Aas 14-276 What Does It Take to Capture an Asteroid?

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AAS 14-276

WHAT DOES IT TAKE TO CAPTURE AN ASTEROID?

A CASE STUDY ON CAPTURING ASTEROID 2006 RH120

Hodei Urrutxua,∗ Daniel J. Scheeres,† Claudio Bombardelli∗, Juan L. Gonzalo∗ and Jes ´usPelaez´ ∗

The population of “temporarily captured asteroids” offers attractive candidates for asteroid retrieval missions. Once captured, these asteroids have lifetimes ranging from a few months up to several years in the vicinity of the Earth.1 One could potentially extend the duration of such temporary capture phases by acting upon the asteroid with slow deflection techniques that conveniently modify their trajectories, in order to allow for an affordable access to an asteroid for in-situ study. In this paper we present a case study on asteroid 2006 RH120, which got temporarily captured in 2006-2007 and is the single known member of this category up to date. We study what it would have taken to prolong its capture and estimate that deflecting the asteroid with 0.27 N for less than 6 months and a total ∆V barely 32 m/s, would have sufficed to extend the capture for over 5 additional years.

MOTIVATION

The growing interest in the study of asteroids, their formation, composition or internal structure, is often hindered by the lack of material samples and the difficulty of measuring certain physical features without in-situ scientific experiments. Hence, the idea of capturing and bringing an asteroid into the vicinity of the Earth has gained popularity in recent years, as proven by many papers attempting to propose asteroid captures,2,3,4,5,6 or most recently the NASA’s Asteroid Retrieval Initiative.7 Such kind of Earth delivery missions would allow an affordable access to asteroids along with a remarkable outcome in their scientific study, the possibility of resource utilization and eventually the development of new exploration technologies for the new-coming era of manned spaceflight. However, according to the Asteroid Retrieval Feasibility Study in Ref.7, accomplishing such a missions comes with a great economical cost; the nominally targeted asteroid is 2008 HU4 of approximately 500 ton mass and 7-8 m diameter, and the idea is to capture it and place it in a high lunar orbit, where it would be energetically in a favourable location to support human exploration beyond cislunar space. Omitting the Earth escape trajectory, heliocentric transfer and rendez-vous with the asteroid, the sole cost of deflecting 2008 HU4 in order to bring it into lunar orbit via a lunar fly-by is estimated in a ∆V of approximately 170 m/s. This arises the question of how much does it really take to bound an asteroid in Earth orbit and whether capturing an asteroid could be done more cheaply by targeting a different kind of asteroid or pursuing a different capture strategy. In this regards, we come to propose a novel conceptual approach to enable an affordable access to asteroids in the Earth vicinity. A special population of small asteroids that become temporarily captured in the Earth-Moon system has recently been discovered. While only a single member of this class of asteroids has been confirmed to date, namely asteroid 2006 RH120, rigorous calculations show that there is a significant flux of these temporary

∗Space Dynamics Group, School of Aeronautical Engineering, Technical University of Madrid (UPM). http://sdg.aero.upm.es/ †Celestial and Spaceﬂight Mechanics Laboratory, Colorado Center for Astrodynamics Research, The University of Colorado at Boulder. http://ccar.colorado.edu/scheeres/Scheeres

1 visitors,1 such that at any given time there are about a dozen half-meter-sized objects and at least two meter- sized objects, and one ∼ 10 meter object would be temporarily captured every 50 years. Once captured, these NEAs can remain bound to the Earth-Moon system for times ranging from a few months up to several years. The mean lifetime of this population is predicted to be 9.5 months, during which time they make almost 3 orbits about the Earth. Hence, the availability of a population of temporarily captured natural satellites of the Earth-Moon system, or “minimoons”, brings into the game new possibilities to provide humanity with access to asteroids in “our own backyard”. The only issue would be for how long we could guarantee them to be bound in the vicinity of the Earth, and if by a convenient modiﬁcation of their orbits we could make them remain close enough to Earth for a longer time, and at what energetic cost or ∆V . This paper contains a numerical study on what it would have taken to extend the temporary capture phase of one such minimoon, namely the asteroid 2006 RH120, during its latest 2006-2007 temporary capture. Though the current study is founded in a past scenario, its results remain representative of the temporarily captured population of asteroids and it serves as a Case Study for any future minimoon of similar characteristics that we might encounter. To support this assessment, the study is also extended to other 9 temporarily captured virtual asteroids.

REPRODUCING THE TEMPORARY CAPTURE OF 2006 RH120

The asteroid 2006 RH120 is a small sized NEA, with a diameter of about 5 meters, that was discovered on 14 September 2006 during its last temporary capture that extended until June 2007. The complexity of the temporary capture of a NEA around the Earth, along with the chaoticity proper of a n- a 1.033271548165328 AU body problem, suggests the use of numerical propagation e 0.02448611507535683 - techniques in order to reproduce the asteroid orbit in both i 0.5953070470775252 deg its heliocentric and geocentric trajectories. However, we Ω 51.14274388312688 deg wanted to find the simplest model that would describe the ω 10.07298372371885 deg dynamics and the outcoming trajectories reliably, hoping M 33.64154746771107 deg that a model as ideal as possible would ease the study by enabling the use of analytical techniques, thus helping us to Table 1: Orbital elements of 2006RH120 at get a better insight in the capture dynamics and how to ex- JD 2456000.5 in heliocentric ecliptic J2000.0 ploit them for our benefit. For that, and taking 2006 RH120 frame, as obtained from JPL 45 ephemeris as testbed, we undertook some numerical experiments in order to draw some conclusions and find out the most con- a 0.9873022869878605 AU venient dynamical setting to balance our interests. e 0.01914526201748496 - For our numerical simulations we have initially consid- i 0.7974471215924634 deg ered the initial conditions for 2006 RH120 given in Table1, Ω 278.6483110626849 deg which are its heliocentric orbital elements on March 14th ω 136.132656110278 deg 2012, after the temporary capture phase. Alternatively, ini- M 74.85730211872003 deg tial conditions were also available at an epoch during the capture, corresponding to February 1st 2007 (see Table2), Table 2: Orbital elements of 2006RH120 at which seem more suitable for our study. Its mass has been JD 2454132.5 in heliocentric ecliptic J2000.0 frame, as obtained from JPL 47 ephemeris estimated assuming a spherical body of 5 meters of diameter and a density ρ = 2000 kg/m3. The very first thing we noticed is that the use of high precision numerical ephemerides is required, since otherwise the temporary capture phase is missed. Any attempt to use simplified ephemerides (analytical ephemerides, circularized orbits, . . . ) would yield the temporary capture phase to disappear. The necessity to use numerical ephemerides has further implications, as the fact that the Jacobi constant is not a constant any more (idem for the Hill-Jacobi constant), as shown in Fig.1. This is very easily understood if we realize that even if the Sun, Earth-Moon Barycenter (EMB) and 2006 RH120 were the only bodies

2 -0.185 -0.85 3.0014 2720 3.0012 -0.19 -0.9 3.001 2700 -0.195 3.0008 -0.95 3.0006 2680 -0.2 3.0004 -0.205 -1 3.0002 2660 3 -0.21 -1.05 2640 2.9998 Jacobi Constant (-) Hill-Jacobi Constant (-) Jacobi Constant (MJ/kg) Constant Jacobi 2.9996 -0.215 Hill-Jacobi Constant (MJ/kg)

2620 2.9994 -1.1 Jacobi Constant (MJ/kg) -0.22 Hill-Jacobi Constant (MJ/kg) Jacobi Constant (-) 2.9992 Hill-Jacobi Constant (-)

2600 2.999 -0.225 -1.15 2000 2002 2004 2006 2008 2010 2012 2006.5 2007 2007.5 2008 Fractional Year Fractional Year

Figure 1: Dimensional and non-dimensional values of the Jacobi constant and the Hill-Jacobi constant for 2006 RH120 during its temporary capture between years 2006 and 2007 considered in our propagation, this wouldn’t be an ideal three-body problem, not even an elliptic three-body problem, because the relative positions of Sun and EMB being given by ephemeris tacitly imply a n-body problem, since the computation of planetary ephemeris inherently considers the perturbations of other third bodies, even if these are being omitted in our problem. Consequently, the Hill constant is not preserved as a constant any longer and proves of little practical use in our case study. Also note that the normalization of the Jacobi and Hill-Jacobi constants is not trivial, since the normalizing magnitudes (distances and angular velocities) also come from ephemerides, so they are not constants but rather vary with time. Consequently, there also seems to be no point in using the Hill equations when 2006 RH120 is inside the Hill sphere of the Earth, since the use of ephemerides invalidates the underlying hypothesis on which the Hill equations are built, and their use results in imprecision in the propagation of the orbit.

0.01 0.01 GMAT Our Propagator

0.005 0.005 Hill Sphere Moon Orbit

0 Hill Sphere 0 Y (AU) Y (AU) Moon Orbit

-0.005 -0.005

JPL 47 Our Propagator -0.01 -0.01 -0.01 0 0.01 -0.01 0 0.01 X (AU) X (AU)

Figure 2: Trajectory of 2006 RH120 during it temporary capture phase. Left: Our numerical propagation is compared to JPL 47 ephemeris, in an Earth centered ecliptic inertial frame. Right: Our numerical propagation is compared to the GMAT solution as integrated from initial conditions of Table2 and plotted in an Earth centered ecliptic synodic frame.

Another very interesting conclusions is the fact that considering the EMB does not give a proper recon- struction of the trajectory, nor the temporary capture phenomenon. Instead, the Earth and Moon should be considered as separate bodies, the position of which must be obtained from precision ephemerides, which agrees with Granvik et al. (Ref.1) whose work “indicates that the presence of the Moon was of critical importance in the capture of 2006 RH120”. According to their statistical study they also claim that “not a single test particle was captured by lowering the geocentric velocity via extremely close lunar ﬂy-by’s”, also,

3 “although the Moon’s orbit is the primary reason for the lack of long-lived temporarily captured objects, it is simultaneously the presence of the Moon that increases the capture probability by allowing faster objects to be captured”. In Fig.3 one can see the different trajectories that arise from considering the EMB instead of the Earth & Moon as separate bodies. As a last remark, we realized that the trajectory

0.01 inside the Earth Hill sphere is extremely sensitive Earth + Moon Earth-Moon Barycenter to the entry trajectory as we enter the Hill sphere coming from the heliocentric phase. If our initial 0.005 Moon Hill Sphere Orbit conditions are given in the future, as taken from Ta- ble1 (year 2012), and we are propagating the trajec- 0 tory backwards in time until the temporary capture Y (AU) phase (year 2006-2007), the heliocentric propaga-

-0.005 tion must be done considering a dynamical model as realistic as possible, in order to ensure that the following trajectory inside the Hill sphere is the right -0.01 one, since propagation error coming from the he- -0.01 0 0.01 X (AU) liocentric arc result in different entry conditions at the Hill sphere, and hence into different geocentric Figure 3: Comparison of the trajectories of 2006 RH120 trajectories during the temporary capture. On the in an Earth centered synodic frame during its tempo- other side, once inside the Hill sphere we have con- rary capture, when propagated using the Earth-Moon Barycenter (EMB) or the Earth and Moon as separate cluded that considering the four-body problem with bodies. Sun, Earth, Moon and 2006 RH120 seems to reproduce the temporary capture with enough reliability, though in trajectories involving very close Earth approaches including J2 might also be reasonable, though omitting it does not introduce noticeable errors.

Finally, we want to note that even if according to Ref.8 “2006 RH 120 is small enough that solar radiation pressure is perturbing its motion perceptibly”, the solar radiation pressure is not taken into account in our simulations, because 1) we veriﬁed that the small error introduced with respect to JPL 47 ephemeris, which do include non-gravitational perturbations, has a negligible impact in the results (see Figure2) and 2) later on we will be proposing the use of a low-thrust deﬂection of the asteroid, so the low-thrust could be adjusted so that it balances the solar radiation pressure force. Hence, we decided to restrain our Full Dynamic Model to a Parameterized Post-Newtonian type model,9 including only gravitational source forces plus relativistic corrections.

So, from a numerical propagation point of view, we Outside Hill sphere Inside Hill sphere have split our propagation in two different kinds of trajectory arcs, depending whether the asteroid is inside or . All 8 Planets + Pluto . Sun + Earth + Moon outside the Earth Hill sphere, and in each case the force . Ceres, Pallas & Vesta . Earth J2 models detailed in Table3 are considered for the propa- . Apparent Sun J2 . Thrust (if applied) gation. The propagation is performed using a Sun cen- . Relativistic Corrections . Thrust (if applied) tered Cowell method in the heliocentric trajectory, and switched to an Earth centered Cowell method inside the Table 3: Description of the different perturba- Hill sphere. In Figure2 we show the goodness of our nu- tions considered by our numerical propagator de- merical propagations as compared to JPL 47 ephemeris pending whether the asteroid is inside or outside and numerical propagations with GMAT software. the Earth Hill sphere.

EXTENDING THE CAPTURE WITH AN IMPULSIVE ∆V

As deﬁned in Ref.1, the two requirements for an asteroid to be temporarily captured are: 1) its planetocentric keplerian energy must be negative, Eplanet < 0, and 2) the planetocentric distance must be less than three Hill radii (approx. 0.03 AU). In such a condition of proximity to the Earth and negative orbital energy, one could expect that applying an

4 instantaneous negative tangential ∆V should provide an even further energy reduction such that the asteroid could remain Earth-bounded for a longer period of time. Eventually, if the applied ∆V is large enough to raise the Jacobi constant beyond C1, so that the zero-velocity curves in the three-body approximation close up at L1 at L2, then the asteroid would mathematically remain captured forever in the vicinity of the Earth, and hence become a permanent satellite of Earth (see Fig.4). This idea is not new, and has already been proposed by Baoyin et al. (Ref. 10), who estimated, as their best results, a total ∆V of 410 m/s to capture asteroid 2009BD.

(a) Before impulsive ∆V (b) After impulsive ∆V

Figure 4: Conceptual schematics on how applying a large enough ∆V could close the zero-velocity curves at L1 and L2 and thus bound the asteroid forever in the vicinity of the Earth. For visual reference, the circle indicates the trajectory of the Moon.

However, the latter approach requires too large a ∆V . Closing up the zero-velocity curves to ensure the permanent capture is indeed mathematically correct, though extremely rigorous for real-life application. The fact is that one does not need to capture the asteroid forever, but instead just for a long enough time, which can be obtained with a much lower ∆V . So, the truth is that even if the zero-velocity curves are not fully closed, but the gaps around the lagrangean points are small enough, then even though mathematically feasible, the escape will not be likely to happen soon, and many years or decades may go by until the adequate three-body geometrical layout is achieved, for the escape to happen.

We have tried to estimate what values of tangential impulsive ∆V would take to deflect asteroid 2006 RH120 enough to guarantee that its temporary capture is prolonged sufficiently to ensure a minimum permanence in the Hill sphere, so that robotic or manned missions to the asteroid could be conceived. An impulsive ∆V is most efficiently applied if its di- rection of application is tangential to the orbit and applied Perigee Julian Date Calendar Date at a perigee passage. In the case of 2006 RH120, the aster- Passage oid goes through up to three perigee passages during its 2006-2007 temporary capture, as listed in table4. #1 2454104.251673 2007 January 03 So, the idea was to perform a numerical exploration of #2 2454185.021830 2007 March 25 what values of tangential impulsive ∆V would prolong #3 2454265.729196 2007 June 14 the residence time of 2006 RH120 inside the Hill sphere, i.e. how many extra days would take 2006 RH120 to Table 4: Julian Dates of perigee passages of leave the Hill sphere after the ∆V was applied at a given 2006 RH120 during its 2006-2007 temporary cap- perigee passage. The exploration shows similar results no ture matter which of the three perigee passages was chosen, though slightly better for the (chronologically) third and last perigee passage, where the distance to the Earth is the minimum of all three perigee passages. Re- sults are shown in Figure5 and reveal that a ∆V of about −36 m/s would be enough to extend the temporary capture for several extra years. Thus, this procedure already provides us with an estimate of what it takes to artificially extend the temporary capture of an asteroid.

5 4500 -1 4500 4000 4000 -1.1 3500 3500 -1.2 3000 3000 -1.3 2500 2500

-1.4 2000 2000

1500 1500 -1.5 Extra days in Hill sphere days Extra Extra days in Hill sphere days Extra 1000 1000 -1.6

500 500 -1.7 0 (-) passage perigee 3rd at Constant Hill-Jacobi 0 0 20 40 60 80 100 0 20 40 60 80 100 dV(m/s) dV(m/s)

(a) Sun + EMB (b) Sun + Earth + Moon

Figure 5: Number of extra days that 2006 RH120 remains inside the Hill sphere, after applying an impulsive tangential ∆V at the third perigee passage, as compared to the unmodiﬁed temp. capture. The value of the Hill- Jacobi constant right after applying the ∆V is also shown in the right vertical axis. Results are shown for two different dynamical models, to point out that considering Earth and Moon as separate bodies introduces a great sensitivity on the results. The propagations were performed just until year 2020.

0.006 0.006

0.004 0.004

0.002 0.002

Y (AU) Y 0 (AU) Y 0

-0.002 -0.002

-0.004 -0.004

-0.006 -0.003 0 0.003 0.006 0.009 -0.006 -0.003 0 0.003 0.006 0.009 X (AU) X (AU)

(a) ∆V = −30 m/s (b) ∆V = −35.7 m/s

Figure 6: Trajectories in Earth centered synodic frame after applying a tangential impulsive ∆V to 2006 RH120 at the third perigee passage. The thick line shows the original unaltered trajectory; the thin line shows the trajectory following the ∆V deﬂection.

One must be cautious with these extended temporary capture phases, since some of the outcoming trajectories might have very close Earth encounters (hence very sensitive to perturbations and uncertainties in the initial conditions) or might even be Earth impact trajectories. Figure6 shows trajectories arising from different values of ∆V applied to 2006 RH120 at its third perigee passage.

EXTENDING THE CAPTURE WITH LOW-THRUST INSTEAD

In the previous section we obtained a ﬁrst estimate of what it takes, in terms of impulsive ∆V , to extend the temporary capture time of 2006 RH120 in the vicinity of the Earth. However, to give such a massive asteroid (∼ 130 tons) an instantaneous ∆V of over 36 m/s might seem pretentious or technologically unre- alistic. Therefore, the next logical step is to study how those estimates vary when the total ∆V is not applied

6 impulsively, but instead distributed throughout a time span ∆T , as a low-thrust acceleration. The values of ∆T and ∆V are related to each other through the mass of the asteroid, mast, and the intensity of the thrust, F , by the relation m · ∆V ∆T = ast . F Also, if the ∆V is not impulsive, it is unclear whether the low-thrust acceleration should begin be- Periapsis fore or after the perigee passage, and how much Passage (JD ) p Hill Sphere before or after. Therefore, we introduce a new pa- Initial Conditions Original rameter, an, which measures the time when we be- (JD0) Trajectory gin applying the low-thrust acceleration, with re- #2 spect to the original or unmodified third perigee pas- #1 sage date, JDp, which corresponds to the epoch #3 JDp + an JD 2454265.729196. Thus, if an < 0, it defines how JDp + an + ∆T many days before perigee passage we start the low- Deflected thrust, so it stands for the anticipation to perigee Trajectory passage of the thrust start date. Else, if an > 0, it de- Figure 7: Sketch describing the numerical integration fines how many days after perigee passage we start procedure carried out, splitting the orbit propagation in the low-thrust, so it stands for the delay to perigee 3 segments passage of the thrust start. For convenience, we are going to measure both, ∆T and an in days. From a numerical implementation point of view, we have to split our orbit in several segments or arcs; some are thrusted and others are not, and depending on the value of an, we may need to integrate some segments backwards in time, and others forward in time. Therefore, we have structured our propagation scheme by splitting it in 3 segments, as detailed in Figure7. For convenience, we pick up the initial conditions for 2006 RH120 from Table2, which are given at the temporary capture phase, at epoch JD0. Hence, the first trajectory segment is propagated, in a geocentric frame, from date JD0 until JDp + an. Then, we begin applying the low-thrust and propagate the segment #2 forward in time for ∆T days. At that time, we stop the low-thrust and Figure 8: Results of three-dimensional grid search continue propagating forward the third trajectory in parameters an (anticipation to perigee passage), ∆T segment until leaving the Hill sphere. As we know (thrust time) and Total ∆V . The results are colored ac- when 2006 RH120 left the Hill sphere in its origi- cording to how many extra days the resulting trajecto- nal unmodified trajectory, and now we can obtain ries remain inside the Hill sphere, compared to the origi- the date when it leaves the Hill sphere after having nal unmodified trajectory (∆V = 0). Propagations were applied a low-thrust arc, then one can check if the carried out until no further than year 2020. Collision tra- low-thrust deflection has extended or reduced the jectories have been omitted, as well as those that remain duration of the permanence in the Hill sphere (and in the Hill sphere a shorter time. presumably the temporary capture phase as well) of 2006 RH120.

For the proper implementation of this scheme one has to be aware that, depending on the values of an and ∆T , the segment #1 may be a forward (if an > JD0 − JDp) or backward (if an < JD0 − JDp) orbital propagation, the segment #3 may not even exist (if the Hill sphere is crossed during segment #2 instead), and Earth colliding trajectories may arise... So, those special cases must be taken care of. Thus, if we are looking for an analogy with Figure5, but for the low-thrust approach, it turns out that

7 (a) Total ∆V = −45 m/s (b) Total ∆V = −55 m/s

(e) Total ∆V = −85 m/s (f) Total ∆V = −95 m/s

Figure 9: Results of two-dimensional grid searches in parameters an (anticipation to perigee passage) and ∆T (thrust time) for different ﬁxed values of Total ∆V . The results are colored according to how many extra days the resulting trajectories remain inside the Hill sphere, compared to the original unmodiﬁed trajectory (∆V = 0). Propagations were carried out until no further than year 2020. Trajectories lying on the right side of the purple line leave the Hill sphere during the trajectory segment #2 (see Fig.7). Collision trajectories have been omitted, as well as those that remain in the Hill sphere a shorter time.

8 (a) Total ∆V = −45 m/s (b) Total ∆V = −55 m/s

(e) Total ∆V = −85 m/s (f) Total ∆V = −95 m/s

Figure 10: Results of two-dimensional grid searches in parameters an (anticipation to perigee passage) and ∆T (thrust time) for different ﬁxed values of Total ∆V , showing only trajectories that remain inside the Hill sphere for over 2000 extra days, compared to the original unmodiﬁed trajectory (with ∆V = 0). Propagations were carried out until no further than year 2020. Collision trajectories have been omitted.

instead of having a single parameter, ∆V , we now have up to three parameters, namely an, ∆T and Total ∆V . Hence, we pursue a grid search in the three-dimensional domain deﬁned by those three parameters by

9 conveniently discretizing each of them, and try to find for how long the resulting trajectories remain inside the Hill sphere before escaping. In Figure8 we show the results of this grid search, where each point represents an orbit propagation with the given combination of an, ∆T an Total ∆V , colored according to how many extra days the trajectory of 2006 RH120 remains inside the Hill sphere. We can really get more insight in the results of the grid search by looking at two dimensional slices. Most representative are the slices of constant Total ∆V , i.e. the intersection of the three-dimensional cube of Figure8 with horizontal planes at given values of Total ∆V . In Figure9 we show a collection of such slices for different values of the Total ∆V . The overall look of the slices is a fractal-like construction, since though there seems to be some kind of macroscopic structure with areas of extended capture time and isles of reduced capture time, the close ups reveal some chaoticity and scatterness of the long-life trajectories. Not in vain, our working problem inside the Hill sphere is a four-body problem, which is highly sensitive to initial conditions and perturbations. However, the wide spectrum of colors in Figure9 (ranging from 0 to nearly 4500 extra days) may be distracting us from the important issue. The fact is that a trajectory lasting for, let’s say, 2000 extra days (over 5 years) is enough to enable a mission to that asteroid. The difference between a trajectory lasting 2000 extra days and another lasting 4000 or more, may simply be a subtle difference of applying the low-thrust acceleration a few hours later or sooner, or for a little longer or shorter time span, and then amplified through many close Earth fly-by’s, due to the chaotic nature of the dynamics. Hence, it might be more representative just to filter long-life trajectories beyond a given threshold, and color them alike. Doing so with a threshold of 2000 extra days, we come to the results of Figure 10. Here, if one tries to fill in the white blanks among the red dot clusters (which are in reality mostly potential long-life trajectories too, just that they were discarded because they impact with the Earth at some point along their orbit), one can see that there is actually some sort of continuum in the existence of long-life solutions. We can see (Figs. 9a& 10a) that a Total ∆V of −45 m/s is already enough to extended the capture for over 2000 extra days in the Hill sphere. However, the lowermost region of these figures (low values of ∆T ) is of doubtful practical use, since it corresponds to too high values of the low-thrust intensity, F . On the other side, if we raise the value of Total ∆V , we would increase the total impulse but we might get a lower thrust intensity for the same ∆T , so it becomes a trade-off between trying to find the lowest Total ∆V trajectories with the lowest thrust intensity values. We also note (Figs. 9f& 10f) that a higher Total ∆V guarantees a higher amount of long-life orbits for many different combinations of an and ∆T .

CLASSIFYING THE RESULTING LONG-LIFE ORBITS At the light of the results of the preceding section, a reasonable next step is to understand why those combinations of an, ∆T and Total ∆V yield long-life orbits, what these orbits are like and what special features they have; in other words, to try to get some insight in what is the physics that causes those orbits to remain longer than others in the Hill sphere. A ﬁrst try could be to take 2D slices of the grid search, like those in Figure9, and overlay contour lines showing the values of selected orbital elements, and try to see how they relate to the an and ∆T parameters. We have done so in Figure 11, by overlaying the values of the orbital elements e, i, Ω and ω at the end of the low-thrust (end of trajectory segment #2) on a 2D slice corresponding to a Total ∆V of −75 m/s. This procedure may not be the most appropriate to gain insight in the structure of the resulting orbits, but it does highlight some interesting features of the orbit of 2006 RH120. For example, we realize that its orbit around the Earth ends up being retrograde with respect to an ecliptic frame, since i ∈ (90◦, 110◦). Similarly, all orbits lie in the range Ω ∈ (100◦, 140◦) and ω ∈ (60◦, 80◦), and the eccentricity, though variable, falls into the range e ∈ (0.5, 0.8) for long-life orbits. Similar observations can be done for the rest of orbital elements and different values of the Total ∆V . A probably more convenient way to visualize it is just by using scatter plots, where each point represents an orbit propagation with a given combination of parameters an, ∆T and ∆V , and the axes may be any of the orbital elements, as in Figure 12. This approach is maybe more intuitive, and underlines not only what the values of orbital elements are, but also how they relate to each other one by one. This approach is also

10 (a) Eccentricity (b) Inclination (deg)

Figure 11: Results of two-dimensional grid searches in parameters an (anticipation to perigee passages) and ∆T (thrust time) for a Total ∆V of −75 m/s. The results are colored according to how many extra days the resulting trajectories remain inside the Hill sphere, compared to the original unmodiﬁed trajectory (∆V = 0), and contour lines have been overlayed showing the ﬁnal values of some orbital elements (in geocentric ecliptic frame) at the end of the low-thrust arc. Collision trajectories have been omitted, as well as those that remain in the Hill sphere a shorter time.

106 106

104 104

102 102

100 100

98 98

96 96 Inclination (deg) Inclination (deg) 94 94

92 92

90 90

88 88 7.2 7.4 7.6 7.8 8 8.2 8.4 8.6 8.8 0.45 0.5 0.55 0.6 0.65 0.7 0.75 5 Semi−Major Axis (km) x 10 Eccentricity

Figure 12: Relation between orbital elements a, e and i for a slice of the grid search in parameters an and ∆T corresponding to a Total ∆V of −70 m/s. Each point represents an orbital propagation and have been colored (from blue to red) according to how many extra days the resulting trajectories remain inside the Hill sphere. Trajectories were ﬁltered so that only those staying over 2000 extra days and requiring a thrust lower than 1 N are shown.

more functional in the sense that shown trajectories may be ﬁltered according to additional criteria, like for example trajectories with a minimum permanence in the Hill sphere, maximum intensity of the low-thrust, and so on. This way we can discard uninteresting trajectories and study the trend or structure of only the most meaningful ones. For instance, in Figure 12 the relations i(a) and i(e) are shown for a Total ∆V of −70 m/s and excluding trajectories lasting less than 2000 extra days or requiring a thrust over 1 N, which could be realistic mission compliant constraints. One can also obtain statistical information from the grid search, as in Figure 13, where the same points (or trajectories) of Figure 12 have been plotted in histograms instead of scatter plots. This way we can get a quick overall view of the types of orbits that arise for the given Total ∆V , independently of the values or parameters an and ∆T . The histograms with statistical counts conﬁrm more straightforwardly some of the

11 70 20 50

60 40 15 50

40 30 10

Counts 30 Counts Counts 20 20 5 10 10

0 0 0 7 7.5 8 8.5 9 0.45 0.5 0.55 0.6 0.65 0.7 0.75 85 90 95 100 105 5 Semi−Major Axis (km) x 10 Eccentricity Inclination (deg)

20 30 25

25 20 15 20 15 10 15

Counts Counts Counts 10 10 5 5 5

0 0 0 100 105 110 115 120 55 60 65 70 75 80 85 0 50 100 150 200 Long. of Ascd. Node (deg) Arg. of Periapsis (deg) True Anomaly (deg)

Figure 13: Statistical counts of the values of the orbital elements for trajectories of the grid search in parameters an and ∆T corresponding to a Total ∆V of −70 m/s. Trajectories were ﬁltered so that only non-impact trajectories staying over 2000 extra days and requiring a thrust lower than 1 N are considered.

Figure 14: Relation between orbital elements a, e and i for every trajectory in the 3D grid search in parameters an ∈ [−190, 30] days, ∆T ∈]0, 190] days and Total ∆V ∈ [−100, −30] m/s. Data was ﬁltered so that only non- impact trajectories staying over 2000 extra days and requiring a thrust lower than 1 N are shown.

conclusions we could draw from Figures 11& 12 on the range of the values of orbital elements of the orbits, and confirm aspects such as most of the orbits being retrograde or most orbits ending up with semi-major axis on the order of twice the lunar distance. So far we have dealt with two-dimensional data slices from the grid search corresponding to a fixed ∆V , but another advantage of visualizing results by means of scatter plots, is that one can plot data for different values of Total ∆V all at once in the same plots, in order to find more general patterns, rather than patterns that might be particular to a given energy level. Hence, we did this including in our analysis each and every trajectory computed in our whole 3D grid search (Fig.8), for values of Total ∆V ranging from −30 to −100 m/s, and again filtering the results that remain over 2000 extra days inside the Hill sphere and require less than 1 N of thrust. The resulting scatter plots are shown in Figure 14, for the exact same combinations of orbital elements as in Figure 12, to ease a direct comparison. Different orbital elements (and their relationships) can be studied

12 using similar scatter plots, which is deﬁnitely a good exercise to do and interesting conclusions can be drawn by the thorough observation of such plots, though the amount of resulting graphics is quite overwhelming, since one can not only see how elements relate to each other, but also how they relate to the parameters an, ∆T or ∆V , which yields huge amounts of plots.

SEEKING LONG-LIFE ‘TARGET ORBITS’

As we saw in the preceding section, different combinations of the parameters an, ∆T and ∆V extend the temporary capture of 2006RH120 in different measure, and the most attractive orbits are those that can be achieved with the lowest possible thrust, F , and the lowest possible Total ∆V , independently of when the thrust is applied, an. Therefore, our goal is to try to find whether well balanced trade-offs of these two parameters are eligible within the results of our grid search. Also, at the light of what we saw in Figures 11- 14, there seems to be a relationship between the duration of these orbits inside the Hill sphere and their orbital elements at the end of the low-thrust phase, as their values are not arbitrary but rather follow some patterns, and cannot be coincidental that most of the trajectories computed in the grid search gather around determined values of their orbital elements, as is clearly observed from the peaks in the statistical counts of Figure 13. These evidences suggest that reaching a certain final or target orbit at the end of the low-thrust might be a sufficient condition to guarantee a long lifetime before escaping the vicinity of the Earth. The next logical step seems then to try to identify what these plausible target orbits are like, and how we can place the asteroid 2006 RH120 there using a low-thrust deflection.

ZONE A ZONE B ZONE A ZONE B

ZONE C ZONE D ZONE C ZONE D

Figure 15: Results of the grid search for a Total ∆V ∈ [−100, −30] m/s expressed as a function of the parameters an and ∆T . Trajectories were ﬁltered so that only non-impact trajectories staying over 2000 extra days and requiring a thrust lower than 1 N are considered. The trajectories are plotted both according to the Total ∆V [left] and the thrust required [right]. Also, zones A, B, C & D have been drawn, enclosing those regions that have a better balance between low Total ∆V and low thrust.

The probably most straightforward way to locate these target orbits is to do it in the domain of the parameters an, ∆T and ∆V , since those are the parameters that we can control to extend the temporary capture phase. If we plot every single trajectory of the grid search lasting over 2000 days and requiring less than 1 N thrust in a scatter plot with axis an and ∆T , and color each point (i.e. each computed trajectory) according to the Total ∆V and the thrust intensity, as in Figure 15, one can visually detect areas or zones in the map that correspond to adequate trade-offs between a low Total ∆V and a low thrust. Doing so we identiﬁed four promising zones, that we catalog as Zones A to D, respectively, and are conveniently illustrated in Figure 15.

Zone A is approximately bounded within the range of parameters an ∈ [−190, −170] and ∆T ∈ [160, 190], and contains the best balanced orbits with both low ∆V and low thrust, although orbits with these fea-

13 1 350 ZONE A 0.9 ZONE B ZONE C 0.8 300 ZONE D

0.7

0.6 250

0.5

Eccentricity 0.4 200

0.3 Long. of Ascd. Node (deg) ZONE A 0.2 150 ZONE B 0.1 ZONE C ZONE D 0 100 2 4 6 8 10 12 14 2 4 6 8 10 12 14 5 5 Semi−Major Axis (km) x 10 Semi−Major Axis (km) x 10

180 350 ZONE A ZONE A ZONE B ZONE B 160 ZONE C ZONE C ZONE D 300 ZONE D 140

250 120

100 200 Inclination (deg)

80 Long. of Ascd. Node (deg)

150 60

40 100 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Eccentricity Eccentricity

400 400 ZONE A ZONE A ZONE B ZONE B 350 350 ZONE C ZONE C ZONE D ZONE D 300 300

250 250

200 200

150 150 Arg. of Periapsis (deg) Arg. of Periapsis (deg)

100 100

50 50

0 0 0 0.2 0.4 0.6 0.8 1 40 60 80 100 120 140 160 180 Eccentricity Inclination (deg)

Figure 16: Relation between different orbital elements for zones A, B, C & D. Only non-impact trajectories staying over 2000 extra days and requiring a thrust lower than 1 N are shown.

tures seem a little bit scattered inside the Zone. Zone B, enclosed in the range an ∈ [−140, −90] and ∆T ∈ [150, 190], also offers an interesting trade off between thrust and ∆V values, with not as low ∆V as Zone A, but with a considerably less scattered and more compact structure. Zone C, located around an ∈ [−190, −170], ∆T ∈ [80, 140], and to some extent also Zone D, an ∈ [−115, −95], ∆T ∈ [95, 120], do not contain well balanced trade-offs, but do exchange higher thrust values for the beneﬁt of a lower Total ∆V , which makes these zones highly interesting too.

In order to check whether the trajectories lying in Zones A to D yield similar orbits at the end of the

14 low-thrust segment, we used the same approach as in Figure 14, i.e. plotting combinations of their orbital elements, but filtering the data such that only trajectories inside Zones A, B, C and D are shown. In Figure 16 we can see what the final orbital elements are for trajectories inside each zone, appropriately colored to distinguish them. We can observe that the final orbits mostly seem to be grouped together into neighboring orbits, which proves our initial intuition that there are in fact orbits, which we could refer to as ‘target orbits’, that seem to guarantee an extended permanence inside the Hill sphere at a good balance between low ∆V and low thrust. As seen in Figure 16, target orbits in Zone C, which contains the lowest ∆V orbits, and Zone D, seem more compactly grouped or closer to each other, which is a clear sign of robustness and a desirable property for target orbits. Zone B is also kind of compact but elongated, which probably indicates that it should be subdivided in smaller sub-zones in order to get more alike target orbits for all points inside the zone, and eccentricities close to unity may suggest that classical orbital elements are not the best suited set of elements to study this zone. Zone A offers the best trade-off of low ∆V and low thrust, but the orbits within this zone show a considerable sparsity too, so target orbits inside Zone A should be looked up carefully.

Total Δ V (m/s) Thrust (N) Total Δ V (m/s) Thrust (N) 0.6 0.6 92.5 92.5

92 92

91.5 91.5 0.55 0.55 91 91

90.5 90.5 Eccentricity Eccentricity

0.5 0.5 Inclination (deg) Inclination (deg) 90 90

89.5 89.5

0.45 0.45 89 89 7 7.5 8 8.5 7 7.5 8 8.5 7 7.5 8 8.5 7 7.5 8 8.5 5 5 5 5 Semi−Major Axis (km)x 10 Semi−Major Axis (km)x 10 Semi−Major Axis (km)x 10 Semi−Major Axis (km)x 10

−68 −66 −64 −62 −60 −58 −0.95 −0.9 −0.85 −0.8 −68 −66 −64 −62 −60 −58 −0.95 −0.9 −0.85 −0.8

Total Δ V (m/s) Thrust (N) Total Δ V (m/s) Thrust (N) 92.5 92.5 82 82

92 92 80 80

91.5 91.5 78 78

91 91 76 76

90.5 90.5 74 74 Inclination (deg) Inclination (deg)

90 90 Arg. of Periapsis (deg) 72 Arg. of Periapsis (deg) 72

89.5 89.5 70 70

89 89 68 68 0.45 0.5 0.55 0.6 0.65 0.45 0.5 0.55 0.6 0.65 102 104 106 108 102 104 106 108 Eccentricity Eccentricity Long. of Ascd. Node (deg) Long. of Ascd. Node (deg)

−68 −66 −64 −62 −60 −58 −0.95 −0.9 −0.85 −0.8 −68 −66 −64 −62 −60 −58 −0.95 −0.9 −0.85 −0.8

Figure 17: Relationship between orbital elements for target orbits of Zone C, after ﬁltering grid search results of Zone C with the additional constraint |∆V | 6 68 m/s

Following compactness and low ∆V criteria, we look in Zone C for our first target orbits for extending the temporary capture of 2006 RH120. Hence, we filter the results again, taking only trajectories from Zone C with a Total |∆V | 6 68 m/s, as we are interested in minimizing the total impulse and fuel for a hypothetical mission. Figure 17 shows how orbital elements relate for these orbits, for which ∆V ranges from 58 to 68 m/s and the thrust intensity, F , ranges from 0.78 to nearly 1 N. As clearly observable in Figure 19, in Zone C the values of ∆V and F are inversely related, i.e. a lower ∆V implies a higher thrust. Therefore, aiming for the lowest ∆V orbits, we make the constraints yet more stringent forcing |∆V | 6 58 m/s. The result of this final filtering encloses in compact clustered groups when plotted in scatter plots, both in the domain of parameters an and ∆T , as in the domain of the orbital elements at the end of the low-thrust. After carefully surveying

15 Total Δ V (m/s) Thrust (N) Total Δ V (m/s) Thrust (N) 0.8 0.8 106 106

104 104 0.75 0.75 102 102

0.7 0.7 100 100

98 98

Eccentricity 0.65 Eccentricity 0.65 Inclination (deg) Inclination (deg) 96 96 0.6 0.6 94 94

0.55 0.55 92 92 8 8.2 8.4 8.6 8.8 8 8.2 8.4 8.6 8.8 8 8.2 8.4 8.6 8.8 8 8.2 8.4 8.6 8.8 5 5 5 5 Semi−Major Axis (km)x 10 Semi−Major Axis (km)x 10 Semi−Major Axis (km)x 10 Semi−Major Axis (km)x 10

−74 −72 −70 −68 −66 −0.7 −0.65 −0.6 −0.55 −74 −72 −70 −68 −66 −0.7 −0.65 −0.6 −0.55

Total Δ V (m/s) Thrust (N) Total Δ V (m/s) Thrust (N) 106 106 60 60

104 104 59 59 102 102

58 58 100 100

98 98 57 57 Inclination (deg) Inclination (deg)

96 96 Arg. of Periapsis (deg) Arg. of Periapsis (deg) 56 56 94 94

92 92 55 55 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 108 110 112 114 116 108 110 112 114 116 Eccentricity Eccentricity Long. of Ascd. Node (deg) Long. of Ascd. Node (deg)

−74 −72 −70 −68 −66 −0.7 −0.65 −0.6 −0.55 −74 −72 −70 −68 −66 −0.7 −0.65 −0.6 −0.55

Figure 18: Relationship between orbital elements for target orbits of Zone A, after filtering grid search results of Zone A with the additional constraint |∆V | 6 75 m/s these data, one can determine that the plausible target orbits in Zone C have their orbital elements bounded. We provide the initial conditions for a sample of Zone C target orbits, collected in Table5 As mentioned before, Zone C provides the lowest Total Δ V (m/s) Thrust (N) ∆V target orbits, though at moderately high thrust 130 130 levels, close to 1 N. If we were trying to find a bet- 125 125 ter trade-off that could reduce the required thrust 120 120 while keeping reasonable values of the Total ∆V , 115 115 then Zone A is a better candidate where to find bet- 110 110 ter balanced target orbits. We noticed that Zone A 105 105 100 100 orbits showed some kind of sparsity or lack of com- Thrust Time (days) Thrust Time (days) pactness; however, if we filter these orbits further 95 95 90 90 just to focus on the lowest ∆V orbits, we do see 85 85 −190 −185 −180 −175 −170 −190 −185 −180 −175 −170 that these orbits locally correlate quite closely to Anticipation (days) Anticipation (days) each other (see Figure 18), enabling us to find new target orbits. So, under the additional constraints −68 −66 −64 −62 −60 −58 −0.95 −0.9 −0.85 −0.8 |∆V | 6 67.7 m/s and F 6 0.55 N we get nice target orbits whose ∆V is around 10 m/s larger than Figure 19: Orbits of Zone C represented in the domain those of Zone C, but the necessary thrust is only of parameters an (anticipation of thrust start to perigee about half. After further filtering and aiming for the passage) and ∆T (thrust time), after filtering grid search lowest ∆V orbits in this zone, we collect in Table6 results of Zone C with the additional constraint |∆V | 6 a few Zone A target orbits. Note that these target 68 m/s orbits require a ∆V of 66.6 m/s but thrust levels below 0.54 N, which may make them more convenient than Zone C orbits for certain purposes.

16 a ∆T Total Thrust Extra a e i Ω ω ν (days) (days) ∆V (m/s) (N) Days (km) (-) (deg) (deg) (deg) (deg) -188.0 88.6 -58.0 -0.9918 2,892 837,240 0.557 92.29 107.31 77.86 254.83 -187.5 88.6 -58.0 -0.9918 2,090 838,403 0.560 92.29 107.27 77.71 257.06 -187.5 88.8 -58.0 -0.9896 4,555 839,013 0.560 92.29 107.27 77.68 258.08 -187.0 88.8 -58.0 -0.9896 4,555 839,973 0.562 92.29 107.21 77.53 260.48 -189.0 89.0 -58.0 -0.9873 4,555 836,083 0.553 92.29 107.40 78.09 252.46 -187.5 89.0 -58.0 -0.9873 4,555 839,595 0.560 92.28 107.26 77.64 259.13

Table 5: A sample of target orbits with low values of ∆V , extracted from Zone C

a ∆T Total Thrust Extra a e i Ω ω ν (days) (days) ∆V (m/s) (N) Days (km) (-) (deg) (deg) (deg) (deg) -183.0 188.5 -66.6 -0.5353 4,555 866,688 0.657 100.65 111.98 59.27 142.39 -183.0 188.8 -66.6 -0.5344 2,442 867,494 0.659 100.72 112.01 59.21 143.02 -182.7 189.0 -66.7 -0.5347 4,555 868,237 0.664 100.86 112.10 59.08 144.17 -183.4 189.2 -66.7 -0.5341 4,555 867,533 0.658 100.67 111.96 59.22 142.99 -182.4 188.5 -66.8 -0.5369 4,555 867,431 0.664 100.85 112.11 59.12 143.86 -183.3 189.4 -66.8 -0.5343 4,555 867,965 0.661 100.75 112.00 59.15 143.68

Table 6: A sample of target orbits with low values of ∆V and thrust, extracted from Zone A

Still the question stands, whether these target orbits Total Δ V (m/s) Thrust (N) are robust or not, in the sense that if when picking an or- 0.575 0.575 bit with infinitesimally different values of the parameter 0.57 0.57 an, ∆T and ∆V , the outcoming target orbit lies grouped 0.565 0.565 next to the others. We checked the response to this ques- 0.56 0.56 tion to be affirmative, by using a denser grid search in 0.555 0.555 the appropriate domain of parameters, as illustrated in Eccentricity Eccentricity 0.55 0.55 Figure 20, which numerically proves that these target orbits (both from Zones A and C) are not over-sensitive to 0.545 0.545 0.54 0.54 changes in their initial conditions. 8.25 8.3 8.35 8.4 8.45 8.25 8.3 8.35 8.4 8.45 5 5 Semi−Major Axis (km)x 10 Semi−Major Axis (km)x 10 It is also interesting to observe how this orbits evolve −58 −57.95 −57.9 −0.995 −0.99 −0.985 in time. We have traced the evolution of different neighboring target orbits like those of Figure 20 or those on Figure 20: Low-∆V target orbits of Zone C ob- Tables5-6, and we have seen that even if all the orbits tained from a finer grid search to test the robustness initially begin the same way, they slowly diverge as time of the target orbits goes by (see Fig. 21). This is obviously in concordance with the four-body dynamics, since all target orbits have 1 tiny differences in their initial conditions at the end of 0.9 the low-thrust, and this differences become more appar- 0.8 ent with time. However, all these orbits do stay in the 0.7 vicinity of the Earth for at least 2000 extra days, this be- 0.6

ing their common feature. This means that even if we Eccentricity missed our exact planned target orbit, any other neigh- 0.5 boring orbit where we end up would still guarantee the 0.4 desired permanence time in the Hill sphere. 0.3

2007 2008 2009 2010 2011 2012 2013 Despite these results are considerably better (in terms Fractional Year of ∆V ) than other asteroid retrieval attempts proposed in Figure 21: Time evolution of the eccentricity for the literature, these low-thrust deﬂections seem a little 19 random target orbits taken from the Zone C

17 ZONE E ZONE E

F ZONE A ZONE B F ZONE A ZONE B ZONE ZONE

ZONE G ZONE G

ZONE H ZONE H ZONE D ZONE D ZONE C ZONE C

Figure 22: Results of the grid search for a Total ∆V ∈ [−80, −30] m/s expressed as a function of the parameters an and ∆T in the extended domain. Trajectories were ﬁltered so that only non-impact trajectories staying over 4000 extra days and requiring a thrust lower than 1 N are considered. The trajectories are plotted both according to the Total ∆V [left] and the thrust required [right]. Also, zones A, to H have been drawn, enclosing those regions that have a better balance between low Total ∆V and low thrust.

upsetting compared to the lower ∆V values we estimated in a preceding section for impulsive deflection maneuvers (about 36 m/s), and makes us wonder if we are missing something. In fact, one can do better than this. Attending at the structure of Figure 15, we realize that low thrust areas obviously exist in the upper part, for higher ∆T values, plus we observe that low ∆V areas, such as Zones C and D, grow to the left-top regions of the diagram following inclined corridors. This suggests that if we moved to the left (for larger anticipations), low ∆V regions could overlap low thrust regions, thus resulting in even better balanced zones than those we have already found. This quest makes sense intuitively, since deflecting the trajectory earlier on should yield lower ∆V maneuvers, while increasing the ∆T reduces the thrust intensity. Therefore, we extend the grid search domain in the parameters an ∈ [−330, 30] and ∆T ∈ [80, 220], and we are satisfied to find new regions in the domain of the parameters an and ∆V that show much better performances. Actually, we find orbits with ∆V values below 35 m/s (comparable to the impulsive deflection) and thrust values below 0.3 N, that we had been missing out in the initial grid search. The extended grid is illustrated in Figure 22, along with the newly classified zones E to H.

Zone E can be bounded within the range of parameters an ∈ [−290, −265] and ∆T ∈ [170, 220], and offers an interesting and appealing combination of ∆V values below 50 m/s with thrust values below 0.5 N. This zone not only covers a wide range in the domain of parameters, but it is also extremely compact, highly dense, and safe in the sense that any neighboring orbit is a long lifetime orbit. As far as our extended grid search goes, other zones can be identified, containing more scattered orbits but with considerably lower ∆V values, namely zones F, G and H. From the whole grid search, Zone F is notoriously the best balanced zone, offering both the lowest ∆V (below 40 m/s) along with the lowest thrust levels (below 0.3 N). Orbits in this zone can be comprised within the parameter range an ∈ [−320, −310] and ∆T ∈ [165, 215]. Orbits in Zone F have also been successfully checked for robustness. One can observe that aiming for one of these target orbits, extending the temporary capture of 2006 RH120 could have been achieved with a Total ∆V of less than 31 m/s and a thrust below 0.28 N with a deflection beginning 315.5 days before the original third perigee passage (see Table4) and lasting for about 172 days. Extending the grid search even further or using a finer grid size in the domain of the thrust parameters, it is reasonable to assume that even slightly better target orbits might be found. However, we believe the current results are already representative enough, both qualitatively and quantitatively, of the cost of extending the temporary capture of 2006 RH120. Another concern about these solutions is their optimality regarding the selection of the thrust control law.

18 The parametric grid search conducted so far makes two important assumptions about the thrust, namely, that it is of constant magnitude and oriented along the tangent to the orbit. This is a reasonable ﬁrst approximation for the control law; not only it is well known that tangential thrust maximizes the instantaneous rate of energy increase for the two-body problem with constant thrust, but it can also be proven to be the asymptotic solution for maximizing the orbital energy increase during a ﬁxed period of time in the low thrust case.11 Nevertheless, the doubt remains whether a noteworthy improvement can be obtained in our more complex scenario by allowing these control variables to change with time.

To perform a preliminary evaluation of the impact of 0.06 the thrust control law in the optimality of the results, the optimal control problem (OCP) of finding the thrust 0.04 orientation law which minimizes the constant thrust, F , 0.02 needed to cover the thrusted arc of a given target orbit is considered. Note that, since ∆T is fixed, this corresponds 0 (deg) to minimizing ∆V . The only constraints are the state of β −0.02 the orbit at the extremes of the arc, given by their val- −0.04 ues in the original, tangential thrust solution. The OCP is posed using a direct transcription method, converting −0.06 it into a Non-Linear Programing (NLP) subproblem in −0.08 0 10 20 30 40 50 60 70 80 which state and control are represented through their val- time (days) ues at the nodes of a suitable grid. The equations from Figure 23: Control law for the the orientation of dynamics are also discretized, using an adequate numeri- the thrust vector, β, for a Zone C target orbit with cal scheme (in this case, an order 2 Implicit Runge-Kutta) an = −188 days, ∆T = 89 days, ∆V = −58 m/s to express them as a set of non-linear defect constraints at and F = −0.9873 N the grid nodes. The NLP subproblem is then solved using a large-scale interior point algorithm, which takes advantage of the sparsity structure of the defect constraints. The resulting orientation control law for a selected target orbit from Zone C is shown in Fig. 23. As expected, the angle β formed by the thrust vector with respect to the tangent of the orbit remains small. Regarding F and ∆V , this control law yields a negligible improvement, which points out that the target orbits obtained in the grid search are already quasi-optimal. Consequently, taking into account the great increase in computational cost of this kind of optimization, its use is not justified for this study.

CONCLUSIONS

In this paper we have exposed a case study on extending the temporary capture phase of asteroid 2006 RH120, which is the single known member of the population of temporarily captured objects of Earth, up to date. According to our simulations, we conclude that for temporarily captured asteroids, the study shows possible to prolong the duration of their capture phase with a slow deﬂection of their trajectory once the capture is begun, and requiring low to moderate values of ∆V , and low thrust.

More particularly, we have estimated that deﬂecting asteroid 2006 RH120 during its past temporary capture in 2006-2007, with 0.274 N for about 172 days and a total ∆V of 31.2 m/s, would have prolonged the duration of the capture over 5 additional years. In this regards (though not shown in the paper due to typesetting constraints), we have also studied the possibility of deﬂecting 2006 RH120 and other alike virtual asteroids, not just during their temporary capture phase, but also early before the begin of their temporary capture, in order to modify or improve the nature of the capture phase and thus turn it into a better behaved or longer lifetime capture. Our simulations show that for most temporarily captured objects, with this strategy it could be possible to extend their capture phases for several years, with ∆V values smaller than 15 m/s and a thrust intensity smaller than 1 N, assuming same mass and size as 2006 RH120. Hence, assuming our test particles are representative of the population of temporarily captured asteroids, these results suggest that

19 1. It should be possible for many temporarily captured objects to considerably extend the duration of their temporary capture phase with a reasonably low ∆V values and low to moderate thrust levels, by slowly deﬂecting their trajectory during the temporary capture. 2. It should be possible for most temporarily captured objects to considerably extend the duration of their temporary capture phase with low ∆V values and low thrust levels, by slowly deﬂecting their trajectory before the begin of their temporary capture.

ACKNOWLEDGMENT Authors wish to acknowledge the Spanish Ministry of Economy and Competitiveness for their support through the research project “Dynamic Simulation of Complex Space Systems” (AYA2010-18796). Hodei Urrutxua acknowledges the Technical University of Madrid (UPM) for supporting his research through their PhD program. Daniel J. Scheeres acknowledges support at the University of Colorado from NASA grant NNX14AB08G. The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement n◦ 317185 (Stardust). The authors also wish to thank Mikael Granvik, William F. Bottke and Robert Jedicke for their careful reading of the manuscript, constructive feedback, and for facilitating the initial conditions for the temporarily captured test particles from Ref.1.

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