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Current bombardment of the Earth-Moon system : emphasis on cratering asymmetries Gallant, John 2006

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Current Bombardment of the Ear th-Moon System Emphasis on Cratering Asymmetries by John Gallant B.Sc, The University of Prince Edward Island, 2004 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF Master of Science in The Faculty of Graduate Studies (Astronomy) The University Of British Columbia August, 2006 © John Gallant 2006 Abstract We calculate the current spatial distribution of projectile delivery to the Earth and Moon using numerical orbital dynamics simulations of candidate impactors drawn from a debiased Near-Earth-Object (NEO) model. Sur-prisingly, we find that the average lunar impact velocity is 20 km/s, which has ramifications in converting observed crater densities to impactor size distributions. We determine that current crater production on the leading hemisphere of the Moon is 1.29 ± 0.01 that of the trailing when considering the ratio of craters within 30° of the apex to those within 30° of the antapex and that there is virtually no nearside-farside asymmetry. As expected, the degree of leading-trailing asymmetry increases when the Moon's orbital dis-tance is decreased. We examine the latitude distribution of impactor sites and find that for both the Earth and Moon there is a small deficiency of time-averaged impact rates at the poles. The ratio between deliveries within 30° of the pole to that of a 30° band centered on the equator is nearly unity for Earth (< 1%) (0.992 ± 0.001) but detectably non-uniform for the Moon (~ 10%) (0.912 ± 0.004). The terrestrial arrival results are examined to determine the degree of A M / P M asymmetry to compare with meteorite fall times (of which there seems to be a P M excess). Our results show that the impact flux of objects derived from the NEOs in the A M hours is ~ 2 times that of the P M hemisphere, further supporting the assertion that meteorite-dropping objects are recent ejections from the main asteroid belt rather than young fragments of NEOs. ii Table of Contents A b s t r a c t i i T a b l e o f C o n t e n t s i i i i L i s t o f F i g u r e s v A c k n o w l e d g e m e n t s vi S t a t e m e n t o f C o - A u t h o r s h i p v i i 1 I n t r o d u c t i o n 1 1.1 Bibliography 4 2 A s y m m e t r i e s 6 2.1 Asymmetry Theory 6 2.1.1 Meteorite fall statistics on Earth 6 2.1.2 Lunar leading hemisphere enhancement 7 2.1.3 Lunar nearside/farside debate 8 2.2 Model and numerical methods 11 2.2.1 The source population 11 2.2.2 Setup of the flyby geometries 11 2.2.3 Integration method 14 2.3 Delivery to Earth 16 2.3.1 Latitude distribution of delivered objects 16 2.3.2 A M / P M asymmetry 20 2.4 Lunar bombardment 24 2.4.1 From impacts to craters 26 2.4.2 Latitude distribution of impacts 27 2.4.3 Longitudinal effects 28 2.4.4 Varying the Earth-Moon distance and the effect of inclination 33 2.5 Bibliography 35 ii i 3 S u m m a r y and conclusions 39 3.1 Bibliography 41 A E a r t h - M o o n Sys tem A s y m m e t r i e s 42 A . l A M / P M 42 A.2 Leading/Trailing and Nearside/Farside 45 B R a y e d Cra te r s 47 C O r b i t a l E lement Convers ion 49 iv List of Figures 2.1 Distribution of incoming projectile directions 9 2.2 Maximum likelihood contours for rayed crater data 10 2.3 Orbital distribution of debiased N E O source population . . . 12 2.4 Schematic of a typical simulation fiyby 15 2.5 Earth delivery velocity histograms 17 2.6 Latitude distribution of Earth deliveries (uncorrected) . . . . 18 2.7 Latitude distribution of Earth deliveries (corrected) 19 2.8 Latitude distribution of Earth deliveries with various restric-tions 21 2.9 Local fall time distribution for Earth deliveries 22 2.10 Afternoon fraction of falls as a function of arrival speed . . . 23 2.11 Lunar impact velocity histograms 25 2.12 Map of lunar impacts 28 2.13 Latitude distribution of lunar impacts 29 2.14 Density of simulated craters as a function of angle away from the apex of motion 30 2.15 Comparison between simulations and observations of rayed craters 32 2.16 The consequences of changing the Earth-Moon distance . . . 34 A . l A M / P M asymmetry schematic 43 A.2 Larger orbit intersecting the Earth's 44 A.3 Leading vs. trailing incoming flux 45 A . 4 Focusing by the Earth 46 B . l Rayed crater distribution 48 C l Four orbital geometries 50 v Acknowledgements I would like to thank my supervisor, Dr. Brett Gladman for being very supportive when things looked bleak and for being extremely helpful in increasing my understanding of the various nuances of this project and in planetary science in general. I could not have asked for a better research supervisor. Thanks to Matija Cuk for helpful discussions and the many coffee breaks which cleared my head. Catherine Johnson of the Earth and Ocean Sciences department at U B C is also due credit as she acted as second reader to this thesis. I thank her for her comments which improved the clarity of this work. I am grateful to the University of British Columbia for having me (though the time was far too short) and for giving me the opportunity to work in an exciting and interesting field of research. vi Statement of Co-Authorship • Identification and design of research program: The main project and research goals were suggested to the author of this thesis, John Gallant, by research supervisor Dr. Brett Gladman. Methodology was left to the discretion of John Gallant • Performing the research: A l l research (i.e., coding and running sim-ulations) was performed by John Gallant. • Data analyses: The results from the simulations were analyzed pri-marily by John Gallant with suggestions from Dr. Brett Gladman and Matija Cuk which improved clarity and efficientcy. • Manuscript preparation: As this is a manuscript thesis, it was written first as a journal article then with additions to meet thesis requrements. Thus, the main body (Ch. 2) has components which are written by John Gallant and Dr. Brett Gladman. Sections 2.1.1, 2.2.1, 2.2.2 and components of Sec. 2.3.2 and Ch. 3 were written mainly by Dr. Brett Gladman, while the remainder of the thesis was authored by John Gallant. vi i Chapter 1 Introduction •"•Craters. Ever since man realized that the landscape of the Moon was pock-eted with these informational gems, a wealth of important science has re-sulted from their study. Craters provide information regarding the original impactor size distribution [17], the age [16] and composition [3] of the sur-face, and, in the case of lunar craters, can even be used in the dating of other surfaces [13]. We wish to ask and answer several questions about the spatial distribution of craters on both the Earth and Moon, thus adding to the diverse library of crater literature. What is the degree of asymmetry between the leading/trailing and near-side/farside hemispheres on the Moon? Studies have been done as early as 1971 examining this concept. Wiesel [20] introduced the idea that the Earth could act as a gravitational lens, focusing incoming trajectories towards the nearside of the Moon, creating an asymmetry between the near and far hemi-spheres. Wood [21] used this concept as well in a rough numerical calculation in the attempt to explain the crustal thickness asymmetry of the nearside and found that under his initial conditions, an enhancement on the near-side was possible. Though state-of-the-art at the time, the numerical setup and initial conditions imposed on the system were very basic and somewhat artificial. Bandermann and Singer [1] posed a purely theoretical study re-futing the gravitational lens concept, claiming if any nearside enhancement were present, the effect would be very small. Since that time, little work has been done in regards to cratering asymmetries with the notable excep-tions of Horedt and Neukum [5], who developed a functional relationship for the expected apex/antapex asymmetry for synchronously rotating satel-lites, and Zahnle et al. [22, 23] who applied that concept to the Gallilean satellites and derived an expression which was consistent with [5]. This area of research has been revitalized recently with high qulity images from the Clementine surveyor [8, 9, 14]. Morota and Furumoto [11], Morota et al. [12] have examined rayed crater counts and found an asymmetry between the 1 A version of this chapter has been submitted for publication consideration. Gallant, J., Gladman, B., and Cuk, M. (2006) Current Bombardment of the Earth-Moon System: Emphasis on Cratering Asymmetries. Icarus. 1 leading and trailing crater densities on the order of a 50% enhancement at the lunar apex compared with the antapex. Numerical work by Le Feuvre and Wieczorek [6, 7] has resulted in confirmation of a nearside/far side asym-metry (factor of four enhancement on average between the hemispheres) as well as latitudinal variation in the impact density. Both [6] and Morota et al. [12] have discussed the possible implications that a non-uniform crater field would have in crater chronology studies, where uniformity is a key assump-tion. If a certain area has a higher density of craters than another, previous studies would claim the over dense region to be older, wheras accounting for crater field location may restrict the fields much closer in age. However, as we show, the crater data suffers from poor statistics and any insights into the impactor distribution are tenuous at best. As well, the numerical work in Le Feuvre and Wieczorek [6] used an artificial impactor population consisting of zero inclination bodies. Our goal with this study is to re-examine the rayed crater data and to perform N-body simulations using a realistic orbital distribution of objects. Doing so will provide an accurate portrait of the current level of cratering asymmetry on the lunar surface. How does reality compare with theoretical predictions? Horedt and Neukum [5] developed a model to describe differential cratering on syn-chronously rotating satellites, but several assumptions, such as the use of an isotropic source population and a single encounter speed for the impactors, were required to make the calculations analytically feasible. We show that with a realistic source population, this assumption breaks down and may cause discrepancies between the theoretical and simulated spacial distribu-tions. Asymmetries may also be present for the Earth and would be most re-liably observed via fireball sightings and meteorite recoveries. Based on ob-servations, many researchers concluded that more daytime fireballs occur in the afternoon hours, causing a morning/afternoon ( A M / P M ) asymmetry [4, 18]. Several studies [4, 10, 19] have been done to understand this P M excess. Halliday and Griffin [4] produced a P M fireball excess, but drew from a highly-selected group of orbital parameters for the meteoroid source pop-ulation. These orbits may not represent the true distribution of objects in near-Earth space. Thus, since we will obtain many simulated Earth impacts in our lunar study, an additional benefit is to reinvestigate the question of the A M / P M asymmetry. The debiased N E O model we use as our source population should give a better approximation of the objects in near-Earth space than the parameters used by Halliday and Griffin [4]. Annual effects are also of interest as at different times of the year certain locations on 2 Earth receive an enhancement in the impactor flux due to the location of the Earth's spin axis relative to the incoming flux direction [4, 15]. To answer these questions, we perform N-body simulations of test par-ticles as they enter the Earth-Moon system. Such an endeavour is possible due to the advent of faster computer processors and advanced numerical techniques. In this study, we numerical integrate trillions of test particles and examine the spatial distribution of their resulting impacts on both the Earth and Moon. To simulate the current conditions in near-Earth space, we use the debiased N E O model from Bottke et al. [2]. In Section 2.1, we give a brief overview on the theory of various asym-metries in the Earth-Moon system. Section 2.2 describes the model for our simulations as well as the methods we have used while implementing it. Our findings begin in Section 2.3, where we first examine Earth impacts in terms of the latitude distribution and then the A M / P M asymmetry. Lunar results are found in Section 2.4 where we examine near-far and leading-trailing hemisphere ratios, as well as the latitude distribution. We end the thesis with Chapter 3, summarizing the key findings of this work and some possible implications. 3 1.1 Bibliography [I] Bandermann, L . W., and Singer, S. F . 1973. Calculation of meteoroid impacts on Moon and Earth. Icarus 19, 108-113. [2] Bottke, W . F., Morbidelli, A . , Jedicke, R., Petit, J . -M. , Levison, H . F. , Michel, P., and Metcalfe, T . S. 2002. Debiased orbital and absolute mag-nitude distribution of the near-Earth objects. Icarus 156, 399-433. [3] Gaddis, L . R., Hawke, B . R., Robinson, M . S., and Coombs, C. 2000. Compositional analyses of small lunar pyroclastic deposits using Clemen-tine multispectral data. Journal of Geophysical Research. 105, 4245-4262. [4] Halliday, I., and Griffin, A . A . 1982. A study of the relative rates of meteorite falls on the Earth's surface. Meteoritics 17, 31-46. [5] Horedt, G. P., and Neukum, G . 1984. Cratering rate over the surface of a synchronous satellite. Icarus 60, 710-717. [6] Le Feuvre, M . , and Wieczorek, M . A . 2005. The asymmetric cratering history of the Moon. 36th Annual Lunar and Planetary Science Confer-ence, March 14-18, 2005, League City, Texas, abstract no.2043. [7] Le Feuvre, M . , and Wieczorek, M . A . 2006. The asymmetric crater-ing history of the terrestrial planets: Latitudinal effect. 37th Annual Lu-nar and Planetary Science Conference, March 13-17, 2006, League City, Texas, abstract no. 1841. [8] McEwen, A . S., Moore, J . M . , and Shoemaker, E . M . 1997. The Phanero-zoic impact cratering rate: Evidence from the farside of the Moon. Journal of Geophysical Research 102, 9231-9242. [9] Moore, J . M . and McEwen, A . S. 1996. The abundance of large, Copernican-age craters on the Moon. Lunar and Planetary Institute Conference Ab-stracts 27, 899. [10] Morbidelli, A . , and Gladman, B . 1998. Orbital and temporal distri-butions of meteorites originating in the asteroid belt. Meteoritics and Planetary Science 33, no. 5, 999-1016. [II] Morota, T., and Furumoto, M . 2003. Asymmetrical distribution of rayed craters on the Moon. Earth and Planetary Science Letters 206, 315-323. 4 [12] Morota, T., Ukai, T., and Furumoto, M . 2005. Influence of the asym-metrical cratering rate on the lunar chronology. Icarus 173, 322-324. [13] Neukum. G. , Ivanov, B . A . , and Hartmann, W . K . 2001. Cratering records in the inner solar system in relation to the lunar reference system. Space Science Reviews 96, 55-86. [14] Pieters, C. M., Staid, M . I., Fischer, E . M . , Tompkins, S., He, G. 1994. A sharper view of impact craters from Clementine data. Science 266, 1844-1848. [15] Rendtel, J . , and Knofel, A . 1989. Analysis of annual and diurnal vari-ation of fireball rates and the population index of fireballs from different compilations of visual observations. Astronomical Institutes of Czechoslo-vakia, Bulletin (ISSN 0004-6248) 40, no. 1, 53-62. [16] Stoffler, D., and Ryder, G . 2001. Stratigraphy and isotope ages of lunar geologic units: chronological standard for the inner solar system. Space Science Reviews 96, 9-54. [17] Strom, R. G. , Malhotra, R., Ito, T., Yoshida, F. , and Kring, D. A . 2005. The origin of planetary impactors in the inner solar system. Science 309, 1847-1850. [18] Wetherill, G. W . 1968. Stone meteorites: Time of fall and origin. Science 159, 79-82. [19] Wetherill, G. W . 1985. Asteroidal source of ordinary chondrites. Mete-oritics 20, 1-22. [20] Wiesel, W . 1971. The meteorite flux at the lunar surface. Icarus 15, 373-383. [21] Wood, J . A . Bombardment as a cause of the lunar asymmetry. Earth, Moon, and Planets 8, 73-103. [22] Zahnle, K . , Dones, L . , and Levison, H . F . Cratering rates on the Galilean satellites. Icarus 136, 202-222. [23] Zahnle, K . , Schenk, P., Sobieszczyk, S., Dones, L . , and Levison, H . F . Differential cratering of synchronously rotating satellites by ecliptic comets. Icarus 153, 111-129. 5 Chapter 2 Asymmetries 2.1 Asymmetry Theory 1 Why search for asymmetries? For the Earth in terms of a morning/afternoon asymmetry, one can glean some information about the origin of the impact-ing population as certain types of orbits will preferentially strike during spe-cific local times. More importantly though, when looking at the the Moon, any spatial variation in the observed crater density will affect how those craters are used to interpret the complex cratering history of the Moon. 2.1.1 Meteorite fall statistics on E a r t h The details concerning an afternoon (PM) excess of meteorite falls on the Earth has been the subject of much debate, creating a large body of litera-ture on the subject (e.g., Halliday [32], Halliday and Griffin [33], Morbidelli and Gladman [43], Wetherill [57, 58] and references therein). This concept was instigated by the fact that chondrites seem to be biased to fall during P M hours. Wetherill [57] quantified the effect as the ratio of daytime P M falls to the total number of daytime falls and arrived at a ratio ~ 0.68, where the convention is to count only falls between 6 A M and 6 P M since human observers are less numerous between midnight and 6 A M . However, even this daytime value may be socially biased as there are more potential observers from noon-6PM than in the 6AM-noon interval. A hypothesis to explain a P M excess is that prograde, low-inclination meteoroids with semi-major axes > 1 A U and pericentres just inside 1 A U preferentially coat Earth's trailing hemisphere (see Appendix A . l ) . Several dynamical and statistical studies [33, 43, 58] support this afternoon enhancement scenario. In our work, we will use the debiased N E O model of Bottke et al. [27] to compute the dis-tribution of fall times from an N E O source population. We can use this to learn about the true orbital distribution of meteorite dropping objects. 1A version of this chapter has been submitted for publication consideration. Gallant, J., Gladman, B., and Cuk, M. (2006) Current Bombardment of the Earth-Moon System: Emphasis on Cratering Asymmetries. Icarus. 6 2.1.2 Lunar leading hemisphere enhancement A leading hemisphere enhancement originates from the satellite's motion about its host planet (see Appendix A.2). As it orbits, the leading hemi-sphere tends to encounter more projectiles, thus enhancing the crater pro-duction on that side. The faster the satellite orbits, the more difficult it becomes for objects to' encounter the trailing hemisphere and the higher the leading-side impact speeds become. In addition to this effect, the size-frequency distribution of craters will be skewed towards larger diameter craters on the leading hemisphere. A crater, of size Dc, at the apex will have been produced from a smaller-sized impactor (on average) than one which makes the same sized crater at the antapex because impact speeds Vimp are generally higher on the leading hemisphere and the crater diameter scales roughly as Dc oc (e.g., Ivanov [35]). Analytic investigations into leading/trailing asymmetries have ranged from the general [34] to the very specific [50, 61], but all analytic treat-ments are forced make the assumption that the impacting population has an isotropic distribution in the rest frame of the planet. In some cases, the gravity of the satellite is ignored, and in the above treatments, impact prob-abilities (depending on the encounter direction and speed) were not included in the derivation of cratering rates. For fixed impactor encounter speed and an isotropic source distribution, the areal crater density F would follow the functional form T(B) = T (1 +a cos B)9 , (2.1) where 3 is the angle from the apex of the moon's motion and F is the value of the crater density at 8 = 90° [46, 62]. The "amplitude" a of the asymmetry is related to the orbital velocity vm\, of the Moon and the velocity of the projectiles at infinity by a = V o r b . (2.2) Note that for large encounter speeds, a —> v0rb A'oo and that a —> l/\/2 as Voo -> 0. The exponent g (which also effects the asymmetry) is expressed as 0 = 2.0 +1.4 V (2-3) where b is the slope of the cumulative mass distribution for the impactors N(> rn) oc m~b . (2.4) 7 Based on current observations [27], b = 0.58 ± 0.03, making g = 2.81 ± 0.05. Although Eq. 2.1 may be fit to an observed crater distribution (see Ap-pendix B and Figure B . l ) , this does not necessarily provide a convenient measure of the asymmetry. Figure 2.1 shows that for our impactor popu-lation (see Sec. 2.2.1) the isotropic assumption fails, so one should expect deviations from the functional form of Eq. 2.1. Note that increasing either a or g raises the leading/trailing asymmetry, introducing a degeneracy in the functional form, making it difficult to decouple the two when determining information about Voo and the size distribution. In fact the observations ac-tually permit a very large range of parameter values. Performing a maximum likelihood parameter determination using the rayed crater data of Morota and Furumoto [44] yields Figure 2.2. Also, determining f with a high degree of precision from a measured crater field is difficult. To minimize these issues when examining the entire lunar surface, we adopt the convention of Zahnle et al. [62] by taking the ratio of those craters which fall within 30° of the apex to those which are within 30° of the antapex. This ratio forms a statis-tic known as the global measure of the apex-antapex cratering asymmetry ( G M A A C A ) . Because of the Moon's small 1.0 km/s orbital velocity and standard literature values of Vcc ~ 10 — 16 km/s [28, 51], one expects a as 1/16 — 1/10 and thus a crater enhancement on the leading hemisphere in the range ~ 1.4 — 1.7, in terms of the G M A A C A . Recent studies of young rayed craters [44, 45] are found to be consistent with these estimates. We will calculate the G M A A C A that current N E O impactors produce and compare with these values. 2.1.3 Lunar nearside/farside debate The concept of a nearside/farside asymmetry has garnered less attention in the lunar cratering literature. Wiesel [59] discussed the idea that the Earth would act as a gravitational lens, focusing incoming objects onto the nearside of the Moon (see Appendix A.2). However, the degree to which this lensing effect occurs is unclear. The amplitude of nearside enhancement has been reported as insignificant [59], a factor of two [55, 60], and most recently, a factor of four compared with the farside in a preliminary analysis by Le Feuvre and Wieczorek [37]. In contrast, Bandermann and Singer [24] used analytic arguments to claim no a priori reason to expect such an asymmetry; the Moon may be in a region of convergent or divergent flux because the focal point of the lensed projectiles depends on the encounter velocity of the objects and the Earth-Moon distance. These latter authors estimated that 8 180 i i i i i i i 0 15 30 45 60 75 90 4000 angle (degrees) Figure 2.1: A scatter plot of the 8 vs. 4> f ° r the source population used in our simulations (a) as well as a distribution of those angles (b). The angle 8 is the polar angle between the relative velocity vector of the potential impactor and the Earth's direction of motion and (j> is an azimuthal location of the relative velocity [56]. From these figures we can see that our distribution of possible impactors is not isotropic, a key assumption in many cratering theories. Note that in (a) only <f> values from 0 — 90° are presented corresponding to post-pericenter encounters at ascending node; higher values of <fi are simply mirror images of the scatter plot shown as all encounters have the same value of 8 for a given object. 9 Figure 2.2: A contour plot of log(likelihood) values showing the degeneracy present in the parameters a and g from Eq. 2.1 using the rayed crater data from Morota and Furumoto [44]. Best fit values for the rayed crater data are shown as the dot at a = 0.063 and g = 4.62. This value for g gives the slope of the impactor mass distribution as b = 1.87, over 3 times the observationally determined value of 0.58. However, a large range of a and g values give acceptable fits to the data, making it impossible to measure the parameters of the projectile population from the crater distribution alone. 10 there should currently be a negligible difference in the near side/far side crater production rate. We will also investigate this issue as the initial conditions used in the previous dynamical studies were somewhat artificial. 2.2 Model and numerical methods 2.2.1 The source population The cratering asymmetry expected on the synchronously-rotating Moon de-pends on the impactor speed distribution that is bombarding the satellite (the scalar UQO distribution and its directional dependence). Therefore, we must have a model of the small-body population crossing Earth's orbit. The potential impactors come from the asteroid-dominated (e.g., Gladman et al. [31]) near-Earth object (NEO) population. The orbital distribution of these objects is best modeled by Bottke et al. [27] who fit a linear combination of several main-belt asteroid source regions and a Jupiter-family comet source region to the Spacewatch telescope's N E O search results. Since the detection bias of the telescopic system was included, this yields a model of the true N E O orbital-element distribution. W . Bottke (private communication, 2005) has provided us with an orbital-element sampling of their best fit distribu-tion, which we have then restricted to 16307 orbits in the Earth-crossing region (Figure 2.3). This will be our source population for the impactors which transit through the Earth-Moon system. We note the common oc-currence of high eccentricity e and inclination i orbits in the N E O sample, which result in high values of UQQ for the bombarding population. Although semimajor axes a in the a=2.0-2.5 A U region are densely populated (since this is the semimajor axis range of the dominant main-belt sources), impact probability is largest for orbits with perihelia q = a(l — e) just below 1 A U or for aphelia Q = a(l + e) just above 1 A U [43]. Since the q ~ 1 population is much larger than the Q ~ 1 population, one might expect the former to dominate the impactors (but see Sec. 2.3.2). 2.2.2 Setup of the flyby geometries The orbital model provides only the (a, e, i) distribution of the N E O pop-ulation and we expect that the 1.7% eccentricity of the Earth's orbit is a small correction to the impactor distribution, so for what follows we have modeled the Earth's heliocentric orbit as perfectly circular. As a result we can, without loss of generality, take all encounters to occur at (1 AU,0,0) in heliocentric coordinates, where we have lost knowledge of the day of the 11 Figure 2.3: The orbital distribution of candidate impactors used in our simulations, based on the debiased near-Earth object (NEO) model of Bottke et al. [27]. We have used only objects whose perihelion q and aphelion Q satisfy q < 1 AU and Q > 1 AU. (a) Semi-major axis a versus eccentricity, e. Note the increase in density near a ~ 2.5 AU due to the 3:1 orbital resonance with Jupiter, (b) Semi-major axis versus inclination i with respect to the ecliptic. year of the encounters (although we can easily average over the year by post-facto selecting a random azimuth for the Earth's spin pole at the time of a projectile's arrival at the top of the atmosphere). Wi th this restriction, each Earth-crossing orbit can have an encounter in one of four geometries (see Figure C . l ) depending on the argument of pericenter co and the true anomaly / , which must satisfy: r = 1 A U = ^ i . (2.5) 1 + e cos / B y construction, the encounter must occur at either the ascending or de-scending node along the x-axis, and so the longitude of ascending node is £1=0 or TT. Taking / and co in [0, 27r), for ascending encounters, / = 2TT—CO for either encounters with pericenters above (co = [0,7r)) or below (co — [TT, 2n)) the ecliptic. If the encounter occurs at the descending node, / = TT — co for post-pericenter encounters and / = 3TT — co for pre-pericenter encounters. Wi th this in mind, we effectively quadruple the number of initial conditions to 65228. Wi th the longitude of ascending node fixed (ie. £7 = 0 or TT), we then construct a plethora of incoming initial conditions based on the orbital el-ements from the debiased N E O model converted to Cartesian coordinates (see Appendix C). For each initial orbit, all four of the encounter geometries are equally likely. We randomly choose a particle for a flyby based on its encounter probability with the Earth as judged by an Opik collision proba-bility calculation [29]. Gravitational focusing by the Earth was not included in the encounter probability estimate as an increased frequency of Earth deliveries will occur naturally during the flyby phase if the Earth's gravity is important. Both Earth and a test particle (TP) are then placed at the nodal inter-section and moved backwards on their respective orbits until the separation between the N E O and the Earth is 0.02 A U . At this point, we create a disk of 10 5 non-interacting test particles, meant to represent potentially-impacting asteroids or comets, centered on the chosen orbit. A short numerical inte-gration is run for each of the 65228 initial conditions and one T P trajectory that results in an arrival at the Earth in each flyby is placed into a table of new initial conditions. This procedure is performed because during the "backup phase" to separate the Earth and T P by 0.02 A U , gravitational focusing was not accounted for. The omission could result in the particle missing the Earth in a forward integration, as the Earth's gravity would be present, modifying the chosen trajectory. This gives us a final set of 65228 initial trajectories that strike the Earth when a forward integration is per-13 formed. For convenience we then convert from heliocentric coordinates to a geocentric frame of reference. For our simulations, one of the new initial conditions is randomly chosen based on a newly calculated encounter probability with the Earth-Moon sys-tem. A new disk, centered on the chosen trajectory, is randomly populated with 10 5 test particles, all given identical initial velocities. The radius of the disk is chosen to be 2.5 lunar orbital radii as testing showed that for all of our initial conditions, a disk of this size spans the entire lunar orbit as it passes the Earth's position. This was done to ensure the crater distribution had no dependence on the lunar orbital phase. Figure 2.4 shows a schematic of the simulations. 2.2.3 Integration method The orbital trajectories of the massive bodies (Earth, Moon and Sun) and the TPs are integrated using a modified version of swift-rmvs3 [39]. The length units are in A U and time units are in years. The initial eccentricity and inclination for both the Earth and Moon are set to zero for the majority of our simulations. In one run we introduce the current 5.15° inclination of the Moon, keeping Earth's e and i as before. Both Earth and Moon are assumed to be perfect spheres. Factors that vary in the simulations include the number of flybys, where a flyby is defined as a disk of 10 5 test particles passing through the Earth-Moon system, and the Earth-Moon distance, REM-We have the Earth as the central body with the Moon acting as an orbit-ing planet and the Sun as an external perturber. The base time step in the simulations is four hours. This is large enough to not be time prohibitive, but small enough such that the integrator can follow the lunar encounters precisely. During the course of the integration, the integrator logs the posi-tions and velocities of all bodies of interest if there is a pericenter passage within the radius of the Moon or Earth. We then use this log as input for a backwards integration using a 6TH-order explicit symplectic algorithm [30] to precisely determine the latitude and longitude of the particle's impact location. This method, along with iterative time steps, enables us to find impact locations to within 5 km of the surface of the Moon and within < 500 m of the top of Earth's atmosphere. This project is computationally intensive. For each simulation, the Earth, Moon and Sun are included as well as 105 test particles. Each of the 65228 initial conditions are used multiple times to improve statistics and in each case the T P locations in the disk are randomly distributed. Thus more than 14 Figure 2.4: A schematic of one 3-dimensional flyby projected onto the eclip-tic in a heliocentric coordinate frame. A disk (thick, solid line) of 10 5 test particles, all with identical initial velocities, is created around a central tra-jectory based on the N E O source population of Bottke et al. [27]. The disk is 2.5 lunar orbital radii in radius and encompasses the entire lunar orbit for any of our possible initial conditions. These test particles are then integrated forward along with the Earth, Moon, and Sun and have an encounter with the Earth-Moon system near (1 AU,0,0). A l l particles are followed for three times the amount of time required for the farthest test particle to reach the Earth. 15 2 x 10 test particles are integrated which results in tens of millions of terrestrial deliveries and hundreds of thousands of lunar ones. 2.3 Delivery to Earth Before turning to the Moon, we examine our numerical results to determine the spatial distribution of Earth arrivals. Our simulations with the Moon at the current orbital distance of 60.R© yielded 21,998,427 terrestrial impacts. Since the crater record on the Earth is difficult to interpret due to geological processes, we choose to examine our results in terms of fireball and meteorite records as the impactors strike the top of the atmosphere. We are thus assuming that the meteorite dropping bodies have a pre-atmospheric orbital distribution similar to the NEOs (but see Sec. 2.3.2). Figure 2.5 shows velocity distributions for the Earth arrivals from our simulations. The delivery speeds should be interpreted as "top of the at-mosphere" velocities, Vimp. The deliveries are dominated by low objects which the Earth's gravitational well has focused and sped up, creating a peak in the distribution at a speed of ~ 15 km/s. The average speed an im-pactor has at the top of the atmosphere is ~ 20 km/s, slightly higher than the often quoted value of 17 km/s [28]. As well, if we compare our Figure 2.5 to the fireball data in Morbidelli and Gladman (1998, Figure 8a), we see a general similarity in the shapes of the Vimp histograms. 2.3.1 Latitude distribution of delivered objects For a uniform spatial distribution of deliveries, one expects the number of arrivals to vary as the cosine of the latitude due to the smaller surface area at higher latitudes. Figure 2.6 shows that to an excellent approximation, the Earth is uniformly struck by impactors (in latitude). To account for the area in each latitude bin, divide by: A B I N = 2TTR%(COS 0I - cos 82), (2.6) where the co-latitudes 9\ > 0% are measured north from the south pole. To correct for the spin obliquity of the Earth, we then choose a random day of fall, and thus a random azimuth for the Earth's spin pole. This then gives us a geocentric latitude and longitude for each simulated deliv-ery. Figure 2.7 shows the spatial density versus geocentric latitude. For the long-term average, we see a nearly uniform distribution of arrivals. If we restrict to azimuths corresponding to northern hemisphere spring or north-ern hemisphere autumn, we see the well-known seasonal variation [33, 49] 16 0.15-a l o . i 0.05 0, l~L impactor v oo r imp L J - 1 source y 00 0 10 20 30 v (km/s) 50 Figure 2.5: Velocity distributions for the Earth deliveries from our simu-lations (impactor and Vimp) as well as the sampled N E O population (source ). Note that the average impact velocity is ~ 20 km/s, higher than the often-quoted value of 17 km/s. One can observe the large effect of the Earth's gravitational well for values < 10 km/s. Here the encounter speeds are "pumped up" to values slightly higher than the Earth's escape velocity of 11.2 km/s. From the distribution of all possible impactors it is clear that the impacts are dominated by low (voo < 20 km/s) objects. 17 • simulation results -- M cos(x) -90 -60 -30 0 30 60 90 ecliptic latitude (degrees) Figure 2.6: A latitude distribution of projectile deliveries to the Earth from one set of simulations. This figure has not accounted for the spin obliquity of the planet (hence "ecliptic latitude"). The points represent the total number of impacts in 10° ecliptic latitude bins and the dashed curve is a cosine multiplied by an arbitrary constant, M. The error bars for the simulation results are smaller than the points. 18 in the fireball flux of roughly 15% amplitude (Figure 2.7). As a measure of the asymmetry between the poles and equator, we take the ratio between polar (within 30° of the poles) and equitorial (a 30° band centered on the equator) arrival densities. We find that when all terrestrial arrivals are con-sidered, the poles receive the same flux of impactors as the equator to < 1% (0.992 ± 0.001). We believe the uncertainty caused by our finite sampling of the orbital distribution is on this level and thus our results are consistent with uniform coverage. 1.1 X 0.9 o.a spring • # yearly average * * fall " -90 -60 -30 0 30 60 90 geocentric latitude (degrees) Figure 2.7: The geocentric latitude distribution, relative to the average spa-tial density of impacts, at different times of the year. The circles represent northern hemisphere spring, when the Earth's spin axis (as viewed from above) is tilted away from the planet's direction of motion. The near mirror-image can be seen in northern hemisphere fall (squares) when the spin axis is tilted towards the Earth's direction of motion. Averaged over a full year (diamonds), the deviation from the expected flat distribution is minimal. To better understand the details of latitudinal asymmetries, we consider 19 the effect of approach velocity, Voo . When our projectile deliveries are filtered such that vcuti < i>oo < vcut2, and restricted to impactors with i < 10° we find results similar to Le Veuvre and Wieczorek (2006, our Figure 2.8a can be compared to their Figure 2). Their Ng parameter is analogous to our cuts. Small encounter velocities produce more uniform coverage because the trajectories are bent towards the poles. Higher encounter velocity tra-jectories are not effected to as great a degree and tend to move in nearly straight lines, leading to a distribution which tends to a cosine-like curve. Obviously when the i restriction is lifted, the poles receive a higher flux which mutes the amplitude of the variation (Fig 2.8b). The preliminary results of Le Feuvre and Wieczorek [38] show a 30% ecliptic latitudinal variation in terrestrial projectile deliveries. Though we are able to produce results similar to theirs under various restrictions (Fig-ure 2.8a), the inclusion of the Earth's spin obliquity is necessary to ac-curately reflect reality. Wi th this inclusion, we find a latitude distribution which is very nearly uniform (Figure 2.8b). 2.3.2 A M / P M asymmetry The ecliptic latitudes and longitudes for the terrestrial arrivals were con-verted into local times via a straightforward method. As stated in the previ-ous section, the Earth's spin-pole azimuth is chosen at random to represent any day of the year and then the arrival location is transformed to these geocentric coordinates. The location relative to the sub-solar direction gives the local time. We looked for a P M excess in our simulation results. However, as evident in Figure 2.9a we see the opposite effect. Previous modeling work produced a near mirror image (reflected through noon) of our result (see our Figure 2.9b and Figure 3 of Halliday and Griffin [33]), but their impactor orbital distri-bution was very different from the debiased N E O model; they chose a small set of orbits with perihelion q in the range 0.62 < q < 0.99 A U with semima-jor axis o obeying 1.3 < a < 3.2 A U . If we restrict our simulation results to approximately the same impactor distribution (by taking only those objects having a and q within ± 0 . 0 1 A U of the entries in Table 1 of Halliday and Griffin [33]), we obtain Figure 2.9b which is very similar to their result. Why is there this discrepancy with our unrestricted case? The origin is not one of method but rather of starting conditions. The debiased N E O model we use is much more comprehensive than the orbits used by Halliday and Griffin [33], where only orbits assumed to represent then-current fireballs were included. The real Earth-crossing population contains a larger fraction 20 -90 -60 -30 0 30 60 90 geocent r ic lat i tude (degrees) 1.2 c -o 1 +-> O S3 •So. > ^ 0.6 no l restriction 12 < v < 17 km/s OO a l l * " * " 3 0 < v < 35 k m / s 1 - Z , (b) -90 -60 -30 0 30 60 90 geocent r ic lat i tude (degrees) Figure 2.8: The terrestrial impact density versus geocentric latitude (a) for different ranges of encounter velocity and restricted to objects with in-clination < 10° and (b) the same with no inclination restriction. For lower ^ 0 0 objects, there is less latitudinal variation because the Earth's gravita-tional field bends incoming trajectories towards higher latitudes. This effect is muted as the velocities become larger - the trajectories move in straighter lines. For these faster objects, the available impact area varies as a cosine so the shape of the distribution for high impactors is expected to do the same. The deviations from a cosine are due to the fact the objects have moderate inclinations and non-infinite velocities. Note that (b) shows the realistic case of all encounter speeds and inclinations; there is very little latitudinal variation. 21 20 <: v. <c 30 km/s imp simulation results 9 12 15 Local Time 24 1.2 <X3 8 ^ 1 o <D 0.8 > 0.6 (b) * i * i i T 1 1 r * restricted to HG82 orbits i i J L J L 0 3 6 9 12 15 18 21 24 Local Time Figure 2.9: The local fall time distribution of arrivals at the top of Earth's atmosphere for both our simulations and radar data (a). The simulation deliveries have been corrected for the Earth's spin obliquity. Clearly there is a large A M excess in the arrivals which is counter to the meteorite fall data which show a P M enhancement. Though a velocity restriction is used (see text), other cuts yield the same general shape for both radar data and simulation results, (b) The local time distribution of Earth deliveries when restricted to orbits similar to the ones used in Halliday and Griffin [33]. This distribution matches well with the time of falls for chondrites. 22 of high-speed orbits, which produce a smaller fraction of P M falls than the shallow Earth-crossers. In fact, our simulated arrivals always show an A M excess (Figure 2.10), even if we apply an upper speed bound (in an attempt to mimic a condition for meteoroid survivability, requiring speeds of less than 20-30 km/s at the top of the atmosphere). Figure 2.10: The fraction of Earth arrivals on the P M hemisphere, as a function of various cutoff speeds in the incoming flux. For the entire popula-tion, the P M ratio is 40%. Applying more and more stringent upper bounds pushes the P M ratio towards 50%, but for reasons discussed in the text we do not believe that the N E O orbital distribution is the same as that of meteorite-dropping fireballs, and thus the A M excess we find (which is not exhibited by the fireball data as a whole) does not create a contradiction with the available data. The apparent conflict with the meteorite data should not be too surpris-ing, as Morbidelli and Gladman [43] argued that that orbital distribution of the 0.1-1 m-scale meteoroids that drop chondritic fireballs must be different than that of the Near-Earth Objects. They showed that to match the ra-23 diant and orbital distributions determined by the fireball camera networks and to also match a P M excess, these sub-meter sized bodies must suffer strong collisional degradation as they journey from the asteroid belt, with a collisional half life consistent with what one would expect for decimeter-scale objects; this produces a match with the fireball semimajor axis distribution, which is dominated by the a > 1.5 A U orbits. In contrast, our simulations show N E O arrivals are much more dominated by a ~ 1 A U objects. We posit this is further evidence that the source region for the majority of the meteorites (the chondrites) is the main belt and not near-Earth space; to use the terminology of Morbidelli and Gladman [43], the 'immediate precursor bodies', in which the meteoroids were located just before being liberated and starting to accumulate cosmic-ray exposure, are not near-Earth objects but must be in the main belt. As a consistency check, we compare our fall time distribution to radar observations [36] of meteoroids arriving at the top of Earth's atmosphere. Figure 2.9a shows that the fall time distribution obtained with our simula-tions is similar to the flux of radar-observed meteors when restricted to the same top-of-the-atmosphere speed range of 20 < Vimp < 30 km/s (though other cuts yield similar results). The typical pre-atmospheric masses of the particles producing the radar meteors is in the micro- to milli-gram range (P. Brown, private communication 2006). The velocity range chosen for Fig-ure 2.9a removes the low-speed fragments which have reached Earth-crossing by radiation forces (unlike the NEOs of the Bottke model, whose orbital dis-tribution is set by gravitational scatterings with the terrestrial planets) and also removes the high-speed cometary component. The match we find permits the hypothesis that the majority of the milligram-particle flux on orbits with these encounter speeds is actually dust that is liberated continuously from NEOs, in stark contrast with the decimeter-scale meteoroids, who must be recently derived from a main-belt source. 2.4 Lunar bombardment Figure 2.11 shows the lunar impact speed distribution from our simula-tions. Because the Moon's orbital and escape speeds (1.02 and 2.38 km/s respectively) are both small compared to typical Voo encounter speeds, the impacts are not as biased towards smaller speeds as for the Earth. As one would expect, since v2mp = v2^ + v2sc, the Vimp and source VQQ distributions are quite similar. The small difference arises from gravitational focusing by 24 the Moon, which increases the speeds of the low Voo population. We compute the average impact speed for NEOs striking the moon to be 20 km/s. This is higher than the often quoted lunar impact velocity of 12-17 km/s [28, 53]. These lower velocities have been derived using only the known NEOs and are therefore biased towards objects whose encounter speeds are lower (which are more often observed in telescopic surveys). 0.1 0.08 § 0.06 > o ti: o.o4 0.02 0, [impactor v0 (dash) r-\ v. (solid) imp v J source v . 0 10 15 20 25 30 35 40 45 velocity (km/s) Figure 2.11: Velocity distributions for the lunar impacts from our simu-lations (impactor and Vimp) and the sampled N E O source population (source Voo ). The average impact velocity, is ~ 20 km/s. The curve showing the Voo of objects striking the Moon closely matches the Vimp distribution as one expects due to the small degree at which the moon's gravity well "speeds up" low Voo objects. The difference between the Vimp and source •Uoo distributions shows that gravitational focusing favours low speed (voo ;$ 20 km/s) objects. These simulation results were for the current Earth-Moon distance with the Moon orbiting in the plane of the ecliptic. The debiased N E O distribution we use has a full suite of high-speed 25 impactors; more than half the impactors are moving faster than vmed = 19.3 km/s when they hit the Moon. This has serious implications for the calculated projectile diameters that created lunar craters since the higher speeds we calculate mean that typical impactor diameters are smaller than previously derived. Strictly speaking, our results apply only when the current N E O orbital distribution is valid, which has likely been true since the post-mare era. However, the generically-higher lunar impact speeds we find are likely true in most cases for realistic orbital distributions, and thus the size-frequency distribution of the impactors must be shifted to somewhat smaller sizes in trying to find matches between the lunar crater distribution and the N E A size distribution (see Strom et al 2005 for a recent example). The reader may be surprised to see that the average lunar impact speed is essentially the same as the average arrival speed at Earth despite the ac-celeration impactors receive as they fall into the deep gravity well of our planet. This (potentially counter-intuitive) result can be understood once one realizes that Earth's impact speed distribution is heavily weighted to-wards low VQO values by the Safronov factor (1 + vlgjv2^). For Earth this so heavily enhances the low encounter speed impactors that the average im-pact speed actually drops to essentially the same as that of the Moon (which does not gravitationally focus the low v^ nearly as well). While the Earth's greater capture cross-section ensures a much larger total flux, the average energy delivered per impact will be similar for both the Moon and Earth. 2.4.1 From impacts to craters To examine crater asymmetries on the Moon, we need to convert our sim-ulated impacts (a sample of which are shown in Figure 2.12) into craters which will account for the added asymmetry resulting from the impact ve-locity Vimp of the impactors. In typical crater counting studies, there is some minimum diameter T which observers are able to count down to due to im-age resolution limitations. There will be more craters larger than T on the leading hemisphere than on the trailing because leading-side impactors have higher impact speeds on average and the commonly-accepted scaling relation for crater size Dc depends on the velocity. where q « 0.28 — 0.33 [41] and Di is the diameter of the impactor. Since both hemispheres receive flux from the same differential impactor size distribution obeying Dc oc v. 2q nZq imp i ' (2.7) dN (2.8) dDi 26 we can integrate "down the size distribution" to determine a weighting factor which transforms our impacts into crater counts. The number of craters N with Dc > T produced by the impacting size distribution is N ^ > r ) = f l b ~ d D i = v^~i D ^ K * " 1 ) / 3 ' ( 2 - 9 ) where Dimin is some minimum impactor diameter and we have substituted Dimin oc (T v72q)1/Zq. Since the differential size index p ~ 2.8 [27, 54], J V ^ uimp-For each simulated impact at a specific latitude and longitude, we assign that impact N = C v\^p craters, where C is an arbitrary proportionality constant. The dependance on p is small as our results are essentially the same when using an older determined value of p = 2. The weak dependance arises from the low orbital speed of the Moon relative to the encounter speeds of the incoming projectiles. Thus for moons such as the Galilean satellites, whose orbital speeds are much higher compared to that of the incoming flux, the value for p becomes more important. Note that Dmin and q are actually irrelevant to our analysis since we are interested in only the crater numbers relative to an average rather than the crater sizes. 2.4.2 Latitude distribution of impacts One expects the departure from uniform density in the lunar latitude distri-bution to be more severe than that of the Earth due to the Moon's smaller mass. Comparing Figure 2.13 to Figure 2.8b shows this is indeed the case. At high Voo cuts, the variation in the latitude distribution tends to the predicted cosine (see Figure 2 in Le Feuvre and Wieczorek, 2006). However, when ex-amining the real case of all lunar impacts we see only a ~ 10% (0.912±0.004) depression at the poles when taking the ratio of the derived crater density within 30° of the poles to the crater density in a 30° band centered on the equator. In contrast, Le Feuvre and Wieczorek [38] (their Figure3) find a polar/equatorial ratio of roughly 60%. The source of this large discrepancy is unclear since the Moon in our simulations had zero orbital inclination and spin obliquity, the same conditions used by Le Feuvre and Wieczorek [38], and the latter also used the Bottke et al. [27] model as an impactor source. Despite the variation we observe being small, researchers should be aware of this spatial variation in the crater distribution when determining ages of surfaces (see Sec. 3). 27 east longitude Figure 2.12: A n equal-area projection of the lunar impacts from our sim-ulations. The Moon was at its current orbital distance of 60.R® with 0° inclination. At the apex, one can see the slight enhancement in the impact density. For clarity, only 7% of the total number of impacts are shown. 2.4.3 Longitudinal effects We wish to compare the results of our numerical simulations with obser-vational data to create a consistent picture of the current level of cratering asymmetry between the Moon's leading and trailing hemispheres. As a mea-sure of this asymmetry, we look at the crater density as a function of the angle away from the apex, ft, which should roughly follow the functional form of Eq. 2.1. In Figure 2.14 we show the results from our simulations as well as a fit to Eq. 2.1 using a maximum likelihood technique assuming a Poisson probability distribution. The best fit parameters resulting from an unrestricted analysis are f = 1.02, a = 0.564, and g = 0.225; this would require b — —1.268. For an impactor diameter distribution following 4|j oc D~b, this value for the slope yields the unphysical situation of having 28 71 1 I 1 I 1 I 1 ' I L J -90 -60 -30 0 30 60 90 ecliptic latitude (degrees) Figure 2.13: The latitude distribution of craters on the Moon from our simulations. Note the scale is different here than in Figure 2.8(b) and that the latitudinal variation is larger for the Moon than the Earth. This is because the Moon's gravity well is not deep enough to significantly modify incoming trajectories to higher latitudes. If the impacts were restricted to impactors which had i < 10° , the variation would be larger as is the case with the terrestrial deliveries in Figure 2.8. Here the Moon's inclination and spin obliquity are not accounted for. 29 more large impactors than small ones. Obviously this cannot be correct and is a consequence of the degeneracy present in Eq. 2.1. We once again con-clude that by fitting Eq. 2.1 to an observed surface distribution, it is virtually impossible to decouple a and g to obtain information about the impactor size distribution and the average encounter velocity of the impactors. Figure 2.14: The spatial density of simulated craters as a function of angle from the apex of motion, 8. The vertical axis is craters/km 2 relative to the mean density over the entire lunar surface. Points represent simulation results while the curve gives a fit using a maximum likelihood technique to the equation T(B) = f (1 + a cos/?) 9, where g = 2 + 1.4 b and b = 0.58 as determined from observations. Wi th this restriction, the best fit values are f = 0.994 and a = 0.0472 with a reduced chi-square of %2 = 2.4. By counting our simulated craters, we find a value for the G M A A C A of 1.29 ± 0.01, significantly lower than the values of 1.4-1.7 estimated in Sec. 2.1.2. This discrepancy can be reconciled by recalling that we find the average impactor Voo to be ~ 20 km/s. Using this value in Eq . 2.2 and the 30 observationally-determined value of b = 0.58, gives a G M A A C A value of 1.32, close to the value we obtain. In Figure 2.14 we notice that the distribution is rather flat for 0° < 3 < 45° and has higher density than the predicted curve (using b = 0.58) for 60° < 3 < 120°. A v a l u e o f 2 - 4 results when the quality of fit is assessed. We believe the origin of this highly-significant departure from the theoretically predicted form lies simply in the fact that both assumptions of an (1) isotropic orbital distribution of impactors with (2) a single VQO value, are violated (see Figure 2.1). Thus, an observed crater field will not follow the form of Eq. 2.1 in detail. Since these assumptions break down for the real impactor population, we will instead directly compare with the rayed crater observations to de-termine if the available data rule out our model. To do this, we scaled our craters down to 222, the same number as in the rayed crater sample used in Morota and Furumoto [44]. We restricted our simulated craters to the same lunar area examined in that study. Mare regions were ignored as it introduces bias in rayed crater observations because they are easier to iden-tify against a darker background surface. In addition, rayed craters on mare surfaces are likely older than their highland counterparts because it takes longer for micrometeorite bombardment to eliminate the contrast. Since we are interested in the current lunar bombardment, it is necessary to eliminate this older population. The area sampled includes latitudes ±41.5° and lon-gitudes 70.5° - 289.5° (see Figure 1 of Morota and Furumoto, 2003). Using a modified chi-square test with the rayed crater observational counts Oi and the expected counts Ei from our simulations, since we are dealing with Poisson statistics. This procedure results in a reduced chi-square value of x2 — 0.67. Thus our model is in excellent agree-ment with the observational data (Figure 2.15). Ideally, we would like to obtain a G M A A C A value for the rayed crater data. However, due to small number statistics and the restricted area (be-cause of Mare Marginis and Mare Smythii, much of the area near the the antapex is excluded), the G M A A C A value is poorly measured by the avail-able data. Regardless, integrating the best fit of the rayed crater data (best fit values: f = 1.53 x l O " 5 k m - 2 , a = 0.063, g = 4.62, with x2- = 0.49) (2.10) where (2.11) 31 Figure 2.15: Density of rayed craters (circles) used in Morota and Furumoto [44] compared with the results from our simulations (squares) for the same lunar area. For reference, the curve is the best fit to the rayed crater data using Eq. 2.1 as a model despite the fact this functional form is based on assumptions which we know to be violated. Best fit parameters are T = 1.53 x 10~5 k m - 2 , a = 0.063, and g = 4.62. The simulation results were scaled to the rayed crater data so that both had 222 craters in the same area. Wi th respect to the observations, the prediction from our numerical calculations has x2. = 0.67 32 from 0-30° and 150-180° to form the G M A A C A ratio gives 1.7±o;6, consis-tent with our result. 2.4.4 Varying the Earth-Moon distance and the effect of inclination Due to tidal evolution, in the distant past the Moon's orbit was smaller. As the orbital distance is decreased, the Moon's orbital speed rises. This increases the impact speeds on the leading hemisphere and makes "catch-ing up" to the Moon from behind more difficult. As Eq. 2.2 suggests, the degree of apex/antapex asymmetry on the Moon is expected to increase as the orbital speed of the satellite does. We examined this by running other simulations with an Earth-Moon separation of a =50, 38, 30, 20, and 10 R®. Figure 2.16 shows our results. Assuming the same impactor orbital distribu-tion in the past, only a mild increase in the apex enhancement is seen (since the orbital speed only increases as l/\/{a)). The increase is that expected based on the resulting change in a caused by the larger vorb (see Eq. 2.2). As discussed in Sec. 2.1.3, the asymmetry between the near and far hemispheres should depend on the lunar distance. Figure 2.16 shows the ratio between nearside and farside craters from our simulations. For all lunar distances we find very little asymmetry. Thus, our results do not support the most recent study which claims a factor of four enhancement on the nearside when compared to the far [37], but are in good agreement with the work done by Wiesel [59] and Bandermann and Singer [24]. We see mild evidence for Bandermann and Singer's (1973) assertion of the Earth acting as a shield for a < 25i?© and little effect outside this distance. In the bulk of our simulations we have used the approximation that the Moon's orbit is in the ecliptic plane. Since we show that the latitudinal de-pendence of lunar cratering is weak (~ 10% reduction within 30° of the poles relative to a 30° band centered on the Moon's equator), we do not expect the inclusion of the moon's orbital inclination to alter our results signifi-cantly, although we expect the polar asymmetry to monotonically decrease with increasing orbital inclination. We have confirmed this by computing the G M A A C A and polar asymmetry ratios for a less-extensive set of sim-ulations with the initial lunar orbital inclination set to its current value of 5.15° and we use the sub-Earth point at the time of impact to compute luno-centric latitudes and longitudes. We find a slight reduction of G M A A C A to 1.24 ± 0.02 from (1.29 ± 0.01) and a crater density within 30° of the pole that is statistically the same as the 0° inclination case (0.914±0.009 instead of 0.912 ±0 .004) . 33 1.5 C/3 O 1.4h Leading / Trailing 1.2 1.3 z o z 1.1 Near / Far 0 1 0.9 10 20 30 40 50 60 Earth - Moon Distance in Earth Radii Figure 2.16: The ratio between total number of craters on the leading ver-sus trailing hemispheres and the same ratio between the nearside and farside hemispheres as a function of lunar orbital distance. As expected, smaller or-bital distances increase the asymmetry between leading and trailing hemi-spheres. This is a result of the increased orbital speed as the Moon is brought closer to the Earth. For all lunar distances there is minimal nearside/farside asymmetry, with some evidence of the Earth shielding the lunar nearside in the very distant past when the lunar semimajor axis was < 25 i?©. 34 2.5 Bibliography [24] Bandermann, L . W. , and Singer, S. F . 1973. Calculation of meteoroid impacts on Moon and Earth. Icarus 19, 108-113. [25] Barricelli, N . A . , and Metcalfe, R. 1975. A note on the asymmetric distribution of the impacts which created the lunar mare basins. The Moon 12, 193-199. [26] Bottke, W . F., Jedicke, R., Morbidelli, A . , Petit, J . -M. , and Gladman, B . 2000. Understanding the distribution of near-Earth asteroids. Science 288, 2190-2194. [27] Bottke, W . F., Morbidelli, A . , Jedicke, R., Petit, J . -M. , Levison, H . F. , Michel, P., and Metcalfe, T . S. 2002. Debiased orbital and absolute mag-nitude distribution of the near-Earth objects. Icarus 156, 399-433. [28] Chyba, C. F. , Owen, T. C , and Ip W . - H . 1994. Impact delivery of volatiles and organic molecules to Earth. Hazards Due to Comets & As-teroids. Space Science Series, Tuscon, A Z : Edited by Tom Gehrels, M . S. Matthews, and A . Schumann. Published by University of Arizona Press, 1994., p.9. [29] Dones, L . , Gladman, B . , Melosh, H . J . , Tonks, W . B . , Levison, H . F. , Duncan, M . 1999. Dynamical lifetimes and final fates of small bodies: Orbit integrations vs Opik calculations. Icarus 142, 509-524. [30] Gladman, B . , and Duncan, M . 1991. Symplectic integrators for long-term integrations in celestial mechanics. Celestial Mechanics and Dynam-ical Astronomy 52, 221-240. [31] Gladman, B . , Michel, P., Froeschle, C. 2000. The near-Earth object population. Icarus 146, 176-189. [32] Halliday, I. 1964. The variation in the frequency of meteorite impact with geographic latitude. Meteoritics 2, no. 3, 271-278. [33] Halliday, I., and Griffin, A . A . 1982. A study of the relative rates of meteorite falls on the Earth's surface. Meteoritics 17, 31-46. [34] Horedt, G. P., and Neukum, G. 1984. Cratering rate over the surface of a synchronous satellite. Icarus 60, 710-717. 35 [35] Ivanov, B . 2001. Mars/Moon cratering rate ratio estimates. Space Sci-ence Reviews 96, 87-104. [36] Jones, J. , Brown, P., Ellis, K . J . , Webster, A . R., Campbell-Brown, M . , Krzemenski, Z., Weryk, R. J . 2005. The Canadian Meteor Orbit Radar: system overview and preliminary results. Planetary & Space Science 53, 413-421. [37] Le Feuvre, M . , and Wieczorek, M . A . 2005. The asymmetric cratering history of the Moon. 36th Annual Lunar and Planetary Science Confer-ence, March 14-18, 2005, League City, Texas, abstract no.2043. [38] Le Feuvre, M . , and Wieczorek, M . A . 2006. The asymmetric crater-ing history of the terrestrial planets: Latitudinal effect. 37th Annual Lu-nar and Planetary Science Conference, March 13-17, 2006, League City, Texas, abstract no. 1841 [39] Levison, H . F. , and Duncan, M . J . 1994. The long-term behaviour of short-period comets. Icarus 108, 18-36. [40] McEwen, A . S., Moore, J . M . , and Shoemaker, E . M . 1997. The Phanero-zoic impact cratering rate: Evidence from the farside of the Moon. Journal of Geophysical Research 102, 9231-9242. [41] Melosh, J . 1989. Impact cratering: A geologic process. Research sup-ported by N A S A . New York, Oxford University Press (Oxford Mono-graphs on Geology and Geophysics, No. 11), 1989, 253 p. [42] Moore, J . M . and McEwen, A . S. 1996. The abundance of large, Copernican-age craters on the Moon. Lunar and Planetary Institute Conference Ab-stracts 27, 899. [43] Morbidelli, A . , and Gladman, B . 1998. Orbital and temporal distri-butions of meteorites originating in the asteroid belt. Meteoritics and Planetary Science 33, no. 5, 999-1016. [44] Morota, T., and Furumoto, M . 2003. Asymmetrical distribution of rayed craters on the Moon. Earth and Planetary Science Letters 206, 315-323. [45] Morota, T., Ukai, T., and Furumoto, M . 2005. Influence of the asym-metrical cratering rate on the lunar chronology. Icarus 173, 322-324. 36 [46] Morota, T., Karuyama, J. , and Furumoto, M . 2006. Lunar apex-antapex cratering asymmetry and origin of impactors in the Earth-Moon system. 37th Annual Lunar and Planetary Science Conference, March 13-17, 2006, League City, Texas, abstract no.1554 [47] Pinet, P. 1985. Lunar impact flux distribution and global asymmetry revisited. Astronomy and Astrophysics 151, 222-234. [48] Pieters, C. M . , Staid, M . I., Fischer, E . M . , Tompkins, S., He, G . 1994. A sharper view of impact craters from Clementine data. Science 266, 1844-1848. [49] Rendtel, J. , and Knofel, A . 1989. Analysis of annual and diurnal vari-ation of fireball rates and the population index of fireballs from different compilations of visual observations. Astronomical Institutes of Czechoslo-vakia, Bulletin (ISSN 0004-6248) 40, no. 1, 53-62. [50] Shoemaker, E . M . and Wolfe, R. A . 1982. Cratering timescales for the Galilean satellites. In Satellites of Jupiter (D. Morrison, Ed.), pp. 277-339. Univ. of Arizona Press, Tucson. [51] Shoemaker, E . M . 1983. Asteroid and comet bombardment of the earth. Annual Review of Earth and Planetary Sciences 11, 461-494. [52] Stofner, D., and Ryder, G . 2001. Stratigraphy and isotope ages of lunar geologic units: chronological standard for the inner solar system. Space Science Reviews 96, 9-54. [53] Strom, R. G. , Malhotra, R., Ito, T., Yoshida, F. , and Kring, D. A . 2005. The origin of planetary impactors in the inner solar system. Science 309, 1847-1850. [54] Stuart, J . A near-Earth asteroid population estimate from the L I N E A R survey. 2001. Science 294, 1691-1693. [55] Turski, W . 1962. On the origin of lunar maria. Icarus 1, 170-172. [56] Valsecchi, G . B. , Jopek, T. J. , and Froeschle, CI. 1999. Meteoroid stream identification: a new approach - I. Theory. Mon. Not. R. Astron. Soc. 304, 743-750. [57] Wetherill, G. W . 1968. Stone meteorites: Time of fall and origin. Science 159, 79-82. 37 [58] Wetherill, G . W . 1985. Asteroidal source of ordinary chondrites. Mete-oritics 20, 1-22. [59] Wiesel, W . 1971. The meteorite flux at the lunar surface. Icarus 15, 373-383. [60] Wood, J . A . Bombardment as a cause of the lunar asymmetry. Earth, Moon, and Planets 8, 73-103. [61] Zahnle, K . , Dones, L . , and Levison, H . F . Cratering rates on the Galilean satellites. Icarus 136, 202-222. [62] Zahnle, K . , Schenk, P., Sobieszczyk, S., Dones, L . , and Levison, H . F . Differential cratering of synchronously rotating satellites by ecliptic comets. Icarus 153, 111-129. 38 Chapter 3 S u m m a r y a n d c o n c l u s i o n s 1 We have used the debiased N E O model of Bottke et al. [63] to examine the bombardment of the Earth-Moon system in terms of various impact and crater asymmetries. For Earth arrivals we find a < 1% variation in the ratio between the areal densities within 30° of the poles and within a 30° band centered on the equator. The local time distribution of terrestrial impacts from NEOs is enhanced during the A M hours. While this fall-time distri-bution corresponds well to recent radar data, it is in disagreement with the chondritic meteorite data and their derived pre-atmospheric orbital distri-butions. This discrepancy thus reinforces the conclusion of Morbidelli and Gladman [64] that the large amount of decimeter-scale material being ejected from the main asteroid belt onto Earth-crossing orbits must be collisionally depleted before much of it can evolve to orbits with a < 1.5 A U . A significant result is that we find the average impact speed onto the Moon to be Vimp = 20 km/s, with a non-negligible higher-speed tail (Fig. 2.11) This combined with quantification of the non-uniform surface cratering has implications for both tracing crater fields back to the size distribution of the impactors and the absolute (or relative) dating of cratered surfaces. First, the higher impact speeds we find mean that lunar impact craters (at least in the post-mare era when we believe the N E O orbital distribution we are using is valid) have been produced by smaller impactors than previously cal-culated. This roughly 10% higher average impact speed corresponds to lunar impactors which are 10% smaller on average than previously estimated; this small correction has ramifications for proposed matches between lunar crater size distributions and impactor populations (e.g., [65]). We find two different spatial asymmetries in current crater production due to NEOs. As expected, due to its smaller mass, the Moon exhibits more latitudinal variation (~ 10%) in our simulations than the Earth. When comparing our simulation results to young rayed craters on the Moon, the surface density variation we predict is completely consistent with available 1 A version of this chapter has been submitted for publication consideration. Gallant, J., Gladman, B., and Cuk, M. (2006) Current Bombardment of the Earth-Moon System: Emphasis on Cratering Asymmetries. Icarus. 39 data; we obtain a leading versus trailing asymmetry of 1.29±0.01 ( G M A A C A value), which corresponds to a 13% increase (decrease) in crater density at the apex (antapex) relative to the average. These results indicate that using a single globally-averaged lunar crater production could give ages in error by up to 10% depending on the location of the studied region. For example, post-mare studies of the Mare Orientale region would overestimate its age by ~ 10% due to its proximity to the apex, assuming that leading point and poles of the Moon have not changed over the last ~4 Gyr. Similarly, the degree of bombardment on Mare Crisium (not far from the antapex) would be lower than the global average. These effects will only be testable for crater fields with hundreds of counted craters so that the Poisson errors are small compared to the 10% variations we find; in most studies of "young" (< 4 Gyr) lunar surfaces the crater statistics are poorer than this (e.g., Stofner and Ryder 2001). When the orbital distance of the Moon was decreased (as it was in the past because of its tidal evolution), the ratio between simulated craters on the leading hemisphere versus the trailing increased as expected due to the higher orbital speed of the satellite at lower semi-major axes. We find virtu-ally no nearside/farside asymmetry until the Earth-Moon separation is less than 30 Earth-radii, which under currently-accepted lunar orbital evolution models dates to the time before 4 Gyr ago (at which point the current N E O orbital distribution may not be a good model for the impactors). Interior to 30 Earth-radii we find that the Earth serves as a mild shield, reducing nearside crater production by a few percent. 40 3.1 Bibliography [63] Bottke, W . F. , Morbidelli, A . , Jedicke, R., Petit, J . -M. , Levison, H . F. , Michel, P., and Metcalfe, T. S. 2002. Debiased orbital and absolute mag-nitude distribution of the near-Earth objects. Icarus 156, 399-433. [64] Morbidelli, A . , and Gladman, B . 1998. Orbital and temporal distri-butions of meteorites originating in the asteroid belt. Meteoritics and Planetary Science 33, no. 5, 999-1016. [65] Strom, R. G. , Malhotra, R., Ito, T., Yoshida, F. , and Kring, D. A . 2005. The origin of planetary impactors in the inner solar system. Science 309, 1847-1850. 41 Appendix A E a r t h - M o o n S y s t e m A s y m m e t r i e s A . l A M / P M What is the physical picture one should have in mind when discussing local fall time asymmetries on the Earth? First, for convienience, assume the Earth has a 0° spin obliquity. Next break the Earth into two hemispheres, with the division being the meridian parallel to the sub-solar direction (see Figure A . l ) . The side facing the direction of motion of the Earth is the morning or A M hemisphere because the Sun will be rising as the Earth rotates. Conversely, the side opposite to the direction of the planet's motion is the afternoon or P M hemisphere, as the Sun will be setting. Wi th these conventions, local noon will be on the dividing meridian on the hemisphere facing the Sun, while local midnight is on the same meridian, but facing away from the Sun. 6 A M and 6 P M are at the leading and trailing points of the Earth's motion, respectively. The addition of the spin obliquity is a trivial matter and only serves to make the calculation of the fall times of the simulated arrivals as realistic as possible. Observing asymmetries in the fall time distribution depends on the in-coming orbits. If there is a significant fraction of material striking the Earth that originates in the Main Asteroid Belt, there will be a P M excess, be-cause this material would have perihelion distances just inside the Earth's orbit and would coat the Earth's trailing hemisphere. However, if there is more material on low semi-major axis or on high-eccentricity orbits, then the leading (AM) hemisphere will receive more flux, leading to a morning excess of arrivals. Figure A.2 shows an orbit with a > 1 A U with perihelion q = 1 A U . From the formula for the velocity at perihelion we see that any ellipse with a > 1 A U and q = 1 A U will have it's velocity enhanced by ((1+e)/(1—e)) a 5 compared to the velocity the Earth has on it's (A. l ) 42 circular orbit ( G M / a ) 0 , 5 . Therefore the relative velocity vector comes in from behind the Earth. If focusing by the Earth's gravity well was unimportant than all objects on orbits with a > 1 A U and q = 1 A U would arrive between noon and midnight. Focusing brings low encounter velocity objects around the limbs of the Earth to the morning hours. Figure A . l : A diagram showing that, although the Earth is rotating, local noon is always the sub-solar point. If more objects encounter the Earth from the Main Asteroid Belt, there will be a P M excess in the local fall time distribution as these orbits tend to "catch up" to the Earth from behind. However, if a greater number of objects with high eccentricities (as to be nearly radial) or low semi-major axes, the morning hemisphere will be coated preferentially, introducing an A M excess. Note that this figure is not to scale. 43 Figure A.2: A n orbit with a > 1 intersecting the Earth's orbit at 1 A U . The velocity of the object VA on the larger orbit is greater than the Earth's VE at the time of intersection. Thus the relative velocity vector makes it appear (from the Earth) that the object is catching up to the Earth. If gravitational focusing was negligible, then all objects on these types of orbits would fall between noon and midnight (see Figure A . l for local time definitions). 44 A .2 Leading/Trailing and Nearside/Farside B y definition, the Moon being a synchronous satellite means that it always presents the same side to the Earth. The leading hemisphere is thus the side corresponding to the Moon's direction of motion and the trailing side is that opposite to the motion. These sides never change, again because of the Moon's synchronous nature. Therefore, one expects the leading hemisphere of the Moon to have a higher crater density than the trailing because it is harder for incoming projectiles to "catch up" with the trailing side. As well, on the leading side, there will be more craters bigger than some minimum diameter. This results from higher impact speeds onto the leading hemispere which make larger craters as seen from the scaling relation D oc v2q D3q CX. U i m p LJi (A.2) where Dc is the crater diameter, Vimp is the impact speed, and Di is the diameter of the impactor. Earth \ leading trailing Moon Figure A.3: Here there is an increased flux of impactors on the leading hemisphere of the Moon. As well, the objects approaching the trailing side have a lower velocity as seen from the Moon. Note that this figure is not to scale. 45 Since the Moon always presents the same side to us, the concept of a nearside/farside asymmetry is straightforward to understand. If a beam of incoming, zero inclination particles enters the Earth-Moon system while the Earth is between the Moon and objects, the gravity well of the Earth can modify the trajectories to hit the nearside of the Moon (see Figure A.4). If the Earth was not present, these trajectories would move in straight lines, missing the Moon. This would also be the case for when the Moon is between the incoming beam of objects and the Earth. Here the incoming objects pass the Moon before being gravitationally modified. As well, if the incoming speeds are large, their trajectories will be modified less, leading to less flux impacting the Moon. Slower objects will have a greater chance of being focused to the nearside. without Earth Figure A.4: Here an incoming object, which would not have hit the Moon if the Earth was not present, has it's trajectory modified by the Earth's gravity and then strikes the Moon on the nearside. The vectors to the left of the Earth represent objects which the Earth shields from the Moon. Note that this figure is not to scale. 46 Appendix B R a y e d C r a t e r s A rayed crater is formed from impact ejecta landing on surrounding terrain. Several prominent examples can be found on the lunar surface (e.g., Tycho and Copernicus). These craters are useful in studies examining the recent bombardment history of the Moon because they are believed to be less than 1 Gyr old. This relatively young age is an approximate upper limit as impact gardening will degrade the rayed structure over time, making the rays indistinguishable from the background surface. However, background features can extend the lifetime. Rayed craters on mare regions will last longer because it takes longer for micro-meteorite bombardment to eliminate the contrast between light highland material and the dark basaltic mare. Therefore, when studying recent cratering, one should restrict to the young sample found on the lunar highlands, where the background terrain is more compositionally uniform. In our study we restrict to the same lunar area and sample size as Mo-rota et al.(2003). Figure B . l shows the locations of rayed craters I 5km in diameter as identified by Morota et aZ.(2003) from Clementine images. Re-gions bounded by empty boxes represent dark mare regions. Most of the nearside has been excluded as well for this reason. The area examined in-cludes 70-290° longitude (measured from the sub-Earth point) as well as ±42° latitude. 47 40 ^ 30 g 201 3 oi -10! •3 -201 ^ -30 -40 •Mare • • ' . . • . • • * » Mare Marrinis • # • | | • • * « * • « • • ^Mare Moscoviense • • • • ... % • • • •• • • • • • 4 Mare Sm^hii Mare • • • Australe South Pole-Aitken 9 9 9 • brientale 75 100 125 150 175 200 225 east longitude (degrees) 250 275 Figure B . l : The distribution of rayed craters larger than 5 km in diameter identified by Morota et aZ.(2003) for longitudes (measured from the sub-Earth point) 70-290° and latitudes ±42° . To compare with this study, we restrict our simulation results to the same area and scale our number of craters down to 222. Appendix C O r b i t a l E l e m e n t C o n v e r s i o n For our sampling of the debiased N E O model of Bottke et al.(2002), we have been given a set of 16307 semi-major axis a, eccentricity e, and inclination i, orbital element triplets. As noted in the text, each of these orbits may be in one of four orientations (see Figure O l with respect to its node crossing at (the arbitrarily chosen) (1 AU,0,0). To account for the different orbital configurations, we "clone" the sample by keeping the original triplet constant and choosing different values for, the argument of pericenter LO, the longitude of ascending node, fl, and the true anomaly, / . For clarity, LO is the angle between a reference direction (usually the Earth's vernal equinox, 7) and the pericenter of the orbit. The true anamoly is the angle between the pericenter and the object's position on it's orbit. Finally, the longitude of ascending node is the angle between 7 and the point at which the orbit crosses the ecliptic from below. Since we've set the node crossing (either ascending or descending) at (1 AU,0,0), / must satisfy r = l A U = ^ 4 (C.l) 1 + e cos / and = 0 or 2TT. For ascending encounters, / = 2ir — LO and LO = [0, IT) or LO = [TT, 2TT) if the pericenter is above or below the ecliptic respectively. For descending encounters, / = TT — LO or / = 3TT — LO for post and pre pericenter encounters, resepctively. Now that all orbital elements are known (a, e, i, LO, fl, and / ) , the con-version to cartesian coordinates can now take place. We begin with defining the x-axis along the unit vector (1 AU,0,0). A set of Euler rotations results in the position vector cos u cos fl — cos i sin fl sin u r = r | cos u sin fi + cos i cos fl sin u | , (C2) sin i sin u where the magnitude of the position vector, r, is obtained from Eq. C . l and where u = LO + /. 49 Figure C . l : The four different orientations that orbits from the N E O sample can have. Node crossings at the x-axis occur at (1 AU,0,0) where the Sun is assumed to be at the origin, (a) Ascending post-pericenter. (b) Descending post-pericenter. (c) Descending pre-pericenter. (d) Ascending pre-pericenter. The dotted portion on each orbit represents being below (z < 0) the ecliptic (the x-y plane). For our N E O distribution all orbits are prograde (counter-clockwise) but this need not be the case in general. 50 To determine the velocity of the object on the orbit, we first obtain the angular momentum vector ( sin Q, sin i \ — cosf is in i , (C.3) cosz / where h?/\JL = a(l — e 2) and u is the reduced mass of the system. For convenience, we create the components of the velocity vector directly in the orbit plane and introduce a right-handed system with f and 6 in the plane of the orbit and h perpendicular to the orbit plane. From 9 = h x f 9 = h/r2 r = T s m / , we have, for the velocity vector v = r06 + ri (C.4) 51 

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