The Rotation Rate Distribution of Small Near-Earth

Home , 2008 HJ, 4769 Castalia, Opposition surge, Rubble pile

THE ROTATION RATE DISTRIBUTION OF SMALL NEAR-EARTH

ASTEROIDS

A thesis presented to

the faculty of

the College of Arts and Sciences of Ohio University

In partial fulﬁllment

of the requirements for the degree

Master of Science

Desire´e Cotto-Figueroa

November 2008

THE ROTATION RATE DISTRIBUTION OF SMALL NEAR-EARTH

ASTEROIDS

DESIREE´ 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

Benjamin M. Ogles

Dean, College of Arts and Sciences Abstract 3

COTTO-FIGUEROA, DESIREE,´ M.S., November 2008, Physics and Astronomy

The Rotation Rate Distribution of Small Near-Earth Asteroids (101 pp.)

Director of Thesis: Thomas S. Statler

Rotation periods or lower limits for 34 Near-Earth Asteroids (NEAs) were obtained through optical light curves. Two codes were developed in order to obtain the true fraction of Fast-Rotating Asteroids (FRAs), F, using Fortran 95 and IDL. The

first code models the shape of an asteroid and simulates its light curve. The second code, uses the results obtained from the observational program and the simulated light curves to obtain the probability density of F, P(F ). The observational and statistical analysis indicates that the population of asteroids with D<150m is almost equally divided between fast and slow rotators, and that the majority of the population of asteroids with D>150m consists of slow-rotators. These results also indicate that selection effects have significantly influenced the currently known distribution of rotation periods of NEAs and therefore that it is not representative of the real population of NEAs.

Approved:

Thomas S. Statler

Professor of Physics and Astronomy 4

To my husband Jos´e and our families. Acknowledgments

First, I would like to acknowledge and thank my advisor, Thomas S. Statler, for all of his support and advice during the completion of this thesis. I would also like to thank the other members of my thesis defense committee, Alexander Nieman and

Joseph C. Shields, for their helpful suggestions. I would also like to thank David

Riethmiller for all of his hard work and contributions to this research. Furthermore I would like to thank Jess Wilhelm, Tomomi Watanabe, and Cristopher Haas for their help with the observations. A special thanks to Mangala Sharma, Kellen Murphy,

Brett Ragozzine, Kyle Uckert, Sajida Khan, all of my friends and everyone else who supported or helped me in any way. Finally, I would like to thank my husband and our families for their endless love, support, and encouragement. 6 Table of Contents Page

Abstract...... 3

Dedication...... 4

Acknowledgments...... 5

ListofFigures...... 8

ListofTables...... 10

1 Introduction...... 11 1.1Near-EarthAsteroids...... 11 1.2 The Rotation Rate Distribution of NEAs ...... 17 1.3TheYORPeﬀect...... 23

2 ObservationalProgram...... 28 2.1Data...... 28 2.2Reductions...... 34 2.3LightCurves...... 36

3 ModelingLightCurves...... 39 3.1Introduction...... 39 3.2TACO...... 40 3.2.1 CodeDescription...... 40 3.2.2 Gaussian Random Spheres ...... 41 3.2.3 BidirectionalReﬂectance...... 43 3.2.4 HapkeParameters...... 48 3.2.5 LibraryofSimulatedLightCurves...... 50 3.3SALSA...... 56

4 ResultsandDiscussions...... 59 4.1LightCurves...... 59 4.2 Rotation Rate Distribution of Small NEAs ...... 63 4.3TruefractionofFRAs...... 69 4.4Discussion...... 72

5 ConclusionsandFutureWork...... 75

Bibliography...... 78 7

A Light Curves of NEAs ...... 84 8 List of Figures 1.1 The orbits of the three classes of NEAs: Amors, Apollos and Atens. . 12 1.2 The orbits of the major planets and the location of asteroids...... 16 1.3 The rubble pile asteroid 25143 Itokawa ...... 17 1.4 The Rotation Rate Distribution for Solar System Bodies...... 18 1.5Thespinlimits...... 20 1.6TheYarvkoskyeﬀect...... 24 1.7TheYORPeﬀect...... 27 1.8Radiationforces...... 27

2.1Anexampleofareduceddataimage...... 35

3.1Gaussianrandomspheres...... 42 3.2Bidirectionalreflectance...... 44 3.3Correctionformacroscopicroughness...... 47 3.4Uniformdistributionofpointsonasphere...... 51 3.5Object1...... 52 3.6LightcurveforObject1inFigure3.5...... 52 3.7Object2...... 53 3.8LightcurveforObject2inFigure3.7...... 53 3.9Object1atadifferentorientation...... 54 3.10LightcurveforObject1inFigure3.9 ...... 54 3.11Object2atadifferentorientation...... 55 3.12LightcurveforObject2inFigure3.11...... 55 3.13 Light curve for one single rotation of a simulated Object ...... 58 3.14 Light curve in Figure 3.13 after been processed with SALSA...... 58

4.1 Light curve for 2008 CP ...... 60 4.2 Light curve for 2006 CL9...... 61 4.3 Light curve for 2006 CL9 folded to the derived period...... 61 4.4 Light curve for 2007 CQ5...... 62 4.5 Distribution of NEAs with our results ...... 65 4.6 Probability density for the true fraction of FRAs ...... 71

A.1 Light curve for 2006 CL9...... 84 A.2 Light curve for 2006 CN10...... 85 A.3 Light curve for 2006 CY10...... 85 A.4 Light curve for 2005 YT55...... 86 A.5 Light curve for 2006 CL10...... 86 A.6 Light curve for 2006 BN55...... 87 A.7 Light curve for 2006 UQ17...... 87 A.8 Light curve for 2006 PA1...... 88 9

A.9 Light curve for 2006 WU29...... 88 A.10 Light curve for 2006 UN216...... 89 A.11 Light curve for 2006 YU1...... 89 A.12 Light curve for 2007 DB61...... 90 A.13 Light curve for 2007 CQ5...... 90 A.14 Light curve for 2007 DD...... 91 A.15 Light curve for 2007 DW...... 91 A.16 Light curve for 2006 VC...... 92 A.17 Light curve for 2006 VD13...... 92 A.18 Light curve for 2007 CO26...... 93 A.19 Light curve for 2007 DF8...... 93 A.20 Light curve for 2007 AH12...... 94 A.21 Light curve for 2007 DD49...... 94 A.22 Light curve for 2007 DY40...... 95 A.23 Light curve for 2007 DS84...... 95 A.24 Light curve for 2007 EV...... 96 A.25 Light curve for 2007 EF...... 96 A.26 Light curve for 2007 EQ...... 97 A.27 Light curve for 2007 WF55...... 97 A.28 Light curve for 2008 AF4...... 98 A.29 Light curve for 2007 PS9...... 98 A.30 Light curve for 2007 TU24...... 99 A.31 Light curve for 2006 JY25...... 99 A.32 Light curve for 2008 CC71...... 100 A.33 Light curve for 2008 CP...... 100 A.34 Light curve for 2008 CR116...... 101 10 List of Tables 1.1 Classes of NEAs ...... 12

2.1 Orbital parameters of NEAs and Observational circumstances. . . . . 30 2.1 Orbital parameters of NEAs and Observational circumstances. . . . . 31 2.1 Orbital parameters of NEAs and Observational circumstances. . . . . 32 2.1 Orbital parameters of NEAs and Observational circumstances. . . . . 33

4.1 Rotation periods or lower limits for each observed NEA ...... 66 4.1 Rotation periods or lower limits for each observed NEA ...... 67 4.1 Rotation periods or lower limits for each observed NEA ...... 68 11 Chapter 1

Introduction

1.1 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. Almost two hundred thousand 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 perihelion1 distance less than 1.3 astronomical units (AU )2.There are three classes of NEAs: the Amors, the Apollos and the Atens (see Figure 1.1).

NEAs are categorized into these classes according to their orbital semi-major axis and their perihelion and aphelion3 distances (see Table 1.1).

The typical lifetime of a NEA is about ten million years (Gladman et al. 2000).

NEAs eventually get destroyed by a collision with an inner planet, are ejected from our Solar System or end up in a Sun-grazing state. As their typical 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

1The perihelion is the point in the orbit where the object is closest to the Sun. 2The mean distance from the Earth to the Sun is one astronomical unit. 1AU =1.5x1011m 3The aphelion is the point in the orbit where the object is furthest from the Sun. 12

Table 1.1. Classes of NEAs

Class Deﬁnition

Atens aphelion distance <1.0 AU, semi-major axis >0.983 AU

Apollos aphelion distance >1.0 AU, perihelion distance <1.017 AU

Amors aphelion distance >1.0 AU, 1.017 AU

Figure 1.1: The orbits of the three classes of NEAs: Amors, Apollos and Atens. The main belt and the orbits of the Earth and Mars are also shown. The orbits are shown for objects with an inclination with respect to the plane of the orbit of the Earth of zero degrees. Figure reproduced from NASA’s Cosmos (http://ase.tufts.edu/cosmos/index.asp) 13 the population of objects with a diameter roughly larger than 1 km in steady state, is 790 ± 200 objects per million year.

Gravitational forces and collisions are considered the primary mechanisms for asteroid delivery (Bottke et al. 2006). After a collision, an object or its fragments will undergo orbital velocity changes that could inject 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 mean motion resonance 3:1 with Jupiter near 2.5 AU (Wisdom 1983; Yoshikawa

1989; Farinella et al. 1993) and the secular resonance ν6 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.

Today, it has been recognized that there are non-gravitational forces that play an important role in the evolution of NEAs. They are even considered a 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 section 1.3, 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. 14

It is widely accepted that NEAs could represent a hazard of global catastrophe for human civilization. Asteroids have impacted the terrestrial planets and the small bodies in the inner solar system and the best evidence of these impacts is the lunar surface, which is covered by craters. 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 Program. As of August 11 of 2008, around 5570 Near-

Earth Objects (NEOs)4 have been discovered and around 960 have been classiﬁed as Potentially Hazardous Asteroids (PHAs)5. Figure 1.2 shows a view of the inner solar system. Around 750 of the NEOs that have been discovered are asteroids with a diameter of approximately 1 km or larger. While asteroids tens of meters in diameter can cause severe local damage, an asteroid larger than 1 km will cause a global disaster. One of the many goals of NASA is to locate at least 90 percent of the NEOs that are larger than 1 km by the end of 2008 5.

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. It is important to characterize them 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. It would be essential to know their size, their material strength and whether they are monolithic or loose aggregates of chunks held together by self-gravity. To study the properties

4NEOs refers to Near-Earth Asteroids and comets with a perihelion distance <1.3 AU. 5Near Earth Object Program (NEOP) (http://neo.jpl.nasa.gov) 15 of these objects over a range of sizes will give us many insights about their material properties and will also put constraints on the evolution of these objects since mechanisms like the Yarkovsky and the YORP eﬀects are strongly size dependent. 16

Figure 1.2: The orbits of the major planets are shown in light blue: the current location of the major planets is indicated by large colored dots. The locations of the minor planets, including numbered and multiple-apparition/long-arc unnumbered objects, are indicated by green circles. Objects with perihelia within 1.3 AU are shown by red circles. 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 from the Minor Planet Center (http://cfa-www.harvard.edu/iau/mpc.html). 17 1.2 The Rotation Rate Distribution of NEAs

The rotation rate distribution of asteroids can give us important information about their material strength and composition. The rotation rates of asteroids obtained from optical light curves has given strong evidence to support the idea that large asteroids are rubble piles (Davis et al. 1979; Harris 1996). Rubble pile asteroids are agglomerations of fragments, ranging in size, that are held together by gravity rather than material strength. An example of a rubble pile asteroid is 25143 Itokawa

(Figure 1.3). From the data obtained by the Hayabusa spacecraft, which was in close proximity to this NEA for about four months in 2005, it was determined that 25143

Itokawa has a low bulk density of 1.9 ± 0.13 g cm−3 (Fujiwara et al. 2006). Due to this low bulk density, a macroporosity of ∼41%, and a shape that is composed of two rounded parts, the asteroid 25143 Itokawa is considered a rubble pile.

Figure 1.3: This image of the rubble pile asteroid 25143 Itokawa was taken by the Hayabusa spacecraft from a range of about 20 km. Figure reproduced from The Internet Encyclopedia of Science (http://www.daviddarling.info) 18

Figure 1.4: The Rotation Rate Distribution for Solar System Bodies. Upper and lower scale indicate approximate diameters. The vertical scale on the right indicates the period in hours. The blue crosses are data from NEAs and the dashed black crosses are data from main-belt asteroids and some main-belt comets. The blue dashed line indicates the rubble pile limit. Modiﬁed from Pravec and Harris (2007b).

The currently known distribution of the rotation periods of asteroids shows a remarkable separation between asteroids larger and smaller than 150 meters, as shown in the period-diameter diagram in Figure 1.4. The blue crosses are the data obtained from NEAs and the dark crosses are data obtained from main-belt asteroids. The dashed line in this ﬁgure indicates the rubble pile limit, which is the maximum rate 19 at which a rubble pile asteroid can spin. If it rotated faster than that, the centrifugal force would overcome gravity and material would either be transported toward or shed from the equator of the object (Harris 1996; Harris and Pravec 2008). The impression from this observed distribution is that there are two populations of asteroids, one that consists of monolithic asteroids (Whiteley et al. 2000) smaller than 150 meters that are fast rotators and a second one that consists of rubble pile asteroids with a diameter greater than 150 meters rotating more slowly than the 2.0 hour rubble pile limit. However, this distinction that is seen in the published data is poorly understood.

Holsapple (2007) has shown 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 one allowed for a rubble pile asteroid. In contrast, for objects with diameters smaller than 10 km, a cohesive and/or tensile strength 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 a rubble pile that requires a small amount of non-gravitational binding. 20

Figure 1.5: The spin limits and data for small Solar System bodies. The dark sloped line assumes a size-dependent strength; it becomes asymptotic to the horizontal red band for materials without cohesion. On the left, the spin limit for cohesive bodies is determined by that cohesive/tensile strength and deﬁnes a strength regime. The horizontal asymptote on the right characterizes a gravity regime where tensile/cohesive strength is of no consequence. Those gravity regime values do depend on shape and friction angle, so average values have been assumed. The data in the upper left triangular region are the fast spinning near-Earth asteroids. Figure and caption reproduced from Holsapple and Michel (2008). 21

Figure 1.5 shows the distribution of asteroids and the derived spin limits for small objects (D <10 km ) that are strength dominated and for larger objects (D >10 km) that are dominated by self-gravity. We can notice from this ﬁgure that the expected shape of the distribution of NEAs from the derived spin limits does not agree with the observed distribution as there seem to be two regions that are almost empty. One of these regions is a transition region where a gradual change in the spin limit as function of size is expected. This transition region is from diameters of 150 meters to a few kilometers with a period less than 2.0 hours. The other region is for diameters less than 150 meters and periods greater than 2.0 hours where a population of small slow rotators may be expected. It is still not clear why there is such a remarkable separation precisely at 150 meters, having then a population of small fast rotators, a population of large slow rotators and two empty regions in the observed distribution when according to the theory (Holsapple 2007) we should be able to ﬁnd large fast- rotators and small slow-rotators as well.

The observed distribution from published data does not necessarily represent the real distribution of NEAs, as there are selection effects that influence the shape of the observed distribution. Objects with a greater diameter are easier to observe which results in a much higher fraction of large objects observed with measured periods compared to small objects. The shape of the object will also affect the detection of its rotation rate, as it is harder to detect a periodic signal from light curves with small amplitudes. Another bias in the observed distribution arises when periodic signals 22 are not detected from the light curves; observers tend to publish data for which they obtain a good period and not publish lower limits.

An important question that then remains unanswered is: What is the true fraction of Fast-Rotating Asteroids (FRAs) as a function of size? To answer this question a more detailed and extensive study of the distribution of rotation periods is necessary.

The study of the distribution of rotation periods will provide essential insight into the apparent separation between the asteroids in the region around and below 150 meters and to the material properties of asteroids, as well as to the radiation recoil torques that modify the spin. 23 1.3 The YORP 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). Radiation recoil forces manifest themselves in two ways: the Yarkovsky effect, which changes the orbit of an asteroid, and the Yarkovsky-O’Keefe-Radzievskii-Paddack (YORP) effect, which changes its spin (Bottke et al. 2006).

There are two components of the Yarkovsky eﬀect, a diurnal and a seasonal component. Figure 1.6a 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 kick 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 this case since the asteroid is drawn as a prograde rotator. The semi-major axis would decrease if it were a retrograde rotator.

Figure 1.6b 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

Sun heats more the North and South hemispheres at points A and C, respectively. 24

Figure 1.6: (a) The diurnal Yarkovsky eﬀect. (b) The seasonal Yarkovsky eﬀect. Reproduced from Bottke et al. (2006).

Due to thermal inertia is at shifted positions of the orbit (points B and D) that the asteroid feel a net kick along its pole due to the thermal photons leaving the surface.

Averaging this force over one orbital period results in a thermal drag that always shrinks the orbit of the asteroid.

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 asymmetry. If the object did not have the two wedges on its equator, no torque will be produced since from any surface element of the sphere the energy that is reemitted will be normal to the surface. Is only when there is an asymmetry 25

that there is a net torque produced. 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 kick 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 to spin up. If the object is rotating in the other sense the torque

will slow it down.

As the YORP eﬀect makes an asteroid spin up, the asteroid could spin so fast

that it can be forced to morph into new shapes or even shed mass if it does not

have the necessary internal strength. Indeed, the YORP eﬀect is considered the pri-

mary mechanism for the formation of binary asteroids (Vokrouhlick´y and Nesvorn´y

2008; Pravec and Harris 2007a). Although the implications of the YORP eﬀect are important for asteroid dynamics, there was no direct detection of this eﬀect until last year. The NEA 2000 PH5 (now renamed 54509 YORP) is the best evidence

to date of the YORP eﬀect, with an increase in the rotation rate that cannot be

explained by gravitational torques (Lowry et al. 2007; Taylor et al. 2007). Another

detection was made on the NEA 1862 Apollo, which also shows a rotational acceler-

ation (Kaasalainen et al. 2007). More recently, a third detection was made on one of

the largest NEAs, 1620 Geographos, which is also spinning up (Durech et al. 2008).

These detections are of extreme importance, since they are supporting evidence for

the signiﬁcance of the YORP eﬀect among NEAs. 26

The Yarkovsky and YORP effects are key elements to explain the rotational and orbital parameters observed for small asteroids and could be even more dominant than collisions and gravitational perturbations. The YORP effect, like the Yarkovsky effect, is sensitive to the size, shape, and material properties of the asteroid. It also depends on how close the asteroid is to the Sun. This effect will vary from object to object, and can spin asteroids either up or down. Therefore, the impression from the observed rotation rate distribution that the great majority of small NEAs are fast rotators is unlikely to be true, as the YORP effect should be equally capable of driving small NEAs to larger or slower rotation rates. What is seen in the published data is not necessarily representative of the real population of NEAs. The true fraction of

Fast-Rotating Asteroids is still unknown. 27

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 to spin up. Reproduced from Bottke et al. (2006).

Figure 1.8: The radiation forces on the wedges. The thick arrows represent the thermal photons leaving the surface. The wedges feel a net kick on the opposite direction. Reproduced from Rubincam (2000). 28 Chapter 2

Observational Program

2.1 Data

In 2006 a program of NEO photometry was initiated by T. Statler at the 2.4-

meter Hiltner telescope at the MDM Observatory1 with the intent of searching for

Fast-Rotating Asteroids (FRAs). The asteroids observed are chosen from the Minor

Planet Center’s list of new objects and from the ESA Spaceguard System’s Priority

list according to the absolute magnitude (H )2 and apparent magnitude (V ) criteria of

H >18 and V <20. Observations are typically in gray to bright (full moon) conditions and made through clear skies or thin clouds. Each object is then observed using an R-band ﬁlter for no more than four hours in one night. Sometimes a follow up is done in the next night depending on the priority of the object according to our objectives. The exposure time is typically 30 seconds and the maximum frame rate is

90 to 100 seconds due to the slow readout time of the CCD. Also random time delays are inserted in order to avoid problems with aliasing. The images are trailed due to the motion of the asteroid during the exposure time. The asteroids typically move about three arc seconds during one exposure.

1MDM refer to the ﬁrst three original members of the consortium: Michigan, Dartmouth and MIT. Today the MDM Consortium consists of ﬁve universities: University of Michigan, Dartmouth College, Ohio State University, Columbia University and Ohio University. 2The absolute magnitude is the apparent magnitude that an object would have if it were at 1 AU from the Sun, at 1 AU from the observer, and at a zero phase angle. 29

To date 34 Near-Earth Asteroids (NEAs) have been observed. The diameters ranges from 22 meters for the smallest object with an H of 25.8 up to 690 meters for a corresponding H of 18.3. The diameter for each asteroid is estimated using the relation:

1329 D = 10−0.2H [km] (2.1) ρ1/2

where ρ is the albedo3, here assumed to be 0.18, which correspond to an average value for S-type asteroids. Table 2.1 shows the orbital parameters and observational circumstances for each observed NEA. The table indicates the absolute magnitude of each object H, the apparent magnitude V at the date of observation, and the phase angle in degrees. It also indicates the date of observation, the observers, and who did the reductions and calibrations for each object.

3The albedo of an object is the fraction of the incident sunlight that is reﬂected. 30 Phase (deg) Date Observers Reductions & Calibrations Table 2.1. Orbital parameters of NEAs and Observational circumstances. HV 21.02 18.6 8.5 02-10-06 T. Statler, T. Watanabe, C. Haas T. Statler 22.733 18.1 38.720.295 19.0 02-10-0620.583 T. Statler, T. 18.7 Watanabe, 40.6 C. Haas21.059 19.4 11.5 02-10-0619.821 T. Statler, T. 18.8 Watanabe, 70.9 C. 02-12-06 T. Haas Statler 21.861 T. Statler, T. 15.8 Watanabe, 27.6 C. 02-12-06 Haas19.688 T. Statler, T. 17.5 Watanabe, 38.9 C. 02-14-06 T. Haas Statler 18.283 T. Statler, T. 18.4 Watanabe, 42.3 C. 12-30-06 T. Haas Statler 41.0 12-30-06 T. Statler T. Statler, J. Wilhelm 12-31-06 T. Statler T. Statler, J. Wilhelm T. Statler, J. Wilhelm T. Statler T. Statler T. Statler 29 9 1 17 10 10 55 55 10 NEA 2006 CL 2006 PA 2006 CL 2006 BN 2006 CN 2005 YT 2006 CY 2006 UQ 2006 WU 31 Table 2.1 (continued) Phase (deg) Date Observers Reductions & Calibrations HV 18.861 17.120.922 18.7 14.923.434 19.2 13.2 01-01-0722.115 T. Statler, J. 19.5 Wilhelm 34.5 01-01-07 T. Statler, J. Wilhelm 34.0 03-05-07 T. Statler, J. T. Wilhelm Statler 03-05-07 T. Statler, J. T. Wilhelm Statler D. Cotto-Figueroa 18.968 17.6 D. Cotto-Figueroa 21.309 17.8 50.0 9.0 03-08-07 03-08-07 T. Statler T. Statler D. Riethmiller D. Riethmiller 1 5 13 26 61 216 NEA 2006 VC 19.904 20.0 40.7 03-07-07 T. Statler D. Cotto-Figueroa 2007 DD 25.754 19.9 20.6 03-05-07 T. Statler, J. Wilhelm D. Cotto-Figueroa 2007 DW 20.336 20.0 51.8 03-06-07 T. Statler, J. Wilhelm D. Riethmiller 2006 YU 2007 CQ 2007 DB 2007 CO 2006 VD 2006 UN 32 Table 2.1 (continued) Phase (deg) Date Observers Reductions & Calibrations HV 19.1 20.3 25.6 01-25-08 T. Statler, D. Riethmiller, D. Cotto-Figueroa D. Cotto-Figueroa 20.316 19.820.498 19.3 33.219.000 19.5 27.5 03-09-0720.805 20.0 41.9 03-09-0720.800 18.6 8.5 03-09-07 45.1 03-10-07 T. Statler 03-10-07 T. Statler T. Statler T. Statler T. Statler D. Cotto-Figueroa D. Cotto-Figueroa D. Cotto-Figueroa D. Riethmiller D. Riethmiller 8 55 49 12 40 84 NEA 2007 EF 20.843 19.3 85.9 03-11-07 T. Statler D. Cotto-Figueroa 2007 EV 25.03 19.1 37.6 03-11-07 T. Statler D. Cotto-Figueroa 2007 EQ 21.133 18.6 44.3 03-11-07 T. Statler D. Cotto-Figueroa 2007 DF 2007 DS 2007 DY 2007 AH 2007 DD 2007 WF 33 Table 2.1 (continued) Phase (deg) Date Observers Reductions & Calibrations HV 19.677 16.823.519 18.8 24.520.306 16.6 4.3 01-25-0820.387 T. Statler, D. 18.6 Riethmiller, 32.2 D. Cotto-Figueroa 01-30-0824.871 T. 17.6 Statler, 48.3 D. 02-13-08 Riethmiller, D. D. Cotto-Figueroa Cotto-Figueroa T. Statler, D. Riethmiller, 15.7 D. 02-13-08 Cotto-Figueroa20.528 T. D. Statler, Cotto-Figueroa D. 17.4 Riethmiller, D. 02-13-08 Cotto-Figueroa D. Cotto-Figueroa T. Statler, D. Riethmiller, 23.8 D. Cotto-Figueroa D. Cotto-Figueroa 02-17-08 D. Cotto-Figueroa T. Statler, D. Riethmiller, D. Cotto-Figueroa D. Cotto-Figueroa 4 9 24 71 25 116 NEA 2008 CP 24.039 18.8 22.4 02-13-08 T. Statler, D. Riethmiller, D. Cotto-Figueroa D. Cotto-Figueroa 2007 PS 2008 AF 2006 JY 2008 CC 2007 TU 2008 CR 34 2.2 Reductions

The data reduction was performed by T. Statler, D. Riethmiller, and the author using the Image Reduction and Analysis Facility (IRAF)4. First, the zero frames are examined statistically and visually. Those zero frames that seem to be bad are removed. The overscan correction and trimming are applied to all the images using

‘ccdproc’. Then the zero frames are combined using ‘zerocombine’ and each flat field bias-subtracted. The flat fields are then inspected. The bad flat fields are removed and the rest are combined using ‘flatcombine’. All the data images are then bias- subtracted and flat-fielded. For some of the data, in which flat-fielding residuals are obvious, dark-sky flats are derived from the asteroid frames and then used to correct the data. An example of a reduced data image is shown in Figure 2.1.

4IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation. 35

Figure 2.1: A reduced data image for 2008 CP. The arrow indicates the position of the asteroid in the frame. Note the trailed image obtained as the asteroid was moving at about 5 arc seconds during one exposure. 36 2.3 Light Curves

The reduced data are analyzed using a suite of IDL routines written by T. Statler.

The ﬁrst step is to identify the asteroid in each frame. This step is done manually; every image is displayed and the user marks the asteroid position. The next step is to

find all the stars in each frame. This is done automatically by first cross-correlating the image with a gaussian point spread function (PSF) model with specified width approximating the true PSF. The routine identifies peaks and makes two lists of objects, with peaks 15σ and 10σ above the noise. Objects in the first list that are within 30 pixels of objects in the second list are removed and each remaining object is

fitted with a gaussian PSF. At this step objects with a discordant fit like galaxies and cosmic rays are rejected. The remaining objects are identified as stars, given an ID number, and are matched across the whole set of frames. At this point if an asteroid did not have a big shift in the position between frames it could be identified as a star and needs to be removed from the list of stars.

Once the stars are identiﬁed through the whole set of frames, another routine is used to ﬁt a two-component Gaussian PSF to all of the stars in each frame. The tail of the PSF must be modeled correctly in order to obtain accurate sky values.

As a check, it is veriﬁed that the sky values are not correlated with the star ﬂuxes.

The next step is to discard PSFs with parameters more than 3σ oﬀ the mean. An average of the stellar PSF is then obtained for each frame, and the stellar photometry is done using the averaged PSFs obtained. We obtain the trail of the asteroid over 37 the duration of each exposure by directly measuring its motion relative to stars. The asteroid photometry is then done using a trailed PSF model which is based on the stellar averaged PSF obtained in each frame.

To calibrate the relative photometry, magnitude offsets for all the frames are calculated such that the magnitude of the stars stay as constant as possible across the whole set of frames. The offsets obtained for each frame from the stellar photometry are also applied to the asteroid. A lightcurve is then obtained with the final magnitude of the asteroid and statistical errors.

We go another step further in the calibrations in order to have the total errors in the magnitude of the asteroid in frame j, σj, which are a combination of statistical

st errors due to photon statistics in the asteroid image, σj , and errors in calibration

cal σj :

2 cal 2 st 2 σj =(σj ) +(σj ) . (2.2)

The IDL routine to calculate the total error was written by the author, with D.

Riethmiller and T. Statler. This routine calculates the average magnitude for each star across the whole set of frames, and the average magnitude for the asteroid. Those stars with average magnitudes that are signiﬁcantly diﬀerent from the average magnitude of the asteroid are excluded. The standard deviations of the stars magnitudes and the average statistical errors for the stars are obtained using a moving (boxcar) average across the whole set of frames. Next, the standard deviations of the stars magnitudes are assumed as the total errors for the stars and the calibration errors, 38

cal σj , are calculated. The total errors in the magnitude of the asteroid in frame j, σj,

st are then calculated using the statistical errors for the asteroid σj , and the calibration

cal errors σj obtained. As a ﬁnal step, a light curve with the total errors is obtained. 39 Chapter 3

Modeling Light Curves

3.1 Introduction

From the observed distribution of NEAs (Figure 1.4), there seems to be a population of small fast rotators, a population of large slow-rotators, and two empty regions.

As discussed in section 1.2, the observed distribution does not necessarily represent the real distribution of NEAs, as there are selection effects that can influence the shape of the observed distribution. Also as discussed in section 1.3, the YORP effect is sensitive to the size and shape of the asteroids and could spin asteroids either up or down. An important question then that remains unanswered is: What is the true fraction of FRAs as a function of size? It is then important not only to report also lower limits on rotation periods as well as measured values, but also to model light curves and obtain statistics of the fraction of FRAs that can be detected in our observational survey. We ask a simple question about the fraction of FRAs because we do not have enough data to get the true distribution of periods. We have developed two codes, the first of which (section 3.2) models the shape of an asteroid and obtain its light curve, and the second of which (section 3.3) calculates the statistical incom- pleteness and obtains the true fraction of FRAs; with a definition of FRA determined 40 partly, by where the rubble pile limit is (2.0 hours) and partly by where our sensitivity drops to zero (2.0 minutes).

3.2 TACO

3.2.1 Code Description

The thermophysical asteroid code TACO1 was ﬁrst written by T. Statler in IDL, and later translated into FORTRAN 95 by the author, D. Riethmiller, and T. Statler.

This code models the surface of an asteroid using a triangular facet representation.

The shape can be assumed ellipsoidal or constructed using the gaussian random spheres method introduced by Muinonen (1998). The surface reﬂectance properties of each facet or tile at a given orientation are calculated using Hapke’s model

(Hapke 1981, 1984, 1986, 1993, 2002). The total brightness of the simulated asteroid is obtained by adding the bidirectional reﬂectance, which is multiplied by the area, of those tiles that are illuminated by the Sun and that are visible from Earth at a given orientation. The object is rotated about its axis of rotation (assumed to be the shortest principal axis) and a light curve is obtained for a complete rotation. This last step is done by a routine written by the author that can obtain one single lightcurve at a given orientation for an object or generate a library of light curves at multiple orientations. 1Thermophysical Asteroid Code, Obviously. 41

3.2.2 Gaussian Random Spheres

The idea that the irregular shapes of asteroids can be modeled using lognormal

statistics, or Gaussian random spheres, was ﬁrst introduced by Muinonen (1998).

Figure 3.1 shows various examples of Gaussian random spheres. The radius r(ϑ,ϕ)is

given by: β2 r(ϑ, ϕ)=a exp s(ϑ, ϕ) − , (3.1) 2

where a is the mean radius and β is the standard deviation of the logradius s(ϑ, ϕ)

which is given by ∞ l s(ϑ, ϕ)= slmYlm(ϑ, ϕ). (3.2) l=0 m=−l

Ylm are the spherical harmonics and slm the spherical harmonic coeﬃcients. The co-

eﬃcients slm are independent gaussian random variables with zero mean and variance given by: 2π Var=(1± δm0) Cl, (3.3) 2l +1

where δm0 is the Kronecker symbol and Cl are the Lengendre coeﬃcients from the

logradius covariance function :

∞ Σs(γ)= ClPl(cos γ). (3.4) l=0 42

Figure 3.1: Gaussian random spheres. Reproduced from Muinonen and Lagerros (1998).

The shapes of different asteroids are generated with TACO using this method of Gaussian random spheres (Muinonen 1998; Muinonen and Lagerros 1998). This routine in TACO, which was written by T. Statler, uses the Legendre coefficients Cl with a cutoff at l=10 as obtained by Muinonen and Lagerros (1998) from the analysis of a sample of seven asteroids with a diameter less than 10 km. 43

3.2.3 Bidirectional Reﬂectance

Bidirectional reﬂectance is the amount of light that is reﬂected in a given direction

by a surface area that is illuminated in a speciﬁc direction. TACO calculates the

bidirectional reﬂectance of each tile of the simulated asteroid at any given orientation

if it is illuminated by the Sun and if it is visible from Earth as it is deﬁned in Hapke

(1981, 1993). Collimated light of intensity J (power per unit area) from the Sun is incident on the upper surface of a tile with an area ΔA in a direction making an angle i with the normal of the tile N as shown in Figure 3.2. The light then interacts with the tile and some of it will emerge from its surface, traveling toward the direction of the Earth in the form of a radiance I (power per unit area per unit solid angle) that makes an angle e with the normal of the tile N. The bidirectional reﬂectance is the

ratio of the outgoing irradiance to the incident intensity:

ID r(i, e, g)= , (3.5) J

where ID means the radiance at the detector which is the Earth. Note that r is a

function that depends on the angles i and e, and on the phase angle g which is the

angle between the directions from the tile to the Sun and to the Earth.

When it is assumed that the light has been scattered by the particles on a medium

only once, an exact solution to the radiance at the detector ID is easily found. But

when multiple scattering from a porous surface is included, it is not such a simple

case. The bidirectional reﬂectance is then obtained using the principle of embedded 44

Figure 3.2: Bidirectional reﬂectance.

invariance which was first developed by Ambartsumian (1958) and which states that the reflectance of a semiinfinite medium does not change if a thin layer is added to the top of the original surface. Using this principle, Hapke (1981) calculates five possibilities for how the light can be scattered due to a thin layer added to the top of the original surface and states that summing over these possibilities the net effect is equal to zero. The resulting equation is then solved for the bidirectional reflectance and the following equation is obtained:

w μ0 r(i, e, g)= [p(g)+H(μ0)H(μ) − 1], (3.6) 4π μ0 + μ where μ0 =cosi, μ =cose and the H(x) function is deﬁned by an integral which

Hapke (1993, 2002) shows is well approximated by:

1 − 2r0x 1+x −1 H(x) ≈ [1 − wx(r0 + ln )] , (3.7) 2 x 45 √ 1−γ − where r0 = 1+γ and γ = 1 w. The term p(g) is the single-particle scattering

function or also known as the phase function, which describes how a particle scatters

light as a function of the phase angle g.

A correction for macroscopic roughness was introduced by Hapke (1984, 1993) to

reduce discrepancies between observations and theory near the poles and illuminated

limb of a low albedo object. The result from this correction is the rough-surface

bidirectional reﬂectance given by:

w μ0 rR(i, e, g)= [p(g)+H(μ0)H(μ) − 1]S(i, e, υ), (3.8) 4π μ0 + μ which is the previous equation obtained for the bidirectional reﬂectance of a smooth surface r(i, e, g) multiplied by a shadowing function S(i, e, υ):

rR(i, e, g)=r(i, e, g)S(i, e, υ). (3.9)

The solution for the shadowing function S(i, e, υ), depends on whether the incident angle i is larger or smaller than the angle of emergence e.Ifi ≤ e, the shadowing function is given by:

μe μ0 χ(θ¯) S(i, e, υ) . (3.10) μe(0) μ0e(0) 1 − f(υ)+f(υ)χ(θ¯)[μ0/μ0e(0)]

If i ≥ e, then the shadowing function is given by:

μe μ0 χ(θ¯) S(i, e, υ) . (3.11) μe(0) μ0e(0) 1 − f(υ)+f(υ)χ(θ¯)[μ/μe(0)]

− υ The parameter υ is the azimuthal angle and f(υ)=exp( 2tan 2 ). The parameter

θ¯, is the mean slope angle that characterizes the distribution function of the tile 46

orientations and the function χ(θ¯)isgivenby:

1 χ(θ¯)= . (3.12) (1 + π tan2 θ¯)1/2

◦ The functions μe(0) and μ0e(0) are the eﬀective cosines at υ =0 and are deﬁned by:

E2(e) μe(0) χ(θ¯)[cos e +sine tan θ¯ ], (3.13) 2 − E1(e)

and

E2(i) μ0e(0) χ(θ¯)[cos i +sini tan θ¯ ], (3.14) 2 − E1(i)

− 2 ¯ − 1 2 ¯ 2 where E1(x)=exp( π cot θ cot x)andE2(x)=exp( π cot θ cot x). The functions

μe and μ0e are the eﬀective cosines at an azimuthal angle υ.Ifi ≤ e the eﬀective cosines are given by:

2 cos υE2(e) + sin (υ/2)E2(i) μ0e χ(θ¯)[cos i +sini tan θ¯ ], (3.15) 2 − E1(e) − (υ/π)E1(i)

and 2 E2(e) − sin (υ/2)E2(i) μe χ(θ¯)[cos e +sine tan θ¯ ]. (3.16) 2 − E1(e) − (υ/π)E1(i)

If i ≥ e, then they are given by:

2 E2(i) − sin (υ/2)E2(e) μ0e χ(θ¯)[cos i +sini tan θ¯ ], (3.17) 2 − E1(i) − (υ/π)E1(e)

and 2 cos υE2(i) + sin (υ/2)E2(e) μe χ(θ¯)[cos e +sine tan θ¯ ]. (3.18) 2 − E1(i) − (υ/π)E1(e) 47

Figure 3.3: The reflectance is plotted versus the longitude along the equator for an object with a low albedo at three different phase angles g. The solid lines indicate the reflectance with the macroscopic surface correction for θ¯ =25◦ and the dashed lines indicate the reflectance without the correction (θ¯ =0◦). Reproduced from Hapke (1984).

Figure 3.3 shows the eﬀects of the macroscopic roughness for an object with a low

albedo (w =0.25) at three diﬀerent phase angles g (10, 80 and 150 degrees). The

solid lines indicate a relative reﬂectance along the equator with a correction for the

macroscopic roughness of θ¯ =25◦. The dashed lines indicate the relative reﬂectance

along the equator without the correction for the macroscopic roughness, θ¯ =0◦.Note

that the bright limb surge is removed by the correction for macroscopic roughness.

A routine in TACO that calculates the bidirectional reﬂectance of a smooth surface

r(i, e, g) was ﬁrst written by T. Statler. This routine was modiﬁed by the author to 48

calculate the rough-surface bidirectional reﬂectance. The equation for bidirectional

reﬂectance was further developed by Hapke (1986, 1993, 2002) in order to include

the shadow-hiding opposition eﬀect BSH(g) and the coherent backscatter opposition

eﬀect BCB(g). The opposition eﬀect is a sharp surge in brightness around zero phase

angle. This eﬀect is relatively unimportant to us, as most of the objects were not

observed at a low phase angle. The opposition eﬀect will be incorporated to TACO

as part of our future work.

3.2.4 Hapke Parameters

A set of ﬁve parameters that are used in the Hapke photometric model (Hapke

1981, 1984, 1993, 2002) are commonly referred to as the Hapke parameters. These

are the single-scattering albedo w, the surface roughness θ¯, the asymmetry factor ξ

and the amplitude B0 and the width h of the opposition surge.

• The single-scattering albedo w is a property of the particles or grains that make

up the surface of the object. It ranges from one to zero being w=1 for an object that

reﬂects all of the sunlight and w=0 for an object that absorbs all of it.

• The surface roughness θ¯, or mean slope angle, is a parameter introduced by the correction for macroscopic roughness (Hapke 1984, 1993). This parameter deﬁnes the degree of the correction for the macroscopic roughness. If θ¯ is set equal to zero, the shadowing function S(i, e, υ) = 1 and no correction is made to the bidirectional 49

reﬂectance of a smooth surface r(i, e, g):

rR(i, e, g)=r(i, e, g)S(i, e, υ). (3.19)

• The asymmetry factor ξ comes from the single Henyey-Greenstein phase function

(Henyey and Greenstein 1941) given by:

1 − ξ2 p(g)= , (3.20) (1 + 2ξ cos g + ξ2)3/2

where ξ = cos θ = − cos g and θ is the scattering angle θ = π − g.TACOusesthe

Legendre polynomial series expansion of this function which is given by:

∞ n p(g)=1+ (2n +1)(−ξ) Pn(cos g). (3.21) n=1

• The amplitude B0 and the width h come from the shadow-hiding opposition

eﬀect BSH(g) and the coherent backscatter opposition eﬀect BCB(g) which are not

included in TACO for this work. As mentioned before, this eﬀects will be included

into TACO later on.

Helfenstein and Veverka (1989) determine the Hapke parameters for an average

S-type asteroid. The values obtained are a single-scattering albedo of w =0.23, a sur-

face roughness or mean slope angle of θ¯ =20◦, an asymmetry factor of ξ = −0.35 and

an amplitude of B0=1.32 and a width of h=0.02 for the opposition surge. We adopt

these parameters, which is standard practice when the exact or approximate values

cannot be derived from photometric analysis (Durechˇ 2002; Hudson et al. 1997). 50

3.2.5 Library of Simulated Light Curves

We generate a sample of twenty asteroid shapes using the Gaussian random sphere method. Each shape, which is a triangular-facet representation, consists of 3184 tiles.

The asteroid is then rotated and a light curve is obtained for one complete rotation at each pole orientation by calculating the bidirectional reﬂectance of each tile that is illuminated by the Sun and visible from Earth. We use 400 possible rotation pole orientations uniformly distributed in an imaginary sphere enclosing the object (Figure

3.4), and a phase angle range of 0 to 90 degrees with an increment of 10 degrees. A lightcurve was obtained for each pole orientation at every phase angle for a total of

4000 light curves for each simulated object, making a total of 80,000 simulated light curves. An example of two diﬀerent objects and their respective light curves at two diﬀerent orientations can be seen in Figures 3.5 to 3.12. 51

Figure 3.4: Uniform distribution of points on a sphere. Each point is determined by the Euler angles ψ and θ and represents a possible pole location of the asteroid. 52

Figure 3.5: Object 1 (Orientation: φ =2◦,θ = 180◦,ψ =0◦, phase =90◦).

Figure 3.6: Light curve for Object 1 in Figure 3.5 (rotated about φ). 53

Figure 3.7: Object 2 (Orientation: φ =2◦,θ = 180◦,ψ =0◦, phase =90◦).

Figure 3.8: Light curve for Object 2 in Figure 3.7 (rotated about φ). 54

Figure 3.9: Object 1 (Orientation: φ =2◦,θ =68◦,ψ = 297◦, phase =40◦).

Figure 3.10: Light curve for Object 1 in Figure 3.9 (rotated about φ). 55

Figure 3.11: Object 2 (Orientation: φ =2◦,θ =68◦,ψ = 297◦, phase =40◦).

Figure 3.12: Light curve for Object 2 in Figure 3.11 (rotated about φ). 56 3.3 SALSA

The author and T. Statler have developed a Simulated Asteroid Lightcurve Sensi- tivity Assessment (SALSA) code, whose purpose is to calculate the probability density of the true fraction of FRAs from the observed sample and the simulated light curve library. SALSA takes as an input a list of light curves obtained for a sample of observed asteroids. We divided the light curves obtained in two subsamples, one for asteroids with diameters smaller than 150 meters and the other one for larger objects. The phase angle for each asteroid is also given as input; it is obtained from the ephemerides of each asteroid at the time of observation. It also takes as inputs two other values, the actual number of FRA detections in the given sample, Nd,andthe number of Monte Carlo trials in each calculation.

A test value of the true fraction F of FRAs in the range from 0 to 1 is selected.

For the given value of F , several steps are made in order to calculate its probability.

In each trial, SALSA will select randomly the number of FRAs (NF ) in the simulated sample. If this number is equal or greater to the number of Nd detections made for that sample then SALSA selects randomly NF real light curves (without repetition) from the sample. If NF

A simulated light curve with the same phase angle as the selected asteroid at the moment of observation is randomly selected for a random object out of our light curve library. A random period chosen uniformly in log P between 2 minutes and 2 hours 57

(based on our sensitivity limit) is then assigned to the simulated light curve. The simulated light curve, which is given in terms of the angle of rotation about the pole, is converted to match the frame times in the real light curve, averaged over the exposure times from the real light curve, and contaminated by random errors consistent with the noise in the real light curve. The simulated “observed” light curve may repeat a number of times depending on the random period that was assigned and the duration of observations in the real light curve. An example of a simulated light curve for one single rotation is given in Figure 3.13 and the same simulated light curve processed to match the sampling and errors of a real light curve is given in Figure 3.14. Note that the simulated observed light curve does not look obviously periodic.

SALSA then looks for a periodicity in the processed simulated light curve using phase dispersion minimization (Laﬂer and Kinman 1965; Stellingwerf 1978). The statistical signiﬁcance of the detection is determined by scrambling the time ordering of the light curve and repeating the phase dispersion minimization analysis 100 times.

If the number of detections (at >99% signiﬁcance) obtained matches the number of the true detections Nd the trial is considered successful. The same procedure is repeated for the given number of trials and a probability for the selected fraction of FRAs is obtained from the number of successful trials divided by the number of trials. The whole procedure is then repeated for each value of the true fraction F . The output obtained from SALSA is the probability for each value of the true fraction of FRAs which as a ﬁnal step is normalized. 58

Figure 3.13: Light curve for one single rotation of a simulated Object .

Figure 3.14: Light curve in Figure 3.14 after been processed with SALSA. The time have been shifted to zero and divided by the assumed period for this example. As can be seen the original light curve repeats about 9 times in the given window of observation. 59 Chapter 4

Results and Discussions

4.1 Light Curves

An example of a light curve for which a detection of a periodic signal is obtained is shown in Figure 4.1. This ﬁgure shows the light curve for 2008 CP, which has an approximate diameter of 48 meters. The period obtained for this object is 0.836 hours.

Searching for periodicities in the light curves was done mainly by T. Statler with the assistance from the author and D. Riethmiller. The methods employed are the periodgram (Lomb 1976), phase dispersion minimization (Laﬂer and Kinman 1965;

Stellingwerf 1978) and folding the light curves and inspecting them visually. Another example in which the underlying periodicity is not obvious is shown in Figure 4.2.

This is the light curve for 2006 CL9, which has an approximate diameter of 88 meters.

The period detected for this object is 0.146 hours. Figure 4.3 shows the light curve in Figure 4.2 folded to the derived period. 60

Figure 4.1: Light curve for 2008 CP .

Figure 4.4 shows an example of a light curve for which no periodic signal is detected. This Figure shows the light curve for 2007 CQ5, which has an estimated diameter of 118 meters. None of the three methods used to ﬁnd a period reveals a periodic signal from this light curve. This object was observed for 3.87 hours and in that amount of time only a positive slope is visible; therefore we estimate a lower limit in the period for this object at 15.48 hours. The light curves for each object can be found in Appendix A. 61

Figure 4.2: Light curve for 2006 CL9.

Figure 4.3: Light curve for 2006 CL9 folded to the derived period. 62

Figure 4.4: Light curve for 2007 CQ5. 63 4.2 Rotation Rate Distribution of Small NEAs

A periodic signal is detected for only ﬁve objects out of the total sample of 34.

One object, 2007 EV, we believe to be FRA but the light curve is ambiguous. We consider this a possible sixth detection. Out of the 34 objects observed, nine have an estimated diameter smaller than 150 meters. For these nine objects, three (possibly four) FRAs are detected. For the other 25 objects, which have a diameter greater than 150 meters, only one FRA detection is made. As we mentioned before is not only important to report good periods that are detected from the light curves but to report lower limits as well. Lower limits are assigned according to what portion of the light curve, which is expected to have two maxima and two minima, is visible in a given window of observation. Table 4.1 shows the periods or lower limits obtained for each observed NEA.

Figure 4.5 shows the rotation rate distribution of small NEAs along with our periods and lower limits from Table 4.1. The results obtained from our observational program confirms that the observed shape of the rotation rate distribution of NEAs has been influenced by selection effects. For objects with diameters smaller than 150 m, if only the objects for which we obtain a good periodic signal were published, they would fall in the apparent population of FRAs. But when the lower limits are included, it can be seen that the impression obtained from the observed distribution that there are no slow rotators in that region is unlikely to be true. 64

Out of the nine objects with diameters smaller than 150 meters, only three (possibly four) are FRAs, leaving six (or five) with lower limits greater than the 2.0 hour rubble pile limit. These six objects fall in the apparent empty region of small slow- rotators. In the other apparent empty region, we find only one FRA out of 25 objects with a diameter greater than 150 meters. Doing simple statistics we find that only

33% (or 44%) of objects smaller than 150 m, and 4% of objects larger than 150 m, are detected as fast rotators. These fractions do not represent conclusive results; a more detailed statistical analysis is neccesary to constrain the true fraction of FRAs.

But just from simple statistics we confirm that the observed distribution of rotation rates is significantly influenced by selection effects. 65

Figure 4.5: Distribution of NEAs in the period-absolute magnitude diagram including our results. The upper scale indicates the estimated diameters assuming an albedo of 0.18. The dots are data from the asteroid light curve database maintained by B. Warner and A. Harris. The large diamonds and the arrows indicate the good periods and lower limits obtained, respectively. 66

Table 4.1. Rotation periods or lower limits for each observed NEA

NEA Period (hrs) Amplitude

2006 CL9 0.146 0.4

2006 CN10 > 8.72 > 0.1

2006 CY10 if P < 6.68 then A < 0.2

2005 YT55 if P < 18.72 then A < 0.2

2006 CL10 if P < 9.72 then A < 0.7

2006 BN55 2.828 0.15

2006 UQ17 if P < 16.68 then A < 0.1

2006 PA1 > 4.58 > 0.1

2006 WU29 if P < 16.6 then A < 0.3

2006 UN216 if P < 16.88 then A < 0.1

2006 YU1 > 15.72 > 0.5

2007 DB61 > 16.32 > 0.1

2007 CQ5 > 15.48 > 0.2 67

Table 4.1 (continued)

NEA Period (hrs) Amplitude

2007 DD 0.037 0.5

2007 DW if P < 9.4 then A < 0.4

2006 VC > 3.72 > 0.9

2006 VD13 > 15.04 > 0.4

2007 CO26 if P < 16.36 then A < 0.1

2007 DF8 if P < 15.92 then A < 0.2

2007 AH12 1.76 0.05

2007 DD49 > 6.12 > 0.5

2007 DY40 if P < 15.76 then A < 0.5

2007 DS84 > 8.42 > 0.15

2007 EV 0.128 0.8

2007 EF > 2.56 > 0.5

2007 EQ if P < 4.76 then A < 0.1 68

Table 4.1 (continued)

NEA Period (hrs) Amplitude

2007 WF55 if P < 3.48 then A < 0.5

2008 AF4 if P < 6.4 then A < 0.15

2007 PS9 > 3.6 > 0.6

2007 TU24 > 3.96 > 0.08

2006 JY25 > 3.8 > 1.0

2008 CC71 if P < 4.32 then A < 1.0

2008 CP 0.836 0.5

2008 CR116 > 15.52 > 0.3 69 4.3 True fraction of FRAs

The probability density for the true fraction of FRAs is obtained using SALSA, which was described in section 3.3. The light curves are divided in two subsamples: nine light curves for asteroids with an estimated diameter smaller than 150 meters and 25 light curves for asteroids with an estimated diameter greater than 150 meters.

The input number of detections Nd of FRAs, is set equal to one for the large-object subsample. For the subsample of smaller asteroids, we adopt both Nd =3andNd =4, to account for the ambiguous object 2007 EV. The number of trials used was 400 in each case.

The probability density of the true fraction F of FRAs, P(F ),isshowninFigure

4.6. The solid line is the result obtained for the large-object subsample. The dashed and dotted lines are the results obtained for the small-object subsample asing Nd =3 and Nd = 4, respectively. For the large-object subsample, a sharply peaked probability density function was obtained. The data indicate that the most probable value of the true fraction is F =0.05 with an expectation value of F =0.095 and a 95% conﬁdence upper limit of F at 0.2.

The probability density function obtained for the small-object subsamples is too wide in comparison with the sharply peaked function obtained for the large-object subsample. This is due to a small sample of only nine objects used in the small-object subsample compared to 25 in the large-object subsample. The data obtained using

Nd = 3 indicate that the most probable value of the true fraction is F =0.4 with an 70

expectation value of F =0.494 and a 95% conﬁdence interval between 0.15 and

0.85. In the case of Nd = 4, the most probable value of the true fraction is F =0.6

with an expectation value of F =0.615 and a 95% conﬁdence interval between 0.25

and 0.95.

The observational and statistical analysis indicates that the majority of the pop-

ulation of objects with D>150 m are slow-rotators, having a 95% conﬁdence upper

limit on the true fraction of FRAs of only 0.2. In the other hand, for asteroids with

D<150 m the results indicate that the population could be almost equally divided between slow and fast rotators, having an expected value for the true fraction of FRAs

+0.36 +0.34 of 0.494−0.34 or 0.615−0.37 (95% conﬁdence). 71

Figure 4.6: Probability density for the true fraction of FRAs. The solid line is the result obtained for the large-object subsample. The dashed and dotted lines are the results obtained for the small-object subsample asing Nd =3andNd = 4, respectively. For the large-object subsample the most probable value of the true fraction is F =0.05 with an expectation value of F =0.095 and a 95% confidence upper limit of F at 0.2. For the small-object subsample, using Nd = 3, the most probable value of the true fraction is F =0.4 with an expectation value of F =0.494 and a 95% confidence interval between 0.15 and 0.85. In the case of Nd = 4, the most probable value of the true fraction is F =0.6 with an expectation value of F =0.615 and a 95% confidence interval between 0.25 and 0.95. 72 4.4 Discussion

It is necessary to continue doing photometry of NEAs and to report lower limits in order to get the real distribution of rotation rates. By obtaining lower limits, we

ﬁnd small slow-rotators in a region that from the published data seemed to be empty.

Interestingly, there is no sign of a cluster of objects at the 2.0 hour rubble pile limit, which is a region for which we have enough sensitivity.

The results obtained for asteroids with diameters smaller then 150 m indicate that the population is almost equally divided between slow and fast rotators. These results are important as they are consistent with predictions of the YORP eﬀect

(Rubincam 2000; Vokrouhlick´yandCapekˇ 2002; Bottke et al. 2006), which states that there should be not only small FRAs but small slow-rotators as well, as this eﬀect will vary from object to object and could make them spin either up or down depending on their size, shape, and other properties.

These results are based on our observationally imposed deﬁnition that a FRA has a period between two minutes and two hours. Therefore, the estimated fraction of 0.6 (or 0.4) for the small slow rotators is a combination of objects with periods greater than two hours and periods shorter than two minutes. The fastest period obtained from our sample of light curves is 2.22 minutes for 2007 DD. We do not consider periods shorter than two minutes because the brightness of the asteroid will be averaged over the exposure time of 30 seconds used in each frame; but that does not means that there are no objects rotating faster than two minutes. A couple of 73 months ago, 2008 HJ was reported by British amateur astronomer Richard Miles as the fastest rotating object yet known 1, with a period of 42.7 seconds. 2008 HJ has an absolute magnitude of 25.76 for an estimated diameter of 24 meters.

A population of objects with diameters smaller than 150 m rotating signiﬁcantly faster than 2008 HJ is also expected from the derived spin limits as a function of size by Holsapple (2007) and from the spin integrations done by Lowry et al. (2007).

The fact that the shortest period detected as of today is only 42.7 seconds can be interpreted as a result of sensitivity limits. If this population of ultra-FRAs does exist, it could be supporting evidence for the YORP eﬀect as the main driver in the evolution of small NEAs, and it could also put new constraints on which parts of the population of NEAs consists of monolithic asteroids and which ones consists of rubble pile asteroids.

For asteroids with diameters greater than 150 m, the most probable value of the true fraction of FRAs is 0.05 with an expectation value of F =0.095 and a 95% conﬁdence upper limit at 0.2. These results indicate that the majority of the population of objects with D>150 m are slow-rotators. A work that is directly comparable to ours and that obtains similar results is the Thousand Asteroid Light

Curve Survey (Massiero et al. 2008). Rotational periods and amplitudes for 925 main belt asteroids down to an absolute magnitude of 18.5 for a corresponding diameter smaller than 1 km are obtained in this survey. Out of the 925 objects, only 2%

1Sky & Telescope (http://www.skyandtelescope.com/news/19353774.html) 74 have a rotation rate shorter than 2 hours. Our results and the results obtained by

Massiero et al. (2008) do not give strong support to the upper envelope derived by

Holsapple (2007) for the spin limits, as the fraction of FRAs among asteroids larger than 150 m is practically negligible. A detailed review of the theory should then be performed in order to determine why the derived spin limits does not seems to represent the real envelope of the distribution.

As a result of this work, we have put constraints on the true fraction of FRAs.

These constraints may change with the addition of more data to our samples. How- ever, the results obtained in this work are more reliable than the currently observed distribution, as the published data sets are very inhomogeneous. 75 Chapter 5

Conclusions and Future Work

Rotation rates or lower limits for 34 NEAs were obtained through optical light curves (Table 4.1). Nine objects out of the 34 objects observed have an estimated diameter smaller than 150 meters. In this sample only three (or possibly 4) detections of a FRA are made, leaving then a majority of small slow-rotators for which lower limits are obtained. For the other 25 objects, which have a diameter greater than 150 meters, only one detection of a FRA is obtained. These results show that the currently known distribution of rotation periods of NEAs has been significantly influenced by selection effects. Therefore it is not representative of the real population of NEAs and the observed fraction of FRAs is not the true fraction.

In order to ﬁnd the true fraction of FRAs two codes were developed. The ﬁrst, is a thermophysical asteroid code (TACO) that simulates the shapes of asteroids using Gaussian random spheres (Muinonen 1998; Muinonen and Lagerros 1998) and simulates the respective light curves at a given orientation following Hapke’s model

(Hapke 1981, 1984, 1986, 1993, 2002). The second, is a Simulated Asteroid Lightcurve

Sensitivity Assessment (SALSA) code that uses the results obtained from the observational program and the simulated light curves to calculate the likelihood of detecting a fraction of FRAs in a given sample. 76

The observational and statistical analysis indicate that for asteroids with D>150

m the most probable value of the true fraction of FRAs is F =0.05 with an expectation

value of F =0.095 and a 95% conﬁdence upper limit of F at 0.2. For asteroids

with D<150 m the results indicate that the most probable value of the true fraction

of FRAs is F =0.4 (or F =0.6) with an expectation value and 95% conﬁdence limits

+0.36 +0.34 of 0.494−0.34 (or 0.615−0.37).

From the results obtained, we conclude that the majority of the population of

asteroids with D>150 m consists of slow-rotators. These results do not give strong

support to the upper envelope derived by Holsapple (2007) for the spin limits. We also

conclude that the population of asteroids with D<150 m is almost equally divided between fast and slow rotators and therefore the currently known distribution of rotation rates is just the result of selection eﬀects. It is necessary to continue doing photometry of NEAs and to consistently report lower limits on rotation periods in order to obtain the real distribution of rotation rates and therefore the true fraction of FRAs. We will continue our observational program at the MDM Observatory in order to increase the dynamical data for NEAs and to obtain the real distribution of rotation rates of small NEAs. The study of the distribution of rotation periods has provided and will continue to provide essential insight into the material properties of asteroids.

The results obtained for the population of asteroids with D<150 m are also supporting evidence for the theory of the YORP eﬀect, as this mechanism should 77

drive small objects not only to spin up but to spin down as well (Rubincam 2000;

Bottke et al. 2006). The YORP eﬀect should also change the obliquities of the aster-

oids, driving the spin axis of the asteroids to obliquity values of 0, 90 and 180 degrees

(VokrouhlickýandCapekˇ 2002). We intend to search for this signature as part of our future work, since as of today, there is no proof of this distinctive signature of the YORP effect. A complete analysis of the spin state distributions of NEAs will help to explain how the YORP effect acts among the NEAs and will contribute to the understanding of the YORP effect, which is a dominant physical process that has been responsible for the evolution of the small NEAs.

We will also continue to develop our codes in order to model the rotational and orbital evolution of NEAs. A model of the evolution of NEAs that includes close encounters, impacts and non-gravitational forces like the YORP and Yarkovsky eﬀects will enhance our understanding not only of their current distribution but also of their sources and the mechanisms that delivered them into their current orbits. NEAs are of great importance in the evolution of the Solar System since they are the pieces left over from the formation of the inner planets, including the Earth. The results presented here and those from our future work will contribute to the understanding of the origin and evolution of NEAs, and therefore to the origin and evolution of our

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Light Curves of NEAs

Figure A.1: Light curve for 2006 CL9. 85

Figure A.2: Light curve for 2006 CN10.

Figure A.3: Light curve for 2006 CY10. 86

Figure A.4: Light curve for 2005 YT55.

Figure A.5: Light curve for 2006 CL10. 87

Figure A.6: Light curve for 2006 BN55.

Figure A.7: Light curve for 2006 UQ17. 88

Figure A.8: Light curve for 2006 PA1.

Figure A.9: Light curve for 2006 WU29. 89

Figure A.10: Light curve for 2006 UN216.

Figure A.11: Light curve for 2006 YU1. 90

Figure A.12: Light curve for 2007 DB61.

Figure A.13: Light curve for 2007 CQ5. 91

Figure A.14: Light curve for 2007 DD.

Figure A.15: Light curve for 2007 DW. 92

Figure A.16: Light curve for 2006 VC.

Figure A.17: Light curve for 2006 VD13. 93

Figure A.18: Light curve for 2007 CO26.

Figure A.19: Light curve for 2007 DF8. 94

Figure A.20: Light curve for 2007 AH12.

Figure A.21: Light curve for 2007 DD49. 95

Figure A.22: Light curve for 2007 DY40.

Figure A.23: Light curve for 2007 DS84. 96

Figure A.24: Light curve for 2007 EV.

Figure A.25: Light curve for 2007 EF. 97

Figure A.26: Light curve for 2007 EQ.

Figure A.27: Light curve for 2007 WF55. 98

Figure A.28: Light curve for 2008 AF4.

Figure A.29: Light curve for 2007 PS9. 99

Figure A.30: Light curve for 2007 TU24.

Figure A.31: Light curve for 2006 JY25. 100

Figure A.32: Light curve for 2008 CC71.

Figure A.33: Light curve for 2008 CP. 101

Figure A.34: Light curve for 2008 CR116.