the sequence of sub-trees obtained by pruning under the weakest link pruning procedure, must
contain Tα . Though, in practice five- to ten-fold cross validation is used to estimate α.
CART classification uses the Gini index as node impurity criterion. Instead of employing stopping rules,
CART generates a sequence of sub-trees by growing a large tree and pruning it back until only the root node
is left. Then it uses cross-validation to estimate the misclassification cost of each sub-tree and chooses the
one with the lowest estimated cost.
4.2
Software implementation
We trained and implemented our decision tree using the scikit-learn package of python [7]. We first prepared a
labeled data set from the results of our orbital simulations. Each data point from the orbital simulation results
consisted of an object’s orbital parameters (specifically semi-major axis a, eccentricity e, and inclination i),
a lead time tlead , a ∆V value, and a label indicating whether the deflection was successful or not. For
each orbit, we generated 100 object diameters, D, from a power law distribution, producing 100 data points
from each single data point in the orbital simulation results. For each of these new points, we generated
a β parameter representing the efficiency of the momentum transfer in the case of a kinetic impactor, as a
proxy for the internal structure and breaking strength of the object. We pulled the β value from a Gaussian
distribution centered at 2.0 with a standard deviation of 0.5. We used the calculations described in Section 3
to determine which of the three technologies we have considered can apply the given ∆V , given the lead time
and object diameter (and the β parameter, in the case of the kinetic impactor). This allowed us to produce a
14
Figure 10: Relative success of the three technologies considered, estimated using the training data set
described in this section. These results are expected to change as the model is improved. In the current
model, nuclear explosives are the most effective due to the large ∆V s they can impart, and the gravity
tractor is the least effective due to its slow application time.
data set where each data point contained six features: a, e, i, D, β, and tlead with four corresponding labels
representing whether the detection was successful, and whether each of the three technologies was capable
of applying the deflection. This produced a training data set with roughly a quarter of a billion points.
4.3
Results
We trained the decision tree on 80% of the training data, then used the remaining 20% as validation data
to test the accuracy of the trained algorithm. We measured the accuracy by inputting the features of
the validation data and comparing the decision tree’s output to the labels we had already produced. We
performed this cross-validation technique ten times, each time randomly selecting 80% of the data set for
training and 20% for validation. The measured accuracy of the trained algorithm was ∼98%, indicating that
this data set is well-suited for classification using the decision tree method.
Now that we have trained a decision tree, we can input a set of hazardous objects characterized by orbit,
size, β, and lead time, and quickly output the distribution of technologies able to deflect these objects. As
observers and theorists continue to refine the simulated distribution of hazardous objects to better reflect
reality, we can quickly run the distributions through the decision tree to produce better predictions of
which technologies will be most useful. As an illustration of this, in Figure 10 we have plotted a histogram
of the successful technologies in our training data. This represents the likely relative success of these three
technologies, given the hazardous object distribution we used to produce our training data. We show this plot
only as an illustration and without quantifying “relative success” due to the limitations in our preliminary
training data. For example, our orbital simulations used randomly chosen values for ∆V and lead time,
while in the event of a real hazardous object threat, humanity would attempt to apply the maximum ∆V
possible at the optimal point along the object’s orbit to achieve a successful deflection. However, as we make
improvements to the model to include these considerations, we will ultimately be able to produce a plot like
Figure 10 that represents an accurate prediction of technology success. Policy and funding decision-makers
can then use these results when considering which technologies to prioritize.
As we refine the model, we will also be able to investigate the sensitivity of the results shown in Figure 10
to the object parameters. For example, the results of the decision tree described here depend 80% on the lead
time, but only 20% on the other object parameters combined. Future improvements to the model, described
15
in the next section, will estimate the lead time as a function of the object’s orbit, which will remove lead
time as an independent variable and may reveal stronger underlying dependencies on the object’s orbital
parameters.
In addition, our analysis can help reveal which types of objects cannot be deflected by currently proposed
technologies. The successful deflections considered in Figure 10 represent less than half of the hazardous
objects in the training data, as the rest of the objects were not deflected successfully in simulations. Again,
this is due to the randomly chosen values for ∆V and lead time: sometimes the values chosen were too
small for a successful deflection. However, future versions of the model that include more realistic choices for
∆V and orbit-dependent values for lead time will give a better indication of what portion of the hazardous
object parameter space represents objects that cannot be deflected. These objects will require new proposed
technologies that can exert higher ∆V s, and/or better hazardous object detection campaigns, to increase
the lead time.
5
Future Work
Having developed the pipeline for producing training data for our machine learning algorithm, our next steps
will be to refine the data set to reduce the number of assumptions and improve the accuracy of our final
results. In this section, we describe our plans for continued work on this project:
1. Detection and Travel Times Our first priority will be to estimate more realistic lead times as a
function of the hazardous object’s orbit. As discussed in Section 2, our current model treats lead
time as an independent variable. This assumptions neglects the relationship between the object’s
orbit and the amount of time humanity would have to address an incoming impact threat. This time
can be considered in two components: the earliest time before impact that humanity would detect
the hazardous object, and the time required to transport the deflection technology to the object for
deployment.
Considering these two contributions to lead time will reveal important dependencies of the choice of
technology on the orbital elements of a hazardous object. For example, the detection time will depend
on the object’s size and the timing of its close approaches to Earth, as well as the approach direction
of the object (from the sunward direction or from opposite the Sun). Calculations of the travel time
will consider the capabilities of humanity’s launch vehicles when estimating where along the object’s
orbit it can be reached by our deflection technologies. For a technology like the kinetic impactor,
whose effectiveness depends on the relative velocity of the technology and the hazardous object, the
spacecraft trajectory calculations will also constrain the ∆V applied by the technology.
2. Comet Population The population of impactors we have used for the training data so far was drawn
from the Near Earth Asteroid population of Aten and Apollo type objects, as described in Section
2.1.1. We discussed a potential method for simulating a population of impacting long-period comets
in Section 2.1.2, but have not yet included this population in our analysis of deflection technologies or
in the machine learning training data. We anticipate that including the comet population, along with
the considerations of detection and travel time described above, will have a significant effect on our
final results. For example, comets will have a much shorter lead time than asteroids due to their origin
far from the Earth and their high velocities, and they tend to be larger than asteroids, so the gravity
tractor will be less effective against these objects.
3. Additional Object Parameters Many object characteristics beyond the object’s size and orbit will
affect the effectiveness of a deflection technology. For example, the risk of fracturing the object with
a nuclear explosive or a kinetic impactor depends on the breaking strength and composition of the
object. Also, some technologies are sensitive to the spin period of the asteroid. We plan to refine the
model by adding object parameters such as breaking strength, composition, spin period, albedo, and
other characteristics, and observing how the results of the machine learning algorithm change.
16
4. Additional Technologies In this work we have considered only the three most plausible deflection
technologies, nuclear explosives, kinetic impactors, and gravity tractors. However, other, more speculative technologies, have been proposed, including thrusters, mass drivers, laser or solar ablation, and
others. We will add these technologies as possible deflection techniques using the methods described
in Section 3.
6
Summary and Conclusions
We have developed a tool to predict, given the parameters of an Earth-impacting object, the deflection
technologies capable of preventing the impact. Our tool uses a decision tree to make this prediction, trained
using a training data set produced by our pipeline, which includes orbital simulations and technology capability calculations. We developed the framework for this model during the six weeks of the FDL. Further
work is required to refine the pipeline to reduce the number of assumptions we have made in producing our
training data, as described in Section 5. Once this is complete, we will be able to retrain the decision tree
and input a synthesized hazardous object population to produce a prediction of which technologies will be
successful for the largest number of the potential impactors.
Once this is complete, we will publish our results and a description of the model. We will make our
pipeline, our data, and the trained decision tree available online. This will allow other researchers to update
our model with improved hazardous object populations and technology specifications, and to use the model
to make predictions. We anticipate that our model will be a valuable tool for researchers in planetary science
and technology development, and ultimately for policy and research funding decision-makers.
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