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Small body orbital dynamics in the Solar System : celestial mechanics and impacts Greenstreet, Sarah 2015

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Small Body Orbital Dynamics in the Solar System:Celestial Mechanics and ImpactsbySarah GreenstreetB.S., Western Washington University, 2007M.S., The University of British Columbia, 2011A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Astronomy)The University Of British Columbia(Vancouver)July 2015c© Sarah Greenstreet, 2015AbstractStudying the orbital dynamics of small body populations in the Solar Systemallows us to understand both their current population and past orbital structure.Planet-crossing populations can also provide impact speeds and probabilities, andwhen coupled to cratering histories of solid bodies can provide planetary surfaceages.The Wide-field Infrared Survey Explorer Near-Earth Object (NEOWISE) de-tections of the near-Earth object (NEO) orbital distributions (Mainzer et al., 2012)are used to illustrate that a pure-gravity NEO orbital model (Greenstreet et al.,2012a) is not rejectable (at > 99% confidence). Thus, no non-gravitationalphysics is required to model the NEO orbital distribution.We discovered in the NEO model numerical integrations the unexpected pro-duction of retrograde orbits from main asteroid belt sources, estimating that∼ 0.1% of the steady-state NEO population is on retrograde orbits. These ret-rograde near-Earth asteroids (NEAs) may answer two outstanding questions inthe literature: the origin of two known MPC NEOs with asteroidal designationson retrograde orbits and the origin of high-strength, high-velocity meteoroids onretrograde orbits.Moving to the outer Solar System, we constructed a Centaur (aJupiter < a <aNeptune) model, supplied from the transneptunian region, to estimate temporaryco-orbital capture frequency and duration with the giant planets, finding that at anytime 0.4% and 2.8% of the population will be Uranian and Neptunian co-orbitals,respectively. This is in agreement with the known fraction of temporary Ura-iinian and Neptunian co-orbitals, respectively. Thus, the scattering transneptunianpopulation provides a self-consistent external source for the unstable giant-planetco-orbitals.In addition to studying the orbital dynamics of small body populations in theSolar System, impact and cratering rates onto planetary surfaces can be deter-mined. The upcoming New Horizons spacecraft fly-through of the Pluto systemin July 2015 will provide humanity’s first data for the crater populations on Plutoand its moons. In principle, absolute ages for these surfaces could be determinedusing the observed surface crater density. However, due to the uncertainty in howthe Kuiper belt size distribution extends to small (d < 100 km) projectiles, abso-lute ages are entirely model-dependent and thus fraught with uncertainty.iiiPrefaceThe text of this dissertation includes modified reprints of previously publishedmaterial as listed below.Chapter 3 (published):• S. Greenstreet and B. Gladman High-inclination Atens are Indeed Rare, ApJ767, L18 (2013).This paper compares the detected orbital element distributions detected by theNEOWISE space telescope and the Bottke et al. (2002) NEO orbital distributionmodel as published by Mainzer et al. (2012) to the newer NEO model fromGreenstreet et al. (2012a). I contributed to this project by taking the publishedbiases for the orbital element distributions from Mainzer et al. (2012) for theNEOWISE detections and applying them to the one-dimensional orbital elementdistributions from the NEO model from Greenstreet et al. (2012a) to compareto the biased distributions from the Bottke et al. (2002) model as published byMainzer et al. (2012). I also executed the Kolmogorov-Smirnov tests to determinethe probability of getting the NEOWISE detected distributions from each model.I wrote the majority of the manuscript, with editing by Brett Gladman, andproduced all the figures and tables.Chapter 4 (published):• S. Greenstreet, B. Gladman, H. Ngo, M. Granvik, and S. Larson, Productionof Near-Earth Asteroids on Retrograde Orbits, ApJ 749, L39 (2012).ivThis paper discusses the unexpected production of near-Earth asteroids(NEAs) onto retrograde (backwards around the Sun) orbits from acceptedmain belt asteroidal sources in the numerical integrations used to produce theGreenstreet et al. (2012a) NEO orbital distribution model. I discovered theseNEAs on retrograde orbits when searching through the numerical integrations.The surprising nature of these objects was pointed out to me by Brett Gladman,which prompted the closer look into their production mechanism and typicalorbital evolutions. I wrote the code used to sift through the integration outputs tofind retrograde behavior as well as extract examples of typical retrograde orbitalevolutions. I also ran one of the two sets of numerical integrations for the best-fitorbits for the two known retrograde NEAs, while Mikael Granvik ran the otherset, and ran the additional integrations for both objects. I produced all the figuresand wrote the majority of the manuscript, with edits done by Brett Gladman,Henry Ngo, Mikael Granvik, and Steve Larson.Chapter 6 (published):• M. Alexandersen, B. Gladman, S. Greenstreet, J. J. Kavelaars, J. M. Petit,S. Gwyn, A Uranian Trojan and the Frequency of Temporary Giant-PlanetCo-Orbitals, Science 341, 994 (2013).This paper examined the frequency of co-orbitals temporarily trapped inthe 1:1 mean-motion resonance with the giant planets in order to explain thediscovery of the first known Uranian Trojan. This was a joint project withPhD student Mike Alexandersen who brought me into the project. Mike’sobservational survey detected the object, tracked it to determine a high-precisionorbit, and then recognized the Trojan character. I set-up and performed thenumerical integrations of the four giant planets and scattering objects as theyentered the Centaur region. I then wrote the code to search through the numericaloutput for co-orbital behavior and computed the fraction of the steady-statepopulation in co-orbital resonance with Uranus and Neptune as well as the mean,median, and maximum durations of the temporary captures. I wrote the followingvsections of the published supplementary material for the manuscript: details ondynamical integrations, co-orbital detection, resonant island classification, andquasi-satellites, since I performed all of the analysis presented in these sections. Ialso provided editing for the manuscript.Chapter 7 (published):• S. Greenstreet, B. Gladman, and W. B. McKinnon, Impact and CrateringRates onto Pluto, Icarus 258, 267 (2015).This paper discusses the impact and cratering rates onto Pluto and its binarycompanion Charon from the various Kuiper belt sub-populations as well as thecatastrophic disruption rate of the four smaller satellites: Styx, Nix, Kerberos, andHydra. The idea for this project came from Brett Gladman and he helped provideme with the background knowledge I needed. Understanding and interpretingthe outcome was a joint effort by the two of us with help from co-author BillMcKinnon. I modified and wrote all the code used in this project. I modifiedan existing piece of collision probability code to produce the impact velocitydistributions and impact probabilities for each Kuiper belt sub-population. Iset-up and ran the numerical integrations used to correct for Pluto’s dynamicaleffects that were not accounted for in the collision probability code. I also wrotethe code to convert the impact rates for each sub-population to cratering rates aswell as the integrated number of craters over the past 4 Gyr. In addition, I wrotethe code to compute the crater density plots and R plots as well as the code tocompute the catastrophic disruption rates for Pluto’s four smaller satellites. Thewriting of the manuscript was divided up as follows: Bill McKinnon wrote thesections on Triton and Secondary craters, Brett Gladman wrote the first threesub-sections of the Discussion section, and I wrote the rest of the manuscript withediting by both Brett and Bill. I also produced all the tables and figures, exceptfigure 2, which was made by Brett.viAppendix A:This appendix discusses a set of integrations that were performed to determinewhether the numerical integrations used to create the Greenstreet et al. (2012a)NEO orbital distribution model had reached convergence at the 4 hour base in-tegration time step used by running the same set of initial conditions with fivedifferent time steps: 2 hours, 4 hours, 8 hours, 16 hours, and 84 hours (3.5 days,used in the Bottke et al. (2002) model integrations). The initial conditions andtime steps chosen for these integrations were decided upon by both myself andBrett Gladman. I performed the integrations and analyzed the orbital distributionof the particles at the end of the 1 Myr integrations to determine if convergencehad reached for the 4 hour time step. I wrote the appendix with the exception ofSection A.4, which was written by Brett. I also produced all the figures.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Symbols and Abbreviations . . . . . . . . . . . . . . . . . . . . . xvAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Small Body Populations . . . . . . . . . . . . . . . . . . 21.1.2 Impact Rates & Cratering in the Solar System . . . . . . . 21.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 An Introduction to Near-Earth Objects and the Main Asteroid Belt 52.1 Near-Earth Objects . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 NEO Dynamical Classification . . . . . . . . . . . . . . . 6viii2.1.2 NEO Orbital Distribution . . . . . . . . . . . . . . . . . . 72.2 Main Belt Dynamics and Near-Earth Object Source Regions . . . 112.2.1 Resonances in the Main Belt . . . . . . . . . . . . . . . . 112.2.2 Jupiter Family Comets . . . . . . . . . . . . . . . . . . . 192.2.3 Past Work with Numerical Integrations . . . . . . . . . . . 192.2.4 Non-Gravitational Forces . . . . . . . . . . . . . . . . . . 232.3 Size Distributions of the Main Asteroid Belt and Near-Earth Objects 242.3.1 Surveys of the Main Asteroid Belt . . . . . . . . . . . . . 262.3.2 Surveys of Near-Earth Objects . . . . . . . . . . . . . . . 292.3.3 Origin of a Wavy Size Distribution . . . . . . . . . . . . . 292.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 High-Inclination Atens are Indeed Rare . . . . . . . . . . . . . . . . 323.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Integration Methods . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.1 Comparison of a, e, i Distributions for the Two Models . . 343.3 Two Models Compared to NEOWISE Aten Detections . . . . . . 353.3.1 Semimajor Axis Distributions . . . . . . . . . . . . . . . 373.3.2 Eccentricity and Inclination Distributions . . . . . . . . . 403.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 Production of Near-Earth Asteroids on Retrograde Orbits . . . . . 444.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Integration Methods . . . . . . . . . . . . . . . . . . . . . . . . . 454.3 Typical Retrograde NEAs . . . . . . . . . . . . . . . . . . . . . . 464.3.1 Sample Retrograde Production . . . . . . . . . . . . . . . 504.3.2 Short-Lived Retrograde Production . . . . . . . . . . . . . 534.3.3 Long-Lived Retrograde Production . . . . . . . . . . . . . 544.3.4 Completeness of Retrograde Population . . . . . . . . . . 564.4 Two Known Retrograde NEAs . . . . . . . . . . . . . . . . . . . 574.5 Estimated Extinct Comet Population . . . . . . . . . . . . . . . . 60ix4.6 High-Strength, High-Velocity Meteoroids on Retrograde Orbits . . 615 A Brief Introduction to the Kuiper Belt and Cratering in the OuterSolar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.1 Current Structure of the Kuiper Belt . . . . . . . . . . . . . . . . 625.1.1 Orbital Classification . . . . . . . . . . . . . . . . . . . . 635.1.2 Mean-Motion Resonances in the Kuiper Belt . . . . . . . 675.2 Giant Planet Migration in our Solar System . . . . . . . . . . . . 735.2.1 Population Decay . . . . . . . . . . . . . . . . . . . . . . 755.3 Size Distributions of Kuiper Belt Sub-Populations . . . . . . . . . 765.3.1 Population Estimates of Kuiper Belt Objects . . . . . . . . 785.4 Cratering in the Outer Solar System . . . . . . . . . . . . . . . . 806 A Uranian Trojan and the Frequency of Temporary Giant-PlanetCo-Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.1 Discovery of the First Uranian Trojan . . . . . . . . . . . . . . . 866.2 Numerical Integrations . . . . . . . . . . . . . . . . . . . . . . . 886.2.1 Co-Orbital Detection in Numerical Integrations . . . . . . 906.2.2 Resonant Island Classification . . . . . . . . . . . . . . . 926.3 Frequency and Duration of Temporary Giant-Planet Co-Orbitals . 926.3.1 Quasi-Satellites . . . . . . . . . . . . . . . . . . . . . . . 946.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 Impact and Cratering Rates onto Pluto . . . . . . . . . . . . . . . . 997.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1007.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097.2.1 Kuiper Belt Population Models . . . . . . . . . . . . . . . 1097.2.2 ¨Opik Collision Probability Code . . . . . . . . . . . . . . 1107.2.3 Impact Rates onto Pluto . . . . . . . . . . . . . . . . . . . 1127.2.4 Cratering Rates onto Pluto . . . . . . . . . . . . . . . . . 118x7.2.5 Number of Craters on Pluto’s Surface . . . . . . . . . . . 1217.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.3.1 Interpretation of Young Surfaces . . . . . . . . . . . . . . 1257.3.2 The Issue of the Size Distribution Below the Break . . . . 1277.3.3 Returning to Crater Retention Ages for Young Surfaces . . 1357.3.4 The Effect of Varying αfaint . . . . . . . . . . . . . . . . 1367.3.5 Secondary Craters . . . . . . . . . . . . . . . . . . . . . . 1407.3.6 Implications of a “Wavy” Size Distribution . . . . . . . . 1407.3.7 Charon . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.3.8 The Four Smaller Moons . . . . . . . . . . . . . . . . . . 1537.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . 1577.5 Some Predictions for the 2015 New Horizons Observations of theCratering Record in the Pluto System . . . . . . . . . . . . . . . . 1618 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180A.1 Close Encounter Scheme of SWIFT-RMVS . . . . . . . . . . . . 180A.2 Convergence Tests for SWIFT-RMVS4 . . . . . . . . . . . . . . . 181A.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182A.4 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . 187xiList of TablesTable 2.1 NEO class percentages . . . . . . . . . . . . . . . . . . . . . . 8Table 3.1 K-S test results . . . . . . . . . . . . . . . . . . . . . . . . . . 40Table 5.1 KBO sub-population estimates for d ≥ 100 km from CFEPS . . 79Table 5.2 Pluto system characteristics . . . . . . . . . . . . . . . . . . . 81Table 7.1 ¨Opik collision probability calculations onto Pluto . . . . . . . . 113Table 7.2 Current primary cratering rates onto Pluto . . . . . . . . . . . 120Table 7.3 Cumulative number of craters on Pluto in past ≈ 4 Gyr . . . . . 123Table 7.4 Current primary cratering rates onto Pluto for varying α . . . . 136Table 7.5 Current primary cratering rates onto Charon . . . . . . . . . . 150Table 7.6 Cumulative number of craters on Charon in past ≈ 4 Gyr . . . 152xiiList of FiguresFigure 2.1 Sample orbits of objects in the four asteroidal NEO classes . . 6Figure 2.2 NEO class distinctions and source regions in a, e space . . . . 9Figure 2.3 RNEO(a, e, i) for NEOSSat-1.0 orbital model . . . . . . . . . 10Figure 2.4 Main asteroid belt orbital distribution . . . . . . . . . . . . . 12Figure 2.5 Approximate (a, i) locations of 3:1, ν5, ν6, and ν16 resonancesfor e = 0 orbits . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 2.6 Kirkwood gaps in the main asteroid belt . . . . . . . . . . . . 17Figure 2.7 Size distribution of known main belt asteroids and NEOs . . . 28Figure 3.1 RNEO(a, e, i) from B02 and G12 . . . . . . . . . . . . . . . . 36Figure 3.2 Fractional NEO a, e, and sin(i) distributions . . . . . . . . . . 38Figure 3.3 Cumulative NEO a, e, and sin(i) distributions . . . . . . . . . 39Figure 4.1 RNEO(a, e, i) for i < 180◦ from G12 . . . . . . . . . . . . . 48Figure 4.2 Retrograde asteroid lifetime vs semimajor axis . . . . . . . . 49Figure 4.3 Sample retrograde asteroid a, e, i history . . . . . . . . . . . . 51Figure 4.4 Zoomed in sample retrograde asteroid a, e, i history . . . . . . 52Figure 4.5 Short-lived retrograde asteroid a, e, i history . . . . . . . . . . 53Figure 4.6 Zoomed in short-lived retrograde asteroid a, e, i history . . . . 55Figure 4.7 Longest-lived retrograde asteroid a, e, i history . . . . . . . . 56Figure 4.8 Observational completeness of 18 < H < 23 NEOs . . . . . . 58Figure 5.1 CFEPS discovered Kuiper belt main classical orbital distribution 64xiiiFigure 5.2 CFEPS debiased Kuiper belt orbital distribution . . . . . . . . 66Figure 5.3 CFEPS debiased classical Kuiper belt orbital distribution . . . 68Figure 5.4 Lagrange points of a planet-Sun system . . . . . . . . . . . . 72Figure 5.5 Schematic of the Pluto system . . . . . . . . . . . . . . . . . 80Figure 6.1 Orbital evolution of temporary Uranian co-orbitals . . . . . . 87Figure 6.2 RKRQ11(a, e, i) for a < 34 AU . . . . . . . . . . . . . . . . . 91Figure 6.3 Orbital evolution of temporary Neptunian co-orbitals . . . . . 95Figure 6.4 Orbital evolution of temporary quasi-satellites . . . . . . . . . 97Figure 7.1 Numerical integration of Pluto’s ecliptic nodal distances . . . 103Figure 7.2 Schematic of three Hg-magnitude differential size distributionscenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Figure 7.3 Impact velocity spectrum onto Pluto . . . . . . . . . . . . . . 111Figure 7.4 Primary crater density on Pluto’s surface versus age (Gyr) . . 126Figure 7.5 Cumulative crater density on Pluto’s surface . . . . . . . . . . 128Figure 7.6 Relative crater frequency plot for Pluto . . . . . . . . . . . . . 132Figure 7.7 Relative crater frequency for Iapetus . . . . . . . . . . . . . . 133Figure 7.8 Cumulative crater density on Pluto’s surface for varying α . . 138Figure 7.9 Relative crater frequency plot for Pluto for varying α . . . . . 139Figure 7.10 Cumulative crater density on Pluto’s surface for M12 and S13 143Figure 7.11 Relative crater frequency plot for Pluto for M12 and S13 . . . 145Figure 7.12 Largest crater in 1 Gyr cumulative crater density on Pluto forM12 and S13 . . . . . . . . . . . . . . . . . . . . . . . . . . 147Figure 7.13 Charon 4 panel cumulative crater density and R plots . . . . . 151Figure 7.14 Catastrophic disruption timescale for Pluto system satellites . 156Figure A.1 Convergence test a, q distributions for high-i . . . . . . . . . 183Figure A.2 Convergence test a, q distributions for low-i . . . . . . . . . . 184Figure A.3 Convergence test a histograms . . . . . . . . . . . . . . . . . 185Figure A.4 Convergence test a, i distributions for high-i . . . . . . . . . . 188Figure A.5 Convergence test a, i distributions for low-i . . . . . . . . . . 189xivList of Symbols and AbbreviationsB02 (Bottke et al., 2002)CFEPS Canada-France Ecliptic Plane SurveyCSA Canadian Space AgencyDES Deep Ecliptic SurveyDRDC Defense Research and Development CanadaEBF Enhanced Bombardment FactorESA European Space AgencyG12 (Greenstreet et al., 2012a)HTC Halley-Type CometIAU International Astronomical UnionIEO Inner Earth ObjectIMC Intermediate Mars CrosserJAXA Japan Aerospace Exploration AgencyJFC Jupiter Family CometKBO Kuiper Belt ObjectxvKRQ11 (Kaib et al., 2011)KS Kolmogorov-SmirnovLHB Late-Heavy BombardmentLPS Palomar-Leiden SurveyLSST Large Synoptic Survey TelescopeMOST Microvariability and Oscillations of Stars space telescopeMPC Minor Planet CenterMPCORB Minor Planet Center Orbit databaseNASA National Aeronautics and Space AdministrationNEA Near-Earth AsteroidNEAR Near-Earth Asteroid RendezvousNEAT Near-Earth Asteroid Tracking programNEO Near-Earth ObjectNEOSSat Near-Earth Object Surveillance SatelliteNEOWISE Wide-field Infrared Survey Explorer Near-Earth Object extendedmissionOMB Outer Main BeltPan-STARRS Panoramic Survey Telescope & Rapid Response SystemPHA Potentially Hazardous AsteroidSFD Size Frequency DistributionxviSKADS Sub-Kilometer Asteroid Diameter SurveySO Scattering ObjectSPL Single Power-LawTNO Trans-Neptunian ObjectWISE Wide-field Infrared Survey ExplorerYMS Yerkes-McDonald surveyAmor - Near-Earth object with 1.017 < q < 1.3 AU.Apollo - Near-Earth object with a > 1.0 AU, q < 1.017 AU.Aten - Near-Earth object with a < 1.0 AU, Q > 0.983 AU.Atira - Near-Earth object with 0.718 < Q < 0.983 AU.Vatira - Near-Earth object with 0.307 < Q < 0.718 AU.Vulcanoid - Near-Earth object with Q < 0.307 AU.Centaur - Object located in the giant planet region (5 < a < 30 AU).AU - Astronomical Unit. Roughly the mean Earth-Sun distance. Defined as149,597,870,700 metres, exactly.a - Semimajor axis of an orbit. Defined as half of the major axis of the ellipse ofthe orbit.e - Eccentricity. Describes an orbit’s ellipticity. A perfect circle has e = 0, anellipse has 0 < e < 1, a parabolic orbit has e = 1, and a hyperbolic orbit hase > 1.q - Pericenter distance. Pericenter (also called perihelion for orbits around theSun) is the point of the elliptical orbit where the object is closest to the centralbody. q can be calculated using a(1− e).xviiQ - Apocenter distance. Apocenter (also called aphelion for orbits around theSun) is the point of the elliptical orbit where the object is farthest from the centralbody. Q can be calculated using a(1 + e).i - The inclination of the plane of an orbit to a reference plane. For objects in ourSolar System, the reference plane is the ecliptic plane, which is the plane throughEarth’s orbit around the Sun.Ω - The longitude of the ascending node in the reference plane is the angle be-tween the reference direction (this is the vernal equinox direction) and where theorbit passes upward through the reference plane.ω - The argument of pericenter is the angle in the orbital plane between where theorbit passes upward through the reference plane and the location of pericenter, theobject’s closest approach to the central body.̟ - The longitude of pericenter is the sum of Ω and ω.M - The mean anomaly is a fictitious angle that describes the average position ofan object along its orbit and is defined by n(t− To), where n is the mean-motion,t is time, and To is the time since the object was last at pericenter.n - The mean-motion is the rate of change of the mean anomalyM and describesthe average angular velocity of an object along its orbit around the Sun, definedby n =√ µa3 .θ - The true anomaly describes the angle in the orbital plane from the pericenterto the location of the body along its orbit.λ - The mean longitude is the sum of Ω, ω, and M.P - The orbital period. P =√4pi2a3GM⊙ .M⊕ - The mass of the Earth. 5.97219×1024 kg.M⊙ - The mass of the Sun. 1.9886×1030 kg.xviiiG - Gravitational constant. 6.67384×10−11 m3 kg−1 s−2.µ - Gravitational parameter. For objects with mobjectM⊙ < 10−11, µ = GM⊙ =1.32712440018x1020 m3 s−2. Above this threshold, µ should by calculated asµ = G(M⊙ + mobject) to maintain the given precision.RH - Hill sphere. For Mplanet/M⊙ ≪ 1, the sphere around a planet wherethe gravitational dominance of the planet exceeds that of the sun. RH =a(Mplanet3M⊙)1/3AU.TJ - Tisserand parameter. This quantity is conserved during planetary close en-counters (in this case with Jupiter) in the circular restricted three-body problem.TJ = aJa + 2√(1− e2) aaJ cos(i).φ11 - The 1:1 mean-motion resonant argument is the difference between the λof a small body of negligible mass and λ of a planet. Libration around 0◦ in-dicates quasi-satellite behaviour, libration around 180◦ indicates horseshoe be-haviour, and libration around 60◦ or 300◦ indicates leading and trailing Trojanbehaviour, respectively.φ31 - The 3:1 mean-motion resonant argument indicates whether an object is inthe 3:1 resonance when the resonant argument librates around 180◦. φ31 = 3λJ −λ− 2̟φ41 - The 4:1 mean-motion resonant argument indicates whether an object is inthe 4:1 resonance when the resonant argument librates around 180◦. φ41 = 4λJ −λ− 3̟L1 - First Lagrange point. Located directly between the Sun and a planet markinga stable gravitational location for a third body of negligible mass.L2 - Second Lagrange point. Located directly opposite the Sun from a planetmarking a stable gravitational location for a third body of negligible mass.xixL3 - Third Lagrange point. Located directly opposite a planet from the Sun mark-ing a stable gravitational location for a third body of negligible mass.L4 - Leading triangular Lagrange point. Located 60◦ ahead of a planet along itsorbit around the Sun marking a stable gravitational location for a third body ofnegligible mass.L5 - Trailing triangular Lagrange point. Located 60◦ behind a planet along itsorbit around the Sun marking a stable gravitational location for a third body ofnegligible mass.R(a, e, i) - Residence time probability distribution. Represents the percentage ofthe steady-state population contained in each bin of a grid of a, e, i cells placedthroughout the Solar System.H-magnitude - Absolute magnitude. The apparent magnitude an object wouldhave if it were located in an equilateral triangle 1 AU from the Sun and the Earthand at zero phase angle φ. m = H + 2.5log10 r2helior2geoP (φ) .φ - Phase angle. Measured between the incident and reflected light directions ofan observed object with values ranging from 0◦ to 180◦, where a 0◦ phase anglerefers to a fully illuminated object.p - Albedo. An object’s reflectivity. Defined as the fraction of incident lightreflected from the surface, where a value of 0 means an object reflects no lightand a value of 1 means an object reflects all light incident upon it.α - Logarithmic “slope” of the power-law differential distribution in number.qslope - Logarithmic “slope” of the power-law differential distribution in diameter.qslope = 5α + 1.d - Impactor or small body diameter in km.D - Crater diameter in km.xxU - Impact velocity in km/s.δ - Volume mass density of an impactor.ρ - Volume mass density of a target at the surface.Q∗RD - Specific energy required for dispersal of a catastrophically disruptedbody, where µ is the reduced mass, MprojectileMtarget/Mtotal. Q∗RD =0.5µU2/Mtotal.xxiAcknowledgementsThank you to my supervisor, Brett Gladman, for providing me with fantastic re-search opportunities and for being a patient teacher over the past six years we haveworked together. You saw potential in me as a new graduate student despite myinexperience and helped me to grow into the scientist I have become.Thank you to the Canadian Space Agency for providing the Science SupportContract that has made the work presented in this thesis and other projects possi-ble.Thank you to my supervisory committee members, Jeremy Heyl, Paul Hick-son, and Catherine Johnson for providing support and guidance along the way.Thank you to Mike Alexandersen, Mikael Granvik, Steve Larson, Bill McKin-non, and Henry Ngo for the expertise they brought to these projects as co-authorsand to Bill Bottke, Clark Chapman, Luke Dones, and Amy Mainzer for their valu-able insights, discussions, and reviews for portions of this thesis.Thank you to every one of my family members (immediate and extended,through blood or marriage) and friends who have provided support through nu-merous conversations about my work and research (especially Jim) that have con-tinued to remind me how lucky I am to be doing something so exciting.Thank you to my good friends Maggie, Steve, Patrick, Belinda, Amanda, Ben,and Nick for providing me with support through fun game nights, delicious home-cooked meals, and a constant fascination with my research that has kept me moti-vated to keep reaching for the stars.xxiiThank you to my husband, Matt, who has been with me every step of thisjourney, providing love and support at every turn, and for always believing in me.Thank you to the wonderful women in my family that have set both an exam-ple and the foundation upon which I have chased after my dreams, to GrandmaSchotzie for her encouragement and expectations of great things from me startingat a young age, to “Ant” Sandy for her constant support and fascination with mywork in astronomy, and to my sister, Amy, for countless hours of conversationsand activities filled with laughter and silliness that have reminded me to never takemyself too seriously.And thank you to my parents for their constant love and support as I haveworked to fulfill my dream of becoming an astronomer.xxiiiDedicationFor my family, with love.xxivChapter 1Introduction1.1 MotivationThe Solar System has three large reservoirs of small bodies: the main asteroidbelt, the Kuiper belt, and the Oort cloud. The Oort cloud is only indirectly relatedto this thesis via its relation to the scattering object population in the Kuiper belt.Each of the populations in the main asteroid belt and the Kuiper belt are dividedinto several sub-populations according to their orbital parameters and dynamicalproperties. The main asteroid belt sits between the orbits of Mars and Jupiter fromroughly 1.8 AU to 4.5 AU and the Kuiper belt is located beyond to the orbit ofNeptune, extending from 30 AU to several hundred AU with the majority of theclassical and resonant populations lying between roughly 30 AU and 85 AU. Ob-jects from these two belts can leave these reservoirs via slow diffusive dynamicalprocesses that can put them onto planet-crossing orbits. The near-Earth object(NEO) population, which have perihelia q < 1.3 AU is an example of one suchpopulation which has its sources in the main asteroid belt. The processes by whichthe orbits of small bodies in the Solar System are perturbed can help us understandthe current orbital parameters of today’s small body populations. Impact proba-bilities and speeds can be computed for planet-crossing populations and cratering1histories can tell us something about the age of planetary surfaces under today’spopulation of potential impactors.1.1.1 Small Body PopulationsSmall body populations in the Solar System are scientifically interesting, becausethey were formed during the formation of the Solar System roughly 4.5 Gyr agoand thus can provide valuable information about the formation process. Dy-namically speaking, their current orbital structure can provide clues about theorbital structure in the early stages of the Solar System. Because these popula-tions are not static, but can move around the Solar System via gravitational andnon-gravitational forces, studying the dynamics of these processes helps us un-derstand the current orbital distribution of today’s populations. The interest infuture manned and unmanned space probe missions to study these small bodieshas fuelled the desire to complete a census of the current population of near-Earthobjects, especially those on Earth-like orbits (Abell et al., 2009; Binzel et al.,2004; Hildebrand et al., 2004). Recent missions to have landed on NEOs includeNASA’s Near-Earth Asteroid Rendezvous (NEAR) to 433 Eros (Veverka et al.,2001) and JAXA’s Hayabusa to 25143 Itokawa (Yano et al., 2006). ESA’s Rosettamission recently achieved the first comet-landing with its Philae probe touchingdown on comet 67P/Churyumov-Gerasimenko. A census of the current smallbody populations can also provide an understanding of the impact and crateringhistory of planetary surfaces and, in particular, can provide the impact threat of po-tentially hazardous asteroids (PHAs) that could impact the Earth (Morrison et al.,1994).1.1.2 Impact Rates & Cratering in the Solar SystemIn addition to understanding the impact threat to Earth, studying impact and cra-tering rates onto planetary surfaces can allow us to date such surfaces either abso-lutely, as in the case of the Moon for which we have samples from the surface, orin a relative sense. The absolute dating of cratered surfaces on the Moon has al-2lowed us to assign a specific time (≈ 3.9 Gyr ago) to the period known as the lunarcataclysm (Tera et al., 1974) when the Moon experienced a spike in its impactinghistory during a violent time in the early stages of the Solar System. Studyingcratered surfaces can also tell us about past resurfacing events and processes suchas geological activity on planetary surfaces. Very few craters on a surface implies,as in the case of Neptune’s moon Triton, that the “planetary” body has had recentgeologic activity that has erased or eroded many of its craters. In addition, thesize distribution of craters on planetary surfaces can be used to infer the size dis-tribution of the impacting populations. For example, Strom et al. (2005) show thatthe crater size frequency distribution on the lunar mare and the size distributionof the near-Earth asteroid population confirms that NEOs are the source of lunarcraters. In the outer Solar System, cratered surfaces can provide insights into thesize distribution of small body populations for which we have less knowledge thanthe main asteroid belt due to their farther distances from the Earth. Particularlyin the case of Pluto, which is the target of the upcoming New Horizons fly-by inJuly 2015, the crater size frequency distribution on its (and its binary compan-ion Charon’s) surface can provide insights into the size distribution of the Kuiperbelt sub-populations. Unfortunately, without knowing the size distribution of theKuiper belt sub-populations down to the smallest sizes (sub-km diameters), littlecan be said about the inferred age of Pluto’s (or Charon’s) surface other than tomake model-dependent predictions (Greenstreet et al., 2015). However, perhapsthe New Horizons fly-through of the Pluto system will provide some clues to theinferred shape of the Kuiper belt size distribution at small diameters.1.2 Thesis OutlineChapter 2 is an introduction to the near-Earth object and main asteroid belt pop-ulations, including their orbital structure, dynamics, and size distributions. Previ-ous work related to the production of a near-Earth asteroid (NEA) orbital model(Greenstreet et al., 2012a) was the subject of my Master’s thesis and is not part of3this PhD thesis. The details of the NEA orbital distribution produced in the modelis discussed, but the production of the model is not.Chapter 3 is based on Greenstreet and Gladman (2013) and compares NEOorbital distribution models to the detected orbital element distributions of Aten-class NEOs detected by the NEOWISE spacecraft (Mainzer et al., 2012).Chapter 4 discusses (based on Greenstreet et al. (2012b)) the production ofretrograde NEAs from main belt asteroidal sources and their connection to high-strength, high-velocity meteoroids on retrograde orbits.At this point, the focus shifts from small body orbital dynamics in the innerSolar System to small body dynamics and cratering in the outer Solar System.Chapter 5 is an introduction to the current and past orbital structure of theKuiper belt as well as its sub-population size distributions and cratering in theouter Solar System.Chapter 6 discusses the capture of scattering Kuiper belt objects into co-orbitalresonance with the icy giant planets, motivated by the discovery of the first Ura-nian Trojan by Alexandersen et al. (2013).Chapter 7 (based on Greenstreet et al. (2015)) focuses on the current impactrate from the Kuiper belt sub-populations onto Pluto and its binary companionCharon as well as their cratering rates over the past 4 Gyr. Model-dependent agesare computed based on assumed extrapolations of the impactor size distribution atsmall sizes (diameter d < 100 km).4Chapter 2An Introduction to Near-EarthObjects and the Main Asteroid Belt2.1 Near-Earth ObjectsThe orbital elements that describe the motions of bodies around the Sun in theSolar System include the semimajor axis a (half the distance of the longest diam-eter of the orbit), the eccentricity e (deviation of the orbit from a perfect circle),and the inclination i (tilt of the body’s orbital plane with respect to the Earth-Sunorbital plane). The pericenter q measures an object’s closest distance to the Sunand is calculated via q = a(1 − e). Similarly, the apocenter Q = a(1 + e) mea-sures an object’s farthest distance from the Sun. Near-Earth objects have periheliaq < 1.3 AUa and are divided into dynamical sub-groups according to their orbitalelements.aThere is no standard upper limit on semimajor axis for the NEO population, although we usea < 4.20 AU from Bottke et al. (2002).5Figure 2.1: Sample schematic orbits of objects in the four traditional NEOclasses. Apollos and Atens cross the orbit of the Earth while Amors lieentirely exterior to the Earth’s orbit and Atiras lie completely interiorto the Earth’s orbit.2.1.1 NEO Dynamical ClassificationNear-Earth objects are traditionally divided into dynamical classes as follows:Amors (1.017 < q < 1.3 AU), Apollos (a > 1.0 AU, q < 1.017 AU), Atens(a < 1.0 AU, Q > 0.983 AU), and Atiras (0.718 < Q < 0.983 AU). In recentliterature, Atira-class asteroids form part of what has been called interior-Earthobjects (IEOs) (Michel et al., 2000), because their orbits lie completely interior toEarth’s orbit (Q < 0.983 AU). Following historical precedent, we adopt the nameAtira for this class of NEO, after its first named member, 163693 Atira.Apollos and Atens are on Earth-crossing orbits, and thus make up the potentialEarth impacting populations. Apollos have orbits larger than the Earth’s orbit andthus spend more time at farther distances from the Sun than the Earth, while Atens6are on orbits smaller than that of the Earth, spending much of their time interiorto Earth’s orbit. Amors have orbits which lie entirely exterior to the orbit of theEarth with perihelia greater than the aphelion of Earth (q > 1.017 AU). Atiras, onthe other hand, lie on orbits completely interior to the Earth’s orbit with aphelialess than the perihelion of Earth (Q < 0.983 AU). Figure 2.1 shows a schematic ofobjects in the four traditional NEO classes. We further expand these into six NEOclasses. As mentioned above, Atira-class NEOs have been referred to as interior-Earth objects (IEOs), which include all objects with orbits interior to that of theEarth (Q < 0.983 AU). We subdivide this region into three orbital classes: Atiras(0.718 < Q < 0.983 AU) are objects decoupled from (interior to the orbit of)Earth but can cross the orbits of Venus and Mercury, Vatiras (0.307 < Q < 0.718AU) are objects decoupled from Venus but can be on Mercury-crossing orbits, andVulcanoids have orbits with Q < 0.307 AU and are decoupled from Mercury. NoVulcanoid or Vatira NEOs are currently known, which is unsurprising due to theirtiny solar elongations (angle from the Sun as viewed from Earth).2.1.2 NEO Orbital DistributionTable 2.1 lists the best estimates for the fractions of NEOs in each orbital classfrom the Greenstreet et al. (2012a) NEO orbital distribution model. Apollos makeup the majority (≈ 63%) of NEOs, however, as is shown in Figure 2.2 of the NEOboundaries in semimajor axis/eccentricity space, the Apollos cover the largest areaof the NEO region. The population of NEOs drops off with decreasing semimajoraxis from the Aten to the Vulcanoid populations, with no objects predicted to existon orbits scattered down to the innermost portions of the Solar System interior tothe orbit of Mercury (see Table 2.1). Objects are gravitationally scattered by aplanet when they pass within a few Hill radii from a planet. The Hill sphereindicates the region around a planet where the planet has gravitational dominance7NEO BestClass Estimate(%)Amor 30.1± 0.8Apollo 63.3± 0.4Aten 5.0± 0.3Atira 1.38± 0.04Vatira 0.22± 0.03Vulcanoid 0.0Table 2.1: NEO class percentages from the Greenstreet et al. (2012a) NEOorbital distribution model. An Atira has 0.718 < Q < 0.983 AU whilea Vatira is a Venus-decoupled object with 0.307 < Q < 0.718 AU. Wedefine Vulcanoids as objects with Q < 0.307 AU. The dynamical modeldoes not produce any NEOs that get scattered down to the innermostportions of the Solar System onto orbits that are completely interior tothe orbit of Mercury (the population we call Vulcanoids).over the Sun, and for Mplanet/M⊙ ≪ 1, can be computed for a given planet via:RH = a(Mplanet3M⊙)1/3(2.1)In order for a main belt asteroid to reach an orbit in the innermost portion of theSolar System (especially near Mercury), an object must experience a sequence offortuitous close encounters that “hand it down” to planets closer to the Sun, whichbecomes increasingly unlikely as an object gets to smaller heliocentric distances.The NEO population estimates, including the dropping trend at smaller semima-jor axes can also be seen in Figure 2.3. This figure shows the residence timeprobability distribution (i.e. the fraction of time NEOs are predicted to spend atany given semimajor axis, eccentricity, and inclination) from the Greenstreet et al.(2012a) NEO orbital distribution model, where the population at low-a drops offas expected.8Figure 2.2: NEO class distinctions and source regions in a, e space. TheNEO population is restricted to orbits with q < 1.3 AU and a <4.2 AU. Amors (1.017 < q < 1.3 AU), Apollos (a > 1.0 AU,q < 1.017 AU), Atens (a < 1.0 AU, Q > 0.983 AU), Atiras(0.718 < Q < 0.983 AU), Vatiras (0.307 < Q < 0.718 AU),and Vulcanoids (Q < 0.307 AU) are the six NEO classes (blue) weadopt. The ν6 secular resonance (red), 3:1 mean-motion resonance(red), intermediate Mars crossers (IMC) (green), and the outer mainbelt (OMB) (green) population constitute the asteroidal source regionsand the Jupiter family comets (JFCs) (green) are the cometary sourceregion for the NEO population.9Inclination i [deg]Semi-major axis a [AU]01020304050607080900 0.5 1 1.5 2 2.5 3 3.5 400.050.10.150.20.250.3Eccentricity eSemi-major axis a [AU]00.10.20.30.40.50.60.70.80.910 0.5 1 1.5 2 2.5 3 3.5 4NEOSSat-1.0modelEccentricity e00.050.10.150.2Figure 2.3: Residence time probability distribution, RNEO(a, e, i), for theGreenstreet et al. (2012a) NEO orbital model. The space is dividedinto a grid of a, e, i cells from a < 4.2 AU, e < 1.0, and i < 90◦ withvolume 0.05 AU x 0.02 x 2.00◦. To create the a, e plot the i bins aresummed and the e bins are summed to create the a, i plot. The colorscheme represents the percentage of the steady-state NEO populationcontained in each bin. Red colors represent cells where there is a highprobability of particles spending their time. The curved lines dividethe NEO region into Amor, Apollo, Aten, and Atira populations aswell as indicating Venus- and Mercury-crossing orbits.10Figure 2.3 shows a high concentration of NEOs between roughly 2 AU and3 AU. This reflects the entry point of most objects into the NEO region fromsources in the main asteroid belt. As shown in Figure 2.2, the asteroidal sourceregions for the NEO population are located just outside the q < 1.3 AU NEOboundary. These NEO source regions are discussed in the next section.2.2 Main Belt Dynamics and Near-Earth ObjectSource RegionsThe main asteroid belt spans semimajor axes between Mars and Jupiter fromroughly 2 AU to 3.5 AU. As can be seen in Figure 2.4 of the main belt (a, e) and(a, i) projections, the majority of the asteroids in the main belt have orbits of mod-erate eccentricity (e < 0.35) and inclination (i < 20◦), with a few high-inclinationgroups. The structure visible in Figure 2.4 is important for the near-Earth objectpopulation, because it marks the source regions that feed the NEO population.2.2.1 Resonances in the Main BeltResonances located in the main asteroid belt also have the ability to mod-ify asteroidal orbits and are often much more powerful than planetary grav-itational scattering. Historically, people began looking at resonances in themain belt as a means to transport meteorites from the asteroid belt to the Earth(Greenberg and Chapman, 1983; Wetherill, 1979). The steady flux of meteoritesarriving at Earth prompted researchers to look at the nearby large reservoir ofsmall bodies, the main asteroid belt, as their source. Because planetary close en-counters are unable to remove objects from the main belt, resonances located inthe main belt began to be explored as the transportation mechanism for gettingmeteorites to the Earth. Further discussion of past work with numerical integra-tions of NEO source regions can be found in Section 2.2.3.11Figure 2.4: Main asteroid belt orbital distribution in a, e and a, i projectionsfor the 422,910 numbered asteroids in the Minor Planet Center Orbit(MPCORB) database on February 5, 2015.12Secular ResonancesIn addition to the orbital semimajor axis, eccentricity, and inclination, the longi-tude of the ascending node Ω (angle projected onto the Earth-Sun orbital planefrom the reference direction of the vernal equinox to the location of the ascendingnode where the orbit plane intersects the Earth-Sun plane), the argument of peri-center ω (angle in the orbital plane from the ascending node to the pericenter), andthe true anomaly θ (angle in the orbital plane from the pericenter to the locationof the body along its orbit) are also used to describe an object’s orbit.The existence of more than two bodies in the Solar System causes the angularorientation of the orbits of all bodies to change over time at a secular precessionrate. This secular precession rate of the longitude of pericenter ̟ (sum of thelongitude of the ascending node and the argument of pericenter ̟ = Ω + ω) foreach planet is made up of a linear combination of the secular precession frequen-cies, or eigenfrequencies gj (j = 1 − 8), of all the planets. Likewise, the secularprecession rate of the longitude of the ascending node Ω for each planet is madeup of a linear combination of the eigenfrequencies sj (j = 1 − 8). An equallycommon naming convention is ν1 − ν8 for g1 − g8 and ν11 − ν18 for s1 − s8.In addition to exciting secular precession in each other’s orbits, the plan-ets induce secular precession in the orbits of small bodies in the Solar System.Eccentricity-type (or inclination-type) secular resonances occur when the preces-sion rate of a body’s longitude of pericenter (or longitude of ascending node)is commensurable with an eigenfrequency, or combination of eigenfrequencies,of the Solar System, i.e. ˙̟ = gj (or Ω˙ = sj). (Michel and Froeschle´, 1997;Wetherill and Faulkner, 1981). Eccentricity-type secular resonances excite the ec-centricities of objects located in the resonance, increasing the object’s chances ofhaving planetary close encounters due to intersecting more planetary orbits as eincreases. On the other hand, inclination-type secular resonances that excite theinclination of objects in the resonance can protect objects from planetary closeencounters as objects are tilted farther out of the ecliptic plane. The ν6 secularresonance, which occurs when ˙̟ ≃ g6, is among the strongest of the eccentricity-13exciting secular resonances in the Solar System. It is located at the inner edge ofthe main belt near a ≈ 2 AU and i < 20◦ (see Figure 2.5). It’s presence can beseen in the top panel of Figure 2.4 where it has sculpted the curved edge of theinner main belt population in a, i space.Objects enter the ν6 resonance from the adjacent main belt as objects randomwalk in a at roughly constant inclination. This random walk is due to the chaoticnature of many overlapping resonances (see Section 2.2.1) located within the mainasteroid belt. Once objects enter the resonance their eccentricities can quicklybe increased to values large enough to put them on Mars-crossing orbits wheresubsequent Martian close encounters can remove them from the resonance and putthem onto a near-Earth orbit (Greenberg and Chapman, 1983; Wetherill, 1979). Ifplanetary close encounters do not kick an object out of the secular resonance, thebody can have its eccentricity increased to unity e = 1, pushing it into the Sun, inonly a few hundred thousand years (Farinella et al., 1994; Gladman et al., 1997).However, if an object leaves the resonance, it can live for tens to hundreds ofmillions of years random walking in a as it experiences a sequence of planetaryclose encounters.Kozai Resonance & Kozai EffectThe Kozai secular resonance (Kozai, 1962) offers an additional protection mech-anism from planetary close encounters. This secular resonance does not relate tothe precession rate of the planets, but that of the asteroid alone. It occurs for asmall body when the precession rate of its longitude of pericenter ˙̟ matches thatof its nodal longitude Ω˙. When these two precession rates are equal, the argumentof pericenter stops precessing, i.e. ω˙ ≃ 0, and begins librating around 90◦ or 270◦(Kozai, 1962). For orbits with a < 2 AU, Michel and Thomas (1996) showed theargument of pericenter can also librate around 180◦. For near-resonant objects,the related Kozai effect results in the argument of pericenter continuing to precess(explore all values between 0◦ − 360◦), but at a highly-variable rate, coupled tolarge e variations.14Figure 2.5: Approximate (a, i) locations of the 3:1 mean-motion resonanceand ν5, ν6, and ν16 secular resonances for e = 0 orbits. As an example,to the right of the ν6 curve an object has a longitude of pericenterprecession rate ˙̟ > g6, while to the left of the curve ˙̟ < g6. Aresonant response occurs when an object is located on a point alongthe ν6 curve where ˙̟ ≃ g6. This resonant response can quicklyincrease the eccentricity of an object to a Sun-grazing orbit within 1Myr (Farinella et al., 1994) unless a planetary close encounter removesthe object from the resonance. A similar resonant response occurs forobjects located along the ν5 secular resonance curve as well as the 3:1mean-motion line, which is discussed in Section 2.2.1. The ν16 secularresonance induces a resonant response in an object’s inclination whenthe longitude of the ascending node Ω˙ ≃ s16.15As the argument of pericenter librates due to Kozai dynamics, thez-component of angular momentum is conserved, and thus the quantity√(1− e2)cos(i) remains constant. This causes the small body’s eccentricity andinclination to oscillate against each other (i.e., high-e corresponds to low-i andvice versa). This mechanism offers protection against planetary close encoun-ters due to the orbital orientations it forces for the asteroid. For objects in theasteroid belt with orbits well inside Jupiter’s orbit and outside Mars’ orbit, closeencounters can only happen at the asteroid’s aphelion or perihelion, respectively.However, because the asteroid’s argument of pericenter ω librates around 90◦ or270◦ and the inclination remains large, the asteroid’s perihelion and aphelion mostoften lie well outside the ecliptic plane, keeping the asteroid from planetary closeencounters (Michel and Thomas, 1996). Thus, objects in Kozai resonance can bestable for tens to hundreds of Myr (Gladman et al., 2000). However, the stabilityof such orbits comes from their high orbital inclinations. Oscillations in e andi can allow orbital configurations when e is very high and i is very low, puttingthe asteroid in a position where fortuitous planetary close encounters could po-tentially dislodge it from the Kozai resonance or other simultaneous resonancessuch as mean-motion resonances (see below), before e gets large enough to pushthe object into the Sun (Gladman et al., 2000). Planetary close encounters remov-ing objects from mean-motion resonances while also in the Kozai resonance isimportant for NEAs on long-lived retrograde (i > 90◦) orbits and is discussed inChapter 4.Mean-Motion ResonancesMean-motion resonances divide the main asteroid belt into the inner, middle, andouter belt. The inner belt is located between ≈ 2.1 AU and ≈ 2.5 AU, the middlebelt lies between ≈ 2.5 AU and ≈ 2.8 AU, and the outer belt covers ≈ 2.8 AU to≈ 3.5 AU. Figure 2.6 depicts the main asteroid belt semimajor axis distributionwhere it can clearly be seen that there are severe gaps in the number of aster-oids located at specific values of a. These gaps are called the Kirkwood gaps16Figure 2.6: Main asteroid belt semimajor axis histogram for the 422,910numbered asteroids in the Minor Planet Center Orbit (MPCORB)database on February 5, 2015. The gaps at roughly 2.5 AU, 2.8 AU,2.95 AU, and 3.3 AU are the Kirkwood gaps corresponding to the 3:1,5:2, 7:3, and 2:1 mean-motion resonances with Jupiter, respectively.(Kirkwood, 1867) and the most obvious ones correspond to the 3:1, 5:2, 7:3, and2:1 mean-motion resonances with Jupiter.The mean-motion, n, of an object describes the average angular velocity of anobject’s orbit around the Sun, and is calculated by:n =√µa3 (2.2)17where µ is the gravitational parameter and a is the semimajor axis. For objectswith mobjectM⊙ < 10−11, µ = GM⊙ = 1.32712440018x1020 m3 s−2. Above thisthreshold, µ should by calculated as µ = G(M⊙ + mobject) to maintain the givenprecision. Mean-motion resonances occur when an asteroid’s orbital period iscommensurable with the orbital period of a planet. For example, the 2:1 mean-motion resonance with Jupiter occurs when the asteroid orbits the Sun twice forevery single orbit of Jupiter, i.e. 2nJ = nast. The repetitious positioning of theasteroid and planet in a mean-motion resonance causes their gravitational interac-tions to be periodic.In principle, this could protect the asteroid from Jupiter’s gravitational per-turbations, which can otherwise kick the asteroid out of the inner Solar System,creating a stable niche in (a, e, i) space. However, when resonances (mean-motionor secular or a combination of the two) overlap, that a, e, i region no longer re-mains stable. The main asteroid belt is quite crowded with numerous overlappingmean-motion resonances with Mars, Jupiter, and Saturn (Nesvorny´ et al., 2002).In the inner main belt, the 3:1 mean-motion resonance located near a ≈ 2.5 AUis one such unstable resonance. The ν6 secular resonance (see Figure 2.5) andseveral mean-motion resonances intersect the 3:1, resulting in its unstable nature.The strong e-pumping mechanism within the unstable 3:1 resonance causes ob-jects located in the resonance to evolve quickly onto at least Mars-crossing orbits(Wisdom, 1985), as depicted in Figure 2.2. Martian close encounters can thencause changes in a and kick the object out of the resonance, leaving it in near-Earth space. If the object stays in the resonance, it can get to Earth-crossingwithout the help of Martian close encounters.Other ResonancesAsteroids can also escape the main belt through mean-motion resonanceswith Mars or three-body mean-motion resonances with the giant planets(Nesvorny´ and Morbidelli, 1998) and reach Mars-crossing orbits (Bottke et al.,2002). Martian close encounters can then scatter these objects onto near-Earth18orbits (Gladman et al., 2000). The outer main belt region (≈ 2.8 AU to≈ 3.5 AU)also has strong mean-motion resonances, such as the 7:3, 5:2, and 2:1 resonanceswith Jupiter, and three-body mean-motion resonances with Jupiter and Saturn(Nesvorny´ and Morbidelli, 1998) which can supply NEOs (Bottke et al., 2002).These resonances can push asteroids to high-e orbits and into the NEO region.2.2.2 Jupiter Family CometsThe last important NEO source region comes from the Jupiter family comets(JFCs). The JFC population is bounded by the Tisserand parameter. In the cir-cular restricted three-body problem, the following quantity is conserved duringplanetary close encounters (in this case with Jupiter):TJ =aJa + 2√(1− e2) aaJcos(i) (2.3)where aJ is Jupiter’s semimajor axis. Jupiter family comets are defined to bethose with 2 < TJ < 3, all of whom are on Jupiter-crossing orbits. Jupiter is thenexcellent at kicking these objects out of the Solar System, however some JFCscan be kicked into the inner Solar System and into the NEO region (Bottke et al.,2002; Levison and Duncan, 1997).2.2.3 Past Work with Numerical IntegrationsThe ability to compute n-body numerical integrations has provided enormous ad-vantages to understanding orbital dynamics and chaotic behavior in the Solar Sys-tem. Numerical integrations showing the chaotic nature of the main asteroid beltas a source for near-Earth objects and meteorites began in the 1970s when Wether-ill developed Monte-Carlo models of injecting collisionally-fragmented objectsinto the ν6 and 3:1 resonances, which slowly raised their eccentricities until theybecame Mars-crossing and Mars could gravitationally remove them from the res-onance (Wetherill, 1979). Martian close encounters could then remove the objectfrom the resonance and cause its orbit to evolve along the perihelion curve at19Mars’ aphelion and eventually the aphelion curve at Mars’ perihelion (similar tothe Earth-crossing curves shown in Figure 2.3), until reaching an Earth-crossingorbit (Greenberg and Nolan, 1993; Wetherill, 1985). Eventually, it was realizedthat resonances alone could move objects from the main asteroid belt to Earth-crossing orbits. Similar models by Greenberg and Chapman (1983) of materialbeing injected into main belt resonances from large-body impacts produced ob-jects being transported onto near-Earth orbits. Both models found ≈ 1 Myr to bethe typical timescale to transport objects from the main asteroid belt to near-Earthspace via resonances, with typical lifetimes of tens of Myr (Greenberg and Nolan,1993). Computational improvements by the 1990s allowed statistically signifi-cant numbers of objects to be numerically integrated with starting positions inmain belt resonances. The turning point in our modern understanding of the ef-ficiency of main belt resonances at supplying NEOs came when Farinella et al.(1994) showed that eccentricities of objects injected into main belt resonancescould not only reach Earth-crossing orbits, but Sun-grazing orbits (e = 1) ontimescales of only≈ 1 Myr. After this, numerical integrations showed typical dy-namical lifetimes of particles placed within many main belt resonances to be onlya few million years, most particles being terminated by becoming Sun grazers orbeing ejected from the Solar System by Jupiter (Gladman et al., 1997).Morbidelli and Nesvorny´ (1999) showed that in addition to the ν6 and 3:1 res-onances, both Mars-crossing asteroids and the outer main belt also efficientlypopulate the NEO region. Many of the Mars-crossing objects are located nearseveral mean-motion resonances with Mars and three-body mean-motion reso-nances, while outer main belt (OMB) objects sit near several mean-motion reso-nances with Jupiter as well as many three-body mean-motion resonances. Chaoticdiffusion of objects migrating in e can both produce and transport Mars-crossingand outer main belt objects into the near-Earth region on timescales of ≈ 25 Myr(Morbidelli and Nesvorny´, 1999).The higher-than-previously-thought number of Sun grazing comets found innumerical integrations by Levison and Duncan (1994) pointed to the short-period20(P < 200 yr) comet population as a non-negligible NEO source. These numericalintegrations found a median lifetime (from the current time) of ≈ 500, 000 yearsuntil most objects were either ejected from the Solar System or became Sun graz-ers. Levison and Duncan (1997) found that≈ 30% of particles evolving out of theKuiper belt reached orbits with q < 2.5 AU at some time during their lives; somehaving a < 4.2 AU and thus could become NEOs. The short amount of time spentin the JFC region by the known JFCs (≈ 0.1 Myr; (Levison and Duncan, 1994)),requires a significant population of objects in the Kuiper belt to supply the knownJFC population (Bottke et al., 2002). This allows the JFC population to supplysome NEOs.The first numerical model of the NEO orbital distribution using these fivemain NEO source regions (the ν6 and 3:1 resonances, Mars-crossers, OMBs, andJFCs) was produced by Bottke et al. (2002). Bottke et al. (2002) fit their inte-grated steady-state orbital distribution for each NEO source to the Spacewatchobservations to determine the contribution of each source region to the overalla < 4.2 AU NEO orbital distribution. Their best-fit parameters for the sourcecontributions were 37 ± 8% from the ν6 resonance, 27 ± 3% from an initiallyMars-crossing population, 20 ± 8% from the 3:1 resonance, 10 ± 1% from theouter-portion of the main belt, and 6± 4% from the JFCs. Their model also brokedown the resulting predicted population into each NEO classb, with Amors con-stituting 31 ± 1%, Apollos 61 ± 1%, Atens 6 ± 1%, and IEOs 2 ± 0.5% of theNEO population.Greenstreet et al. 2012 NEO ModelThe most recent numerical model of the NEO orbital distribution was pro-duced by Greenstreet et al. (2012a). The numerical integrations used in theGreenstreet et al. (2012a) model were similar to those of Bottke et al. (2002) withseveral improvements due to recent increases in computational power (see Chap-bBottke et al. (2002) normalized their population fractions only to the Q > 0.983 AU re-gion. Greenstreet et al. (2012a) have included the near-Sun populations and renormalized thesefractions.21ter 3 for a discussion of these improvements). The updated model was motivatedby the needs of Canada’s microsatellite NEOSSat (Near-Earth Object SurveillanceSatellite).Canada’s NEOSSat is a joint project between the Canadian SpaceAgency (CSA) and Defense Research and Development Canada (DRDC)(Hildebrand et al., 2004). The science mission via CSA is to search for and trackNEOs, specifically those on orbits with a < 1.0 AU. NEOSSat is designed sim-ilarly to the Microvariability and Oscillations of Stars (MOST) space telescope(Walker et al., 2003). NEOSSat features an attached baffle allowing the satelliteto look as close as 45◦ to the Sun, and was launched in February 2013. In order tooptimize an efficient pointing strategy for NEOSSat to maximize the number ofdetections as well as reach orbits for discoveries that are good enough to no longerneed follow-up observations (fractional uncertainty, δ a/a, is small), a model ofthe NEO orbital distribution with good statistics in the a < 1.0 AU region wasneeded.Though the dominant population regions (Amors and Apollos) of the NEOorbital distribution are well represented in the Bottke et al. (2002) model, it wasobvious that the uncertainty in the a < 1.0 AU region was too large to plan anoptimal pointing strategy for NEOSSat to discover and track Atens and Atiras.There was also concern that the exclusion of Mercury from the Bottke et al. (2002)integrations could have caused the a < 1.0 AU populations to inaccurately rep-resent the intrinsic orbital distribution. These reasons motivated the computationof the new steady-state NEO orbital distribution model, which have better statis-tics and greater integrator accuracy than the previous model. Figure 2.3 depictsthe resulting orbital distribution of the NEO population from the Greenstreet et al.(2012a) model. Table 2.1 lists the best estimates for the fractions of NEOs in eachorbital class from the Greenstreet et al. (2012a) NEO orbital distribution model,which are in agreement with the previous model’s estimates to within their esti-mated uncertainties. The better accuracy of this NEO orbital distribution modelfor the a < 1.0 AU orbital element distribution and agreement with the NE-22OWISE detections (Mainzer et al., 2012) is the subject of Chapter 3. In addi-tion, the surprising discovery of main belt sources producing near-Earth objectson retrograde orbits within the Greenstreet et al. (2012a) numerical integrations(Greenstreet et al., 2012b) is discussed in Chapter 4.2.2.4 Non-Gravitational ForcesThe dynamical lifetime of objects inside many main belt resonances is muchshorter than the age of the Solar System, requiring a stable source to resupplythese asteroidal source regions within the main asteroid belt. Possible supplymechanisms include collisions (Farinella et al., 1993), semimajor axis drift drivenvia the Yarkovsky effect (Bottke et al., 2001; Farinella and Vokrouhlicky´, 1999),and chaotic dissipation (Carruba et al., 2003; Morbidelli and Nesvorny´, 1999).It has long been thought that collisional fragments or the break-up of aster-oids in the main asteroid belt can push debris into the strong e-pumping ν6 and3:1 resonances and supply asteroids to near-Earth orbits as well as meteorites tothe Earth (Farinella et al., 1993; Greenberg and Chapman, 1983; Wetherill, 1985).Collisions at closer distances to the Sun where the volume is smaller could alsochange the orbital distribution of NEAs at low semimajor axes (Grun et al., 1985).Tidal disruptions due to planetary close encounters could also break apart weakrubble-pile (non-monolithic structure consisting of coalesced pieces of rock fromgravitational forces) asteroids (Richardson et al., 1998) that could also migrateinto resonances.The Yarkovsky effect affects small, rotating asteroids, causing them to driftin semimajor axis due to the anisotropic emission of radiation absorbed by theasteroid. For a rotating body, the absorbed solar energy it re-radiates will not bein a direction opposite to incoming solar radiation due to thermal properties of thebody that produce a lag between the absorption of sunlight and the re-radiation ofheat. This causes a net force on the body along the direction of motion. This effectis known as the diurnal Yarkovsky effect. Prograde rotators (those that rotatein the counter-clockwise direction from a top-down view of the Solar System)23have a net force in the direction of motion along their orbit, and so have theirsemimajor axis slightly increased, causing the object to drift outward from theSun. Retrograde rotators drift inward. The effect of semimajor axis drift due toYarkovsky as well as direct radiation pressure effects is very slow compared tothe gravitational forces discussed above. Thus, non-gravitational forces have nottraditionally not been included in models of NEO orbital dynamics.2.3 Size Distributions of the Main Asteroid Beltand Near-Earth ObjectsAnother well-studied aspect of the main asteroid belt and NEO populations is theirsize frequency distributions (SFDs) (cumulative number of objects as a functionof size). The SFD of asteroids can tell us about their collisional evolution overthe age of the Solar System, the impact strength of asteroids, the “original” massin the main asteriod belt, and the cratering rate onto the terrestrial planets, amongother things (Jedicke and Metcalfe, 1998). SFD determination is done throughtelescopic surveys of apparent brightness distributions, which are then translatedto absolute magnitudes (directly related to an object’s size).The H-magnitude (absolute magnitude) of a Solar System object is defined tobe the apparent magnitude an object would have if it were located in an equilateraltriangle 1 AU from the Sun and the Earth and at zero phase angle φ. The phaseangle is measured between the incident and reflected light directions of an ob-served object. Phase angles range from 0◦ to 180◦, where a 0◦ phase angle refersto a fully illuminated object. The H-magnitude of a Solar System body can becomputed from its observed (from Earth) apparent magnitude by:m = H + 2.5log10r2helior2geoP (φ) (2.4)where rhelio is the heliocentric distance of the body, rgeo is the body’s geocentricdistance, and P (φ) is the phase function with P (0) = 1. The H-magnitude relates24to an object’s diameter D, in km, via:Dkm =1329√p 10−0.2H (2.5)where the object’s visual albedo p, or reflectivity, is defined as the fraction ofincident light reflected from the surface. An albedo of 0 means an object reflectsno light and a value of 1 means an object reflects all light incident upon it. TypicalNEO albedos range from 0.05-0.25. Using equation 2.5, a NEO with H = 18 hasa diameter range of roughly 0.7 km to 1.5 km for the usual albedo range.The differential number of objects N as a function of H-magnitude can bemodeled as a power-law (equation 2.6), where α is the logarithmic slope (hereafterreferred to simply as the slope) and allows mapping to the differential distributionin diameter d given in equation 2.7 by qslope = 5α + 1.dNdH ∝ 10(α∗H) (2.6)dNdD ∝ D(−qslope) (2.7)The size distribution of the main asteroid belt has collisioinally evolved overthe past ≈ 4.5 Gyr. Dohnanyi (1969) theoretically examined the size frequencydistribution of objects undergoing collisions, assuming the strength of an asteroidper unit mass is independent of size. He found the SFD of such a collisionallyevolved population should follow a single power-law with slope α = 0.5 (qslope =3.5) at all sizes. O’Brien and Greenberg (2003) examined the collisional evolutionof the main belt size distribution analytically and found the same result only forobjects of constant strength independent of their size. As O’Brien and Greenberg(2003) discuss, this does not hold, however, when an object’s strength depends onits size (see Section 2.3.3).Observationally, the main asteroid belt SFD is difficult to determine due toobservational biases of magnitude limited surveys. At opposition, a main belt as-teroid can be at a large variety of distances from the Earth (0.3 to 4.9 AU) and25express apparent brightness fluctuations up to ∼ 6.5 magnitudes depending onits location along its orbit (Jedicke and Metcalfe, 1998). Thus, debiasing aster-oid surveys is key to understanding the intrinsic population. This observationaldifficulty is also present for the near-Earth asteroid population.2.3.1 Surveys of the Main Asteroid BeltThe earliest systematic magnitude survey of main belt asteroids performed on pho-tographic plates was the Yerkes-McDonald survey (YMS) by Kuiper et al. (1958),photographing the ecliptic plane to latitudes of 20◦ and to a limiting photographicmagnitude of ∼ 16.5 (Jedicke and Metcalfe, 1998). They discovered 1,550 aster-oids, determing magnitudes for roughly two-thirds.The Palomar-Leiden Survey (LPS) (van Houten et al., 1970) extended themagnitude frequency distribution to a photographic magnitude of roughly 20,discovering > 2, 000 main belt asteroids, of which they used ∼ 1, 800 todetermine the magnitude frequency distribution (Jedicke and Metcalfe, 1998).van Houten et al. (1970) were the first to measure a change in slope from theα = 0.5 slope predicted and observationally confirmed for the largest asteroidsto a shallower slope now measured to be α = 0.3 for d < 30 km (Gladman et al.,2009).The advances in both CCD technology and computers in the 1990s greatlyaided the ability to perform automated scanning and searching algorithms formain belt asteroids at sizes smaller than tens of km. The University of Ari-zona’s Spacewatch system located on Kitt Peak was the first to successfully useCCD technology to systematically find main belt asteroids and near-Earth objects.Jedicke and Metcalfe (1998) used the Spacewatch observational data of 59,226 as-teroids found between 23 September 1992 and 8 June 1995 to investigate the or-bital and absolute magnitude distribution of main belt asteroids. They found thata single power-law slope in the range 8 < H < 16 (roughly 2.5 km to 105 km foran assumed albedo of 10%) did not fit the observed distribution well, finding a dis-tinctive break in the power-law at roughly H = 13 (corresponding to a diameter of26roughly 6.5 to 15 km, depending on the assumed albedo) (Jedicke and Metcalfe,1998). They found α ≈ 0.3 ± 0.2 best fit the 8 < H < 13 (roughly 10.5 km to105 km for p = 10%) range, α ≈ 0.5 ± 0.2 for 13 < H < 15 (roughly 4 km to10.5 km for p = 10%), and a slightly less steep slope (α ≈ 0.22 − 0.26 ± 0.04)for 15 < H < 17 (roughly 1.5 km to 4 km for p = 10%). Thus, a “wavy”size distribution with multiple slope changes was beginning to emerge from theobservational data. Figure 2.7 shows the size distribution in H-magnitude of theknown (biased) main belt asteroids and NEOs for objects with diameter d & 1 kmassuming p = 10%, where the gradual change in slope at smaller diameters canbe seen.The (S)ub-(K)ilometer (A)steroid (D)iameter (S)urvey, or SKADS, conductedby Gladman et al. (2009) used observations from the Kitt Peak National Obser-vatory, confirming the power-law slopes found by previous surveys (Ivezic´ et al.,2001; Jedicke and Metcalfe, 1998; Wiegert et al., 2007; Yoshida and Nakamura,2007; Yoshida et al., 2003) for H < 15 (corresponding to d & 4 km for p = 10%)and found a shallower slope for 15 < H < 18 (corresponding to roughly d = 1 kmto d = 4 km for p = 10%) of α = 0.30 ± 0.02, also roughly consistent with pre-vious surveys. By this point, it became fairly clear that the main asteroid belt sizedistribution was “wavy”, however not all surveys agreed on the slope for sub-km-sized asteroids (H > 18) (Gladman et al., 2009).The most recent large-scale space-based observational platform for detectingmain belt asteroids and near-Earth objects is NASA’s Wide Field Infrared Explorer(WISE)/NEOWISE thermal infrared space telescope. Launched in 2009, it per-formed an all-sky survey from 15 January 2014 to 5 August 2014, when its coolantdepleted (Mainzer et al., 2011). During that time it identified almost 130,000main belt asteroids. Masiero et al. (2011) used these main belt asteroid obser-vations to measure its SFD, confirming the 13 . H . 18 slope of α ≈ 0.5 fromGladman et al. (2009) as well as the kink at H = 13 from Jedicke and Metcalfe(1998) and the shallower than 0.5 Dohnanyi (1969) slope for 10 < H < 13 fromvan Houten et al. (1970).27

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