Distribution and Orientation of Boulders on

Asteroid 25143 Itokawa

Sara Mazrouei-Seidani

A Thesis submitted to the Faculty of Graduate Studies in Partial Fulfillment of the Requirements for the Degree of Master of Science

Graduate Program in Earth and Space Science, York University, Toronto, Ontario

September 2012

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In the research of early planetary system evolution processes, the study of primitive aster­ oids is key. The objective of this thesis is to confirm and update any previously identified trends in the global and regional distribution of boulders on asteroid 25143 Itokawa, as well as, to discover new findings to better understand the history of this asteroid. Itokawa is a near-Earth object, and the first asteroid that was targeted for a sample return mission. Trends in boulder population should provide new insights in regards to Itokawa’s current appearance following the disruption of a possible parent body, and how its surface might have changed since then. The population of boulders will also be analyzed with respect to latitude for both Itokawa’s current state, and a hypothetical parent body. In particular, boul­ der distribution over the surface of the asteroid might provide a means to test the hypothesis of whether or not Itokawa is a contact binary. Dedication

This thesis is dedicated to my parents, for giving me motivation and strength in moments of despair and discouragement. I would not be where I am today had it not been for their love and sacrifices. Acknowledgments

It would not have been possible to write this thesis without the support of the kind people around me, to only some of whom it is possible to give particular mention here. First and foremost, I offer my sincerest gratitude to my supervisors Professors Michael Daly and Olivier Barnouin for their constant help, advice, and support. I would like to thank Carolyn Ernst for her input regarding this project, Eliezer Kahn for his support with the SBMT, and Mike Ilnicki for all his help with IDL programming. In my daily work, I have had the opportunity to work with the best group of people, so thank you to all the members of the Planetary Instrumentation Laboratory. I would also like to thank Paul Delaney for he has been invaluable on both an academic and a personal level, for which I am extremely grateful. Last but definitely not least, I would like to thank my beloved parents for their uncon­ ditional love and support as well as Shiva and Shahroukh for their endless motivation and encouragement. Table of Contents

Contents

Abstract ii

Dedication iii

Acknowledgments iv

Table of Contents v

List of Tables viii

List of Figures ix

1 Introduction and Background 1 1.1 The Hayabusa M issio n ...... 1 1.2 Asteroid 25143 Itokawa ...... 5 1.2.1 How Did Itokawa E volve? ...... 13

2 Tools and Datasets 15 2.1 Small Body Mapping Tool (SBMT) ...... 15

2.2 Gaskell Shape M o d el ...... 17 2.3 A M IC A ...... 22 2.4 LIDAR ...... 25

3 Distribution of Boulders on Asteroid Itokawa 29

3.1 Motivation and background ...... 29

3.2 M ethodology ...... 31

v 3.3 Results and A n aly sis ...... 37 3.3.1 Overall Distribution ...... 37

3.3.2 Head vs. B o d y ...... 41 3.3.3 East vs. W e st ...... 43 3.3.4 Conclusion ...... 45 3.4 Mission Considerations ...... 48 3.4.1 Introduction to OSIRIS-REx ...... 48 3.4.2 Boulder Distribution on the Muses Sea Lowlands ...... 49 3.4.3 Boulder Distribution on the Sagamihara L ow lands ...... 51

' 3.4.4 Conclusion ...... 54

4 Itokawa Now and Then 55 4.1 Motivation and Background ...... 55

4.2 M ethodology ...... 55 4.2.1 Volume and Centre of Mass C alculations ...... 57 4.2.2 Inertia Tensor Calculations ...... 60 4.3 Analysis and R e su lts ...... 65 4.3.1 The Difference between the rotation poles ...... 66

4.3.2 Latitude vs. Boulder D istribution ...... 68

5 Conclusion and Future Work 76

References xiv

Appendices xxii

Appendix A - Chi-square Error A nalysis ...... xxii

Appendix B - P lo ts ...... xxiii Cumulative Boulder Distribution Plots for the H e a d ...... xxiii Cumulative Boulder Distribution Plots for the B o d y ...... xxiv Cumulative Boulder Size Distribution Plots for the East S id e ...... xxv

Cumulative Boulder Size Distribution Plots for the West S id e ...... xxvi Appendix C - Derivations for Rotation and Transformation ...... xxvi List of Tables

List of Tables

1 Hayabusa Spacecraft Specifications ...... 4 2 Asteroid Itokawa Characteristics ...... 12 3 AMICA Instrument Specifications [ 1 ] ...... 24 4 LIDAR Instrument Specifications [2] 27 5 Power-index Values for Cumulative Boulder Size Distribution over the En­ tire Surface of Itokawa ...... 39 6 Power-index Values for Cumulative Boulder Size Distribution on the Head vs. Body ...... 43 7 Power-index Values for Cumulative Boulder Size Distribution on the East vs. W est...... 45 8 Power-index Values for Cumulative Boulder Size Distribution - Muses Sea . 51 9 Power-index Values for Cumulative Boulder Size Distribution - Sagamihara 54

viii List of Figures

List of Figures

1 Hayabusa Spacecraft [3] 2 2 Asteroid Itokawa’s Orbit Around the Sun [4] 6 3 Itokawa’s Reflectance Spectrum [5] ...... 7 4 Asteroid 25143 Itokawa, provided by JAXA The circles outlining the head and body of Ito k a w a ...... 8 5 Itokawa’s Slope Profile (Northern view on the left, Southern view on the

right) [6] 11 6 Itokawa’s Gravitational Potential Profile (Northern view on the left, South­

ern view on the right) [ 6 ] 11 7 Small Body Mapping Tool (Itokawa Tab) ...... 16 8 Implicitly Connected Quadrilateral Model and Grids [7] 20 9 Representation of Implicitly Connected Quadrilateral Labels [8 ] ...... 21

10 AMICA Image Data Storage [ 9...... ] 23 11 Example AMICA Image (st_2566271576_v.fit) ...... 25

12 LIDAR Path Plots (coloured: lidar path, black: spacecraft position) ...... 28

13 Cumulative Mass Distribution of Fragments (Aluminum Impact Target)

[ 1 0 ] ...... 30 14 Results of a Catastrophic Disruption by Impact [11] 31

ix 15 Example of an AMICA Image Overlayed on the Shape Model and Sample

Mapped Boulders (blue: craters, purple and yellow: boulders, purple line: semi-major axis, yellow line: semi-minor axis) ...... 32 16 Boulder Distribution on the Head (red) vs. Body (black) of the Asteroid (on the left) Boulder Distribution on the East (black) vs. west (red) sides of Itokawa (on the right) ...... 34 17 Mapped Boulders on the Head (left) and Body (right) of Asteroid Itokawa (blue: craters, purple and yellow: boulders, purple line: semi-major axis, yellow line: semi-minor axis) ...... 35 18 Mapped Boulders on the East (upper) and West (lower) Sides of Asteroid Itokawa (blue: craters, purple and yellow: boulders, purple line: semi-major axis, yellow line: semi-minor axis) ...... 36 19 Cumulative Boulder Size Distribution Per unit area on the Entire Surface of Asteroid Itokawa Size defined as long axis (left) and as short axis (right) Black: Data Used for Power-index Calculation, Red: Omitted D ata ...... 38

20 Cumulative Boulder Size Distribution Per unit area on the Entire Surface

of Asteroid Itokawa Size defined as average (left) and as RSS (right) Black: Data Used for Power-index Calculation, Red: Omitted D ata ...... 38

21 Cumulative Boulder Size Distribution per unit area Black and red: two different boulder populations, green: data points omitted 40

x 22 Cumulative Boulder Size Distribution per unit area, bin size 10 cm

Black and red: two different boulder populations, green: data points omitted 41 23 Cumulative Number of Boulders per unit area on the Head (red) and Body (black) of Itokawa Size defined as long axis (left) and as short axis (right) Black and red: two different boulder populations, green: data points omitted 42 24 Cumulative Number of Boulders per unit area on the Head (red) and Body (black) of Itokawa Size defined as average (left) and as RSS (right) Black and red: two different boulder populations, green: data points omitted 42 25 Cumulative Number of Boulders per unit area on the East (red) and West (black) of Itokawa Size defined as long axis (left) and as short axis (right) Black and red: two different boulder populations, green: data points omitted 44 26 Cumulative Number of Boulders per unit area on the East (red) and West (black) of Itokawa Size defined as average (left) and as RSS (right) Black and red: two different boulder populations, green: data points omitted 44 27 Difference in Slopes ...... 46 28 Possible Evolution of Ito k a w a ...... 47 29 Results of Numerical Simulations [1 2 ] ...... 48 30 Selected Area in the Muses Sea ...... 50

31 Cumulative Number of Boulders per unit area on a selected area of the Muses Sea

Size defined as long axis (left) and as short axis (right) Black: Data Used for Power-index Calculation, Red:Omitted D ata ...... 50

xi 32 Cumulative Number of Boulders per unit area on a selected area of the

Muses Sea Size defined as average (left) and as RSS (right)

Black: Data Used for Power-index Calculation, Red: Omitted Data .... 51 33 Selected Area in Sagam ihara ...... 52 34 Cumulative Number of Boulders per unit area on a selected area of the Sagamihara Size defined as long axis (left) and as short axis (right)

Black: Data Used for Power-index Calculation, Red: Omitted Data .... 53 35 Cumulative Number of Boulders per unit area on a selected area of the Sagamihara Size defined as average (left) and as RSS (right) Black: Data Used for Power-index Calculation, Red: Omitted Data .... 53 36 Itokawa Shape Model (128 p la te s ) ...... 56 37 Ellipsoid Representing Itokawa Then ...... 56 38 ICQ Model to Triangular Plate M o d e l ...... 57 39 Triangular-Based Tetrahedron ...... 58 40 Transformation g and g_1[1 3 ] ...... 61 41 The Different Poles of Rotation ...... 67

42 Rotation A ngles ...... 68 43 Latitude vs. # of boulders/km2(Itokawa N o w ) ...... 71

44 Latitude vs. # of boulders/km2(Itokawa T h en ) ...... 72 45 Itokawa N o w ...... 73

46 Itokawa T h e n ...... 74

47 Cumulative Number of Boulders per unit area on the Head

Size defined as long axis (left) and as short axis (r ig h t) ...... xxiii

xii 48 Cumulative Number of Boulders per unit area on the Head Size defined as average(left) and as RSS (rig h t) ...... xxiii

49 Cumulative Number of Boulders per unit area on the Body Size defined as long axes (left), Size defined as short axes (rig h t) ...... xxiv 50 Cumulative Number of Boulders per unit area on the Body Size defined as average (left) and as RSS (right) ...... xxiv 51 Cumulative Number of Boulders per unit area on the East Side Size defined as long axis (left) and as short axis ( rig h t) ...... xxv 52 Cumulative Number of Boulders per unit area on the East Side Size defined as average(left), and as RSS (right) ...... xxv 53 Cumulative Number of Boulders per unit area on the West Side

Size defined as long axis (left) and as short axis (r ig h t) ...... xxvi 54 Cumulative Number of Boulders per unit area on the West Side Size defined as average (left) and as RSS (right) ...... xxvi

xiii 1 Introduction and Background

Primitive asteroids provide critical insights to the formation and evolution of the early solar system. Some of these insights, however, are obscured by the collisional evolution these asteroids undergo, the geological consequences of which are not well understood. Boulders on asteroids provide a means to detangle this collisional history, and provide an important opportunity to study the physical properties and geological evolution of asteroid surfaces.

1.1 The Hayabusa Mission

The Hayabusa Spacecraft, Japanese for Peregrine falcon, and formerly known as Muses-C, was developed by the Japanese Aerospace Exploration Agency (JAXA) and launched on May 9th 2003 using the M-V launch vehicle. This spacecraft rendezvoused with the near- Earth asteroid 25143 Itokawa on September 12th 2005 [14]. This mission was designed to acquire several tens of grams of asteroidal surface material and return them to Earth on June 13th 2010. Due to the small size of Itokawa and its weak gravity, the spacecraft did not orbit the asteroid but hovered near two stations on a plane between the asteroid and Earth. This spacecraft was in the “Gate Position”, at a range of about 18 kilometers until September 30th, and then moved to the “Home Position”, which was about seven kilometers from the surface of the asteroid. In order to move away from the equator and obtain polar data, as well as to gather varying illumination conditions, Hayabusa made several journeys to higher phase locations between October 8th and October 28th. It should also be mentioned that on November 4th, 9th, and 12th, the spacecraft made approaches to Itokawa to prepare for touchdowns on November 20th and 26th [15]. At the second touchdown, the spacecraft was programmed to fire small projectiles at the surface of Itokawa using its deployable collection hom and to then collect the resulting spray. An artist’s rendition of the Hayabusa spacecraft is shown in Figure 1. Figure 1: Hayabusa Spacecraft [3]

Hayabusa was used to study the asteroid’s shape, spin, colour, composition, topography, density, and history. Previously, other missions such as Galileo [16] and NEAR Shoemaker [17], sent by NASA had visited asteroids. However, the Hayabusa mission was the first to attempt an asteroid sample return to Earth for analysis, as well as being the first space­ craft designed to deliberately land on an asteroid and take off again. Hayabusa’s major scientific objectives include: 1) obtaining global surface mineralogical, geomorphologi- cal, and chemical information using orbiter remote sensing instruments; 2) gathering local microscopic information using micro-rover instruments; 3) to determine the bulk density of asteroid Itokawa through orbiter in-situ measurements; and 4) to understand the min­ eralogical, chemical, and isotropic abundance of surface materials via collected samples returned to Earth [14]. Hayabusa was also used for experimental research on new engi­ neering technologies used for planetary sample collections and their return back to Earth, such as: electrical propulsion, sampler, autonomous navigation, and reentry capsule. The mini-lander that was to be used to achieve objective 2 was called “Minerva”, short

2 for Micro/Nano Experimental Robot Vehicle for Asteroid. Minerva weighed around 591 grams and was solar-powered. This vehicle was designed to benefit from Itokawa’s very low gravity by using an internal flywheel assembly to hop around the surface of the asteroid. One of Minerva’s main objectives was to relay images from its camera to Hayabusa when­ ever the spacecraft and the rover were in sight of each other [18]. This micro-rover was deployed on November 12th 2005, and the lander release command was sent from Earth. However, Hayabusa’s altimeter measured its distance from the asteroid to be around 44 meters and began an automatic altitude safe keeping sequence before the release command arrived. Therefore, the probe was ascending at a higher altitude than initially intended and away from Itokawa when the release command arrived and Minerva was released. As a re­ sult, Minerva escaped Itokawa’s gravitational pull and disappeared into space and objective 2 was not met [14]. The Hayabusa mission, just like any other mission, was modified several times before and after launch. Hayabusa was originally scheduled to launch in July 2002 to a different asteroid called 4660 Nereus, with asteroid 1989 ML being an alternative target. A failure in the M-V launch vehicle in July 2000 caused a delay in the launch and therefore put both asteroid Nereus and 1989 ML out of reach. Consequently, asteroid Itokawa was then chosen as the target asteroid. In addition to Minerva, Hayabusa was to deploy another small rover called Muses-CN developed by Jet Propulsion Laboratory (JPL) and the National Aeronautics and Space Administration (NASA). This rover was canceled in November

2000 by NASA due to budget constraints. The launch was once again postponed from December 2002 to May 2003 because the O-ring of the reaction control system was found to be made from a different material than the one specified. The arrival of Hayabusa at Itokawa was delayed from June 2005 to September 2005 because of a solar flare that damaged the solar cells of the spacecraft in

2003. The damaged solar cells caused a reduction in the electrical power, which reduced the capability of the ion engines and thus, resulted in the delay in arrival. As a result of this delay the spacecraft had a shorter amount of time to spend at Itokawa, and therefore the number of landings on Itokawa were reduced from three to two. On November 9th, Hayabusa made a descent to 70 meters from 7.5 kilometers to test the landing navigation and the laser altimeter. Next, Hayabusa moved to a higher position and then descended to 500 meters and released a target marker into space to test the spacecraft’s tracking ability. Through analysis of the closeup images, the Woomera Desert site, which was one of the two selected sample sites, was found to be too rocky for a safe landing. Therefore, the sample sites were reduced from two to one, and the Muses Sea was chosen for the landing site [19]. The first touchdown did not result in a proper sample collection and only had the probability that some dust was whirled up into the sampling horn when it touched the asteroid. The second touchdown, was also unsuccessful as the pellets were not fired due to a leak in the thruster system. Table 1 provides some details on Hayabusa.

Table 1: Hayabusa Spacecraft Specifications Launch Date May 9th, 2003 Launch Vehicle M-V Launch Location Kagoshima Space centre (Uchinoura) Spacecraft Weight 510 kilograms Type of Orbit Heliocentric Telescopic Camera(AMICA) Light Detection And Ranging (LIDAR) Near Infrared Spectrometer(NIRS) Wide-view Camera(ONC-W) Scientific Instruments Onboard X-ray Fluorescence Spectrometer(XRS) Sampler and Reentry Capsule Small Rover MINERVA Target Marker Ion Thrusters

From mid-September to late-November, scientific observations were done over asteroid

4 Itokawa. This asteroid’s shape, surface altitude distribution, terrain, reflectance, gravity, mineral composition, and other properties were observed by the scientific instruments at altitudes ranging from 20 kilometers to 3 kilometers. These observations provided new information and insight into asteroid formation processes.

1.2 Asteroid 25143 Itokawa

Asteroid 25143 Itokawa, an Apollo and Mars-crosser asteroid, was the first target asteroid for a sample return mission. This asteroid was discovered by the Lincoln Near-Earth Aster­ oid Research (LINEAR) project [20] in 1998, and was called 1998 SF36. In August 2003, Minor Planet Circular reported that this asteroid was officially named after Hideo Itokawa, a Japanese rocket scientist. This small asteroid has dimensions of 535 x 294 x 209 meters [21]. It has an orbital path that ranges from 0.9530 astronomical units (AU), just inside the Earth’s orbit, and extends to 1.6947 AU, just outside of the Mars’ orbit [22], Figure 2 shows the orbit of asteroid Itokawa around the Sun. This orbit allows close encounters with Earth which provide frequent ground-based observation opportunities. This near-Earth asteroid has a semi-major axis of 1.324 AU, an eccentricity of 0.280, and an inclination of 1.622°

[14].

5 25143 Itokawa

514arftokaw

3 to + L tS '

Figure 2: Asteroid Itokawa’s Orbit Around the Sun [4]

Itokawa is an S-type asteroid, which means it has a stony composition and is mainly made of iron and magnesium-silicates. S-type asteroids are the second most common types of asteroids after C-types, and they compose about 17% of the asteroid population [23]. The albedo of S-type asteroids ranges from 0.10 to 0.22 which makes them moderately bright asteroids. These type of asteroids are the dominant type in the inner asteroid belt, within 2.2 AU, frequent in the central belt, within approximately three AU, and become rare further out. Normally, these types of asteroids have spectra that have a fairly steep slope at wavelengths shorter than 0.7 pm. In addition, they also have moderate to weak absorption features around one pm, an indicator of the presence of silicate, and two pm, indicator of

Pyroxene and Olivines. These features can be observed in Figure 3, which represents the reflectance spectrum of asteroid Itokawa integrated over the whole asteroid. Similar spectra were obtained using Hayabusa’s near-infrared spectrometer (NIRS) [24]. The composition of S-type asteroids is also similar to those of stony meteorites. The surface mineralogy of Itokawa is mainly dominated by silicates, pyroxene and olivine, and iron.

6 (25143) Itokawa 1.6

1.5

1.4

cu u 1.3 rac inu cc 1.2 cca> oj >

0.9

0.8 0.4 0.6 0.8 1.4 1.6 2.2 2.4 2.6 Wavelength (microns}

Figure 3: Itokawa’s Reflectance Spectrum [5]

Similar to any other celestial body, photometric and radar observations provided the initial shape model for Itokawa. The rotational period of this asteroid is about 12.1 hours. It has a rotation axis that is almost perpendicular to the ecliptic plane, and has a retrograde rotation direction [22]. Hayabusa data confirmed the pre-arrival results that were gathered by ground-based observations. The pole position was determined by tracking the ground control points on an Asteroid Multi-band Imaging Camera (AMICA) image, and no short­ term precession of the rotation pole was observed [25], Itokawa is similar in shape to a sea otter. This asteroid is composed of two parts, the “head” and the “body”, which are its small and large components respectively as shown in Figure 4.

7 Figure 4: Asteroid 25143 Itokawa, provided by JAXA The circles outlining the head and body of Itokawa

The surface of Itokawa shows a variety of features that suggest a complex evolutional history. The surface is divided into rough terrain which is roughly 80% of the whole sur­ face, mainly composed of numerous boulders, and smooth terrain [26], The smooth terrain on Itokawa is divided into two main parts: the Muses Sea, extending around the “neck” which is the area between the head and body and Sagamihara, which is the north polar region. In this paper, boulders are defined as rocks and features with distinctive positive relief that are larger than a few meters in size. Regolith, refers to any loose fine grained material on the surface, which range in size from granules to pebbles to cobbles to very small boulders [27]. Rubble is defined as a mixture of boulders that range in size from a few tens of centimeters to meters that are mostly unconsolidated. The largest boulder on Itokawa is called Yoshinodai and is located near the end of the body. There are several other large boulders with sizes larger than tens of meters on the west side. There is a black boulder at the end of the head which marks the body’s prime meridian. It is worth noting that there are no significant differences in the mineralogy of

8 the different areas of the asteroid. LIDAR observations indicate a fairly large amount of regolith on the surface of Itokawa, which is exceptional given the small size of this asteroid and its small gravity field [27]. From a scaling argument, and assuming Itokawa to be a weak object, it was predicted that there is about one meter of regolith on the surface of Itokawa [28]. The particle sizes observed in close-up images range from centimeters to several tens of meters. Smooth terrains such as Muses Sea are home to some of the smallest particles. These particles, along with other powdery materials are thought to exist as a result of impact processes. However, there could be other scenarios that explain their existence - such as possessing a higher ejection velocity upon impact and therefore not reaccumulating, segregation of boulders into the interior of Itokawa, and/or having been electrostatically raised up and removed by solar radiation pressure [29]. The presence of boulders on an asteroid provides the opportunity to study the geological evolution of the asteroid surface and its collisional history. These boulders are formed by either impact cratering or catastrophic disruption of the parent body or a combination of both. The boulder distribution on the martian satellites, as well as asteroid 243 Ida and asteroid 433 Eros, suggests that their boulder populations are attributed to impact cratering [30]. During an impact, only the ejecta with velocities lower than the escape velocity can fall back and settle on the surface under the influence of the asteroid’s gravity. The evidence of many large boulders in the AMICA images of the surface of Itokawa, instead of a few, indicates that they were not formed by one single impact on the surface of the asteroid. Rather, they originate from the disruption of a parent body of Itokawa.

Using different methods, the mass of Itokawa was determined from the Hayabusa track­ ing and navigation data. At first, the mass of Itokawa was estimated from the range and Doppler data of the Hayabusa spacecraft moving from the Gate Position to the Home posi­ tion. This estimated value was 3.51 x 1010 kg with an uncertainty of 15%, comprising er­ rors that were due to the effect of the radiation pressure that was much larger than Itokawa’s gravity [22], When Hayabusa was closer to the surface of Itokawa at about three kilome­ ters, the altitude maneuvers were stopped intentionally to allow for mass calculations using LIDAR and optical data in addition to the range and Doppler data. This time, the mass was estimated to be 3.43 x 1010 kg with an uncertainty of 5%. The mass estimates were calcu­ lated again, once when Hayabusa was 1400 to 800 meters from Itokawa, and again when Hayabusa was approximately 800 to 100 meters away, using LIDAR and navigation data.

The estimated masses were 3.58 x 1010 kg with an uncertainty of 5% and 3.54 x 1010 kg with an uncertainty of 6%, respectively. The weighted mean of these four mass estimates provide Itokawa with a mass estimate of 3.51 x 1010 kg with an uncertainty of 3% [22] . The volume of Itokawa was estimated using three independent methods. The easiest and fastest method is the limb profiling method that integrates the outlines of images. How­ ever, the disadvantage of this method is that it cannot reproduce concave surface areas. The second method is the geometric method where a shape model is constructed using stereo­ metric views of selected points from the surface. The accuracy of this method depends on how well the selected points are extracted. The last method combines stereo and photo- clinometry. This method solves for geometric stereo and surface brightness to determine slope in a scene. This is done by using a relative estimate of the scattering properties of the surface of the body [8]. The approximated volume from these three different methods is

1.84 x 107m3 with an uncertainty of 5% and yields with the mass estimate a bulk density of 1.95 g/cm3 [31]. To compute the surface slopes, accelerations, and potential, a polyhedral shape model of Itokawa and the total estimated mass of the body are combined. Orbital data indicates that a constant density model for Itokawa is consistent with the mass observations and can be used to evaluate the gravity field potential and surface acceleration. Those combined with centripetal acceleration and potential help calculate the ameliorated surface slope, potential, and total acceleration [32]. The potential is higher towards the two end regions

10 and lowest near the neck and northern areas. Similarly, the slope is the lowest near the north pole and the Muses Sea, which also happen to be the smooth terrain areas. Figure 5 shows the slope profile and Figure 6 shows the gravitational potential across the surface of Itokawa provided by the Small Body Mapping Tool [6],

r '- f . .

Slope (deg) 43.40.0324

Figure 5: Itokawa’s Slope Profile (Northern view on the left, Southern view on the right) [6]

Gravitational Potential (J/kg) ■0.0156..... ~ - 0.0143 - 0.0131 r .T7wmmmr--n - 0.0118 - 0.0106

Figure 6: Itokawa’s Gravitational Potential Profile (Northern view on the left, Southern view on the right) [6]

Table 2 provides more details on asteroid Itokawa.

11 Table 2: Asteroid Itokawa Characteristics Properties Values discovered by LINEAR discovery date September 26, 1998 dimensions 535 x 294 x 209 m effective body diameter 0.33 km volume 1.84 x 107m3 mass 3.51 x 10lokg density 1.9 g/cm3 semi-major axis 1.3238 AU eccentricity 0.2801 perihelion distance 0.9530 AU aphelion distance 1.6947 AU rotational period 12.1324 hrs orbit inclination 1.6218 longitude of the ascending node 69.0859 argument of perihelion 162.7961 mean anomaly 176.4789 orbital period 556.595 absolute magnitude 19.2 mag gravitational constant (GM) 2.1 x 10"9km3/s2 average orbital speed 25.37 km/s equatorial surface gravity ~ 0.0001 m/s2 escape velocity ~ 0.0002 km/s albedo 0.53 temperature ~ 206 K

Discoveries found by Hayabusa regarding Itokawa, such as its boulder rich surface with a rubble-pile structure, may in fact be common for small S-type asteroids. When considering parameters such as orbital elements, rotational period, and spectra, Itokawa is not part of any specific population. There is a larger number of elongates and binary asteroids in the near-Earth asteroid population, and these characteristics are consistent with these objects evolving into contact binary systems. The Hayabusa results and data from asteroid Itokawa help with the understanding of the general properties of small asteroids.

12 1.2.1 How Did Itokawa Evolve?

There are different hypotheses for how Itokawa was formed into its current shape. The first is the contact binary hypothesis, which implies the head and body of Itokawa had formed separately and then later came into contact with each other at a slow speed. In this case, it is assumed that the original shape of both bodies were retained and there was only some mass movement, leading to the shape we see today. The depressed part of the neck, viewed in the southern side, is one of the high-slope regions of Itokawa, and this slope could suggest that the material from the head has not yet completely merged into the neck and towards the body. Some also believe that the Muses Sea is a result of the collision of the head and body. The two different orientations of the principal axes of the head and body could also imply that these two bodies were separate and Itokawa is the product of a slow collision between the two [26]. The origin of the contacted body also gets further divided into different scenarios. The first one is the idea of formation by capture of two independent asteroids. This scenario is however unlikely, because encounters at relative velocities lower than the crushing velocity have very low probabilities [33]. There is also no solid evidence, such as different min­ eralogies, suggesting that the head and body have originated from two different asteroids. The second is the mutual capture of two fragments after a catastrophic disruption to the parent body. Numerical modeling results have shown that this scenario might be possible, however capture scenarios do not explain the ellipsoidal shapes of the head and the body. The third scenario deals with a catastrophic breakup of the parent body and assumes that the flying fragments accumulated after impact to produce some rubble-pile bodies. Two of these rubble-pile bodies then co-orbited each other and, after the development of their ellipsoidal shapes, came into contact [33], Another proposal is a rotational fission due to an excessive spin rate. This method does not seem very realistic when observing the current

13 shape of Itokawa [34]. The last possibility, is mass shedding of the asteroid due to tidal disruptions of a possible close planetary encounter [22]. The other hypothesis for Itokawa’s evolution and the formation of the depressed neck region is a possible large impact. This scenario assumes one original body which went through a large impact, creating the depression in the neck and the presence of the head and body. Impact experiments show that a large crater impact on the lateral side of an elongated body could produce a saddle-like depression [35]. An impact could have caused a large breakage around the neck, or even separated the parent body into two, the head and the body, with the possibility of the two bodies settling back together and forming Itokawa’s shape today. This thesis aims to reveal more insight into the formation of Itokawa by evaluating the different evolution hypotheses through data analysis and experiments. Since the distribu­ tion of boulders on an asteroid can give insight into the origin and evolution of small bod­ ies, the primary objective is to determine the distribution of boulders on asteroid Itokawa. Next, the boulder distributions are studied on both global and local scales. The boulder population is also evaluated based on latitudes both for the body of Itokawa today, and for a hypothetical ellipsoid body.

14 2 Tools and Datasets

2.1 Small Body Mapping Tool (SBMT)

The main tool used to map the outline of boulders and craters on asteroid Itokawa is an asteroid Geographic Information System (GIS), called the Small Body Mapping Tool (SBMT), created by the Johns Hopkins University Applied Physics Laboratory [6]. The SBMT facilitates the analysis of small body data by allowing the user to quickly search through and visualize the data. This tool incorporates modern interactive 3D visualization algorithms and graphics and runs on major operating systems such as Mac OSX, Windows, and Linux. It is written in the Java programming language and uses the Visualization Toolkit (VTK) for 3D visualization and modeling. Remote servers are used by the SBMT, via the internet to fetch data rather than storing all the data on the user’s machine. This is highly beneficial due to the large amounts of data that can be generated. In order to avoid repetitive downloads, the required data files are downloaded and cached as needed. The SBMT started as a visualization tool for asteroid Eros’ shape model, and mainly focused on visualizing data from the Near Earth Asteroid Rendezvous (NEAR) mission [17]. Currently, the SBMT provides a variety of Gaskell [36], Thomas [37], Stooke [38], and Hudson (Radar) [39] Shape Models, as well as, the option for users to import their own shape models. For the purpose of this project, the SBMT is used to view the Gaskell Shape Model of asteroid Itokawa, described in Section 2.2, in 3D. This tool allows easy interaction with the Itokawa shape model by using the mouse to rotate, spin, pan, and zoom in and out of the model. The left panel of the SBMT is divided into four main sections: Itokawa, AMICA, LIDAR, and Structures. Under the Itokawa tab, the Itokawa shape model can be viewed at four resolution levels: low resolution (49152 plates), medium resolution (196608 plates), high resolution (786432 plates), and very high resolution (3145728 plates). There is also the option to colour Itokawa by slope, elevation,

15 gravitational acceleration, or gravitational potential, as well as overlaying the coordinate grid which represents the latitude and longitude system. Figure 7, shows an outline of the

Itokawa tab.

Figure 7: Small Body Mapping Tool (Itokawa Tab)

Next to the Itokawa tab, there is the AMICA tab that allows the user to find data from the Asteroid Multi-band Imaging Camera (AMICA [22]) archived at the NASA Planetary Data System, and correctly geolocate them on the Itokawa shape model [8]. Users can search for images by number, date, or by selecting a region on the shape model. After the image is selected, the user can choose to map the image boundary or the full image, as well as having the option to centre the image in the window, view the frustum, generate backplanes, save original FITS image, and view some properties of the image. The Properties window gives the option to change the contrast of the image for better viewing results, in addition to seeing the spacecraft distance and the time the image was captured. Filter types, as well as , incidence, emission, and phase angles are also viewable. Details regarding the

AMICA datasets are provided in Section 2.3. Another tab is the LIDAR tab which was not directly used for this project. Similar to the AMICA tab, this tab permits the user to search

16 and overlay LIDAR data. Further information about the LIDAR datasets are provided in

Section 2.4. The last tab, Structures, is divided into four sections: Paths, Circles, Ellipses, and Points. For this project, the Ellipses Structure was used to outline boulders and craters and map them from the images onto the asteroid, automatically providing the correct scale of the features outlined. The file format for the Ellipses Structure is a tab separated ta­ ble with seventeen columns consisting of: Identifier; name; x-coordinate; y-coordinate; z-coordinate; latitude; longitude; distance from the centre of body; slope; elevation; grav­ itational acceleration; gravitational potential; size (twice the semi-major axis); flattening (ratio of semi-major axis to semi-minor axis); angle; color; and ellipse angle. The Paths Structure was used to draw the major and minor axes of the boulders and craters in the same fashion as the ellipses. The file format for the Paths Structure is XML, which consists of a series of path elements containing one element per path. Each element is comprised of an ID, color, length, name, and vertices, where each vertex is specified as a 3D point in spherical coordinates represented by latitude, longitude, and radius. The structures can be saved and loaded to and from the disk as text files.

2.2 Gaskell Shape Model

The shape model of asteroid 25143 Itokawa, the Gaskell Itokawa Shape Model, was created by Robert Gaskell and derived from 775 AMICA images taken from September 11th 2005 to November 12th 2005. The shape and topography of Itokawa, the spacecraft’s position, as well as data to set the global scale, such as radar or laser ranging, is required to create a shape model. A set of control points defined as fixed points on the surface that can be identified in several images are used for navigating a spacecraft when it is close to the small body. Through stereographic analysis, the body-fixed location of the control points

17 are found given that the spacecraft positions are known for each image. Initially, locations are chosen for the control points and their image space positions are predicted. Then, the differences between the predicted and observed locations in the images are minimized in the least squares fashion to estimate their actual locations. For navigation purposes, the location of the body-fixed control points are assumed and a spacecraft state is chosen for an image. The difference between the predicted and observed control point locations are minimized in order to estimate the actual spacecraft state. It should be noted that the control point locations and the spacecraft states, both have uncertainties, and therefore, many images are iterated to reach a best fit [8]. The control points can be chosen using a few different methods. Some use a technique that automatically identifies features and correlates them between various images for stere- ographic analysis [40]. The down side to this method is that it requires images with similar resolution, illumination, and balanced stereo separation, which results in large absolute un­ certainties. The NEAR mission made use of craters as landmarks [41], but this method is only useful if craters do exist. However, due to the small size and number of craters on

Itokawa, this method was not possible for the Hayabusa mission. In both cases many pixels are required to specify a single surface point, and therefore, there is a significant loss of resolution relative to the imaging data. The Gaskell model uses a new technique in which a control point or landmark is defined as the centre of a small L-map (a digital topography and albedo map) determined from multiple images with stereophotoclinometry (SPC) [8]. The main constituent of the L-map construction is SPC. With this technique, the albedo and slope are determined at each pixel using linear estimation. This is done in order to minimize the summed square brightness residuals at each of the pixels and can be done for anywhere from three to hundreds of images, each with varying viewing angles and illumination. In order to produce a height distribution, the slopes are integrated with some sampled heights from external sources,

18 such as limbs and other maps, in order to restrain the integration. The L-maps are distributed over multiple viewing geometries, ranges, and illuminations as they are in three dimensions, and therefore, help create artificial images that are very close to their corresponding real imaging data. They are used to correlate diverse images and to help locate their common control points and landmarks. Two images can be co­ registered even if they do not contain any common points due to the rigid structure of the L-maps. Another advantage of L-maps is that they can provide nearly 90 degrees of stereo separation and can easily be correlated with topography from other L-maps. The control points can be located anywhere on the body because they can represent surface types, therefore, the L-maps can accurately characterize the surrounding albedo variations and topography to approximate the resolution of the best images. The L-maps can also allow other data types such as laser or radar to be associated into the solution. The extracted imaging data used for the construction of the L-maps suffers from errors in spacecraft position, camera pointing, control point location, and rotational kinematics that further result in distortions. In order to reduce the errors in the construction of the L-maps, the first landmarks to be used are usually the most obvious ones, such as large craters and rocks. The stereo separation is kept small to reduce the image sampling errors. The future plan is to improve this method and set it apart from others by using an iterative approach to improve geometry solutions and make smaller-scale L-maps that would allow for larger stereo separations, hence more precise topography and geometry solutions [8]. Another aspect in the construction of the Gaskell Shape Model, is the composition of the global topography model (GTM), which provides the basics for initial gravity estimates, as well as providing effective estimates for new L-maps allowing for almost immediate convergence. The Itokawa shape model is constructed from 871 L-maps, which are three dimensional illustrations of parts of the surface with control points as their centers, with each L-map representing about 10,000 surface points. The traditional models use trian­

19 gular plate formats, whereas the Gaskell model uses quadrilateral cells whose vertices are connected using a simple labeling format. The advantage of the quadrilateral cell model is that it allows for a denser model and halves the spacing between points at each step which results in faster computation time. It should be noted that this model can easily be converted into a triangular form using a simple algorithm. The Gaskell model, the global topography model (GTM), is in the Implicitly Connected Quadrilateral (ICQ) format and provided in four different resolution levels. The vertices are labeled in a grid points format on the faces of a cube. Each face of the cube, referred to as F - contains (Q+l)2 vertices, where Q represents the resolution (128 in this project). The vertices are represented as v (I,

J, F), where I represents the rows of the grid and ranges from I = 0 to I = Q and J represents the columns of the grid and ranges from J = 0 to J = Q. Each face also has Q2 facets shown as f(I,J,F), where I ranges from I = 0toI = Q- l, and J goes from J = 0toJ = Q- l. Figure 8 shows the unfolded cube containing six faces on the right, and a representation of the grids on the left.

0 ------— I ------Q 0

' I I I J . . . . F ...... | 5 | 4 | 3 2

Q •

Figure 8: Implicitly Connected Quadrilateral Model and Grids [7]

The vertices of the facets are represented as: v (I, J, F), v (I, J+l, F), v (1+1, J+l, F), and v (1+1, J, F). Figure 9 represents the overall ICQ format and labels.

20 Figure 9: Representation of Implicitly Connected Quadrilateral Labels [8]

The faces share common vertices at each of the 12 cube edges - for example, the last row of face one has the same vertices as the first row of face two. The eight corners of the cube also have shared vertices from all three of the joined faces. The file structure containing the data is in a format where the first line is the value of Q, and the rest of the file is consistent of 6(Q+1)2 lines holding the vertices. A FORTRAN code, similar to the following could be used to read the file [7]: READ (10,*) Q DO F = 1, 6 DO J = 0, Q DO I = 0, Q READ (10,*) (V (K, I, J, F), K =1, 3)

ENDDO ENDDO ENDDO The vertices are represented by three vectors, and sometimes other components con­

21 taining different surface characteristics are added. It should be noted that the quadrilateral facets are not always flat in this shape model, as not all four vertices are necessarily copla- nar; the cross product of the diagonals of the facets define their normals. The coordinate system of this shape model is a right-handed Cartesian body-fixed frame in which the z-axis goes through the rotation pole and the x-axis goes through a feature on the surface called the “black rock” at zero longitude [7].

2.3 AMICA

The Asteroid Multi-band Imaging Camera (AMICA) is one of the optical navigation cam­ eras onboard the Hayabusa spacecraft and consists of three components; three charge- coupled device (CCD) camera heads; an analog signal processing unit; and a digital signal processing unit. The AMICA camera head flight model is a charge-coupled device camera (CCD) and a refractive telescope containing radiation-resistant and antireflection-coated lenses providing an effective field of view (FOV) of 5.83 x 5.69 degrees [1]. Itokawa was mapped using AMICA prior to the “touch-and-go” sampling phase to find suitable sample sites that are free of any major obstacles, such as any large craters and boulders that would affect the spacecraft’s landing [42]. AMICA obtained over 1660 images starting shortly after launch, until November 19th 2005, after asteroid Itokawa was encountered. The CCD device has a frame-transfer configuration that contains an image area and a frame storage area. The distance between the camera and the observed objects in focus, also referred to as the depth of field, is designed to be from 75 meters to infinity [9].

The AMICA instrument is installed at the bottom panel of Hayabusa and has its optical axis boresighted with the near-infrared spectrometer (NIRS) [43] and the LIDAR [44], To reduce light from sources outside of the FOV, a cylindrical stray light baffle is attached in front of AMICA. The effective FOV, which is 5.83 x 5.69 degrees, is covered by 1024 x

22 1000 pixels. Therefore, the instantaneous FOV (IFOV) corresponds to 20.49”/pixel which corresponds to about seventy centimeters from Itokawa’s surface at home position. Each image data was stored in a 1024 x 1024 array with two 12 column wide masked areas on the left and right edges of each array for monitoring the zero-signal level, as shown in Figure

10.

12 pixel 1000 pixel 12 pixel

( 0, 0)'

-$c o - 1024 pixel 1' > 'sc V * sc

H address

Figure 10: AMICA Image Data Storage [9]

The exposure time for AMICA is controlled electronically and it has no mechanical shutter. There are thirty different exposure times ranging from 5.44 ms to 178 seconds, as well as, a “zero-second” exposure time which is less than 1 ps. The large exposure time is used to image both the bright asteroid and the surrounding dim stars.

AMICA is an eight filter system in which seven of the bands are compatible with the Eight Color Asteroid Survey (ECAS) [45] which are the standard for ground-based taxon­ omy. There is also a neutral density filter (ND) due to the fact that the target was changed from a low-albedo asteroid (4660 Nereus) to a high-albedo one (Itokawa). The v-band data is calibrated using in-flight stellar observations, while the other band data is calibrated by

23 comparison to the ground-based observations in order to avoid uncertainty and error of solar radiation in those bands. Table 3 provides a short summary of the AMICA specifications.

Table 3: AMICA Instrument Specifications [1] Parameters Values Effective Lens Diameter 15 mm (F/8.0) Focal Length 120.80 mm (fixed) Field of View 5.83 x 5.69 deg CCD Dimensions 1024 x 1000 pixels Pixel Size 12 micron square Pixel Resolutions 20.490 arcsec/pix Electronic Shutter Exposure Time 5.33 ms - 178 s Image Memory Storage 16 frames Filters 7 bands Polarizers 4 position-angle glass polarizers

For the purposes of this study, the AMICA dataset, “HAYABUSA AMICA V1.0” was obtained from the NASA Planetary Dataset System (PDS) website. This dataset contains all the images from the mission in addition to pre-flight flat field images. The data directories are divided into folders named in the yyyymmdd format, and they are further divided into the flight images (FITS files) and the labels (LBL files), grouped into the daily directories based on the date the images were taken. The file names follow the general format of [T][C]_[nnnnnnnnnn]_[filter].fit for the FITS files, or [T][C]_[nnnnnnnnnn]_[filter].lbl for the label files where: [T] is the data type, either “S” for scientific or “N” for navigation; [C] indicates the name of the instrument, where T stands for AMICA; [nnnnnnnnnn] represents the mission elapsed time when the image is generated on the digital signal processing unit; and [filter] is the filter identifier (ul, b, v, w, x, p, and zs) [1]. Figure 11, shows an example AMICA image. This image was taken when Hayabusa was roughly a kilometer away from the surface of Itokawa and in the Touchdown Phase.

24 Figure 11: Example AMICA Image (st_2566271576_v.fit)

2.4 LIDAR

The light detection and ranging instrument, LIDAR, aboard the Hayabusa Spacecraft helps provide information on the shape, mass, and surface topography of asteroid Itokawa. The primary objective of the Hayabusa LIDAR, is to establish the range between the spacecraft and the asteroid for navigation purposes during surveying and collection phases of the mission, as well as to provide a precise estimate of the location of the spacecraft with respect to the asteroid. The secondary objective of the LIDAR, is to determine the mass of

Itokawa, as well as measuring its global surface elevation and roughness [46]. The LIDAR determines the round-trip time of flight for the laser light to travel from the spacecraft to the asteroid to measure the distance. The LIDAR measures the stop time by filtering a pulse received from the surface of the asteroid and measuring its time of peak intensity which allows the LIDAR to average the topography within the LIDAR footprint on the surface, approximately 5 by 12 meters, at a 7 kilometer altitude with normal incidence [31]. The pulse detection technique allows for identifying features that are smaller than the LIDAR footprint. It should be noted that the centre-to-centre along-track footprint spacing is highly variable throughout the mission, and the range measurements have an effective resolution of 50 cm, which are then converted to asteroid radius profiles and topographic height after the orbit and pointing errors correction. The shot locations are determined to within 10 meters in the cross-track and along-track directions. The LIDAR, which is one of the orbiter remote sensing instruments on Hayabusa Space­ craft, consists of a pulse generator, Nd:YAG laser transmitter with a 1.064 micrometer wavelength laser pulse and a pulse energy of 10 mJ, a silicon avalanche photodiode detec­ tor, and a 0.126 meter diameter Cassegrain telescope as the receiver [2]. It began operating from September 10th 2005, at a distance of about 49 meters from the asteroid, through November 25th 2010, where a total of 4,107,104 shots were fired and 1,665,548 returns were detected [31]. It should be noted that the number of returns were significantly re­ duced after October 2th 2005, when the spacecraft positioning became less accurate due to a failure in two out of the three reaction wheels. Table 4 provides more details about the LIDAR instrument.

26 Table 4: LIDAR Instrument Specifications [2] Parameters Values Volume 0.0137 m3 Mass 3.56 kg Total Power 22.00 W Laser Type Q-switched, diode-pump Nd:YAG Wavelength 1.064 pm Laser Energy 10 mJ pulse"1 Laser Power Consumption 17 W Pulse Width 14 ns Pulse Repetition Rate 1 sec"1 Beam Divergence 1.7 x 1.7 mrad Telescope Type Cassegrain Mirror Composition SiC Telescope Diameter 0.126 m Detector Type Silicon avalanche photodiode (Si APD) Receiver Type Filtered peal trigger Time Resolution 3.0 nsec Range Resolution 0.5 m Range Accuracy (at 50 km) 10.0 m Footprint Size (at 7 km) 12.0 x 4.9 m

The LIDAR dataset, “HAYABUSA LIDAR V1.0” [2], was obtained from the NASA PDS, and contains the Hayabusa LIDAR data for all mission phases. The LIDAR data is divided into two categories: the Experimental Data Record (EDR) and the Calibrated Data Record (CDR). The EDR is the source of the science data, while the House Keeping Experiment Data Record (HKEDR) and telemetry provide the data needed for determining the position of the spacecraft relative to asteroid Itokawa. The CDR consists of the LIDAR science and telemetry data that has been converted to engineering and physical units, as well as the resulting orbit, geometric, and calibration data that can be used to determine the location of the LIDAR boresight on the surface of the asteroid. The CDR data, is further divided into two sets, the unfiltered (UF) version of the data, and the filtered (F) which does not include data with estimated surface points located more

27 than 10 meters from the predicted intersection of the vector defining the pointing of the LIDAR boresight and the shape model of the asteroid [2]. In addition to the LIDAR data, the CDR files contain information on the boresight of the Near Infra-Red Spectrometer (NIRS) aboard the spacecraft. The filtered and unfiltered data are then further divided into three sets over three months of data collection. The first set, corresponds to the data acquired during the Gate Position Phase when the Hayabusa spacecraft was at a distance of about 20 km from the surface of Itokawa. The second set, corresponds to the data acquired during the Home Position Phase, when the spacecraft was at a distance of 3-7 km from the surface of the asteroid. The last set corresponds to the Touchdown Phase, when Hayabusa attempted to sample the surface of asteroid Itokawa. Figure 12 shows the LIDAR path in colour and the spacecraft position in black from the October 2005 data on the left, and the LIDAR path from the November 2005 data on the right.

03

Figure 12: LIDAR Path Plots (coloured: lidar path, black: spacecraft position)

The LIDAR data helps in understanding the orbit and position of the spacecraft during the different phases. These datasets were also used to improve the shape model and the

SBMT.

28 3 Distribution of Boulders on Asteroid Itokawa

3.1 Motivation and background

The distribution of boulders on an asteroid can give insight into the origin and evolution of small bodies (e.g., [47]). This chapter focuses on the distribution of boulders on aster­ oid Itokawa, using recent improvements in the geolocation of images from the Hayabusa spacecraft. The objective is to confirm and update any previously identified trends in the global and regional distributions of boulders on Itokawa. Trends found should provide new insights into how Itokawa formed into its current state as well as how its surface might have changed and evolved over time. In particular, boulder distribution over the surface of the asteroid might provide a means to test the hypothesis of whether or not Itokawa is a contact binary. When two bodies collide there is a spectrum variety of possible outcomes, varying from simple inelastic rebound to the complete shattering of both colliding bodies. Collisions in the Solar System involve different material types for both the target and the projectile, rang­ ing from weak icy bodies to strong metallic ones. There are some laboratory experiments on collisional disruptions and impact fragmentations presenting the break-down of different materials after impact. In a study [10] at the Ames Vertical Gun Range (AVGR), a series of impact experiments were done using gas, powder, and compressed air guns. The spherical targets were hung in the impact chamber which was evacuated to a pressure of roughly six millibars. All shots were centrally aimed at the spherical mortar targets that are made of a mixture of cement, sand, and water. These mixtures ranged from being weak or strong depending on the ratio of the ingredients. The projectiles are made of steel, aluminum, and

Pyrex. They are shot at the targets with the gun arbitrarily set to an angle of 15 degrees from the horizontal. Figure 13 shows the cumulative mass distribution of fragments for a strong aluminum projectiles, each containing two fits to best represent the data distribution.

29 100 0.69 1.5 0.39 0.30 2.6 0.46 o.06 3.0 aso z a: mid 2 z> z LU > 10 3 2 3 O

ALUMINUM PROJECTILES STRONG TARGETS

NORMALIZED MASS M/M t

Figure 13: Cumulative Mass Distribution of Fragments (Aluminum Impact Target) [10]

As seen in the impact experiments [10], the fragment distribution follows a more or less linear pattern on a log-log plot, showing a larger number of fragments at smaller sizes and a smaller number of fragments are larger sizes. The break in the slope is an indicator of a disrupted body. Other studies show fragments in catastrophic dismptions by impact [11], in which a basalt sphere was destroyed by a central impact of a cylindrical polycarbonate projectile shot via a two-stage light-gas gun. Figure 14 shows selected frames from the high-speed film capturing this collision.

30 Figure 14: Results of a Catastrophic Disruption by Impact [11]

This catastrophic impact has resulted in a dispersed body. Here a dispersed body is defined as one that has been broken into pieces without leaving any traces of the parent body. On the other hand, a disrupted body is referred to as one that has gone through a breakup, but part or parts of the parent body are still intact. The current state of asteroid Itokawa, suggests that it went through a strong enough collision to produce a disrupted body but not strong enough to result in a dispersed body [48],

3.2 Methodology

To understand the boulder distribution on asteroid Itokawa, a good knowledge of the num­ ber of boulders is required to create an unbiased survey for analysis. The first step for this analysis is to map all the boulders that are bigger than five meters in length along the

31 long axis. Due to different image resolutions, boulders of at least five meters long were chosen to avoid introducing any bias in the boulder count. The Small Body Mapping tool

[6] facilitates the mapping of boulders and craters on the surface of Itokawa by allowing the user to overlap correctly geo-located Hayabusa AMICA [22] images onto the Itokawa shape model. The Ellipses function is used to outline boulders which can then be mapped from the images onto the asteroid, automatically providing the correct scale of the features outlined. Figure 15 shows an example.

Figure 15: Example of an AMICA Image Overlayed on the Shape Model and Sample Mapped Boulders (blue: craters, purple and yellow: boulders, purple line: semi-major axis, yellow line: semi-minor axis)

In the SBMT, multiple images can be viewed, to handle any biases that might occur as a result of the lighting and viewing geometry. It should be mentioned that through extensive use of the SBMT, it was found that there were some misalignments in the projection of the AMICA images. The author of this thesis helped with debugging this tool by observing

these differences and suggesting improvements in the geo-location of the AMICA images.

32 To ensure the entire surface of Itokawa was covered, over one hundred AMICA images [22] were used, providing more than one chance to verify the boulder sizes and locations.

These images were chosen from the Home and Touchdown phases which provided better resolution. A total of 820 boulders, along with 37 craters or potential craters [42] were measured over the entire surface area of 0.4011 km2 in order to evaluate any correlation between these craters and boulders. This total boulder count is about sixty percent higher than a similar previous study [30]. Following a previous study [42], the craters are identified based on three surface fea­ tures: quasi-circular depression, quasi-circular features with smooth and flat floors, and quasi-circular facets. The quasi-circular features refer to those that are usually results of hypervelocity impacts, similar to impact craters observed on other bodies. However, in this case the presence or absence of rims are ignored. Quasi-circular features with smooth and flat floors are the depressions that are buried with fine materials. There exists a transition between the first and second type of craters as a result of the gravitational motion of finer materials. The third type of surface feature used in identifying craters is the quasi-circular facets which do not have bowl-like feature. These craters could however still be formed by large impact events. Laboratory experiments have shown that the curvature of the crater floor is highly determined by the curvature of the original surface [35], The size-frequency statistics of the boulders are done for the asteroid on a global scale, as well as regional scales comparing the head and the body of the asteroid, and comparing the east and west sides of Itokawa. Figure 16 demonstrates how the boulder distributions are divided. The head and the body boulders are shown in two different colours and divided around the neck of Itokawa. The boulders are also separated into two equal degrees of longitude that align with the east and west sides of Itokwa.

33 Figure 16: Boulder Distribution on the Head (red) vs. Body (black) of the Asteroid (on the left) Boulder Distribution on the East (black) vs. west (red) sides of Itokawa (on the right)

In order to normalize the results, the cumulative number of boulders are divided by the area of each section. In order to calculate each section’s area, an Interactive Data Language (IDL) code is used to read and extract the area file from the SBMT’s website [6]. In this thesis, all the analysis are done using the shape model containing 128 plates, which represents medium resolution. In this file, the area of each plate, along with its corresponding latitude and longitude, are given. The different sections, such as the head and body of the asteroid, are then separated using latitude and longitude values, and the corresponding plate areas are added up. Figures 17 and 18 show the mapped boulders on the head versus body and on the east side versus the west side, respectively. The blue ellipses represent the outline of the craters, purple ones represent the outline of boulders from high resolution images, and yellow ones represent the outline of boulders from low resolution images.

34 Figure 17: Mapped Boulders on the Head (left) and Body (right) of Asteroid Itokawa (blue: craters, purple and yellow: boulders, purple line: semi-major axis, yellow line: semi-minor axis)

35 Figure 18: Mapped Boulders on the East (upper) and West (lower) Sides of Asteroid Itokawa (blue: craters, purple and yellow: boulders, purple line: semi-major axis, yellow line: semi-minor axis)

The boulder size distribution analyses are performed by defining the size in four dif­ ferent manners: the long axis, the short axis, the average of the long and short axis, and the root-sum square (RSS), which is the best method for this type of analysis. The ratio between the long and short axes of the boulders vary and therefore RSS average reduces bias if the ratio is large. Using RSS values of the boulder sizes are also very similar to using the volume of boulders which is the best estimate. In this thesis the analysis focus on the results from the RSS values. It should be noted that the SBMT outputs the “size” of

36 the ellipses outlining the boulders, which are in fact the long axes of the boulders, as well as providing the flattening of each ellipse. In the SBMT, flattening is defined as the ratio between the long and short axes, meaning if the ratio is one if the ellipse is in the shape of a perfect circle. The analysis and plots are done using IDL.

3.3 Results and Analysis

An IDL program is written to determine the size-frequency distribution of boulders on asteroid Itokawa. The plots have the size of the boulders in meters on their x-axes and the number of boulders per unit area on their y-axes. In order to compute the number of boulders per unit area, a histogram function with one meter bin sizes is used.

3.3.1 Overall Distribution

To understand the distribution and origin of boulders on asteroid Itokawa, a cumulative boulder size distribution over the surface area of 0.4011 km2 is produced. Figure 19 shows the analysis for when the boulder size is defined as the long axis or the short axis, and Figure 20 shows the analysis for the average boulder size and RSS size. It should be noted that all the boulder distribution analyses are done on log-log plots and the histogram bin sizes are set to one meter.

37 10000 10000

| 1000 | 1000 2o £3 100 o 100

po»ver-indv\: -3.33 ± 0.05

100 1 10 100

Figure 19: Cumulative Boulder Size Distribution Per unit area on the Entire Surface of Asteroid Itokawa Size defined as long axis (left) and as short axis (right) Black: Data Used for Power-index Calculation, Red: Omitted Data

soooo 10000

2 5 1000 - gc i 100 o 100 .* 33 oi powor-inclcx: -3.27± 0.07 power-index: -2.98 ± 0.1

1 100 1 100

Figure 20: Cumulative Boulder Size Distribution Per unit area on the Entire Surface of Asteroid Itokawa Size defined as average (left) and as RSS (right) Black: Data Used for Power-index Calculation, Red: Omitted Data

The power-index for the cumulative boulder size distribution plots for long-axis, short- axis, average, and RSS boulder sizes are: -2.98 + 0.09, -3.33 + 0.05, -3.27 ± 0.07, and -2.98

± 0.08, respectively.

38 Table 5: Power-index Values for Cumulative Boulder Size Distribution over the Entire Surface of Itokawa Defined Size Power-index Value Long-axis -2.98 ± 0.09 Short-axis -3.33 ± 0.05 Average -3.27 ± 0.07 RSS -2.98 ± 0.08

It should be noted that the red data points in the plots are not used for calculating the power-index, as they are either outliers or insignificant for the purposes of this analysis. The power-index, or slope, of -2.98 ± 0.08 agrees with the smaller sampling from [30] which was -3.1 ± 0.1. The power-index can exhibit minor variance due to histogram bin sizes or the definition of the boulder size. IDL’s LINFIT function is used to fit the data to a linear model of y = A + Bx by minimizing the chi-square error statistic, explained in Appendix A. A closer look at the cumulative boulder size distribution reveals a break in slope around boulder sizes of about fifteen meters, as shown in Figure 21. It should be pointed out that the points shown in green were omitted for slope calculations for two reasons. The repetitive points, such as those between boulder sizes 40 to 60 meters, have no meaning since they show a void of boulders, and therefore do not play a significant role. The green points, less than eight meters in boulder size, were not used for the purpose of this representation to avoid introducing bias in the data. Since there is a lack of information on boulder sizes smaller than five meters the trail off is not clear and the slope might be modified.

39 10000

* « % 1000

%l~ 5 3 O CD o 100 * * s p

Power-index; - 2.4 ± 0.1 Power-index; - 3.5± 0.1 l 1 10 100 Boulder Size [m]

Figure 21: Cumulative Boulder Size Distribution per unit area Black and red: two different boulder populations, green: data points omitted

The break in slope suggests that there might be two different boulder populations with unique origins. Boulder sizes roughly between seven and fifteen meters have a power-index of -2.4 ±0.1, and boulders greater than fifteen meters, have a power-index of -3.5 ±0.1. To show the break in slope more evidently, a smaller bin size of ten centimeters is used, as shown in Figure 22.

40 10000

| 1000

T3a> 3 O CD 100

Power-index: -2.4 ± 0.1 Power-index: -3.5 ± 0.1

1 10 100 Boulder Size [m ]

a

Figure 22: Cumulative Boulder Size Distribution per unit area, bin size 10 cm Black and red: two different boulder populations, green: data points omitted

3.3.2 Head vs. Body

For further studies, Itokawa was divided into two parts; the head and the body. Figure 23 shows the cumulative boulder size distribution per unit area on the head (red) and body (black) of asteroid Itokawa for long and short axis boulder sizes. Figure 24 shows the same plots for average and RSS boulder sizes.

41 10000

| 1000 £ tooo v> © T> £3 £ © too o too

©

3 63 u.3 power-index (body): -3.12 ± 0.09 power-index (body): -3.60 ± 0.11 power-index (head): -2.49 * 0.13 power-index (head): -2.65 * 0.06

1 100 1 100

Figure 23: Cumulative Number of Boulders per unit area on the Head (red) and Body (black) of Itokawa Size defined as long axis (left) and as short axis (right) Black and red: two different boulder populations, green: data points omitted

10000 10000

I 1000 ¥ 1000 I s o 100100 ,»© 3 3 E <3 power-index (body): -3.44 ± 0.09 power-index (body): -3.2 ± 0.09 power-index (head): -2.84 ± 0.11 power-index (bead): -2.45 £• 0.11

1 10 100 1 10 100 Boulder size [m] Boulder size [m]

Figure 24: Cumulative Number of Boulders per unit area on the Head (red) and Body (black) of Itokawa Size defined as average (left) and as RSS (right) Black and red: two different boulder populations, green: data points omitted

The power-index for the cumulative boulder size distribution plots of head versus body for long axis, short axis, average, and RSS boulder sizes are: -2.49 ± 0.13 vs. -3.12 ± 0.09, -2.65 ± 0.06 vs. -3.60 + 0.11, -2.84 ± 0.11 vs. -3.44 + 0.09, and -2.45 ± 0.11 vs. -3.20 ±

0.09; respectively. The individual plots of each care are presented in Appendix B.

42 Table 6: Power-index Values for Cumulative Boulder Size Distribution on the Head vs. Body ______Defined Size Power-index (Head) Power-index (Body) Long-axis -2.49 + 0.13 -3.12 + 0.09 Short-axis -2.65 + 0.06 -3.60 + 0.11 Average -2.84 + 0.11 -3.44 + 0.09 RSS -2.45 + 0.11 -3.20 + 0.09

It is evident that the cumulative number of boulders per unit area on the body has a steeper slope than the head. Shallower slopes imply greater variation in size of the boul­ ders. This means that the body has boulders that are closer in size whereas the size of the boulders on the head have a larger variation. The significance of these differences is further explained in Section 3.3.4.

3.3.3 East vs. West

Further analysis where the measured boulder population is separated into two equal de­ grees of longitude that align with the east and the west sides of Itokawa shows additional differences. Figure 25 shows the cumulative boulder size distribution per unit area on the east (red) and west (black) of asteroid Itokawa for long and short axis boulder sizes. Figure 26 shows the same plots for average, and RSS boulder sizes.

43 10000 10000

2 | 1000 5 1000 2v

100 o 100

.8

power-index (west): -2.55 ± 0.08 pcmer-intlcx (west): -3.13 ± 0.09 power-index (east): -3.26 ± 0,15 power-index (east): -3.43 * 0.16

1 10 100 1 100 Boulder size fm}

Figure 25: Cumulative Number of Boulders per unit area on the East (red) and West (black) of Itokawa Size defined as long axis (left) and as short axis (right) Black and red: two different boulder populations, green: data points omitted

10000 10000

| 1 0 0 0 1000

100 100

10

power-index (west)* -2.90 ± 0.08 power-index (west): -2.56 ± 0.07 power-index (east): -3.53 ± 0.15 power-index (east): -3.15 ± 0.14 1 1 100 1 10 100 8oulder size [m]

Figure 26: Cumulative Number of Boulders per unit area on the East (red) and West (black) of Itokawa Size defined as average (left) and as RSS (right) Black and red: two different boulder populations, green: data points omitted

The power-index for the cumulative boulder size distribution plots of east side versus west side for long axis, short axis, average, and RSS boulder sizes are: -3.26 + 0.13 vs. -2.55 + 0.08, -3.43 ± 0.16 vs. 3.13 + 0.09, -3.53 + 0.15 vs. -2.90 ± 0.08, and -3.15 + 0.15 vs. -2.56 + 0.07 respectively. The individual plots are provided in Appendix B.

44 Table 7: Power-index Values for Cumulative Boulder Size Distribution on the East vs. West Defined Size Power-index (East) Power-index (West) Long-axis -3.26 + 0.13 -2.55 + 0.08 Short-axis -3.43 + 0.16 3.13 ±0.09 Average -3.53 + 0.15 -2.90 + 0.08 RSS -3.15 + 0.15 -2.56 + 0.07

This difference can be accounted for by the higher number of large boulders at high surface elevations where no regolith coverage exists, thereby resulting in a steeper slope. The east side of Itokawa contains the majority of the Muses Sea lowland and the west side includes most of Sagamihara lowland, where the regolith dominates and has probably submerged the boulders.

3.3.4 Conclusion

The surface of asteroid Itokawa is covered with at least 820 boulders greater than five meters in the long-axis and 37 craters or potential craters. The break in slope of boulder sizes versus the cumulative number of boulders, suggests two different boulder populations with different origins. The two different populations are further supported by the relatively large difference in the power-indices of the head versus body. In addition, a steeper slope, in other words a greater power-index, could imply that the surface is relatively older, as demonstrated in Figure 27.

45 Fresher Surface Older Surface

Figure 27: Difference in Slopes

A boulder distribution with a steeper slope suggests an older surface. This is so, be­ cause as the surface evolves and gets affected by different impacts, the larger boulders break down, resulting in a higher number of smaller boulders and a lower number of large boulders [10]. The break in slope of the cumulative boulder size distribution on Itokawa suggests that Itokawa has gone through disruption, as well as, evolution. One should keep in mind that asteroids normally go through many collisions and are shaped via processes of accretion and fragmentation [49]. The power-index of plotted boulders varies on different regions of Itokawa. The difference in the power-index of the head and body of this asteroid supports the contact binary hypothesis and demonstrates that the two have gone through separate evolutionary paths. When comparing the west and east regions of Itokawa, the power-index difference appears to reflect differences that can be associated with surface el­ evation. Larger boulders at high elevations are not affected by the presence of fine-grained regolith that might cover these boulders up. At low elevations, the boulders are hidden by this regolith, which dominates these low regions. Figure 28 demonstrates possible evolu­ tionary stages of Itokawa based on the above analysis and results.

46 Figure 28: Possible Evolution of Itokawa

Figure 28 shows the parent body of Itokawa, still containing some regolith, and another body before collision in frame 1. Frame 2 of the same figure demonstrates the state of the two bodies after impact, in which part of the parent body remains unchanged and still containing regolith. Attention should be drawn to the fact that the ejecta particles with velocities higher than 10 to 20 cm/s would easily escape Itokawa’s surface [29]. Later, some of the larger particles, or possibly one big particle (the “head”), orbits the “body” while sweeping some of the smaller particles in its orbit. In frame 4, the head and body finally come together. The larger particles or boulders settle on the head and body of Itokawa and the fragments re-accrete to form Itokawa [49]. The catastrophic disruption and re-accretion scenario, previously mentioned in Section 1.2.1, explains the head and body shape of Itokawa that further supports that Itokawa is a contact binary. Two results from a previous study’s simulations are presented in Figure 29. Figure 29: Results of Numerical Simulations [12]

Previous numerical simulations [12] have shown that fragments can accrete into aggre­ gates that orbit one another briefly and then come together at a low velocity, as presented in frames 3 and 4 of Figure 28.

3.4 Mission Considerations

The distribution of boulders on a small body can give insight into the formation and origin of that body. This distribution could also be used in future mission planning, especially those that involve a sample return where boulders represent a hazard. For sample return missions, or any mission that involves the spacecraft to land on the surface of a body, temporarily or long term, a good knowledge of the surface is required. An example of an asteroid sample return mission is the OSIRIS-REx (Origins, Spectral Interpretation, Resource Identification, and Security-Regolith Explorer) Mission [50].

3.4.1 Introduction to OSIRIS-REx

The OSIRIS-REx Asteroid Sample Return Mission, chosen by NASA as the third New

Frontiers mission, will launch in September 2016. This mission will thoroughly character- ize the near-Earth asteroid 1999 RQ36, which is the most accessible carbonaceous asteroid, as well as the most potentially hazardous asteroid known [51]. The detailed characteriza­ tion of RQ36 and return of immaculate samples will enhance the knowledge of the early stages of planet formation in addition to the sources of organics that may have led to the origin of life. The OSIRIS-REx has five main mission objectives, the first one being to successfully return and analyze a sample of pristine carbonaceous asteroid regolith in order to study the history, nature, and make up of the organic material and minerals [50]. In order for OSIRIS-REx, or any other sample return mission to succeed in sample collection, the spacecraft needs to land in a location that makes it easy for regolith collec­ tion. If the spacecraft lands in an area filled with boulders it may not only have a rough landing, but it could also result in unsuccessful sample collection as boulders are not as easy to sample. Having some knowledge of the anticipated boulder density distribution on asteroids in general, and on RQ36 in particular, could provide better constraints on the size and selection of a safe landing site. Unlike Itokawa’s boulder filled surface, RQ36 is believed to have a smoother surface. Thus, the local boulder distributions in smooth areas of Itokawa, such as within the Muses Sea region or Sagamihara could be analog to what might be present at RQ36, and are chosen for further detailed analysis.

3.4.2 Boulder Distribution on the Muses Sea Lowlands

One of the areas chosen for the simulation of asteroids, such as RQ36 is the lowland regions near the neck of asteroid Itokawa, also known as Muses Sea region. Figure 30 shows the outlined boulders in the selected area.

49 Figure 30: Selected Area in the Muses Sea

A total of 68 boulders greater than ten centimeters in length in long axis were counted in an area of approximately 0.00099 km2. Three high resolution images from the November AMICA [9] dataset were used to outline and confirm the location of these boulders. Figure 31 shows the cumulative boulder size distribution per unit area for long and short axis boulder sizes. Figure 32 shows the same plots for average, and RSS boulder sizes.

10s

S « e 2 s3 3 &to* £ 10* O o j 3 6a <->a 10* power-index: -2.36 ± 0.10 - power-index: -2.21 ± 0.10

10 1000 10 1000

Figure 31: Cumulative Number of Boulders per unit area on a selected area of the Muses Sea Size defined as long axis (left) and as short axis (right) Black: Data Used for Power-index Calculation, Red: Omitted Data

50 j t %J t e •D £ 2 to* o o

a 3 E E 3 o 105 power-index: -2.X5 ± 0.J0 10* power-index: -2.26 ± 0.08

10 100 1000 10 100 1000 Boulder Size (cm] Boulder Size (cm]

Figure 32: Cumulative Number of Boulders per unit area on a selected area of the Muses Sea Size defined as average (left) and as RSS (right) Black: Data Used for Power-index Calculation, Red: Omitted Data

The power-index for the cumulative boulder size density for size defined as long axis, short axis, average, and RSS are: -2.36 + 0.10, -2.21 + 0.10, -2.15 + 0.10, and -2.26 ± 0.08 respectively.

Table 8: Power-index Values for Cumulative Boulder Size Distribution - Muses Sea Defined Size Power-index Long-axis -2.36 + 0.10 Short-axis -2.21 ± 0.10 Average -2.15 + 0.10 RSS -2.26 ± 0.08

3.4.3 Boulder Distribution on the Sagamihara Lowlands

Another area chosen for simulation of boulder distribution on asteroids with smoother sur­ faces is an area in the Sagamihara lowlands, shown in Figure 33.

51 Figure 33: Selected Area in Sagamihara

A total of 45 boulders greater than ten centimeters in length in long axis were counted in an area of about 0.00123 km 2 . Similar to the Muses Sea methodology, three high res­ olution images from the November AMICA [9] dataset were used to outline the boulders and eliminate any bias presented by angled geometry or lighting. Figure 34 shows the cu­ mulative boulder size distribution per unit area for long and short axis boulder sizes. Figure 35 shows the same plots for average, and RSS boulder sizes. « 10000 o s 10000 3

© 1 £3 E o3 1000 power-index: -2.50 ± 0.13 1000 power-index: -2.07±0.12

10 100 1000 10 100 1000 Boulder Site {cm]

Figure 34: Cumulative Number of Boulders per unit area on a selected area of the Sagami­ hara Size defined as long axis (left) and as short axis (right) Black: Data Used for Power-index Calculation, Red: Omitted Data

I Nj t I 10000 ® 10000 3 23 o CD s © o

•Io 3 E o3 1000 power-index: -2.22 ± 0.11 1000 power-index: -2.50 ±0.13

10 100 1000 10 100 1000 Boulder Size [cm] Boulder Size [cm]

Figure 35: Cumulative Number of Boulders per unit area on a selected area of the Sagami­ hara Size defined as average (left) and as RSS (right) Black: Data Used for Power-index Calculation, Red: Omitted Data

The power-index for the cumulative boulder size density for size defined as long axis, short axis, average, and RSS are: -2.50 ± 0.13, -2.07 + 0.12, -2.22 ± 0.11, and -2.30 + 0.10 respectively.

53 Table 9: Power-index Values for Cumulative Boulder Size Distribution - Sagamihara Defined Size Power-index Long-axis -2.50 + 0.13 Short-axis -2.07 + 0.12 Average -2.22 + 0.11 RSS -2.30 + 0.10

3.4.4 Conclusion

The power-index for the cumulative number of boulders per unit area, assuming an RSS average for the size, is -2.26 + 0.08 for the Muses Sea area and -2.30 ± 0.10 for the Sagami­ hara, which is lower than the slope of -2.98 + 0.08 for the cumulative boulder size density on the entire surface of Itokawa. However, these power-indices are comparable with the shallower slope of -2.4 ± 0.10 previously shown in Figure 21. This similarity shows that these results do not change by much even when going to boulder sizes as small as ten cen­ timeters. These areas represent the surfaces of asteroids and other small bodies that are mainly covered with regolith and not as densely populated by boulders as Itokawa. The touch-and-go Sample Acquisition Mechanism (TAGSAM) onboard the OSIRIS- REx spacecraft, is capable of carrying up to two kilograms of material with grain sizes ranging from dust up to two centimeters [52], The analysis in this section shows that there are approximately two boulders, each with an RSS of greater than fifty centimeters, per 25 m2 in an area similar to Muses Sea; and about one boulder per 25 m 2 on an asteroid with a surface similar to the Sagamihara lowland. In the missions to RQ36 and other bodies with surfaces similar to the lowlands of Itokawa, considering these probabilities is an asset. These analyses and statistics provide data points from another asteroid that can help plan a safe landing as well as locating suitable locations for sampling.

54 4 Itokawa Now and Then

4.1 Motivation and Background

Section 3.3.4 details the possibility that the parent body of Itokawa has gone through catas­ trophic disruption. With that hypothesis, the current state of Itokawa was formed when the impact fragments re-accumulated. If a part of the parent body was not disrupted by the impact, it is possible that the orientation of boulders today is related to the parent body. After a body is formed, the body tries to spin into a disc while the majority of particles and boulders either settle on or around the equator, or gradually shift to these areas [49]. To test the hypothesis, the boulder distribution per degree latitude should be produced both for Itokawa’s surface today, from this point on referred to as Itokawa Now, as well as a hypothetical ellipsoid, referred to as Itokawa Then. To do this, the volume of both shapes, as well as their centers of mass, need to be calculated. To produce a new coordinate system and define new degrees of latitude and longitude on Itokawa Then, the tilt and rotation of the axis of rotation needs to be calculated. This chapter will explain the volume, centre of mass, inertia tensor, and axis of rotation calculations in addition to comparing the boulder distributions based on latitude for both bodies.

4.2 Methodology

For the purpose of this thesis, only a rough approximation is used to create the “Itokawa

Then” ellipsoid as well as rough calculations for the tilt and rotation of the spin axis. As previously discussed in Section 2.2, the global topography models (GTM) are presented in an implicitly connected quadrilateral (ICQ) format, and the vertices are labeled as though they are grid points on the faces of a cube. The first step is to read in the Gaskell Shape

Model [7] using the “read_ascii” function of IDL, which reads data from an ASCII file into

55 an IDL structure variable. Once the data is organized in the right format, the Itokawa shape model (128 plates) is plotted as seen in Figure 36.

Figure 36: Itokawa Shape Model (128 plates)

In order to create an ellipsoid to represent “Itokawa Then”, the cube side representing Itokawa’s head is removed and replaced with the cube side parallel to the head. The new cap is then reduced in size to fit the rest of the body better, as a rough approximation, representing an ellipsoid. The resulting body, Itokawa Then, is shown in Figure 37.

Figure 37: Ellipsoid Representing Itokawa Then

For the purposes of easier analyses and comparisons, the shape model is converted from an ICQ format to a triangular plate format by dividing each quadrilateral in two. Each

56 quadrilateral has four vertices, here refereed to as PI, P2, P3, and P4; and each vertex is associated with its own x, y, and z. Figure 38 shows how the transformation between the

ICQ format and triangular plate format. The ICQ plate model is described in details in

Section 2.2.

P1(i,j) l*2{i+l,j)

Figure 38: ICQ Model to Triangular Plate Model

The plate model vertices can be written as [53]:

/15 n= 1,... ,1V (1)

Where each vertex, vn is a Cartesian vector in asteroid-body fixed coordinates and is numbered by the index n for the total of N vertices.

4.2.1 Volume and Centre of Mass Calculations

To calculate the volume of each shape model, the volume of each plate needs to be calcu­ lated. In the triangular plate model, each individual volume increment is in the shape of a triangular-based pyramid, as shown in Figure 39.

57 V I

vk

Figure 39: Triangular-Based Tetrahedron

The general formula for the volume of a triangular-based pyramid, also referred to as a tetrahedron, is as follows:

V = l-A h (2)

where A is the area of the base and h is the height. To determine the volume of each plate, the centroid and the outward normal vector need to be calculated. The centroids of the plates are calculated using [53]:

Ri = (vi + Vj + vk)/3, n=l,...,Ap (3)

where vertices i, j, and k form the triangular plate, n represents the plate number, and Np the total number of plates. The outward normal vector of each plate can be calculated

58 using the following [53]:

Nn - (vj - v,-) x (v* - v,) (4)

in which Nn is defined as a surface element with dimensions of area and with a magni­ tude twice the area of the plate. From there the volume of the shape model which approxi­ mates the volume of the asteroid or the ellipsoid can be calculated using [53]:

V = Y,(Nn.Rn)/6 (5) n Using the above formula the volume of asteroid Itokawa (Itokawa Now) is calculated to be approximately 1.77 x 10 7 m3, and the volume of the ellipsoid (Itokawa Then) is roughly 1.57 x 107 m3. It should be noted that the volume of the ellipsoid is a very rough approximation, as the cap replacing the head of Itokawa is not a perfect fit. In order to determine the shift in the axis of rotation of the two shape models, the difference between their centers of mass need to be calculated. Having the volume and assuming a uniform density, the following formula can be used to calculate the coordinates centre of mass vector [53]:

XCM = ^ Y . Nn(Rn-Rn) (6) ^ n The coordinates of the centre of mass of Itokawa (Itokawa Now) are calculated to be: (0.0988, -0.0097, -0.0488), and the coordinates of the centre of mass of the ellipsoid

(Itokawa Then) are: (-33.2033, -0.172, 4.457). Section 4.3.1 shows how the rotation axis shifts using the difference in the centers of mass.

59 4.2.2 Inertia Tensor Calculations

To find the axis of rotation of each body, the greatest moment of inertia needs to be calcu­ lated, as the spin axis is perpendicular to the plane with the greatest inertia. Most objects tend to rotate in the direction of their major axis to conserve angular momentum and min­ imize rotational energy. Since each shape model is divided into tetrahedra, the first step is to calculate the inertia tensor for each individual tetrahedron. Lets assume the tetrahedron vertex coordinates are represented by: A*= (xj, y i, z;), where i = 1,..., 4. For a rigid body D, the inertia tensor E q with respect to the three axes x, y, and z centered at Q has components given by [13]:

Eq — —b' b —a! (7)

^ —c' —a! c y

Having p represent density:

(8) D

(9) D

( 10) D

j jiyzdD ( 11) D

60 b ' — J[ixzdD (12) D

c = J fixydD (13) D

In the inertia tensor E q , quantities a, b, and c are the moments of inertia with respect to the x, y, and z axes, respectively. The elements a’, b’, and c’ on the other hand, are the products of inertia. The eigenvalues of E q represent the principal moments of inertia and the eigenvectors provide the principal directions [54], To calculate the above integrals more easily, an affine coordinate system transformation g is assumed, as shown in Figure 40. An affine transformation being defined as any transformation that preserves collinearity and ratios of distances [55].

9 K

A Ag= (0,1.0) V

9

Figure 40: Transformation g and g-1[13]

The x, y, and z then become [13]:

61 x = x[+(x2-xi).^ + (x3-xi).ri + (x4~xi).C

y = yi + (y2-yi)-Z + (y3-y\)-n + (y4-y\)-C (14)

Z = Zl+{z2-Zl) .^+(Z3~Z\) .rj + (z4-Zl) -C

Next, the determinant of the Jacobian is calculated since a change of variable in eval­ uating a multiple integral is made. The magnitude of the Jacobian determinant is used as a multiplicative factor within the integral to accommodate for the change of coordinates. The calculations are done using the following [13]:

8x 8x 8x x2 -X l x3-X\ X4 -XI 54 8r] 54 Sy 8y 8y DET(J) - )>3 y4 = 6 . volume (15) 54 Srj 54 yi - y i - y \ - y i 5z 5z 5z Z2 — Z\ 23 24 - 2 1 54 5 77 54 -21

in which volume represents the volume of the rigid body D. In this case, calculating the determinant of the Jacobian can be avoided, because the volume has already been calculated in Section 4.2.1. To normalize triangular-based pyramid D into a triangular-based pyramid D’, transformation g"1, shown in Figure 40, needs to be performed. If f (x, y, z) is a generic function continuous on D, where g is regular and D is bounded by regular surfaces, the integral of f (x, y, z) can be written as [13]:

J f{x,y,z)dD = J f[x(^,T],Q,y(^,'q,Q,z(^,'n,Q]. \DET(J)\dD' (16) D D' It should be noted that D’ is normal with respect to the £r)-plane, and the projection of D’ on the $r)-plane is normal with respect to the £-axis. Therefore the above integral can further be simplified as:

62 [f(x,y,z)dD=\DET(J)\. f\% [' * dr) [' * J Jo Jo Jo D (17) Once again, assuming a uniform density [i, the integrals of the inertia tensor could be written in the above format. An example is shown below [13]:

a = n [{y2+z2)dD = fi.\DET{J)\. [ dE, [ dr] [ \y2(^,ri,Q+z2(^,ri,Q}dC J Jo Jo Jo D (18) The solution to the above integral is then:

a = ii. \d e t (j ) | . (yj+ yiy2+y2 + y m + ym + yi + y m + y2y4 + j w (19) +y} + Z2 + ZlZ2 + Z2 + Z1Z3 + Z2Z3 + Z3 + Z 1Z4 + Z2Z4 + Z3 Z4 + z 2)/60

The rest of the integrals of the inertia tensor can be re-written in the same format as Equation 18, and their solutions are shown below [13].

b = fl. \DET(J)\ . {x\ +x\x2 +X 1X 3 +x2X3 +x% +X1X4 +x2X4 + X3X4 (20 )

+ X 4 - h z f + Z 1Z2 + 22 + 2123 + 2223 + 23 + 2 l 24 + Z2 Z4 + 2324 + Z 4 ) /6 0

c = /i . |D£T(/)| . (xf + X 1X 2 +X2 +X1X3 +X2X3 +X3 + X 1 X 4 + X2X4 + X3X4 (21)

+X4 +y\+y\y2 +y2+ y m + y m + y i+ y m + yiy4 + y 3>4 + y i )/60

63 a' = ii. \DET(J)\ . (2ym +yiz\ +yiz\ +y*z\ + y\Z 2 + 2y2Z2 + y3Z2 (22) +y4Z2 + y iZ 3 + y 2 Z 3 + 2 y 3 Z 3 + y 4 Z 3 + y \Z 4 + y2Z4 + y3Z4 + 2 y4Z120 4 )/

b' = ll. \DET(J)\ .(2xiZl+X2Zl+X3Z\ +X4Zl+X\Z2+2x2Z2+X3Z2 (23) + X 4Z2 + X 1Z3 + X2Z3 + 2 X3 Z3 + X4Z3 +x\Z4+X 2Z4 + X 3 Z4 + 2 x 4za ) /120

c' = 1 1 . \DET(J)\ . (2xiyi +x2yi +x3yi +x4y{ +xiy2 + 2 x2^2 + ^ 2 (24) +2:4^2 + x \ y 3 +x2y3 + 2 x 3>-3 +x4y3 +xi y4 +x2y4 +x3y4 + 2 x4^4)/120

As mentioned previously, the determinant of the Jacobian is equal to six times the volume, and since the volume has been calculated, the DET(J) value in the above equations is replaced with six times the volume calculated in Section 4.2.1. The density, p, is assumed to be homogenous for the whole asteroid and the ellipsoid shape models, which is 1950 kg/m3 [22], An IDL program is used to calculate the inertia tensors for all the tetrahedra in the shape models using the above formulae. The greatest inertia tensor for Itokawa Now is shown below:

7.88650 x 10 9 6.93351 x 109 -3.02766 x 10 9

6.93351 x 109 9.55556 x 10 10 8.26055 x 108 (25)

-3.02766 x 10 9 8.26055 x 108 9.01960 x 1010

The greatest inertia tensor for Itokawa Then is:

64 3.82985 xlO9 -1.95677 x 10 9 9.89294 x 109

-1.95677 x 10 9 2.67011 x 10 10 7.28984 x 108 (26)

9.89294 x 109 7.28984 x 108 3.02423 x 1010

4.3 Analysis and Results

As mentioned in Section 4.2.2, in order to calculate the axis of rotation of a rigid body the greatest eigenvalue and its corresponding eigenvector need to be calculated from the inertia tensors, as they produce the greatest principal moment of inertia and the greatest principal direction respectively. The eigenvalues and eigenvectors of all the inertia tensors are calculated using IDL’s EIGENQL function. The EIGENQL function is used when working with n-by-n real symmetric arrays, such as the inertia tensors. It should be noted that to confirm the methodology and calculations, all the steps were tested for a simple oblate spheroid with known dimensions. In the case of Itokawa Now, the eigenvalues corresponding to the greatest inertia tensor are: (9.61592 x 1010, 9.02519 x 1010, 7.22696 x 109), providing the greatest moment of inertia of 9.61592 x 1010. The relating eigenvectors are:

0.0745197 -0.992232 -0.0996135

0.0446714 0.0964695 -0.0994333 (27) -0.996218 0.0785473 -0.0371356

The eigenvector representing the axis of rotation is therefore:

-0.0745197 Vecton = 0.0446714 (28)

0.996218

65 The eigenvalues relating to the greatest inertia tensor of Itokawa Then is calculated in the same manner, providing: (3.35376 x 1010, 2.68657 x 1010, 3.6993 x 108), resulting in the greatest moment of inertia of 3.35376 x 1010. The correlating eigenvectors are:

-0.315282 -0.010944 -0.948935 -0.078421 0.996814 0.014559 (29) 0.945752 0.0790071 -0.315136

The axis of rotation vector from the greatest eigenvector is:

-0.315282 Vecton = -0.078421 (30)

0.945752

4.3.1 The Difference between the rotation poles

The Gaskell shape model is in the right-handed Cartesian body-fixed frame coordinate system, and defines the z-axis as the rotation pole [7]. This spin axis is derived from light curves and tracking landmarks as the asteroid rotates [25]. The calculated Vectori in equation 28, should theoretically represent that, however, there are some discrepancies due to errors in the shape model and rough calculations. It should be noted that a constant density was assumed for these calculations, so the discrepancies in the rotation poles could suggest internal density asymmetries [56]. When calculating the angle difference between the rotation pole of Itokawa Now and Itokawa Then, the difference between Vectori and the defined rotation pole is taken into consideration. Figure 41 shows a rough sketch of the different rotation poles.

66 Z-axis: defined spin axis Calculated spin axis for Itokawa Now New spin axis for Itokawa Then (ellipsoid)

Figure 41: The Different Poles of Rotation

Vectori and Vector show a rotation in the pole of rotation in both the x and z axes. In order to calculate the change in the rotation pole, as well as creating the new coordinate system for the ellipsoid, two angles need to be calculated. One is the complimentary angle to that between the vector and the x-axis, also referred to as the angle between the vector and the xy-plane gamma (y). The other, is the angle from the x-axis to the projection of the vector on the xy-plane, referred to as alpha (a). The two angles are shown in Figure 42.

67 V e c t o r

X

Figure 42: Rotation Angles

The two angles y and a are calculated using the following equations:

(31) sm y= M

Vr cos a (32) |V|cosy

For Vectori , angle y is calculated to be 85.02° and angle a is 30.94°. Vector on the other hand has an angle y of 71.60° and an angle a of 166.03°. The pole has shifted about 13.4° in the z-axis and about -135° in the x-axis.

4.3.2 Latitude vs. Boulder Distribution

The new Cartesian coordinate system for Itokawa Then, is created by rotating the Cartesian coordinate system of Itokawa Now, which is the same as rotating the boulders and the area

68 plates. However, both the location of the boulders, as well as the file containing the area measurements, are with reference to their respective latitudes, cp, and longitudes, X. The configuration of the matrix transformations and rotations are shown below.

R X y R'

§ -> y / -> ¥ X z z' X'

The transformation from spherical to Cartesian coordinates in terms of latitude are:

X — r .cos((j)) .cos(X) (33)

Y = r .cos((p) .sin(X)

Z = r .sin()

Next is to perform the Cartesian coordinates rotation about the x and z axes using the 3 x 3 matrix rotations. The rotation along the x-axis is done using:

i 0 0 1 0 0

Rr = 0 cos(90 - r ) — sin (90 — 7) = 0 sin 7 — cos 7 (34)

0 sin(90 -y ) cos (90 — 7) 0 cos 7 sin 7 The rotation about the z-axis is done using:

cos (90 - a ) - sin(90 - a) 0 sin a — cos a 0

Rz = sin (90 - a ) cos (90 — a) 0 = cos a sin a 0 (35)

0 0 1 0 0 1

69 The matrix multiplication of the rotation along x and z axes can be written and used in the following format:

sin a — cos a 0

R = cos a sin 7 sin a sin 7 — cos 7 (36)

cos a cos 7 cos 7sin a sin a

The new resulting Cartesian coordinates from Equations 33 and 36 are shown below:

X ' = r . cos 0 sin (a — X) (37)

Y' = r . [— cos7sin + cos 0 sin 7cos(a — A)]

Z' — r . [sin 0 sin 7+ co s0 cos7cos(a: — A,)]

The derivations for the above equations are provided in Appendix C. The final step for creating the new coordinate system is to go back to spherical coordinates from Cartesian using the following expressions:

7 ! (j>r = arcsin(—) (38) R

X' — arctan2(7/,X/)

Resulting in:

(j)' = arcsin|sin0sin7+cos0cos7cos(a — A)] (39)

70 X' = arctan2[(—cosysin^ + cos 0 sinycos(a — X ) ) , cos0 sin(a — X)} (40)

The location of boulders, as well as the area plates, are rotated and transformed using the above equations. The boulder distribution per ten degrees of latitude for Itokawa Now, is shown in Figure 43. It should be noted that Itokawa Now only includes the body of Itokawa to avoid bias.

1 J ■ 1 1 .r ..“T“

*

3000 I o I - o

- $ - 3 * COO I » » * * » - ■ r i - ❖ « m k t 100 •50 0 50 Latitude

Figure 43: Latitude vs. # of boulders/km2(Itokawa Now)

The boulder distribution per ten degrees of latitude for Itokawa Then is shown in Figure

44.

71 100

50

0 Lothude

“ 50

Figure 44: Latitude vs. # ofboulders/km2(Itokawa Then)

0 -1 0 0 1 0 0 0 3000 2 0 0 0 4000

jvUJH/sjspinoe jo | The bar graphs representing the same data for Itokawa Now and Itokawa Then are shown shown in Figures 45 and 46, respectively. # of Boulders/km T 1 ------1 ------Figure 45: Itokawa Now Itokawa 45: Figure 1 ------1 ------1 ------1 ------73 Latitude 1 ------1 ------1 ------1 ------1 ------1 ------1 ------1 ------1 ------T 4000 r - 1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------1------r

Latitude

Figure 46: Itokawa Then

From Figure 46, it is evident that there is a peak around +10 degrees latitude. It was predicted that the majority of boulders would gather around the equator of the asteroid. The shift of about 10 degrees could be due to the fact that the Itokawa Then ellipsoid is not a perfect shape model. Comparing the boulder distribution around +50 and -50 degrees, a lack of boulder presence is observed around +50 degrees. This could be due to the Sagamihara lowland around +50 degrees. One of the other sources of error is the rough approximation of the cap closing the ellipsoid. This analysis could further support the contact binary hypothesis, implying that the head and body of Itokawa had formed separately and then came into contact with each

74 other later on. In this hypothesis it is also assumed that both bodies maintain their original shape and only some mass movement has created Itokawa’s shape today. The boulder population peak around the equatorial region of Itokawa Then provides more evidence for this scenario.

75 5 Conclusion and Future Work

The surface of asteroid Itokawa is covered with at least 820 boulders greater than five meters in size along the long-axis, 60% higher than a previous study [30], and 37 potential craters. The overall power-index for the cumulative boulder distribution has a slope of -2.98 + 0.08, which is in agreement with the previous study [30], which had a slope of -3.1 + 0.1. Unlike earlier studies, this thesis analyzes the plot of boulder size versus the cumulative number of boulders per km 2 closely which reveals a break in the slope, something not previously identified. This break suggests that Itokawa is a result of a disrupted parent body, consistent with impact experiment results. This disruption was then followed by surface evolution, meaning the surface rocks subsequently broke further either by impact or other erosional processes, as evident from the curves steepening relative to the overall slope. To further investigate the evolution of Itokawa, the boulder distribution is analyzed on regional scales divided into the head, body, east side, and west side of the asteroid. The power-index of plotted boulders varies along different regions of Itokawa. The head of the asteroid has a power-index of -2.45 ± 0.11, while the body has a power-index of -3.2 ± 0.09. The shallower slope of the head suggests that it has a fresher surface in comparison to the body. This difference provides evidence that the head and body of Itokawa went through different evolutionary paths before coming together, supporting the contact binary hypothesis. When comparing the west and east regions of Itokawa, the east and west have power- indices of -3.15 + 0.15 and -2.56 ± 0.07 respectively. This difference can be accounted for by the higher number of boulders identified at high surface elevations where no regolith coverage exists, thereby resulting in a steeper slope. The east side of Itokawa contains the majority of the Muses Sea lowland and the west side includes most of the Sagamihara

76 lowland where the regolith dominates and has probably submerged the boulders. This thesis finds through the analysis of global and regional boulder distributions that the parent body of Itokawa has gone through a disruption and the head and the body have come together at a later point. To further test this scenario, an innovative approach is taken by analyzing the boulder distribution based on latitude both for the current state of Itokawa, referred to as Itokawa Now, and for a hypothetical intermediate ellipsoid body, referred to as Itokawa Then. To do so, the rotation axes as well as the corresponding latitudes and longitudes were calculated using the volume of the bodies and the their inertia tensors. When comparing the rotation axis of Itokawa Now and Itokawa Then, a difference of about 13.5° in the z-axis is observed. The boulder population with respect to latitude shows a larger concentration around the equatorial region of Itokawa Then. This analysis further supports the results from regional boulder distribution on the head and body of Itokawa. Both experiments support the hypothesis that Itokawa is a contact binary asteroid and a result of a disrupted parent body. The flying fragments that accumulated after impact likely formed rubble-piles. Two of the rubble-pile bodies later on co-orbited each other and evolved separately. After forming their ellipsoidal shapes, the two rubble-piles then came together at a slow speed to form Itokawa’s current shape. In addition to finding new support for Itokawa’s contact binary hypothesis, this thesis provides useful information for future missions. The boulder distribution density is ana­ lyzed in the lowlands of Itokawa to provide an analog for planetary objects with a smoother surface. Further statistical analysis of these data will provide different missions, such as the OSIRIS-REx [50], with necessary information for selecting appropriate sample collection or landing sites. In the future, to further support this hypothesis or just to simply gain more insight into the formation of Itokawa, the boulder distribution can be analyzed based on elevation to understand if it plays a role in there being a higher concentration of different boulder sizes

77 in different areas. The study of boulder population and orientation in the neck region of Itokawa could also provide more information about how the head and body of Itokawa have come into contact with one another. Furthermore, establishing any connection between the observed distributions, size, and aspect-ratio of boulders with any of the possible cratering processes that might have occurred on Itokawa are useful in order to gain more insight into surface evolution of this asteroid.

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xxi Appendices

Appendix A - Chi-square Error Analysis

For any experiment in which a value x is being measured and the measurement is repeated N times, the possible results of x can be divided into bins, k = 1, ... , n. The number of observations that fall into each bin k is referred to as Ok. Assuming that the measurements are indeed governed by the expected distribution, the expected value, Ek, in the kth bin can be calculated. Evidently, there is not a perfect agreement between Ek and Ok after any finite number of measurements. However, the different values for Ok should have an average value of Ek and would be expected to fluctuate around Ek with a standard deviation on the order of y/Ek- The two numbers to be compared are therefore the deviation of

Ok — Ek and the expected size of its fluctuations \fE~k- A considered ratio should then be:

Ok — Ek .... ■ W (41) For some bins the above ratio will be positive, while for the others it will be negative. To standardize this variation, it is natural to square the number for each k and sum over all bins k = 1,... , n. The value produced in this manner is referred to as the chi squared, which is defined as [57]:

X 2 = '£ - ~J ~ — (42) fetl Ek

It should be noted that if X 2 < n, the agreement between the observed and expected distributions are acceptable. However, if X 2 >>n, then there is a significant difference between the two.

xxii Appendix B - Plots

Cumulative Boulder Distribution Plots for the Head

10000 10000

2 | 1000 | 1000 e e •o © 3 3 (Bo

© .1 > s 53 3 E 3 £ o 3 power-index: -2.65 ± 0.06

1 100 1 10 100

Figure 47: Cumulative Number of Boulders per unit area on the Head Size defined as long axis (left) and as short axis (right)

Power-index (long axis): -2.49 ± 0.13, Power-index (short axis): -2.65 ± 0.11.

1000010000

? E 1000 | 1000

©' © 3 3 <23

© .*■ .1 3 2 a a £ £ 3 3 power-index: -2.84 ± O.U power-index; -2.45 ± 0.10

1 100 1 10 100 Boulder Size [m]

Figure 48: Cumulative Number of Boulders per unit area on the Head Size defined as average(left) and as RSS (right)

Power-index (average): -2.84 ±0.11, Power-index (RSS): -2.45 ±0.11.

xxiii Cumulative Boulder Distribution Plots for the Body

10000 10000

2 | 1000 1000 s© a> o3 CD <2 O too O too

©> o 3 s E E O3 o3

1 10 100 i 10 ioo Boulder size fm] Boulder s«2e [m]

Figure 49: Cumulative Number of Boulders per unit area on the Body Size defined as long axes (left), Size defined as short axes (right)

Power-index (long axis): -3.12 ± 0.09, Power-index (short axis): -3.60 ± 0.11.

toooo 10000

| 1000 iooo

« •o I O 100 100

1 .5' 33 E o3 power-index: -3.44 ± 0.09

1 10 100 1 10 100 Boulder she [m] Boulder sl2e [m]

Figure 50: Cumulative Number of Boulders per unit area on the Body Size defined as average (left) and as RSS (right)

Power-index (average): -3.44 + 0.09, Power-index (RSS): -3.20 ± 0.09.

xxiv Cumulative Boulder Size Distribution Plots for the East Side

10000 fOOOO

| 1000

<8 100 o 100 o

© © > o o s 9 E E o o3

1 1001 10 100

Figure 51: Cumulative Number of Boulders per unit area on the East Side Size defined as long axis (left) and as short axis (right)

Power-index (long axis): -3.26 ± 0.15, Power-index (short axis): -3.43 ± 0.16.

10000 10000

| 1000 tOQO in 4)£ 5 2 cB3 I o 100 o 100 .1 3 a 3 3 aE aE

1 100 1 10 100 Boulder size [m]

Figure 52: Cumulative Number of Boulders per unit area on the East Side Size defined as average(left), and as RSS (right)

Power-index (average): -3.53 + 0.15, Power-index (RSS): -3.15 ± 0.15.

xxv Cumulative Boulder Size Distribution Plots for the West Side

10000 1 0 000

| 1000 | 1000

A>e © 3 3 33 2 2 o o 100

o > I© 2 3 3 E E o3 a

1 100 1 10 100

Figure 53: Cumulative Number of Boulders per unit area on the West Side Size defined as long axis (left) and as short axis (right)

Power-index (long axis): -2.55 + 0.08, Power-index (short axis): -3.13 ± 0.09.

ioooo 10000

$ I tooo ¥ 1000

«* I© 3 23 o too© too i 2 2 3 3 E 2 u power-index: -2.90 ± 0.1 power-index: -2.56 ± 0.07

1 10 100 1 10 100 Boulder size [mj Boulder size [mj

Figure 54: Cumulative Number of Boulders per unit area on the West Side Size defined as average (left) and as RSS (right)

Power-index (average): -2.90 + 0.08, Power-index (RSS): -2.56 + 0.07.

Appendix C - Derivations for Rotation and Transformation

To perform the rotation in the Cartesian coordinate system, the following matrix multipli­ cation is done:

xxvi X' sin a — cos a 0 r.cos((j>) .cos(X) r = cos a sin y sin a sin y — cosy X r.cos(

X' = ( r . sin a cos 0 cos X) + (—r. cos a cos 0 sin X) + 0

X' = r. cos (p (sin a cos X — cos a sin X) Using the angle sum and differences identities, the equation is simplified to:

X' — r. cos0sin(a — X)

Y' = (r. cosasinycos0cosA) + (r. sinasinycos^sinA) + (—r. cosysin^)

Y' = r[— cosysin0 + cos0siny(cosacosA + sinasinZ)]

Y' — r [— cos y sin 0 + cos

Z' = ( r . cos ct cos y cos (j> cos X) + (r. cosy sin a cos 0 sin A) + (r. sinysin^) Z' = r[sinysin0 + cos0cosy(cosacosZ + sinasinA)] Z' = r(sinysin0 + cos 0 cosy cos (a — A,))

xxvii