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The orbital distribution of near-Earth objects inside Earth's orbit Greenstreet, Sarah 2011

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The Orbital Distribution of Near-Earth Objects Inside Earth’s Orbit by Sarah Greenstreet B.Sc., Western Washington University, 2007 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Astronomy) The University Of British Columbia (Vancouver) September 2011 c© Sarah Greenstreet 2011 Abstract Canada’s Near-Earth Object Surveillance Satellite (NEOSSat), set to launch in early 2012, will search for and track Near-Earth Objects (NEOs), tuning its search to best detect objects with semimajor axis a < 1.0 AU. In order to construct an optimal pointing strategy for NEOSSat, we needed more detailed information in the a < 1.0 AU region than the best current model (Bottke et al., 2002) provides. We present here the NEOSSat-1.0 NEO or- bital distribution model with larger statistics that permit finer resolution and less uncertainty, especially in the a < 1.0 AU region. We find that Amors = 30.1 ± 0.8%, Apollos = 63.3 ± 0.4%, Atens = 5.0 ± 0.3%, Atiras (0.718 < Q < 0.983 AU) = 1.38 ± 0.04%, and Vatiras (0.307 < Q < 0.718 AU) = 0.22 ± 0.03% of the steady-state NEO population, where Q is the orbit’s aphelion distance. Vatiras are a previously undiscussed NEO popu- lation clearly defined in our integrations, whose orbits lie completely interior to that of Venus. Our integrations also uncovered the unexpected produc- tion of retrograde orbits from main-belt asteroid sources; this retrograde NEA population makes up ≃ 0.10% of the steady-state NEO population. The relative NEO impact rate onto Mercury, Venus, and Earth, as well as the normalized distribution of impact speeds, was calculated from the NEOSSat-1.0 orbital model under the assumption of a steady-state. The new model predicts a slightly higher Mercury impact flux. ii Preface • Identification and design of research project: The production of a steady-state NEO orbital distribution model was a project suggested to Sarah Greenstreet by Dr. Brett Gladman. The details of the con- struction of this model were decided upon by both Sarah Greenstreet and Dr. Gladman. • Performing the research: Sarah Greenstreet inherited the numerical integrations consisiting of the orbital evolutions at 300-year output intervals of test particles initialized in the asteroid main-belt used to construct the steady-state NEO orbital distribution model which is analyzed in this thesis. In 2008, Dr. Brett Gladman built the ini- tial conditions for the numerical integrations by perturbing those used previously by Bottke et al 2002. The integrations were managed by Henry Ngo from 2008-2009. Sarah Greenstreet finished the numeri- cal integrations from 2009-2010. These numerical integrations can be thought of as a data set inherited by Sarah Greenstreet and the anal- ysis and scientific interpretation of these integrations as well as the construction of the steady-state NEO orbital distribution performed by Sarah Greenstreet with the help of Dr. Gladman are the subject of this thesis. • Data analysis: The numerical integrations produced a 0.5 Terabtye data set which recorded (in a condensed binary format) the time his- tories of near-Earth objects from their initial locations until their elim- ination. Analysis of the numerical integrations dominantly inherited by Sarah Greenstreet was done using pieces of code which extract and analyse the orbital evolution histories of the integrated test particles, construct the residence time distributions of test particles, estimate uncertainties in the residence time distributions, compute the impact speed distributions for Mercury, Venus, and Earth as well as the col- lision probability distribution for the terrestrial planets. Preliminary versions of some of these codes were written by Henry Ngo. Sarah Greenstreet expanded these codes to extract the analysis presented in iii Preface this thesis. Further information concerning the contributions made by Sarah Greenstreet can be found in Appendix B. • Manuscript preparation: The majority of this thesis has been submit- ted to the scientific journal Icarus for publication. This thesis was writ- ten entirely by Sarah Greenstreet with the exception of Section 3.1.2 and portions of Section 3.2, which were written by Dr. Brett Gladman. iv Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Near-Earth asteroid orbital classes . . . . . . . . . . . 1 1.1.2 Sources of near-Earth objects . . . . . . . . . . . . . 4 1.1.3 NEOSSat: Near-Earth Object Surveillance Satellite . 7 1.1.4 Past work with numerical integrations . . . . . . . . . 9 2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1 Recomputation of the Bottke et al. (2002) integrations . . . 12 2.2 NEOSSat-1.0 NEO orbital model . . . . . . . . . . . . . . . 19 2.2.1 Accuracy of the NEOSSat-1.0 NEO orbital model . . 24 2.3 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 Degeneracy between source regions . . . . . . . . . . 25 2.3.2 Gravitational effects only included . . . . . . . . . . . 27 3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1 New populations discovered within the NEOSSat-1.0 NEO orbital model . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.1 Population of objects decoupled from Venus . . . . . 28 3.1.2 Vulcanoids . . . . . . . . . . . . . . . . . . . . . . . . 32 3.1.3 Retrograde NEAs . . . . . . . . . . . . . . . . . . . . 33 v Table of Contents 3.2 Impact speeds and rates for Earth, Venus, and Mercury . . . 35 4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . 41 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Appendices A Supplementary Information . . . . . . . . . . . . . . . . . . . 47 B Additional Notes on Co-Authorship . . . . . . . . . . . . . . 52 vi List of Tables 1.1 Glossary of acronyms and symbols . . . . . . . . . . . . . . . 2 2.1 Number of integrated test particles . . . . . . . . . . . . . . . 12 2.2 NEO class percentages and fractional errors . . . . . . . . . . 15 2.3 RNEO(a, e, i) from each source region and planet crossing impact rate contributions for NEO classes . . . . . . . . . . . 21 3.1 Impact flux ratios . . . . . . . . . . . . . . . . . . . . . . . . . 39 vii List of Figures 1.1 Sample orbits of objects in the four asteroidal NEO classes . 3 1.2 Approximate (a, i) locations of 3:1 and ν6 resonances for e = 0 orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 NEO class distinctions and source regions in a, e space . . . . 14 2.2 RNEO(a, e, i) for B02 recomputation . . . . . . . . . . . . . . 16 2.3 Even/odd split of RNEO(a, e, i) for B02 recomputation . . . 17 2.4 RNEO(a, e, i) for NEOSSat-1.0 orbital model . . . . . . . . . 20 2.5 RNEO(a, e, i) for a < 1.0 AU region of NEOSSat-1.0 orbital model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6 Even/odd split of RNEO(a, e, i) for NEOSSat-1.0 orbital model 26 3.1 First sample Vatira evolution . . . . . . . . . . . . . . . . . . 30 3.2 Second sample Vatira evolution . . . . . . . . . . . . . . . . . 31 3.3 RNEO(a, e, i) for inclinations up to 180◦ for NEOSSat-1.0 NEO orbital model . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4 Impact speed distribution . . . . . . . . . . . . . . . . . . . . 36 A.1 Rν6 (a, e, i) for NEOSSat-1.0 NEO orbital model . . . . . . . 47 A.2 R3:1 (a, e, i) for NEOSSat-1.0 NEO orbital model . . . . . . 48 A.3 RIMC (a, e, i) for NEOSSat-1.0 NEO orbital model . . . . . 49 A.4 ROMB (a, e, i) for NEOSSat-1.0 NEO orbital model . . . . . 50 A.5 RJFC (a, e, i) for NEOSSat-1.0 NEO orbital model . . . . . 51 viii Acknowledgements I would like to thank Dr. Brett Gladman, my supervisor, who provided me with support, guidance, and patience as I worked to understand the many concepts of Solar System orbital dynamics associated with this project and planetary sciences in general. I would also like to thank Henry Ngo for the work he put into monitoring the numerical integrations and the initial analysis he performed for this project as well as the comments and insights he provided for portions of this thesis. I also offer my thanks to Jeremy Heyl who acted as second reader for this thesis. Thanks to Bill McKinnon, Alan Hildebrand, and Bill Bottke for their valuable input and discussions of portions of this thesis. Production of this model was made possible by a Science Support Contract from the Cana- dian Space Agency. About half of the orbital computations were performed on resources provided by WestGrid and Compute Canada. A portion of the computations were also performed on the GPC supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of On- tario; Ontario Research Fund - Research Excellence; and the University of Toronto. ix Chapter 1 Introduction 1.1 Motivation 1The Near-Earth Object (NEO) population consists of minor planets (aster- oids and de-volatilized comets) with perihelia q < 1.3 AU.2 NEOs originate in the asteroid main-belt and reach near-Earth space via resonances and planetary close encounters. Near-Earth asteroids are scientifically interest- ing because they were formed 4.5 billion years ago during the formation of the Solar System and can provide valuable information about this pro- cess (Hildebrand et al., 2004). Unlike main-belt asteroids, NEOs, especially those with Earth-like orbits, are close enough to be potential targets for fu- ture manned and unmanned space probe missions (Abell et al., 2009; Binzel et al., 2010; Hildebrand et al., 2004). Recent missions to have landed on NEOs include NASA’s Near-Earth Asteroid Rendevous (NEAR) to 433 Eros (Veverka et al., 2001) and JAXA’s Hayabusa to 25143 Itokawa (Yano et al., 2006). The NEO population is the source of impact cratering on the terres- trial planets and a better understanding of the NEO orbital distribution will allow tightening impact chronology (Le Feuvre & Wieczorek, 2011; Marchi et al., 2009). NEOs with orbits which cross that of the Earth (perihelia less than Earth’s aphelion q < 1.017 AU and aphelion greater than Earth’s perihelion Q > 0.983 AU) can pose a collisional threat to the Earth, thus an understanding of the NEO orbital distribution also serves a non-scientific interest. 1.1.1 Near-Earth asteroid orbital classes According to Kepler’s First Law, the Sun acts as a focal point for the ellip- tical orbits of all Solar System objects. The characteristics which describe 1A version of this chapter has been submitted for publication consideration. Green- street, S., Ngo, H., and Gladman, B. (2011) The Orbital Distribution of Near-Earth Objects Inside Earth’s Orbit. Icarus. 2There is no formal upper limit on semimajor axis for the NEO population although we use a < 4.20 AU. 1 Chapter 1. Introduction Acronym/Symbol Definition 3:1 a main-belt mean-motion resonance a semimajor axis AU astronomical unit Amor NEO with 1.017 < q < 1.3 AU Apohele alternate name for Atira Apollo NEO with a > 1.0 AU, q < 1.017 AU Aten NEO with a < 1.0 AU, Q > 0.983 AU Atira NEO with 0.718 < Q < 0.983 AU B02 (Bottke et al., 2002) recomputation e eccentricity H absolute magnitude HTC Halley-type comet i inclination IEO NEO with Q < 0.983 AU IMC intermediate Mars crossing asteroid JFC Jupiter-family comet LPC long-period comet n mean-motion of an orbiting object NEO near-Earth object (q < 1.3 AU) NEOSSat Near-Earth Object Surveillance Satellite NEA near-Earth asteroid OMB outer main-belt asteroid P orbital period q perihelion distance Q aphelion distance R(a, e, i) residence time probability distribution Vatira NEO with 0.307 < Q < 0.718 AU Vulcanoid NEO with Q < 0.307 AU α source region weighting factor for R(a, e, i) µ gravitational parameter, G(MSun +mobject) ν6 a main-belt secular resonance Table 1.1: Glossary of acronyms and symbols. these orbits are called orbital elements. The semimajor axis, a, measures half the distance of the longest diameter of the orbit, the eccentricity, e, describes the deviation of the orbit from a perfect circle, and the inclina- tion, i, provides the tilt of the asteroid’s orbital plane with respect to the 2 Chapter 1. Introduction Earth-Sun orbital plane. The closest distance to the Sun of an object’s orbit (perihelion distance) can be calculated from its a and e via q = a ∗ (1 − e). Similarly, an object’s furthest distance from the Sun (aphelion distance) can be computed via Q = a ∗ (1 + e). Figure 1.1: Sample schematic orbits of objects in the four traditional as- teroidal NEO classes. Apollos and Atens cross the orbit of the Earth while Amors lie entirely exterior to the Earth’s orbit and Atiras lie completely interior to the Earth’s orbit. These orbital elements allow the asteroidal component of the NEO popu- lation to be divided into the four traditional named classes: 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 now Atiras (0.718 < Q < 0.983 AU), where Q is the or- bit’s aphelion distance. In recent literature, Atira-class asteroids form part of what has been called Interior-Earth Objects (IEOs) (Michel et al., 2000), because their orbits lie completely interior to Earth’s orbit. The name Apo- heles, which was suggested by the discoverers of 1998 DK36, is Hawaiian for ‘orbit’ and was chosen as another name for this NEA class for its simi- larity to the words ‘aphelion’ and ‘helios’; unfortunately this object became unobservable and is thus not able to be formally named. The NEO 2003 CP20 was discovered by the LINEAR survey and upon being tracked to 3 Chapter 1. Introduction a high-quality orbit was named 163693 Atira (Schmadel, 2009). Following historical precedent, we thus adopt the name Atira for this class of NEO, after its first named member. Atens and Apollos are Earth-crossing asteroids. Atens have orbits smaller than the Earth’s and thus spend much of their time interior to the Earth’s orbit while Apollos have orbits larger than the Earth’s and spend more time outside the Earth’s orbit. These classes make up the potential Earth- impactors. Amors have orbits entirely exterior to the Earth’s with perihelia greater than the aphelion of Earth (q > 1.017 AU). Atiras, on the other hand, lie on orbits competely interior to the Earth’s orbit with aphelion less than the perihelion of Earth (Q < 0.983 AU). Atiras are in some sense the symmertric opposite of Amors, having Q touching the Aten population just as Amors have a q boundary on the Apollo interface. Figure 1.1 shows sample schematic orbits of objects in each of the four traditional asteroidal NEO classes. We further expand these into six asteroidal NEO classes (Sec- tion 2.1). 1.1.2 Sources of near-Earth objects NEOs originate in the asteroid main-belt and Jupiter-family comet (P < 20 years) regions (or other cometary reservoirs). There are several paths which can transport objects from either the asteroid main-belt or the Jupiter- family comet region to near-Earth space. Mean-motion resonances, secular resonances, and planetary close encounters are a few of these paths. Mean-motion resonances Mean-motion resonances located within the asteroid main-belt are an important supply mechanism for NEOs. The mean-motion, n, of an object describes the average angular velocity of an object’s orbit about the Sun. It is calculated by: n = √ µ a3 (1.1) where µ is the gravitational parameter and a is the semimajor axis. For objects with mobject MSun < 10−11, µ = GMSun = 1.32712440018 x 10 20 m3 s−2. Above this threshold µ should by calculated as µ = G(MSun + mobject) to maintain the given precision. Mean-motion resonances occur when the asteroidal mean-motion and the planetary mean-motion form a simple frac- tion. For example, the 2:1 mean-motion resonance with Jupiter occurs when the asteroid orbits the Sun two times for every single orbit of Jupiter, i.e. 2nJ = na. Due to the repetition of the positions of the asteroid and planet in a mean-motion resonance, the gravitational interactions between the two 4 Chapter 1. Introduction objects is periodic. This can protect the asteroid from Jupiter’s gravitational perturbations which can kick the asteroid out of the inner Solar System and create a stable niche in (a, e, i) space. Other mean-motion resonances are unstable due to intersecting mean-motion resonances located in the aster- oid main-belt. The 3:1 mean-motion resonance located near a≃2.5 AU is one such unstable resonance. The strong e-pumping mechanism within the resonance causes objects located in the resonance to evolve quickly onto Mars-crossing orbits (Wisdom, 1985). Martian close encounters can then kick the object out of the resonance and cause changes in a which place the object in near-Earth space. Secular resonances If the Solar System consisted of only the Sun and one other object, the two orbits would follow the same path around their center of mass indefi- nitely. However, since there are more than two objects in the Solar System, this is not true. The gravitational perturbations of each object in the Solar System causes the location in (a, e, i) space of the pericenter (the longi- tude of the pericenter) of each object to precess with time. The precession rate of each planet is a linear combination of the eigenfrequencies associated with the orbital dynamics of the Solar System. A secular resonance occurs when the precession rate of an asteroid’s longitude of pericenter matches that of one of the fundamental eigenfrequencies of the Solar System (Michel & Froeschlé, 1997; Wetherill & Faulkner, 1981). The ν6 secular resonance, which occurs when the precession rate of the asteroid’s longitude of peri- helion matches that of the sixth secular frequency of the Solar System, is among the strongest of the secular resonances in the Solar System. The ν6 resonance is located at the inner edge of the asteroid main-belt near a≃2 AU and i < 15◦. Figure 1.2 shows an a, i plot of the approximate location of the ν6 and 3:1 resonances for e = 0 orbits. Objects are fed into the ν6 resonance by material in the adjacent main- belt (area to the right of the ν6 curve in Figure 1.2) as objects random walk in a at roughly constant inclination. Once objects enter the resonance they can quickly have their eccentricities increased to Mars-crossing space where subsequent Martian close encounters can remove the object from the resonance and put it onto a near-Earth orbit. Lifetimes inside mean-motion and secular resonances are a few hundred thousand years before objects are pushed into the Sun (Gladman et al., 1997). However, planetary close encounters can kick objects either into or out of resonances. If an object leaves the resonance, it can live for tens to hundreds of millions of years random walking in a as it experiences a sequence of planetary close encounters. 5 Chapter 1. Introduction Figure 1.2: Approximate (a, i) locations of the 3:1 mean-motion resonance and ν6 secular resonance for e = 0 orbits. To the right of the ν6 curve an object has an orbital frequency less than the sixth secular frequency of the Solar System while to the left of the curve an object’s orbital frequency is greater than the sixth secular frequency. If an object is locacted on a point along the ν6 curve that object’s orbital frequency is equal to the sixth secular frequency of the Solar System and a resonant response occurs. This resonant response can quickly increase the eccentricity of an object to a Sun-grazing orbit within 1 Myr (Farinella et al., 1994) unless a planetary close encounter removes the object from the resonance. A similar resonant response occurs for objects located along the 3:1 mean-motion line shown above. Intermediate Mars-crossing population Asteroids can also escape the main-belt through mean-motion resonances with Mars or three-body mean-motion resonances with the giant planets and reach orbits which cross that of Mars. Martian close encounters can then put objects onto near-Earth orbits. These low-inclination Mars-crossing objects 6 Chapter 1. Introduction with q > 1.3 AU are known (Bottke et al., 2000a) as the Intermediate Mars Crossers (IMC source). This population is located above the ν6 resonance (i > 15◦) and split by the 3:1 resonance (located near a≃2.5 AU) (Bottke et al., 2002). Objects in the IMC population can enter either the ν6 or 3:1 resonances and vice versa. The size of this population depends on the eccentricity of Mars which changes on a period of 2 Myr (Bottke et al., 2002). The outer main-belt (OMB) region also has strong mean-motion resonances and three-body resonances which can supply NEOs (Bottke et al., 2002). Additional potential NEO sources Jupiter can scatter Jupiter-family comets (JFCs) onto near-Earth or- bits. Jupiter, however, is much more efficient at throwing objects out of the Solar System, so NEOs produced from the JFCs only account for ≃ 6% of the total steady-state NEO population (Bottke et al., 2002). Other sources could also feed the NEO population at a minor level. The Hungaria and Phocaea families are both located near the ν6 secular resonance and the 3:1 mean-motion resonance. However, their inclinations are large enough to keep them from entering the ν6 resonance and the planetary close encounters which could push them into near-Earth space are less frequent at high incli- nations. Halley-type comets (20 < P < 200 years) and long-period comets (P > 200 years) can also make their way into near-Earth space as they encounter resonances and experience plantary close encounters. However, because these objects spend most of their time far from the Earth, deter- mining the size of the populations is difficult. As of 2002 Spacewatch had not discovered any HTCs or LPCs on orbits with q < 1.3 AU (Bottke et al., 2002). For this reason these sources are currently not considered primary suppliers of the NEO population. It is a long process to reach near-Earth space from the asteroidal and cometary source regions. It is even more arduous to obtain orbits with a < 1.0 AU, making Atens and Atiras the least populous of the NEO classes. This is in part the reason the number of observed Aten and Atira class asteroids is small compared to the number of Amors and Apollos. Atens and Atiras also spend less time at high solar elongations, making them more difficult to observe from ground-based telescopes. These seldom-observed asteroid populations are the focus of NEOSSat’s observational goals. 1.1.3 NEOSSat: Near-Earth Object Surveillance Satellite Ground-based telescopes have been quite successful in observing NEOs. In 1989, the Spacewatch program at the Univeristy of Arizona’s Lunar and Planetary Laboratory made the first automated discovery of an NEO (1989 7 Chapter 1. Introduction UP) (Gehrels & Jedicke, 1996). Within the next 7 years, Spacewatch discov- ered ≈ 75 NEAs (Gehrels & Jedicke, 1996). Spacewatch has since increased this number by roughly an order of magnitude. Other surveys, like LIN- EAR (Stuart, 2001) and Catalina have since discovered thousands3 of NEOs. However, ground-based telescopes have several disadvantages when it comes to increasing the census of Atens and Atiras. From the ground, telescopes observing during night hours can only see objects which are located at high enough solar elongations. Although observations at solar elongations less than 60◦ are possible from the ground, this is only during a brief time win- dow after or before twilight. Space-based telescopes are unconstrained by the daylight cycle and are not affected by light pollution and weather con- ditions on the ground. From orbit, their longer baselines allow for parallax distance measurements of objects as well as tracking of fast-moving objects. Canada’s NEOSSat is a joint project between the Canadian Space Agency (CSA) and Defense Research and Development Canada (DRDC) (Hilde- brand et al., 2004). The science mission via CSA is to search for and track NEOs, specifically those on orbits with a < 1.0 AU. NEOSSat is designed similarly to the Microvariability and Oscillations of Stars (MOST) space telescope (Walker et al., 2003). In addition to the advantages listed above of a space-based telescope, NEOSSat features an attached baffle allowing the satellite to look as close as 45◦ to the Sun. The currently scheduled launch is set for early 2012 and nominal lifetime is 1 year, although MOST is still operating in 2011 after a 2003 launch (Sarty et al., 2011). At only 75 kg NEOSSat is both lightweight and compact with dimen- sions of 1.0 m x 0.8 m x 0.4 m. It will be in a Sun-synchronous, low-Earth orbit at an altitude of ≈ 650 km. NEOSSat will have a 0.15m telescopic mirror with a 0.86◦ x 0.86◦ field of view to detect objects via reflected visi- ble light. A 100-second exposure time will reach ≈ 20th magnitude in the V band (Hildebrand et al., 2004). It will have two operational modes during science mission operation: a survey mode and a tracking mode. The goal during survey mode is to find and inventory new objects. The software pack- age OpenOrb (Granvik et al., 2009) will be used to determine preliminary orbital parameters and classification of new objects as well as plan recovery observations to determine the orbit of each object. Follow-up will occur dur- ing NEOSSat tracking mode and with the help of ground-based telescopes. In order to optimize an efficient pointing strategy for NEOSSat to maximize the number of detections, a model of the NEO orbital distribution with good statistics in the a < 1.0 AU region is needed. 3See the JPL NEO Discovery Statistics website: http://neo.jpl.nasa.gov/stats/ . 8 Chapter 1. Introduction 1.1.4 Past work with numerical integrations Once it was discovered that objects in the main-belt can be fed into the NEO region through resonances, several numerical integrations have been done to model this process and the resulting NEO orbital distribution. During the 1970’s, Wetherill developed Monte-Carlo models of collisionally-fragmented objects injected into the ν6 and 3:1 resonances which slowly raised their ec- centricities until Mars could gravitationally remove them from the resonance (Wetherill, 1979). Further Martian close encounters caused random walking in a until these objects reached Earth-crossing space. Greenberg & Chap- man (1983) produced similar models of cratering debris from large-body im- pacts reaching near-Earth space after being injected into resonances. Both models found the path to near-Earth space via resonances took ≈ 1 Myr to reach barely Earth-crossing orbits (Greenberg & Nolan, 1993) with typical lifetimes of tens of Myr. By the 1990’s computers had become powerful enough to perform numerical integrations of statistically significant num- bers of objects starting in these resonances. Farinella et al. (1994) showed that on a timescale of ≈ 1 Myr, eccentricities of objects injected into orbital resonances can be increased not only to Earth-crossing orbits, but to Sun- grazing orbits. This marks the turning point of our modern understanding that main-belt resonances are efficient direct NEA producers. Numerical integrations (Gladman et al., 1997) showed typical dynamical lifetimes of particles placed within many main-belt resonances to be only a few million years, most particles being terminated by becoming Sun grazers or being ejected from the Solar System by Jupiter. Because this dynamical lifetime is much shorter than the age of the Solar System, there must be a stable source resupplying these asteroidal source regions within the main aster- oid belt. Collisions (Farinella et al., 1993), semimajor axis drift driven by Yarkovsky (Bottke et al., 2001; Farinella & Vokrouhlický, 1999), and chaotic dissipation (Carruba et al., 2003; Morbidelli & Nesvorný, 1999) are possible supply mechanisms for the main-belt orbital resonances. Morbidelli & Nesvorný (1999) showed that in addition to the ν6 and 3:1 resonances, Mars-crossing asteroids also efficiently populate the NEO region. This population of Mars-crossing asteroids is produced through chaotic dif- fusion of objects migrating in e, typically living for ≈ 25 Myr (Morbidelli & Nesvorný, 1999). The IMC population (q > 1.3 AU, 2.06 ≤ a≤ 2.48 AU or 2.52 ≤ a≤ 2.80 AU, and i ≤ 15◦) is populated by asteroids leaving mean-motion resonances with Mars and three-body mean-motion resonances (Morbidelli & Nesvorný, 1999) or removed from the 3:1 and ν6 resonances due to Martian close encounters (Bottke et al., 2002). Several mean-motion 9 Chapter 1. Introduction resonances with Jupiter are located in the outer part of the main asteroid belt along with many three-body resonances. These e-pumping resonances can also produce chaotic diffusion of outer main-belt (OMB) asteroids onto near-Earth orbits (Morbidelli & Nesvorný, 1999). Numerical integrations of the orbits of known short-period comets (P < 200 years) performed by Levison & Duncan (1994) gave a median lifetime (from the current time) of ≈ 500,000 years until most objects were either ejected from the Solar System or became Sun grazers. The number of Sun-grazing short-period comets found by these numerical integrations was significantly larger than previous analytical estimates (Levison & Duncan, 1994). This result pointed to the short-period comet population as a non- negligible source of NEOs after a large number may lose their cometary aspect due to de-volitalization4 . Levison & Duncan (1997) found ≃ 30% of the particles evolving out of the Kuiper belt reached orbits with q < 2.5 AU at some time during their lives. Some of these have a < 4.2 AU and thus would be NEOs. Because the time spent in the JFC region by the known JFCs is short (≈ 0.1 Myr; (Levison & Duncan, 1994)), there must be a significant population of objects in the Kuiper belt to supply the known JFC population (Bottke et al., 2002). For this reason the JFC population is also considered a primary NEO source. The most recent comprehensive set of numerical integrations used to model the NEO population was computed by Bottke et al. (2002). Five source regions were included in these computations: the ν6 secular reso- nance, the 3:1 mean-motion resonance, the intermediate source Mars crosser (IMC) population, the outer main-belt (OMB) population, and the Jupiter- family comet (JFC) population. The integrations of particles initialized in the Kuiper Belt (the JFC source region) used by Bottke et al. (2002) were those of Levison & Duncan (1997). To determine the contribution of each source region to the overall a < 4.2 AU NEO orbital distribution, Bottke et al. (2002) fit their integrated steady-state for each source to the Spacewatch observations. Their best-fit parameters for the source contri- butions were αν6 = 0.37 ± 0.08, αIMC = 0.27 ± 0.03, α3:1 = 0.20 ± 0.08, αOMB = 0.10±0.01, and αJFC = 0.06±0.04 which can be interpreted as the fraction of the steady-state NEO population from each source. Note that in a given sub-region of orbital parameter space these fractions can vary widely; for example, more than 37% of the Atens come from the ν6 since the JFC and OMB sources contribute no Atens. This model also broke 4We adopt the terminology that a q < 1.3 AU object with coma is a comet and not an NEO, otherwise the object is termed a near-Earth comet (NEC) and not considered here. 10 Chapter 1. Introduction down the resulting predicted population into each NEO class5, with Amors constituting 31 ± 1%, Apollos 61 ± 1%, Atens 6± 1%, and IEOs 2 ± 0% of the NEO population, where we assume the final IEO uncertainty was not zero, but rather < 0.5%. We will further break the IEO population into 3 subgroups in Section 2.1 with Atiras as the most populous of these groups. Though the dominant population regions (Amors and Apollos) of the NEO orbital distribution are well represented in the Bottke et al. (2002) model, we concluded that the uncertainty in the a < 1.0 AU region was too large to plan an optimal pointing strategy for NEOSSat to discover and track Atens and Atiras. We also had concern that the exclusion of Mercury from their integrations could have caused the a < 1.0 AU populations to be distributed differently than in their orbital model. For these reasons we were motivated to compute a new steady-state NEO orbital distribution, with more test particles and greater integrator accuracy. In this paper, we first present a recomputation of the Bottke et al. (2002) integrations in order to prove reproducibility and understand the effect of small-number statistics present in the model. We then present a new set of numerical integrations to model the steady-state NEO orbital distribution with better statistics in the a < 1.0 AU region. The NEOSSat-1.0 NEO orbital model consists of approximately six times as many test particles as the Bottke et al. (2002) model. We show the uncertainty in the a < 1.0 AU NEO populations of the NEOSSat-1.0 model are significantly improved. A description of two previously undiscussed populations seen in the NEOSSat- 1.0 model can be found in Section 3.1. Section 3.2 discusses the relative NEO impact rate onto Mercury, Venus, and Earth along with the normalized distribution of impact speeds computed for the NEOSSat-1.0 model. 5Bottke et al. (2002) normalized their population fractions only to the Q > 0.983 AU region. We have included the near-Sun populations and renormalized these fractions. 11 Chapter 2 Methods 2.1 Recomputation of the Bottke et al. (2002) integrations 6Due to recent increases in computational power, we present here new nu- merical integrations similar to those of Bottke et al. (2002) with several im- provements. Our first set of integrations was a simple recomputation of the Bottke et al. (2002) integrations with similar initial conditions, using only a smaller timestep, in order to prove reproducibility. In fact, we obtained the precise initial conditions used by the previous integrations. Although chaotic dynamics would have in any case generated rapidly-diverging tra- jectories from the previous study, we decided to slightly perturb the initial orbits. To generate new initial conditions for our integrations, the value of Source Bottke et al. B02 NEOSSat-1.0 region (2002) Recomputation Model ν6 3519 3000 27000 3:1 2354 987 19740 IMC 2977 5568 11136 OMB 1964 1964 3928 Total 10814 11519 61804 Table 2.1: Number of test particles integrated for each source region for the Bottke et al. (2002) model, our B02 recomputation, and the NEOSSat-1.0 model. the mean anomaly for particles starting in the ν6 resonance, IMC popula- tion, and OMB population were re-randomized from that used by Bottke et al. (2002). For particles in the 3:1 mean-motion resonance, both the lon- gitude of perihelion and mean anomaly were changed, but in a way that 6A version of this chapter has been submitted for publication consideration. Green- street, S., Ngo, H., and Gladman, B. (2011) The Orbital Distribution of Near-Earth Objects Inside Earth’s Orbit. Icarus. 12 Chapter 2. Methods preserved the value of the resonant argument, keeping the particles in the resonance. Table 2.1 shows the number of particles integrated for each of the four asteroidal source regions for the Bottke et al. (2002) model, our Bottke et al. (2002) (B02) recomputation, and the NEOSSat-1.0 NEO orbital model (discussed in the next Section). The total number of test particles integrated by Bottke et al. (2002) and for our B02 recomputation are comparable, though we integrated less than half as many particles initialized in the 3:1 resonance and twice as many in the IMC source region. For the initial orbital elements of particles originating in each source region see Bottke et al. (2002) Section 2 and their Figure 1. These source regions are also shown schematically in Figure 2.1. The N-body code SWIFT-RMVS4 was used to perform this computa- tion7 with a default timestep of ≈ 6 hours. This tiny timestep allowed high- speed encounters with Earth and Venus to be correctly resolved. A record of the orbital elements of all test particles was logged every 1000 years. The gravitational perturbations of Venus-Neptune were included. Mercury was not present in these integrations for the purpose of the recomputation. Once a test particle reached the NEO region it was tracked until it either struck the Sun, struck a planet, travelled outside 19 AU from the Sun, or the final integration time was reached. Test particles were monitored for 150 Myr in each asteroidal source region. Construction of the steady-state NEO orbital distribution was the pri- mary goal. This is expressed via a grid of a, e, i cells throughout the inner Solar System covering a < 4.2 AU, e < 1.0, and i < 90◦ with volume 0.05 AU x 0.02 x 2.00◦. This is 12.5 times the resolution used by Bottke et al. (2002). The cumulative time spent by all particles in each cell was normal- ized to the total time spent by all particles in all cells of the q < 1.3 AU and a < 4.2 AU NEO region. This determines the steady-state NEO orbital distribution supplied by each source region. These residence time probabil- ity distributions, Rx(a, e, i), can be interpreted as the steady-state fraction of NEOs from each source region x in that cell. Each source region is then weighted by the best-fit parameters8 of Bottke et al. (2002) to make the 7The rmvs4 variant of the SWIFT package (Levison 2008, private communication) maintains the planetary positions to be the same regardless of timestep changes induced by test-particle proximity. 8αν6 = 0.37, αIMC = 0.27, α3:1 = 0.20, αOMB = 0.10, αJFC = 0.06 13 Chapter 2. Methods Figure 2.1: NEO class distinctions and source regions in a,e space. We restrict the NEO population 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) we adopt in this paper. The ν6 secular resonance (red), 3:1 mean-motion resonance (red), intermediate Mars crossers (IMC) (green), and the outer main-belt (OMB) (green) population constitute the aster- oidal source regions and the Jupiter family comets (JFCs) (green) are the cometary source region used in the B02 recomputation and the NEOSSat-1.0 NEO orbital integrations. 14 Chapter 2. Methods combined residence time distribution, RNEO(a, e, i), of the NEO region. RNEO(a, e, i) = αν6Rν6 +αIMCRIMC +α3:1R3:1+αOMBROMB +αJFCRJFC (2.1) Though no particles of cometary origin were integrated for this recomputa- tion, RJFC(a, e, i) from Bottke et al. (2002) was available and used. Bottke et al. B02 NEOSSat-1.0 (2002) Recomputation Model NEO Best Best Fractional Best Fractional Class Estimate Estimate Error Estimate Error (%) (%) (%) Amor 31± 1 32.0± 0.1 0.003 30.1± 0.8 0.025 Apollo 61± 1 57.7± 3.0 0.053 63.3± 0.4 0.007 Aten 6± 1 5.5± 1.5 0.28 5.0± 0.3 0.05 Atira −− 3.9± 1.2 0.31 1.38± 0.04 0.03 Vatira −− 0.9± 0.4 0.47 0.22± 0.03 0.15 Vulcanoid −− 0.0± 0.0 −− 0.0± 0.0 −− IEO 2± 0 4.8± 1.6 0.34 1.6± 0.1 0.05 Table 2.2: NEO class percentages and fractional errors for both our B02 recomputation and the NEOSSat-1.0 NEO model. An Atira has 0.718 < Q < 0.983 AU while a Vatira is a Venus-decoupled object with 0.307 < Q < 0.718 AU. We define Vulcanoids as objects with Q < 0.307 AU. The uncertainties in our percent class estimates and the resulting fractional errors are computed using the even/odd splits of the residence time distributions. For comparison, the NEO class percentages computed by Bottke et al. (2002) and renormalized to 100% are also given. Bottke et al. (2002) combine all Q < 0.983 AU objects into the IEO population. The IEO class percentages for our B02 recomputation and the NEOSSat-1.0 NEO model are simply the sum of the Atira, Vatira, and Vulcanoid class percentages. The best estimates for our B02 recomputation and the NEOSSat-1.0 model agree (within our estimated errors) with the Bottke et al. (2002) NEO class best estimates. Figure 2.2 shows two projections of RNEO(a, e, i) for our B02 recom- putation. In order to estimate how much of this residence time is due to small-number stastistics fluctuations, some of which are due to single parti- cles, two residence time distributions were computed by splitting the even- and odd-numbered particle contributions of the asteroidal source regions. The test particles were numbered via the sequential incrementation of the 15 Chapter 2. Methods I n c l i n a t i o n  i  [ d e g ] Semi-major axis a [AU] 0 10 20 30 40 50 60 70 80 90 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.05 0.1 0.15 0.2 0.25 0.3 E c c e n t r i c i t y  e Semi-major axis a [AU] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 recomputation B02 E c c e n t r i c i t y  e 0 0.05 0.1 0.15 0.2 Figure 2.2: Residence time probability distribution, RNEO(a, e, i), from our recomputation of the Bottke et al. (2002) integrations using a smaller timestep. To monitor the orbital evolution of each particle, a grid of a, e, i cells was placed throughout the inner Solar System from a < 4.2 AU, e < 1.0, and i < 90◦ with volume 0.05 AU x 0.02 x 2.00◦ (Bottke et al., 2002). To create the a, e plot the i bins are summed and the e bins are summed to create the a, i plot. The color scheme represents the percentage of the steady-state NEO population contained in each bin. Red colors represent cells where there is a high probability of particles spending their time. The curved lines divide the NEO region into Amor, Apollo, Aten, and Atira populations as well as indicate Venus and Mercury crossing orbits. 16 Chapter 2. Methods E c c e n t r i c i t y  e Semi-major axis a [AU] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 recomputation B02 even-numbered particles E c c e n t r i c i t y  e 0 0.05 0.1 0.15 0.2 E c c e n t r i c i t y  e Semi-major axis a [AU] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 recomputation B02 odd-numbered particles E c c e n t r i c i t y  e 0 0.05 0.1 0.15 0.2 Figure 2.3: In order to determine how much of the residence time proba- bility distribution displayed in Figure 2.2 is due to small-number stastistics fluctuations, some of which are due to single particles, the residence time was computed by splitting the even- and odd-numbered particle contributions of the asteroidal source regions. Two projections of the resultingRNEO(a, e, i) are shown here. The even-numbered particle contribution is shown in the top plot while the odd-numbered particle contribution is shown in the bottom plot. The discrepancies between these two plots show the potential impor- tance of small-number statistics in our B02 recomputation. Particularly, the a < 1.0 AU regions show large discrepencies in the NEO population. The fractions of NEOs in each class vary by a factor of 2-3 between the even and odd particle splits. 17 Chapter 2. Methods initial conditions for each source region; thus, splitting the even- and odd- numbered particle contributions gives two comparable NEO distributions with very similar initial conditions. The resulting residence time probabil- ity distributions are shown in Figure 2.3. The discrepancies between these two plots give estimates of the importance of small-number statistics in our B02 recomputation. Particularly, the a < 1.0 AU regions of Figure 2.3 show large discrepencies in the NEO populations, varying by factors of 2-3. Table 2.2 shows that the best estimates for the NEO class fractions for our B02 recomputation agree (within our estimated errors) with the conclusions of Bottke et al. (2002). Note that Bottke et al. (2002) combined all Q < 0.983 AU objects into the IEO population (Table 2.2) while we split this region into three subpopulations: Atiras (0.718 < Q < 0.983 AU), Vatiras (0.307 < Q < 0.718 AU) which are objects decoupled from Venus, and Vulcanoids which we require to have orbits with Q < 0.307 AU (the perihelion of Mercury). For a graphical representation, Figure 2.1 shows the NEO class boundaries we adopt in this paper. The IEO class percentage given for the B02 recomputation and the NEOSSat-1.0 model are simply the sum of the Atira, Vatira, and Vulcanoid class percentages. Vatiras and Vulcanoids are discussed in greater detail in Section 3.1. The best estimates given in Table 2.2 are an average of the even/odd split NEO class fractions. The percent errors given for the best estimates are half of the difference in the NEO class fractions for the even/odd splits of RNEO(a, e, i)9. The fractional errors given in Table 2.2 are percent errors for each NEO class divided by the fractional best estimate10. The Bottke et al. (2002) uncertainties were originally computed by varying their best-fit parameters for the source regions within their estimated uncertainties and recomputing the NEO class fractions (Bottke 2011, private communication), but this neglects the uncertainties due to small-number statistics in the integrations themselves. The fractional errors of 0.3 – 0.5 for the a < 1.0 AU region show the potential importance of small-number statistics in the B02 recomputation. Since this is the region of phase space NEOSSat will be focusing its observations, we were motivated to compute a new set of numerical integrations to model the steady-state distribution of the NEO population with better statistics in the a < 1.0 AU region. 9For example, the even split of the B02 recomputation gives Atens= 7.0% of the NEO population while the odd split gives Atens= 4.0%. Therefore, the percent error is (7.0%− 4.0%)/2 or 1.5%. 10For example, the 1.5% error given for the Aten estimate of the B02 recomputation divided by 5.5% (the best estimate) is 0.27, so the relative fractional uncertainty is about 1 part in 4. 18 Chapter 2. Methods 2.2 NEOSSat-1.0 NEO orbital model For the numerical integrations of the NEOSSat-1.0 orbital model, we used SWIFT-RMVS4 with a base timestep of ≈ 4 hours. Note that the code adaptively reduces the timestep down to ≈ 10 minutes in deep planetary encounters. Due to computational limitations a decade ago, Bottke et al. (2002) were unable to consider including the gravitational effects of Mercury in their integrations. The 4-hour timestep allows us to resolve even the highest-speed Mercury encounters in our computation. The inclusion of Mercury is important in our efforts to better model the a < 1.0 AU NEO population. We include the gravitational perturbations of Mercury-Saturn. Only point-mass Newtonian gravity is included. The orbital elements of all particles were recorded every 300 years. Particles were tracked from their starting points in one of the source regions until they reached one of the following sinks: hit the Sun, hit a planet, reached transuranian space (Q > 19 AU), or the final integration time was reached. Almost no particles remained with q < 1.3 AU at the end of the integrations. The same perturbed initial condition algorithm for each asteroidal source region used in the B02 recomputation was used, but roughly six times as many particles were tracked (Table 2.1). 27000 particles starting in the ν6 secular resonance were followed for 200 Myr. 19740 particles starting in the 3:1 mean-motion resonance were followed for 200 Myr. 11136 particles starting in the IMC population were followed for 150 Myr and 3928 particles starting in the OMB population were followed for 100 Myr. The variation in total integration time for the 4 source regions reflects the need to integrate until > 99% of the q < 1.3 AU particles were eliminated. Altogether, these integrations took more than 300 core years of computation to complete using the fastest cores available in 2009. The output of the entire integration set is available upon request. The same bins (a < 4.2 AU, e < 1.0, and i < 90◦ with volume 0.05 AU x 0.02 x 2.00◦) were used to express RNEO(a, e, i) for the NEOSSat- 1.0 model as our B02 recomputation. We continue to use RJFC(a, e, i) computed by Bottke et al. (2002) to compute RNEO(a, e, i). Figure 2.4 shows the combined residence time distribution of the five source regions, again using the best-fit weightings of Bottke et al. (2002). The particle evolutions do not show surprises compared to previous in- tegrations, except for the rare outcomes mentioned in Section 3.1. Largely, particles which enter the q < 1.3 AU NEO region scatter gravitationally off the planets and migrate around in (a, e, i) space. Most are removed quickly when they encounter a strong resonance (usually the ν6 or 3:1) and end in 19 Chapter 2. Methods I n c l i n a t i o n  i  [ d e g ] Semi-major axis a [AU] 0 10 20 30 40 50 60 70 80 90 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.05 0.1 0.15 0.2 0.25 0.3 E c c e n t r i c i t y  e Semi-major axis a [AU] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 NEOSSat-1.0 model E c c e n t r i c i t y  e 0 0.05 0.1 0.15 0.2 Figure 2.4: Residence time probability distribution, RNEO(a, e, i), for the NEOSSat-1.0 NEO orbital model. This figure is constructed the same as Figure 2.2. 20 Chapter 2. Methods a Sun-grazing state. These paths to high-e via the ν6 and 3:1 resonances can be seen in Figure 2.4 at a≃ 2 AU and a≃ 2.5 AU, respectively. The median eccentricity of objects in the a > 2 AU NEO region is e≃ 0.5 and rises slightly at lower a (Figure 2.4). In particular, this produces a concen- tration of Mercury-crossing Apollos at a≃ 1.2 AU. The median inclination for NEOs with a > 2 AU is ≃ 15◦ with slight deficits at a≃ 2 AU and a≃ 2.5 AU where the ν6 and 3:1 resonances do not allow objects to increase much in inclination before being pushed into the Sun. The median inclination rises to > 25◦ at a < 1.5 AU. Vatira Atira Aten Apollo Amor % ν6 81.3± 7.5 61.5± 0.5 52.1± 1.8 38.5± 0.2 30.0± 0.2 % 3:1 9.6± 4.3 15.3± 0.2 20.7± 2.5 21.3± 0.5 17.5± 0.5 % IMC 9.1± 3.3 23.2± 0.7 27.2± 0.7 24.5± 0.2 32.4± 0.8 % OMB 0 0 0 9.8± 0.9 12.7± 1.6 % JFC 0 0 0 6.0 7.3 % Earth Impact −− −− 20.3± 0.9 79.7± 0.9 −− % Venus Impact −− 16.5± 0.1 19.8± 0.6 63.7± 0.5 −− % Mercury Impact 3.2± 0.9 13.1± 0.2 26.4± 0.2 57.3± 0.9 −− Table 2.3: The top portion gives the fraction of RNEO(a, e, i) from each source region for each NEO class. Note the increasing relative importance of the ν6 source for NEO classes nearer the Sun. The bottom portion gives the planet crossing impact rate contributions from NEO classes (Sec. 3.2). This can be interpreted as Atens contribute 20.3% of the Earth crossing impact rate and Apollos contribute 79.7%, etc. The uncertainties for both portions are computed using the even/odd particle splits of the residence time dis- tribution, expect for the JFC population where an uncertainty estimate is unavailable. The NEOSSat-1.0 model predicts Amors to be 30.0±0.8% of the steady- state NEO population and Apollos to be 63.3 ± 0.4% of the population (Table 2.2). These best estimates agree (within our estimated uncertainties) with those of Bottke et al. (2002). In addition, the NEOSSat-1.0 model also allows the computation of the fraction of RNEO(a, e, i) from each source region for each NEO class (top portion of Table 2.3). For example, 21% of the Apollos come from the 3:1 resonance. Note that RNEO(a, e, i) gives the relative NEO class fractions, not the absolute NEO population. To order of magnitude, given there are ∼ 1000 NEOs with H < 18 (Bottke et al., 2002; Stuart, 2001), these estimates imply there should be of order 300 Amors and 600 Apollos of this size at any time, with more at smaller diameters. 21 Chapter 2. Methods For H < 21, there are ∼ 20,000 NEOs, implying of order 6,000 Amors and 13,000 Apollos exist of this size at any time. The H-magnitude (absolute magnitude) of a Solar System object is de- fined to be the apparent magnitude that object would have if it were located in an equilateral triangle 1 AU from the Sun and the Earth (observer) and at zero phase angle. The phase angle refers to the angle between the in- cident and reflected light of an observed object. The phase angle ranges from 0◦ to 180◦, where a 0◦ phase angle refers to a fully illuminated ob- ject. H-magnitude can be used to estimate the diameter of an object. The relationship between H-magnitude and diameter involves the albedo (re- flectivity) of an object. The albedo refers to the fraction of incident light reflected from the surface of an object. An albedo of 0 means an object reflects no light and a value of 1 means an objects reflects all light incident upon it. The typical albedo range of NEOs is 0.05–0.25. The diameter of an object can be calculated from the H-magnitude and albedo by: D = 1329km√ A 10−0.2H (2.2) An NEO with H∼18 has a diameter range of roughly 0.67 km to 1.5 km for the estimated albedo range. NEOSSat’s observations will be optimized to maximize a < 1.0 AU de- tections, particularly seeking Atira-class asteroids. Figure 2.5 shows two projections of RNEO(a, e, i) rescaled for a < 1.0 AU from the NEOSSat- 1.0 orbital model. The median eccentricity for Atens, Atiras, and Vatiras is ≃ 0.4. Figure 2.4 shows that the median inclination climbs steadily from ≃ 10◦ at a≃ 2 AU to ≃ 30◦ at a≃ 1.2 AU and drops to ≃ 25◦ for Atens, Ati- ras, and Vatiras (Figure 2.5). The median inclination uncertainty is about a degree, based on the inclination bin size of RNEO(a, e, i) and the even/odd particle split. Figure 2.5 shows clear population drop-offs between Atens and Atiras as well as between Atiras and Vatiras. The typical path of NEAs moving from the main-belt to a < 1.0 AU orbits occurs via planetary close encounters and resonances. Once objects enter Earth-crossing space via resonances they often approximately begin following lines of constant Tisserand parameter roughly parallel to q≃ 1 AU until planetary close encounters put them onto a < 1.0 AU orbits (Michel et al., 2000). If NEAs succeed in reaching the Aten and Atira regions, their dynamical lifetimes from that instant then increases due to their orbital distance from the resonances which can push them into the Sun (Gladman et al., 2000). Protection mechanisms, such as 22 Chapter 2. Methods I n c l i n a t i o n  i  [ d e g ] Semi-major axis a [AU] 0 10 20 30 40 50 60 70 80 90 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 E c c e n t r i c i t y  e Semi-major axis a [AU] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 E c c e n t r i c i t y  e 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Figure 2.5: Same as Figure 2.4 except displaying only the a < 1.0 AU region of the NEOSSat-1.0 model. The right edge of the plots mark the Aten/Apollo boundary. Clear population drop-offs are visible at Q = 0.983 AU (aphelion at Earth, to the right of which are Atens) and Q = 0.718 AU (aphelion at Venus, to the left of which are Vatiras, with Atiras occupying the region between the curved lines). 23 Chapter 2. Methods Kozai, also increase their lifetimes while on a < 1.0 AU orbits (Michel & Thomas, 1996; Michel et al., 2000). Table 2.3 makes it clear that the NEO classes have different contribu- tions from the five source regions. The OMB and JFC sources supply es- sentially no objects to the a < 1 AU region. It is also apparent that the inner belt (ν6 and IMC) sources become progressively more dominant for NEO classes with lower and lower semimajor axis. For example, the ν6 source contributes nearly two thirds of the Atira asteroids but only about one third of the Apollos. Because the source regions are likely feeding as- teroids of different spectral types into the NEO population, a difference in average colors (for example) between Atens and Apollos does not necessarily imply an evolutionary effect with time. The NEOSSat-1.0 orbital model predicts Atens to be 5.0 ± 0.3% of the steady-state NEO population, Atiras to be 1.38 ± 0.04%, and Vatiras to be 0.22 ± 0.03% of the population (Table 2.2). These estimates are smaller than those computed for the B02 recomputation, but agree within the larger estimated errors of the latter. The uncertainty for these predictions are significantly lower for the NEOSSat-1.0 model. In addition to computing source region fractions for each NEO class, the fraction of RNEO(a, e, i) from each source region can also be done at an individual cell level, to determine the likelihood that an NEO came from a given source region. For example, the cell containing NEO 163693 Atira (a = 0.741 AU, e = 0.322, i = 25.6◦) has 55 ± 6% of RNEO(a, e, i) coming from the ν6 resonance, 22 ± 6% from the 3:1 resonance, 24 ± 6% from the IMC population and 0% from the OMB and JFC populations. This can be interepreted as Atira having a ≃ 55% chance of originating in the ν6 resonance, etc. To determine the variation in this estimate, the mean fraction of RNEO(a, e, i) coming from each source was found by moving up and down one cell each in a, e, and i. The uncertainties given are the standard deviation of this variation. This source region mapper for the NEOSSat-1.0 model is available upon request. To convert from relative NEO class estimates to absolute values, given there are ∼ 1000 NEOs with H < 18, these estimates imply there should be of order 50 Atens, 15 Atiras, and 2 Vatiras of this size at any time, with more at smaller diameters. For H < 21, there are ∼ 20,000 NEOs, implying of order 1,000 Atens, 275 Atiras, and 40 Vatiras exist of this size at any time. 2.2.1 Accuracy of the NEOSSat-1.0 NEO orbital model Our usage of roughly six times as many particles as the Bottke et al. (2002) integrations allows finer resolution. To graphically illustrate the reduced 24 Chapter 2. Methods importance of small-number statistics, we split the NEOSSat-1.0 orbital in- tegrations into even- and odd-numbered particle contributions in Figure 2.6. The discrepancies between these two plots are clearly now much smaller, es- pecially in the a < 1.0 AU region, compared to theRNEO(a, e, i) projections shown in Figure 2.3, as would be expected given the small fractional errors. Estimates (Table 2.2) for the steady-state NEO class percentages for both the NEOSSat-1.0 NEO orbital model and our B02 recomputation are all within our estimated uncertainties for both. The new NEO class contri- butions are within twice our estimated uncertainties of the original Bottke et al. (2002) values, but since we are using their same weighting factors this may not be surprising. The fractional error estimates for the Apollo popula- tion is now a factor of eight smaller. We did not refit our integrations to the Spacewatch data because our analysis above shows that in the Apollo and Amor regions, which dominate the Spacewatch detections, the NEOSSat-1.0 and Bottke et al. (2002) models are in agreement, and thus the weightings will not change. It would be more beneficial to fit our integrations to a new NEO survey, such as the NEOWISE data (Mainzer et al., 2011). The fractional errors make it clear we succeeded in decreasing the uncertainty in the fractional populations of the a < 1.0 AU objects by a factor of more than five. The number of Atens and Atiras for our B02 recomputation var- ied by a factor of 2-3 between the even- and odd-numbered particle splits. However, the fractional errors of these two populations for the NEOSSat- 1.0 model are now less than 0.10. Since this is the region NEOSSat will be focusing its observations, we are confident that the NEOSSat-1.0 orbital model gives a better representation of the NEA steady-state distribution for the Aten and Atira regions than previous models. The variation in the population of Mercury-crossing Apollos at a≃ 1.2 AU in the NEOSSat-1.0 model (Figure 2.6) is much smaller than before and thus this feature is a robust property of the NEO distribution. 2.3 Caveats Two possible issues with the NEOSSat-1.0 NEO orbital model which may affect the interpretation of the model include degeneracy between source regions and the exclusion of non-gravitational effects in the integrations. 2.3.1 Degeneracy between source regions One possible issue with the weightings found by Bottke et al. (2002) and used for the NEOSSat-1.0 NEO model integrations is the degeneracy between the 25 Chapter 2. Methods E c c e n t r i c i t y  e Semi-major axis a [AU] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 3.5 4 NEOSSat-1.0 model even-numbered particles E c c e n t r i c i t y  e 0 0.05 0.1 0.15 0.2 E c c e n t r i c i t y  e Semi-major axis a [AU] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 3.5 4 NEOSSat-1.0 model odd-numbered particles E c c e n t r i c i t y  e 0 0.05 0.1 0.15 0.2 Figure 2.6: Same as Figure 2.3 except computed for the NEOSSat-1.0 model. The discrepancies between these two plots are clearly much smaller than in our B02 recomputation (Figure 2.3). 26 Chapter 2. Methods ν6 resonance and IMC populations. These two source regions produce simi- lar residence time probability distributions because they partially overlap in phase space. Bottke et al. (2002) recognized this degeneracy and attempted to avoid the problem by integrating a large enough set of test bodies within the clearly defined boundaries of the IMC and ν6 resonance source regions. It is also unclear if the IMC population is correctly bounded and accurately populated in the inital conditions of the integrations. To ensure their best- fit parameters for each source region were as accurate as possible, Bottke et al. (2002) computed the flux rate of material feeding each of the source regions using the number of objects in the steady-state population of each source region and the mean lifetime spent by objects in each source region from their integrations. These flux rates are important in understanding the steady-state populations of both the source regions and the NEOs they supply (Bottke et al., 2002). We accept the method used by Bottke et al. (2002) for computing their weighting factors is sufficient in decreasing the degeneracy problem. To better address the issue, the NEOSSat-1.0 NEO orbital model should be fit to a new NEO survey, such as the NEOWISE data (Mainzer et al., 2011) and the population of the IMC region checked against the refitting. The roughly six times increase in the number of parti- cles integrated in the NEOSSat-1.0 NEO orbital model should combat this degeneracy problem further. 2.3.2 Gravitational effects only included The NEOSSat-1.0 orbital integrations include only point-mass Newtonian gravitational effects. We believe the inclusion of other physical effects will change RNEO(a, e, i) by less than the current statistical uncertainties. The RNEO(a, e, i) computations assume the NEA size distribution does not vary over orbital element space, ignoring the possible effects of collisions, tidal disruptions, or thermal breakup. Collisions (Farinella et al., 1993) and chaotic dissipation (Carruba et al., 2003; Morbidelli & Nesvorný, 1999) may play a role in the supply mechanisms from the main-belt but are of little importance for km-scale NEAs in the NEO region. The effect of Yarkovsky drift is believed to cause main-belt asteroids to migrate to either larger or smaller a, allowing them to drift into the supply resonances (Bottke et al., 2000b; Farinella & Vokrouhlický, 1999), but again its effect on the large-scale NEO orbital distribution with q < 1.3 AU should be minor. 27 Chapter 3 Results 3.1 New populations discovered within the NEOSSat-1.0 NEO orbital model 11Analysis of the NEOSSat-1.0 orbital model showed the presence of two new rare types of NEAs, which have little or not at all been discussed in the literature. 3.1.1 Population of objects decoupled from Venus Inspection of RNEO(a, e, i) revealed the existence of NEAs with orbits entirely inside that of Venus. While it is obviously plausible that a Venus encounter could pull an Aten orbit’s aphelion below 0.983 AU and thus decouple an NEO from the Earth to become an Atira, a similar process for decoupling from Venus has not been discussed. In the a < 1.0 AU region of RNEO(a, e, i) (Figure 2.5), there are clear population drop-offs at Q = 0.983 AU (aphelion at Earth) and Q = 0.718 AU (aphelion at Venus). The 0.718 < Q < 0.983 AU population are the Atira class NEOs. We have provisionally12 named objects with 0.307 < Q < 0.718 AU Vatiras, because they are Atiras which are decoupled from Venus. The Q > 0.307 AU cut is to distinguish Vatiras from the Mercury-decoupled Vulcanoid population, which we define as objects with Q < 0.307 AU. Figure 2.1 shows the NEO class boundaries we adopt in this paper. The Vatira population is rare and makes up only ≃ 0.22% of the steady-state NEO orbital distribution (Table 2.2). Most of the particles reaching the Vatira region spend little time in this state integrated over their lifetimes, ∼ 0.25 Myr. Particles reach the Vatira region via gravitational scattering in (a, e, i) space due to planetary close 11A version of this chapter has been submitted for publication consideration. Green- street, S., Ngo, H., and Gladman, B. (2011) The Orbital Distribution of Near-Earth Objects Inside Earth’s Orbit. Icarus. 12Provisional because it will be abandoned once the first discovered member of this class will be named. 28 Chapter 3. Results encounters with Earth, Venus, and Mercury. Objects in the Vatira region typically have i≃ 25◦ and e≃ 0.4 (Figure 2.5). Only a few examples exist in our integrations of particles long-lived (> 1 Myr) in the Vatira region. The orbital evolution of one long-lived Vatira is shown in Figure 3.1. This particle comes out of the ν6 resonance and rapidly drops to the Vatira state within ≈ 10 Myr via a set of Earth and Venus encounters; reaching the Vatira state this quickly is rare. About 5 Myr later, it experiences a close encounter which recouples it to Venus. It then repeats this short decouple/recouple evolution before fully decoupling from Venus at around 45 Myr into its lifetime. This decouple/recouple process is dominated by Mercury close encounters. During this evolution, the particle’s inclination reaches ≃ 40◦. It then remains in the Vatira state for ≈ 75 Myr during which time it random walks in a due to planetary close encounters with Mercury before colliding with Mercury at ≈ 125 Myr. At termination, i≃ 80◦ and at terminal impact is at a speed of ≃ 61 km/s. Figure 3.2 shows a second example of a long-lived Vatira. This particle came out of the ν6 resonance and spends most of the first 80 Myr as an Apollo. Plantery close encounters then lower a below that of Venus and it spends ≈ 30 Myr oscillating between the Aten and Atira classes due to its aphelion oscillating quickly. This rapid oscillation is due to Kozai which occurs when the argument of pericenter librates around either 90◦ or 270◦ instead of precessing (Kozai, 1962). For orbits with a < 2 AU, Michel & Thomas (1996) showed the argument of pericenter can also librate around 180◦. By 115 Myr the particle’s a drops further due to Mercury encounters and it oscillates between the Vatira and Atira classes. Though the particle shown in Figure 3.2 appears to stay coupled to Venus after ≈ 110 Myr since its aphelion oscillates near 0.718 AU, the nodal distances are never outside 0.718 AU for the remainder of the particle integration because Kozai pre- vents the node from approaching the aphelion distance. It stays alive for another ≃ 120 Myr due to the high inclinations (between roughly 30◦ and 80◦) reached during its Kozai oscillations, putting it out of harm’s way of a planetary collision with Venus, but is pushed into the Sun 228 Myr after leaving the main-belt. These long-lived Vatiras are rare. With typical lifetimes in the Vatira region of a few hundred thousand years, most NEAs entering the Vatira region do so several times as they experience planetary close encounters which decouple and recouple the particle from Venus before evolving back out in a. The median time after injection into the NEA region at which objects first enter the Vatira region is ≈ 17 Myr. The median lifetime of objects after their first injection into the Vatira region is ≈ 21 Myr. 42% of 29 Chapter 3. Results Figure 3.1: This figure shows the aphelion (red), semimajor axis (black), and perihelion (green) versus time of a particle from the NEOSSat-1.0 orbital integrations. This particle rapidly reaches the Vatira state and remains there for the last ≈ 75 Myr, ≈ 60% of its lifetime. It is a rare occurance for a particle to reach the Vatira state so quickly. This particle strikes Mercury with an impact speed of ≃ 61 km/s at ≃ 125 Myr. 30 Chapter 3. Results Figure 3.2: Same as Figure 3.1 except of a particle from the NEOSSat- 1.0 orbital integrations which originates in the ν6 resonance, spending the majority of the first 80 Myr as an Apollo. Close encounters move it to an Aten orbit. Its aphelion then begins oscillating quickly due to Kozai as it fluctuates between the Aten and Atira states. Roughly 30 Myr later, the a drops further due to a planetary close encounter causing it to oscillate between the Atira and Vatira classes. This particle lives long due to the high inclinations (between roughly 30◦ and 80◦) during its Kozai oscillations putting it out of harm’s way of a planetary collision with Venus. It becomes a Sun-grazer 228 Myr after leaving the main-belt. 31 Chapter 3. Results these objects are terminated because they don’t stay decoupled from Venus and eventually collide with Venus. 16% collide with Earth, 10% collide with Mercury, and ≃ 1% collide with Mars. 20% evolve onto Sun-grazing orbits, ≃ 9% are still alive at the end of the integration, ≃ 3% are thrown out of the Solar System, and ≃ 1% hit the giant planets. The ≈ 21 Myr time scale is longer than a typical Apollo due to the orbital distance from resonances which can push them onto Sun-grazing orbits (Gladman et al., 2000). Objects in the Vatira region are nearly unobservable from Earth’s surface because they never reach solar elongations > 45◦. The Messenger spacecraft could observe them from its position in orbit around Mercury, but given the small-number of these objects that exist, we calculate the probability of Messenger finding a Vatira is extremely small given the aperture and field of view of its imaging system. During rare “eclipse seasons” where Earth’s dark limb blocks the Sun, NEOSSat will be able to image this close to the Sun, but again a detection is extremely unlikely. While one may think this population of objects could affect the impact chronology on Mercury, we calculate that only ≃ 3.2% of the Mercury impact rate comes from Q < 0.718 AU orbits (Table 2.3). 3.1.2 Vulcanoids The existence of a non-negligible population of Venus-decoupled Vatiras thus begs the question as to whether any objects reach orbits entirely interior to that of Mercury. Accepted convention would likely to be to call such an object a Vulcanoid, although the term is usually intended to mean an object which has been resident inside Mercury for the entire lifetime of the Solar System (Stern & Durda, 2000). The most obvious production path would be a Vatira or Atira NEO that suffers a close enounter with Mercury which converts the NEO’s orbit to one with aphelion at close to the encounter point. With no planet interior to Mercury, a close encounter cannot be then used to decouple from the planet, but the object could interact with the border of a mean-motion resonance with Mercury (or even high-order resonances of more-distant Venus or Earth) and have its eccentricity reduced. With the Vatira evolutions as example, it is also clear that a large-i Mercury crosser could temporarily decouple as part of a Kozai oscillation. We examined the NEOSSat-1.0 integrations to search for the appearance of Vulcanoids. Although we found several examples of particles reaching or- bits with aphelia Q smaller than Mercury’s aphelion, there were no particles logged with Q < qMerc = 0.307 AU, which we have adopted as the defini- tion of a Vulcanoid (an orbit entirely inside that of Mercury). The smallest 32 Chapter 3. Results observed Q was Q = 0.348 AU, and the smallest NEO semimajor axis ob- served was a = 0.344 AU for the same particle, an orbit entirely inside Mercury’s semimajor axis but not perihelion. The fraction of RNEO(a, e, i) with Q < QMerc is 0.006%, so we take this as a strong upper limit to the possible value of the Vulcanoid fraction with Q < 0.307 AU. Given this we would not expect a Vulcanoid from an NEO source to exist larger than H≃21 on average. If a Mercury-decoupled object is ever found, the possibility should be considered that it is a NEO brought down through the Atira and Vatira regions. The future lifetime of such an object will be limited by Yarkovsky drift (Vokrouhlický et al., 2000) and collisional grinding, but survival of 100-meter scale NEAs for hundreds of Myr is plausible. 3.1.3 Retrograde NEAs Further analysis of the NEOSSat-1.0 integrations uncovered the unexpected production of retrograde orbits from main-belt asteroid sources. We find that NEAs originating in any of the four asteroidal source regions can even- tually become retrograde, and typically begin their evolution in the q < 1.3 AU region random walking in a due to planetary close encounters. The ma- jority of particles which become retrograde (regardless of initial source) do so via the 3:1 mean-motion resonance after they reach it by a random walk in a. Once in the resonance, they often experience Kozai oscillations in e and i which can pump their inclinations up to 80◦. It is clear that Kozai alone does NOT result in the inclination passing through 90◦, because only a tiny fraction (if any) become retrograde outside a resonance even if very high i’s are reached. A dynamical phenomenon in the resonance then causes the in- clinations to pass through 90◦; the nature of this mechanism is unclear, but it is not planetary close encounters. About 86% of the retrograde particles stay in the resonance and terminate within 10,000 years when the resonance pushes the high-e particles into the Sun. 98% of the retrograde NEAs are eventually eliminated from the integrations due to Sun-grazing (Farinella et al., 1994), with only ≃ 2% thrown out of the Solar System. Because of their high relative encounter speeds caused by the retrograde orbit, gravi- tational focussing is negligible and planetary collisions are rare (< 1%). A minority are kicked out of the retrograde-inducing resonance due to a plan- etary close encounter, and then can in some cases live tens of millions of years. The details of this mechanism are the subject of another paper; here we provide only the orbital-element distribution of the retrograde NEAs. Figure 3.3 shows R180(a, e, i) for the region a < 4.2 AU, e < 1.0, and 33 Chapter 3. Results Figure 3.3: Residence time probability distribution, RNEO(a, e, i), for inclinations up to 180◦ for the NEOSSat-1.0 NEO orbital model. The color scheme represents the logarithm of relative density of residence time spent in any given cell in relation to the amount of residence time spent in all cells. The dashed line divides retrograde from direct orbits. The retrograde NEA population makes up ≃ 0.10% of the steady-state NEO population. Two known retrograde NEOs are shown. 34 Chapter 3. Results i < 180◦ (with cell volume 0.05 AU x 0.02 x 2.00◦). This figure shows the logarithm of the normalized fraction of time spent by particles in each cell. A total of ≃ 0.10% of R180(a, e, i) is in the retrograde NEA population. Although most of the retrograde objects flip while in the 3:1 resonance, R180(a, e, i) shows most of the power for the retrograde objects near a≃ 2 AU. This is due to a single particle which flips in the 3:1 resonance early in its lifetime and then spends ≈ 200 Myr near 2 AU. There are two known retrograde NEAs: 2007 VA85 (a = 4.226 AU, e = 0.736, i = 131.769◦) and 2009 HC82 (a = 2.528 AU, e = 0.807, i = 154.519◦). These are plotted in Figure 3.3. 3.2 Impact speeds and rates for Earth, Venus, and Mercury Under the steady-state assumption, one can calculate the relative NEO im- pact rate onto Mercury, Venus, and Earth, as well as the normalized distri- bution of ‘top of the atmosphere’ impacts speeds. Our simulations cannot provide this information for Mars due to the existence of q > 1.3 AU Mars- crossing asteroids not modelled here. This calculation is best performed using the detailed orbital histories of all integrated particles. The intrinsic collision probability and average impact speed onto each of the terrestrial planets were computed by a method described by (Dones et al., 1999; Farinella & Davis, 1994; Wetherill, 1967). This method gives the expected number of impacts that should be recorded in the simulation, given the particle histories. This was first done separately for each particle in each of the four asteroidal source regions, since different numbers of particles were computed from each source. We find extremely good agreement between the expected number of impacts diagnosed by the collision probability code and the number of impacts directly recorded in the simulations13. This agreement gives us confidence in the impact speed distribution simultaneously derived from the collision probability algorithm. The impact speed distribution from each source region was normalized to the number of particles from that source which entered the q < 1.3 AU NEO region. Since each source region is weighted differently in its contribution to the overall NEO population, the resulting normalized impact speed probabilities were weighted by the source region fractions from Bottke 13 For example, from the ν6 source the expected number of impacts for Mer- cury/Venus/Earth was 106/571/519 while those directly observed in the simulation were 91/563/517; the Poisson errors of the latter are consistent with the former. 35 Chapter 3. Results et al. (2002). The resulting distributions were used to create the impact speed distributions for the terrestrial planets shown in Figure 3.4. This figure shows only the speed distribution resulting from asteroidal source regions (neglecting the JFCs). Figure 3.4: ‘Top of the atmosphere’ impact speed distribution for asteroidal material reaching Earth, Venus, and Mercury. The mean impact speed for Earth is 20.6 km/s, for Venus is 24.7 km/s, and for Mercury is 37.9 km/s. Each planet’s distribution is separetely normalized. As a check, the impact speed distributions were also calculated using the final residence time probability distributions. The orbital elements of each (a, e, i) cell of volume 0.05 AU x 0.02 x 2.00◦ were used to determine im- pact probabilities and mean speed, for the cell center. For each planet, the fractional residence time was multiplied by that cell’s collision probability. Since the residence time probability distributions are already normalized, the resulting impact speed distribution could be directly compared to Fig- ure 3.4. Because the residence time distribution has much less detailed information than the particle integrations, the former produces noisier, but otherwise similar, impact speed distributions. This allowed us to compare these distributions with and without the JFC source included in the steady- 36 Chapter 3. Results state orbital distribution. Unsurprisingly, because the JFC source is only ≃ 6% of the NEO population and the impact probability per JFC is small compared to most NEAs, the impact speed distributions are nearly identical with and without the JFC source included. We thus present only the as- teroidal impact speed distribution; workers wishing to include the cometary speed distribution should probably also include all other cometary sources (Halley-type and long-period comets as well). The NEA impact speed distribution onto Earth is extremely similar to that shown in Gallant et al. (2009) at lower speed resolution, with a strong peak near 15 km/sec and long tail out to 45 km/sec. We use 1 km/sec bins and thus have twice the speed resolution of the most recent work (Gallant et al., 2009). The venusian impact speeds (which should be interpreted as ‘top of the atmosphere’ speeds) peak only slightly faster (≈ 16 km/sec), with a tail extending to higher speeds (55 km/s) due to the greater heliocentric speeds in closer to the Sun. When converted to cumulative form, the venu- sian speed distribution is (perhaps amazingly) similar to that approximated by McKinnon et al. (1997). In the case of Venus the interaction of the im- pactor with the massive atmosphere of course has a critical further role in determining if the surface is reached. Marchi et al. (2009) published a Mercury impact speed distribution in comparison with the Earth-Moon system based on a fraction of the Bottke et al. (2002) simulations; one must compare our results with their large (L) distribution shown in their Figure 11 which has not been modified from the pure-gravity model. The importance of small-number statistics in the integrations is evident in the impact speed distribution of Marchi et al. (2009); although there is a broad distribution with maximum probability in the 40-50 km/sec range, the two narrow peaks (at ≃ 29 and 41 km/sec) in their distribution are almost certainly due to two single particles that get lodged in long-lived Mercury-crossing orbits; in comparison, our impact speed distribution for Mercury is much smoother. Recently Le Feuvre & Wieczorek (2011) produced an improved Mercury impact speed distribution using the complete Bottke et al. (2002) model, which retains the broad maximum at 40-50 km/sec but in which these two peaks are absent. In contrast, both of these earlier impact speed distributions lack the strong Mercury impact speed peak we see near 27 km/sec, that is a robust feature of the NEOSSat-1.0 calcluation, produced by the well-populated ‘finger’ near a≃ 1.2 AU visible in Figure 2.4. The much finer timestep and the presence of Mercury in our integrations means that our q < 0.45 AU region is more accurately defined; we feel confident in our result, especially because the Earth and Venus speed distributions (less sensitive to the larger timestep) 37 Chapter 3. Results agree well with previous estimates. We thus predict a Mercury impact flux more dominated by low-speed impactors than both these workers derived from the Bottke et al. (2002) integrations. In addition to the impact speed distributions, the NEOSSat-1.0 inte- grations also permit the computation of relative impact rates on Mercury, Venus, and Earth. While it would be most desirable to have crater forma- tion rates as a function of crater diameter, this depends not only on the speed distribution but also the size distribution of the impactors and the cratering scaling law used. We prefer to remain as close as possible to the direct measurements we can draw from the integrations; our impact speed distributions are available on request for workers who wish to assert their favored impactor size distribution and generate cratering-rate estimates. In- stead, we present only the relative ‘top of the atmosphere’ impact rate, for fixed size impactor. This requires the assumption that the NEO size distri- bution does not vary in different regions of orbital element space. This is almost certainly untrue for comets, which may split for low perihelia, and may also be false for asteroids. Asteroids on orbits with higher planetary encounter probablility may be prone to tidal disruption (Richardson et al., 1998) potentially altering the size distribution as a large object turns into many smaller ones. However, because comets are a minor fraction of the impact flux and because the rest of the literature also operates under the assumption that the NEA size distribution does not vary over orbital ele- ment space, we proceeded to compute impact rates. When we weight the source regions as described above, we find that the impact rate (per NEO per square kilometer of target) ratio is 1.13 ± 0.01 for Venus/Earth and 1.21 ± 0.04 for Mercury/Earth (Table 3.1). Le Feuvre & Wieczorek (2008, 2011) found these ratios to be 1.11 and 1.15 respectively. The roughly 5% higher Mercury impact rate we find would translate into younger surfaces on Mercury when dated via impact-cratering chronology (eg. Marchi et al. (2009)) if rate was the only factor; because we simultaneously also argue that mercurian impact speeds are slightly slower on average, the effect will be reduced. Note that the various source regions give different planetary impact flux ratios (Table 3.1). Due to their ability to more efficiently deliver long-lived asteroids to high-e orbits, the ν6 and 3:1 provide relatively more Venus and Mercury impactors than the other source regions. If future recalibration of the source fractions (α) occur the planetary impact ratios will also change accordingly. More interestingly, Table 2.3 listed the fraction of each planet’s im- pact rate which is contributed by NEOs of the various dynamical classes. 38 Chapter 3. Results Impact Ratio ν6 3:1 IMC OMB Weighted Steady Even Odd Venus/Earth 1.22 1.19 1.10 0.77 1.13 1.12±0.01 1.11 1.12 Mercury/Earth 1.40 1.31 1.04 0.78 1.21 1.26±0.04 1.22 1.30 Table 3.1: Impact flux ratios (per unit surface area of target) for Venus/Earth and Mercury/Earth for the individual asteroidal source inte- grations as well as their weighted average. The asteroidal steady-state ratios (computed from the residence time distribution) and the even/odd splits are given in order to estimate the uncertainties. The quantities measured via the integrations should be more accurate and thus we quote those as best estimates but use the uncertainties from the odd/even splits of RNEO(a, e, i). Due to geometrical factors and shorter orbital periods, the low-a classes always produce a proportionally larger contribution to each planet’s im- pact flux than their abolute number fraction would imply. For example, Atens make up only 5.0/(5.0+63.3)=7.4% of the Earth crossing popula- tion, but contribute 20.3% of the impact flux due to their higher collision probability; this fact is previously discussed in the literature (see, for ex- ample Bottke et al. (1994)). We can extend this analyis to even lower semimajor axes. Apollos/Atens/Atiras make up 86.3%/10.7%/3.0% of the Venus-crossing orbits, respectively, but the Atiras contribute 16.5% of the venusian impact rate. The situation is even more extreme for Mercury, whose Apollo/Aten/Atira/Vatira fractions are 82.3%/13.2%/3.4%/0.8% of the Mercury-crossing NEOs but for which Apollos make up only 57% of the impact rate while Vatiras contribute four times their weight, at 3.2% of the mercurian impact rate. Even still, neglecting the Vatira population would produce only a small error in the mercurian cratering rate but ignoring the Atira-class asteroids, which supply 13% of the mercurian impact rate, could produce a serious underestimate of the flux. 39 Chapter 4 Future Work The NEOSSat-1.0 orbital model was computed to optimize an efficient point- ing strategy for NEOSSat to maximize the number of detections, particularly those in the a < 1.0 AU region. The completion of this model will allow us to perform simulations of the search strategy for NEOSSat as well as its tracking abilities. This can be generalized to any space-based observing platform wishing to observe NEAs, especially those at low solar elongation. The unexpected discovery of the production of retrograde orbits from as- teroidal main-belt sources and the mechanism responsible for flipping orbits through 90◦ is another topic to be explored. We have established that most asteroids flip while in the 3:1 mean-motion resonance with Jupiter, however the nature of the mechanism causing the flip in unclear. Since many objects which flip to retrograde orbits experience Kozai oscillations which can bring their inclinations up to 80◦, whether Kozai is responsible for further in- creasing inclinations through 90◦ is an interesting question. However, since almost all particles flip while in mean-motion resonances, Kozai alone can- not be the mechanism which puts objects onto retrograde orbits. Whether retrograde orbits are only produced while objects are within mean-motion resonances and not secular resonances is another topic to be explored. Another future goal is the investigation of tidal effects which can re- move small amounts of debris from surfaces of objects which experience very close encounters with Earth and Venus. It is of interest whether the post-encounter orbital distribution of NEOs suffering these close planetary encounters is the same for objects with a just inside and outside Earth’s orbit, because the observational color distribution of NEOs appears to be different in these two regions (Binzel et al., 2004). Once NEOSSat begins providing observational data, provided the neces- sary circumstances are available, the debiasing of the NEOSSat data using the NEOSSat-1.0 orbital model will be possible. Refitting the NEOSSat-1.0 orbital model to the NEOWISE data (Mainzer et al., 2011) is also possible now that the model is complete. However, the NEOSSat-1.0 model is not yet pulbic. This will allow further constraint of the best-fit parameters for the NEO source regions. 40 Chapter 5 Summary and Conclusions 14We present the NEOSSat-1.0 orbital model of the steady-state NEO distri- bution with larger statistics that permit finer resolution and less uncertainty in the a < 1.0 AU region, the focal point of Canada’s NEOSSat’s observa- tions. Although this model was produced for the purpose of optimizing the NEOSSat pointing strategy to discover and track NEOs with a < 1.0 AU, the model is not dependent upon the satellite in any way. The Atira asteroid class represents ≃ 1.4% of the NEO population, and these objects typically have i∼ 25◦ and will thus be found most often at high ecliptic latitudes when viewed from Earth or Earth orbit. Analysis of our integrations pro- vided an accurate characterization of the population of objects decoupled from Venus. This population, which we call Vatiras, constitutes ≃ 0.22% of the steady-state NEO orbital distribution. We put an upper bound on the potential Vulcanoids produced via an NEA supply chain. We also show a supply path for the production of retrograde orbits from main-belt asteroid sources. The population of retrogade NEAs is estimated to be ≃ 0.10% of the steady-state NEO orbital distribution and is dominantly supplied via the 3:1 mean-motion resonance with Jupiter. We also calculate the relative NEO impact rate onto Mercury, Venus, and Earth, as well as the normal- ized distribution of impacts speeds and impact flux ratios for Venus/Earth and Mercury/Earth, showing that the Mercury impact rate (at fixed size) is higher and typical impact speeds lower than previous estimates. Upon request, the following products of the NEOSSat-1.0 NEO orbital model are available: 1. The full set of integrations (0.5 terabyte) at 300 year output intervals including the gravitational perturbations of Mercury-Saturn. 2. Residence time probability distribution, R(a, e, i), for each of the five asteroidal and cometary source regions as well as the combined RNEO(a, e, i) for a < 4.2 AU, e < 1.0, and i < 180◦. 14A version of this chapter has been submitted for publication consideration. Green- street, S., Ngo, H., and Gladman, B. (2011) The Orbital Distribution of Near-Earth Objects Inside Earth’s Orbit. Icarus. 41 Chapter 5. Summary and Conclusions 3. The fraction in each cell of RNEO(a, e, i) that came from each source region. 4. ‘Top of the atmosphere’ asteroidal impact speed distributions for Mer- cury, Venus, and Earth. 5. A realization of the NEO population consisting of 30,000 NEOs ex- tracted from the NEOSSat-1.0 NEO orbital model. 42 Bibliography Abell, P. A., Korsmeyer, D. J., Landis, R. R., Jones, T. D., Adamo, D. R., Morrison, D. D., Lemke, L. G., Gonzales, A. A., Gershman, R., Sweetser, T. H., Johnson, L. L., & Lu, E., 2009, Meteoritics and Planetary Science 44, 1825 Binzel, R. P., Morbidelli, A., Merouane, S., DeMeo, F. E., Birlan, M., Vernazza, P., Thomas, C. A., Rivkin, A. S., Bus, S. J., & Tokunaga, A. T., 2010, Nature 463, 331 Binzel, R. P., Perozzi, E., Rivkin, A. S., Rossi, A., Harris, A. W., Bus, S. J., Valsecchi, G. B., & Slivan, S. M., 2004, Meteoritics and Planetary Science 39, 351 Bottke, W. F., Jedicke, R., Morbidelli, A., Duncan, M., Petit, J. M., & Gladman, B., 2000a, Science 288, 2190 Bottke, W. F., Morbidelli, A., Jedicke, R., Petit, J. M., Levison, H. F., Michel, P., & Metcalfe, T. S., 2002, Icarus 156, 399 Bottke, W. F., Nolan, M., Greenberg, R., & Kolvoord, R., 1994, in T. Gehrels (ed.), Hazards Due to Comets and Asteroids, pp 337–357, Univer- sity of Arizona Press Bottke, W. F., Rubincam, D. P., & Burns, J. A., 2000b, Icarus 145, 301 Bottke, W. F., Vokrouhlický, D., Broz, M., Nesvorný, D., & Morbidelli, A., 2001, Science 294, 1693 Carruba, V., Burns, J. A., Bottke, W., & Nesvorný, D., 2003, Icarus 162, 308 Dones, L., Gladman, B., Melosh, H. J., Tonks, W. B., Levison, H. F., & Duncan, M., 1999, Icarus 142, 509 Farinella, P. & Davis, D. R., 1994, Icarus 97, 111 43 Bibliography Farinella, P., Froeschlé, C., Froeschlé, C., Gonczi, R., Hahn, G., Morbidelli, A., & Valsecchi, G. B., 1994, Nature 371, 315 Farinella, P., Gonczi, R., Froeschlé, C., & Froeschlé, C., 1993, Icarus 101, 174 Farinella, P. & Vokrouhlický, D., 1999, Science 283, 1507 Gallant, J., Gladman, B., & Ćuk, M., 2009, Icarus 202, 371 Gehrels, T. & Jedicke, R., 1996, Earth, Moon and Planets 72, 233 Gladman, B., Michel, P., Cellino, A., & Froeschl’e, C., 2000, Icarus 146, 176 Gladman, B. J., Migliorini, F., Morbidelli, A., Zappalà, V., Michel, P., Cellino, A., Froeschlé, C., Levison, H. F., Bailey, M., & Duncan, M., 1997, Science 277, 197 Granvik, M., Virtanen, J., Oszkiewicz, D., & Muinonen, K., 2009, Mete- oritics and Planetary Science 44, 1853 Greenberg, R. & Chapman, C. R., 1983, Icarus 55, 455 Greenberg, R. & Nolan, M. C., 1993, in J. S. Lewis, M. S. Matthews, & M. L. Guerrieri (eds.), Resources of near-Earth space, pp 473–492, Univer- sity of Arizona Press Hildebrand, A. R., Carroll, K. A., Tedesco, E. F., Faber, D. R., Cardinal, R. D., Matthews, J. M., Kuschnig, R., Walker, G. A. H., Gladman, B., Pazder, J., Brown, P. G., Larson, S. M., Worden, S. P., Wallace, B. J., Chodas, P. W., Muinonen, K., & Cheng, A., 2004, Earth, Moon, and Planets 95, 33 Kozai, Y., 1962, The Astronomical Journal 67, 591 Le Feuvre, M. & Wieczorek, M. A., 2008, Icarus 197, 291 Le Feuvre, M. & Wieczorek, M. A., 2011, Icarus 214, 1 Levison, H. F. & Duncan, M. J., 1994, Icarus 108, 18 Levison, H. F. & Duncan, M. J., 1997, Icarus 127, 13 44 Bibliography Mainzer, A., Bauer, J., Grav, T., Masiero, J., Cutri, R. M., Dailey, J., Eisenhardt, P., McMillan, R. S., Wright, E., Walker, R., Jedicke, R., Spahr, T., Tholen, D., Alles, R., Beck, R., Brandenburg, H., Conrow, T., Evan, T., Fowler, J., Jarrett, T., Marsh, K., Masci, F., McCallon, H., Wheelock, S., Wittman, M., Wyatt, P., DeBaun, E., Elliott, G., Elsbury, D., Gautier, T., I., Gomillion, S., Leisawitz, D., Maleszewski, C., Micheli, M., & Wilkins, A., 2011, The Astrophysical Journal 731, 4936 Marchi, S., Mottola, S., Cremonese, G., Massironi, M., & Martellato, E., 2009, The Astronomical Journal 137 McKinnon, W. B., Zahnle, K. J., Ivanov, B. A., & Melosh, H. J., 1997, in S. W. Bougher, D. M. Hunten, & R. J. Phillips (eds.), Venus II, pp 969–1014, University of Arizona Press Michel, P. & Froeschlé, C., 1997, Icarus 128, 230 Michel, P. & Thomas, F., 1996, Astronomy and Astrophysics 307, 310 Michel, P., Zappalà, V., Cellino, A., & Tanga, P., 2000, Icarus 143, 421 Morbidelli, A. & Nesvorný, D., 1999, Icarus 139, 295 Richardson, D. C., Bottke, W. F., & Love, S. G., 1998, Icarus 134, 47 Sarty, G. E., Szalai, T., Kiss, L. L., Matthews, J. M., Wu, K., Kuschnig, R., Guenther, D. B., Moffat, A. F. J., Rucinski, S. M., Sasselov, D., Weiss, W. W., Huziak, R., Johnston, H. M., Phillips, A., & Ashley, M. C. B., 2011, Monthly Notices of the Royal Astronomical Society 411, 1293 Schmadel, L. D., 2009, Dictionary of Minor Planet Names: Addendum to 5th Edition: 2006-2008, Springer-Verlag Berlin Heidelberg Stern, S. A. & Durda, D. D., 2000, Icarus 143, 360 Stuart, J. S., 2001, Science 294, 1691 Veverka, J., Farquhar, B., Robinson, M., Thomas, P., Murchie, S., Harch, A., Antreasian, P. G., Chesley, S. R., Miller, J. K., Owen, W. M., Williams, B. G., Yeomans, D., Dunham, D., Heyler, G., Holdridge, M., Nelson, R. L., Whittenburg, K. E., Ray, J. C., Carcich, B., Cheng, A., Chapman, C., Bell, J. F., Bell, M., Bussey, B., Clark, B., Domingue, D., Gaffey, M. J., Hawkins, E., Izenberg, N., Joseph, J., Kirk, R., Lucey, P., Malin, M., McFadden, L., Merline, W. J., Peterson, C., Prockter, L., Warren, J., & Wellnitz, D., 2001, Nature 413, 390 45 Bibliography Vokrouhlický, D., Farinella, P., & Bottke, W. F., 2000, Icarus 148, 147 Walker, G., Matthews, J., Kuschnig, R., Johnson, R., Rucinski, S., Pazder, J., Burley, G., Walker, A., Skaret, K., Zee, R., Grocott, S., Carroll, K., Sinclair, P., Sturgeon, D., & Harron, J., 2003, The Publications of the Astronomical Society of the Pacific 115, 1023 Wetherill, G. W., 1967, Journal of Geophysical Research 72, 2429 Wetherill, G. W., 1979, Icarus 37, 96 Wetherill, G. W. & Faulkner, J., 1981, Icarus 46, 390 Wisdom, J., 1985, Nature 315, 731 Yano, H., Kubota, T., Miyamoto, H., Okada, T., Scheeres, D., Takagi, Y., Yoshida, K., Abe, M., Abe, S., Barnouin-Jha, O., Fujiwara, A., Hasegawa, S., Hashimoto, T.and Ishiguro, M., Kato, M., Kawaguchi, J., Mukai, T., Saito, J., Sasaki, S., & Yoshikawa, M., 2006, Science 312, 1350 46 Appendix A Supplementary Information The residence time probability distribution, R(a, e, i), for each of the five source regions is provided here. Each source region is combined via Equa- tion 2.1 to produce the master residence time distribution, RNEO(a, e, i), shown in Figure 2.4. In cl in at io n i [d eg ] Semi-major axis a [AU] 10 20 30 40 50 60 70 80 0 0.5 1 1.5 2 2.5 3 3.5 4  0  0.05  0.1  0.15  0.2  0.25  0.3 Ec ce nt ric ity  e Semi-major axis a [AU] 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4  0  0.05  0.1  0.15  0.2 Figure A.1: Residence time probability distribution, Rν6 (a, e, i), for the NEOSSat-1.0 NEO orbital model. This figure is constructed the same as Figure 2.4 except it only displays the ν6 source. 47 Appendix A. Supplementary Information In cl in at io n i [d eg ] Semi-major axis a [AU] 10 20 30 40 50 60 70 80 0 0.5 1 1.5 2 2.5 3 3.5 4  0  0.05  0.1  0.15  0.2  0.25  0.3 Ec ce nt ric ity  e Semi-major axis a [AU] 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4  0  0.05  0.1  0.15  0.2 Figure A.2: Residence time probability distribution, R3:1 (a, e, i), for the NEOSSat-1.0 NEO orbital model. This figure is constructed the same as Figure 2.4 except it only displays the 3:1 source. 48 Appendix A. Supplementary Information In cl in at io n i [d eg ] Semi-major axis a [AU] 10 20 30 40 50 60 70 80 0 0.5 1 1.5 2 2.5 3 3.5 4  0  0.05  0.1  0.15  0.2  0.25  0.3 Ec ce nt ric ity  e Semi-major axis a [AU] 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4  0  0.05  0.1  0.15  0.2 Figure A.3: Residence time probability distribution, RIMC (a, e, i), for the NEOSSat-1.0 NEO orbital model. This figure is constructed the same as Figure 2.4 except it only displays the IMC source. 49 Appendix A. Supplementary Information In cl in at io n i [d eg ] Semi-major axis a [AU] 10 20 30 40 50 60 70 80 0 0.5 1 1.5 2 2.5 3 3.5 4  0  0.05  0.1  0.15  0.2  0.25  0.3 Ec ce nt ric ity  e Semi-major axis a [AU] 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4  0  0.05  0.1  0.15  0.2 Figure A.4: Residence time probability distribution, ROMB (a, e, i), for the NEOSSat-1.0 NEO orbital model. This figure is constructed the same as Figure 2.4 except it only displays the OMB source. 50 Appendix A. Supplementary Information I n c l i n a t i o n  i  [ d e g ] Semi-major axis a [AU] 0 10 20 30 40 50 60 70 80 90 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.05 0.1 0.15 0.2 0.25 0.3 E c c e n t r i c i t y  e Semi-major axis a [AU] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 4 E c c e n t r i c i t y  e 0 0.05 0.1 0.15 0.2 Figure A.5: Residence time probability distribution, RJFC (a, e, i), for the NEOSSat-1.0 NEO orbital model. This figure is constructed the same as Figure 2.4 except it only displays the JFC source. 51 Appendix B Additional Notes on Co-Authorship This appendix contians additional information concerning the specific con- tributions made by Sarah Greenstreet to the various components of the analysis of the NEOSSat-1.0 orbital model. Analysis of the numerical inte- grations dominantly inherited by Sarah Greenstreet was done using pieces of code which monitor the orbital evolution histories of the integrated test par- ticles, construct the residence time distributions of test particles, estimate uncertainties in the residence time distributions, compute the impact speed distributions for Mercury, Venus, and Earth as well as the collision proba- bility distribution for the terrestrial planets. Sarah Greenstreet contributed the following data analysis and scientific interpretation of the NEOSSat-1.0 NEO orbital integrations to the construction of the steady-state NEO orbital distribution model discussed in this thesis. • Orbital evolution histories: The analysis of the Vatira population: – computation of the smallest aphelion distance of each test particle – extraction of orbital histories of particles upon becoming Vatiras the retrograde NEA population: – computation of the semimajor axes particles have at the instant they become retrograde – extraction of orbital histories of particles upon becoming retro- grade and particle orbital evolutions: – extraction of the median inclination trend from 4.2 AU to Sun – computation of the region of (a, e, i) space contributing to the dominant features in the impact speed distribution 52 Appendix B. Additional Notes on Co-Authorship was done by Sarah Greenstreet with suggestions made by Dr. Glad- man. • Residence time distributions, RNEO(a, e, i): – expanding the residence time distributions for a < 4.2 AU, e < 1.0, and i < 90◦ to a < 4.2 AU, e < 1.0, and i < 180◦ – computaion of the fraction of RNEO(a, e, i) from each source region for each NEO class – rebinning RJFC(a, e, i) used by (Bottke et al., 2002) to match our cell volume (0.05 AU x 0.02 x 2.00◦) – and the source region mapper code were all written by Sarah Greenstreet. • Uncertianty estimates for residence time distributions: – computation of RNEO(a, e, i) for even- and odd-numbered par- ticle contributions was done by Sarah Greenstreet with suggestions made by Dr. Glad- man. • Collision probability code: This code computes the collision probabil- ity distribution for any planet as well as the impact speed distribution. – modification of the collision probability code to run on the resi- dence time distributions in addition to the numerical integration particle histories – expansion of the collision probability code to run on each indi- vidual source region – additional code to compute the planet crossing impact rate con- tributions from each NEO class (Table 2.3) – the computation of the impact flux ratios (per unit surface area of target) for Venus/Earth and Mercury/Earth for: ∗ the individual asteroidal source integrations ∗ the weighted average of the impact flux ratios for the indi- vidual asteroidal source integrations ∗ the asteroidal steady-state ratios (computed from the resi- dence time distribution) 53 Appendix B. Additional Notes on Co-Authorship ∗ and the even/odd particle splits in order to estimate uncer- tainties was done by Sarah Greenstreet with suggestions made by Dr. Glad- man. • Additional analysis: – a realization of the NEO population consisting of 30,000 NEOs extracted from the NEOSSat-1.0 NEO orbital model was computed by Sarah Greenstreet with suggestions made by Dr. Gladman. The figures as they are shown in this thesis were constructed by Sarah Green- street except for Figure 1.1, which was made by Henry Ngo, and Figure 2.1, which was made by Dr. Gladman. 54

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