NEAR-EARTH OBJECT (NEO) ANALYSIS OF TRANSPONDER TRACKING AND GRAVITY
TRACTOR PERFORMANCE

Submitted to B612 Foundation

  -----------------------------------------------------------------------
  Final Report

  -----------------------------------------------------------------------

JPL Task Plan No. 82-120022

September 22, 2008

**Study carried out and report written by D.K. Yeomans, S. Bhaskaran,
S.B. Broschart,**

**S.R. Chesley, P.W. Chodas, M.A. Jones, and T.H. Sweetser**

**Table of Contents**

[1 Scientific/Technical Work 1-0](#scientifictechnical-work)

[1.1 Introduction 1-0](#introduction)

[1.2 Objectives 1-1](#objectives)

[1.2.1 Study Objectives 1-1](#study-objectives)

[1.3 Technical Approach and Methodology
1-2](#technical-approach-and-methodology)

[1.4 Work Plan 1-2](#work-plan)

[1.4.1 Statement of Work and Results
1-2](#statement-of-work-and-results)

[1.4.2 Subtask 1 Results 1-3](#subtask-1-results)

[1.4.3 Subtask 2 Results 1-17](#subtask-2-results)

[1.4.4 Subtask 3 Results 1-48](#subtask-3-results)

[1.4.5 Subtask 4 Results 1-55](#subtask-4-results)

[1.4.6 Subtask 5 Results 1-55](#subtask-5-results)

[1.4.7 Key Milestones and Products 1-55](#key-milestones-and-products)

[1.5 Study Conclusions 1-56](#study-conclusions)

[2 References 2-59](#references)

# Scientific/Technical Work

## Introduction

This report is submitted to the B612 Foundation as part of JPL Task Plan
Number 82-120022. The report outlines the tasks originally planned in
the proposal as well as the work that was actually performed, the
conclusions, and the plan for possible future work.

Lu and Love (2005) outlined a gravity tractor concept in which the
gravitational attraction between an Earth-threatening asteroid and a
nearby thrusting spacecraft could be used to slightly adjust the motion
of the asteroid years before the potential collision. Although the very
small acceleration imparted to the asteroid by the gravity tractor would
most likely be insufficient for most primary deflection attempts, the
gravity tractor could be used to slightly alter or trim the motion of
the asteroid following a deflection attempt by another technique (e.g.,
a kinetic energy impact by a colliding spacecraft), should the primary
deflection attempt inadvertently push the asteroid through a "keyhole"
into a resonant return trajectory that would lead to a subsequent Earth
impact. A keyhole is defined on the target plane (also called the
b-plane), which is the plane perpendicular to the incoming asymptote of
an asteroid's trajectory during a close approach to a planet. The Earth
close approach can perturb an asteroid's trajectory onto a path that
brings it back to an Earth impact an integral number of years later. A
"keyhole" is the small region in the target plane that leads to a
perturbed trajectory, impacting during a subsequent close approach. The
target plane of a close approach can have numerous such keyholes.

The work outlined in this study addresses the scenario in which a
near-Earth object (NEO) is on a possible collision course with the Earth
and a gravity tractor spacecraft carrying a radio transponder (t-GT) is
dispatched to rendezvous with the object. The mission's first objective
would be to verify the possible impact via precision tracking of the
spacecraft. If the NEO is confirmed to be on a collision course and the
impulse required to avoid collision is beyond the capability of the
gravity tractor, another deflection technique (such as a kinetic
impactor) would be employed to effect a primary deflection. After the
deflection, the t-GT would be used to precisely determine the new
trajectory of the NEO, and it would be available (if necessary) to
slightly alter or trim the NEO's motion to ensure it is not headed for a
near-term resonant return (keyhole) and a resulting secondary Earth
impact trajectory.

The objectives of this study were to analyze the two primary
capabilities of the t-GT spacecraft in the above scenario.  First, we
examined the spacecraft requirements for accurately determining the
trajectory of the NEO, both before and after a deflection.  This is the
\"transponder\" part of the t-GT mission, which would be used both to
verify that the primary deflection successfully averted the primary
impact and to determine whether the NEO might pass through a keyhole. 
Secondly, we studied the requirements for the \"gravity tractor\"
portion of the mission, which is the capability of \"towing\" the NEO
trajectory so that it passes sufficiently far away from a keyhole.

There were 5 subtasks identified in the original proposal and this
report is organized in order of these subtasks:

1.  Select a strawman NEO orbit and a timeline for the case study.

    a.  Select a hypothetical Earth impactor "2016 NM4" with an impact
        in 2049.

    b.  Analyze the orbit of 2016 NM4 and its Earth and Venus close
        approaches.

    c.  Simulate a primary kinetic impactor deflection in 2028 to avoid
        the 2049 impact.

    d.  Analyze the "keyholes" in the 2049 Earth target plane.

    e.  Simulate an inadvertent deflection into the 6:5 keyhole in the
        2049 Earth target plane -- leading to potential impact in 2054.

    f.  Simulate gravity tractor operations beginning in July 2032 and
        lasting \~200 days.

2.  Study the gravity tractor spacecraft requirements to accelerate a
    NEO in a pre-determined direction and quantity.

    a.  Select a representative NEO physical model (i.e., 2016 NM4 is a
        scaled down Itokawa shape model)

    b.  Demonstrate that a spacecraft control law is possible so that it
        can thrust and hover closely in front of an irregularly shaped,
        rotating asteroid.

3.  Study NEO tracking using radiometric tracking of the nearby
    spacecraft:

    a.  How accurately can the pre-deflection orbit of 2016 NM4 be
        determined?

    b.  Following the primary deflection via kinetic impactor, how long
        an interval of spacecraft tracking does it take to determine the
        resultant change in velocity (ΔV) on the asteroid to sufficient
        accuracy?

    c.  Following the gravity tractor phase, how long an interval of
        spacecraft tracking does it take to determine the acceleration
        imparted to the asteroid to sufficient accuracy?

4.  Perform preliminary analysis of a mission design and implementation.
    (Work on Subtask 4 was descoped because the customer did not wish to
    pursue the Team X study at this time.)

5.  Perform Team X study to develop full mission concept design and cost
    estimates (descoped).

## Objectives 

### Study Objectives

The overarching objectives of this study were to determine the
feasibility of using the gravity tractor concept for use in NEO impact
mitigation and to build credibility for the concept, assuming it is
concluded to be viable. In order to gain insight into the feasibility of
the gravity tractor concept, the following two focus areas were
addressed by the Guidance, Navigation and Control team at JPL:

1\. Determine the performance characteristics of the gravity tractor as
a function of NEO proximity and provide control guidelines for station
keeping and navigation in close proximity to an irregularly-shaped,
rotating NEO, such that the spacecraft can change the trajectory of the
NEO into a non-threatening trajectory without jeopardizing the
spacecraft's safety.

2\. Determine whether radio tracking of the transponder/gravity tractor
spacecraft could improve the trajectory of the NEO sufficiently to
ensure that the NEO would not enter a keyhole within a specified degree
of uncertainty and interval of time.

Building the credibility of the concept was begun by demonstrating its
viability through attaining positive results from the models used to
study the focus areas noted above. Additional enhancement of the
concept's integrity could be achieved if the customer chose to pursue
this option or some variation of this option, through a Team X study
that would

3\. Develop a rough full mission concept and design.

4\. Obtain a cost estimate for the mission concept.

[]{.mark}

## Technical Approach and Methodology

The approach taken for this study involved breaking up the objectives
into two sections: (1) concept feasibility; and (2) credibility of the
implementation concept. A team from JPL's Guidance Navigation and
Control (GNC) group was formed to address the feasibility issues of
station keeping and navigation near a small body and accurately tracking
a NEO with a transponder. Various models were used to address each of
the subtasks noted in the statement of work (section 1.4.1). Results
from each of these subtasks directly address objectives 1 and 2 above.
The option of providing additional details to increase the credibility
of the implementation concept was also added as subtask 4. If desired,
the customer could choose one of the two Team X study options at the end
of this study.

## Work Plan

### Statement of Work and Results

***Demonstration of Concept Feasibility***

Work outlined in Subtasks 1-3 of the original proposal focused upon
demonstrating feasibility of the concept and directly addresses
objectives: (1) station keeping and navigation at the NEO; and (2)
tracking accuracy enabled by the transponder. This work was completed
over a seven month time period from the start of the contract by the JPL
GNC team.

**Subtask 1: Selection of a strawman NEO, orbit, and timeline for case
study**.

A strawman target NEO was selected with the following characteristics:

-   V~∞~ with respect to Earth in the range: 5-10 km/s

-   Effective diameter: 140 m

-   Bulk density: 2.0 g/cm^3^

-   Axial ratios: 1:1:2

-   Rotation period: 12 hr

-   Rotation pole obliquity: 45°

-   Size and spin state may be varied parametrically for a portion of
    the study

    The existing Itokawa shape model (pre-Hayabusa) was scaled to the
    desired effective diameter for use in the analysis outlined in
    subtask 2. It should be noted that the selected target's obliquity
    and rotation period used in our analysis (135 degrees and 6 hours)
    differ from what was proposed (45 degrees, 12 hours). See Table 2
    for more details about the selected target's properties. 

### Subtask 1 Results

***Target Selection Results: Virtual Impactor 2016 NM4***

The hypothetical asteroid, provisionally designated 2016 NM4 (Chesley
and Chodas, 2008), is discovered in July 2016, shortly after the
initiation of the Large Synoptic Survey Telescope (LSST) survey, and is
headed for a collision with Earth 33 years later, in September 2049. The
primary impact location, chosen at random, is in the South Pacific,
about 5400 km due south of Los Angeles and 1500 km NW of Easter Island
(Figure 1).

![](media/image1.jpeg){width="4.459722222222222in"
height="2.9770833333333333in"}

Figure 1. Impact location for 2016 NM4 on 2049 Sept. 12.

The asteroid is an Aten-class object on what we believe is a rather
typical impactor trajectory, a low-inclination, shallow-crossing orbit
largely interior to Earth. Figure 2 shows the orbit of the asteroid
along with the orbits of Earth and Venus. The hazardous close approaches
with the Earth occur at the asteroid's descending node, indicated on the
Figure. Figure 3 shows the motion of the asteroid in a frame rotating
with the Earth. Because the object is very near the 13:10 mean motion
resonance with Earth, its orbit path in Figure 3 comes very close to
repeating after 10 years (13 revolutions of the asteroid). It therefore
has particularly good observing circumstances every 10 years starting
with the 2016 discovery apparition. In addition, since the orbit is also
fairly close to the 4:3 resonance, the asteroid can be observed three
years before and after the favorable apparitions, when it passes through
the lobes above and below the Earth in Figure 3. Table 1 details the
planetary close approaches within 0.1 AU for 2016 NM4 from the time of
discovery through impact. We note the Earth close approach in 2046,
three years prior to the primary impact. This encounter has a minimum
distance of about one lunar distance, and leads to a 100-fold separation
of nearby trajectories between 2046 and 2049. As a result, the 2046
encounter provides improved leverage for the deflection effort, at the
expense of a similar increase in prediction uncertainty in 2049, the
ramifications of which are discussed later.

The dynamical model used in these orbit calculations includes a standard
set of perturbations, including the gravitational accelerations of the
Sun, Moon, 8 planets and 4 asteroids, along with the effects of solar
relativity. A more complete dynamical model, including for example the
Yarkovsky effect, would be adopted as needed.

**Table 1. Planetary and lunar close approaches (within 0.1 AU) of 2016
NM4 from discovery through impact. The last line represents the impact,
since the close approach distance is less than the radius of the
Earth.**

Date Body Dist.(AU)

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

2016 Aug 27.94383 Earth 0.042706

2017 Oct 30.31594 Venus 0.080965

2020 Dec 5.38243 Venus 0.062307

2024 Jan 7.88372 Venus 0.046527

2026 Aug 15.82433 Earth 0.055507

2027 Feb 5.14327 Venus 0.042289

2030 Mar 5.01357 Venus 0.040493

2033 Mar 29.28549 Venus 0.052710

2036 Apr 17.59255 Venus 0.073968

2036 Aug 31.82278 Earth 0.033996

2039 May 11.61451 Venus 0.081424

2042 Jun 4.36096 Venus 0.087426

2045 Jun 26.60054 Venus 0.094500

2046 Sep 13.67208 Earth 0.002772

2049 Sep 11.98767 Moon 0.001902

2049 Sep 12.35720 Earth 0.000019

While simulations show that most impactor discoveries surpass 99% impact
probability very early in their second apparition, 2016 NM4 does not
reach that threshold until the fourth apparition, fully ten years after
discovery. By the end of the discovery apparition the impact probability
has reached only 0.2%. Three years later it is recovered and the impact
probability shoots to 10%, topping 48% by the end of the apparition in
Nov. 2019. It is re-observed in 2023 and reaches 85% probability by the
end of that observing window. It finally exceeds 99% at the outset of
the fourth apparition, in 2026. This somewhat slow rise in impact
probability for 2016 NM4 is due to the poor observing circumstances (low
solar elongations and large geocentric distances) encountered during the
second and third apparitions, as well as the presence of a 1.1 lunar
distance encounter in 2046, which amplifies orbit uncertainties three
years prior to impact. Note that the impact probabilities stated here
neglect the possibility of radar measurements, despite the fact that
radar observations would very likely be made in late August 2016, when
the Arecibo SNR is of order 10^3^ and the impact odds would be around 1
in 2500. See Figure 4 for a graph showing the rise of computed impact
probability of the asteroid. While we have not considered radar
astrometry in 2016, it is likely that the impact probability would
exceed 99% by 2023, or even 2019, if they were included.

We note there is some discrepancy between the assumed albedo (0.143) and
the assumed bulk density (2 g/cm^3^). This is because, based on our
limited current understanding, smaller objects with relatively high
albedos (0.1-0.2) have somewhat higher densities, in the 2.5-3.5 g/cm^3^
range, whereas darker asteroids tend to have lower densities. This is
not a particularly severe contradiction, nor is it a critical concern
for the study, but it could easily be resolved by assuming a fairly high
macro-porosity, which is entirely consistent with the rather long
rotation period. Alternatively, reducing the albedo to a number more
consistent with the assumed density (say 0.06) and increasing the
absolute magnitude to 23 would also lead to reasonable parameter values.
None of this affects the mass or size of the target asteroid and is
therefore of no importance to the outcome of this study.

Table 2 lists most of the relevant impact, orbital, and physical
parameters for our target asteroid 2016 NM4

![](media/image2.png){width="4.05in" height="3.7868055555555555in"}

Figure 2. Orbits of Earth, Venus and 2016 NM4. The positions are noted
for the July 2016 discovery of 2016 NM4.

![](media/image3.png){width="3.558333333333333in"
height="3.3291666666666666in"}

Figure 3. Orbit of 2016 NM4 relative to Earth in the Earth rotating
frame.

![](media/image4.png){width="4.695138888888889in"
height="3.5215277777777776in"}

Figure 4. Rise of computed impact probability for 2016 NM4 from
discovery in July 2016 through 2028.

**Table 2. Relevant Parameters for Asteroid 2016 NM4.**

+----------------------------------------+-----------------------------+
|                                        |                             |
+----------------------------------------+-----------------------------+
| Discovery Date                         | July 8, 2016                |
+----------------------------------------+-----------------------------+
|                                        |                             |
+----------------------------------------+-----------------------------+
| Earth Impact information               |                             |
+----------------------------------------+-----------------------------+
| > Date                                 | 2049-09-12.35               |
+----------------------------------------+-----------------------------+
| > Location                             | 269.233º E, -14.514º S      |
+----------------------------------------+-----------------------------+
| > Velocity                             | 12.3 km/s                   |
+----------------------------------------+-----------------------------+
|                                        |                             |
+----------------------------------------+-----------------------------+
| Orbital Properties                     |                             |
+----------------------------------------+-----------------------------+
| > Orbital Period                       | 280.756 day (0.7687 yr)     |
+----------------------------------------+-----------------------------+
| > Eccentricity                         | 0.2490                      |
+----------------------------------------+-----------------------------+
| > Inclination                          | 5.185º                      |
+----------------------------------------+-----------------------------+
| > Perihelion dist.                     | 0.6198 AU                   |
+----------------------------------------+-----------------------------+
| > Semi-major axis                      | 0.8254 AU                   |
+----------------------------------------+-----------------------------+
| > Aphelion dist.                       | 1.0309 AU                   |
+----------------------------------------+-----------------------------+
| > V~∞~ S/C impact velocity             | 5.1 km/s                    |
+----------------------------------------+-----------------------------+
|                                        |                             |
+----------------------------------------+-----------------------------+
| Physical Properties                    |                             |
+----------------------------------------+-----------------------------+
| > Absolute Magnitude                   | 22.0                        |
+----------------------------------------+-----------------------------+
| > Albedo                               | 14.3%                       |
+----------------------------------------+-----------------------------+
| > Effective radius[^1]                 | 70 m                        |
+----------------------------------------+-----------------------------+
| > Shape                                | Scaled Itokawa radar        |
|                                        | model[^2]                   |
+----------------------------------------+-----------------------------+
| > Bulk density                         | 2.0 g/cm^3^                 |
+----------------------------------------+-----------------------------+
| > Mass                                 | 2.868 × 10^9^ kg            |
+----------------------------------------+-----------------------------+
| > GM                                   | 1.914 × 10^-10^ km^3^/s^2^  |
+----------------------------------------+-----------------------------+
| > Obliquity                            | 135º                        |
+----------------------------------------+-----------------------------+
| > Rotation period                      | 6 hours                     |
+----------------------------------------+-----------------------------+
| > Spin axis orientation                | (α,δ) = (121º, -29.3º)      |
|                                        |                             |
|                                        | (λ,β) = (132.4º, -48.3º)    |
+----------------------------------------+-----------------------------+

**Deflection Scenario Timeline**

By the end of 2023 the impact probability would have risen to about 85%
using only optical observations and, with the incorporation of radar
observations, probably in excess of 99%. Therefore we assume that the
decision would be taken at that time to launch both a rendezvous t-GT
and a kinetic energy Impactor Deflection Mission (IDM). We assume both
missions would be launched on the same booster in mid-2026 around the
time of the asteroid's close Earth approach, which provides for a
particularly favorable opportunity for a quick rendezvous. Thus the t-GT
spacecraft should be on station only months after launch, by December
2026, and we assume the IDM spacecraft takes two additional years to
gain orbital energy for a higher velocity impact that takes place on
July 4, 2028, a few weeks before the asteroid's perihelion passage.
Gravity tractor operations would start in July 2032 to trim the
trajectory in order to avoid a secondary impact after 2049. This
"tractoring" phase would be expected to take several months.

**Impactor Deflection**

The IDM would collide with 2016 NM4 with a relative velocity of about 5
km/s. The mission design for the IDM is beyond the scope of the present
study; we simply assume that the spacecraft would be maneuvered in order
to effect the desired collision. The orbit of the concept is shown in
Figure 5. The spacecraft would follow an exterior tangential orbit at
the point of collision, so that the impact would result in a purely
along-track increase in asteroid velocity (positive ΔV). This constrains
the IDM spacecraft orbit to that given in Table 3:

**Table 3. IDM Spacecraft Orbit Constraints.**

  ----------------------------------- -----------------------------------
  Eccentricity                        0.5329

  Perihelion dist.                    0.6545 AU

  Semi major axis                     1.401 AU

  Aphelion dist.                      2.148 AU

  Perihelion date                     2028 Jul 13.28669

  Long. Asc. Node                     170.379º

  Arg. Perihelion                     316.555º

  Inclination                         5.429º

  Period                              1.659 yr
  ----------------------------------- -----------------------------------

The deflection imparted by the IDM is given by:

ΔV = *β* (*m/M*) V~∞~

where *m* is the spacecraft mass, *M* is the asteroid mass, V*~∞~* is
the spacecraft relative velocity at impact, and *β* is the momentum
enhancement factor. For *β *= 1, we have a plastic collision, with the
impactor absorbed into the body without producing ejecta. For *β *= 2,
the collision is elastic, with the ejecta momentum equal and opposite to
the impactor momentum. Super-elastic collisions (*β *\> 2) are
considered likely, and sub-plastic collisions (*β *\< 1), where the
material spalled from the back of the asteroid carries more momentum
than that of the crater ejecta, are considered very unlikely. We assume
here that 1 \< *β *\< 5, and that, nominally, *β *= 2.

We assume an IDM mass of 1150 kg at impact, which leads to a ΔV/*β* = 2
mm/s. The nominal momentum enhancement of *β *= 2 produces ΔV = 4 mm/s,
while the full range of *β* values produces deflections in the range
2-10 mm/s. This range of deflections produces a range of miss distances
in 2049 of 4.7-30.3 R~⊕~, with a nominal value of 11.0 R~⊕~ .

![](media/image5.png){width="4.625in" height="4.144444444444445in"}

Figure 5. Orbit of Impactor Deflection Mission (IDM) spacecraft (in
blue) and asteroid 2016 NM4 (in red), as well as the orbits of Earth and
Mars. The positions of the objects two months before the primary
deflection are also shown, along with the location of the primary
deflection.

For the purposes of this study, we assume that the asteroid is
successfully deflected away from the 2049 impact, but happens to be
deflected onto a new impact trajectory, with the secondary collision
occurring sometime within ten years after the date of the 2049 primary
impact. There are a number of resonances available for this scenario,
and we select the 6:5 resonance, with an orbital period of 0.8333 yr,
corresponding to an impact in September 2054. This secondary impact is
obtained for a near-nominal IDM deflection of ΔV = 4.6889 ± 0.00015
mm/s, which implies a very realistic *β *= 2.34. A list of these
secondary impact possibilities is given in Table 4. Note that a one-year
return to impact is possible for the unlikely case of a sub-plastic
collision. Other than that, the earliest possible return for positive ΔV
is after four years, in 2053, with the 5:4 and 3:4 resonances (periods
0.8 yr and 1.33 yr, respectively).

A systematic search over the ΔV range of ±10 mm/s with 0.1 μm/s
resolution reveals that resonant returns to impact within 25 years
account for a few parts in 10,000 of the ΔV range. Almost half of this
keyhole density is associated with the 7:5 resonance near -8 mm/s, which
is not of concern for our ΔV \> 0 scenario. Many more theoretically
possible low order resonances do not appear in the Table 4 because they
do not appear to support an impact. This is likely due to the presence
of intervening approaches to Venus, Earth or the Moon that disrupt the
resonant return (see Table 1). Moreover, later returns to impact tend to
be eliminated through so-called MOID-sweeping, in which the asteroid's
node precesses to the point that an impact is not possible, regardless
of the arrival time. Some of the keyholes have been detected for
tertiary impact years; these cases pass through a keyhole in a secondary
b-plane. Near the selected 6:5 resonance, the numerical search for
keyholes was reduced in step size from 0.1 μm/s to 0.01 μm/s to increase
confidence that no significant keyholes were being missed. The locations
of the major keyholes in Table 4 are depicted on the target plane in
Figure 6 along with the corresponding circles predicted by
Öpik-Valsecchi theory (Valsecchi et al. 2003). In this idealized model,
the keyholes are located at the intersections of the resonant return
circles with the line of variations. The actual miss distances in 2049
as a function of ∆V are depicted in Figure 7.

**Table 4. List of keyholes through 2074 in the 2049 target plane of
2016 NM4\
as a function of primary deflection ΔV in the range +/-10 mm/s.**

+---------------+---------------+------------+------------------------+
| ∆V            | Impact Year   | Width      | Resonance              |
|               |               |            |                        |
| (mm/s)        |               | (km)       |                        |
+===============+===============+============+========================+
| -8.0391       | 2062          | 0.2        | Tertiary from 2054     |
+---------------+---------------+------------+------------------------+
| -8.0217       | 2057          | 0.7        | Tertiary from 2054     |
+---------------+---------------+------------+------------------------+
| -8.0097       | 2054          | 49         | 7:5                    |
+---------------+---------------+------------+------------------------+
| -8.0050       | 2056          | 0.5        | Tertiary from 2054     |
+---------------+---------------+------------+------------------------+
| -7.9976       | 2065          | 0.07       | Tertiary from 2054     |
+---------------+---------------+------------+------------------------+
| -7.9905       | 2070          | 0.14       | Tertiary from 2054     |
+---------------+---------------+------------+------------------------+
| -4.6571       | 2065          | 10.3       | 23:16                  |
+---------------+---------------+------------+------------------------+
| -3.4484       | 2062          | 7.5        | 19:13                  |
+---------------+---------------+------------+------------------------+
| -1.2559       | 2070          | 1.1        | 31:21                  |
+---------------+---------------+------------+------------------------+
| -1.2314       | 2068          | 1.0        | 28:19                  |
+---------------+---------------+------------+------------------------+
| -1.2025       | 2066          | 1.5        | 25:17                  |
+---------------+---------------+------------+------------------------+
| -1.1649       | 2064          | 0.9        | 22:15                  |
+---------------+---------------+------------+------------------------+
| -1.1223       | 2073          | 0.09       | Tertiary from 2062     |
+---------------+---------------+------------+------------------------+
| -1.1073       | 2073          | 0.5        | 35:24                  |
+---------------+---------------+------------+------------------------+
| -1.0817       | 2060          | 1.0        | 16:11                  |
+---------------+---------------+------------+------------------------+
| -1.0202       | 2074          | 0.03       | Tertiary from 2058     |
+---------------+---------------+------------+------------------------+
| -0.9884       | 2065          | 0.4        | 23:16                  |
+---------------+---------------+------------+------------------------+
| -0.9020       | 2061          | 0.4        | 17:12                  |
+---------------+---------------+------------+------------------------+
| -0.8836       | 2066          | 0.1        | 24:17                  |
+---------------+---------------+------------+------------------------+
| -0.8732       | 2071          | 0.2        | 31:22                  |
+---------------+---------------+------------+------------------------+
| -0.7542       | 2068          | 0.1        | 26:19                  |
+---------------+---------------+------------+------------------------+
| -0.7261       | 2063          | 0.06       | 19:14                  |
+---------------+---------------+------------+------------------------+
| -0.6724       | 2052          | 0.3        | 4:3                    |
+---------------+---------------+------------+------------------------+
| -0.6427       | 2071          | 0.05       | 29:22                  |
+---------------+---------------+------------+------------------------+
| -0.6307       | 2065          | 0.04       | 21:16                  |
+---------------+---------------+------------+------------------------+
| -0.6207       | 2062          | 0.06       | 17:13                  |
+---------------+---------------+------------+------------------------+
| 0.6403        | 2073          | 0.06       | 17:24                  |
+---------------+---------------+------------+------------------------+
| 0.6688        | 2056          | 0.1        | 5:7                    |
+---------------+---------------+------------+------------------------+
| 0.6955        | 2074          | 0.07       | 18:25                  |
+---------------+---------------+------------+------------------------+
| 0.7051        | 2067          | 0.06       | 13:18                  |
+---------------+---------------+------------+------------------------+
| 0.7267        | 2060          | 0.09       | 8:11                   |
+---------------+---------------+------------+------------------------+
| 0.7664        | 2068          | 0.06       | 14:19                  |
+---------------+---------------+------------+------------------------+
| 0.7764        | 2072          | 0.02       | 17:23                  |
+---------------+---------------+------------+------------------------+
| 0.8165        | 2053          | 0.3        | 3:4                    |
+---------------+---------------+------------+------------------------+
| 0.8608        | 2070          | 0.02       | 16:21                  |
+---------------+---------------+------------+------------------------+
| 0.8711        | 2066          | 0.06       | 13:17                  |
+---------------+---------------+------------+------------------------+
| 0.8877        | 2062          | 0.07       | 10:13                  |
+---------------+---------------+------------+------------------------+
| 0.9187        | 2058          | 0.2        | 7:9                    |
+---------------+---------------+------------+------------------------+
| 0.9476        | 2063          | 0.09       | 11:14                  |
+---------------+---------------+------------+------------------------+
| 0.9986        | 2054          | 0.3        | 4:5                    |
+---------------+---------------+------------+------------------------+
| 1.0644        | 2060          | 0.2        | 9:11                   |
+---------------+---------------+------------+------------------------+
| 1.1201        | 2055          | 0.3        | 5:6                    |
+---------------+---------------+------------+------------------------+
| 1.1685        | 2062          | 0.04       | 11:13                  |
+---------------+---------------+------------+------------------------+
| 1.2101        | 2056          | 0.3        | 6:7                    |
+---------------+---------------+------------+------------------------+
| 1.2804        | 2057          | 0.3        | 7:8                    |
+---------------+---------------+------------+------------------------+
| 1.3001        | 2074          | 0.1        | 22:25                  |
+---------------+---------------+------------+------------------------+
| 1.3375        | 2058          | 0.2        | 8:9                    |
+---------------+---------------+------------+------------------------+
| 1.3847        | 2059          | 0.2        | 9:10                   |
+---------------+---------------+------------+------------------------+
| 1.4029        | 2070          | 0.03       | 19:21                  |
+---------------+---------------+------------+------------------------+
| 1.6165        | 2069          | 0.05       | 19:20                  |
+---------------+---------------+------------+------------------------+
| 1.6302        | 2070          | 0.1        | 20:21                  |
+---------------+---------------+------------+------------------------+
| 1.8981        | 2050          | 2.4        | 1:1                    |
+---------------+---------------+------------+------------------------+
| 2.2556        | 2069          | 0.2        | 21:20                  |
+---------------+---------------+------------+------------------------+
| 2.3030        | 2067          | 0.3        | 19:18                  |
+---------------+---------------+------------+------------------------+
| 2.3314        | 2066          | 0.2        | 18:17                  |
+---------------+---------------+------------+------------------------+
| 2.5600        | 2061          | 0.3        | 13:12                  |
+---------------+---------------+------------+------------------------+
| 2.6416        | 2060          | 0.6        | 12:11                  |
+---------------+---------------+------------+------------------------+
| 2.6439        | 2070          | 0.02       | Tertiary from 2060     |
+---------------+---------------+------------+------------------------+
| 2.8798        | 2058          | 0.6        | 10:9                   |
+---------------+---------------+------------+------------------------+
| 4.3017        | 2065          | 1.5        | 19:16                  |
+---------------+---------------+------------+------------------------+
| 4.3058        | 2070          | 0.07       | Tertiary from 2065     |
+---------------+---------------+------------+------------------------+
| 4.4001        | 2070          | 0.9        | 25:21                  |
+---------------+---------------+------------+------------------------+
| [4            | [2054]{.mark} | [5         | [6:5]{.mark}           |
| .6887]{.mark} |               | .5]{.mark} |                        |
+---------------+---------------+------------+------------------------+
| 4.6916        | 2058          | 0.01       | Tertiary from 2054     |
+---------------+---------------+------------+------------------------+
| 4.6955        | 2063          | 0.03       | Tertiary from 2054     |
+---------------+---------------+------------+------------------------+
| 5.0677        | 2073          | 0.07       | Tertiary from 2068     |
+---------------+---------------+------------+------------------------+
| 5.9940        | 2071          | 0.11       | Tertiary from 2062     |
+---------------+---------------+------------+------------------------+
| 6.0033        | 2067          | 0.02       | Tertiary from 2062     |
+---------------+---------------+------------+------------------------+
| 6.0074        | 2062          | 5.3        | 16:13                  |
+---------------+---------------+------------+------------------------+
| 6.0147        | 2066          | 1.1        | Tertiary from 2062     |
+---------------+---------------+------------+------------------------+
| 7.2471        | 2066          | 0.08       | Tertiary from 2053     |
+---------------+---------------+------------+------------------------+
| 8.9226        | 2068          | 0.18       | Tertiary from 2064     |
+---------------+---------------+------------+------------------------+
| 8.9367        | 2064          | 10.7       | 19:15                  |
+---------------+---------------+------------+------------------------+

![](media/image6.png){width="6.495138888888889in"
height="5.0472222222222225in"}**Figure 6. Selected resonant returns from
the 2049 encounter on the 2049 target plane. The dotted vertical line
marks the line of variations, i.e., the locus for kinetic ΔV in the
range ±10 mm/s. The large black circles mark the loci for resonant
returns under the idealized Öpik-Valsecchi model (Valsecchi et al.
2003). The red symbols mark the actual intersections of the resonant
return and line of variation under the complete dynamical model. The
blue dot marks the extent of the Earth, while the gray region marks the
impact cross-section.**

**Gravity Tractor Phase**

Although the probability of a kinetic deflection directly into a keyhole
is only around 10^-4^, the likelihood that the post-deflection
uncertainty would intersect a keyhole is significantly larger. In the
scenario contrived for this study, the gravity tractor would indeed be
necessary to pull the asteroid away from the 6:5 resonance keyhole, and
thereby avoid a secondary impact in 2054. There would be a relatively
close approach to the Earth in Sept. 2046, at about 1 lunar distance
that amplifies the 2049 b-plane deflection by a factor of 100 relative
to that obtained on the 2046 b-plane. This is not a particularly unusual
scenario, but it would allow the gravity tractor to be more effective
than it would be in the absence of close encounters.

The analysis of this effort takes place entirely in the 2049 b-plane,
where the objective is simply to avoid the keyhole, rather than in the
2054 b-plane, where the analysis is more numerically challenging. In our
simulation, the t-GT is capable of generating an acceleration of 2.5 x
10^-12^ m/s^2^ or about 6.5 μm/s /month, and given the mass of the
asteroid, the corresponding force is about 7.2 mN.

![](media/image7.png){width="4.247222222222222in"
height="3.185416666666667in"}

**Figure 7. The miss distance in Earth radii on Sept. 12, 2049 vs. the
tangential deflection in mm/s on July 4, 2028.**

We allow at least a year after the mid-2028 IDM collision to
re-establish the asteroid's orbit, determine that passage through the
6:5 keyhole is a significant possibility, and then make the decision to
start tractoring well before the September 2049 keyhole passage.
However, this would push the tractoring schedule into a period of time
where deflections of any sort have a very low effectiveness, and this
would force a delay in the tractoring for a few years. In this
particular case, a three-year time interval centered on mid-April 2030
is inefficient to the point that it was avoided altogether. (See Figure
8.) This period of ineffectiveness is due to the presence of intervening
close approaches between the deflection and the 2049 close approach.
From Table 1, we note the asteroid makes a number of close approaches to
Venus, Earth and the moon. Of particular interest for us is the
relatively shallow 2036 encounter with Earth, at 0.034 AU. The geometry
of this encounter is such that it tends to undo deflections that would
be attempted in the months surrounding April 2030. Figure 8 shows how
long the tractoring effort should last to effect a deflection of 750 km
(three times the expected one-sigma uncertainty, as described in subtask
3). From that Figure we observe that the most efficient time to apply
the low thrust deflection would be around 2037, when 750 km could be
obtained in only about 50 days, although that same effect could be
accomplished with less than 180 days of tractoring if the effort is
initiated in 2032-2044. For the purposes of subtask 3, and before
knowing what those results might be, we elect to start tractoring on
July 4, 2032 for 200 days, which would yield a 2049 B-plane deflection
of 650 km. (See Fig. 9.) As we shall see in the final analysis of
Subtask 3, the 2049 b-plane one-sigma uncertainty is 260 km, so that the
650-km deflection is successful at the 650 km/260 km = 2.5 sigma level.
An additional forty days beyond the simulated 200 days would bring the
deflection to the 3 sigma level. Longer durations would provide
ever-increasing statistical confidence in the miss. In practice, the
start and length of the tractoring interval would be carefully selected
to obtain the desired confidence in the primary objective of impact
avoidance.

![](media/image8.png){width="5.247222222222222in"
height="3.935416666666667in"}

**Figure 8. Tractoring efficiency as a function of start time. The plot
shows the tractoring duration in days required to cause a deflection of
750 km on the 2049 b-plane. A shallow minimum exists from 2032 to
2043.**

Inadvertently tractoring into an adjacent keyhole is obviously
undesirable. However, the closest keyhole for a secondary encounter is
over 6,000 km away in the 2049 b-plane, and this could only be reached
after several years of tractoring. As shown by Table 4, our keyhole
search has confirmed the existence of two nearby keyholes to tertiary
impacts, and indeed there may be additional such keyholes below the
resolution of our search. These keyholes are tiny -- only tens of meters
in width -- and lie within 200 km of their associated primary keyhole.
This second layer of nearby keyholes could be avoided by lengthening the
tractoring time by another few months.

One might ask what the effectiveness of the gravity tractor would be if
the 100:1 leverage afforded by the 2046 Earth encounter was not present.
Under this alternate, hypothetical scenario, we assume that the size of
the keyhole to be avoided is the same, only now it would be 6 km in
2046, rather than in 2049. The question is whether it would be
practicable to steer the uncertainty footprint clear of the keyhole
under this scenario, or equivalently, could the gravity tractor be
effective in the absence of strong gravitational interactions. As we
shall see later, from Subtask 3, the prediction uncertainty with a
modest amount of post-tractoring tracking is about 3 km in the 2046
b-plane and 260 km in 2049 b-plane. Thus the gravity tractor would have
to effect a deflection of up to the sum of 3 km (the semiwidth of the
keyhole) and 9 km (to keep the 3-sigma ellipse clear of the keyhole. The
absolute effectiveness of the gravity tractor in 2046 can be derived by
reducing the computed 2049 deflection by a factor of 100. With this
rescaling, Figure 9 reveals that the desired 12 km deflection in 2046
could be obtained with one year of tractoring starting on July 4, 2032,
an amount not dramatically different from the duration required to clear
the actual 2054 keyhole on the 2049 b-plane. The implication is that the
strong dynamical effects of close planetary encounters (in the years
between a deflection effort and the target plane) do not necessarily
translate into correspondingly large reductions in the challenge of
effecting a particular deflection. Rather, the leverage enlarges the
uncertainty region at a rate comparable to the deflection effectiveness,
and so the net effectiveness remains about the same.

Thus we have shown that the gravity tractor is a viable concept for
providing a trim maneuver in our adopted strawman scenario. And, in all
likelihood, it would be a viable concept for the majority of impact
scenarios, although further study is needed to confirm this. Conversely,
there might be possible scenarios with keyholes up to 2 orders of
magnitude larger (or more), in which case the gravity tractor concept
might not be adequate. Future threat possibilities would probably have
to be analyzed individually and a customized deflection approach
developed for each.

![](media/image9.png){width="3.359027777777778in"
height="2.5256944444444445in"}

**Figure 9. Tractoring efficiency as a function of start time. The plot
shows the size of the 2049 deflection along the line of variations after
200 days of tractoring for various tractoring start times.**

**Subtask 2: Study the gravity tractor spacecraft requirements to
accelerate a NEO in a pre-determined direction and amount.**

**Subtask 2 Objectives**

1.  Choose a representative NEO model for all work that represents the
    median case (i.e., median of NEO size, aspect ratio, density,
    rotation rate, and orbit parameters as in above strawman target
    characteristics).

2.  Demonstrate that a control law is possible that allows the
    spacecraft to hover in close proximity as the irregular NEO rotates
    beneath it.

3.  Carry out a parametric analysis of station keeping performance in
    order to derive first-level requirements. Investigate the hovering
    and control distances to determine viability of station keeping. Run
    each case for 1 orbital period. Establish whether or not hovering
    and control are possible at close distances.

    a.  Parameters (can be changed by joint agreement of JPL and B612
        Foundation if results of initial analyses make it clear that
        parametric values are way off):

        1.  S/C mass (500, 1000, 1500 kg)

        2.  S/C range from NEO center of mass, keeping a
            to-be-determined margin of safety for NEO shape variations
            (parameters depend on NEO size and aspect ratio). Determine
            minimum acceptable stand-off distance for each case.

        3.  Station-keeping box size and location

        4.  Available thrust (use 5 engines and determine thrust levels
            required for each case; compare to available thrusters)

    b.  Outputs: (for each case listed above)

        1.  S/C accumulated delta-V/ thrust history required for
            maintenance

        2.  Accumulated NEO movement as a function of time

### Subtask 2 Results

**1.4.3.1 Choose a representative NEO physical model**

The representative NEO parameters we have selected for 2016 NM4 are
largely covered as part of Subtask 1 in Table 2, including bulk density,
rotation period, effective diameter, trajectory, and rotation pole.

The contribution to the definition of the representative NEO from this
subtask is the shape and gravity model to be used. The shape chosen is
the "rough model" of Itokawa defined by Ostro et al.(2005), scaled such
that the volume of the shape is equivalent to the volume of a 70 m
radius sphere (i.e., scaled to a 140 m effective diameter). Ostro et
al.'s "rough model" of Itokawa *does not* include the shape information
derived from the Hayabusa mission; it is however representative of those
real data. The axial ratio of this model (1.95 x 1.08 x 1.00) is very
close to the target value outlined in the proposal (2 x 1 x 1). Images
of the shape model being used (with scale information) are shown in
Figure 10.

![](media/image10.jpeg){width="2.922222222222222in"
height="2.191666666666667in"}![](media/image11.jpeg){width="2.922222222222222in"
height="2.191666666666667in"}![](media/image12.jpeg){width="2.922222222222222in"
height="2.191666666666667in"}

Figure 10. Images of the baseline asteroid shape. These are principal
axis views where x is the axis corresponding to the minimum moment of
inertia, y corresponds to the intermediate moment of inertia, and z
corresponds to the maximum moment of inertia (i.e., the spin axis).

The gravitational potential model being used is defined by assuming the
above defined shape has a uniform density value of 2.0 g/cm^3^ (GM =
1.914 × 10^-10^ km^3^/s^2^) as described in Table 2. Accelerations
derived from this model are computed using the method of Werner and
Scheeres (1997).

**1.4.3.2 Demonstrating that a control law is possible**

The second objective of Subtask 2 was to demonstrate that it would be
possible to safely maintain sufficiently close proximity between a
gravity tractor spacecraft and asteroid 2016 NM4 to exert the ΔV on the
asteroid required to avoid impact in 2054 via a keyhole passage in 2049.
To address this objective, we have created a high-fidelity software
simulation of the translational spacecraft dynamics in the vicinity of
our baseline model of 2016 NM4[^3]. A thruster configuration and control
law were developed to implement the stationary trajectory required for
gravity tractoring to work. We have simulated the spacecraft dynamics
under this thrust control for gravity tractoring durations of up to six
months. The remainder of this subsection describes the spacecraft
properties we have assumed, the simulation software, the thrust control
law, and the results obtained in representative simulations of one and
six month durations.

*Preliminary Definitions:*

The following coordinate frames are defined here to aid in understanding
the discussion in this section.

"Asteroid Body-Fixed" coordinate frame

This non-inertial reference frame rotates with the asteroid such that
the coordinates of any position on the asteroid surface are constant.
The Z axis of this frame is aligned with the inertially-fixed rotation
pole of the asteroid, which is parallel to the principal axis
corresponding to the maximum moment of inertia of the 2016 NM4 shape
model. The asteroid body-fixed coordinate frame rotates uniformly around
its Z axis with respect to inertial space. The X axis of this frame is
aligned with the principal axis of the asteroid shape corresponding to
its minimum moment of inertia, i.e., the X axis is more-or-less aligned
with the "longest" extent of the body. The Y axis completes the
right-handed coordinate frame.

"Spacecraft" coordinate frame

This frame is defined to help us understand the orientation of the
thrusters with respect to the desired gravity-tractor towing direction.
The positive X axis of this frame points along the asteroid velocity
vector; this is the desired towing direction. The positive Y axis is
defined by the cross product of the X unit vector with the asteroid
rotation pole direction. This vector points along the longitudinal
direction with respect to the asteroid rotation pole. The positive Z
axis is defined by the cross product of the X unit vector and the Y unit
vector, completing the right-handed coordinate system. This vector
points in the direction of changing latitude with respect to the
asteroid rotation pole.

*Gravity Tractor Spacecraft Properties:*

We have attempted to use realistic parameters in our simulation to model
our gravity tractor spacecraft concept, though we haven't gone as far as
doing a detailed spacecraft design. The following values were chosen for
the spacecraft parameters with direct impact on the translational
dynamics. The mass of the spacecraft at the beginning of the gravity
tractor phase was chosen to be 1000 kg. The area of the spacecraft
surface projected onto a plane perpendicular to the Sun direction was
chosen to be 25 m^2^. These values give a mass-to-area ratio of 40,
which is an important parameter for computing the acceleration due to
solar radiation pressure (SRP).

Given the power required to operate many solar electric propulsion (SEP)
thrusters simultaneously, the 25 m^2^ area would probably not be
consistent with a spacecraft that is powered entirely by solar panels;
we are in effect implicitly assuming a radioisotope thermoelectric
generator (RTG) power or some other similar power source. If the mission
were to be done with only solar power, there would be impacts on the
spacecraft dynamics and control system, including increased demands on
the attitude control system, decreased thrust accuracy (due to flexing
of the solar panels), and some loss of tractoring ΔV efficiency due to
larger SRP accelerations. These issues would increase the spacecraft
development cost, but would probably not alter the feasibility of the
mission.

*Gravity Tractor Spacecraft Thruster Configuration:*

We have assumed the gravity tractor spacecraft has 5 throttle-able
fixed-direction SEP thrusters (not including backups) of which any three
could be operated simultaneously. Each thruster is modeled after the ion
thrusters used on the Dawn spacecraft, providing a maximum of 90 mN of
thrust at a specific impulse (Isp) of 3100 seconds (Dawn spacecraft
website, 2008).

The fixed orientation of the thrusters is defined with respect to the
spacecraft coordinate frame (see "*Preliminary Definitions*" in this
section) and is shown in Figure 11. Each thruster is represented by a
red conical "nozzle" on the grey square spacecraft. Acceleration is
applied to the spacecraft center of mass in the direction of the apex of
the red nozzle when the thruster is fired. With this fixed thruster
orientation, the spacecraft has the capability to apply control thrust
in any direction in three-dimensional space.

The pair of thrusters labeled 'T1' and 'T2' apply force in the XZ plane
of the spacecraft frame at an angle β away from the positive X
direction. The angle β is the "thruster canting angle," which is further
discussed in the part of this section on "*Selection of Tractoring
Range.*" These thrusters generate most of the thrust required for the
gravity tractor trajectory (since gravitational acceleration is
primarily nulled by these thrusters) and act to control motion away from
the nominal tractoring position in the negative X and +/-Z directions.
The 'T3' thruster accelerates the spacecraft in the negative X
direction. This thruster could be used as a safety net when the
spacecraft gets much further from the body than the desired tractoring
range and combines in a vector sense with 'T1' and 'T2' to create
control thrusts that are purely along the Z direction. Finally, a pair
of thrusters could be used to accelerate the spacecraft in the positive
and negative Y directions, 'T4' and 'T5.' These thrusters would point
roughly perpendicular to the average towing direction and are used for
countering acceleration from solar radiation pressure, controlling
unexpected perturbations in the Y direction, and countering the
open-loop component of gravity that varies along the Y direction as the
asteroid rotates.

![](media/image13.jpeg){width="4.833333333333333in"
height="3.7916666666666665in"}

**Figure 11: Diagram of the fixed thruster orientations on the gravity
tractor spacecraft in the spacecraft coordinate frame. The heavy dark
arrow originating at the asteroid represents its rotation pole.**

*Gravity Tractor Thrust Control Law:*

In order to change the orbit of 2016 NM4, an acceleration must be
applied to the asteroid that results in a secular change in its orbit
elements. We have decided for this study that the most effective method
of moving away from the 2049 keyhole is to change the asteroid orbit's
semi-major axis. A quick examination of Gauss's equations governing the
evolution of orbit elements under a low-thrust acceleration tells us the
best way to change semi-major axis in a secular way is to apply
acceleration along the asteroid's direction of motion (or in the
opposing direction). Based on the workings of the gravity tractor
concept (Lu and Love, 2005), the most efficient approach to changing the
asteroid's semi-major axis is then to maintain a fixed spacecraft
position along the asteroid's velocity vector[^4].

We have selected a relatively simple dead-band control law to
demonstrate the feasibility of tractoring for an extended period of time
near 2016 NM4. The "dead-band" control approach allows us to define a
bounding box around the (essentially) fixed tractoring position that we
desire to which the spacecraft motion would restricted by thruster
firings. This control concept has been successfully demonstrated in
flight by JAXA's Hayabusa mission (Kominato et al., 2006). This type of
control would be a good choice because it is robust against the fairly
large amount of uncertainty in key environmental parameters (such as
asteroid density and gravitational harmonics) that a mission could
expect to have upon arrival at an asteroid.

The total spacecraft thrust applied under our control law is the vector
sum of the contributions from two independent controller components. The
first component is an open-loop thrust which acts to null the
[expected]{.underline} acceleration on the spacecraft at the nominal
tractoring position. This component of thrust varies in both direction
and magnitude with time as the asteroid rotates beneath the spacecraft
and as the asteroid moves around the Sun.

The second component is the dead-band controller. This controller would
confine the spacecraft motion to the region inside a specified dead-band
"box" around the nominal tractoring position. The principal axes of the
box are aligned with the basis vectors of the spacecraft coordinate
frame (see discussion earlier in this section called "*Preliminary
Definitions*"). When the spacecraft position is inside the box, no
thrust is applied (from the dead-band component of the controller). When
the spacecraft position is outside of the box along any of the
spacecraft frame axes, thrust is applied along the dimension of the box
that has been violated in the opposite direction of the violation as to
accelerate the spacecraft back towards the box. If more than one
dimension of the box is exceeded simultaneously, thrust is applied along
all the dimensions that are violated.

The position dead-band controller described above has no damping and in
the presence of uncertainty and finite duty cycles, spacecraft velocity
could increase over time and eventually override the authority of the
controller (Broschart and Scheeres, 2007). To stop this from happening,
we place a restriction on the dead-band control law based on spacecraft
velocity. As the spacecraft is accelerated back towards the dead-band
box (when the dead-band thrust is active), the dead-band thrust would
turn off when the spacecraft returns to the box or a maximum allowable
speed in that direction is exceeded. This approach is effective at
keeping the spacecraft velocity bounded over long tractoring durations.

*Selection of Tractoring Range:*

One of the most important decisions that must be made when designing a
gravity tractor mission is at what range from the asteroid center of
mass the spacecraft should be placed during the tractoring phase.
Because the acceleration applied to the asteroid increases as the
inverse square of the range, it is desirable to remain as close as
possible in order to minimize the required tractoring duration to
deflect the asteroid. Of course, the spacecraft must not be positioned
so close that impact with the surface is a risk.

The maximum extent of our model for 2016 NM4 is \~117.3 m from the
center of mass. This maximum occurs at roughly -4.2 deg latitude in the
asteroid body-fixed coordinate frame (see frame definition above in the
"*Preliminary Definitions*" part of this section). Though the spacecraft
could be positioned inside of this distance for high-latitude tractoring
positions, this distance would serve as a minimum threshold for all
tractoring at all latitudes in the results reported here.

Another factor in determining the tractoring range might be avoidance of
resonant excitation phenomenon. The synchronous orbit radius for our
baseline asteroid is 131 m. It has been demonstrated that lateral (i.e.,
non-radial) motion of a spacecraft hovering (i.e., keeping the
spacecraft position at a fixed position as we propose for tractoring)
near this distance can be excited by this resonance (Broschart and
Scheeres, 2005). Figure 12 shows the magnitude of the largest eigenvalue
of the monodromy matrix corresponding to lateral motion after one
rotation of the asteroid as a function of radial hovering range and
latitude. In this plot, values greater than unity indicate instability
in the lateral directions of motion, which in this case corresponds to
resonant excitation. This is the case for tractoring locations in the
triangular region on the left half of the plot. A gravity tractor
spacecraft positioned at a location in the regions where there is no
resonant excitation (right of the triangular region) should exhibit
stable lateral motion. Tractoring in regions with the resonant
excitation requires station-keeping control in all directions. Care must
be taken to avoid excitation that could cause the spacecraft velocity to
increase to the point it overwhelms the authority of the station-keeping
control. There is some loss of ΔV efficiency when tractoring in the
resonant area, but we will show in the simulation results later in this
section that highly efficient use of ΔV would not be required to deflect
2016 NM4 away from the 2049 resonant-return keyhole. So we may conclude
that this phenomenon is not a driver on the tractoring range unless the
spacecraft is restricted to apply thrust along a single axis only, which
would never be the case for this type of mission.

The final factor to consider when selecting a tractoring distance is the
thruster canting angle required to avoid thruster plume impingement on
the asteroid surface. The thruster canting angle β is defined as half of
the angle between the two thrusters used to null gravitational
acceleration on the spacecraft that point towards the asteroid (see the
schematic representation in Figure 11; the canting angle is measured
between 'T1' and 'T2'). The thrusters must be canted away from the
asteroid direction so that the momentum of the particles coming out of
the thruster would not be transferred to the asteroid surface; the
resulting acceleration on the asteroid would work against the efforts of
the gravity tractor. It is desirable for the thruster canting angle to
be as small as possible to minimize cosine losses incurred when both
thrusters are firing simultaneously (which would usually be the case).

![](media/image14.jpeg){width="5.997222222222222in"
height="5.0152777777777775in"}

**Figure 12: The magnitude of the largest eigenvalue of the lateral
motion of a hovering spacecraft after one rotation period of the small
body. Colors other than blue indicate resonant excitation.**

To determine the minimum allowable thruster canting angle as a function
of tractoring position, an oblate tri-axial ellipsoid (129.705 m x
129.705 m x 63.741 m) was first defined that circumscribes the baseline
2016 NM4 shape. This circumscribing shape must be an ellipsoid of
rotation (oblate) because of the asteroid's rotation relative to the
tractoring position. For the purpose of this analysis, the required
thruster canting angle is computed as half of the angle between opposing
vectors with base at the spacecraft that are tangent to the oblate
ellipsoid surface plus a 5 deg margin. In general, the smallest possible
minimum canting angle for a particular tractoring position will
correspond to a thruster pair oriented in the XZ plane of the spacecraft
frame. The largest possible minimum canting angle at a fixed tractoring
position will correspond to a thruster pair oriented in the XY plane in
the spacecraft frame. This is because the smallest dimension of the
circumscribing ellipsoid lies in the XZ plane of the spacecraft frame.
Figures 13 and 14 show the smallest and largest minimum canting angles
allowable (respectively) as a function of tractoring latitude with
respect to the asteroid equator and distance from the center of mass.
Note that the canting angle required for thrusters in the XZ plane of
the spacecraft frame is always smaller than the requirement in the XY
plane. Larger canting angles correspond to reduced towing efficiency, so
the thrusters should be canted in the XZ plane to maximize efficiency.

![](media/image15.jpeg){width="4.376388888888889in"
height="3.9430555555555555in"}

**Figure 13: Minimum allowable thruster canting angle (degrees) for a
thruster pair in the XZ plane of the spacecraft frame as a function of
radial range from the asteroid center of mass and latitude with respect
to the asteroid equator. This thruster orientation minimizes the
required canting angle.**

![](media/image16.jpeg){width="4.3805555555555555in"
height="3.285416666666667in"}

**Figure 14: Minimum allowable thruster canting angle (degrees) for a
thruster pair in the XY plane of the spacecraft frame as a function of
radial range from the asteroid center of mass and latitude with respect
to the asteroid equator. This thruster orientation results in the
largest possible minimum canting angle for tractoring at a given
location. (Values computed using a minor approximation. The value given
is an upper bound on the true minimum.)**

*Gravity Tractor Trajectory Simulation Software and Inputs:*

We have integrated the translational dynamics of the gravity tractor
spacecraft under the control law specified above in the "*Gravity
Tractor Thrust Control Law*" discussion using software written in
Matlab©. The spacecraft attitude dynamics were not considered in this
study. It was assumed that the attitude controller could maintain
spacecraft alignment with the spacecraft coordinate frame without
impacting the translational dynamics in any serious way.

The gravity tractor spacecraft dynamics were integrated using Matlab's
'ode45' integrator with the spacecraft equations of motion defined in an
inertial reference frame. Forces on the spacecraft arising from the
constant-density polyhedron gravity model (see section 1.4.3.1 and Table
2), solar radiation pressure (based on the model from Scheeres and
Marzari, 2002), solar gravity, and the spacecraft's thrusters were
included. The asteroid's position was specified by the ephemeris
generated by Chesley and Chodas (2008). The asteroid's path was not
modified by the spacecraft in the course of the simulation, but the
forces induced on the asteroid by the spacecraft were recorded. The
small motion of the asteroid relative to its original ephemeris would be
a second-order effect in this simulation that would not be expected to
have any significant bearing on the performance of the gravity tractor.

For these simulations, we have defined the desired tractoring position
to be 155 meters from the asteroid center-of-mass along the asteroid's
velocity vector. This will produce a tug on the asteroid that is roughly
along the velocity vector, which will increase the semi-major axis of
the asteroid's orbit around the Sun. The nominal 155 meters was chosen
based on the criteria in the "*Selecting Tractoring Range*" discussion
above as a reasonable compromise between towing force, spacecraft
safety, and thruster efficiency (i.e., canting angle).

The size of the dead-band box boundary was +/- 25 meters from the
desired tractoring position in the Y and Z directions (in the spacecraft
frame). In the X direction, the "bottom" of the box is defined as 10
meters below the desired position and the "top" was 60 meters above the
desired position. As an intermediate stage, if the spacecraft exceeded
10 meters above the desired position, all thrust in the X direction
(including the open-loop component) was turned off. This was done for
purposes of efficiency; the asteroid's gravity can be used to accelerate
the spacecraft in the --X direction, saving fuel. The high limit of +60
meters in the X direction serves as a safety net, which has never been
activated in any of the simulations we have run using the controller
discussed here. The threshold velocity at which the dead-band thrust is
turned off was selected to be 5 mm/sec in the X direction and 8 mm/sec
in the Y and Z directions.

We have included uncertainties in the navigation solutions for position
and velocity in the simulations. These navigation uncertainties were
included in the simulation by the following methodology. At a fixed time
interval, the on-board navigation solution was updated. This consisted
of a random draw from a normal distribution for each component of
position and velocity around the integrated trajectory solution. Each
update of the filter was uncorrelated to the previous solution. The
erroneous solution was then propagated forward as the "on-board
estimate" along with the true solution. All thrust calculations made
using the thrust control law were based on the on-board estimate. There
were no differences between the dynamics used to propagate the on-board
estimate and the true spacecraft state.

In the simulation data shown here, we assumed a 1-sigma uncertainty in
each component of position of 1 meter and a 1-sigma uncertainty in each
component of velocity of 1 mm/sec. These uncertainties are consistent
with the performance expected from a spacecraft navigation system that
incorporates altimetry, optical navigation, and radiometric
measurements. Estimates of the spacecraft state were updated every 15
minutes.

**Table 5: Summary of Gravity Tractor Simulation Parameters for the
Spacecraft and Thrust Controller**

  ----------------------------------- -----------------------------------
  Spacecraft Initial Mass:            1000 kg

  Spacecraft Area (normal to          25 m^2^
  Sun-line, fixed value)              

  Number of Thrusters                 5

  Maximum Thrust (per thruster)       90 mN

  Thruster Isp (constant)             3100 seconds

  Thruster Canting Angle              50 degrees

  Tractoring Range                    155 meters

  Position Dead-band Limit (X         +/-10 meters, + 60 meters
  direction)                          

  Position Dead-band Limit (Y and Z   +/-25 meters
  direction)                          

  Dead-band Thrust Cutoff (velocity   5 mm/sec
  threshold in X direction)           

  Dead-band Thrust Cutoff (velocity   8 mm/sec
  threshold in Y and Z directions)    

  Position Uncertainty (1 sigma in    1 meter
  each component)                     

  Velocity Uncertainty (1 sigma in    1 mm/sec
  each component)                     

  Navigation Filter Update Frequency  Every 15 minutes
  ----------------------------------- -----------------------------------

*Gravity Tractor Trajectory Simulation Results:*

Using the setup described above, spacecraft trajectories subject to our
control law were integrated for periods of 1 day, 5 days, 1 month (8
independent runs), and 6 months (2 independent runs). The relative
tolerance used was 1x10^-4^ and the absolute tolerance was 1x10^-6^. In
this document, we will show the results from a single one month period
and a single six month period. Both periods begin on July 4, 2032. All
of the runs we have done exhibit the same qualitative properties as the
data shown here. Data from other runs can be obtained by request. In all
cases, our control law was successful in keeping the position of the
spacecraft near the desired tractoring position and exerting the
expected level of acceleration on 2016 NM4.

Figure 15 shows the spacecraft trajectory (blue) for the 1-month and
6-month integrations relative to the 2016 NM4 center-of-mass in an
inertial coordinate frame aligned with the asteroid fixed frame at the
initial epoch. Note that in this frame, the desired tractoring position
migrates around an inclined circle as the asteroid moves around the Sun
(i.e., as the asteroid velocity vector rotates in inertial space). The
spacecraft trajectory tracks this slow drift in the desired tractoring
position well, as shown by the "smear" of the blue trajectory around
roughly half a "tractor orbit" relative to the body. The "tractor orbit"
is inclined due to the obliquity of the asteroid's pole with respect to
its orbit around the Sun. Also note that the asteroid shape is rotating
in this frame. The shaded area around the asteroid represents the
circumscribing oblate ellipsoid filled by the asteroid as it rotates.

![](media/image17.jpeg){width="5.125in" height="3.3090277777777777in"}

![](media/image18.jpeg){width="4.9375in" height="3.1125in"}

**Figure 15: Spacecraft trajectory relative to 2016 NM4 (inertial
coordinate frame).**

**Top: 1 month integration, Bottom: 6 month integration.**

Figure 16 shows the same spacecraft trajectories as in Figure 15 in the
asteroid body-fixed coordinate frame. Each time the asteroid rotates
beneath the spacecraft in an inertial frame (every 6 hours), the
trajectory would trace out an "orbit" in this frame. Over the course of
the 2016 NM4 orbit around the Sun, the latitude of the desired
tractoring position shifts with respect to the fixed asteroid equatorial
plane (see Figure 15), which causes the latitude of these "orbits" to
drift. Over the course of one-half of 2016 NM4's heliocentric orbit, the
tractoring trajectory traces out a spherical surface segment from +45
deg latitude to -45 latitude in this frame (as shown in the 6 month
result).

Figure 17 shows both spacecraft trajectories relative to the desired
tractoring position. This figure may be most helpful is assessing the
effectiveness of our control law at keeping the spacecraft near the
desired tractoring position. In this figure, the blue line represents
the actual spacecraft trajectory and the red dashed line shows the
on-board estimate of position. Note how the on-board estimate makes
sudden changes; this corresponds to times when the estimate is updated.
[It can be seen here that our controller does an excellent job of
keeping the spacecraft in the desired region around the desired
tractoring position.]{.underline} The negative X, negative Y, and
positive Y boundaries of the position dead-band control are clearly
visible (-10, -25, and + 25 meters respectively). Excursions beyond the
dead-band boundary are minimal, which tells us that our station-keeping
thrusters have sufficient thrust to do their job well. Remember that at
the positive X boundary, thrust is only turned off instead of reversed,
so the excursions outside the box are a little larger on that side. Also
note that the "safety net" dead-band at +60 meters in the X direction is
never needed. The perspective of these figures only shows the X and Y
deviations, but the Z direction is similarly well bounded (as Figure 18
shows). Also, if you look closely at the 1 month figure, you can see
that the trajectory "web" is denser around the area where the Y
deviation is zero. This corresponds to motion early in the trajectory
that is primarily up and down relative to the asteroid. Eventually
perturbations set in and the motion becomes more irregular, filling the
entire box (Evidence of this effect will also be seen in Figure 20,
where the difference between the tractoring ΔV and the ΔV applied to the
spacecraft is minimal near the beginning of the simulations).

Figure 18 shows each component of the spacecraft deviation from the
desired tractoring position as a function of time. This is the same data
as shown in Figure 17 plotted as a function of time. The red line
represents the deviation along the spacecraft frame X coordinate, the
blue line shows deviation along the Y coordinate, and the green line
shows deviation along the Z coordinate. Again, you can see that the
position dead-band boundaries in the X, Y, and Z directions are well
respected by the spacecraft trajectory. Note that when the trajectory
goes from the maximum to the minimum and back repeatedly in any
particular coordinate, that means that the primary oscillation of the
spacecraft around the desired tractoring point is in that direction at
that time. For pretty much the entire duration, the trajectory bounced
back and forth between the X boundaries; this is because the trajectory
is unstable in that direction. At times, the spacecraft also traverses
the entire width of the dead-band box in the Y or Z directions
repeatedly as well.

![](media/image19.jpeg){width="5.125in" height="3.3465277777777778in"}

![](media/image20.jpeg){width="5.125in" height="3.225in"}

**Figure 16: Spacecraft trajectory relative to 2016 NM4 (asteroid
body-fixed coordinate frame). Top: 1 month integration, Bottom: 6 month
integration.**

![](media/image21.jpeg){width="4.9375in" height="3.46875in"}

![](media/image22.jpeg){width="4.9375in" height="3.46875in"}

**Figure 17: Spacecraft trajectory relative to the desired tractoring
position (spacecraft coordinate frame). Top: 1 month integration,
Bottom: 6 month integration.**

![](media/image23.jpeg){width="5.125in" height="3.6in"}

![](media/image24.jpeg){width="5.125in" height="3.5902777777777777in"}

**Figure 18: Components of spacecraft position relative to the desired
tractoring position (spacecraft coordinate frame) as a function of time.
Red: X coordinate, Blue: Y coordinate, Green: Z coordinate. Top: 1 month
integration, Bottom: 6 month integration.**

Figure 19 shows each component of spacecraft velocity as a function of
time. It can be seen that the spacecraft velocity remains well bounded
over time and does not grow in a monotonic way. This is a good thing
because if velocity was increasing over time, the dead-band control
authority would eventually be overwhelmed and control of the spacecraft
would be lost. This demonstrates that the return velocity threshold
included in the dead-band portion of the control law is effective at
damping any excess energy that arises from uncertainties. Recall though,
that the limit on velocity is only imposed when the dead-band thrust is
active along that particular axis. Therefore, the speed in any
particular direction could be larger than the threshold when the
position dead-band thrust is not active. This can be seen here on many
occasions. The point of the velocity threshold is not to hold the
spacecraft speed below some threshold, but to keep the kinetic energy of
the spacecraft bounded.

Figure 20 shows the ΔV used by the simulated spacecraft thrusters as a
function of time. The red line shows the total ΔV expended by the
spacecraft, defined as the sum of the ΔV used by each thruster. The blue
line shows the applied ΔV, defined as the integral over time of the
total thrust vector magnitude. The green line shows the tractoring ΔV,
defined as the ΔV required to counter the component of gravitational
acceleration along the velocity vector of the asteroid. This quantity
can be thought of as the ΔV used to move the asteroid along its velocity
vector.

Efficiency measures can be defined from these quantities. Let the
"thruster efficiency" be defined as the applied ΔV divided by the total
ΔV expended. This number describes the loss due to applying thrust in a
vector direction with a fixed thruster orientation. This loss is
primarily driven by the thruster canting angle required to avoid having
the thruster plume impinge on the asteroid surface. Let the gravity
tractor efficiency be defined as the tractoring ΔV divided by the total
expended ΔV. This tells us the fraction of the total expended ΔV that is
being used to accelerate the asteroid along its velocity vector. This
number includes losses due to thrust vectoring, station-keeping thrust,
and oscillation of the pull direction due to the irregular shape of the
body. Figure 20 shows that for the one month period shown, the simulated
spacecraft expends a total of 40.91 m/s of ΔV; 63.2 percent of which is
used to accelerate the spacecraft (thruster efficiency) and 48.4 percent
of which is used to pull the asteroid (gravity tractor efficiency). For
the 6 month case, 261.3 m/s of ΔV is expended with a thruster efficiency
of 63.6 percent and a gravity tractor efficiency of 46.8 percent. Note
that all ΔV costs grow at a consistent rate over time. Also, these
efficiency numbers were consistent to within a few percentage points
across the simulations performed.

![](media/image25.jpeg){width="5.125in" height="3.6in"}

![](media/image26.jpeg){width="5.125in" height="3.5902777777777777in"}

**Figure 19: Components of spacecraft velocity (spacecraft coordinate
frame) as a function of time. Red: X coordinate, Blue: Y coordinate,
Green: Z coordinate. Top: 1 month integration, Bottom: 6 month
integration.**

![](media/image27.jpeg){width="5.125in" height="3.6in"}

![](media/image28.jpeg){width="5.125in" height="3.5902777777777777in"}

**Figure 20: Spacecraft ΔV consumption versus time (Red is total ΔV
consumed, blue is total ΔV applied to spacecraft, green is total ΔV used
to accelerate the asteroid).**

**Top: 1 month integration, Bottom: 6 month integration**

Figure 21 shows the spacecraft mass as a function of time. Because of
the high Isp of the ion propulsion thrusters and the small accelerations
experienced by the spacecraft at 2016 NM4, the rate of propellant mass
consumption (for translational thrust) averages only about 1.4 kg per
month of tractoring. Over the 6 month period beginning on July 4, 2032,
the spacecraft consumes only 8.6 kg of propellant to perform gravity
tractoring. Note that over the 6-month period, some small variations
would be expected in the mass consumption rate due to variations in
tractoring latitude with respect to the asteroid and variance in the use
of the lateral station-keeping thrusters. This effect is present, but
its magnitude is minimal.

Figure 22 shows the time variation in magnitude of the simulated
spacecraft acceleration due to thrust. The vector magnitude is shown in
red and the sum of the thrust magnitudes for each thruster is shown in
blue. Each individual thruster is limited to a maximum thrust of 90 mN,
as is the total thrust vector magnitude. However, when more than one
dead-band boundary is violated simultaneously, the sum of the thrust
magnitudes can be larger (blue line). Note that a thrust of 90 mN
roughly corresponds to an acceleration of 0.9x10^-4^ m/s^2^ for a 1000
kg spacecraft. The low level of thrust that would be active for the
majority of the time corresponds to the open-loop control thrust
component.

Figure 23 shows the total ΔV applied to the asteroid along its velocity
vector as a function of time. This plot tells us how well the simulated
gravity tractor is working. It can be seen here that the rate of
increase in the total ΔV applied to 2016 NM4 is very consistent; the
primary variations (which are not really visible) are due to the
seasonal change in tractoring latitude with respect to the asteroid.
This is because the ΔV on the asteroid is only a function of spacecraft
positions (and the masses), which would vary very little when the
tractoring controller is working well (see Figures 17 and 18). The small
variations in position around the desired tractoring position have a
negligible effect in this simulation. It can be seen from these
simulation results that a total ΔV of about 7 μm/s is applied to the
asteroid each month in the scenario we have simulated here. This means
that our spacecraft can exert sufficient ΔV to divert 2016 NM4 from the
hazardous 2049 keyhole in a matter of months under this control law (see
Table 4).

Figure 24 shows the angular deviation of the simulated spacecraft from
the asteroid velocity vector as a function of time. The goal of our
controller was to maintain this angle close to zero, which was achieved.
To get a truly optimal gravity tractoring control law, this would vary
somewhat so that the gravitational acceleration vector acting on the
spacecraft and asteroid was aligned with the asteroid velocity vector,
which is not necessarily the same as what we've done because of the
irregular shape of the asteroid. This would allow for small gains in
efficiency, but as seen by the modest total propellant mass expended in
Figure 19, the savings of a few grams of fuel are not that important.
The key here is to keep this angle small so that the cosine loss of the
actual gravitational acceleration is small.

![](media/image29.jpeg){width="5.3125in" height="3.73125in"}

![](media/image30.jpeg){width="5.4375in" height="3.8152777777777778in"}

**Figure 21: Simulated spacecraft mass versus time. Top: 1 month
integration, Bottom: 6 month integration.**

![](media/image31.jpeg){width="5.125in" height="3.84375in"}

![](media/image32.jpeg){width="4.9375in" height="3.702777777777778in"}

**Figure 22: Spacecraft acceleration from thrust versus time.Top: 1
month integration, Bottom: 6 month integration.**

![](media/image33.jpeg){width="4.9375in" height="3.702777777777778in"}

![](media/image34.jpeg){width="4.9375in" height="3.702777777777778in"}

**Figure 23: ΔV applied to 2016 NM4 along its velocity vector versus
time. Top: 1 month integration, Bottom: 6 month integration.**

![](media/image35.jpeg){width="5.125in" height="3.6in"}

![](media/image36.jpeg){width="5.125in" height="3.6in"}

**Figure 24: Angle between the spacecraft position and the asteroid
velocity vector versus time. Top: 1 month integration, Bottom: 6 month
integration.**

*Summary and Discussion*

Using a high-fidelity numerical simulation, we have demonstrated the use
of a thrust control law that could safely maintain the spacecraft
positioning required for the gravity tractor phase of our mission when
there is navigation uncertainty in the spacecraft state. The same
simulations have shown that tractoring for a matter of months near 2016
NM4 at a range of 155 meters from the center-of-mass is sufficient to
safely accelerate the asteroid by the amount required to avoid a keyhole
in 2049. An effort was made to make realistic assumptions about the
spacecraft properties and capabilities throughout (though a detailed
design was not done).

One item of particular interest was the very small amount of propellant
mass required to do the job. This suggests that for a mission with
similar requirements to our baseline mission, SEP propulsion might not
be required; chemical options might be sufficient.

In future work, the robustness of our control could be further tested by
utilizing an imperfect on-board dynamics model. This would create error
in the open-loop portion of the thrust control law, which would probably
decrease the ΔV efficiency, but not make the controller ineffective
unless the error was very large (orders of magnitude larger than the
nominal acceleration itself). Adding an imperfect dynamics model would
also cause the on-board state estimate error to grow larger as it is
propagated forward between updates. This would probably not break the
control law either. Error in the thrust application could be added as
well. If thruster application error (which could reflect attitude
uncertainty as well) and dynamics uncertainty were included, in addition
to navigation uncertainty, the tractoring control law could be put
through a rigorous test of its robustness.

**1.4.3.3 Carry out a parametric analysis of station keeping
performance**

In this section, some questions put forth in the proposal document are
addressed. They are addressed here as discussion topics in light of the
simulation results and experiences.

*Viability of Tractoring at Close Proximity*

A discussion of tractoring distances was given in the discussion
entitled "*Selection of Tractoring Range*" in Section 1.4.3.2. The
conclusions were that tractoring should occur at no less than the
maximum radius of the circumscribing ellipsoid (\~130 meters for 2016
NM4) from the center-of-mass. Some margin is required on that number
depending on the size of allowable deviations from the nominal
tractoring position, uncertainty in the asteroid shape, and dimensions
of the spacecraft. Outside of this minimum radius, our simulations and
analytical data suggest that any range is doable, though the higher
required canting angles for low altitude tractoring positions decreases
the ΔV efficiency. Figure 12 also suggests a more simplistic and more
efficient controller might suffice beyond a range of about 160 meters.
This notion however was not tested via simulation.

*Varying the Spacecraft Mass*

These results are very nearly spacecraft mass independent. That is to
say, they scale very cleanly with initial spacecraft mass within a
reasonable range. The ΔV applied to the asteroid would be changed from
the values given here as the ratio of the new mass divided by 1000 kg.
The fuel consumed by the spacecraft would change in the same way (though
the percentage of spacecraft mass consumed by the thrusters will be the
same). Given the same thruster performance capabilities, the control law
would become less effective as the spacecraft mass increases; it would
allow deviations further and further outside of the box until it becomes
a problem. The mass would probably have to increase by at least an order
of magnitude before that happened. In that case, larger thrusters or
banks of thrusters would remedy the situation.

*Varying Range from the Center-of-Mass*

All else equal, varying the target tractoring range from the
center-of-mass would vary the towing force (and therefore, the applied
ΔV) by roughly the inverse square of the ratio of the distances.
Assuming the control law keeps the spacecraft near the desired position,
the only parameter that affects the ΔV on the asteroid is the spacecraft
position through the gravitational acceleration, which has roughly an
inverse square scaling.

Regarding the spacecraft ΔV efficiency, there will be some improvement
as the tractoring range is increased since the required canting angle
would be reduced and a simpler control law with less station-keeping
thrust might be possible.

*Varying Station-Keeping Box Size*

In the simulation data shown here, the station-keeping box was set to
allow a maximum deviation of 10 or 25 meters depending on the direction.
This is a fairly small value, but appropriate due to the close proximity
to the surface and the danger of resonant phenomenon arising during long
ballistic segments (for close tractoring). Because of our focus on
tractoring so close to the body, it might not be practical to vary this
parameter much as part of a parametric analysis. A smaller box would
probably decrease the efficiency somewhat and could become a problem
with respect to uncertainty in spacecraft state and dynamics.

*Comments on the Required Thrust versus Available Thrusters*

For the baseline scenario studied here, the limitations imposed by the
Dawn ion thrusters had no adverse impact on controller performance. As
mentioned above, if the spacecraft were much more massive, larger
thrusters might be required.

**Subtask 3: NEO Tracking.**

The key question to be addressed in this subtask is:

What is the time evolution of NEO ephemeris errors that would be
obtained by a station-keeping spacecraft equipped with a standard X-band
transponder in the immediate vicinity of a NEO? This question is to be
understood in the context of the following scenario:

Suppose a NEO is on course to potentially impact the Earth at a future
date (the impact epoch). Approximately 20 years before the impact epoch,
a rendezvous t-GT spacecraft, equipped with altimetry, optical
navigation, and radio transponder instruments, is maintaining station in
the vicinity of the threatening NEO and verifies whether or not the
potential impact would occur. If the potential impact is confirmed, we
suppose that a deflection attempt is carried out. The t-GT spacecraft
would then be used to verify that the deflection maneuver was sufficient
to prevent the Earth impact, and furthermore to determine whether the
NEO is headed for any keyholes in the impact plane that would lead to a
subsequent Earth impact. If passage through a keyhole is a possibility,
the t-GT spacecraft could be used to "tow" the NEO to move its
trajectory away from the keyhole. The spacecraft would move in very
close to the NEO to perturb it onto a non-threatening trajectory via the
mutual gravitational attraction of the spacecraft and NEO. This GT mode
would be maintained until the NEO was confirmed to be on a safe
trajectory.

Specific questions to be addressed through this subtask are:

1.  Considering that the *a priori* uncertainty in the effectiveness of
    the deflection technique is very large (i.e., the momentum
    multiplier, β, is poorly known), how soon after the deflection epoch
    could the primary deflection be confirmed?

2.  When would the uncertainties on the target plane at the 2049 impact
    epoch be comparable to the scale of the secondary impact keyholes
    that are to be avoided? Similarly, if the NEO is indeed deflected
    onto a secondary impact trajectory, how soon would this be
    recognized as likely?

3.  Assuming that a secondary impact is found to be very possible, and
    the rendezvous spacecraft is then employed as a gravity tractor, how
    quickly could the GT deflection be measured? Could the deflection be
    adequately inferred through the thruster history? Could it be
    explicitly measured more or less in real time?

The nature of the NEO trajectory and in particular its close encounter
history during the years from the deflection epoch to the impact epoch
is crucial in determining the uncertainty at the impact epoch. This task
attempts to examine this issue by subdividing the analysis into two
parts: first, the NEO ephemeris uncertainty at the time of the
observations, and second, the uncertainty mapped to the impact epoch.

**Overview**

The purpose of this subtask is to quantify the ability to precisely
determine the orbit of the asteroid by tracking a spacecraft in orbit or
hovering near the asteroid using standard radiometric and optical
navigation techniques. The scenario assumed for this study is as
follows. First, an orbiter spacecraft would arrive at the asteroid in
early January 2028, about 6 months prior to the arrival of an impactor
spacecraft. During this 6 month period, the orbiter would be tracked
from the Earth to obtain a very precise orbit of the asteroid prior to
impact. Next, the orbiter would be maneuvered to a safe position and the
impactor would hit the asteroid on July 4, 2028, imparting a change in
velocity ΔV to the asteroid. Soon thereafter, the orbiter would be
maneuvered back to the asteroid and then tracked in order to measure the
ΔV that was imparted. After a period of time in which a high accuracy
measurement of the ΔV is obtained, the gravity tractoring portion of the
mission begins. Following the tractoring phase, the orbiter would once
again be tracked to determine the net acceleration imparted to the
asteroid. The three primary questions to be addressed in this subtask
are:

1.  How accurately could the pre-impact orbit of the asteroid be
    determined? This accuracy could be parameterized by the amount of
    time prior to impact that the spacecraft is tracked, as well as the
    combinations and accuracies of the data types.

2.  How many days/weeks/months of spacecraft tracking following the
    impact would it take to determine the impact ΔV to a sufficient
    accuracy?

3.  How many days/weeks/months of spacecraft tracking following the
    gravity tractoring phase does it take to determine the acceleration
    imparted by the tractor to a sufficient accuracy?

In all cases, "sufficient" accuracy is defined as the level needed to
ascertain whether or not the asteroid is on an Earth impacting
trajectory. In principle, the most obvious way to determine the orbit of
the asteroid *in situ* would be to land a transponder on the surface and
track the radio signal of this beacon from the Earth. This method,
however, has some operational problems that might be difficult to
overcome, including power and thermal issues for a landed spacecraft
encountering extended and recurring periods of darkness due to the
asteroid's spin and orientation, and the question of whether the
transponder would survive the deflection collision of the impactor
spacecraft. We propose an alternate scenario in which the orbiting or
hovering spacecraft would be tracked from the Earth through standard
radiometric methods while the spacecraft in turn images the asteroid to
provide its position and velocity relative to the asteroid (Figure 25).
The Earth-based radiometric data are Doppler and range, which measure
the radial velocity and position, respectively, of the spacecraft with
respect to the tracking station, and Delta Differential One-way Range
(DDOR), which measures the angular location of the spacecraft relative
to two tracking stations. The key to this methodology is the imaging
data taken from the spacecraft from which landmark locations on the
asteroid could be measured to determine the spacecraft's position
relative to the asteroid with fairly high accuracy. This capability was
demonstrated on the Japanese Hayabusa mission (Kominato et al., 2006;
Gaskell et al., 2006). Further numerical experiments have shown that,
with reasonable assumptions about the camera and the ability to perform
the landmark detection, the accuracy of the asteroid orbit determination
would be nearly identical to that possible using a surface transponder,
and might even be better if occultation effects of a landed transponder
are considered.

![](media/image37.jpeg){width="5.8in" height="3.6222222222222222in"}

**Figure 25: Schematic of data types**

**Procedure**

In order to address the above questions, a covariance analysis was
performed to quantify the uncertainties in the asteroid orbit in all the
phases. The covariance analysis included the following steps:

1.  Numerically integrate a spacecraft trajectory hovering around the
    asteroid from January 5, 2028 (arrival of the orbiter) to August 4,
    2028 (a month after the impact). The control of the spacecraft is
    not addressed in this study, but other studies, as well as the
    results from Subtask 2 in this report show that it would be feasible
    to maintain a pre-defined location near the asteroid. For this
    covariance analysis, we assumed the spacecraft was in a hovering
    position roughly 10-15 km from the asteroid's center, in a location
    perpendicular to the sun line. The results that will be presented,
    however, are largely independent of the particular hovering
    location.

2.  Simulate Doppler and range tracking data at a reasonable frequency
    during the time span. In principle, the spacecraft could be tracked
    continuously for the entire time period using the three DSN tracking
    complexes around the globe. In practice, however, conflicts from
    other spacecraft would preclude this level of support. For this
    study, we assumed a background schedule of 3 DSN tracks per week
    over the entire time span. The Doppler data were weighted at 0.1
    mm/s, and the range at 2 m.

3.  Simulate DDOR data at a reasonable frequency over the time span.
    DDOR data require a nearby quasar to track, as well as simultaneous
    viewing from two DSN complexes. For the quasars, we simulated
    fictitious quasar locations near the geocentric direction of the
    asteroid during the tracking phase. The DDOR data were scheduled
    during Goldstone/Canberra and Goldstone/Madrid overlaps at a rate of
    two baseline pairs per week. The DDOR data were weighed at a value
    of 2 nsec.

4.  Simulate optical tracking data by scattering landmarks across the
    asteroid at some reasonable spacing, and take optical measurements
    of these landmarks at some reasonable frequency. For the landmarks,
    we assumed a spherical shape (for simplicity) of the asteroid with a
    diameter of 140 m, and placed landmarks at an equal spacing of
    roughly 10 degrees over the entire surface. Images of the asteroid
    were taken at a rate of one per week, except for the period from two
    weeks prior to impact to four weeks after impact, during which
    images were taken at a rate of four per day. We assumed an onboard
    CCD camera with a focal length of 500 mm and a chip size of 1024
    square pixels, which gives a total field-of-view (FOV) of 1.4 deg.
    For any given picture, only landmarks that were in the camera FOV
    and illuminated by the sun were used in the filter. The optical data
    were weighted at a value of 0.25 pixels.

5.  Set up an estimation filter to process combinations of all three
    data types, with appropriate weights on the data and a priori
    uncertainties on the filter parameters. Table 6 lists the complete
    set of filter parameters and their settings that were used for this
    analysis. Note that the parameters are divided into the "estimated"
    and "consider" categories; the "estimated" parameters are ones that
    are adjusted in the filter, and whose *a priori* uncertainties are
    therefore reduced as the data are processed, while the "consider"
    parameters are those which are too weak to be estimated, but whose
    errors contribute to the final covariance. The output of the filter
    is a post-fit covariance matrix which provides uncertainties on the
    spacecraft orbit, asteroid orbit, and ancillary parameters in the
    filter. The quantity of primary interest is the uncertainty in the
    asteroid orbit, which is provided as a fully correlated covariance
    on the Cartesian asteroid state at an epoch time, and this could
    then be mapped to different coordinate frames and times, such as the
    time of the 2049 impact. All uncertainty results in this subtask are
    quoted as 1-sigma values.

### Subtask 3 Results

The first set of results concerns the 6-month period of time from the
orbiter spacecraft arrival in January 2028 through the kinetic
deflection in July, 2028. During this time, the spacecraft would be in a
stable hovering position near the asteroid and could commence taking
optical images of the landmarks while it is simultaneously tracked using
Earth-based radiometric data. Prior to this time, the knowledge of the
asteroid's orbit would have been obtained solely from Earth-based
optical images taken of the asteroid over a period of many years. The
primary question to be answered is the level of improvement that could
be obtained from these in situ spacecraft measurements. The results are
presented in two forms; first in the six classical orbital elements of
the asteroid, and then in terms of the major and minor axis of the
uncertainty ellipse after mapping the uncertainties to the b-plane of
the 2049 encounter with the Earth.

**Figure 26. A priori 1-sigma uncertainties in orbital elements (blue
lines) and the improved uncertainties (red lines) due to 6 months of
tracking prior to the arrival of the kinetic impactor in July
2028.**![](media/image38.jpeg){width="6.35in" height="4.65in"}

Figure 26 shows plots of the 1-sigma asteroid ephemeris uncertainties in
the 6 classical asteroid orbital elements as functions of time in days
after arrival on January 5, 2028 and ending just prior to the close
Earth encounter on September 13, 2046. In each plot, the blue line
represents the *a priori* knowledge of the asteroid's orbit (i.e.,
without tracking the orbiter spacecraft), while the red line shows the
uncertainties after 6 months of tracking are included. The sudden
changes past day 3000 are caused by the distant Earth flyby on August
2036, but the net effect on the uncertainties is only slight. It can be
seen that the ground-based ephemeris knowledge is quite good; for
example, the semi-major axis has an uncertainty of only 500 m. The
results are not mapped beyond the 2046 encounter because this Earth
flyby has a large dispersing effect on the uncertainties. This can be
seen more clearly in Figures 27 and 28, which plot the asteroid error
ellipse uncertainties at the time of the 2046 Earth flyby and the 2049
impact, in their respective b-planes. The horizontal axis in this case
represents the number of days of tracking starting on January 5, 2028.
Thus, at time 0, no spacecraft tracking has been obtained and the values
represent the *a priori* knowledge. Here, it can be seen that at the
2046 flyby, the error ellipse is only 9 x 3.5 km, while in 2049, the
major axis has grown considerably so that the ellipse has dimensions of
about 800 x 3.5 km.

Next, the radiometric and optical tracking data are added to see how
they improve the asteroid ephemeris uncertainty. In Figures 27 and 28,
the result is shown as a function of the number of days of tracking for
the 2046 and 2049 Earth b-planes, respectively. After 8 days, the impact
major axis ellipse has been reduced to about 400 km for the 2049 epoch
(the effect of this amount of tracking on the orbital elements is
plotted by the red line in Figure 26, which shows a factor of 2-4
improvement in these parameters). Beyond this point, adding more data
doesn't result in further improvement, probably due to the presence of
the consider parameters and the fact that the asteroid orbit is already
well determined. Consider parameters are those that are not solved for
within the orbit solution but whose uncertainties do affect the solution
uncertainties. Various numerical experiments with different assumptions
on the filter parameters indicate that the 8-10 day period appears to be
an inflection point where the majority of improvement using the *in
situ* data has been reached, at least for this asteroid and time period.

The next step is to examine how well the orbit of the asteroid could be
re-determined after the impact on July 4, 2028. It is assumed that the
impact would deliver a nominal ΔV to the asteroid of 4.7 mm/s. However,
the uncertainty on this value might be large since the blowback momentum
is poorly known. Thus, in the covariance analysis, the asteroid
ephemeris is assumed to have an applied impulsive ΔV of 4.7 mm/s with an
uncertainty of 1 cm/s. This effectively corrupts the ground-based
knowledge such that the b-plane uncertainty ellipse semi-major axis at
the time of the 2046 and 2049 encounters increases to about 2,000 and
200,000 km, respectively (the semi-minor axis shows little change).

It would take several years to pin down the new asteroid orbit after the
deflection if only Earth-based observations were used, as the change to
the asteroid's orbit would be relatively small, but *in situ*
measurements provide a very powerful way of determining the post-impact
orbit. This is seen in Figure 29, which shows the effect on the 2046 and
2049 encounter b-plane uncertainty ellipses of the post-deflection
tracking. After only one day of tracking, the semi major axes have been
reduced from 2,000 and 200,000 km to roughly 6 and 510 km. Further
tracking doesn't have much effect for the next couple of weeks until
after about 21 days, when the results improve to values of about 3 km
and 125 km, which seems to represent the best that could be done for
this time period. It is not entirely clear why, at this time period, the
asteroid's orbit could be determined to this level of accuracy whereas
in January, the floor value was about 400 km, but it is likely that the
relative viewing geometries of the Earth and the asteroid play a major
role.

The final objective of this task is to determine the level of accuracy
that the asteroid's orbit could be determined following gravity
tractoring. The tractoring phase was set to be 200 days long beginning
on July 2, 2032 and ending on January 20, 2033. Because this is long
after the original tracking period in 2028, and to simplify the set up
for the analysis, we reverted to the original ground based *a priori*
covariance of the asteroid as a starting point. We then added a
transverse acceleration uncertainty to the asteroid during the tracking
phase of 2.5 x 10^-12^ m/sec^2^. Unlike in the case of the instantaneous
deflection, the net effect on the Earth b-plane uncertainties in 2049
from this disturbance is negligible because of its relatively small
magnitude. Thus, at the beginning of the post-tractor phase tracking
period on January 20, 2033, the b-plane uncertainty ellipse in 2049 is
800 km, as before.

Figure 30 plots the results of the radiometric and optical tracking of
the spacecraft following tractoring. Here, it can be seen that only one
day of tracking is needed to bring the uncertainties down to floor
values of 3.1 and 260 km for each of the Earth encounters. Adding more
tracking has little effect, and it appears that the limit of knowledge
has been reached.

**Subtask 3 Conclusions**

The specific conclusions of this subtask for the scenario analyzed are:

-   The asteroid's pre-impact trajectory was determined to a value of
    about 400 km in the semi major axis of the b-plane uncertainty
    ellipse at the Earth encounter in 2049. This level is reached after
    only eight days of tracking the spacecraft once it arrives at the
    asteroid.

-   Following deflection, the asteroid's orbit was determined to its
    pre-impact levels after only one day of tracking. The ellipse
    semi-major axis was determined to an accuracy of 125 km after three
    weeks of tracking (1-sigma).

-   Following the tractoring in early 2033, the asteroid's orbit was
    determined to its pre-tractoring levels again after only one day of
    tracking. The ellipse semi-major axis knowledge reached a floor
    value of about 260 km (1 sigma).

Several general conclusions can also be drawn from this study:

-   The combination of radiometric tracking of an orbiting or hovering
    spacecraft with optical imaging of the asteroid from the spacecraft
    is sufficient to dramatically improve the knowledge of an asteroid's
    orbit. No transponder on the surface of the asteroid is needed.

-   Tracking the asteroid via the spacecraft improves the orbit accuracy
    by a factor of from 2 to 5 over the accuracy that can be obtained
    from Earth-based observations of the asteroid. The size of the
    improvement is dependent on the relative viewing geometry, and hence
    the time period over which the spacecraft is tracked.

-   The length of time it takes to realize improvements in the knowledge
    of the asteroid's ephemeris is measured in days to weeks. A
    spacecraft need not be in place for months or years for significant
    orbit knowledge improvements to take place.

-   A close flyby, such as the one that occurs in 2046 for this
    scenario, magnifies uncertainties by a large factor for subsequent
    flybys. ![](media/image39.jpeg){width="4.5in"
    height="3.1625in"}![](media/image40.jpeg){width="4.116666666666666in"
    height="3.2375in"}

Figure 28: Asteroid uncertainties mapped to the Earth b-plane in
September 2049 as a function of the number of tracking days
(pre-deflection impact).

![](media/image41.jpeg){width="5.45in" height="2.0375in"}

![](media/image42.jpeg){width="5.5in" height="2.075in"}

Figure 29: Asteroid uncertainties mapped to the Earth b-plane in
September 2046 (top) and September 2049 (bottom) following deflection,
as a function of the number of tracking days (post-deflection impact).

![](media/image43.jpeg){width="5.55in" height="2.1in"}

![](media/image44.jpeg){width="5.5in" height="2.0625in"}

Figure 30: Asteroid uncertainties mapped to the Earth b-plane in
September 2046 (top) and September 2049 (bottom) following tractoring as
a function of the number of tracking days.

**Subtask 4: Preliminary Planning for Analysis of Mission Design and
Implementation.**

This subtask was to develop a preliminary assessment and set of
requirements for the t-GT NEO mission. The results would provide a
preliminary scoping of the mission and provide requirements that would
be addressed during the Team X study. It would include a pre-session
with the Team X leader to develop the necessary documentation of the
customer specifications for the NEO mission and reach agreement on Team
X study products. The specific tasks are:

1.  Develop Level 1 requirements for the t-GT mission.

2.  Identify, where possible, existing S/C that would roughly match
    parameters from Subtask 2 above.

3.  Provide a cost by analogous mission of the gravity tractor mission
    including launch vehicle, based upon Subtask 2 parameters. Provide
    an annual operations cost (for scaling purposes). Assume a mission
    could launch to the NEO and use optimal launch and arrival times
    (i.e., could launch anytime in next 20 years, as opposed to
    non-optimal launch/ arrival combinations associated with "little
    warning time to impact" scenarios).

### Subtask 4 Results

Work on Subtask 4 was descoped as it directly related to preparing for a
Team X session. The customer did not express interest in holding a Team
X session in the near future.

***Analysis of Mission Design and Implementation***

**Subtask 5: Team X study to develop full mission concept design and
cost estimate.**

The options outlined in subtask 5 focus upon demonstrating a viable
mission and spacecraft design. The options outline conducting an
Advanced Project Design Team (Team X) study to develop a rough full
mission concept and cost estimate. The two options presented here
describe both a full concept study with a cost estimate and a minimal
study that outlines a full mission concept with no cost estimate. The
work outlined in this subtask, specifically options 1 and 2, are
additional services that the customer can select if desired. While
results from subtasks 1-4 do not depend upon results from options in
subtask 5, pursuing an option in subtask 5 provides mission and
spacecraft design details that would enhance the credibility of the
concept significantly. If feasibility can be established by doing the
groundwork in subtasks 1-3, viable mission and spacecraft design
concepts could be achieved by having Team X develop an end-to-end
mission design for a t-GT mission consisting of visuals such as the
spacecraft configuration, details on each of the spacecraft subsystems
including propulsion, as well as a cost estimate based upon the
specifics of the mission designed by the team (only for option 1). Work
in subtask 5 will be completed by Team X and will be scheduled near the
end of the study period so that technical information gained by results
from any or all of subtasks 1-4 can be fed directly into the Team X
study if the customer selects the baseline or basic option in subtask 5
(see proposal for description of options.

### Subtask 5 Results 

Should the B612 Foundation decide to award Subtask 5 at later date, JPL
will at that time submit a formal Space Act Agreement Revision and Cost
Estimate for those activities that will be performed by JPL subject to
NASA approval.

### Key Milestones and Products

The subtasks (1-4) outlined in the statement of work were initially to
take place over 5 work months from the initiation of the contract,
February through June. However, delivery of the final report was delayed
due to preparation of this material for the Asteroid, Comets, and
Meteors Conference in mid-July. The customer agreed to a final report
due date of September 1, 2008 at the June 23, 2008 face-to-face meeting
held at JPL.

## *Study Conclusions*

1.  This study has shown that a relatively simple and robust thrust
    control law can keep a gravity tractor spacecraft in close proximity
    to the station-keeping location required to effectively tow an
    irregularly shaped, rotating near-Earth asteroid. For our test case,
    the spacecraft could be kept within a 20 x 50 x 50 meter box at a
    nominal distance of 155 meters from the asteroid's center-of-mass
    for a total translational ΔV monthly cost of 40 - 45 m/s with a
    corresponding monthly fuel consumption of only \~ 1.4 kg.

2.  This modest fuel consumption suggests that for a mission with
    similar requirements to our baseline, SEP propulsion may not be
    required; chemical options may be sufficient.

3.  About 48% of the expended ΔV was used to pull the asteroid (this was
    the gravity tractor efficiency).

4.  For our simulated test case, where the tractoring spacecraft was
    about 1000 kg, the total ΔV imparted to the asteroid in one month
    was about 7 µm/s.

5.  The furthest extension of the rotating NEA was \~117 meters.

6.  The 2049 impact probability was essentially at 100% prior to the
    time of the t-GT spacecraft arrival in 2026.

7.  The asteroid's pre-deflection trajectory was determined to a 1-sigma
    value of about 400 km in the semi-major axis of the b-plane
    uncertainty ellipse at the Earth encounter in 2049. This level was
    reached after only eight days of tracking the spacecraft once it
    arrived at the asteroid in 2028. This indicates that the 2049 Earth
    impact location can be predicted with comparable precision.

8.  A 140-m sized, coherent NEA could be deflected from an Earth impact
    via a 1150 kg kinetic impactor striking the NEA in 2028, some 21
    years in advance of the potential Earth impact in 2049.

9.  A 4.00 mm/s ΔV in 2028 produced an asteroid deflection of 80,000 km
    in 2049, more than enough to prevent impact, but a 4.69 mm/s ΔV in
    2028 would move the asteroid trajectory into the 6:5 keyhole for a
    subsequent impact in 2054. Our simulations assumed that the
    rendezvous spacecraft (t-GT) would be used as a gravity tractor to
    tow the asteroid an amount sufficient to miss this keyhole in 2049.

10. Following the kinetic impact deflection, the asteroid's orbit was
    determined to its pre-impact levels after only one day of tracking.
    The ellipse semi-major axis in the 2049 impact plane was determined
    to an accuracy of 125 km after three weeks of tracking in 2028.

11. Following the completion of tractoring in early 2033, the asteroid's
    orbit was determined to its pre-tractoring accuracy levels after
    only one day of tracking. The uncertainty ellipse semi-major axis
    knowledge in 2049 reached a floor value of about 260 km (1 sigma).

12. In 200 days of tractoring, starting in mid-2032, the asteroid's
    uncertainty region could be moved completely off the keyhole with
    650 km/260 km = 2.5 sigma level of confidence. This corresponds to
    an upper bound for the impact probability of 2%. Tractoring could be
    continued after the 200 days to reach higher confidence levels of
    avoiding the keyhole. For example, simple scaling indicates that one
    year of tractoring should effectively clear the keyhole at the 4.6
    sigma confidence level (impact probability less than10^-5^).

13. The 6:5 keyhole in 2049 is only 6 km wide and the other six keyholes
    are similarly narrow, and the likelihood of a passage through any of
    the keyholes is only \~ 0.02%. But, due to fairly large
    uncertainties in the asteroid's 2049 ephemeris positions (\~260 km,
    1-sigma) even after the spacecraft tracking that would follow the
    July 2028 impactor, the likelihood of having to worry about a
    keyhole passage is \~100 times larger, or a few percent.

14. Because of intervening approaches to Earth and Venus, the dynamics
    of our test case was very non-linear and the optimal time for
    tractoring is [not]{.underline} as soon as possible after
    confirmation that passage through the 6:5 resonant return keyhole is
    possible. The optimal time for tractoring would be around 2037 when
    200 days of tractoring provides \~1500 km motion on the 2049 impact
    plane -- more than enough to ensure the NEA's 3σ uncertainty ellipse
    would be moved off the relatively tiny keyhole. Other impactor
    cases, however, could be very different.

15. The one lunar distance encounter in 2046, three years prior to the
    primary impact in 2049, serves to amplify any deflections obtained
    before that time, but similarly amplifies the mapped uncertainty.
    The net effect of the close approach is modest: the gravity tractor
    can achieve the necessary deflection even without that close
    approach.

16. A gravity tractor could be useful for the possible case in which a
    primary deflection technique such as a kinetic impactor happens to
    move the asteroid trajectory into a keyhole; the gravity tractor
    could shift the asteroid's trajectory enough to miss a secondary
    impact keyhole. At the same time, tracking of the gravity tractor
    spacecraft could provide precision orbit information for the
    asteroid before and after the primary deflection attempt and after
    the gravity tractor trim maneuver.

17. While the gravity tractor in our simulation example was a viable
    method for towing the asteroid away from the 2049 keyhole, and hence
    avoiding a 2054 Earth collision, there might be other impacting
    scenarios for which it would not be viable. Each threat scenario
    would have to be analyzed individually to determine whether a
    gravity tractor could be used to move an asteroid trajectory away
    from a keyhole.

Additional general study conclusions include the following:

-   The most threatening NEAs are those on Earth similar orbits.

-   Simulations show that most actual Earth impactor discoveries surpass
    99% impact probability very early in the second optical apparition
    -- or after optical and radar data are obtained during the discovery
    apparition.

-   The primary deflection techniques (e.g., kinetic energy impactor)
    provide relatively uncertain amounts of deflection (e.g., the
    momentum multiplier β is unknown)

-   Secondary impact possibilities (keyholes) must be carefully examined
    for each specific case.

-   Determining potential keyholes during Earth encounters and
    determining optimal times for tractoring to avoid a keyhole passage
    requires fully perturbed, non-linear numerical analysis (two-body
    analyses do not suffice).

-   The combination of radiometric tracking of a nearby spacecraft with
    optical imaging of the asteroid from the spacecraft is sufficient to
    significantly improve knowledge of an asteroid's orbit. It is not
    necessary to place a transponder on the surface of the asteroid to
    achieve precise asteroid tracking.

-   The asteroid orbit accuracy improvements provided by the spacecraft
    range from factors of 2 to 5 over the knowledge which can be
    obtained using only Earth-based observations of the asteroid. The
    size of the improvement is dependent on the relative viewing
    geometry and hence the time period over which the spacecraft is
    tracked.

-   The amount of time it takes to realize these improvements in the
    knowledge of the asteroid's ephemeris is measured in days to weeks.
    A spacecraft need not be in place for months or years for the
    improvements to take place.

-   A close flyby, such as the one that occurs in 2046 for this scenario
    can magnify the asteroid's position uncertainty for subsequent
    flybys by a large factor.

# References

Broschart, S.B., and D.J. Scheeres (2005), "Control of Hovering
Spacecraft near Small-Bodies: Application to Asteroid 25143 Itokawa",
*Journal of Guidance, Control, and Dynamics*, Vol.28, No. 2, pp.
343-354.

Broschart, S.B., and D.J. Scheeres (2007), "On the Implementation of
Spacecraft Hovering under Reduced-order Dead-band Control", paper
presented at the 2007 AAS/AIAA Astrodynamics Specialists Meeting,
Mackinaw Island, MI, August 2007, AAS Paper 07-397.

Chesley, S., and P. Chodas (2008), "Gravity Tractor Study Target
Selection", memo to the Gravity Tractor Study team, March 7, 2008.

Dawn spacecraft website (2008),
<http://dawn.jpl.nasa.gov/mission/ion_prop.asp>, retrieved on

August 26, 2008.

R. Gaskell, O. Barnoin-Jha, D. Scheeres, T. Mukai, N. Hirata, S. Abe, J.
Saito, M. Ishiguro, T. Kubota, T. Hashimoto, J. Kawaguchi, M. Yoshikawa,
K. Shirakawa, T. Kominato, "Landmark Navigation Studies and Target
Characterization in the Hayabusa Encounter with Itokawa", paper
presented at the 2006 AAS/AIAA Astrodynamics Specialist Conference,
Keystone, CO, August 2006, AAS Paper 06-6660.

Kominato, T., M. Matsuoka, M. Uo, T. Hashimoto, and J. Kawaguchi (2006),
"Optical Hybrid Navigation in Hayabusa -- Approach, Station Keeping &
Hovering", presented at the AAS/AIAA Spaceflight Mechanics Meeting,
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Lu, E.T., and S.G. Love (2005), "A Gravitational Tractor for Towing
Asteroids", *Nature,* Vol. 438, No. 2., pp. 177-178.

Ostro, S.J., L.A.M. Benner, C. Magri, J.D. Giorgini, R. Rose, R.F.
Jurgens, D.K. Yeomans, A.A. Hine, M.C. Nolan, D.J. Scheeres, S.B.
Broschart, M. Kaasalainen, and J. Margot (2005), "Radar Observations of
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Scheeres, D.J., and F. Marzari (2002), "Spacecraft Dynamics in the
Vicinity of a Comet", *Journal of the Astronautical Sciences*, Vol. 50,
No. 1, pp. 35-52.

Werner, R.A., and D.J. Scheeres (1997), \"Exterior Gravitation of a
Polyhedron Derived and Compared with Harmonic and Mascon Gravitation
Representations of Asteroid 4769 Castalia,\" Celestial Mechanics and
Dynamical Astronomy Vol 65, pp.313-44.

Valsecchi , G.B., A. Milani, G.F. Gronchi, S.R. Chesley.(2003).
"Resonant Returns to Close Approaches: Analytic Theory." *Astronomy and
Astrophysics*, Vol408, pp. 1179-1196.

[^1]: Radius of sphere of equivalent volume

[^2]: See Ostro et al. (2005).

[^3]: Note that attitude dynamics were not simulated here. We have
    assumed that an attitude control system exists on the spacecraft
    that would be capable of maintaining the desired attitude for
    gravity tractoring. The desired attitude remains aligned with the
    velocity vector of the asteroid, so it changes quite slowly over
    time.

[^4]: Technically, this exact position is generally not the optimal
    tractoring position when the asteroid has a non-spherical gravity
    distribution. However, for the purpose of argument and
    demonstration, it is close enough.