A dissertation presented to. the faculty of. the College of Arts and Sciences of Ohio University. In partial fulfillment

Transcription

1 Radiation Recoil Effects on the Dynamical Evolution of Asteroids A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Desireé Cotto-Figueroa December 2013 c 2013 Desireé Cotto-Figueroa. All Rights Reserved.

2 2 This dissertation titled Radiation Recoil Effects on the Dynamical Evolution of Asteroids by DESIREÉ COTTO-FIGUEROA has been approved for the Department of Physics and Astronomy and the College of Arts and Sciences by Thomas S. Statler Professor of Physics and Astronomy Robert Frank Dean, College of Arts and Sciences

3 ABSTRACT 3 COTTO-FIGUEROA, DESIREÉ, Ph.D., December 2013, Physics and Astronomy Radiation Recoil Effects on the Dynamical Evolution of Asteroids (111 pp.) Director of Dissertation: Thomas S. Statler The Yarkovsky effect is a radiation recoil force that results in a semimajor axis drift in the orbit that can cause Main Belt asteroids to be delivered to powerful resonances from which they could be transported to Earth-crossing orbits. This force depends on the spin state of the object, which is modified by the YORP effect, a variation of the Yarkovsky effect that results in a torque that changes the spin rate and the obliquity. Extensive analyses of the basic behavior of the YORP effect have been previously conducted in the context of the classical spin state evolution of rigid bodies (YORP cycle). However, the YORP effect has an extreme sensitivity to the topography of the asteroids and a minor change in the shape of an aggregate asteroid can stochastically change the YORP torques. Here we present the results of the first simulations that self-consistently model the YORP effect on the spin states of dynamically evolving aggregates. For these simulations we have developed several algorithms and combined them with two codes, TACO and pkdgrav. TACO is a thermophysical asteroid code that models the surface of an asteroid using a triangular facet representation and which can compute the YORP torques. The code pkdgrav is a cosmological N-body tree code modified to simulate the dynamical evolution of asteroids represented as aggregates of spheres using gravity and collisions. The continuous changes in the shape of an aggregate result in a different evolution of the YORP torques and therefore aggregates do not evolve through the YORP cycle as a rigid body would. Instead of having a spin evolution ruled by long periods of rotational acceleration and deceleration as predicted by the classical YORP cycle, the YORP effect is self-limiting and stochastic on aggregate asteroids. We provide a statistical description of the spin state evolution which lays out the foundation for new simulations of a coupled

4 4 Yarkovsky/YORP evolution. Both self-limiting YORP and to a lesser degree a stochastic YORP provide a viable means to explain why the Near-Earth Asteroid (NEA) population seems to remember their initial spin states at the time of delivery from the Main Belt. The YORP effect drives the obliquity of most objects that follow the YORP cycle to the values of 0, 90 and 180 degrees. NEAs could complete a YORP cycle on timescales much shorter than their typical dynamical lifetime. Therefore, one should expect the obliquity distribution of the population of NEAs to be concentrated about those values if they follow the YORP cycle. But to obtain a direct measurement of the obliquity distribution will require radar observations or multiple lightcurves at different illumination and orbital phases for each NEA. Instead of obtaining a direct measurement, the obliquity distribution can be inferred if the distribution of semimajor axis drift rates due to the Yarkovsky effect can be measured. From the linear heat diffusion theory for a spherical body, the semimajor axis drift rate varies linearly with cosine obliquity. Previous studies have attempted to infer the obliquity distribution taking advantage of this simple dependence. However, those results should be considered only approximate because of the neglect of the dependence of the semimajor axis drift rate on density, thermal properties, and shape. Here we seek to obtain the obliquity distribution of NEAs using a better approach based on Bayesian inference that takes into account our prior knowledge of the distributions of the physical parameters on which the semimajor axis drift rates depend. A preliminary obliquity distribution of the NEA population has been estimated to be a V-shaped model that lacks a concentration of objects at an obliquity of 90 degrees and which suggests that the most probable value of the fraction of retrograde rotators is 70.0%. Once the obliquity distribution is obtained, it can in turn be used to test YORP predictions and constrain YORP evolution.

5 To my parents, who have always believed in me. 5

6 ACKNOWLEDGMENTS 6 First, I would like to gratefully acknowledge support from the NASA Harriet F. Jenkins Predoctoral Fellowship Program (JPFP), the NASA Ohio Space Grant Consortium (OSGC) Fellowship and the NASA Planetary Geology & Geophysics Program. I would also like to acknowledge and express my sincere appreciation and gratitude to my advisor, Thomas S. Statler, for all of his support and advice during the completion of this dissertation. I am also grateful to the other members of my dissertation committee, Markus Boettcher, Keith Milam and Joseph C. Shields for their helpful suggestions. A special thanks to our collaborators Derek C. Richardson, Paolo Tanga, Steven R. Chesley and Davide Farnocchia for their contributions to this research. Furthermore, I would like to thank everyone who have supported me or helped me in any way not forgetting my dear friends, who were always there for me. Finally, I am deeply grateful to my daughter Iadara Nicole, my husband José, and our families for their endless love, support, and encouragement.

7 TABLE OF CONTENTS 7 Page Abstract Dedication Acknowledgments List of Tables List of Figures General Introduction Near-Earth Asteroids Aggregate Asteroids Yarkovsky effect YORP effect Open Questions Obliquity Distribution of NEAs Introduction Bayesian Approach Results Discussion Aggregate Dynamics Introduction Code Development Tiling Sensitivity Transformation The Spin State Evolution Simulations Aggregate Objects Choice of Simulation Parameters Results and Discussions End States

8 3.5.2 Stochastic and Self-Limiting YORP Effect Mass Loss and Binary Formation Axis Ratio Evolution The Statistical Spin and Obliquity Evolutions Conclusions and Future Work References

9 LIST OF TABLES 9 Table Page 1.1 Classes of NEAs Taxonomic Types Aggregate Objects Comparison of End States Mass Loss Episodes Mass Loss Spin Rates

10 LIST OF FIGURES 10 Figure Page 1.1 The orbits of the three classes of NEAs: Amors, Apollos and Atens The orbits of the major planets and the location of asteroids The aggregate asteroid Itokawa The rotation rate distribution for asteroids and trans-neptunian objects The Yarkovsky Effect Detection of the Yarkovsky effect The YORP effect The radiation forces on the wedges The YORP torques for Pseudo-gaspra The spin state evolution for Pseudo-gaspra Predicted Obliquity Distribution Detection of the YORP Effect Typical variation in the YORP torques caused by an identical single boulder randomly placed on an identical object Identical views of the two objects from Figure 1.13 with the most positive and most negative spin torques The de-biased Fast-Rotator Fraction F plotted against absolute magnitude H and nominal diameter Snapshots of a simulation where changes in the shape and mass loss of an aggregate asteroid can be observed as a result of applying constant YORP torque Inferred obliquity distribution based on 136 objects [Farnocchia et al., 2013] Contours of the posterior density of p 1 and p 2, P(p 1, p 2 ) for the V-shaped model applying all filters to the data Contours of the posterior density of p 1 and p 2, P(p 1, p 2 ) for the V-shaped model applying Filters 1 and 3 to the data Contours of the posterior density of p 1 and p 2, P(p 1, p 2 ) for the V-shaped model applying Filters 2 and 3 to the data Contours of the posterior density of p 1 and p 2, P(p 1, p 2 ) for the V-shaped model applying Filter 3 to the data The preliminary estimated best solution for the obliquity distribution of NEAs Triangular tiling obtained for the equivalent ellipsoid with the same mass and inertia tensor as the aggregate object Triangular tiling obtained the aggregate object

11 3.3 Normalized average absolute deviation (NAAD) in the change in the spin torques obtained for the object composed of 921 spheres Normalized average absolute deviation (NAAD) in the change in the spin torques obtained for the object composed of 4,281 spheres Normalized average absolute deviation (NAAD) in the change in the obliquity torques obtained for the object composed of 921 spheres Normalized average absolute deviation (NAAD) in the change in the obliquity torques obtained for the object composed of 4,281 spheres Tiling of the original object plotted in orange and the new tiling obtained after a movement over plotted in yellow Numerical and analytical solutions obtained for the spin state evolution given an initial value of the obliquity and the spin and given a set of coefficients A and B that satisfy B = A 3.9 Numerical and analytical solutions obtained for the spin state evolution given an initial value of the obliquity and the spin and given a set of coefficients A and B that satisfy B = A 3.10 Python script Continuation of the Python script The semi-axis ratio distribution of the initial shapes A triangular tiling representation of the original aggregate object in simulation The obliquity and the spin torques obtained for the object in simulation The obliquity and the spin evolutions obtained if the object in simulation 6 was a rigid-body A triangular tiling representation of the evolved aggregate object in simulation 6 at a time of T=3.248 Myrs The obliquity and the spin torques through which the original object in Figure 3.13 has evolved in Myrs The obliquity and spin evolutions of the object in Figure 3.13 if it was a rigidbody along with the actual evolution of the aggregate object The obliquity and spin evolutions of the object in simulation The obliquity and spin evolutions of the object in simulation The obliquity and spin evolutions of the object in simulation The obliquity and spin evolutions of the object in simulation The distribution of all mass loss events The shape of the aggregate object in simulation 11 before having the first mass loss episode The shape of the aggregate object in simulation 27 before having the last mass loss episode The spin evolution obtained for the object in simulation The obliquity evolution obtained for the object in simulation

12 3.28 The shape of the aggregate in simulation 8 at the last point of contact before splitting and forming a binary asteroid The shape of the aggregate object at the end of simulation The spin evolution obtained for the object in simulation The obliquity evolution obtained for the object in simulation The evolution of the 28 objects in the triaxiality and semi-axis ratio c space a 3.33 The obtained distribution of the statistical spin evolution α s Contours of the estimated density function f (α s,δt) A simulated stochastic and self-limiting spin evolution A simulated self-limiting spin evolution A simulated stochastic spin evolution The obtained distribution of the statistical obliquity evolution ζ s

13 1 GENERAL INTRODUCTION Near-Earth Asteroids Asteroids are rocky and metallic objects, left over pieces from the formation of the Solar System about 4.6 billion years ago. Hundreds of thousands of asteroids have been discovered within the Solar System and the vast majority is found within the Main Belt between the orbits of Mars and Jupiter. Near-Earth Asteroids (NEAs) are asteroids with perihelion 1 distance less than 1.3 astronomical units (AU) 2. There are four classes of NEAs: the Amors, the Apollos, the Atens and the Atiras (see Figure 1.1 ). NEAs are categorized into these classes according to their orbital semi-major axis and their perihelion and aphelion 3 distances (see Table 1.1). The typical dynamical lifetime of an NEA is about ten million years [Gladman et al., 2000]. NEAs eventually get destroyed by a collision with an inner planet or the Sun, or are ejected from the inner Solar System. As their typical dynamical lifetime is less than the age of our Solar System, it is thought that NEAs are objects from the Main Belt constantly delivered to their current orbits by various mechanisms. Bottke et al. [2002] estimated that the replenishment rate from the Main Belt, in order to have the population of objects with a diameter roughly larger than 1 km in steady state, is 790 ± 200 objects per million years. Gravitational forces and collisions were once considered the sole mechanisms for asteroid delivery [Bottke et al., 2006]. After a collision, an object or its fragments will undergo orbital velocity changes that could place them in a variety of powerful and diffusive resonances [Morbidelli et al., 2002]. Resonant gravitational perturbations are capable of transporting objects to Mars-crossing or Earth-crossing orbits by changing the eccentricity and/or the inclination of the object. The most powerful resonances are the 1 The perihelion is the point in the orbit where the object is closest to the Sun. 2 The mean distance from the Earth to the Sun is one astronomical unit. 1AU = m 3 The aphelion is the point in the orbit where the object is furthest from the Sun.

14 14 Figure 1.1: The orbits of the four classes of NEAs: Atiras, Amors, Apollos and Atens. The Sun and the orbit of the Earth are shown for reference. Figure reproduced and modified from Near Earth Object Program (NEOP) ( mean motion resonance 3:1 with Jupiter near 2.5 AU [Wisdom, 1983, Yoshikawa, 1989, Farinella et al., 1993] and the secular resonance ν 6 at the inner edge of the Main Belt near 2.1 AU [Wetherill, 1979, Yoshikawa, 1987, Morbidelli, 1993]. The object can then be extracted from these resonances by close encounters. Table 1.1: Classes of NEAs. Class Atiras Amors Apollos Atens Definition aphelion >1.0 AU, perihelion <0.983 AU aphelion >1.0 AU, 1.02 AU <perihelion <1.3 AU aphelion >1.0 AU, perihelion <1.02 AU aphelion < AU, semi-major axis >0.983 AU

15 15 Today, it has been recognized that there are non-gravitational forces that play an important role in the evolution of NEAs. They are considered an even more dominant mechanism in the evolution of the smaller objects (D <40 km) than gravitational forces and collisions [Bottke et al., 2006]. The Yarkovsky and the YORP effects, which are explained in more detail in sections 1.3 and 1.4, are a thermal radiation force and torque, respectively, which can modify the semi-major axis and the spin state of the object [Bottke et al., 2006, Rubincam, 2000]. These two effects can also contribute to the delivery of small NEAs from the Main Belt. It is widely accepted that Near-Earth Objects (NEOs) 4 could represent a hazard of global catastrophe for human civilization. NEOs have impacted the terrestrial planets and the small bodies in the inner Solar System, including Earth. On October 7, 2008 an NEA with an estimated diameter of 4 meters impacted Earth in the Nubian Desert of northern Sudan only 20 hours after discovery as predicted by early orbital solutions [Jenniskens et al., 2009]. A year later, on October 8 of 2009 an NEA with an estimated diameter of 10 meters exploded over the town of Bone, Indonesia without advance warning 5. The NEA exploded in the atmosphere at an altitude of kilometers above the surface of the Earth with an energy of about 50 kilotons of TNT. On February 15 of 2013 an asteroid with an estimated diameter of 17 meters exploded over Chelyabinsk, Russia with an energy of about 470 kilotons 6. This is the most powerful event since the Tunguska explosion of June 30, 1908, which resulted in a flattened forested area of over 2,000 km 2 near the Podkamennaya Tunguska River in Siberia, Russia. Today there are several programs whose purpose is to detect and characterize asteroids that could represent a hazard to Earth, such as the NASA Near Earth Object 4 NEOs are small objects in the Solar System, such as asteroids and comets, whose orbits bring them close to the Earth s orbit. 5 Asteroid Impactor Reported over Indonesia ( 6 What Exploded over Russia? (

16 16 Figure 1.2: The orbits of Mercury, Venus, Earth and Mars are shown in light blue. Main Belt and Near-Earth asteroids are indicated by green and red circles, respectively. Objects observed at more than one opposition are indicated by filled circles, objects seen at only one opposition are indicated by outline circles. Numbered periodic comets are shown as filled light-blue squares. Other comets are shown as unfilled light-blue squares. Figure and caption reproduced and modified from the Minor Planet Center ( Program 7. As of June 21, 2013, 10,004 NEAs have been discovered and 1,408 have been classified as Potentially Hazardous Asteroids (PHAs) 8. Figure 1.2 shows a view of the 7 Near Earth Object Program (NEOP) ( 8 PHAs are asteroids that have the potential to make threatening close approaches to Earth.

17 17 inner Solar System. Around 860 of the NEAs that have been discovered are asteroids with a diameter 1 km. While asteroids tens of meters in diameter can cause severe local damage, an impact of an asteroid larger than 1 km could cause a global disaster. The continuous discovery of unknown NEAs is very important, but it is also of great importance to characterize these objects and to understand their origin and their evolution. This is essential in order to develop a correct strategy in the future to deflect a threatening object away from Earth in case of an imminent impact and to understand how the object would respond. 1.2 Aggregate Asteroids The rotation rates of asteroids obtained from optical light curves have given strong evidence to support the idea that large asteroids are aggregates [Davis et al., 1979, Harris, 1996]. Aggregates are agglomerations of fragments, ranging in size, that are held together by gravitational forces and/or material strength. An example of an aggregate asteroid is Itokawa (Figure 1.3). Data collected by the Hayabusa spacecraft was used to calculate a low bulk density of 1.9 ± 0.13 g cm 3 for this NEA[Fujiwara et al., 2006]. This along with an estimated macroporosity of 41% and a shape that is composed of two rounded parts, suggests that Itokawa is an aggregate asteroid. Figure 1.4 shows the distribution of NEAs, Main Belt asteroids and trans-neptunian objects in the rotation period-diameter plane. The asteroid Itokawa has a mean diameter of km and period of about 12 hours, placing it below the gravity limit along with the majority of objects. The gravity limit is the maximum rate at which an aggregate object can spin and still retain loosely bound material at its equator. Giving equal gravitational and centrifugal forces at the equator of a spherical body we have: GMm R 2 = mω 2 R (1.1)

18 18 Figure 1.3: This image of the aggregate asteroid Itokawa was taken by the Hayabusa spacecraft from a range of about 5 km. Figure reproduced from the Japan Aerospace Exploration Agency ( where G is the gravitational constant, M is the total mass of the object, m is the mass of a fragment near the surface, R is the asteroid radius and ω is the angular rotation rate. Asuming an object with constant density we can rewrite the mass M in terms of its density: G4πρR 3 m = mω 2 R 3R 2 (1.2) G4πρ = ω 2 3 (1.3) Rewriting ω in terms of the period we have: G4πρ 3 = ( ) 2 2π (1.4) P Gρ 3 = π P 2 (1.5)

19 19 Figure 1.4: The rotation rate distribution for asteroids and trans-neptunian objects. The horizontal scale indicates approximate diameters and the vertical scale indicates the period in hours. The dark sloped line shows one of the theoretical spin limits derived by Holsapple [2007] assuming a size-dependent strength. Objects with a diameter greater than 10 km have their spin limit determined purely by gravity while for objects with diameters smaller than 10 km, a cohesive and/or tensile strength can be sufficient to allow a higher spin rate. Figure reproduced from Holsapple and Michel [2008]. Having then a gravity limit at a period of: P = ( ) 1 3π 2 (1.6) Gρ If it rotated faster than the period given in equation 1.6, the centrifugal force would overcome gravity and material could be shed from the equator of the object. An average rock density of 2.5 gcm 3 gives a minimum rotation period of about 2.2 hours. Note that the maximum rotation rate is independent of size for objects of the same shape and density.

20 20 The dark sloped line in Figure 1.4 shows one of the theoretical spin limits derived by Holsapple [2007] based on a strength model for geological materials. Holsapple [2007] showed that an object with a diameter greater than 10 km has its spin limit determined purely by gravity since the presence of a cohesive and/or tensile strength does not permit a higher rotation rate than that allowed by the gravity limit. In contrast, for objects with diameters smaller than 10 km, a very small cohesive and/or tensile strength similar to that of moist sand, can be sufficient to allow a higher spin rate. Therefore an asteroid with a diameter less than 10 km that rotates very fast does not necessarily have to be monolithic but indeed can be also an aggregate. 1.3 Yarkovsky effect Radiation recoil forces are caused by the anisotropic emission of thermal photons from the surface of a rotating object that is heated by sunlight. Ivan O. Yarkovsky, a Russian engineer, was the first one to propose the effect that these recoil forces could have on the motion of objects in space [Opik, 1951]. There are two components of the Yarkovsky effect, a diurnal and a seasonal component. Figure 1.5a shows the diurnal component of the Yarkovsky effect on an asteroid with the spin axis normal to the orbital plane. The Sun heats the surface of the asteroid and after a time delay, due to thermal inertia, this energy is reemitted as thermal radiation. The thermal photons impart a net thrust on the asteroid by momentum conservation as they leave the surface. The thermal force, which is in the direction of the orange arrows, has a component parallel to the orbit because of thermal inertia. This force causes the semi-major axis to increase in the case of prograde rotation. The semi-major axis would decrease in the case of retrograde rotation. Figure 1.5b shows the seasonal component of the Yarkovsky effect on an asteroid with the spin axis in the orbital plane. As the asteroid travels around its orbit, the strength of the recoil force varies due to thermal inertia.

21 21 Figure 1.5: (a) The diurnal Yarkovsky effect. (b) The seasonal Yarkovsky effect. Reproduced from Bottke et al. [2006]. Averaging this force over one orbital period results in a thermal drag that always shrinks the orbit of the asteroid. The Yarkovsky effect requires nonzero thermal conductivity and depends on several parameters, such as the size of the object, the distance from the Sun, the spin and the obliquity. For a fixed thermal conductivity, the magnitude of the Yarkovsky effect increases with spin at slower spin rates and becomes neglible in the limit of infinitely higher spin rates [Bottke et al., 2006]. If the object has a very fast rotation period, the temperature differences between the day and the night side of the object vanish. The diurnal and the seasonal components of the Yarkovsky effect are proportional to the cosine and to the sine of the obliquity, respectively [Vokrouhlický et al., 2000]. The diurnal effect is typically the dominant component. The Yarkovsky effect has been detected by fitting orbits to precise measurements of the position on the sky and/or measurements of the distance from Earth that are obtained

22 from optical astrometry or radar ranging. In 2003, radar ranging from Arecibo 9 revealed the Yarkovsky effect on the NEA 6489 Golevka, which has a diameter of 0.5 km [Chesley et al., 2003]. Using only astrometric measurements from optical data to obtain the orbital fit of the NEA 1992 BF, which has an estimated diameter of 0.4 km, a second detection of the Yarkovsky effect was made in 2008 [Vokrouhlický et al., 2008]. More recently, a third detection was obtained also using radar ranging from Arecibo and Goldstone 10 on the NEA Bennu (1999 RQ 36 ), which has a diameter of 0.5 km and is the target of the OSIRIS-REx 11 sample return mission [Chesley et al., 2012]. Figure 1.6 shows the gravitational and the Yarkovsky predictions of the observations in the radar delay and the Doppler space obtained for the NEA 6489 Golevka. The Yarkovsky effect must be included in order to match the obtained radar observations. A model with only gravitational perturbations is not able to produce a correct prediction, not even accounting for the respective uncertainties in astrometric measurements and small body and planetary masses. The Yarkovsky effect has become an essential tool for understanding several aspects of asteroid dynamics such as meteorite ages [Farinella et al., 1998], the characteristics exhibited by asteroid families [Bottke et al., 2001] and the delivery of NEAs from the Main Belt [Vokrouhlicky and Farinella, 1998]. Previous studies concerning the latter conclude that 37% of the NEAs have come through the secular resonance ν 6 at the inner edge of the Main Belt, generally only by retrograde rotation, while the rest have come through other resonances with an equal fraction of retrograde and prograde rotators [Bottke et al., 2002, La Spina et al., 2004]. Therefore, the majority of NEAs are thought to 9 The world s largest and most sensitive radio telescope located in Arecibo, Puerto Rico. ( 10 Goldstone, located in California, is a set of antennas established to communicate with spacecrafts that are also used for Solar System Radar (GSSR). ( 11 The OSIRIS-REx spacecraft will travel to the NEA (101955) Bennu, study it in detail, and bring back a sample to Earth. ( 22

23 23 Figure 1.6: Predicted Yarkovsky-induced offset with 90% confidence ellipses in the space of radar delay (range) and Doppler (range rate) measurements on 26 May :38 UTC. Ellipses labeled OBS, SBM, PLM, and YRK represent the dispersions caused by uncertainties in astrometric measurements, small body masses, planetary masses, and Yarkovsky modeling, respectively. The SUM1 ellipse, which is the combination of the OBS, SBM, and PLM uncertainties, depicts the 90% confidence region for a non- Yarkovsky prediction. Similarly, the SUM2 ellipse, which includes the added uncertainty of the Yarkovsky modeling, represents the 90% confidence region for the prediction with Yarkovsky accelerations. The actual Arecibo radar delay and derived Doppler measurement at this epoch is shown by a diamond with error bar. Figure and caption reproduced from Chesley et al. [2003]. have been delivered to their current orbits having a retrograde rotation. Indeed, recent studies that estimate the semimajor axis drift rates and uncertainties due to the Yarkovsky effect using available radar and optical astrometry for a sample of NEAs find that the majority of the objects have negative drifts, which indicates retrograde rotation [Chesley

24 24 et al., 2008, Farnocchia et al., 2013]. In Chapter 2, we peform a superior analysis of the estimated semimajor axis drift rates in order to obtain the actual fraction of retrograde vs prograde rotators. 1.4 YORP effect The Yarkovsky effect depends on the spin state of the object, which is modified by the YORP effect. The thermal reemission from irregularly shaped bodies results in a torque that can change the rotation rate and the orientation of the spin axis [Rubincam, 2000]. Figure 1.7 shows a rotating sphere with two wedges attached to its equator to represent an object with some reflection asymmetry. If the object were reflection-symmetric, no torque would be produced since when summed over all surface elements and averaged over a rotation and orbit, the secular torques would cancel out. A net torque is only produced when reflectional asymmetry is present. The faces of the two wedges are not parallel to the surface of the sphere and when the energy is reradiated in the directions of the thick arrows shown in Figure 1.8, the net thrust that the wedges feel are in different directions. This will produce a net torque about the rotation axis that in this case will make the object spin up. If the object is rotating in the other sense the torque will slow it down. The YORP effect changes not only the rotation rate of object but also the orientation of its spin axis by changing its obliquity, which is the angle between the rotational axis and the normal to the orbital plane. Figure 1.9 shows the net torques obtained for an hypothetical asteroid shape based on a spherical harmonic expansion of the real shape of the asteroid 951 Gaspra [Rubincam, 2000]. The torques are averaged over a rotation around its spin axis and about a circular orbit around the Sun. From the obtained torques we can consider how the object would evolve if it were a rigid body given an intitial value of the spin and the obliquity. The obliquity torque obtained is always positive for obliquity values between 0 and 90,

25 25 Figure 1.7: The radiation forces on the wedges will produce a net torque about the rotation axis that in this case will make the object spin up. Reproduced from Bottke et al. [2006]. Figure 1.8: The thick arrows represent the thermal photons leaving the surface. The wedges feel a net kick on the opposite direction. Reproduced from Rubincam [2000].

26 26 Figure 1.9: The thermal torques divided by the moment of inertia C for Pseudo-Gaspra as a function of obliquity. Pseudo-Gaspra is an hypothetical asteroid shape based on a spherical harmonic expansion of the shape of the asteroid 951 Gaspra. The object speed up for obliquities smaller than 55 and slow down for obliquities greater than 55. The obliquity torque is always positive for obliquity values between 0 and 90, so that it always tip over further and further as time progresses. Figure and caption reproduced and modified from Rubincam [2000]. therefore at any obliquity value in this range, the obliquity torque would increase the obliquity with time until it reaches the stable value of 90. In the range of 90 to 180, the torques are simply inverted. Therefore at any obliquity value in the range of 90 to 180, the obliquity torque would be negative decreasing the obliquity until it reaches 90. In the case of the spin torque, it is positive in the range of 0 to 55 and negative from 55 to 90. Therefore the object will have a rotational acceleration for obliquities smaller than 55 and it will slow down for obliquities greater than 55. As in the case of the

27 27 Figure 1.10: The evolution of spin and obliquity for Pseudo-gaspra, assuming Gaspras present obliquity and rotation rate (indicated by dashed line). Eventually it slows down, at which point it either starts its evolution all over (the YORP cycle) or perhaps tumbles randomly. Figure and caption reproduced from Rubincam [2000]. obliquity torques, the spin torques will be inverted in the range of 90 to 180, slowing down in the range of 90 to 135 and accelerating in the range of 135 to 180. The expected evolution of the spin state of rigid bodies, as determined by their respective YORP torques, is known as the YORP cycle (see Figure 1.10)[Rubincam, 2000]. The object would continue to spin up while increasing its obliquity until a certain point as determined by the YORP torques. Then the object would start to slow down until it changes its spin again or it would rotate so slowly that it will end up tumbling until it reestablishes its principal axis of rotation to start the YORP cycle all over. The timescales that it takes the objects to complete a YORP cycle can be shorter than the typical dynamical lifetime of an asteroid. The YORP effect completely dominates collisions for

28 28 smaller objects due to the YORP timescale dependence on R 2 [Rubincam, 2000]. The YORP timescales are in the order of millions of years for kilometer-sized objects but only decades for meter-sized objects. Vokrouhlický and Čapek [2002] computed the YORP torques on a sample of 500 simulated asteroid shapes approximated by Gaussian random spheres [Muinonen, 1998, Muinonen and Lagerros, 1998] and found that about 40% of the sample had a shape of the torques as in Figure 1.9 while about another 40% have inverted shapes of the curves in the range of 0 and 90. With the inverted curves, the obliquity torque will be negative in the range of 0 and 90 and positive in the range of 90 to 180 making the asteroid evolve toward the asymptotic values of 0 and 180 instead. The remaining 20% of the sample had a variation of the YORP torques where the obliquity torque will go from positive to negative in the range of 0 and 90. The obliquity distibution obtained for the whole sample is shown in Figure The change in the obliquities of rigid bodies due to the YORP effect should leave a distinctive signature, driving the spin axis of most objects to obliquity values of 0, 90 and 180 degrees [Vokrouhlický and Čapek, 2002]. The NEA 2000 PH 5 (now renamed YORP) is the best evidence to date of the YORP effect, which as shown in Figure 1.12, has an increase in the rotation rate that cannot be explained by gravitational torques [Lowry et al., 2007, Taylor et al., 2007]. A second detection was made in the same year on the NEA 1862 Apollo [Kaasalainen et al., 2007] and a third detection was made in the following year on one of the largest NEAs, 1620 Geographos [Durech et al., 2008]. More recently in 2012, a possible fourth detection was made on the NEA 3103 Eger [Ďurech et al., 2012]. All of these NEAs for which the YORP effect has been detected show a rotational acceleration. However, the YORP effect will vary from object to object and should be equally capable of driving asteroids to faster or slower rotation rates.

29 29 Figure 1.11: Distribution p(ɛ) of the asymptotic obliquities (ɛ) of the YORP evolution for the sample of 500 Gaussian random spheres. Reproduced from Vokrouhlický and Čapek [2002]. Prior to the Hayabusa spacecraft encounter, using pre-encounter shape models of the NEA Itokawa, Vokrouhlický et al. [2004] predicted a rotational acceleration due to the YORP effect. Scheeres [2007a] found that the first post-encounter shape model implied a rotational deceleration; thus it was impossible to correctly predict even the sign of the effect from pre-encounter models. Moreover, Scheeres [2007a] found that the predicted YORP torques did not converge as they incorporated progressively higher resolution models from the later data obtained with the Hayabusa Spacecraft. Recently Statler [2009] has shown that the YORP effect has an extreme sensitivity on the topography of the asteroids. Using identical objects and adding the same crater or boulder in different positions Statler [2009] showed that the resulting torques could differ by

30 30 Figure 1.12: Observed YORP effect on asteroid (54509) 2000 PH5. The sidereal rotation period P is steadily decreasing. Solid symbols represent the observed period change (the formal 1s uncertainty on the period is smaller than the plotted data points). The solid curve is a theoretical prediction for YORP strength computed from different shape variants of the asteroid and recalibrated by a factor of 0.16 to Open circles (with 1s uncertainties) indicate the expected change of P during annual close encounters with Earth (arbitrarily offset for clarity), which is negligible relative to the observed period change. Reproduced from Lowry et al. [2007]. factors of order unity or even change sign (see Figures 1.13 and 1.14). These results imply not only that we can expect most predictions of the YORP effect to be incorrect by an order 100% due to the unresolved topography of the asteroid, but also that the evolution driven by the YORP effect is poorly understood. A minor change in the surface of an asteroid could cause a different YORP torque and therefore an asteroid may not necessarily evolve through the YORP cycle as was previously thought. The results found

31 31 Figure 1.13: Typical variation in torques caused by an identical single boulder randomly placed on an identical object. Top and bottom curves correspond to the objects shown in Figure Note the possibility of reversing the sign of the spin torque at all obliquities simply by repositioning one boulder. Figure and caption reproduced and modified from Statler [2009]. by Statler [2009] can explain why the predictions of the YORP torques failed to converge in the case of Itokawa. A similar conclusion was reached by Scheeres and Gaskell [2008] who found that internal density variations can alter the YORP torques.

32 32 Figure 1.14: Identical views of the two objects from Figure 1.13 with the (left) most positive and (right) most negative spin torques. Line of sight is in the plane containing the short and middle axes of the body, 45 from each, and the illumination is from the right at 30 phase angle. Moving the boulder by only twice its own diameter reverses the sign of the spin torque. Figure and caption reproduced from Statler [2009]. 1.5 Open Questions During this last decade our understanding of the Yarkovsky and the YORP effects has significantly improved and we have been succesful in obtaining proof of these effects by direct detections. However, it is essential to further develop our knowledge on how these effects act and how they have affected the evolution of NEAs. The Yarkovsky Effect is an essential tool for understanding the delivery of NEAs from the Main Belt [Vokrouhlicky and Farinella, 1998] and therefore is an essential mechanism to understand the orbital evolution of NEAs. The Yarkovsky effect depends on the spin state of the object, which is modified by the YORP effect. The YORP effect changes not only the rotation rate of an object but also the orientation of its spin axis. This effect drives the obliquities of the objects to values of 0,

33 33 90 and 180 degrees [Vokrouhlický and Čapek, 2002]. Since the time that it takes the objects to complete a YORP cycle could be much less than their typical dynamical lifetimes, one should expect the obliquity distribution of the population of NEAs to be concentrated about the values of 0, 90 and 180 degrees. It is therefore neccessary to search for the signature of the YORP effect in the obliquity, not on individual objects, but rather in the population of small NEAs. Obtaining the obliquity distribution of NEAs is one of the main goals of this research and the results are presented in Chapter 2. If the obliquity distribution of NEAs shows that the obliquities of the asteroids tend towards those values, it would be supporting evidence for the significance of the YORP effect as the main physical process in the spin evolution of small NEAs. As the asteroids complete the YORP cycle and spin down to zero, the memory of the previous spin state is forgotten as they tumble until reestablishing their principal axis rotation with their spin axis pointing in a random direction. Therefore, if the YORP effect is a dominant mechanism in the evolution of NEAs and the time required for small NEAs to complete a YORP cycle is much less than their typical dynamical lifetime, it is unexpected that the population of NEAs seems to remember their initial spin state at the time of delivery from the Main Belt to their current orbits. The YORP effect should have randomized the distribution of prograde vs retrograde rotators. Also, the impression from the observed rotation rate distribution (see Figure 1.4 on page 19) that the great majority of small NEAs are fast rotators is not what we would expect either. Recently, Statler et al. [2013] debiased their observed sample of 83 NEAs and obtained the fraction of Fast-Rotating Asteroids (FRAs) as function of absolute magnitude (see Figure 1.15). They find that the spin rate distribution changes rapidly with size, having essentially nothing larger than 170 meters in diameter as an FRA while nearly everything smaller than 60 meters is an FRA with a transiton happening over less than a factor of 3 in diameter. The YORP effect will vary from object to object and should be equally capable of driving

34 34 small NEAs to faster or slower rotation rates. Despite that radiation recoil forces seem to be indeed a dominant mechanism in the orbital evolution of small NEAs, in terms of the spin evolution, our understanding of these effects seems to be poor. Our current knowledge of these effects cannot explain why the YORP torques somehow become negligible and do not dominate the spin state evolution as expected. Previous simulations of the dynamical evolution of asteroids under the influence of the YORP effect have assumed a monotonic YORP torque that continuously spins up the object past the point where mass shedding and possible re-accumulations of the sheded mass occur without taking into consideration the consequences of the changes in the shape (see Figure 1.16) [Walsh et al., 2008, Richardson et al., 2009]. As a result, the YORP effect has been a preferred candidate for the formation of binary asteroids. However, from the results found by Statler [2009] we know that this scenario is not realistic. Statler [2009] has shown that the YORP effect has an extreme sensitivity on the topography of the asteroids and a minor change in the surface of an aggregate asteroid can stochastically change the YORP torques. Moreover, Holsapple [2010] showed that if an object is spun up and allowed to deform continuously, the deformation increases the moment of inertia sufficiently, such that the increase in the angular momentum results in a decreasing spin rate. The time scales over which mass reconfigurations occur are much shorter than the time scale over which the YORP effect changes the spin rates and obliquities. Statler [2009] predicts that shape changes may result in a self-limiting or stochastic YORP effect. If the continuous reconfiguration leads to a shape of the aggregate that is nearly symmetric, the YORP torques could become negligibly small or even vanish. On the other hand, due to the extreme sensitivity of the YORP effect on the topography of asteroids the reconfiguration of an asteroid that for example is spinning up could lead to a change in sign of the YORP torques and make the asteroid spin down. Both cases would imply a

35 35 Figure 1.15: The de-biased (completeness-corrected) fast-rotator fraction F is plotted against absolute magnitude H (lower scale) and nominal diameter (upper scale, assuming spherical objects with albedo of 0.17). Vertical dotted lines indicate boundaries of bins containing nearly equal numbers of observed objects. Grayscale indicates probability density P(F) in each absolute magnitude bin. Solid line connects the most probable value of F in each bin; dashed lines enclose 68% of the integrated probability, corresponding to a1σ error region. Note the abrupt transition, spanning less than a factor of 3 in diameter, between the small objects, the strong majority of which are fast rotators, and the larger objects, almost none of which are fast rotators. Figure and caption reproduced from Statler et al. [2013]. self-limitation in the evolution of the spin state due to the YORP effect and the objects would not follow the classical YORP cycle. Moreover, subsequent reconfigurations could lead to random walk in the YORP torques making the evolution of the spin state completely stochastic.

36 36 Figure 1.16: Snapshots of a simulation where changes in the shape and mass loss of an aggregate asteroid can be observed as a result of applying constant YORP torque. The particles in the surface shown in orange, move toward the equator and are eventually shed as the shape becomes more oblate. Figure reproduced from Walsh et al. [2008]. An extensive analysis that self-consistently models the YORP effect on dynamically evolving aggregates is then crucial. Until such an analysis is done, assumptions concerning the consequences of the YORP effect on the evolution of the NEA population are very limited and potentially erroneous. Among various aspects, we will not comprehend in its totality the role of the YORP effect on the formation of binary asteroids and neither can we provide a correct interpretation of the obliquity distribution of NEAs, which we obtain in Chapter 2. We present the results of the first simulations that self-consistently model the YORP effect on dynamically evolving aggregates in Chapter 3. Such analysis will let us test whether the YORP acceleration is self-limiting and whether the shape changes interrupt the YORP cycle and make the spin evolution stochastic. The results obtained will help to explain how the YORP effect acts among the NEAs and therefore will contribute to the understanding of the dynamical evolution of the NEA population.

37 2 OBLIQUITY DISTRIBUTION OF NEAS Introduction The YORP effect drives the obliquity of most objects that follow the YORP cycle to the values of 0, 90 and 180 degrees [Vokrouhlický and Čapek, 2002]. But to obtain a direct measurement of the distribution of NEA obliquities is not an easy task. To determine directly the rotation poles of asteroids will require radar observations or multiple lightcurves at different illumination and orbital phases for each NEA. Once the light curves are obtained, one can follow the Kaasalainen et al. [2001] method of light curve inversion to find the rotation poles of the asteroids. So far, there are only about 20 NEAs for which a rotation pole has been determined [Kryszczyńska et al., 2007]. There are ongoing efforts to try to help determine the rotation poles of asteroids. A Database of Asteroid Models from Inversion Techniques (DAMIT) was launched in 2010, where existing data and models are available to the scientific community [Durech et al., 2010]. Also the Minor Planet Bulletin is continuously releasing lists of target asteroids for photometry opportunities for which additional light curves will help to determine their shape and rotation pole via light curve inversion or for which additional light curves can help constrain the estimated solutions [Warner et al., 2013]. Here, we intend to search for the signature of the YORP effect in the obliquity, not of individual objects, but rather in the population of small NEAs by constraining the obliquity distribution. If the distribution peaks at obliquities of 0, 90 and 180 degrees, it would be supporting evidence for the significance of the YORP effect as the main physical process in the spin evolution of small NEAs and for the basic correctness of the YORP cycle picture. Instead of obtaining a direct measurement, the obliquity distribution can be inferred if the distribution of semimajor axis drift rates can be measured. The semimajor axis drift

38 38 due to the Yarkovsky effect is given by: da dt = A 2 ( r0 a ) 2 2a 3 2 k(1 e 2 ), (2.1) where a is the semimajor axis, e is the eccentricity, r 0 is a normalizing parameter arbitrarily chosen to be 1 AU and k = AU 3 2 /d is the Gaussian gravitational constant. The acceleration parameter A 2, which is determined from orbital fits, depends only on properties of the object. From the linear heat diffusion theory for a spherical body [Vokrouhlicky, 1998, Vokrouhlický, 1999], A 2 is given by A 2 = 4α 9 Φ(1AU)F(Θ) cos ɛ, (2.2) where ɛ is the obliquity, the distribution of cos ɛ is given by the assumed model, α is the absorptivity, Φ(1AU) is the radiation force factor at 1 AU and F(Θ) is a function of the thermal parameter Θ. The absorptivity is given by α = 1 A, where A is the Bond albedo. The radiation force factor at 1 AU is given by: Φ(1AU) = 3F(1AU), (2.3) 2Dρc where F(1AU) = 1371Wm 2 is the mean solar radiation flux at 1 AU, D is the diameter, ρ is the bulk density and c is the speed of light. The function of the thermal parameter Θ is given by: F(Θ) = and the thermal parameter is given by: 0.5Θ 1 + Θ + 0.5Θ 2, (2.4) Θ= Γ εσt 3 o 2π P, (2.5) where ε is the emissivity, σ is the Stefan-Boltzman constant, P is the rotation period, T o is the subsolar temperature given by: ( ) 1 αf(1au) 4 T o = (2.6) εσa 2

39 39 and Γ is the thermal inertia given by: Γ= ρκc, (2.7) where κ is the thermal conductivity and C is the specific heat capacity. Note that (for small observational uncertainties), objects that have negative drifts indicate a retrograde rotation and an obliquity value greater than 90 and less or equal than 180. On the other hand, objects with positive drifts indicate prograde rotation and an obliquity value equal or greater than 0 and less than 90. If an object has a small uncertainty and a semimajor axis drift rate close to zero, the obliquity value may be 90, or it could be due to a combination of physical parameters that makes the Yarkovsky acceleration small, such as high mass and low thermal inertia. Therefore, is not possible to infer individual obliquities. Taking advantage of the simple dependence between the drift rate and cos ɛ, previous studies have attempted to infer the obliquity distribution [Chesley et al., 2008, Farnocchia et al., 2013]. Figure 2.1 shows a distribution resulting from 136 measurements of the Yarkovsky drifts [Farnocchia et al., 2013]. These authors stress that this result should be considered only approximate because of the neglect of the dependence of A 2 on density, thermal properties, and shape. Moreover, they adopted the unfortunate strategy of placing objects with unphysical computed values of cos ɛ > 1 into the leftmost and the rightmost bins in the figure. Here we seek to obtain the obliquity distribution of NEAs using a better approach based on Bayesian inference that takes into account our prior knowledge of the distributions of the physical parameters on which the semimajor axis drift rates depend. Our approach guarantees physical values of cos ɛ for all NEAs in the sample and can be equally applied to the whole data set or to subsets of the data, giving us some insight as well about the weight that assumptions made about the quality of the data have on the final

40 40 Figure 2.1: Inferred obliquity distribution based on 136 objects. Farnocchia et al. [2013] represents the obliquity with the symbol γ instead of ɛ. Reproduced from Farnocchia et al. [2013]. results and about the quality of the data itself. The obliquity distribution can in turn be used to test YORP predictions and constrain YORP evolution. 2.2 Bayesian Approach We want to identify the intrinsic obliquity distribution that is most consistent with the semimajor axis drift rates recently measured for a sample of 4042 NEAs by Steven R. Chesley and Davide Farnocchia using the Jet Propulsion Laboratory (JPL) Comet and Asteroid Precision Orbit Determination Package [Farnocchia et al., 2013]. We have developed a code that can test a wide variety of models of the obliquity distribution. The

41 41 code takes as an input a list of the obtained acceleration parameter A 2, the respective uncertainty σ A2, the semimajor axis a and the absolute magnitude H, for the sample of observed NEAs. The acceleration parameter A 2 and its uncertainty are direct outputs from the orbital fits and the absolute magnitude is known from photometric observations. We choose a parameterized model of the obliquity distribution to be tested. For each observation we run a large number of Monte-Carlo trials to obtain the distribution of the expected values of the acceleration parameter, A 2EXP. The acceleration parameter A 2 is related directly to the semimajor axis drift as shown in equations (2.1) and (2.2). In order to obtain A 2EXP, we need the semimajor axis a, the absolute magnitude H, the obliquity ɛ, and a set of other important physical parameters. The semimajor axis a and the absolute magnitude H are obtained from the observations, the obliquity ɛ from the model to be tested, and the set of other physical parameters are given by our prior knowledge of their distributions. Since we do not intend to constrain the prior distributions of the other physical parameters, they are considered nuisance parameters in Bayesian terminology. The set of nuisance parameters are the absorptivity α, the albedo p v, the bulk density ρ, the thermal inertia Γ and the period P. The absorptivity is given by α = 1 A, where A is the Bond albedo. The Bond albedo is the total incident solar radiation that is reflected back into space by the object and it is related to the geometrical albedo p v and the slope parameter G by the emprirical relation A = ( G)p v [Schläppi et al., 2008]. The slope parameter describes the change in the brightness of the object at different illumination phase angles. A Gaussian distribution with a mean of 0.18 and a σ of 0.13 is chosen from the observationally detemined values of G in the JPL Small Bodies Database [Farnocchia, 2013]. The geometric albedo p v and the bulk density ρ are given by randomly selecting a taxonomic type from the current estimates of the percentage of the NEA population that belong to the S, C and X types [Chesley et al., 2002] and then selecting their characteristics according

42 42 to the mean and standard deviation values shown in Table 2.1 [Thomas et al., 2011, Farnocchia, 2013]. The thermal inertia Γ distribution is given by Γ= d 0 D ξ [Delbo et al., 2007] where d 0 = 300 ± 45, ξ = 0.36 ± 0.09 and D is the diameter, which is determined from D = H [km]. (2.8) ρ 2 1 The period P distribution is taken to be flat in logp between limits of 30 seconds and 30 hours, which is reasonably representative of the entire population of small NEAs down to approximately 10 m in diameter. A more sophisticated treatment would have a size dependent period distribution but we neglect this complication for this first effort. Table 2.1: Taxonomic Types. Taxonomic type Percentage ρ(gcm 3 ) p v S ± ± C ± ± X ± ± Each trial involves randomly choosing a value of each nuisance parameter according to its prior distribution and a value of the obliquity according to the distribution set by our parameterized model in order to obtain A 2EXP. A normalization of the distribution of the N = 30, 000 values of A 2EXP obtained gives us then the probability of A 2EXP, P(A 2EXP ). The probability P(A 2EXP ) is then convolved with a Gaussian of width σ A2 to obtain the probability density for the measured A 2, P (A 2 ). We evaluate this distribution at the actual measured value A 2 = A 2M. Multiplying all the values of P (A 2M ) gives us the likelihood of the data in the model obliquity distribution. The model obliquity distribution is itself a parameterized function of cos ɛ with parameters p 1, p 2... Repeating the whole process

43 43 over the parameter space gives us the likelihood function L(p 1, p 2...). We assume no prior knowledge of the distribution of parameters that describe the model other than flat over the parameter space. Then L(p 1, p 2...) is normalized so that the integral over the parameter space is one to obtain the posterior density P(p 1, p 2...). Once the posterior density P(p 1, p 2...) is obtained, the value(s) of the parameters that best fit the data can be determined. A significant, probably large, but unknown fraction of the drift rate measurements are probably spurious, arising from systematic errors in astrometry rather than from the Yarkovsky effect. In order to test the robustness of our results, and to gain some information on the likely fraction of spurious measurements, we apply three different filters to exclude objects that may have a greater chance of spurious drifts. - Filter 1 [Farnocchia et al., 2013] is based on the result for 1999 RQ36, which is the largest measured drift rate and the most reliable detection. Because this object is both a binary and a space mission target, it is extremely well studied, and its mass, obliquity, albedo, density, and thermal inertia are all well constrained. Furthermore, the values of these parameters are such that the measured rate is close to the maximum possible Yarkovsky drift for an object of this size. Because A 2 scales as D 1 (all other parameters being assumed equal), Filter 1 excludes all objects for which the measured A 2 is greater than 1.5A 2,max where A 2,max = A 2,RQ36 (D RQ36 /D). Applying this Filter leaves 491 objects in the sample. - Filter 2 is a more stringent version of Filter 1, based on the ability of the data and the orbit fit to detect a drift rate near the maximum given the uncertainty σ A2 determined by the fit. Filter 2 excludes objects for which σ A2 /A 2,max < 1.5, leaving 125 objects remaining. - Filter 3 is unrelated to expected values of the drift rate, and instead is based solely on the quality of the astrometric data and the orbit fit. Filter 3 excludes all objects for

44 44 which the uncertainty in the fitted a is greater than AU, which leaves 352 objects in the sample. We test a V-shaped model (Figure 2.6) of the distribution of obliquities f (cos ɛ) with two parameters, p 1 and p 2. The first parameter p 1 is defined as the fraction of retrograde rotators F R, which can range from 0 to 1. The second parameter p 2 describes the behavior of the function at cos ɛ = 0 and also ranges from 0 to 1. A value of p 2 = 0 forces f (0) = 0 while a value of p 2 = 1 gives f (0) its maximum allowed value given the value of p 1 and the requirement that f (cos ɛ) 0. We also test a 2-bin model with one bin in the range of cos ɛ from 1 to 0 and the other from 0 to 1, corresponding to the regions of retrograde and prograde rotators. The single parameter p 1 is the fraction of retrograde rotators F R. More elaborate models with three or more parameters can, and as we show, should, be used, but for these prelimary results we confined the discussion to these two simple models. 2.3 Results Figure 2.2 shows the 68.3%, 95.4% and 99.7% highest posterior density contours corresponding to the boundaries of 1σ,2σ and 3σ error regions in the parameter space of the posterior density P(p 1, p 2 ). All filters were applied to the data in this case. Even using all the filters, the constraint on the model parameters p 1 and p 2 is very confined. In fact, the coarse grid of parameters used only marginally resolves the sharp peak. The effects of Filters 1 and 2, which are based on the values of A 2, can be seen in Figures 2.3 and 2.4. In these preliminary results, the peaks are completely unresolved. Nonetheless one can see that Filter 1 is responsible for the extension of the peak towards lower values of p 1 and Filter 2 for the extension towards higher values of p 1.

45 45 Figure 2.2: 1, 2 and 3 σ contours of the posterior density of p 1 and p 2, P(p 1, p 2 ) for the V- shaped model applying all filters to the data. The parameter p 1 is the fraction of retrograde rotators F R and the second parameter p 2 describes the behavior of the obliquity distribution function at cos ɛ = 0. A value of p 2 = 0 forces f (0) = 0 while a value of p 2 = 1 gives f (0) its maximum allowed value given the value of p 1 and the requirement that f (cos ɛ) 0. Filter 3, which filters the data without regard to the value of A 2, produces a single unresolved peak (Figure 2.5) at p 1 = 0.7 and p 2 = 0.0. This best fit distribution is shown in Figure 2.6. Because the peak is unresolved, we can say that the 3σ error in p 1 is no larger than 0.05, this being the resolution of the grid. The fraction of retrograde rotators is 0.70, the fraction of prograde rotators is 0.30 and the value of the function at cos ɛ = 0 is equal to zero. This result is consistent with the result of Farnocchia et al. [2013] (see Figure 2.1) who find a retrogade fraction of 0.69.

46 46 Figure 2.3: 1, 2 and 3 σ contours of the posterior density of p 1 and p 2, P(p 1, p 2 ) for the V-shaped model applying Filters 1 and 3 to the data. The parameter p 1 is the fraction of retrograde rotators F R and the second parameter p 2 describes the behavior of the obliquity distribution function at cos ɛ = 0. A value of p 2 = 0 forces f (0) = 0 while a value of p 2 = 1 gives f (0) its maximum allowed value given the value of p 1 and the requirement that f (cos ɛ) 0. For the case of a simpler 1 parameter 2-bin model, we find similar unresolved peaks between retrograde fractions of 0.70 and The 2-bin model assumes a uniform distribution in the range of cos ɛ from 1 to 0 for retrograde rotators and another uniform distribution in the range of cos ɛ from 0 to 1 for prograde rotators. Obtaining similar values for the fraction of retrograde rotators using models with such significant difference in the distributions, a uniform distribution in contrast with the more constrain V-shaped distribution, suggests that the results are robust.

47 Figure 2.4: 1, 2 and 3 σ contours of the posterior density of p 1 and p 2, P(p 1, p 2 ) for the V-shaped model applying Filters 2 and 3 to the data. The parameter p 1 is the fraction of retrograde rotators F R and the second parameter p 2 describes the behavior of the obliquity distribution function at cos ɛ = 0. A value of p 2 = 0 forces f (0) = 0 while a value of p 2 = 1 gives f (0) its maximum allowed value given the value of p 1 and the requirement that f (cos ɛ) 0. 47

48 Figure 2.5: 1, 2 and 3 σ contours of the posterior density of p 1 and p 2, P(p 1, p 2 ) for the V-shaped model applying Filter 3 to the data. The parameter p 1 is the fraction of retrograde rotators F R and the second parameter p 2 describes the behavior of the obliquity distribution function at cos ɛ = 0. A value of p 2 = 0 forces f (0) = 0 while a value of p 2 = 1 gives f (0) its maximum allowed value given the value of p 1 and the requirement that f (cos ɛ) 0. 48

49 49 Figure 2.6: The preliminary estimated best solution for the obliquity distribution of NEAs using a V-shaped model with f (0) = 0 and having a fraction of retrograde rotators of F R = Discussion The 2-parameter V-shaped model and the 1-parameter 2-bin model result in essentially the same values for the retrograde fraction, implying that this result is robust and does not strongly depend on the detailed shape of the distribution. Clearly all of the models should be re-run on a much finer grid in order to fully resolve the peaks in the probability distributions. Applying only Filter 3 results in the more constrained solution. Filter 3 rejects objects with σ a > AU, which may signal a problem with the orbital fit. This suggests that the additional applications of filters based on assumptions of what should be

50 50 the expected values of the semimajor axis drifts should be avoided, as actual probable values of the Yarkovsky drifts are being excluded. This is actually not surprising due to the dependence of the Yarkovsky effect on several physical parameters of the NEAs, which are unknown for the great majority. Assuming an expected value of the semimajor axis drift locks a certain combination of the physical parameters of the NEAs, when in reality there is vast range of possible combinations. The results presented here demostrate the power and flexibility of the Bayesian approach. We intend to investigate more sophisticated multi-parameter models for the obliquity distribution before preparing this work for publication. The publication would be in collaboration with Steven R. Chesley and Davide Farnocchia from the Jet Propulsion Laboratory (JPL), who are employing a different methodology with the same data.

51 3 AGGREGATE DYNAMICS Introduction The rotation rates of asteroids obtained from optical light curves have given strong evidence that most asteroids are aggregates. These objects may not remain rigid bodies throughout their evolution. A minor change in the shape of an asteroid may significantly change the YORP torques. This chapter presents the first effort to self-consistently model the YORP effect on dynamically evolving aggregates, which can change in shape. For these simulations we have combined two codes, TACO 12 [Statler, 2009] and pkdgrav [Stadel, 2001, Richardson et al., 2000]. TACO is a thermophysical asteroid code developed by Thomas S. Statler, the author and other students at Ohio University. TACO models the surface of an asteroid using a triangular facet representation, which is the typical method used due to its geometrical simplicity [Vokrouhlický and Čapek, 2002, Ćuk and Burns, 2005]. Shadowing is handled by calculating an horizon map for each tile, which gives the maximum elevation of all visible parts as a function of azimuth. The reflectance properties of the surface are represented by a Hapke model for the bidirectional reflectance [Hapke, 2002], adopting parameters representative of S class asteroids [Helfenstein and Veverka, 1989], which are the most abundant type. The thermal emission is assumed to be Lambertian (i.e., isotropic into the sky hemisphere), with a correction for partial blockage of the sky by an elevated horizon. The code computes the torques from both the reflected and emitted radiation, though the latter dominates for typically dark asteroids. For these simulations we work in the limit of zero thermal inertia, so that absorbed radiation is re-emitted instantaneously. In general a non-zero thermal inertia changes the obliquity torques, but not the spin torques, so this simplification is a reasonable strategy for obtaining statistically representative results. The code pkdgrav is a 12 Thermophysical Asteroid Code, Obviously.

52 52 cosmological N-body tree code modified to simulate the dynamical evolution of asteroids represented as aggregates of spheres using gravity and collisions by Richardson et al. [2000]. We use the hard-sphere discrete element method in which collisions are treated as instantaneous events between rigid spheres. The spheres are constantly bouncing by a distance much smaller than the radius of the spheres. Energy dissipation is parameterized using a normal and a tangential coefficient of restitution. A normal coefficient of restitution equal to one or zero defines a completely inelastic or elastic collision, respectively. This is the same code used in the binary formation simulation shown in Chapter 1 (Figure 1.16). 3.2 Code Development We have developed several algorithms in order to interact correctly between TACO and pkdgrav Tiling Each code, TACO and pkdgrav, uses the most appropiate representation of an asteroid to do its calculations. Spheres are convenient to simulate the dynamical evolution of aggregates but the triangular tiling is the method of choice for calculating the YORP torques due to its geometrical simplicity. We have developed an algorithm that fits a triangular tiling over an object created for pkdgrav, composed of spheres, in order to be used with TACO to compute the YORP torques. Assuming all spheres have the same uniform density, the total mass of the object is obtained. Then the coordinates of the center of mass are determined and the coordinates of all spheres are shifted so that the center of mass is at the origin. The inertia tensor for the object is then calculated and diagonalized. Using the diagonalized inertia tensor we solve for the values of the three axes a, b and c of an equivalent ellipsoid with the same mass and inertia tensor. Then we

53 53 Figure 3.1: The aggregate object used for the pkdgrav (spheres) dynamical simulations along with the triangular tiling obtained for the equivalent ellipsoid with the same mass and inertia tensor as the aggregate object using TACO (colored lines). create a tiling of the equivalent ellipsoid as shown in Figure 3.1. The original object is rotated to the principal axis orientation. At this stage the ellipsoidal tiling is close to the object and the goal is to now adjust the vertices to fit the tiling tightly around the object. We define the function G(x, y, z) R 2n N [ (x x j ) 2 + (y y j ) 2 + (z z j ) 2] n G0, (3.1) j=1 where R is the radius of each sphere (assumed to be the same for all spheres), x j, y j and z j are the coordinates of the center of sphere j, and n and G 0 are constants. The value of G 0 is chosen so that the surface G(x, y, z) = 0 tightly surrounds the object. We have found by trial and error that the choice n = 2 and G 0 = 1.25 places the tiling tightly around a variety of aggregate shapes. Each vertex of the ellipsoidal tiling is moved in or out in the direction normal to the ellipsoid, to place it on the surface G(x, y, z) = 0 as shown in Figure 3.2. Last, the final tiling is rotated back to the true orientation of the original object.

54 54 Figure 3.2: Each vertex of the ellipsoidal tiling obtained in Figure 3.1 is moved in or out in the direction normal to the ellipsoid, to place it on the surface Sensitivity It will be necessary to recompute the YORP torques and modify the tiling every time there is a movement of material as we dynamically evolve the aggregate objects using pkdgrav. From the results found by Statler [2009], we know that the YORP effect has an extreme sensitivity on the topography of the asteroids. Therefore, it is of great concern how sensitive the torques are to how the tiling is positioned in the surface of the objects. Redoing the whole mesh does not guarantee that the tiles will be at the exact position that they were before in those regions where there was no material movement. We carry out some tests in order to determine how a shifted tiling on the same object would affect the YORP torques. We do a test using two different aggregate objects. The objects are composed of 921 and 4,281 spheres. For each object, we create a grid of 16 rotations that will shift the tiling. We use the angle subtended by the diameter of the most distant sphere as the upper limit angle. The upper limits for the two objects tested were 3 and 8 degrees. Each rotation was also done using 8 different resolutions, ranging from having 1,784 tiles up to

55 19,984 tiles. A total of 128 tilings is then obtained for each object. We obtain the torques of each shifted tiling and compare it with the torques obtained at the original orientation of the object. Figures 3.3 to 3.6 show the normalized average absolute deviation obtained for the spin and the obliquity torques at the different resolutions obtained for the two objects used. The normalized average absolute deviation is defined as: 55 NAAD = m 1 m i=1 T m T original T original (3.2) at each input obliquity, where m is the number of rotations, and T m and T original are the torques for a given rotation and for the original orientation of the object, respectively. NAAD is obtained for each resolution at each value of obliquity and is the difference between the torques averaged over the 16 rotations. Each color in Figures 3.3 to 3.6 represent a different resolution. As expected, due to the extreme sensitivity of the YORP effect on the topography of the asteroids, the resulting torque depends on the shift of the tiling at all resolutions. There is no ideal resolution that minimizes the sensitivity, and there is nothing to be gained by using an extremely fine tiling that over-resolves the spheres (which are themselves idealizations). This implies that the exact results of our simulations will depend on arbitrary choices of parameters related to the resolution and tiling. As a consequence, the most significant results that can be obtained from the simulations would be those that are generic, such as what types of behavior result form the self-consistent YORP, and those that are statistical, such as what fraction of objects exhibits a particular behavior of the YORP effect. However, it is essential to guarantee that the tiling stays the same in those regions where there was no movement, so that any difference in the torques is only due to a new shape of the object and not as a result of a shifted tiling. We then need to perform a transformation of the object at the time t = T back to its original position at time t = 0 every time that the tiling needs to be recalculated. This guarantees that those spheres that

56 Figure 3.3: Normalized average absolute deviation (NAAD) in the change in the spin torques obtained for the object composed of 921 spheres. Each color represents a different resolution as indicated in the legend. The lowest resolution is n = 15, which consist of 1, 784 tiles and the highest resolution is n = 50, which consist of 19,984 tiles. The maximum NAAD for resolutions of n = 30, 40 and 50 is about 39, 57 and

57 Figure 3.4: Normalized average absolute deviation (NAAD) in the change in the spin torques obtained for the object composed of 4,281 spheres. Each color represents a different resolution as indicated in the legend. The lowest resolution is n = 15, which consist of 1, 784 tiles and the highest resolution is n = 50, which consist of 19,984 tiles. The maximum NAAD for resolutions of n = 40 and 50 is about 29 and

58 Figure 3.5: Normalized average absolute deviation (NAAD) in the change in the obliquity torques obtained for the object composed of 921 spheres. Each color represents a different resolution as indicated in the legend. The lowest resolution is n = 15, which consist of 1, 784 tiles and the highest resolution is n = 50, which consist of 19,984 tiles. The maximum NAAD for the resolution of n = 20 is about 15 at an obliquity of 0 and about 3.5 at an obliquity of

59 59 Figure 3.6: Normalized average absolute deviation (NAAD) in the change in the obliquity torques obtained for the object composed of 4,281 spheres. Each color represents a different resolution as indicated in the legend. The lowest resolution is n = 15, which consist of 1, 784 tiles and the highest resolution is n = 50, which consist of 19,984 tiles. The maximum NAAD for resolutions of n = 15, 20 and 25 is about 296, 78 and 135. do not move will be at the same exact position that they were initially and therefore will result in the same tiling Transformation A movement of material is defined as when one or more spheres move by more than a quarter of the sphere s radius. If the spheres have not moved, the object at time t = T should be a rotated and translated copy of the object at the original orientation at time t = 0 except for differences due to the allowed bouncing of spheres. Our transformation

60 60 routine must return the part of the object that did not move to its original orientation in order to guarantee the placement of the tiling in the same exact positions. The spheres are at (x, y, z) at t = 0 and at (x, y, z ) at t = T. We want to find the Euler angles (α, β, γ) and displacements x, y and z that give the best fit to the transformation from the (x, y, z ) to the (x, y, z) coordinates. The Euler rotation matrix is given by: R = cos α cos γ cos β sin α sin γ cos γ sin α + cos α cos β sin γ sin β sin γ cos β cos γ sin α cos α sin γ cos α cos β cos γ sin α sin γ cos γ sin β sin α sin β cos α sin β cos β (3.3) For any given sphere the original position is given by: x = (cos α cos γ cos β sin α sin γ)x + (cos γ sin α + cos α cos β sin γ)y + sin β sin γz + x (3.4) y = ( cos β cos γ sin α cos α sin γ)x +(cos α cos β cos γ sin α sin γ)y +cos γ sin βz + y (3.5) z = sin α sin βx cos α sin βy + cos βz + z (3.6) For each sphere we have 3 equations that relate the x, y, and z positions at time t = 0 with the positions x, y, and z at a later time t = T. We have 6 unknown parameters for each transformation and a number of equations equal to 3 times the number of spheres that can be used to find the best fit for the angles α, β and γ and the displacements x, y and z. We use the LMDIF routine from the MINPACK [More et al., 1980, 1984] library of Fortran 90 routines in order to find the values of these six parameters that minimizes the sum of the squares of the residuals. The transformation routine searches for the α, β, γ, x, y and z values that result in the best fit allowing a tolerance of 25% of the sphere radius in order to account for the bouncing of the spheres. After the initial fit, all the spheres that have moved by more than

61 61 Figure 3.7: Tiling of the original object plotted in orange and the new tiling obtained after a movement over plotted in yellow. The line of sight is the spin axis and the middle axis points up. the allowed tolerance are excluded and the fit is obtained again. The exclusion and repetition of the process is continued until none of the remaining spheres has moved by more than the allowed tolerance. The obtained α, β, γ, x, y and z values are then used to transform the whole object to its initial orientation. Figure 3.7 shows the tiling obtained for an original object along with the tiling obtained after there was movement of the spheres over-plotted The Spin State Evolution The rate of change of the obliquity ɛ and the rotation angular velocity ω are given by: dɛ dt = T ɛ Cω (3.7)

62 and dω dt = T ω C where C is the moment of inertia about the rotation axis and T ɛ and T ω are the obliquity and spin torques, averaged over a rotation about the principal axis and an orbit around the Sun. For convenience a circular orbit at 1 AU is assumed. 62 (3.8) Equations (3.7) and (3.8) for the obliquity and spin evolutions are solved numerically using a fourth-order Runge Kutta integrator. For simple cases of rigid-body evolution, some enlightening exact solutions can be derived. As an example, we can use a simple approximation of T ɛ C and T ω C : and T ɛ C T ω C = A sin 2ɛ, (3.9) = B cos 2ɛ, (3.10) where A and B are constants. These simple forms mimic the typical behavior of the torques shown in the Pseudo-Gaspra example in Chapter 1 (Figure 1.9). Equations (3.7) and (3.8) then become: and dɛ dt dω dt = A sin 2ɛ ω (3.11) = B cos 2ɛ. (3.12) Combining equations (3.11) and (3.12) we then obtain: dω dɛ = Bω cos 2ɛ A sin 2ɛ = B ω cot 2ɛ (3.13) A

63 The integration of equation (3.12) gives us the rotation angular velocity as a function 63 of the obliquity: ω dω ω 0 ω = B ɛ cot 2ɛdɛ, A ɛ 0 (3.14) ln(ω) ln(ω 0 ) = B 2A [ln(sin(2ɛ)) ln(sin 2ɛ 0)], (3.15) ω = ω 0 ( sin 2ɛ sin 2ɛ 0 ) B 2A. (3.16) Substituting equation (3.16) into (3.11) we have: dɛ dt = A sin 2ɛ ω 0 ( ) B sin 2ɛ0 2A, (3.17) sin 2ɛ which can be solved analytically for particular values of B to obtain the obliquity as 2A function of time. The case B = 0 corresponds to a zero spin torque (Equation (3.10)). Equation (3.12) 2A then gives ω = ω 0 = constant and equation (3.17) integrates to: ɛ = tan 1 ( e 2At ω 0 tan ɛ 0 ). (3.18) The object evolves at constant spin rate toward an asypmtotic obliquity of 90 in the limit t. where For the case B = 1 we obtain: 2A ω = ω 0 sin 2ɛ sin 2ɛ 0, (3.19) ɛ = ɛ 0 + A sin 2ɛ 0 ω 0 t. (3.20) The object will spin up until reaching an obliquity of 45, then spin down to zero at an obliquity of 90 after a time: ω 0 t = (90 ɛ 0 ). (3.21) A sin 2ɛ 0

64 64 For the case B = 2 we obtain: 2A ( ) 2 sin 2ɛ ω = ω 0, (3.22) sin 2ɛ 0 where ( ɛ = cos 1 cos 2ɛ 0 2A ) sin2 2ɛ 0 t. (3.23) ω 0 The object will spin up until reaching an obliquity of 45, to then spin down to zero at an obliquity of 90 after a time: ω 0 t = (cos 2ɛ 0 cos 90 ). (3.24) 2A sin 2 2ɛ 0 These simple analytic solutions help us test our numerical integration of the spin state evolution. Figure 3.8 shows the numerical solutions obtained from equations (3.11) and (3.12) and the analytic solutions obtained using equations (3.22) and (3.23) choosing a set of the coefficients A and B that correspond to the case of B = 2. Initial values of the 2A obliquity and the spin are also assumed. The numerical solutions are shown as black solid lines. The analytic solution for the obliquity and the spin are shown as red and green dashed lines, respectively. Notice the typical behavior of a rigid-body evolving through the YORP cycle. Note also that, the numerical and the analytic solutions are in excellent agreement. For other cases of B, equation (3.17) cannot be solved explicitly to obtain the 2A obliquity as a function of time. For example, in the cases of B = 3 and 4, we obtain: 2A and respectively. 4ɛ sin 4ɛ = 8A sin3 2ɛ 0 ω 0 t + 4ɛ 0 sin 4ɛ 0, (3.25) cos 6ɛ 9 cos 2ɛ = 24A sin4 2ɛ 0 ω 0 t + cos 6ɛ 0 9 cos 2ɛ 0, (3.26)

65 65 Figure 3.8: The numerical and analytical solutions obtain given an initial value of the obliquity and the spin and given a set of coefficients A and B that satisfy B = 2. The 2A numerical solutions are shown as black solid lines and analytical solutions are shown as colored dashed lines (Obliquity = Red, Spin=Green). Figure 3.9 shows the numerical solutions obtained from equations (3.11) and (3.12) and the analytic and numerical solutions for the non-explicit solution of the obliquity in equation (3.25) obtained for the case of B = 3. The spin and obliquity evolutions are 2A shown as a dashed and solid line, respectively, on left side of the plot. The right plot shows the numerically and the analytic non-explicit solution of the obliquity in equation (3.25) shown as a solid black line and as a red dashed line, respectively. Note the excellent agreement between the numerical and the analytic non-explicit solutions as well.

66 66 Figure 3.9: The left plot shows the numerical solutions obtained for the obliquity and the spin as a function of time (Obliquity = Solid, Spin=Dashed) given a set of coefficients A and B that satisfy B = 3 and given an initial obliquity and spin. The right plot shows the 2A numerical and the analytical solutions for the non-explicit solution of the obliquity as given in equation equation (3.25) (Numerical solution = Black solid line, Analytical solution= Red dashed line) Simulations Changes in the centrifugal force experienced by a sphere or agglomerations of spheres, could result in a movement of material. The timescales at which minor changes in the shape occur are much shorter than the timescales required for an object to evolve through the YORP cycle. Minor changes would occur on timescales of hours to days while the YORP timescales take thousands to millions of years. Running a continuous pkdgrav simulation for 10 9 dynamical times is not feasible. But we can exploit the timescale difference by recognizing that the episodes of dynamical motion and settling,

67 67 when the object reconfigures, will be short compared with the YORP timescale; therefore these episodes take place effectively at constant angular momentum. This fact allows us to adopt a two-step process in which we track the spin state evolution at constant shape until the spin changes enough that it may cause a dynamical event, and then check whether this is true by running the aggregate dynamically for a few rotations. If a dynamical event (mass motion) has occurred, we recompute the torques for the new shape and resume the spin state integration. This back-and-forth algorithm is the key to making these simulations possible. In order to perform these simulations, the author has developed a script using python software, which makes it possible for the codes TACO, pkdgrav, and the three new routines explained in the previous sections to interact. Figures 3.10 and 3.11 shows a flowchart of the whole exact procedure in the script. The python script merges the routines of TACO that compute YORP torques (Blue) with those of pkdgrav that simulate the dynamical evolution of aggregates (Orange), and the three new routines to evolve the spin and obliquity with time (Yellow), to transform the object to the original orientation (Purple) and to tile the object (Green). We start by dynamically evolving the object for several rotations at the initial spin rate using pkdgrav. Then we obtain the tiling and YORP torques for the original object and evolve the obliquity and spin in time until the spin rate changes by 0.5% or reaches an extremum. The object is run dynamically with pkdgrav at the new spin rate for several rotations and if there is a movement of spheres we let it evolve for several more rotations until there are no more movements. If at any movement of spheres, more than half of the spheres moved, we take that as an indication that our increment is spin rate may have been too large. In that case we go back and increment the spin by 1 5 of the previous increment instead. Once the object has settled down after a movement, we transform it to the orientation of the original object and obtain a new tiling and YORP torques. We follow

68 Figure 3.10: Python script: TACO (Blue), pkdgrav (Orange), Spin State Evolution (Yellow), Transformation (Purple) and Tiling (Green). 68

69 Figure 3.11: Continuation of the Python script: TACO (Blue), pkdgrav (Orange), Spin State Evolution (Yellow), Transformation (Purple) and Tiling (Green) 69

70 70 the evolution of aggregate objects computing the sequence of spin states and YORP torques through which they evolve as the changing spin alters their shape, which subsequently changes the YORP torques. 3.3 Aggregate Objects We use the end state objects from Tanga et al. [2009], who created ellipsoidal aggregates with arbitrary values of the semi-axis ratios, normalized spin and angular momentum and let them evolve dynamically allowing reshaping until they reached a stable configuration. The objects have a natural disordered packing, are composed of 1000 spheres and have a sphere density of 2.96 gcm 3. The bulk density and the diameter of the objects are in the range of 1.5 to 1.7 gcm 3 and 1.3 to 2.6 km, respectively. Table 3.1 shows the initial parameters for the sample of 28 aggregate objects that are used for our simulations. The set of 28 objects was chosen from an initial collection of 144 objects [Tanga et al., 2009]. We first required that the spin axis of the object must be aligned with the z-axis, as the spin rate of the aggregate is incremented along the z-axis using pkdgrav. This requirement discarded 41 objects from the initial sample set. For the remaining 103 objects, we obtained their tilings and computed the respective YORP torques using TACO. Assuming a rigid body evolution, a total of 9 objects were selected based on the spin state at the end of the YORP cycle, which we refer to as the rigid-body end state. We selected the only 9 cases from the sample of 103 objects where the object would spin up at all obliquities, having a rigid-body end state that would spin up in an infinite time while reaching an asymptotic obliquity. We apply this selection based on our interest in understanding how agregates would evolve under the preferred method for the formation of binary asteroids, where the spin up due to the YORP effect is assumed to spin the objects past the point where mass can be shed from the object.

71 71 Table 3.1: Aggregate Objects. Simulation Semi-axis ratio Semi-major axis a Bulk Density Period Initial Obliquity b/a c/a (km) (gcm 3 ) (hours) (degrees) The remaining 19 objects from the sample of 28 where chosen by placing a grid on the semi-axis ratio space and chosing randomly an object in each region of the grid in order to keep a representative distribution. A total of 28 simulations was chosen based on timeline constraints. The number could be increased to improve our statistical results by selecting more objects from the sample or by running the simulations with different initial obliquity values that would result in a different end state. Figure 3.12 shows the initial distribution of the 28 objects in the semi-axis ratio space. The initial distribution in the

72 72 Figure 3.12: The semi-axis ratio distribution of the initial shapes. The horizontal axis is the semi-axis ratio b and the vertical axis is the semi-axis ratio c. The dashed line indicates a a b = c. semi-axis ratio space is representative of the distribution of known asteroid shapes approximated by triaxial ellipsoids [Tanga et al., 2009, Kryszczyńska et al., 2007]. In our simulations, we first run the objects for several rotations at their initial spin. Some objects have movement of spheres at their initial spin rates. This does not necessarily imply that the given object is not in a stable configuration but rather that it needs to be dynamically evolved for a greater number of rotations in order to completely settle down. We then consider the rigid body evolution of the settled object. Table 3.2 (page 80) shows the rigid-body end states obtained for the 28 objects, given an initial obliquity. The initial obliquities were chosen to require the object to evolve through a wide range of obliquities and spin rates based on the expected rigid-body evolution

73 73 determined by the initial torques. Table 3.2 indicates the spin rate and the obliquity at the end of the YORP cycle, along with the time required to complete the cycle. As a result of some of them not being settled down initially, only 5 cases have an actual rigid-body end state that would spin up in an infinite time. 3.4 Choice of Simulation Parameters We use the hard-sphere discrete element method in pkdgrav and set the normal and tangential coefficients of restitution to 0.2 and 0.5 for all simulations. A normal coefficient of restitution equal to one/zero defines a completely inelastic/elastic collision. A tangential coefficient of restitution equal to one means that there is no surface friction between the spheres. The obliquity and spin evolutions are obtained using a fourth-order Runge Kutta integrator with a 10 3 year step size. The resolution used to obtain the tilings is n=10, which uses 784 tiles. We choose a rather low resolution with the intent of reducing computational time since there is no ideal resolution that minimizes the sensitivity of the YORP torques. A sensitivity assesment of the results that will include using higher resolutions will be conducted in the future but is out of the scope of the work presented here. The initial obliquities were chosen to maximize the range of obliquities through which the rigid object would evolve, normally at values of 5 for objects whose rigid-body evolution evolves to an asymptotic value of 90, and at 85 for objects whose rigid-body evolution evolves to an asymptotic value of 0. Ideally, for future simulations a random distribution of initial obliquities would be the method of choice in order to avoid any biases.

74 Results and Discussions The continuous changes in the shape of an aggregate cause a different evolution of the YORP torques and therefore aggregates do not evolve through the YORP cycle as a rigid body would. Figure 3.13 shows the tiling obtained for the original aggregate object in simulation 6 as an example. The respective spin and obliquity torques are shown in Figure Given initial values of the spin and obliquity, we can evolve them with time. At an initial obliquity of 5, the obliquity torque is positive. Therefore if the object were a rigid body, it would increase its obliquity with time until it reached the asymptotic value of 90. At an initial obliquity of 5, the spin torque is also positive. Therefore if the object were a rigid body, it would spin up in time until it reached the obliquity value of 55 at which point it would begin to spin down to zero. It is believed that the object will end up tumbling as it begins to rotate too slowly, only to restablish its principal axis of rotation at a later time. Figure 3.15 shows the respective spin and obliquity evolutions assuming a rigid-body evolution. The spin rate and obliquity at the end of the YORP cycle, which we refer to as the end state, is then an asymptotic value of 90 and spinning down to zero. The end states for the rigid bodies on all simulations are characterized by reaching an asymptotic value of obliquity and either spinning down in a finite time or spinning up in an infinite time. However, as we dynamically evolve the aggregate object using pkdgrav, the change in spin due to the YORP effect alters its shape, which subsequently changes the YORP torques resulting in a different evolution. Figure 3.16 shows the shape of the object at a time of T=3.248 Myrs. At this time the changes in the spin have made the object evolve through 50 different shapes with 50 different respective torque curves. Figure 3.17 shows the spin and obliquity torques through which the object has evolved in Myrs while the spin and obliquity evolutions are shown in Figure Each different color

75 75 Figure 3.13: A triangular tiling representation of the original aggregate object in simulation 6 (Time= 0.0 Myrs). The spin axis is along the line of sight. corresponds to a different shape. We continue to evolve dynamically the aggregate object using pkdgrav while computing the sequence of YORP torques to obtain the final spin and obliquity evolutions.

76 76 Figure 3.14: The obliquity (left) and the spin (right) torques obtained for the object in simulation 6 (Figure 3.13). Figure 3.15: The obliquity (left) and the spin (right) evolutions obtained if the object in simulation 6 (Figure 3.13) was a rigid-body with the spin torques shown in Figure 3.14.

77 Figure 3.16: A triangular tiling representation of the evolved aggregate object in simulation 6 at a time of T=3.248 Myrs. The spin axis is along the line of sight.. The shape continuously changes and the current shape of the object at T=3.248 Myrs is noticeably rounder than the original object shown in Figure

78 Figure 3.17: The obliquity (left) and the spin (right) torques trough which the original object in Figure 3.13 has evolved in Myrs. The horizontal axis is the obliquity in degrees and the vertical axis shows the respective torques over the moment of inertia C in units of radians per second square 78

79 79 Figure 3.18: The obliquity (left) and the spin (right) evolution obtained if the object in Figure 3.13 was a rigid-body is shown in white while the actual evolution of the aggregate object is shown in a color sequence. Each different color represents a new shape with a corresponding torque (see Figure 3.17) End States Table 3.2 indicates the end states of the spin and obliquity and the time that it takes to complete a YORP cycle, assuming the objects evolve as rigid bodies, along with the actual end states of the aggregates obtained from the simulations. We refer to an end state as spin up or spin down, if the object is increasing or decreasing its spin rate at the end of the YORP cycle. The time is indicated in units of the rigid body YORP cycle, if the rigid body spins down to zero in a finite time. Otherwise, a time in Myrs is given if the rigid body spins up in infinite time. Nearly all simulations were run until the spin and obliquity evolutions reached 15 Myrs, based on the fact that the typical dynamical lifetime of NEAs is around 10 Myrs. After reaching that time, the simulations were continued if the time evolved was not greater than the time required to complete the YORP cycle. Some

80 80 simulations were run less than 15 Myrs, if the objects were spinning down to zero with slow rotation periods of over 20 hours. Table 3.2: Comparison of End States. Rigid-Body End State Aggregate End State Simulation Obliquity Spin YORP Cycle Obliquity Spin N YORP Cycles (deg) (Myrs) (deg) or Time 1 90 Down Wander Down Wander Down 1.64 Wander Down Down 6.78 Wander Wander Down Down Down Wander Down Wander Down Wander Down Down 9.30 Wander Down Down Wander Down 6.85 Wander Wander Down Down Down Wander Down Down 3.62 Wander Wander Down Down Up Infinite Wander Wander Myrs Down Wander Up Infinite Wander Wander Myrs Down Down Down Wander Up Infinite Wander Wander Myrs Down 2.86 Wander Wander Down Wander Down Wander Down 6.15 Wander Wander Up Infinite 90 Wander Myrs Up Infinite 90 Down Down Down 0.62

81 81 The end states for the aggregate objects are either reaching an asymptotic value or wandering in obliquity and either spinning down in a finite time or wandering up and down in spin. Out of the 28 simulations, given an initial obliquity, 23 objects would have spun down in a finite time as rigid-bodies while the other 5 would have spun up in an infinite time. Twenty objects would have evolved toward an asymptotic value of 90, while only one towards 0 and the other seven towards other different values. In the case of the aggregate objects, only 10 objects will spin down to zero in a finite time, assuming that there are no more changes in the shape which could reverse the YORP torques. For the remaining 18 objects, 10 are still wandering in spin but their obliquities have evolved to a near-constant value. In 12 simulations, the objects reach and asymptotic obliquity of 90 while other 4 reach and asymptotic obliquity of 0. The remaining 12 simulations continue to wander in obliquity. While the majority of aggregates wander in spin and/or in obliquity, some cases could be described as following a shape-modified YORP cycle instead of the classical YORP cycle Stochastic and Self-Limiting YORP Effect As shown in the previous section, aggregate objects under the influence of the YORP effect do not follow the classical YORP cycle. The continuous reconfiguration of an aggregate object leads to two different kinds of behavior of the YORP effect, which we call self-limiting and stochastic. Figure 3.19 shows an example of self-limiting evolution of the spin in simulation 22. In self-limiting YORP, consecutive reversals in the sign of the torques occur repeatedly, resulting, in this case, in a spin that remains close to a value of almost 5 cycles per day for over 10 Myrs. In some other cases, the continuous reconfiguration leads to a set of YORP torques that become negligibly small and nearly vanish such as the case of simulation 16 which is shown in Figure Generally in a self-limiting evolution, the spin of the object remains close to its initial value.

82 82 Figure 3.19: The obliquity (left) and spin (right) evolutions of the object in simulation 22. The expected evolution if it was a rigid-body is shown as a black line while the actual evolution of the aggregate object is shown in a color sequence. Each different color represents a new shape with a corresponding torque. The spin evolution is self-limiting. The constant reversals in the YORP torques do not allow the object to spin up or down continuosly. The other behavior of the YORP effect on aggregates is stochasticity. Figure 3.21 shows an example of stochastic evolution of the spin in simulation 25. Stochastic YORP evolution is characterized by unpredictably long segments, each similar to part of a classical YORP cycle, but never completing. An aggregate object can also exhibit both behaviors. Figure 3.22 shows simulation 21, an example of spin evolution that is sometimes stochastic and sometimes self-limiting. The evolution consists of unpredictably long segments similar to a classical YORP cycle evolution as well as consecutive rapid reversals of the YORP torques.

83 83 Figure 3.20: The obliquity (left) and spin (right) evolutions of the object in simulation 16. The expected evolution if it was a rigid-body is shown as a black line while the actual evolution of the aggregate object is shown in a color sequence. Each different color represents a new shape with a corresponding torque. The spin evolution is self-limiting. The continuous reconfiguration of the objects leads to a set of YORP torques that become negligibly small and nearly vanish. We have described the self-limiting and the stochastic behaviors in terms of the spin evolution. Notice however, that the obliquity evolution can also be described by these two behaviors, and it does not neccesarily exhibit the same behavior as the spin evolution. The obliquity shows a self-limiting behavior in simulation 16 (Figure 3.20), a stochastic behavior in simulations 22 and 25 (Figures 3.19 and 3.21), and is self-limiting and stochastic in simulation 21 (Figure 3.22).

84 Figure 3.21: The obliquity (left) and spin (right) evolutions of the object in simulation 25. The expected evolution if it was a rigid-body is shown as a black line while the actual evolution of the aggregate object is shown in a color sequence. Each different color represents a new shape with a corresponding torque. The spin evolution is stochastic. The object goes through a random walk in the spin torques resulting in unpredictably long segments similar to a classical YORP cycle evolution. 84

85 85 Figure 3.22: The obliquity (left) and spin (right) evolutions of the object in simulation 21. The expected evolution if it was a rigid-body is shown as a black line while the actual evolution of the aggregate object is shown in a color sequence. Each different color represents a new shape with a corresponding torque. The spin evolution is stochastic and self-limiting. The evolution consist of unpredictably long segments similar to a classical YORP cycle evolution and of consecutive rapid reversals of the YORP torques as well Mass Loss and Binary Formation From the set of 28 simulations, 9 experienced mass loss episodes. Table 3.3 shows the number of mass loss events, the total percent of mass loss from the initial mass and the average time between events for the 9 simulations. Figure 3.23 shows the distribution of events in a period and mass loss percentage diagram. The minimum percentage of the initial mass that an aggregate object loses in a single event is 0.1% ( this being 1 sphere) while the maximum is 2.9%. The mass loss episodes can occur as isolated events or as a

86 86 chain of events. The average time between events can be as short as 0.01 Myrs for consecutive events and more than 1 Myr for isolated events. Table 3.3: Mass Loss Episodes. Simulation Number of Total % of Average time # Events Mass Loss between events (Myrs) The average period at which a mass loss event occur is at 4.39 hours, while the minimum and maximum periods occur at 3.80 and 5.30 hours, respectively. The period at which the centrifugal force would overcome gravity, is inversely proportional to the square root of the density of the object, ρ 1 2 (Equation (1.6)). Therefore, the periods would be greater than 2.2 hours for those objects with densities lower than the average rock density of 2.5 gcm 3. Besides having lower densities, evolving to more extreme shapes can result in mass loss episodes at slower rotation rates. For the case of a prolate spheroid having a semi-major axis a and a semi-minor axis b, the period given in Equation (1.6) would increase by a factor of (a/b) 1 2 in comparison with a sphere with a radius of a [Harris, 1996]. Table 3.4 shows the mean density of the objects and the periods of the estimated spin limits for a sphere and for a prolate spheroid using the mean densities and also the mean semi-axis ratio of all the shapes before losing mass. The objects loose mass at longer periods than those obtained using the simple approximations of a prolate spheroid. Figures 3.24 and 3.25 show examples of the aggregate shapes in simulations 11

87 87 Figure 3.23: The distribution of all mass loss events. The horizontal axis is the period in hours and the vertical axis is the percent of the initial body mass that is lost. The minimum percentage of the initial mass that an aggregate object loses in a single event is 0.1% while the maximum is 2.9%. The minimum and maximum periods at which a mass loss episode occurs are at 3.80 and 5.30 hours, respectively. The average period from all events is 4.39 hours. and 27 before having a mass loss episode. The mass is commonly lost from one end of the objects as their shape becomes elongated. At this point it is still unclear why mass loss occurs at slower spin rates than predicted. We can speculate that it may be caused by the pointiness of the object and the inadequacy of the prolate spheroid approximation, or by the dynamical motion of the material close to the tip. The spheres are removed from the simulations after being shed from the main object. We do not track their evolutions since the aggregate shapes become very prolate and Scheeres [2007b] has shown that they would not reach a stable configuration to form a

88 88 Table 3.4: Mass Loss Spin Rates. Simulation Mean density Sphere-Spin limit a/b Prolate Spheroid-Spin Actual Spin # gcm 3 Period (hours) limit Period (hours) Period (hours) Figure 3.24: The shape of the aggregate object in simulation 11 before having its first mass loss episode, in which it looses 3 spheres. The z-axis is along the line of sight. Material is lost from the right side.

89 89 Figure 3.25: The shape of the aggregate object in simulation 27 before having its last mass loss episode. The z-axis is along the line of sight. The aggregate has shed 50 spheres. Material is lost from the bottom right side. binary asteroid. However, we do encounter a case of binary formation in one of the simulations. Figures 3.26 and 3.27 show the spin and obliquity evolution of the aggregate object in simulation 8. White squares indicate when a mass loss episode has occured. The objects splits into two close to a time of about 2.5 Myrs, at which point the simulations is stopped. At the moment of splitting into two, the object had already lost 6.1% of its initial mass, was increasing in angular momentum but decreasing in spin rate because of its evolution towards an elongated shape. Figure 3.28 shows the aggregate shape at the last point of contact before splitting into two. The primary body of the binary asteroid contains 52.7% of the initial mass while the secondary contains 41.2%. Simulation 17 shows a similar shape evolution reaching the final shape shown in Figure 3.29, however, the object remains stable and does not split. Figures 3.30 and 3.31 show the spin and

90 obliquity evolution of the aggregate object in simulation 17. The object remains stable at a constant shape for over 30 Myrs. 90 Figure 3.26: The spin evolution obtained for the object in simulation 8. The horizontal axis is the time in million years and the vertical axis shows the spin in cycles per day. The expected evolution if it was a rigid-body is shown in black while the actual evolution of the aggregate object is shown in a color sequence. Each different color represents a new shape with corresponding new torque curves. Black squares in the aggregate evolution indicate that a mass loss episode has occured. At the end of the simulation the object splits into a binary asteroid.

91 Figure 3.27: The obliquity evolution obtained for the object in simulation 8. The horizontal axis is the time in million years and the vertical axis shows the obliquity in degrees. The expected evolution if it was a rigid-body is shown in black while the actual evolution of the aggregate object is shown in a color sequence. Each different color represents a new shape with corresponding new torque curves. Black squares in the aggregate evolution indicate that a mass loss episode has occured. At the end of the simulation the object splits into a binary asteroid. 91

92 92 Figure 3.28: The shape of the aggregate in simulation 8 at the last point of contact before splitting and forming a binary asteroid. The z-axis is along the line of sight. Figure 3.29: The shape of the aggregate object at the end of simulation 17. The z-axis is along the line of sight.

93 Figure 3.30: The spin evolution obtained for the object in simulation 17. The horizontal axis is the time in million years and the vertical axis shows the spin in cycles per day. The expected evolution if it was a rigid-body is shown in black while the actual evolution of the aggregate object is shown in a color sequence. Each different color represents a new shape with corresponding new torque curves. Black squares in the aggregate evolution indicate that a mass loss episode has occured. 93

94 94 Figure 3.31: The obliquity evolution obtained for the object in simulation 17. The horizontal axis is the time in million years and the vertical axis shows the obliquity in degrees. The expected evolution if it was a rigid-body is shown in black while the actual evolution of the aggregate object is shown in a color sequence. Each different color represents a new shape with corresponding new torque curves. Black squares in the aggregate evolution indicate that a mass loss episode has occured Axis Ratio Evolution Figure 3.32 shows the evolution of the 28 objects in the triaxiality and semi-axis ratio c space, where the triaxiality parameter is given by: a T = b2 c 2 a 2 c 2. (3.27) Values for the initial objects are shown as black diamonds while the values for the objects at the end of the simulations are shown in red. Lines connect the initial values with the respective final values. Objects that exhibit mass loss episodes are indicated with blue

95 95 Figure 3.32: The evolution of the 28 objects in the triaxiality and semi-axis ratio c a space. The horizontal axis is the triaxiality parameter and the vertical axis is the semi-axis ratio c a. Initial and final values are shown as black and red diamonds, respectively. Lines connect the initial values with the respective final values. Blue lines indicate the objects with mass loss episodes. lines. There are objects that are still evolving in spin state at the end of the simulations and therefore may not have reached their final shape yet. There is not an apparent correlation between the initial and the final shapes. Similar shapes evolve in the triaxiality and semi-axis ratio c a space towards different directions. Objects with mass loss episodes, experience in most cases, the more extreme shape evolutions The Statistical Spin and Obliquity Evolutions The constant changes in the shape of an aggregate results in a self-limiting and stochastic spin state evolution due to the YORP effect. Simulations that model very important aspects of asteroid dynamics, such as the delivery of NEAs to their current

96 96 orbits and the orbital distribution of asteroid families, include an estimation of the spin state evolution due to YORP. However, the estimations have been based on the only previously known behavior of YORP, the classical YORP cycle. Since the majority of asteroids are most probably aggregates, it is important then to be able to describe the self-limiting and stochastic spin state evolution statistically. We break the evolution into discrete intervals bounded by material movements (shape changes). The shape is constant (except for small bouncing of the spheres) during each interval. Consider one such interval of duration δt, over which the change in spin rate is δω. We define the statistical spin evolution, α s, by α s = ± δω δt, (3.28) where the sign is determined by whether the sign of δω is the same as (+) or opposite to ( ) that in the previous interval. With this definition, rapidly alternating spin-up and spin-down behavior (self-limiting YORP) would be characterized by predominantly negative values of α s, while stochastic YORP would be characterized by predominantly positive values. Similarly, we define the statistical obliquity evolution, ζ s, in terms of the change of obliquity during the interval according to ζ s = ± δɛ δt, (3.29) with the sign determined by comparison with the previous interval, as above. Figure 3.33 shows the joint distribution in (α s,δt) for all intervals in all simulations; this is as close as we can come to a statistical description of the spin evolution of a population of aggregates, or of a probabilistic description of the evolution of one aggregate with unknown properties. A by-hand optimization of a reasonable fitting function gives us the following analytic approximation:

97 97 Figure 3.33: The obtained distribution of α s. Time intervals δt has units of Myrs and are plotted in a logarithm base 10. where δt is in log scale, A is a normalization factor given by: f (α, δt) = A σ e α σ , (3.30) and σ is given by: δt e A = 2e 2, (3.31) σ = e δt 40. (3.32) Figure 3.34 shows the contours of f (α s,δt) along with the obtained distribution of α s as black diamonds. A simulated distribution of α s with respective time intervals δt that satisfy f (α s,δt) is overplotted as red diamonds.

98 98 Figure 3.34: The contours of the estimated density function f (α s,δt) along with the obtained distribution of α s as black diamonds. A simulated distribution of α s with respective time intervals δt that satisfy f (α, δt) is overplotted as red diamonds. We can use the fit to reconstruct simulated spin evolutions interval by interval by randomly choosing sets of (α s,δt) values and laying the segments end to end. Figure 3.35 shows the obtained simulated evolution using all possible values of α s, which allows for both stochastic and self-limiting behavior. Consecutive reversals in the spin, characteristic of self-limiting behavior, can be observed along with unpredictably long segments similar to a classical YORP cycle that never completes, characteristic of stochastic behavior. Using predominantly negative or positive values of α s results in a self-limiting or stochastic spin evolution as shown in Figures 3.36 and 3.37, respectively.

99 99 Figure 3.35: A simulated stochastic and self-limiting spin evolution obtained using a set of positive and negative values from the simulated distribution of α s with respective time intervals δt that satisfy f (α s,δt). Figure 3.36: A simulated self-limiting spin evolution obtained using a set of predominantly negative values from the simulated distribution of α s with respective time intervals δt that satisfy f (α s,δt).

100 100 Figure 3.37: A simulated stochastic spin evolution obtained using a set of predominatly positive values from the simulated distribution of α s with respective time intervals δt that satisfy f (α s,δt). The statistical spin (α s ) and obliquity (ζ s ) distributions provide a description of the spin state evolution of aggregates, one that is self-limiting and stochastic, and not deterministic as previously assumed. The density functions can be incorporated into models of the orbital evolution of asteroids that include the Yarkovsky effect, to take into account the qualitatively different spin state evolutions of aggregates and rigid bodies.

101 Figure 3.38: The obtained distribution of ζ s. Time intervals δt has units of Myrs and are plotted in a logarithm base

102 4 CONCLUSIONS AND FUTURE WORK 102 Previous studies about the delivery of NEAs from the Main Belt conclude that 37% of the NEAs will come through the secular resonance ν 6 at the inner edge of the Main Belt, generally only by retrograde rotation, while the rest will come through other resonances with an equal fraction of retrograde and prograde rotators [Bottke et al., 2002, La Spina et al., 2004]. Therefore, 68.5% of NEAs are expected to be delivered to their current orbits via retrograde rotation. Indeed, the obtained fraction of retrograde rotators by Farnocchia et al. [2013] and the most probable value of the fraction of retrograde rotators estimated in this work are 69.1% and 70.0%, respectively. This suggests, that the whole population of NEAs remember their initial spin state. However, the YORP effect is expected to be a dominant mechanism driving the spin state evolution. It is expected that, once the asteroids spin down to zero, they forget their initial spin state. Therefore, if the timescales required to complete a YORP cycle are much shorter than the typical dynamical lifetimes of NEAs, we should expect a 50% retrograde fraction if deterministic YORP cycle evolution dominates. But by self-consistently modeling the dynamical evolution of aggregates, we find that about two thirds of the population is prevented from spinning down to zero. Both self-limiting YORP and to a lesser degree a stochastic YORP provide a viable means to explain why the NEA population seems to remember their initial spin states at the time of delivery from the Main Belt. Assuming then that the asteroids are aggregates, about 25% out of the 37% of NEAs that come through the secular resonance ν 6 will be prevented from forgetting their initial spin states. Adding half of the 12% of the ν 6 objects that do forget plus half of the 63% that come through other resonances we expect a 62.5% retrograde rotator factor. This is not inconsistent with the data but current constraints on the obliquity distribution do not allow us to determine what fraction of the NEAs are aggregate asteroids. Dynamical

103 103 simulations that include the spin state evolution of both monolithic and aggregate asteroids are required in order to properly understand the evolution of the population as a whole. The expected end state obliquity distribution for the YORP cycle obtained by Vokrouhlický and Čapek [2002] (Figure 1.11) peaks at values of 0, 90 and 180. This predicted distribution does not agree with the obliquity distributions obtained from the Yarkovsky effect (Figures 2.1 and 2.6) which lack a concentration of objects at ɛ = 90. Vokrouhlický and Čapek [2002] predict that about 40% of the population should evolve toward that value. However, the predicted end state obliquities of the aggregate objects used here, treated as rigid bodies, result in a different distribution dominated by obliquities of 90 (Table 3.2). A likely reason for the disagreement is the difference in the simulated asteroids used. Vokrouhlický and Čapek [2002] uses gaussian random spheres while in this work we use equilibrium configurations of spherical particles [Tanga et al., 2009], which are less asymmetric. This implies that a correct prediction of the obliquity distribution cannot even be made for rigid bodies without taking into account all possible shapes and asymmetries. In the dynamical evolution of the aggregate objects, we observe not only mass movements at the surface but movements throughout the interiors of the objects as well. When an object reconfigures to a new shape, it is the entire object that reconfigures. This kind of behavior is difficult to quantify or to describe statistically but is important. In particular two aspects deserve attention. A particular type of self-limiting we observe is repeated switching between two nearly identical configurations, one having a positive spin torque and the other a negative spin torque. The object in simulation 17 exhibits this particular behavior. The object reaches the shape shown in Figure 3.29 at about a time of 18 Myrs. After that, the object repeatedly toggles between the two configurations for a period of over 25 Myrs, which results in the consecutive reversals of the torque curves and therefore also of the rotational

104 104 acceleration as it is shown in Figure Similar switching behavior is also seen in simulation 22 (Figure 3.26). The second type of behavior that we observe is partial crystal packing of the objects as the identical spheres fall into close-pack configuration. The rigidity produced by crystallization can be regarded as a partially realistic representation of aggregates, as the crystallized fragments in the aggregates can range in size. Having a part of the object that becomes rigid would represent a bigger fragment in comparison to those represented by the spheres. However, we believe that how our objects evolve toward rigidity is not realistic. In order to prevent crystallization, the simulations could be done using aggregates composed of spheres that have small variations in size, instead of the identical spheres used here. The use of aggregates with some spheres that are substantially larger than the rest could result in an aggregate spin state evolution that could follow more closely a YORP cycle evolution as in the case of monolithic asteroids. Sánchez and Scheeres [2013] have shown that the smaller spheres tend to evolve toward the void spaces between the larger fragments, providing some cohesion between the larger fragments and resulting in a more rigid object. A natural next step is to verify the robustness of the results to the dynamic behavior of the spheres. While we have applied here the pkdgrav hard sphere method, a soft sphere discrete element method has been recently developed [Schwartz et al., 2011]. While the hard sphere method treats collisions as instantaneous events between rigid spheres, the soft sphere method allows the spheres to penetrate each other. The hard sphere method has been criticized for possibly exhibiting excessively fluid like behavior. The change in dynamics in the soft sphere method could perhaps produce a different outcome in some aspects of the results. While we expect YORP would still be found to be self-limiting and stochastic, mass loss and shape change results could differ from those obtained using the

105 105 hard sphere method. Therefore, we plan to conduct simulations using the soft sphere method for comparison. The influence of the YORP effect on aggregate asteroids results in a self-limiting and stochastic spin state evolution. The statistical description of the spin (α s ) and obliquity (ζ s ) evolutions lay out the foundation for new simulations of a coupled Yarkovsky/YORP evolution. These simulations will impact other aspects of asteroid dynamics, such as the spreading of collisional families. The results obtained here help to explain how the YORP effect acts among the NEAs and therefore contribute to the understanding of the dynamical evolution of the NEA population.

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