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Earth's Temporarily-Captured Natural
Satellites { The First Step on the Ladder to
Asteroid Resources
Mikael Granvik
Department of Physics, University of Helsinki, Finland
Robert Jedicke and Bryce Bolin
Institute for Astronomy, University of Hawaii, USA
Monique Chyba and Geo� Patterson
Department of Mathematics, University of Hawaii, USA
Gautier Picot
Department of Mechanical Engineering, University of Hawaii, USA
November 2, 2012
Abstract
The Earth is surrounded by a cloud of small asteroids that have
been temporarily captured from the near-Earth-object population into
Earth-centric orbits. These temporarily-captured Earth orbiters (TCOs)
form a natural �rst target for any project that aims to take advantage
of material resources available in asteroids. We review our current
knowledge of the characteristics of the TCO population and their de-
tectability. We also discuss the feasibility of a space mission to a TCO
in terms of possible transfer orbits and highlight some of the remaining
challenges.
1
1 Introduction
For the purpose of utilizing resources available in asteroids it is important to�rst carry out accurate remote prospecting. Detailed mineralogies of aster-oids can in some cases be derived from spectrometric observations (Ga�eyet al. 2002) but in general it is not straightforward to link meteorite types andasteroid classes. It was, for example, assumed for a long time that M-classasteroids are primarily made of Fe-Ni metal. It was later realized that theircomposition is more complex and contains substantial amounts of silicates(see, e.g., Ockert-Bell et al. 2010). The mineralogy of the two major asteroidtaxonomical classes, the S and C, is thought to be understood.
Spectrometric techniques for understanding asteroid mineralogies can bevalidated by comparing telescopic observations of an object with the laboratory-de�ned mineralogy of meteorites from the same object. This type of val-idation is a prime motivation for asteroid sample-return missions such asJAXA's successful Hayabusa mission and the planned Hayabusa II, OSIRIS-REx (NASA), and MarcoPolo-R (ESA) missions. But it will take decadesand substantial amounts of funding before a large-enough set of samples havebeen obtained to calibrate a meaningful fraction of all asteroid taxonomicalclasses. Fortunately, there are cheaper alternatives.
In 2008, a small asteroid, 2008 TC3, was discovered on a trajectory thatwould lead to a collision with the Earth some 22 hours later. Astrometric,photometric and spectrometric follow-up observations using ground-basedtelescopes were scheduled and executed in the time before impact. The as-teroid entered Earth's atmosphere over northern Sudanese desert and me-teorites corresponding to the object were later found in the region. It waslater established that the asteroid's F taxonomic class corresponds to darkcarbon-rich anomalous ureilites, initially thought to originate in S-class as-teroids (Jenniskens et al. 2009). Due to the fragility of the material it didnot exist in meteorite collections prior to the 2008 event.
The rate of discovery of Earth-impacting small asteroids will increase asthe capabilities of telescopic surveys improve but events similar to 2008 TC3
will nevertheless remain relatively rare. First of all the object needs to belarge enough not to completely vaporize in the atmosphere upon impact.Let us assume that 2008 TC3, with an e�ective diameter of about a fewmeters, de�nes the lower size limit for objects that can produce macroscopicmeteorites that survive the passage through the Earth's atmosphere. Objectsof this size impact the Earth a few times every year (Brown et al. 2002).
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About 70% of all meteorites will end up in oceans and a substantial fractionof the remaining 30% will occur in areas that are di�cult to reach or otherwisenot well suited for meteorite collection such as rain forests, taiga, mountains,etc. Thus, an event like 2008 TC3 is not likely to occur more than onceevery few years even if future surveys could discover all such objects beforethey enter the Earth's atmosphere. Finally, a major drawback of waitingfor a small asteroid impact is that we cannot choose the taxonomic class ofthe impactor | if we simply wait for impactors it require several decadesto calibrate a reasonable fraction of all taxonomic classes. We suggest thatthe calibration rate can be improved because a suitable source of calibrationtargets can be found in Earth orbit.
Granvik et al. (2012a) predict that the Earth is surrounded by a cloud ofsmall temporarily-captured asteroids. These temporarily-captured orbiters(TCOs) originate in the near-Earth-object (NEO) population and are tem-porarily captured by the gravity-well formed by the Earth and Moon. Theypredict that the largest object in orbit around Earth at any given moment(other than the Moon) is about 1{2 meters in diameter (Sect. 2). The num-ber of particles is inversely proportional to the size such that there are about1,000 0.1-meter-diameter TCOs in orbit around Earth at any given time.
Only one TCO, 2006 RH120, has ever been discovered but recent resultssuggest that we will start discovering more as survey technology improves(Bolin et al. 2013, Sect. 3). When this happens, TCOs will serve as easily-accessible calibration targets for remote prospecting (Sect. 4). Analyzingthe mineralogy of objects of di�erent taxonomic classes would also alloweconomically sound choices to be made when selecting targets for miningmissions. TCOs could also be utilized as test beds for the technology to bedeveloped for mining operations on asteroids such as high-precision automaticnavigation and resource mapping using remote sensing technology. Studyingtheir interior structure would increase our knowledge of the interior structureof asteroids in general. It has, for instance, recently been suggested that evensmall asteroids|previously assumed to be monolithic|may be rubble pileskept together by cohesive forces (D. Scheeres, private communication).
3
2 Predicted population characteristics for Earth's
temporarily-captured natural satellites
Granvik et al. (2012a) compute the TCO size-frequency distribution (SFD)using a two-step process. First, the capture probability is estimated as afunction of the NEOs' heliocentric orbital elements. Second, the captureprobability is multiplied by the NEO orbital element distribution and scaledusing the best available NEO SFD.
The capture probability as a function of heliocentric semimajor axis ah,eccentricity eh and inclination ih was computed by integrating 10 milliontest particles through the Earth-Moon system (EMS) that have a reasonablepossibility of being captured. A particle is classi�ed as a TCO if it makes oneor more revolutions around Earth in a coordinate system co-rotating with theSun while being energetically bound to Earth and within three Hill radii ofthe Earth's center.
Particles that will become TCOs have speeds of less than 2:2 km s�1 priorto capture at a distance of 4{5 Hill radii from the Earth. The pre-captureTCOs' heliocentric orbits are extremely Earth-like with a � 1AU, e � 0 andi � 0 (Fig. 1).
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Figure 1: TCO capture probability as a function of pre-capture heliocentricorbital elements. Modi�ed from Granvik et al. (2012a).
Once captured, typically through the Sun-Earth L1 or L2 point, TCOs spendmost of their time between 1 and 10 lunar distances from the Earth. TheTCOs' geocentric speeds are typically . 1:5 km s�1 but reach Earth escapespeed during close Earth encounters (Fig. 2).
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Figure 2: TCO residence time as a function of geocentric distance and speed(rgandvg respectively). The implication is that TCOs will be di�cult todetect from the ground because they are moving fast when they are close toEarth and bright. Modi�ed from Granvik et al. (2012a).
TCOs are captured for 286 � 18(rms) days and make 2:88 � 0:82(rms)revolutions around the Earth during this time. Both distributions have verylong tails: the longest simulated capture event lasted about 900 years duringwhich the particle made almost 15,000 revolutions around Earth.
Figure 3 shows the resulting TCO SFD assuming Bottke et al.'s (2002)NEO orbit distribution and Brown et al.'s (2002) Earth-impactor SFD. Thelatter is generally thought to accurately re ect the SFD of non-impactingsmall NEOs.
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Figure 3: The cumulative steady-state TCO SFD based on Bottke et al.'s(2002) NEO orbit distribution and Brown et al.'s (2002) NEO SFD. Themaximum size at which at least one object is captured at any time is H � 32(or a diameter of approximately 1{2m). The frequency of capturing a TCOwith H � 30 is about once every decade. The conversion from H magnitudeto diameter assumes a geometric albedo of 0.15. The line width re ects theuncertainty in the TCO SFD. Modi�ed from Granvik et al. (2012a).
Most of the uncertainty in the TCO SFD estimate is due to the underlyingNEO SFD. Granvik et al. (2012a) prefer the NEO SFD by Brown et al. (2002)because it was measured for objects in the relevant size range and is consistentwith the occurrence of a 2006 RH120-like event once every decade. A steeperNEO SFD such as the one by Rabinowitz et al. (2000) can nevertheless notbe ruled out. Brown et al.'s (2002) SFD explains the observed TCO SFDassuming that the survey's detection e�ciency for 2-meter-diameter TCOs is100%. Reducing the assumed detection e�ciency increases the steady-statenumber of TCOs at every size.
NASA's WISE spacecraft detected more low-inclination Atens than pre-
7
dicted by the Bottke et al. (2002) NEO model predicts (Mainzer et al. 2012)|the objects most likely to be captured by the EMS. The reason appears to bethat too large an integration step was used in Bottke et al.'s (2002) residence-time integrations (Greenstreet et al. 2012). There is still some inconsistencyin the inclination distribution even after accounting for the integration ac-curacy (S. Greenstreet, private communication). The likely source for theresidual discrepancy is the uniform inclination distribution used for the initialconditions by both Bottke et al. (2002) and Greenstreet et al. (2012). In re-ality the inclination distribution is skewed towards low inclinations (Granviket al. 2012b). A new NEO model currently in development will soon shedlight on this issue.
Furthermore, the integrations for the capture probability only take intoaccount gravity. Future work will need to account for non-gravitational forcessuch as atmospheric drag and solar radiation pressure that will a�ect theTCO SFD.
The uncertainties in the NEO orbit distribution and SFD suggests that-Granvik et al.'s (2012a) TCO SFD may be interpreted as a lower limit.
3 Detecting Earth's temporarily-captured nat-
ural satellites
3.1 Detectability
Detecting a 1{2-meter-diameter TCO is not easy. There are about two million1{2 m diameter NEOs that pass within one lunar distance of Earth eachyear. Yet only 13 NEOs with 30 < H < 33 are recorded in the MPC NEOcatalogue as of 29 Oct 2012 (i.e., those corresponding roughly to about 1{2 mdiameter). Given that modern CCD asteroid surveys have been operatingfor about two decades it seems that the annual probability of detecting a1{2 m scale NEO within one lunar distance is about 10�7. Thus, it maybe somewhat surprising that the few meter diameter TCO 2006 RH120 wasdetected at all. However, even though larger TCOs are considerably morerare they are also much easier to detect. Even so, all else being equal, thediscovery of a �2 m diameter object that becomes a TCO is exceedinglyunlikely. Thus, the �nal contributing factor to the discovery of 2006 RH120
is that TCOs approach Earth slowly and remain in orbit for relatively longtime intervals compared to close approach ybys of objects of the same size.
8
Figure 4: Normalized TCO sky-plane residence distribution at time of cap-ture. There are no constraints on the TCO's apparent magnitude, distanceor rate of motion.
Thus, TCOs are more likely to be identi�ed because there is more time todiscover them.
Bolin et al. (2013) studied several modes of detecting TCOs while inEarth orbit including 1) optical ground-based a) wide area and b) targetedsurveys 2) infrared space-based surveys and 3) radar. Considering that thereare currently no infrared space-based surveys we ignore discussion of thatopportunity here. Furthermore, considering that we are interested here indetecting TCOs with a maximum amount of time available for a spacecraftmission we concentrate on options 1a and 3 that both attempt to detect theTCOs close to their time of capture when they are passing near L2.
Figure 12 of Granvik et al. (2012a) suggests that at the moment whenTCOs become energetically bound to the EMS they are all located at roughlythe same distance, moving at roughly the same speed, and in roughly thesame direction. Indeed, Bolin et al. (2013) show that they are at a geocentricdistance of 6:3�1:4(rms) lunar distances (LD) moving at 0:6�0:1(rms) km/sat a position angle of 94� � 25�(rms). At 6.3 LD and 10� from opposition1m/0.25m diameter objects have apparent V magnitudes of �24.7/27.7 |the largest objects being barely detectable with the largest aperture tele-scopes if the objects are not trailed too much. Figure 4 shows the sky-planedistribution of TCOs at the moment of capture. The distribution on the leftand right side of the �gure is in the direction of the Sun and not detectable inoptical surveys but the distribution towards opposition (center of the �gure)
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Figure 5: Normalized TCO sky-plane residence distribution with no restric-tions on apparent magnitude or rate of motion.
is detectable with a targeted survey with Hyper Suprime-Cam (HSC) on theSubaru telescope (Takada 2010). Since there are � 145 TCOs &25 cm diam-eter in orbit at any time and since their typical lifetime is about 9 monthswe expect that there are about 100 of these objects captured each year nearL2. Thus, a targeted survey near opposition with enough time on a largetelescope could detect the largest TCOs at a rate that might be suitable forspacecraft missions.
Figure 5 shows the unconstrained (by apparent magnitude, rate of motion,distance) TCO sky plane distribution. There are strong enhancements atquadrature where the objects tend to be further from Earth (Bolin et al. 2013)and therefore spend more time. These enhancements might be exploited indetecting asteroids with radar. A 100cm/25cm asteroid can be detected withthe Arecibo radar facility if it has a good enough orbit, approaches within� 4/0:25 LD and is rotating very slowly. Thus, it is in theory possible forthe Arecibo radar system to detect TCOs. The greatest uncertainty in theradar capability is the TCO rotation rate distribution. The rotation ratedistribution of meteoroids in this size range is entirely unconstrained withestimates ranging over orders of magnitude. If the TCOs rotate too quicklythe re ected radar signal is spread over too wide a frequency range, droppingthe returned S=N below the detection threshold.
The most straightforward way to detect TCOs is with existing and futureground-based optical surveys as illustrated in Fig. 6. There is a strong en-
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Figure 6: Skyplane number distribution for TCOs with H < 38, apparentmagnitude V < 20 and rate of motion < 15 deg/day.
hancement towards opposition and along the ecliptic as occurs for most solarsystem small body populations as viewed from Earth. The enhancementstoward quadrature from Fig. 5 have disappeared because the phase anglee�ect and because the objects are more distant in that direction. A surveysystem with the constraints imposed in Fig. 6 would not be e�ective at reg-ularly discovering TCOs but an LSST-like system with a limiting magnitudeof V > 24 might discover 1 TCO/month (Bolin et al. 2013).
In summary, the most likely prospects for detecting TCOs suitable forspacecraft mission targets is probably a concerted targeted survey with alarge aperture telescope and wide �eld camera. Alternatively, it is possiblethat a clever survey strategy from a space-based IR or ground-based radarfacility could be optimized for TCO discovery.
3.2 Collapsing the orbit uncertainty
Once a TCO is detected it is important to understand how much follow-up astrometry is necessary to ensure i) that the object is orbiting Earth ii)to allow continued recovery and physical characterization and iii) that theTCO's orbit is known accurately enough to allow the planning of a spacemission (Sect.4). Recent work by Granvik et al. (2012c) suggest that a TCOorbit is well-constrained after a few consecutive nights of astrometric follow-up observations (Fig. 7).
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Figure 7: Evolution of a synthetic TCO's orbital uncertainty as a functionof increasing observational timespan and number of observations; (top left)3 detections spanning one hour, (top right) 6 detections spanning 25 hours,(bottom left) 9 detections spanning 49 hours, and (bottom right) 12 detec-tions spanning 73 hours. The black curve shows the true orbit in the XYand XZ planes in an ecliptic coordinate system co-rotating with the Sun sothat Earth is always in the center (0,0,0) and the Sun is always at about(1,0,0). The gray shaded area shows the extent of all acceptable trajectoriesand the black dots mark the locations of the synthetic TCO at the times ofobservation. All trajectories are shown for 500 days beyond the observationepoch.
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4 Spacecraft missions to Earth's temporarily-
captured natural satellites
Designing and executing any space mission is extremely ambitious yet historyhas proven that the challenge can be met successfully. There are currentlyeight active missions involving spacecraft navigation beyond Mars but withinour solar system.
Rendezvous missions with asteroids, comets or Earth's temporarily-capturednatural satellites present even greater challenges for the spacecraft than ex-ploring the outer planets. One of the problems is that the small body'sgravitational �eld is minimal and cannot be used in the design of the space-craft trajectory. As a consequence, missions to asteroids and comets mustrely on years of orbit adjustment to match their position and velocity to thetarget.
NASA's Near Earth Asteroid Rendezvous (NEAR) and JAXA's Hayabusa(formerly known as MUSES-C) are completed missions that involved a ren-dezvous with an asteroid. NEAR's target was the large NEO 433 Eros, anirregularly shaped asteroid about 30 km in its longest dimension, while theHayabusa mission targeted the �600 m long NEO 25143 Itokawa. Both mis-sions managed safe landings on their S-class targets.
Spacecraft rendezvous missions with TCOs will provide unique challengesbeyond those of the NEAR and Hayabusa missions. The �9 month averagecapture duration is a major criterion when designing a TCO mission. Theprimary di�culty is probably the non-elliptical nature of the TCOs' trajec-tory as shown in Fig. 7 (note that we often do not refer to their paths as`orbits'). All spacecraft missions to date have targeted objects that were onelliptical orbits so a TCO mission requires new methods for designing thespacecraft intercept. Even if the TCO orbit is known well enough to launchthe spacecraft, the meter-scale targets may not be easy to identify in thespacecrafts imaging sensors. Navigating close to the TCO may require real-time control from ground-based operators and, once close to the TCO, theymay be in a rapid or tumbling rotation state making contact with the objectmore problematic than large asteroids.
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4.1 Low-thrust propulsion transfers
The �rst step in designing a TCO rendezvous missions is to determine if itis possible given the time constraints and the use of a low-thrust propulsionspacecraft such as electro-ionic propulsion engines (Ferrier & Epenoy 2012,Geo�roy et al. 1996). We will focus on the initial stage of the mission, i.e., thetransfer of the spacecraft to the rendezvous location with the satellite, andassume that the spacecraft is parked in a geostationary Earth orbit awaitingdiscovery of a TCO to initiate the mission.
For simplicity, we �rst impose the transfer departing position of the space-craft on the geostationary orbit to be the point located on the line betweenthe center of the Earth and Moon. Clearly we have limited control on thelocation of the spacecraft at the time of detection and a complete study in-cluding any departing position is part of the mission design. The selectionof the rendezvous location on the TCO orbit is also a large component ofthe mission design. Our initial work is based on known results for low-thrust2-dimensional transfers to the Earth-Moon L1 point (Picot 2012). We se-lected the 100 TCOs with the smallest absolute perpendicular coordinate (z)to the Moon's orbital plane at the time they are nearest the Earth-MoonL1 point from the 18,096 TCO trajectories of Granvik et al. (2012a). Thischoice is motivated by the fact that near the selected rendezvous locationthe two-dimensional projection of the TCO's orbit on the Moon's orbitalplane provides a good approximation to the three-dimensional orbit for asigni�cant interval of time. However, the �nal computed transfers are fullythree-dimensional trajectories.
Our numerical computations of low-thrust time-minimal transfers fromthe geostationary orbit to TCOs rely on fundamental mathematical resultsfrom modern optimal control theory. The motion of a spacecraft in the Earth-Moon system can be approximated as a solution of the restricted three-bodyproblem due to the small eccentricity of the Moon's orbit (Szebehely 1967).This model describes the motion of an object with negligible mass under thein uence of the gravitational �elds of two planets revolving circularly aroundtheir center of mass. Control terms are added to the equation of motion torepresent the thrust of the spacecraft that are bounded by the constraintson the engine propulsion power. Taking time as the criterion to minimize,the design of a transfer mission from geostationary orbit to the rendezvous
14
location on the TCO'S orbit can be phrased as an optimal control problem
8<:
_q = F0(q) + F1(q)u1 + F2(q)u2 + F3(q)u3minu(�)2B
R3(0;�)
R tf
t0dt
q(0) 2 Og; q(tf ) = qrend:
: (1)
where q(:) represent the position and velocity of the spacecraft, u(:) is abounded measurable function de�ned on [0; t(u)] � R
+, � is determined bythe constraints on the thrust, Og is the geostationary orbit and qrend is therendezvous point with the TCO.
The Pontryagin Maximum Principle (Pontryagin et al. 1962) gives �rst-order necessary conditions for a transfer from the geostationary orbit to therendezvous location to be time-minimal in optimal control problems. Usingso-called indirect methods in optimal control we numerically identify theexistence of a time-minimal transfer among the set of candidate transfers,called extremal curves (solutions of the Pontryagin Maximum Principle).Extremals can be numerically computed by means of a shooting methodbased on a Newton algorithm. However, Newton algorithms are known tobe very sensitive to the initial guess. The use of a smooth continuationmethod is an e�cient way to overcome this di�culty and provide an accurateinitialization for the shooting method to converge (Bonnard et al. 2011).The local optimality of the computed extremal curves is then veri�ed usingthe condition of second order which is related to the geometric concept ofconjugate time. All our computations are carried out using the Hampathsoftware (Caillau et al. 2012).
The transfer time duration is directly correlated to the engine thrust |the more powerful the thrust the faster we can reach the rendezvous location.This simple physical statement translates mathematically into a faster/easierconvergence of the Newton method. This motivates the use of a continuationmethod on the maximum thrust �. Based on those techniques, and using theplanar time-minimal transfer from the geostationary orbit to the Earth-MoonL1 point computed by Picot (2012) as a reference initial guess, we computea three-dimensional reference extremal corresponding to a maximum thrustof 1N departing from the geostationary orbit to the rendezvous location foreach of the selected TCOs. A discrete continuation method on the parameter� is then used to determine low-thrust time-minimal transfers. Additionally,at each step of the continuation algorithm the �rst conjugate time along everygenerated extremal is computed to ensure that it was locally time-optimal
15
dE(qrend � L1) [LD] trend [days] Tcapt [days] nrev
0.0540 133.3 214.4 1.4137
Table 1: Characteristics of the synthetic TCO used for the simulated ren-dezvous mission. Column 1 gives the Euclidean distance dE(qrend � L1) be-tween qrend and the Earth-Moon L1 point and column 2 gives the time trendin days after initial capture that qrend occurs. Column 3 displays the totalamount of time Tcapt the TCO is captured and column 4 gives the number ofrevolutions nrev it makes around the Earth.
according to the second order condition. The algorithm does not alwaysconverge. In those cases we �rst transit through a two-dimensional transferby modeling the Earth-Moon system using the planar restricted three-bodyproblem where the motion is restricted to the fz = 0g plane. The projec-tion of the TCO on the Moon's orbit plane is calculated at every time stepby taking into account the position of the Moon in its orbit. The resultingtwo-dimensional transfer is expressed as a trajectory in the rotating frameof the planar restricted three-body problem. Then, either we run a contin-uation method on the thrust for the two-dimensional transfer to obtain aninitial guess for the low-thrust three-dimensional transfer, or we �rst designa three-dimensional transfer corresponding to 1N thrust before applying thecontinuation method for a �nal low-thrust three-dimensional transfer.
4.2 Orbital transfer simulations
Our methodology was successful in de�ning orbit transfers for a signi�cantportion of our selected TCO sample. Moreover, some preliminary re�nementon the initialization of the algorithm suggests that the method can be usedto create transfers to the majority of the simulated TCOs. Here we presenta representative simulation of a rendezvous mission from the geostationaryorbit to a TCO. Table 1 provides information about the synthetic TCO inthe simulated rendezvous mission and Table 2 provides information aboutthe simulated mission.
The conjugate times are important to guarantee that our algorithm con-verges and guarantee local time optimization but they do not play a rolein the mission practicality. As would be expected, the transfer time withthe low-thrust propulsion is about �ve times longer than with a 5� morepowerful engine. We can re�ne our search for possible transfers once the
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Thrust [N] tf [days] tc [days] trend � tf [days]1.0 13.5 21.4 119.80.2 62.0 69.5 71.3
Table 2: Characteristics of the simulated TCO spacecraft mission. Column1 gives the available thrust, column 2 gives the transfer time tf , column 3gives the corresponding conjugate time tc, and column 4 gives the di�erencebetween the TCO rendezvous time and the transfer time.
mission's spacecraft capabilities are determined. More generally, we foundthree-dimensional 1N TCO transfer durations were in the range of 10 to 20days (e.g., Fig. 8) while the durations of the three-dimensional 0.2N transferswere between 55 and 81 days (e.g., Fig. 9).
The transfer time tf is less than the time it takes the TCO to evolvefrom its point of capture to the rendezvous point (trend) for a majority ofthe three-dimensional 0.2N transfers. This is crucial from a practical standpoint since it suggests that it may be feasible to detect a TCO enough inadvance to launch a low-thrust time-optimal rendezvous mission.
4.3 Other technological challenges
The technological challenges of an asteroid-return mission to a TCO have notyet been assessed in detail but there are some challenges that are common forall asteroid sample-return missions. For robotic exploration the main issueis that the current positional uncertainty for autonomous navigation is onthe order of tens of meters. For a meter-sized object that uncertainty wouldneed to be reduced by 1{2 orders of magnitude. Of course, in Earth-orbit itmight be possible to implement real-time control for a TCO rendezvous.
As mentioned above, and although observational selection e�ects havenot been ruled out, there appears to be a lack of small asteroids that rotateslowly according to data obtained with the Arecibo and Goldstone planetaryradars (P. Taylor, private communication). Thus, an asteroid sample returnmission would probably need to be equipped with a tool for grabbing and de-spinning fast-spinning objects. E.g., 2008 TC3 was in a state of non-principal-axis rotation with main rotation periods of 49 and 97 seconds (Jenniskenset al. 2009). Analytical calculations have also shown that even meter-classasteroids may be rubble piles rather than monolithic objects (D. Scheeres,private communication). Grabbing and de-spinning a rubble pile is clearly
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)Earth
Moon at rendezvous time
Path of Moon during transfer
L1 at rendezvous time
Path of L1 during transfer
Craft start
Path of craft
Rendezvous point
TCO capture point
Path of TCO
TCO escape point
(a) TCO 3813, 1N, tf=12.6 days (wide view)
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
q1 (LU)
q2 (
LU
)
Earth
Moon at rendezvous time
Path of Moon during transfer
L1 at rendezvous time
Path of L1 during transfer
Craft start
Path of craft
Rendezvous point
Path of TCO
(b) TCO 3813, 1N, tf=12.6 days (close view)
Figure 8: (Top) A wide and (bottom) close up view of a locally time-minimalthree-dimensional transfer with 1N thrust to a TCO in the inertial frame.
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−3
−2
−1
0
1
−1
0
1
2
3
4
5
−2
−1
0
1
q1 (LU)
q2 (LU)
q3 (
LU
)
Earth
Moon at rendezvous time
Path of Moon during transfer
L1 at rendezvous time
Path of L1 during transfer
Craft start
Path of craft
Rendezvous point
TCO capture point
Path of TCO
TCO escape point
(a) TCO 3813, 0.2N, tf=64.2 days (wide view)
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
q1 (LU)
q2 (
LU
)
Earth
Moon at rendezvous time
Path of Moon during transfer
L1 at rendezvous time
Path of L1 during transfer
Craft start
Path of craft
Rendezvous point
Path of TCO
(b) TCO 3813, 0.2N, tf=64.2 days (close view)
Figure 9: As in �g. 8 but with a maximum thrust of 0.2N.
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an even greater challenge.
5 Conclusions
Although the small size of a typical temporarily-captured natural Earth satel-lite would make commercial mining operations unpro�table, we have madethe case that studying this population and sending spacecraft to these ob-jects is a natural �rst step for any project that aims to take advantage of theenergy and material resources available in asteroids.
First of all, the TCO population can be utilized in validating orbit andSFD models of NEOs on Earth-like orbits. Bringing an entire TCO to Earth-based laboratories would allow detailed analysis of its interior structure whichwould allow us to test our current theories of the interior structure of aster-oids. A mineralogical analysis of the object would allow the calibration ofremote-prospecting methods. TCOs also provide small-scale platforms fortesting the technologies that need to be developed such as accurate auto-mated navigation, and asteroid de-spinning or, alternatively, anchoring meth-ods.
Some of the unknowns|such as detectability of TCOs and the existenceof viable orbital transfer paths to the chaotic TCO trajectories|have alreadybeen assessed in a quantitative manner though numerous aspects still remainto be studied in greater detail. Examples of things that can be studied fairlyeasily are the lead time from discovery to the escape of the object fromthe Earth-Moon system and a detailed assessment of the frequency of TCOdiscoveries produced by the LSST.
There are also several technological challenges that need to be solved be-fore launching a space mission to retrieve a TCO. One of the major questionsis whether it is realistic to expect that a mission either be launched withinmonths of a TCO discovery or be parked in, e.g., a geosynchronous orbituntil a suitable object is discovered. The obvious technological challengesalso include the need for a major improvement in the accuracy of automatednavigation and the tools to grab and de-spin rotating bodies.
We remain optimistic that these aspects will be studied and solved in thenot-too-distant future owing to the current interest in space-based resourcesin the scienti�c community, space agencies, commercial entities and, last butnot least, the general public.
Finally, the scienti�c reward of bringing an entire small asteroid to ground-
20
based laboratories is immense and would most likely lead to advances thatwould also bene�t the commercial entities aiming to take advantage of theenergy and material resources in asteroids.
Acknowledgments
MG was funded by grant #137853 from the Academy of Finland. BB andRJ were supported by NASA NEOO grant NNXO8AR22G.
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