VALIDATION AND APPLICATION OF A ...

(Preprint) AAS 12-228

VALIDATION AND APPLICATION OF A PRELIMINARY TARGET SELECTION ALGORITHM FOR THE DESIGN OF A NEA HOPPING MISSION Michael V. Nayak* and Bogdan Udrea† This paper describes the setup and results of an algorithm employed for preliminary asteroid target selection for a manned multi-asteroid prospecting mission. The algorithm is used to determine the order of transfer between viable targets, keeping delta-V at a minimum, minimizing the computational burden of optimization and maximizing the number of targets visited within the mission timeline. Based on distance from the originating asteroid, inclination change and planetary perturbation effects, the algorithm is used to support a ‘decision-tree’ approach to target selection. Results are validated using Satellite Tool Kit v.9.1’s Design Explorer Optimizer and used to plan a five-asteroid, six-target hopping mission.

INTRODUCTION Focus and purposes of this study The study presented here is focused on the prospecting of Near-Earth Asteroids (NEAs). Using current commercially available technology, the goal of this study is to estimate the feasibility of a human spaceflight (HSF) mission to the maximum number of NEAs possible, before the middle of the 21st century, with the ultimate goal of returning to Low Earth Orbit (LEO) with mined materials of commercial and scientific value. Commercial value. Private space missions would be driven by a cost-versus-benefit model, which would naturally gravitate toward NEAs composed of materials of intrinsic value. The assumption is that with the change from government-led space exploration to private industry initiative, such missions may become interesting for venture capitalists. For example, Kargel 1 offers the history of Aluminum and its alloys as an example of use expansion, and speculates that the same would be true if future missions created an abundance of metals in the groups and periods neighboring Platinum, specifically Ruthenium, Rhodium, Palladium, Osmium, and Iridium, all of which are available on several large NEAs. Kargel1,2 estimates that just one highly-enriched metallic NEA of 1 km diameter could yield up to 400,000 tons of precious metals, which at 1994 market prices holds an estimated value of $5,091 billion US dollars. Such tonnage would represent a ten-fold increase in the current global production rate, but even deflated over 20 years, the

*

Satellite Flight Test Engineer, Space Development and Test Directorate, United States Air Force, Albuquerque, NM. [email protected]. † Assistant Professor, Dept. of Aerospace Engineering, Embry-Riddle Aeronautical University, Daytona Beach, FL

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intrinsic value of the return from a single NEA could be as much as $323 billion1. While mass and vehicle limitations of space-based mining present a significant deterrent to such return tonnage, more general studies3, 4 of the near-Earth space, to include NEAs, seem to imply that a pathfinder mission such as the one proposed in this study would fuel NEAs to be a continued and profitable target for commercial exploitation. Scientific Value. As revealed by the Hayabusa (JAXA) mission, which returned to the Earth’s surface in June 2010 with a capsule containing a sample of the asteroid Itokawa, and the NEAR (NASA) mission, which soft-landed on Eros, the short range imaging and in-situ study of the asteroids opens new doors to our understanding of the fundamental processes that led to the formation of the Solar System itself. Therefore, the scientific value of the proposed prospecting mission is almost as important as its commercial value. It is proposed that high-resolution closerange multispectral mapping of the NEAs and accurate gravitational measurements be an integral part of the mission. Further, some of the samples returned to Earth would be of high value for study by planetary geologists. The architecture defined by this design could be used or adapted by private launch enterprises in the near future for similar missions. Mission Outline This paper focuses on a multiple-NEA trajectory, however a high-level outline of choices made for the major mission hardware is presented for completeness: Launch Vehicle. The mass and volume of subsystems for such a mission, to include living quarters, radiation shielding, electric power, propulsion, EVA equipment and asteroid rendezvous vehicles would preclude construction on Earth and launch in completed form, given current commercially available vehicles. The final assembly, followed by departure, would take place in LEO. Multiple Space-X Falcon 9 and Falcon 9 heavy launches are proposed. Spacecraft Structure. A combination of rigid structures similar to the International Space Station (ISS) modules, inflatable habitats from Beagle Aerospace, and Dragon capsules from SpaceX is desired. Propulsion. Various propulsion and power system options are being traded, to include nuclear thermal options as well as Ad-Astra’s Variable Specific Impulse Magnetoplasma Rocket (VASIMR), specifically the VX-200 or its derivative. According to the Ad-Astra’s mission statement, the VASIMR engine is projected to be in full operational space deployment by 2014 and will provide “primary propulsion for...ready access to space resources, including asteroids and comets”5. A mission timeline is therefore desired by which time either nuclear thermal engines or the VASIMR engine has reached a high level of maturity sufficient for human flight. Electric power. The VX-200’s main hindrance is the large amount of power required, and investigations into two propulsion systems are underway. The first option is Solar Electric Power (SEP), where the power required is generated by photovoltaic panels similar to those of the ISS with battery charge and discharge units from Space Systems Loral and Boeing. The second option is Nuclear Electric Power (NEP), which utilizes nuclear fission reactors and thermal-to-electric energy conversion to generate the electric power needed by VASIMR. Reactors similar to the Los Alamos National Lab’s space nuclear reactor SAFE-400 are proposed. At the time of this writing the SAFE-400 reactor is not in commercial production; however the know-how required for its fabrication, spacecraft integration and safe operation are well understood and somewhat mature.

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Human factors analysis. Such an unprecedented, long-duration deep-space flight requires a thorough evaluation of life support needs. Based on mission goals versus available HSF engineering, physiological and psychological data from US and former-Soviet LEO station missions, an optimal mission duration of 24 months, with a not-to-exceed duration of 36 months has been selected, to balance astronaut health and support issues with maximum mission return 6,7. In addition to equipment and consumable needs, the astronaut schedules will be optimized to balance long travel periods between NEAs with round-the-clock prospecting operations during each rendezvous. Trajectory Design Requirements Private enterprise goals and HSF mission considerations bring several previously unencountered issues into play. This paper’s specific focus is the trajectory design involved to meet these mission requirements. -

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Mission time must be minimized while maximizing return. This is achieved by visiting the maximum possible number of NEAs within the given timeframe (“hopping”). Exposure to radiation must be minimized, eliminating certain epochs and regions of space for travel. Since we do not anticipate that the private space industry will be fully capable of funding, building and supporting such a complex manned mission until at least ten years from this writing, the earliest epoch of departure from LEO was fixed at 01 Jan 2022, with a mission timeline no greater than 36 months. Another necessary constraint was the size of the target asteroid, to maximize the potential for return, and therefore the appetite of private industry to invest in such a mission. It was therefore decided to restrict NEA targets to those with diameter 0.25 km. or greater. This size selection also serves to eliminate NEAs that either do not have characterized orbits or are subject to detectability issues when approaching from the Sun23, both concerns for manned space flight. Although not currently modeled, inertial holding around NEAs is propellant-intensive.

TARGET SELECTION FOR ASTEROID HOPPING Statistical Target Elimination The Jet Propulsion Laboratory’s (JPL) HORIZONS Small Body Database (SBDB) 8 holds information on 574,456 asteroids and 3,118 comets as of this writing, and was the starting point for asteroid target selection. Imposing the diameter restriction of 0.25 km., 2605 matching objects were found. Eliminating comets and asteroid-belt objects left 126 candidate NEAs. To refine such a wide array of possible targets, using the orbital data of these 126 targets from SBDB, a basic algorithm was created with MATLAB to statistically estimate the transfer ΔV. These values of ΔV will not consider phasing of asteroid orbits, which could increase or decrease the transfer ΔV significantly. These values are therefore used only for statistical comparison and trend determination. A Hohmann transfer trajectory and circular orbit of the Earth around the sun is sufficient for such a preliminary analysis. Since we are already assumed to be in LEO, we will need to impart an impulsive ΔV to the spacecraft to exceed its LEO velocity and cause it to fly-by the NEA in question. This is denoted by ΔVdep. Following the analysis of Stacey and Connors9, 10, if escape velocity is denoted by Vinf and the speed of the spacecraft in parabolic orbit by Vpar:

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Vinf 2  VLEO  Vdep   Vpar 2 2

(1)

Normalizing distances to astronomical units (AU) and velocities to that of Earth’s heliocentric travel (29.77 km/sec) 11, we find that the normalized velocity of the spacecraft in LEO is 0.26 for a circular orbit and 0.37 for a parabolic orbit. The escape velocity can also be written in terms of the nodal distance d to the target asteroid:

Vinf 2 

2d 8d  1 1 d 1 d

(2)

The distance d to the outer (negative sign in denominator) node and inner (positive sign) node is defined in terms of the asteroid’s eccentricity e, argument of perihelion ω and semi-major axis a as:

d

a 1  e2 

1 e cos 

(3)

Using Equation (3) in Equation (2) and substituting to Equation (1), the fly-by ΔV can be calculated. To drop into the target NEA’s orbit will involve an inclination change ΔVinc described by Equation (4):

Vin c

2

2 2  2  d  1 2  2a 1  e      cos i d 1  d  a d  d 1  d    

(4)

The total transfer ΔV is the sum of ΔVdep from Equation (1) and ΔVinc from Equation (4).

Figure 1. Scatter plot of NEA Transfer ΔV.

Fig. 1 shows the ΔV variation across the 126 target NEAs, where the lesser of the outer and inner node transfer ΔV is plotted. Though our objective is refinement to a smaller subset of targets, we bias ourselves towards inclusion rather than exclusion. Therefore we eliminate only those NEAs with statistical transfer ΔVs higher than 50 km/sec. Epochs for rendezvous that have potentially unacceptable dynamical properties such as Venus or Mars swing-bys are also eliminated.

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We do not wish to face a rendezvous at a distance too far from the Sun for SEP power reasons, or too far from the Earth in the event of an emergency that necessitates a quick return. Further, Adamo23 suggests that from an HSF standpoint, the best accessibility occurs near the NEA orbit apsis, i.e., we wish to rendezvous as close to aphelion as possible. Therefore we eliminate NEAs with aphelions of Q > 5 AU. Fig. 2 shows a plot of the aphelion of the remaining target NEAs against the statistical transfer ΔV.

Figure 2. NEA Aphelion vs. ΔV.

Figure 3. NEA Period vs. ΔV

Next, all NEAs with periods of greater than 1825 days (5 years) are eliminated, as seen from Fig 3. The final step is to eliminate those NEAs that have an inclination of greater than 10 degrees from the plane of the ecliptic, in accordance with the findings of the NEO HSF Accessible Targets Study23 (NHATS). Table 1 below lists the 50 target NEAs that have an aphelion distance of less than 5 AU, period of less than 5 years, a statistical ΔV of less than 50 km/sec (to either the inner or outer node) and an inclination of less than 10 degrees from the plane of the ecliptic. We shall carry these forward for further consideration. Hopping Algorithm Desirables Nomenclature. Departure from one target, cruising to the next target and coasting with it for prospecting and exploring will be referred to henceforth as a “phase”. The period while in orbit around a target NEA and astronaut operations are progressing on the surface is referred to as an “operations hold”. Hopping trajectories generated by our algorithm are referred to as “itineraries”. At first glance, close approach data from HORIZONS8 seems to make choosing the Phase I target simple. However from that point on, mere intuition or a trial-and-error approach will not suffice to determine successive targets. A given target NEA could be cheap and quick to get to, but have no other targets in its vicinity following the completion of operations, which would lead to a costly return to Earth or an unacceptably long or expensive future rendezvous. In such a case, it would be better to choose a Phase I target that is more expensive to arrive at but has an abundance of nearby targets from Table 1 – this will result in greater mission productivity and overall lesser propellant consumption. Therefore the driving reason behind the creation of our algorithm was to optimize not one leg of the trajectory, but multiple.

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Table 1. Candidate NEAs.

Horizons SPK-ID

Radius (km)

e

a (AU)

i (deg)

Perihelion distance (AU)

Aphelion distance (AU)

Period (years)

887 Alinda

2000887

2.1

0.567316

2.477866

9.356876

1.072134

3.883599

3.900543

1627 Ivar

2001627

4.56

0.396934

1.863103

8.447932

1.123574

2.602632

2.5431

1685 Toro

2001685

1.7

0.435958

1.367259

9.380219

0.771191

1.963326

1.598763

1862 Apollo

2001862

0.75

0.559846

1.470091

6.353509

0.647066

2.293116

1.782479

1943 Anteros

2001943

1.15

0.255846

1.430324

8.704995

1.064381

1.796267

1.710645

2061 Anza

2002061

1.3

0.53748

2.264128

3.774618

1.047204

3.481052

3.406902

2101 Adonis

2002101

0.3

0.764062

1.874861

1.332504

0.44235

3.307371

2.567212

2201 Oljato

2002201

0.9

0.712438

2.173649

2.523232

0.625059

3.722239

3.204737

2340 Hathor

2002340

0.15

0.449792

0.844208

5.855274

0.46449

1.223926

0.775679

2368 Beltrovata

2002368

1.15

0.413964

2.104661

5.236012

1.233407

2.975915

3.053384

3102 Krok

2003102

0.8

0.448111

2.151514

8.41955

1.187398

3.11563

3.155909

3288 Seleucus

2003288

1.4

0.457466

2.031858

5.934384

1.102353

2.961363

2.896331

3361 Orpheus

2003361

0.15

0.322743

1.209679

2.685067

0.819264

1.600095

1.330496

3362 Khufu

2003362

0.35

0.468638

0.989466

9.917301

0.525765

1.453167

0.984259

3551 Verenia

2003551

0.45

0.487042

2.092568

9.50651

1.073399

3.111738

3.027106

3757

2003757

0.25

0.445535

1.834668

3.868582

1.017259

2.652077

2.485104

3908 Nyx 4015 WilsonHarrington

2003908

0.5

0.458787

1.927448

2.181651

1.043159

2.811737

2.675977

2004015

2

0.624139

2.639888

2.783928

0.99223

4.287546

4.2893

4179 Toutatis

2004179

2.7

0.629364

2.529975

0.44617

0.9377

4.122249

4.024227

4660 Nereus

2004660

0.165

0.359914

1.488784

1.431665

0.95295

2.024618

1.816585

4688

2004688

0.3

0.515027

2.235353

6.377407

1.084085

3.38662

3.34216

4769 Castalia

2004769

0.7

0.48318

1.063178

8.888374

0.549471

1.576884

1.096268

5604

2005604

0.275

0.405293

0.926872

4.794061

0.551217

1.302527

0.892356

5646

2005646

2.15

0.436616

2.143533

7.913113

1.207633

3.079434

3.138365

5797 Bivoj

2005797

0.2

0.443829

1.893832

4.186539

1.053295

2.734369

2.606275

6063 Jason

2006063

0.7

0.766144

2.212672

4.918862

0.517446

3.907898

3.291423

6178 8013 GordonMoore

2006178

1.15

0.58416

2.81596

4.308438

1.170989

4.460931

4.725502

2008013

1.15

0.431667

2.199539

7.568878

1.25007

3.149008

3.262163

8014

2008014

0.35

0.455769

1.746292

1.864636

0.950386

2.542199

2.307723

NEA Name

9856

2009856

0.5

0.622351

2.247415

9.773139

0.848735

3.646096

3.36925

14827 Hypnos

2014827

0.45

0.664894

2.843049

1.980996

0.952722

4.733376

4.793853

19356

2019356

0.455

0.56392

2.501398

3.003861

1.090811

3.911986

3.956238

25143 Itokawa

2025143

0.165

0.280306

1.324035

1.621908

0.952901

1.69517

1.523553

52760

2052760

0.5

0.620438

2.411997

2.431243

0.915502

3.908492

3.746049

85182

2085182

0.55

0.780073

2.215747

3.183913

0.487302

3.944193

3.298288

86039

2086039

1.11

0.579196

1.759478

7.122084

0.740396

2.77856

2.333909

6

99942 Apophis

2099942

0.135

0.191076

0.9223

3.33196

0.746071

1.09853

0.885761

136564

2136564

0.2

0.394382

1.865594

2.980996

1.129837

2.601351

2.548202

139056

2139056

0.16

0.328224

1.885768

4.73146

1.266813

2.504722

2.589647

153591

2153591

1

0.478167

1.986786

6.684182

1.036771

2.936802

2.800496

162000

2162000

0.2

0.462576

1.678428

1.095411

0.902028

2.454827

2.174514

168318

2168318

0.45

0.54238

2.162902

6.489792

0.989787

3.336017

3.180999

217628 Lugh

2217628

0.7

0.702837

2.551459

4.01893

0.7582

4.344718

4.075596

(1988 TA)

3001644

0.2

0.478627

1.541499

2.54214

0.803696

2.279302

1.913917

(1989 UP)

3001825

0.15

0.47274

1.864306

3.857389

0.982974

2.745639

2.545565

(1989 VB)

3001835

0.2

0.460924

1.864889

2.133661

1.005316

2.724462

2.546759

(1990 UA)

3002495

0.2

0.525622

1.6405

0.926465

0.778216

2.502783

2.101224

(2005 YU55)

3309689

0.2

0.428948

1.142716

0.513456

0.65255

1.632882

1.221563

(2008 EV5)

3404781

0.225

0.083598

0.958285

7.436761

0.878174

1.038395

0.938102

Some of the desirables are: 1. Arrival at the target as far from perihelion as possible, to minimize rendezvous ΔV. 2. Maximize astronaut operation time in the vicinity of a target, while ensuring that this length of stay does not increase the transfer ΔV to a successive target or to Earth. 3. Minimal overall ΔV expenditure with minimum time spent in flight. 4. Maximize the number of targets visited within the mission timeline without compromising astronaut safety. Therefore, for every target we choose, we must evaluate the impact on the overall mission ΔV, total time in flight and the reachable number of future targets. If any of these are sub-optimal or unacceptable, we must retrace our steps, select a new target and repeat this analysis. In other words, a branching decision-tree type approach is necessary. These are demanding goals; even with modern optimization tools, without a good baseline approach to eliminate unreasonable and unapproachable targets, a lot of effort and computing time will be wasted. Refining the statistical limits for implementation Given a Phase I target, we refine the statistical limits imposed earlier for this implementation. The hopping algorithm used to determine which NEA would be the optimal target for Phase II works on the assumption that the transfer ΔV is controlled largely by the following six factors. Global boundary parameters (GBPs). The GBPs for this algorithm are an overall mission ΔV of 100 km/sec and an overall mission timeframe of less than 36 months, selected to maintain our posture of inclusion (rather than exclusion) for potential transfer trajectories. Further, targets that have already been visited are eliminated from further consideration along an itinerary. When the GBPs are exceeded on a given itinerary, the algorithm breaks and proceeds to calculate the next case. Distance from the originating asteroid. Attempting to reduce ΔV required to return to Earth at the end of the hopping mission, the maximum acceptable travel distance is capped at 0.5 AU (74.78 million km). Work by Adamo23 suggests that minimal ΔV and Δt occur during windows when the target NEA approaches the Earth to within 0.1 AU, but we have chosen a wider distance

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to accommodate the relatively low number of selectable targets from Table 1 approaching the Earth (and subsequent NEAs) within 0.1 AU, and keeping in mind the high accessibility that would potentially be afforded to such a mission by the VASIMR engine13. Inclination change from the origination asteroid’s ecliptic plane. The maximum acceptable inclination change from the plane crossing through the center of the originating NEA and the Sun is reduced from the earlier max of 10 degrees suggested by Barbee24 to 7.5 degrees. Planetary perturbation from nearby bodies. Using ephemerides data for the departure NEA, the other 49 target NEAs and the planetary bodies from HORIZONS 8, the algorithm calculates the influence of Venus, Mars and the Sun and plots in MATLAB as a function of time after the departure epoch. This is later extended using Satellite Tool Kit (STK) models to include third body effects of all the planets, the Jupiter moon system and solar radiation pressure. An example of the importance of this factor is 4769-Castalia. Following a close encounter on 21 Apr 2039 at 0.338 AU from Earth, Castalia proceeds to cross Venus’ orbit on 2 Aug 2043 within 0.134 AU; it would be necessary to account for this perturbation adding or relieving ΔV burdens while considering transfers between 2039 and 2043. Slope of the transfer graph. A positive slope on a distance transfer graph implies increasing distance between the originating asteroid and the target, i.e. the spacecraft is “chasing” its target, which is not favorable to minimum delta-V. The higher the slope, higher the rendezvous ΔV. Ideally, a minima would be present within the transfer window. Should the distance and inclination criteria be met, the algorithm will use the slope and perturbation effects to predict favorable transfers. Example outputs are presented in the next section. If favorable for rendezvous, the target will be promoted to an operations hold and further propagation. Grid approach to hold time and Time of Flight (TOF). The maximum TOF between NEAs is capped at 150 days (~5 month window). An embedded grid approach similar to that outlined by Barbee et al24 is utilized to parameterize the problem and scan forward to compute all possible trajectories within the GBPs. The grid size was initially kept variable to locate a best fit. Ten day grid propagation was found to be the best compromise between computational time and realistic ΔV values. Smaller grid sizes were used with almost identical results but longer computation times. The “hold time” for operations at a NEA is held at a minimum of 120 hours, or 5 days, and a maximum of 75 days. An identical grid approach is used, with a grid size of five days. Through a MATLAB/STK implementation of the above logic, our hopping algorithm outputs the trajectory, targets achieved, ΔV cost, total operations hold time and total TOF. The algorithm is programmed to scan and propagate the top 3 targets forward at every step, thus outputting several potential itineraries for review. Example output: Transfer forward from Castalia To elucidate the algorithm’s operation, for example purposes the first target is chosen to be 4769-Castalia. According to HORIZONS, Castalia’s close approach to the Earth occurs on 2 Mar 2038. Assuming a rendezvous close to this date and a minimum of seven days for round-the-clock

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prospecting and mining operations, the mission is ready to proceed to Phase II, departure from Castalia, on 19 Mar 2038. Fig. 4 shows a plot of 4769-Castalia versus 1862-Apollo, both in the inertial frame, from 19 Mar 2038 to 29 Jul 2038. Based on the slope, the algorithm decides that a transfer to Apollo is infeasible – the slope of the graph is increasing, and at the end of four months Apollo is almost 344 million km (2.3 AU) away from Castalia and still increasing. At five months the algorithm reaches the maximum TOF restriction, resets the grid to a new target NEA and begins reevaluating.

Figure 4. 4769-Castalia vs. 1862-Apollo.

Figure 5. 4769-Castalia vs. 1566-Icarus

Fig. 5 shows another sample output, between Castalia and 1566-Icarus, during the same transfer window. The trend is in a favorable direction (decreasing), however at the end of our window Icarus is still 208 million km (1.39 AU) away; outside the 0.5 AU limit and therefore an unfavorable transfer choice.

Figure 6. 4769-Castalia vs. 1865-Cerberus.

Figure 7. 4769-Castalia vs. 3544-Amun

Figs. 6 and 7 are sample output examples of favorable trends but unfavorable transfers. 1865Cerberus exhibits a local minima right within our desired transfer window (10 Jun 2038) and would involve an overall inclination change of 7.2 degrees (barely within the inclination cut-off criteria), but at its closest is 122 million km (0.82 AU) away from Castalia. We therefore expect a relatively high transfer ΔV, which was confirmed with STK to be 22.8 km/sec, therefore validating the 0.5 AU distance cut-off.

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Fig 7 shows the other case: 3554-Amun approaches Castalia inside 17 million km (0.1 AU) on 7 Apr 2038, but two factors cause the algorithm to reject this as a viable transfer: the high inclination change of 14.47 degrees required (7.5 degree max) and the extremely short time between departure from Castalia and desired arrival at Amun. Imposing a requirement of covering 17 million km. in 18.8 days drives the steep slope following the minima. This tells us that it will be propellant intensive to “chase” Amun to a rendezvous, as opposed to “intercepting”; the transfer ΔV was confirmed with STK to be 34.9 km/sec. Fig. 8 below shows an example of the output we would like to see. 1943-Anteros is 0.26 AU away from 22-28 Apr 2038, which gives us a flight time of 40.4 days. The second criterion imposed shows an inclination change of 0.184 degrees. Note the relatively large, shallow minima.

Figure 8. 4769-Castalia vs. 1943-Anteros

Based on these observations our algorithm can conclude that Anteros would be an excellent candidate for Phase II of the NEA-Hopping trajectory if Castalia is selected as the Phase I target. The algorithm will add varying ops times according to the grid approach (5 – 75 days in five-day grid increments) and continue to propagate forward with the top 3 choices, as long as they do not violate the GBPs. Proceeding in a similar manner for all other target NEAs, implementing the algorithm gives the data shown in Table 2. Note the unacceptably large close approach distances (Castalia – 4015Wilson-Harrington) and that we appear to have a better candidate than Anteros in 6178, which approaches closer to Castalia and provides a longer flight duration, which could translate to potential ΔV savings.

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Table 2. Output for favorable outcomes expected from Castalia Close approach between

Distance

1 Apr 2038 – 29 Jul 2038

(mil km)

Investigate?

Castalia – Amun:

7 Apr 2038

17

Yes – Expect high ΔV

Castalia – Icarus:

Decreasing past 29 Jul 2038

210

No

Castalia – Apollo:

Increasing past 29 Jul 2038

344

No

Castalia – Daedalus:

Decreasing past 29 Jul 2038

157

No

Castalia – Cerberus:

10 Jun 2038

122

No

Castalia – Anteros:

22 – 28 Apr 2038

39

Yes – Option #2

Castalia – Seleucus:

Increasing past 29 Jul 2038

537

No

Castalia – Syrinx:

Increasing past 29 Jul 2038

584

No

Castalia – Hephiastos:

Increasing past 29 Jul 2038

628

No

Castalia– Wilson-Harrington:

13 May 2038

422

No

Castalia – 5646:

Increasing past 29 Jul 2038

383

No

Castalia – Jason:

Decreasing past 29 Jul 2038

507

No

Castalia – 6178:

26 May 2038

Castalia – Gordonmoore:

Increasing past 29 Jul 2038

508

No

Castalia – 85182:

Increasing past 29 Jul 2038

616

No

Castalia – Lugh:

Decreasing past 29 Jul 2038

463

No

Castalia – Geographos:

Increasing past 29 Jul 2038

379

No

Castalia – Toro:

Increasing past 29 Jul 2038

277

No

Castalia – Oljato:

Increasing past 29 Jul 2038

354

No

34.96

Yes – Option #1

APPLICATION TO OPTIMAL NEA-HOPPING TRAJECTORY DESIGN Results of Simulation In terms of computation time, the selection of Anteros as the Phase II target was trivial in nature. However if there is a lack of suitable targets forward from Anteros, the true flexibility of the algorithm lies in the ability to easily backstep and select an alternate, suitable target from among the top three, even back to the Phase I target of Castalia, if necessary. This simple and computationally inexpensive method is the key to implementing a ‘decision tree’ approach to target selection and design of a hopping trajectory that meets the design goals. For example, a best-transfer ΔV to Castalia was found; with an optimized departure date of 12 Mar 2038 at 6.13 km/sec (the statistical ΔV calculated using Equations (1) – (4) was 60.15 km/sec, highlighting the very restricted capability in which we can apply those formulae). However in that time frame, even up to the maximum hold time of 75 days in orbit around it, the best Phase II transfer was to Anteros for 22.7414 km/sec, which was rejected in the name of feasibility. Fig 9 illustrates the ‘decision-tree’ approach that our algorithm makes possible.

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Figure 9. NEA-hopping itinerary: Iteration 1

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Figure 10. NEA-hopping itinerary: Iteration 2

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Earth return. The algorithm allows us the flexibility to look ‘into the future’ and examine which target NEA has a greater number of future potential targets to travel to (without the computational burden of trajectory optimization at every step), as well as whether future targets will allow a feasible return to the Earth within the GBPs to keep HSF mission considerations feasible. The boxes in Fig 9 that display “Earth NF” denote that a “not feasible” return to Earth within the imposed GBPs was found. An itinerary in Fig 9 that does not show branches did not have any second or third choices that met the global parameters. As shown by the two branches headed to Nereus, nine years apart, multiple launch windows are calculable as well. Backtracking. Use of this algorithm helps decide whether a given set of target NEAs (“itinerary”) meet the goals of maximizing the number of NEAs visited and minimizing crew travel time. Fig 9 clearly illustrates the capability to backtrack across two or more Phases to select an alternate option. We can easily determine whether extra ΔV to get to one particular target NEA over another would be an acceptable trade toward optimizing across the global mission timeline. For example, at first it would seem apparent that the 34-day, 6.99 km/sec transfer from Earth to Apophis would be the ideal Phase I transfer. However, the itinerary is forced to return to Earth (Earth-1) after visiting just two targets, due to no viable transfers being found from Beltrovata onward. Therefore we can back-track, and select an itinerary that transfers to Itokawa or Nereus at a higher initial ΔV, but an overall smaller per-NEA ΔV burden, with a higher number of targets visited, thus fulfilling our goals of maximizing mission productivity. Modification for Earth return NEA-hopping is intrinsically a propellant-expensive, time-intensive endeavor, which is why we are striving to maximize productivity. The ratio of ops to flight time is one indicator, as shown in Fig 9, as are the number of NEAs visited. As previously mentioned, the available itineraries are easily tailorable to meet the goals of the company sponsoring the mission, to either minimize mission time and guarantee productivity (Earth-1, best ops to flight ratio, but only 2 NEAs visited), or NEAs visited (Earth-4, five NEAs, but unrealistic mission ΔV cost of 91 km/sec). To accomplish both it was realized that the global boundary parameters had to be relaxed to allow greater flexibility on the Earth return. Increasing the overall mission timeline was not an option due to HSF considerations. However, additional time/allowable distance from the Earth on the final leg might increase the number of NEAs that could be visited within the mission timeline. Therefore, the maximum allowable transfer flight time was relaxed from 150 days to 250 days, and the maximum operations hold time from 75 days to 100 days. As shown in Fig 10, three new itineraries now emerge, two of them allowing for hopping between five target NEAs. Earth-7 immediately stands out. This itinerary visits the maximum of five NEAs, and compares to Earth-4 with 30% less ΔV for only 99 extra mission days, without a significant drop in the ops to flight ratio. Earth-6 shows a higher ΔV than Earth-4 with less mission productivity. Earth-7 has the longest mission timeline, coming in just under the 36 month boundary (34.8 months), but in return shows the most time spent in operations (195 days). This would justify the cost and risk of such a mission to the private enterprise, which is one of the goals of this study. RESULTS AND CONCLUSIONS Final NEA-Hopping Trajectory Using this algorithm and our GBPs, and balancing the mission goals against manned mission risks, the NEA-hopping trajectory itinerary shown in Fig. 11 (Earth-7) was selected as optimal. The overall mission ΔV does not account for Trajectory Correction Maneuver (TCM) allowances,

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or station-keeping costs in the NEA vicinity; these have yet to be modeled. With human factors astronaut schedule optimization allowing for round-the-clock operations, this itinerary boasts 4682.2 hours of mining, experimentation and sample collection time. Earth (16 Feb '31)

162000 (23 Sep '31)

Nereus (25 Jun '31)

2008EV5 (28 Aug '32)

Apophis (11 Mar '32)

Earth (17 Oct '33)

GordonMoore (9 Feb '33)

Figure 11. Final NEA-Hopping Itinerary

Fig. 12 below shows this final trajectory in STK. The dotted lines represent target NEA orbits. The solid green trajectories are transfers, and the solid dark blue trajectories are operation holds around target NEAs.

Figure 12. Final NEA-Hopping Trajectory in STK

The resulting ΔVs are shown in Table 3 below, together with the operations hold time in the vicinity of each NEA.

Phase 1 2 3 4 5 6

Table 3. ΔV values for the optimized itinerary Hold Itinerary Departure Arrival time (UTCG) (UTCG) (days) Earth – Nereus 16 Feb 2031 26 Apr 2031 60 Nereus – 162000 25 Jun 2031 03 Sep 2031 20 162000 – Apophis 23 Sep 2031 11 Jan 2032 60 Apophis– 2008EV5 11 Mar 2032 08 Aug 2032 20 2008EV5–Gordon-Moore 28 Aug 2032 05 Jan 2033 35 Gordon-Moore– Earth 09 Feb 2033 17 Oct 2033 n/a

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Phase ΔV (km/sec) 10.72 10.74 11.06 7.11 10.88 18.327

Mission ΔV (km/sec) 10.72 21.46 32.52 39.63 50.51 68.837

Physical Properties of Chosen NEAs Now that we know which NEAs we wish to target, it is appropriate to comment on their physical properties8, which are summarized in Table 4. #

Table 4. Physical properties of the target asteroids. NEA Class Diameter Rotation Abs. Mag(km) Period nitude (hr) (mag)

4660-Nereus 1

Apollo

0.33

15.1

18.2

162000 2

Apollo

0.40

2.536

19.295

99942-Apophis 3

Aten

0.27

30.4

19.7

2008 4 EV5

Aten

0.45

3.725

20.005

2.30

6

16.6

8013 6 Gordon-Moore Amor

Some of our target NEAs are already subjects of studies. Numerous publications have already identified Nereus16, 17, 18 and Apophis19, 20 as potential targets for unmanned missions. The commercial value will supplement the already high scientific interest in these NEAs. For example, as a member of the Tholen E-class, Nereus is composed of oldhamite, troilite and other pyroxeneolivine compounds21. The taxonomy of Gordonmoore, on the other hand, is not well characterized, which sets it high on the scientific value scale. The rotation period listed in Table 4 is based on fragmentary light curves that have been approximated between two maxima and two minima14; therefore the corresponding period could be incorrect. Similarly, information is scarce on 162000 except through type or class generalizations, and the theorized potential of satellites in orbit about this NEA would make it an interesting target for scientific exploration. Radar, broadband visual and infrared photometry and spectrophotometry studies for these targets would be highly desirable. Further study will be needed to determine their commercial value, although extrapolations may be made from NEAs in the same class with similar characteristics. Discovered relatively recently (2008), EV5 presents a more interesting case from the commercial viewpoint. Using a combination of observed near-surface bulk densities and optical albedos, EV5 can be inferred to be composed primarily of silicate-carbon and silicate-metal mixtures15. EV5 has been classified as a Potentially Hazardous Asteroid (PHA). However, considering the mass of the object, between 1011 and 1.5x1011 kg, as long as mining is restricted to the equatorial concavity that extends all the way around the NEA, it is not anticipated that any significant dynamical change to the slow-spinning asteroid will occur as a result of activity on the surface. Limitations Approximation to finite maneuvers with impulsive burns. Ultimately, the goal of this trajectory design was to create a NEA-hopping mission centered around a spacecraft powered by the VASIMR engine14. The current simulation must be converted from impulsive maneuver modeling to finite maneuvers with an accurate constant thrust engine model built in to assess the flexibility and capability of this engine in achieving the mission’s goals. The results of the impulsive maneuvers described here will be used to “seed” the finite maneuvers.

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Optimization of operations hold-time. While the approach laid out in this paper takes the brunt of the computational work out of hunting for feasible NEA-hopping itineraries, the grid approach to hold-time and TOF results in a sub-optimal trajectory. The selected itinerary is only a baseline, upon which we will expand. We expect the overall ΔV to decrease further when the hold time at each NEA is optimized globally across the mission timeline. Initial simulations in this area have proven to be extremely promising, and results will be documented in a follow-up study. Inertial or body-centered holding above target NEAs. During operations, either inertial or body-centered hovering* will have to be performed, to allow ease of travel between the spacecraft and the asteroid. Body-centered hovering appears to be the better option from a crew operations point of view; however, it will be more propellant-intensive. Realistic figures for these stationkeeping operations will need to be added to the mission’s ΔV budget. Heliocentric propagation in the vicinity of target NEAs. Currently, all modeled trajectories use a sun-centered heliocentric propagator in MATLAB. Third-body influences are limited to those of Earth, Mars and Venus; incorporating a full-system third-body perturbator model and solar radiation pressure models will be used for a better estimation. When modeling the trajectory close to the asteroid, uneven gravity gradients caused by irregularly shaped NEAs will have to be approximated as well. Asteroid orbit perturbations due to surface operations. Surface operations, ranging from the astronauts touching down on the asteroid to asteroid mass removal, will have the effect of exchanging the rotational and translational momentum and energy of the asteroid. Propagated years into the future, these changes may have unintended consequences. This is especially relevant in the much-studied case of Apophis. Though our rendezvous is after the “keyhole” event 25 in 2029, future Earth-impact should still be a consideration. Once a few surface operation scenarios have been generated, our investigation will seek to quantify the effect of those changes on the orbit of each asteroid, by modifying an approach laid out by Izzo et al22, to determine a safe amount of surface operations such that the asteroid will not collide with Earth in the near future. CONCLUSION The outlined NEA-hopping algorithm is designed to isolate those NEAs that would present good candidates for low ΔV transfers, making the search and optimization of transfer trajectories simpler and more efficient through ‘best-guess’ parameters based on relative inclination change, distance and undesirable dynamical qualities. This approach, when coded in MATLAB, greatly simplified our ability to isolate workable targets and trajectories, and allowed them to be bridged to STK through Connect for further optimization and refinement. The guesses were proven substantial by using STK 9.1’s Design Explorer Optimizer for validation. The paper also reported on overall mission considerations and impacts of trajectory design on various subsystems involved in the design of a manned NEA-hopping mission.

* Body-centered holding involves fixing the position of the spacecraft relative to the rotating NEA. This requires compensating for the gravity gradients of the NEA, motion of the NEA about its own axis, centrifugal force and solar radiation pressure [12]

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ACKNOWLEDGMENTS Thanks to Ms. Jaclyn R. Beck, University of Wisconsin, Madison. For initial funding support, the authors extend their thanks to Dr. Geoffrey Kain and the Embry-Riddle Aeronautical University Honors Program as well as the Space Development and Test Directorate, Space and Missile Systems Center, United States Air Force. REFERENCES 1. Kargel, J. (1994). Asteroids: Sources of Precious Metals. Space Resource News , 3 (12). 2. Kargel, J. (1994). Metalliferrous asteroids as potential sources of precious metals. J. Geophys. Res. , 99 (E10) (21), 129-21, 141. 3. Lewis, J. M. (1994). Resources of Near-Earth Space. (M. Guerrieri, Ed.) University of Arizona Press. 4. Lewis, J. (1997). Mining the Sky: Untold Riches from the Asteroids, Comets and Planets (Helix Book). Basic Books. 5. Mission Statement of the Ad-Astra Company. (2008). Retrieved March 2010, from VASIMR VX-200: http://www.adastrarocket.com/aarc/MissionStatement 6. Eckart, P. (1996). Spaceflight life support and biospherics. Torrance, CA: Microcosm Press. 7. Larson, W. L. (2000). Human spaceflight: Mission Analysis and Design. New York: McGraw-Hill Companies, Inc. 8. Jet Propulsion Laboratory (JPL). (n.d.). Retrieved March 2010, from Solar System Dynamics: HORIZONS Small Body Database (SBDB): http://ssd.jpl.nasa.gov 9. Stacey, R. M. Delta-V requirements for Earth co-orbital rendezvous missions. Journal of Planetary and Space Science , 57, 822-829. 10. Berinde, S. Searching for gravity-assisted trajectories to accessible near-Earth asteroids. In Z. M. Knezevic (Ed.), Proceedings of IAU Colloquium, 197. 11. Callister, J. (1990). Brief Review in Earth Science. New York: Prentice Hall. 12. Nathues, A. H. (2010). ASTEX: An in-situ exploration mission to two near-Earth asteroids. Advances in Space Research , 45, 169-182. 13. Squire, J. P. (September 5-6, 2008). VASIMR Performance Measurements at Powers Exceeding 50 kW and Lunar Robotic Mission Applications. International Interdisciplinary Symposium on Gaseous an Liquid Plasmas. 14. Hoffman, Martin. "Photometry of 1990KA." Minor Planet Bulletin 18 (1991): 10-11. Print. 15. Busch, Michael W., et al. "Radar Observations and the Shape of Near-Earth ASTEROID 2008 EV5." Icarus 212 (2011): 649-60. Jet Propulsion Laboratory. Web. . 16. D'Arrigo, P. & Barucci, M. A., Lagerkvist, Claes-Ingvar. “The ISHTAR mission”. Proceedings of Asteroids, Comets, Meteors - ACM 2002. International Conference, 29 July 2 August 2002, Berlin, Germany. Ed. Barbara Warmbein. ESA SP-500. Noordwijk, Netherlands: ESA Publications Division, ISBN 92-9092-810-7, 2002, p. 95 – 98. 17. D'Arrigo, P. & Barucci, M. A., Lagerkvist, Claes-Ingvar. “The ISHTAR Mission: Executive Summary for ESA Publication”. Available online at: 18. Helin, Eleanor F.; Hulkower, Neal D.; and Bender, David F. "The discovery of 1982 DB, the most accessible asteroid known". Icarus 57 (1984): 42–47.

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19. "China Reveals Solar Sail Plan To Prevent Apophis Hitting Earth in 2036 - Technology Review." Technology Review: The Authority on the Future of Technology. Web. 02 Jan. 2012. . 20. Noland, David. "5 Plans to Head off the Apophis Killer Asteroid - Popular Mechanics." Automotive Care, Home Improvement, Tools, DIY Tips - Popular Mechanics. Web. 02 Jan. 2012. . 21. Brozovich, M. et al. " Radar observations and a physical model of Asteroid 4660 Nereus, a prime space mission target" Icarus 201 (2009): 153-166. Jet Propulsion Laboratory. Web. < http://echo.jpl.nasa.gov/asteroids/4660_Nereus/brozovic.etal.2009.nereus.pdf>. 22. Izzo, Dario, A. Bourdoux, R. Walker, F. Ongaro. "Optimal trajectories for the impulsive deflection of near earth objects." Acta Astronautica 59 (2006): 294-300. Print. 23. Adamo, Daniel R., and Brent W. Barbee. "Why Atens Enjoy Enhanced Accessibility for Human Space Flight." American Astronautical Society 11-449 (2011). Print. 24. Barbee, B.W., T. Esposito, E. Pinon III, S. Hur-Diaz. R.G. Mink, D. R. Adamo. "A Comprehensive Survey of the Near-Earth Asteroid Population for Human Mission Accessibility." Proc. of American Institute of Aeronautics and Astronautics, Guidance, Navigation and Control Conference, Toronto, 2010-8368 (2010). Print. 25. Gennery, Donald B. "Scenarios for Dealing with Apophis." Proc. of Planetary Defense Conference, Washington DC. 2007. Print.

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