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A carefully characterised trans-neptunian survey Alexandersen, Mike 2015

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A carefully characterised Trans-NeptuniansurveybyMike AlexandersenB. Astronomy, University of Copenhagen, 2007M. Astronomy, University of Copenhagen, 2009a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoralstudies(Astronomy)The University of British Columbia(Vancouver)August 2015c© Mike Alexandersen, 2015AbstractThe Trans-Neptunian Objects (TNOs) preserve evidence of planet buildingprocesses in their orbital and size distributions. While all populations showsteep size distributions for large objects, recently a relative deficit of Neptu-nian Trojans and scattering objects with diametersD < 100 km was detected.We have investigated this deficit with a 32 square degree survey, in which wedetected 77 TNOs brighter than a limiting r-band magnitude of 24.6. OurPlutino sample (18 objects in 3:2 mean motion resonance with Neptune)also shows a deficit of D < 100 km objects. We reject a single power-law sizedistribution at > 99% confidence. The fact that three independent samplesof three different populations show this trend suggests that it is a real fea-ture, possibly shared by all hot TNO populations as a remnant of “born big”planetesimal formation processes. We surmise the existence of 9000 ± 3000Plutinos with absolute magnitude Hr ≤ 8.66 and estimate 37000+12000−10000 Pluti-nos with Hr ≤ 10.0 (95% confidence ranges). Our survey also discovered onetemporary Uranian Trojan, one temporary Neptunian Trojan and one stableNeptunian Trojan. With these discoveries, combined with our survey char-acteristics, we derive populations of 110+500−100, 210+900−200 and 150+600−140 for thesepopulations, respectively, with Hr ≤ 10.0. With such approximately equalnumbers, the temporary Neptunian Trojans cannot be previously stable Tro-jans that happen to be escaping the resonance now; they must be capturedfrom another reservoir. Our population estimate also reveals that the Nep-tunian Trojans are less numerous than the main belt asteroids (semi-majoriiaxis 2.06 < a < 3.65), which has 592 asteroids with Hr ≤ 10.0. As the biasagainst detection of objects grows with larger semi-major axis, our discoveryof three 3:1 resonators and one 4:1 resonator adds to the growing evidencethat the high-order resonances are far more populated than can currently beexplained theoretically.iiiPrefaceThis thesis involves observations of outer Solar System minor bodies, as wellas modelling of their distributions. It is based on a single large observationalproject, and contains partial and/or modified reprints of previously publishedmaterials and materials that have been submitted for publication, as listedbelow. All figures and tables not explicitly stated below were created bymyself.Chapter 1, 2, 3, 4 and 5 contains material from this manuscript:• M. Alexandersen, B. Gladman, J. J. Kavelaars, J.-M. Petit, S. Gwyn,and C. Shankman. A carefully characterised and tracked Trans-Nep-tunian survey, the size-distribution of the Plutinos and the number ofNeptunian Trojans. Submitted Nov 2014, in revision, 2015This manuscript was submitted on November 27th 2014. A referee re-port was received on January 8th 2015. This report was generally positive,although it called for a significant amount of additional work. It has not beenpossible to revise and resubmit the manuscript in parallel with writing thisthesis, although many of the referee’s comments have been addressed in thisthesis. The manuscript will be revised and resubmitted in the fall of 2015.This manuscript presents the bulk of my thesis work, describing both themulti-year observational program and the subsequent analysis of the discov-ered TNOs. I was the Principal Investigator (PI) on the project; as such,ivI planned the observations, carried out the data-verification and character-isation, and conducted the analysis. The telescope time was proposed forby M. Alexandersen, J.-M. Petit, B. Gladman and S. Lawler (Lawler sub-sequently left the project). JJ. Kavelaars and S. Gwyn joined later andhave played a major role in data-reduction and processing. The observa-tions were chiefly performed by queue observers with MegaCam at the 3.6-m Canada-France-Hawaii Telescope (CFHT) during 2011-2013, with a fewadditional observations by queue observers with Gemini Multi-Object Spec-trograph (GMOS) at Gemini North in 2013. I planned and programmed allobservations on both telescopes. The obtained images from CFHT were re-duced and processed by JJ. Kavelaars and S. Gwyn; they used sophisticatedalgorithms to reduce the images (remove systematic noise) and to align theimages to an unprecedented accuracy, allowing us to measure unusually self-consistent astrometry. JJ. Kavelaars also ran the discovery images (threeimages of each field in a single night; a small portion of all the images fromthe many months) through an existing and well-tested automated MovingObject Pipeline (MOP) [Petit et al., 2004] which detected thousands of can-didate moving objects (most of which had been artificially implanted in theimages, also done by JJ. Kavelaars.). I subsequently inspected every singlecandidate detection, as the MOP produces many false detections - this in-spection process was very time consuming. Once the candidates had beeninspected, all real objects were tracked in follow-up observations. I searchedfor every object in our follow-up images (very time consuming) and used theadditional measurements to plan further observations.With almost all objects tracked to orbits sufficiently accurate for mod-elling, B. Gladman ran dynamical integrations to classify the objects usingan existing process [Gladman et al., 2008]. I performed modelling of variouspopulations of TNOs found in the survey, including the Plutinos and Trojans,focusing especially on the size distribution of the Plutinos. I wrote my owncode (using Python, Bash and Awk) for some analysis and all visualizationvof the results; for the vetting of candidate detections, I made extensive addi-tions and modifications to code originally by B. Gladman and R.L. Jones. Ialso used and edited the Survey Simulator originally written by J.-M. Petit,JJ. Kavelaars and B. Gladman (Fortran code), as well as the object-trackingsoftware originally written by JJ. Kavelaars (Perl script). For statistical anal-ysis, I combined and modified Fortran code written by R.L. Jones, S. Lawlerand J.-M. Petit; I also translated this code into Python code, but that versionwas not used in this paper, as Python and its packages are not particularlyforward/backward compatible, and my Python statistics script would there-fore not run on most of the systems that I used.The majority of the submitted paper was written by myself, with editingby the co-authors. The figures in the paper are all original and created bymyself.Chapter 4, Appendix A and A.4, and a few sentences in Chapter 1 and2, contain material from this (published) paper:• M. Alexandersen, B. Gladman, S. Greenstreet, J. J. Kavelaars, J.-M.Petit, and S. Gwyn. A Uranian Trojan and the Frequency of Tem-porary Giant-Planet Co-Orbitals. Science, 341:994–997, aug 2013a.doi:10.1126/science.1238072The published text of the paper was written by myself, although withmajor contributions from B. Gladman and S. Greenstreet. B. Gladman per-formed the dynamical integration that confirmed the nature of 2011 QF99 as atemporary Uranian Trojan. S. Greenstreet performed the dynamical integra-tions that produced the residence time distribution for scattering objects anddetermined the orbital distribution of temporary co-orbitals. I used these re-sults to perform survey simulations, which determined the biases for/againstdetecting co-orbital objects and wrote the manuscript. Figure 4.4, A.2 andA.3 were produced by B. Gladman; Figure A.1 was produced by S. Green-street; all other figures were produced by myself.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xvList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Terms, parameters and co-ordinates . . . . . . . . . . . . . . . 21.2 Structure of the trans-Neptunian region . . . . . . . . . . . . . 51.3 Mean-motion resonances . . . . . . . . . . . . . . . . . . . . . 81.3.1 Co-orbitals . . . . . . . . . . . . . . . . . . . . . . . . 131.4 Planet migration . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Magnitude distribution (a proxy for the size distribution) . . . 161.5.1 Discussion of albedo variations . . . . . . . . . . . . . . 25vii1.6 Data-sets used in this thesis . . . . . . . . . . . . . . . . . . . 271.7 The CFEPS L7 model . . . . . . . . . . . . . . . . . . . . . . 291.8 The survey simulator . . . . . . . . . . . . . . . . . . . . . . . 322 Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1 Survey design . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.1.1 Why CFHT? . . . . . . . . . . . . . . . . . . . . . . . 382.2 Discovery observations . . . . . . . . . . . . . . . . . . . . . . 392.3 Tracking observations . . . . . . . . . . . . . . . . . . . . . . . 412.3.1 Gemini observations . . . . . . . . . . . . . . . . . . . 432.4 Accurate astrometry . . . . . . . . . . . . . . . . . . . . . . . 462.5 Characterisation methods . . . . . . . . . . . . . . . . . . . . 482.5.1 Efficiency of search . . . . . . . . . . . . . . . . . . . . 512.5.2 False positive rate . . . . . . . . . . . . . . . . . . . . . 562.5.3 Tracking fraction and lost objects . . . . . . . . . . . . 562.6 Orbit classification . . . . . . . . . . . . . . . . . . . . . . . . 582.6.1 Discovery summary . . . . . . . . . . . . . . . . . . . . 603 Plutinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.1 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.2 Changes to the L7 model and survey simulator . . . . . . . . . 723.3 Testing previously published models . . . . . . . . . . . . . . . 733.3.1 Comparison of our 18 Plutinos with published models . 733.3.2 Comparison using 42-Plutino combined sample . . . . . 773.3.3 Two-bin test . . . . . . . . . . . . . . . . . . . . . . . . 823.3.4 Maximum likelihood analysis . . . . . . . . . . . . . . 903.3.5 Number of detections required for rejection. . . . . . . 1023.4 Investigating allowable parameters . . . . . . . . . . . . . . . . 1053.4.1 Orbital distribution using Hr < 8.3 sample . . . . . . . 1063.4.1.1 Comparison of new and old statistical method 1103.4.2 Magnitude distribution for Hr < 11 . . . . . . . . . . . 112viii3.4.3 Population estimate . . . . . . . . . . . . . . . . . . . 1154 Co-orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.1 Stable Trojans . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.1.1 About 2012 UV177 . . . . . . . . . . . . . . . . . . . . 1214.1.2 Population estimate . . . . . . . . . . . . . . . . . . . 1214.2 Temporary co-orbitals . . . . . . . . . . . . . . . . . . . . . . 1234.2.1 About 2011 QF99 . . . . . . . . . . . . . . . . . . . . . 1264.2.2 About 2012 UW177 . . . . . . . . . . . . . . . . . . . . 1304.2.3 About 2004 KV18 . . . . . . . . . . . . . . . . . . . . . 1324.2.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.2.5 Population estimates . . . . . . . . . . . . . . . . . . . 1374.2.5.1 Neptunian co-orbitals . . . . . . . . . . . . . 1374.2.5.2 Uranian co-orbitals . . . . . . . . . . . . . . . 1384.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395 Population estimates for the 3 : 1 and 4 : 1 resonances. . . . 1406 Comparison with cosmogonic models. . . . . . . . . . . . . . 1467 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1528 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159A Details on dynamical integrations . . . . . . . . . . . . . . . 176A.1 Steady state residence distribution . . . . . . . . . . . . . . . 176A.2 Co-orbital detection . . . . . . . . . . . . . . . . . . . . . . . . 180A.3 Resonant island classification . . . . . . . . . . . . . . . . . . 181A.4 Quasi-satellites . . . . . . . . . . . . . . . . . . . . . . . . . . 183B Combined table of all population estimates . . . . . . . . . . 185ixC Observation log-book . . . . . . . . . . . . . . . . . . . . . . . 187D Python code used in Section 3.3.4 . . . . . . . . . . . . . . . 194xList of TablesTable 2.1 Field coverage of our survey. . . . . . . . . . . . . . . . . . 39Table 2.2 Efficiency function parameters and characterisation limits. . 55Table 2.3 List of all objects detected in our survey. . . . . . . . . . . 61Table 3.1 List of all Plutinos detected in our survey. . . . . . . . . . . 68Table 3.2 Population estimates for the Plutinos. . . . . . . . . . . . . 117Table 4.1 Population estimates for various co-orbital populations. . . 120Table 5.1 Population estimates for the 3:1 and 4:1 resonances. . . . . 141Table 6.1 Expected number of discovered objects in various resonances.149Table B.1 Combined table of all population estimates. . . . . . . . . . 186xiList of FiguresFigure 1.1 Schematic definition of common orbital elements. . . . . . 3Figure 1.2 Top down view of the known outer Solar System. . . . . . 6Figure 1.3 Distribution of orbital elements of known TNOs. . . . . . . 7Figure 1.4 Schematic of how resonant protection operates. . . . . . . 10Figure 1.5 Schematic to explain libration of resonant objects. . . . . . 11Figure 1.6 Formation of a single power-law size distribution. . . . . . 18Figure 1.7 Formation of a divot size distribution. . . . . . . . . . . . 23Figure 1.8 Schematic of the different H-distribution models. . . . . . 24Figure 2.1 Sky coverage of our survey. . . . . . . . . . . . . . . . . . 40Figure 2.2 Illustration of our observing strategies. . . . . . . . . . . . 42Figure 2.3 Screenshots showing how objects are recovered. . . . . . . 44Figure 2.4 Evolution of the ephemeris uncertainty. . . . . . . . . . . . 45Figure 2.5 Discovery field and first pointed recoveries of low-lat field. 46Figure 2.6 Comparison of orbit residuals for CFEPS and our survey. . 49Figure 2.7 Screenshot showing the layout of the inspection process. . 51Figure 2.8 Triplet postage stamps of a candidate moving object. . . . 52Figure 2.9 Examples of invalid candidates. . . . . . . . . . . . . . . . 53Figure 2.10 Screenshots of the inspection process. . . . . . . . . . . . . 54Figure 2.11 Efficiency functions for our survey. . . . . . . . . . . . . . 55Figure 2.12 Orbital element uncertainty after 1 and 2 years. . . . . . . 59Figure 3.1 Scatter plot of the a-e elements of the 42 Plutinos. . . . . 67xiiFigure 3.2 Cumulative distributions of detections from our survey. . . 76Figure 3.3 Cumulative distributions for combined Hr < 11.0 sample. . 78Figure 3.4 Cumulative distributions for combined Hr < 9.0 sample. . 79Figure 3.5 Cumulative distributions for combined Hr < 8.3 sample. . 80Figure 3.6 Plutino histogram, visibility and two-bin test results. . . . 83Figure 3.7 Demonstration of the efficacy of two-bin test. . . . . . . . 87Figure 3.8 Tweaking the knee parameters. . . . . . . . . . . . . . . . 89Figure 3.9 Tweaking the divot parameters. . . . . . . . . . . . . . . . 91Figure 3.10 The Plutino completeness of our surveys. . . . . . . . . . . 93Figure 3.11 The value of log f as a function of Hr. . . . . . . . . . . . 95Figure 3.12 The Plutino completeness as function of e, i and Hr. . . . 100Figure 3.13 Number of detections needed to distinguish models. . . . . 103Figure 3.14 Rejectability in ce, we and αb space, for Hr ≤ 8.3. . . . . . 107Figure 3.15 Cumulative distributions for different orbit distributions. . 109Figure 3.16 Rejectability in ce, we and αb space using CFEPS method. 111Figure 3.17 Rejectability in Ht, c and αf space. . . . . . . . . . . . . . 113Figure 3.18 Cumulative distributions of the least rejectable models. . . 116Figure 4.1 The future evolution of 2012 UV177. . . . . . . . . . . . . . 122Figure 4.2 The visibility of Neptunian co-orbitals. . . . . . . . . . . . 124Figure 4.3 The short-term motion of 2011 QF99. . . . . . . . . . . . . 127Figure 4.4 The future evolution of 2011 QF99. . . . . . . . . . . . . . 129Figure 4.5 The future evolution of 2012 UW177. . . . . . . . . . . . . 131Figure 4.6 Results of survey simulations of scattering objects. . . . . 136Figure A.1 Residence time probability distribution. . . . . . . . . . . 179Figure A.2 Orbital evolution of temporary Neptunian co-orbitals. . . . 182Figure A.3 Orbital evolution of temporary quasi-satellites. . . . . . . . 184Figure C.1 Log-book of observations for August and October, 2011. . 188Figure C.2 Log-book for November 2011 to January 2012. . . . . . . . 189Figure C.3 Log-book for February 2012. . . . . . . . . . . . . . . . . . 190xiiiFigure C.4 Log-book for August and Septemberm 2012. . . . . . . . . 191Figure C.5 Log-book for October and November, 2012. . . . . . . . . 192Figure C.6 Log-book for February, 2013. . . . . . . . . . . . . . . . . 193xivList of AbbreviationsBelow are listed abbreviations used in this thesis, in alphabetical order.AD Anderson-Darling, a goodness of fit test [Anderson andDarling, 1954].ALMA The Atacama Large Millimeter/submillimeter ArrayAU Astronomical Unit; roughly the average distance between theSun and the Earth, although defined as exactly149 597 870 700 m.CCD Charge-Coupled DeviceCFEPS the Canada-France Ecliptic Plane SurveyCFHT the 3.6-m Canada-France-Hawaii TelescopeDES the Deep Ecliptic SurveyDec Declination; similar to latitude on the Earth, but measures acelestial body’s angular position North or South of thecelestial equator. Usually measured in degrees, arcminutesand arcseconds, but is sometimes represented as degrees withdecimals.GMOS Gemini Multi-Object Spectrograph, which can also be used asan imager.xvIAU International Astronomical UnionIQ Image QualityKS Kolomogorov-Smirnov, a goodness of fit test.LSST the Large Synoptic Survey Telescope, which will image theentire sky to a depth similar to our survey every few days.MOP Moving Object PipelineMPC Minor Planet Center, the International AstronomicalUnion (IAU) clearinghouse for Solar System minor bodiesOSSOS the Outer Solar System Origins Survey, a large (∼ 170 sq.deg.)survey currently in progress at CFHT, with a very similardesign to the one of our survey (although much larger).PI Principal InvestigatorRA Right Ascension; similar to longitude on the Earth, butmeasures a celestial body’s angular position along the celestialequator, measured from the direction of the Vernal Equinox.Usually measured in hours, minutes and seconds, but issometimes represented as degrees with decimals (360◦= 24 hr).RMS Root-Mean-Squared, the square root of the mean of square ofvalues.TNO Trans-Neptunian ObjectxviList of SymbolsThis list contains most of the important symbols used in this thesis, sortedapproximately by the order the symbols appear in the thesis. This list is notexhaustive. For additional details on nomenclature, see Chapter 1, particu-larly Section 1.1.a Semi-major axis of an elliptical orbit. Half the distance frompericentre to apocentre (through the focus, not along orbit).e Eccentricity of an orbit; e = 0 for a perfect circle, 0 < e < 1 foran ellipse, e = 1 for a parabola and e > 1 for a hyperbola.i Inclination of the plane of an orbit relative to the ecliptic plane(the Earth’s orbit plane).Ω The longitude of ascending node; the angle between the VernalEquinox direction and the ascending node of an orbit (the pointon the orbit where the object passes through the ecliptic planein the South-to-North direction).ω The argument of pericentre; the angle between the ascendingnode and the point of pericentre.f The true anomaly; the angle (measured from the focus) betweenthe point of pericentre and the object somewhere along its orbit.M The mean anomaly; the angle (measured from the focus) be-tween the point of pericentre and the fictional point in the orbitwhere an object would be if it was on a circular orbit. Thus, Mis basically 360◦multiplied by the fraction of an orbital periodxviithat has passed since pericentre.E The eccentric anomaly; a fictional angle, used as intermediatestep in relating M and f .λ The mean longitude = Ω + ω +M .φpq The resonant angle for an object in p : q mean motion resonancewith a planet. φpq = pλ− qλPlanet − (p− q)(Ω + ω). An objectis said to be resonant if φpq oscillates (as opposed to exploringall 360◦ of possible values (circulating)).q Pericentre distance. The distance from pericentre to the focus,the closest an object on an elliptical orbit gets to the focus.Q Apocentre distance. The distance from apocentre to the focus,the farthest an object on an elliptical orbit gets to the focus.D Diameter of an object.q The differential index of a power-law size distribution.Hr The r-band Solar System absolute magnitude.α The differential exponent of an exponential absolute magnitudedistribution, α = (q − 1)/5.αb The exponent of the absolute magnitude distribution for brightobjects in distributions with a transition.αf The exponent of the absolute magnitude distribution for faintobjects in distributions with a transition.Ht The absolute magnitude (in r-band) at a transition in the ab-solute magnitude distribution.c The contrast between the two sides of a discontinuity (a divot)in the absolute magnitude distribution.AD The value of the Anderson-Darling statistic.xviiiAcknowledgementsI can tell that you’re doing something that you love,by the fact that it is clearly completely useless.— Joe AlexandersenI would like to thank the following:R. Pike for useful discussions, suggestions and help with theoretical under-standing; for love and support during paper writing, thesis writing and al-ways; for feeding me and putting up with my stressed zombie-like state forthe last several months; for putting up with all my old junk and for wantingto move halfway across the world to Taiwan with me.C. Shankman for endless arguments, witty banter, support and friendshipthroughout my Ph.D process.S. Lawler for valuable feedback on my paper manuscripts and constructiveinteractions during our time as office-mates.All of the above for willingly and voluntarily reading and providing con-structive feedback on both of my papers, even though they are mostly notco-authors (Shankman is on one) and thus have no obligation to do so. Thequeue service observers at CFHT for their help and effort in making thisproject a success.Dr. Brett Gladman for giving me this opportunity to engage in the kind ofresearch that got me interested in astronomy in the first place; for teachingxixme a valuable range of observational and analytical skills; for providing mewith great opportunities for learning and networking at conferences; for be-ing so flexible, accommodating and understanding; and for putting up withme for five years.The administrative staff, the kitchen staff and every resident of St. John’sCollege between August 2010 and December 2014 for making those yearssome of the best years of my life. I will forever miss the camaraderie andcommunity, the board game nights and the parties, the dinner conversationsand the food... well, maybe not the food, but being fed at least.Devron Waldgula and Deepak Azad for repeatedly lending me a place tosleep.Wreck Beach for helping me make it through all of those papers that I hadto read.S. & F. Michaud, F. Glass, H. Broekhoven, C. Shankman, Y. Mottiar,D. Azad and A. Hoekstra for being amazing friends and for helping me moveall my stuff when I moved.Special thanks to my parents, Gitte and Soren Alexandersen, whose manyyears of support (emotional and financial) and never ending encouragementwas crucial for me to get to where I am today.M. Alexandersen and B. Gladman were supported by the National Sci-ences and Engineering Research Council of Canada.This work is based on observations obtained with MegaPrime/MegaCam,a joint project of CFHT and CEA/IRFU, at the CFHT which is operated bythe National Research Council of Canada, the Institut National des Sciencede l’Univers of the Centre National de la Recherche Scientifique of France,and the University of Hawaii.This work uses observations obtained at the Gemini Observatory, whichis operated by the Association of Universities for Research in Astronomy,xxInc., under a cooperative agreement with the NSF on behalf of the Geminipartnership: the National Science Foundation (United States), the NationalResearch Council (Canada), CONICYT (Chile), the Australian ResearchCouncil (Australia), Ministe´rio da Cieˆncia, Tecnologia e Inovac¸a˜o (Brazil)and Ministerio de Ciencia, Tecnolog´ıa e Innovacio´n Productiva (Argentina).This thesis makes use of data products from the Two Micron All SkySurvey, which is a joint project of the University of Massachusetts and theInfrared Processing and Analysis Center/California Institute of Technology,funded by the National Aeronautics and Space Administration and the Na-tional Science Foundation.xxiChapter 1IntroductionThe agreement of this law with naturewill be better seen by the repetition of experimentsthan by a long explanation.— Hans Christian ØrstedThe outer Solar System (beyond ∼ 5 Astronomical Unit (AU)) containsfour giant planets, thousands of Jovian Trojans, at least four dwarf planetswith diameters D > 1000 km and thousands of smaller Trans-Neptunian Ob-jects (TNOs) whose numbers generally rise dramatically with decreasing size.In recent years, it has become increasingly clear that the trans-Neptunian re-gion contains a lot of structure, with many different sub-populations. Theorbital and size distribution of TNOs, the remnants of planetesimal forma-tion, provide important information about the early Solar System, allowingtesting of Solar System formation and evolution models.This thesis will discuss our observational survey designed to search forTNOs and our subsequent modelling of the TNO populations. Chapter 2will describe our observational survey in detail. Our analysis of the Plutinoand co-orbital populations are covered in Chapter 3 and 4. Lastly, Chapter 5will describe some work with the 3 : 1 and 4 : 1 resonances and Chapter 61compares our observations to cosmogonic models from the literature, beforeChapter 7 concludes the thesis.1.1 Terms, parameters and co-ordinatesAn object’s motion can be uniquely predicted if six independent values areknown (as well as the time at which those values apply). In some uses, thethree dimensional location and three dimensional velocity in a Cartesian/-spherical co-ordinate system, is the most practical. However, for describingthe motion of objects (asteroids, comets, satellites, TNOs, planets) in orbitaround a central mass (like the Sun or planets), a more convenient selectionof parameters can be chosen, such that only one parameter changes on shorttime-scales. These “orbital elements” describe the shape, size and orienta-tion of an elliptical orbit, and the location on the orbit of an object at agiven time. The orbital elements are: the semi-major axis a of an ellipse;the eccentricity e of the ellipse; the inclination i of the orbit relative to theecliptic plane; the angle along the ecliptic plane between a reference direc-tion (usually the Vernal Equinox) and the location where the object movesthrough the ecliptic plane in the south-to-north direction, known as the lon-gitude of ascending node Ω; the angle along the orbit plane (as seen fromthe focus) between the ascending node and the pericentre (closest approachto the ellipse focus), known as the argument of pericentre ω; and the anglealong the orbit (as seen from the focus) from the pericentre to the currentlocation of the orbiting object, known as the true anomaly f . See Figure 1.1for a visual representation of these orbital elements. The date that f belongsto is called the epoch of the orbital elements. Sometimes the time of lastpericentre Tperi is used instead of f , which reduces the amount of requiredparameters by explicitly stating the time at which f = 0◦. The mass of thecentral object being orbited (or the period of the orbit) is also required inorder to use the orbital parameters to calculate an object’s future location.As f does not increase at a constant rate (because objects move faster near2SunFigure 1.1: Schematic definition of common orbital elements. Theorbital elements are: the semi-major axis a of an ellipse; the eccentricitye of the ellipse; the inclination i of the orbit relative to the ecliptic plane;the angle along the ecliptic plane between a reference direction (usuallythe Vernal Equinox) and the location where the object moves throughthe ecliptic plane in the south-to-north direction, known as the longitudeof ascending node Ω; the angle along the orbit plane (as seen from thefocus) between the ascending node and the pericentre (closest approachto the ellipse focus), known as the argument of pericentre ω; and theangle along the orbit (as seen from the focus) from the pericentre to thecurrent location of the orbiting object, known as the true anomaly f .3pericentre of an elliptical orbit), the mean anomaly M is often used instead.The mean anomaly M is the value that f would have, if the eccentricity waszero. The conversion between from f to M isM = E − e sinEwhere E (often called the eccentric anomaly) can be expressed asE = arctan(√1− e2 sin fe+ cos f).However, the conversion from M to f is non-trivial and does not have ageneral solution. An approximate solution (to fifth order in eccentricity) isf = M +(2e− 0.25e3 +596e5)sin(M)+(54e2 −1124e4)sin(2M)+(1312e3 −4364e5)sin(3M)+10396e4 sin(4M)+1097960e5 sin(5M))+O(e6).A common parameter in celestial dynamics is the longitude of pericentre$ = Ω + ω; this is not a physical angle, as it is the sum of two angles indifferent planes. Another common parameter is the mean longitude, λ =Ω + ω + M , which, as it can also be expressed as λ = $ + M is the sumof two fictional angles. However, both $ and λ are often useful properties,especially when considering resonant interactions (as in Section 1.3).41.2 Structure of the trans-Neptunian regionThe trans-Neptunian region consists of many sub-populations. The clas-sical belt consists of a dynamically cold (low mean eccentricity and incli-nation, typically e < 0.1 and i < 10◦) component and a dynamically hot(large mean eccentricity and inclination) component. Additionally, thereare numerous populations in mean-motion resonance with Neptune (see Sec-tion 1.3); these objects can have high eccentricities, as their integer-periodratios with Neptune can allow resonant phenomena that protects them fromclose encounters. TNOs whose semi-major axis change more than 1.5 AU in10 Myr (due to interactions with giant planets, primarily Neptune) are re-ferred to as “scattering” objects; the subset of scattering objects that havesemi-major axis less than that of Neptune are often classified as Centaurs.Objects at large semi-major axes and pericentre distances are referred to as“detached” (sometimes also ”scattered”, as their orbital evolution is negligi-bly affected by Neptune. The trans-Neptunian disk is bounded on the insideby Centaurs, planet-crossers with a < an, and on the outside by the Oortcloud, with the inner Oort cloud defined as beginning at a = 2000 AU byGladman et al. [2008]. Objects with sufficiently large eccentricity to havepericentre q < 7.35 AU are typically not called Centaurs or TNOs Gladmanet al. [2008], especially if they have visible outgassing, in which case they arecalled comets [Gladman et al., 2008].Figure 1.2 shows a top-down view of the Solar System, with all knownobjects (as of February 27 2015) with a > 18 AU and heliocentric distance <60 AU. Figure 1.3 shows the orbital elements a, e and i for the same objects.Comparing the two figures, it is clear that there is a bias towards detectingobjects closer to the Sun, both in preferentially detecting low semi-major axisand in only detecting high semi-major axis objects near pericentre. An objectbecomes fainter with increased distance from the Sun, because TNOs are onlyseen due to the Sun-light that they reflect, with the apparent magnitudeincreasing by ∼ 3 for a doubling in distance. It is thus tempting to think50°45°90°135°180°225°270°315°Figure 1.2: Top down view of the known outer Solar System, asseen on February 27 2015. Radial distance is the heliocentric distancer, while the angle is the longitude angle along the ecliptic, measuredfrom the Vernal Equinox (right). Dashed circles show distances of 10,20, 30, 40, 50 and 60 AU. The green and blue solid lines represent thesemi-major axes of Uranus and Neptune; the location of those planetsare marked with a yellow diamond and square, respectively. All otherobjects plotted are from the MPC’s lists of “Centaurs and Scattered-Disk Objects” (green stars), “Transneptunian Objects” (cyan circles),“Trojan minor planets of Uranus” (one magenta hexagon), “Trojan mi-nor planets of Neptune” (red octagons), filtered to only show objectswith q > 5.2 AU, a < 80 AU, r < 80 AU, e < 0.75, i < 60◦ and obser-vations in at least two oppositions. These lists were downloaded fromhttp://www.minorplanetcenter.net/iau/lists/MPLists.html on February 272015. 1:1 4:3 3:2 5:3 2:1 5:2 3:1 4:1q=5.2 AUq=30.1 AUq=47.7 AU10 20 30 40 50 60 70 80Semi−major axis (AU)01020304050Inclination(deg)U1:1 1:1 4:3 3:2 5:3 2:1 5:2 3:1 4:1Figure 1.3: Distribution of orbital elements of known TNOs and Cen-taurs. Eccentricity versus semi-major axis (top) and inclination ver-sus semi-major axis (bottom) for the same objects as were plotted inFigure 1.2 (colours/symbols defined in that figure’s caption). Dashedvertical lines show the locations of some of the major mean-motion res-onances and the ones of interest to this thesis. Unfortunately, the MPCdoes not provide lists of objects in resonances (apart from Trojans), butthe over-densities near the resonance locations is clear, especially for thePlutinos (3:2). The ’classical belt’ (40 AU . a . 48 AU) is also verynoticeable. The dashed green curves show lines on which objects wouldhave a pericentre distance q of 5.2 AU, 30.1 AU and 47.7 AU, correspond-ing to the semi-major axis of Jupiter, Neptune and the 2:1 mean-motionresonance with Neptune.7that the apparent outer “edge” seen in Figure 1.2 is purely an observationalartifact and that the trans-Neptunian disk continues to be highly populatedbeyond ∼ 50 AU. However, closer inspection of Figure 1.3 makes it clear thatthis is not the case. Three curves of constant pericentre, with q = 5.2 AU,30.1 AU and q = 47.7 AU are plotted, equal to the semi-major axis of Jupiter,Neptune and the 2:1 resonance (with Neptune), respectively. Any objectalong one of these lines should be equally detectable at pericentre, with theonly bias being that at higher semi-major axis, a decreasing fraction of theperiod is spent at small distances. It is clear that there are many knownobjects with 45 AU < a < 47.7 AU and low eccentricity, but hardly anyknown objects beyond a = 47.7 AU with pericentres 45 AU < q < 47.7 AU.The 2:1 resonance therefore represents an outer edge of the classical belt,and a sharp drop in object numbers.1.3 Mean-motion resonancesThe large number of TNOs occupying mean-motion resonances is strongevidence that the gas giants underwent a period of migration in the earlyyears of the Solar System [Chiang and Jordan, 2002, Hahn and Malhotra,2005, Malhotra, 1995]. Mean-motion resonances are regions of phase-space,where the motion of a small object (such as a TNO) is coupled to the motionof a planet in such a way that strengthens (resonates) the planet’s influenceon the object. Mean-motion resonances exist at semi-major axes where theperiod of the TNO and the planet form a ratio of small integers, r : p (wherer > p represent a resonance exterior to that of the planet1), while the specialcase of r = p is the case of the co-orbitals (see Section 1.3.1). Severalresonances (particularly the 1 : 1, 3 : 2, 2 : 1 resonance with Neptune)are noticeable in Figure 1.3, from their abundance of objects relative to1This convention is standard for outer Solar System scientists; inner Solar Systemscientists typically use the opposite convention, such that an outer resonance with r > pwould be written p : r.8surrounding phase-space.Having a semi-major axis near the resonant semi-major axis is not suffi-cient to ensure resonant behaviour, as eccentricity and inclination are impor-tant factors in the stability of a resonant object. The only way to accuratelydetermine whether an object is in a resonance is to run a dynamical integra-tion of its motion and observe the evolution of the resonant angle:φrp = rλ− pλP − (r − p)$where λ and λP are the mean longitudes of the object and planet, respectively,and $ is the longitude of pericentre of the object. If the resonant anglecirculates (experiences every possible value from 0◦ to 360◦), the object isnot resonating in the r : p resonance. Only if φrp oscillates (“librates”)around an equilibrium value can the object be said to be resonant. Theequilibrium values of φrp depends on the values of r and p [see Gladmanet al., 2012, Murray-Clay and Chiang, 2005, for details]. In the case of the3 : 2 resonance, there is only one equilibrium value φ32 = 180◦, while the r : 1resonances have three equilibrium values [Murray-Clay and Chiang, 2005]and the co-orbitals have four equilibrium values (see Section 1.3.1)Resonance occupation can protect an object from close encounters withthe planet, even when the object’s orbit is so eccentric as to cross the orbitof the planet. This happens due to the equilibrium values of φrp, combinedwith the near integer period ratio, being such that when the small object isat pericentre, the planet is never nearby, and when the planet and objectare at inferior conjunction (as seen from the minor object), the minor objectis far from pericentre and thus far from the planet. See Figure 1.4 for aschematic representation of this for the case of a Plutino with φ32 = 180◦,which never has a close approach with Neptune, because conjunction onlyoccurs when the Plutino is at apocentre. From inferior conjunction (seenfrom the Plutino) to superior conjunction, the Plutino has completed onefull orbit, while Neptune has completed one and a half orbits; during this9Figure 1.4: Schematic of how resonant protection operates, showingthe path of Neptune (red) and a Plutino (a TNO in the 3:2 resonance;blue) in the ideal case of φ32 = 180◦. Each frame shows the motions thatoccur during 34 of the period of Neptune, equal to half the period of thePlutino. The four frames together thus shows three orbits of Neptuneand two orbits of the Plutino, at which point they are both back intheir initial configuration. Circles indicate the starting location of theobjects while arrows indicate the final location of the object as well as thedirection of motion. Notice that when the Plutino comes to pericentre,Neptune is always 90 degrees away, allowing the Plutino to approachclose to, or even inside, the orbit of Neptune. In the left two frames,Neptune is “ahead” of the Plutino, and in the right two frames, Neptuneis “behind” the Plutino. Thus, over the full three Neptunian periods, theperturbations on the Plutino roughly average to zero, keeping it in theresonant configuration. This figure is inspired by Figure 3 in Murray-Clay and Chiang [2005], which contains a very pedagogical explanation(although they consistently use the term “opposition” incorrectly) ofresonance dynamics using the 2:1 resonance as example.whole time, Neptune was “ahead” of the Plutino. However, from superiorto inferior conjunction (another period of the Plutino and 1.5 for Neptune),Neptune is “behind”; due to the symmetry and weak forces (which don’tchange the Plutinos orbit significantly over the course of one orbit), the netforce on the Plutino over this cycle averages to zero. For resonant objects onorbits so eccentric that they cross the orbit of the planet, the same mechanismensures that the planet is never nearby when the resonant object crosses the10Figure 1.5: Schematic to explain libration of resonant objects. Thisfigure is similar to Figure 1.4 (which had φ32 = 180◦), but for the caseof φ32 ≈ 280◦. Notice that because inferior conjunction (seen from theTNO) occurs after the Plutino’s apocentre, the Plutino moves throughmore than half of its elliptic orbit in its first half-period (left frame).While Neptune is “ahead” (left two frames), Neptune and the Plutinoare therefore on average closer than they are in the φ32 ≈ 180◦ case.Subsequently, when Neptune is “behind” (right two frames), the separa-tion is on average larger than in the φ32 ≈ 180◦ case. The Plutino thusgets a net acceleration forward, increasing its semi-major axis slightly.With many orbits, the period ratio slowly grows, so if the period ratiowas exactly 3 : 2 before, the increase means that the position of infe-rior conjunction drifts backwards, and the value of φ32 decreases, until itreaches a minimum value (≈ 80◦ in this case), at which point it will startincreasing again, due to a similar argument. This Plutino would thusbe librating around φ32 = 180◦with a libration amplitude of L32 = 100◦.Resonant libration thus prevent Plutinos from precessing in such a waythat inferior conjunction occurs near the Plutino’s pericentre.planet’s orbit; additionally, any difference in longitude of ascending node alsomeans that the orbits don’t actually cross each other perfectly (more like tworings passing through each other).However, few (if any) resonant objects have φrp exactly equal the equi-librium value all the time. If the period ratio is not exactly 3 : 2 whenφ32 = 180◦, the position of conjunction will drift slowly (over many orbits)and φ32 will change. In the case of the Plutinos, if the period ratio is slightly11less than 3 : 2, φ32 will slowly increase. When φ32 > 180◦, conjunction occursafter the Plutino’s apocentre, as can be seen in Figure 1.5. In this case, be-cause Neptune and the Plutino are closer at inferior conjunction (seen fromthe Plutino) and the Plutino moves faster (due to being closer to pericentre),so they continue to be closer than they were in the case when conjunctionoccurred when the Plutino was at apocentre. Therefore, over the first orbitof the Plutino (when Neptune is “ahead”), Neptune is on average closer tothe Plutino than in the φ32 = 180◦ case, and inversely, in the second orbit ofthe Plutino (when Neptune is “behind”), the separation is on average greaterthan in the φ32 = 180◦ case. The Plutino therefore experiences a net force inthe forward direction, increasing its semi-major axis and its period, slightlydecreasing the period ratio. Once the period ratio reaches 3 : 2 exactly,φ32 stops increasing; however as φ32 > 180◦ still, the above effect continuesto happen, increasing the period ratio, which now causes the conjunctionposition to drift backwards and φ32 to decrease. When φ32 = 180◦ again,the period ratio will reach its maximum value; as the period ratio is above3 : 2, φ32 continues to decrease. Now that φ32 < 180◦, Neptune is on aver-age closer during the “behind” phase, so the Plutino experiences a net forcebackwards, decreasing its semi-major axis and the period ratio. Once theperiod ratio is again exactly 3 : 2, φ32 stops decreasing and again increases.When φ32 = 180◦ again, the period ratio and Plutino-Neptune configurationhas returned to what they were at the start of this paragraph (ignoring per-turbations from other planets) and the cycle repeats. The value of φ32 thuslibrates, centred on 180◦; the maximum and minimum values of φ32 reachedare roughly symmetrical around 180◦.Half of the difference between the minimum and maximum values of φrpis called the “libration amplitude” Lrp and the average time to go from onemaximum to the next is called the “libration period”. The ratio of the orbitalperiod of a resonant object to that of the planet is thus not exactly r : palways, but oscillates (with a fairly small amplitude), averaging to r : p over12the libration period. The known Plutinos have libration amplitudes fromL32 =∼ 20◦ to ∼ 130◦. Libration periods are typically thousands of years.1.3.1 Co-orbitalsCo-orbital objects are in the 1:1 mean-motion resonance with a planet (thushaving, on average over time, the same orbital period and semi-major axisas the planet) and a librating resonant angle φ11 = λ − λP . φ11 roughlymeasures how far ahead in orbital phase the object is relative to the planet.For co-orbital motion, φ11 librates around one of four values: 0◦, 60◦, 180◦,300◦ [Mikkola et al., 2006]. Quasi-satellites librate around 0◦; in the co-orbitalframe these move like retrograde satellites, despite being outside the planet’sgravitational dominance. Leading and trailing Trojans librate around theL4 (60◦ ahead of planet) and L5 (300◦ ahead = 60◦ behind planet) Lagrangepoints, respectively. Objects on horseshoe orbits execute librations aroundL3 (180◦ from the planet) with high-amplitudes that encompasses the L3,L4 and L5 Lagrange points. Of these, only Trojans are generally long-termstable (resonant on Gyr time-scale) [C´uk et al., 2012, Mikkola et al., 2006].There are known objects in co-orbital motion with several of the planetsin the Solar System, both as long-term stable, presumably primordial (bywhich we mean> 4 Gyr lifetimes) populations and also as temporary captures(which will scatter out of co-orbital motion on timescales much shorter thanthe age of the Solar System). Working outwards from the Sun2:• Venus has one known temporary quasi-satellite, 2002 VE68 [Mikkolaet al., 2004].• Earth has multiple unstable co-orbital companions. 3753 Cruithne[Wiegert et al., 1998] is on a complex orbit, a combination of horseshoe-quasi-satellite, due to its substantial inclination and eccentricity. 20022This list was up to date as of May 2013; since then, Parker [2014] have discovered oneadditional long-term stable Neptunian Trojan, and Alexandersen et al. [2015] has addedone temporary and one long-term stable Neptunian Trojan.13AA29 [Connors et al., 2002] exhibits periods of both temporary horse-shoe and quasi-satellite behaviours. 2003 YN107 [Connors et al., 2004]is currently a quasi-satellite while 2010 TK7 [Connors et al., 2011] is atemporary L4 Trojan. The most stable (longest duration of resonanceoccupation) known co-orbital of the Earth is 2010 SO16 [Christou andAsher, 2011], which remains in a horseshoe orbit for more than 100 kyr.• Mars has eight known Trojans, all of which have been shown to bestable on at least Gyr time scales [de la Fuente Marcos and de la FuenteMarcos, 2013b, Scholl et al., 2005].• Jupiter has almost 6000 known long-term stable Trojan asteroids(MPC database, 2013 June 11) in its Trojan clouds, which are be-lieved to outnumber the main asteroid belt [Levison et al., 1997]. Interms of temporary co-orbitals, only a few very short-term, < 1 kyr,captures have been identified [Karlsson, 2004].• Saturn does not have any known co-orbitals and its co-orbital phase-space is believed to be highly unstable due to overlapping resonanceswith the other giant planets, especially Jupiter [Nesvorny´ and Dones,2002]. Orbital simulations [Horner and Wyn Evans, 2006] show tem-porary captures are possible.• Uranus has one known temporary (∼ 20 kyr) horseshoe companion [dela Fuente Marcos and de la Fuente Marcos, 2013a] and one temporaryL4 Trojan [Alexandersen et al., 2013a] which is described in more detailin Chapter 4 of this thesis. The Uranian co-orbital region is thought tobe mostly unstable due to overlapping resonances with the other giantplanets [Nesvorny´ and Dones, 2002] although some stable niches mayexist [Dvorak et al., 2010].• Neptune was recently discovered to have a large stable Trojan pop-ulation which might outnumber the Jovian Trojans [Chiang and Lith-14wick, 2005]. Neptune currently has nine known Trojans, of which 6are known to be stable over the age of the Solar System [Parker et al.,2013, Sheppard and Trujillo, 2006, 2010a]. The first discovered Nep-tunian Trojan, 2001 QR322 [Chiang et al., 2003], as well as 2008 LC18,have orbital uncertainties straddling the boundary between long-termstable and temporary liberators and may be short-lived [Brasser et al.,2004, Guan et al., 2012, Horner and Lykawka, 2010, Horner et al., 2012]although primordial orbits are also possible. The first known Neptu-nian Trojan that is certainly unstable on a short time scale (2004 KV18,Figure A.2 left column) was discovered in the Canada-France EclipticPlane Survey (CFEPS) [Gladman et al., 2012, Horner and Lykawka,2012, Petit et al., 2011]. Recently others [de la Fuente Marcos andde la Fuente Marcos, 2012a,b] have run short numerical integrationsof known Centaurs and identified several potential temporary Neptu-nian co-orbitals: a temporary quasi-satellite, three temporary Trojansand a temporary horseshoe. Although some of these classifications arestill insecure, the number of known transient Neptunian co-orbitals isnow, maybe surprisingly, of order the same as the long-term stableco-orbitals.1.4 Planet migrationIt is generally believed that the giant planets did not form in their current lo-cations [Hahn and Malhotra, 2005, Kaula and Newman, 1992, Kortenkampet al., 2004, Levison et al., 2008, Malhotra, 1993, 1995, Murray-Clay andChiang, 2005, Thommes et al., 2002, Tsiganis et al., 2005], the reason beingthat the large amount of structure seen in the trans-Neptunian region cannotbe explained by any models where everything has formed in place with littleinteraction. Without migration, the trans-Neptunian disk should be a diskof low-eccentricity, low-inclination objects. The odd orbit of Pluto was ten-tatively explained by having Neptune migrate by Malhotra [1993], proposing15that as Neptune migrated outwards, Pluto got stuck in the 3 : 2 resonanceand got “swept” out onto an orbit with a higher semi-major axis and eccen-tricity; the eccentricity increase is caused by the fact that Pluto’s semi-majoraxis was forced to increase, without an input of angular momentum. Morerecently, the observed fraction of resonant objects in the trans-Neptunianbelt is strong evidence in support of this idea of resonance sweep-up duringa migration phase of Neptune. However, the exact time, extent and pace ofthe migration, as well as the original configuration of the planets and plan-etesimals, have a large impact on the final product. It is therefore importantto study the distribution of TNOs, especially the relative numbers in variouspopulations and their orbital distribution, as this allows discrimination be-tween various migration models. Testing cosmogonic models of Solar Systemformation and evolution is not the primary goal of this thesis, however we domake tentative comparisons of two models with observations in Chapter 6.1.5 Magnitude distribution (a proxy for thesize distribution)As the sizes of TNOs are not directly measurable, the best proxy for studyingthe size distribution is the absolute magnitude distribution, as these twoare equivalent if one assumes all TNOs have the same albedo (reflectivity).Section 1.5.1 discusses the propriety of this assumption. The Solar Systemabsolute magnitude, H, is defined as the apparent magnitude that an objectwould be observed with if the object were 1 AU from both the Sun and theobserver, and there were no phase-effects (ie. the apparent magnitude of anobject at 1 AU from an observer standing on the Sun).Minor body populations can typically be well represented, over somerange of absolute magnitudes H, by an exponential H-magnitude distribu-tion. The distribution has the differential form dNdH ∝ 10αH where dN/dHis the number of objects per H-mag, and α (and the constant of propor-tionality) is specific to the dynamical population of interest (typically these16constants are positive). The cumulative absolute magnitude distribution canbe described by integrating this to get N(< H) ∝ 10αH where N(< H) isthe cumulative number of objects brighter than H; note that both distribu-tions have the same exponent (but the constant of proportionality differs bya factor of α ln(10)). Assuming a single albedo value for all objects and con-verting H-magnitude to diameter, the exponential magnitude distributioncan easily be converted to a power-law diameter distribution dN/dD ∝ D−q,where q = 5α+ 1 is the differential size index (not to be confused with q, thedistance of pericentre)3.The TNO populations have been shown to feature steep (high-α, α & 0.6)distributions for D > 100 km [Bernstein et al., 2004, Elliot et al., 2005,Fraser and Kavelaars, 2008, Fuentes and Holman, 2008, Gladman et al., 2001,Jewitt et al., 1998, Petit et al., 2006]. These steep distributions are thoughtto be imprints of planet accretion as D < 10 km objects accumulate intolarger objects through collisions, known as ‘bottom-up formation’ [Kenyonand Luu, 1998, Weidenschilling et al., 1997]. A cartoon plot of how anpower-law size distribution might form is shown in Figure 1.6. Kenyon andLuu [1999] investigate numerically many different scenarios in which theysimulate collisions of a primordial disk of sub-km objects. Despite usingvarious different parameters for the typical impact velocity and likelihoodof merger (instead of catastrophic destruction), they find that most of theirsimulations produce large planetesimals with a power-law size distributionwith q ≈ 4 in the 1− 500 km range. Similar numerical accretion calculationsby others often produce q ≈ 4 (α = 0.6) [Kenyon and Luu, 1998, Schlichtinget al., 2013]. The reason the size distribution is often seen to be a power-lawis that the accretion/collision processes have no preferred scale; the samenumber of 1 km objects is needed to form a 10 km object as the numberof 10 km objects needed to form a 100 km object. The numerical studies3Note that while the cumulative diameter distribution has a power law ∝ D1−q, thecumulative absolute magnitude distribution retains the exponential exponent α.17Figure 1.6: Formation of a single power-law size distribution. Thiscartoon shows how a single power-law size distribution typically formsin numerical simulations of accretion from sub-km objects. The hori-zontal axis has decreasing sizes going right, in order to be oriented thesame way as absolute magnitude. Here, an initial population of sub-kmobjects (solid black line) evolve collisionally in a low impact velocityscenario, such that objects tend to stick together after a collision ratherthan disintegrating catastrophically. Thus, with time, larger and largerobjects are produced. After about 100 Myr (dotted blue line) a singlepower-law with differential q = 4 (α = 0.6) has formed for objects withD ≈ 0.5 km to D ≈ 500 km. (This cartoon was created with inspirationfrom Figure 13 of Kenyon and Luu [1999].)18are supplemented with analytic work, such as that by Dohnanyi [1969] andO’Brien and Greenberg [2003], which conclude the equilibrium index to beq = 3.5 and q = 3.04 − 3.66 (α = 0.5 and α = 0.408 − 0.532), respectively;however, both of these analytic analyses make very simple assumptions aboutthe dependence of object strength with size (Dohnanyi [1969] assumes nodependence). Observationally, even steeper q = 5− 7 (α = 0.8− 1.2) slopesare measured for D > 100 km [Adams et al., 2014, Bernstein et al., 2004,Elliot et al., 2005, Petit et al., 2011]. Other Solar System populations suchas the asteroid belt and the Jovian Trojans have q = 4.5 and q = 5.5 ± 0.9for D & 100 km, respectively [Bottke et al., 2005, Jewitt et al., 2000], whileD < 100 km main belt asteroids show q = 2.2 to 4.1 [Gladman et al., 2009c,Jedicke et al., 2002]. A likely explanation for how these populations canhave size distributions that do not agree with the predicted equilibrium stateis that the populations are not in accretional/collisional equilibrium; thepopulations may have formed quickly with a steep size distribution, andthen planetary migration scattered the planetesimals, lowering the spatialdensity such that the populations are not collisionally active and have notreached collisional equilibrium.While the diameter distribution is the real, physical property, we oftendo not know the albedo and thus do not know the size of observed objects,only their absolute magnitude; we therefore exclusively use the magnitudedistribution. Assuming a TNO r-band albedo of 5% (as used by Gladmanet al. [2012], Petit et al. [2011], Sheppard and Trujillo [2010b], although thisis in the low end of recently measured TNO albedos from Lacerda et al.[2014], Mommert et al. [2012]), D ' 100 km is equivalent to absolute r-bandmagnitude Hr = 8.9. This was calculated using a conversion similar to thatused by Ivezic´ et al. [2001]:Hr = mSun,r − 2.512 log(pr ×D2/(4× (1 AU)2)) (1.1)where pr is the r-band albedo, D is the diameter and mSun,r = −26.91±0.0219is the apparent magnitude of the Sun. The apparent magnitude of the Sunwas calculated usingr = V − 0.46 ∗ (B − V ) + 0.11 (1.2)from Jester et al. [2005], with VSun = −26.75 [Allen, 1973] and (B−V )Sun =0.642±0.016 [Holmberg et al., 2006]4. The size distribution of observationallyaccessible 100 km < D < 1000 km TNOs have been the subject of intensestudy [Fuentes and Holman, 2008, Petit et al., 2008], while D < 100 kmhas only recently started to be investigated [e.g. Fraser and Kavelaars, 2009]due to the need to probe moving targets with apparent mags mr > 24.Interpreting these results is complicated by the recent realization that thereare multiple sub-populations present with different magnitude distributionexponents and orbital distributions [Bernstein et al., 2004, Elliot et al., 2005,Fraser et al., 2010, Petit et al., 2011].Fuentes and Holman [2008] investigated the luminosity function of allTNOs together, and found that a single exponential cumulative magnitudedistribution could not continue past mr ∼ 25, determining a bright objectslope of αb = 0.75+0.12−0.08, with a transition at mr = 24.9+0.5−0.9 to a faint slope ofαf = 0.23+0.07−0.14. However, they themselves point out that the trans-Neptunianregion is made up of multiple populations, and upon splitting their sampleinto a “classical” and “excited” sub-sample, find that while the two popula-tions independently reject a single exponential as well, when implementinga transition, they favour significantly different slopes and transition param-eters. Of course, investigating the luminosity function, that is the distribu-tion of objects as a function of apparent magnitude, rather than the absolutemagnitude distribution, means that any features of the distribution will besmeared out by the fact that objects are at a range of distances. Objectsused by Fuentes and Holman [2008] typically have distance uncertainties of4Other values from Holmberg et al. [2006] that the reader might find helpful include(U −B)Sun = 0.173± 0.064, (V −R)Sun = 0.354± 0.010 and (g − r)Sun = 0.45± 0.0220at least 2.5 AU, causing the uncertainty in the calculated absolute magnitudeto be at least 0.25 magnitudes, often even larger.Kenyon et al. [2008] perform accretion models while simultaneously try-ing to account for how dynamically excited the trans-Neptunian region is,compared to flat, low-eccentricity disks usually used in accretion simulations.They find that once 10-100 objects with D ∼ 600 − 2000 km are produced,accounting for ∼ 10% of the disk mass, these bodies will excite the remainingdisk sufficiently to halt accretion by raising the average collision speed in thedisk. This causes the small objects to reach collisional disruption equilibriumwith q = 3.5 (α = 0.5), while large objects retain q = 2.7−4 (α = 0.34−0.6).While the transition size is model-parameter dependent, they find the tran-sition between the accretion and disruption slopes to be D ∼ 40 − 80 km.Over 4.5 Gyr, collisional and dynamical processes expel planetesimals fromthe region, but the size-distribution remains more or less fixed. A transitionin the size distribution as described by Fuentes and Holman [2008] could thusbe produced by processes as described in Kenyon et al. [2008].Recent work, which were able to both probe past 100 km and determinethe distance accurately, has found a dramatic change in the magnitude dis-tribution of Neptunian Trojans [Sheppard and Trujillo, 2010b] and scatteringobjects [Shankman et al., 2013], with a significant lack of D < 100 km ob-jects. This might indicate that planetesimals were “born big” [Morbidelliet al., 2009], skipping the intermediate sizes, in sharp contrast with bottom-up formation, which may explain how α > 0.6 for D > 100 km. In theborn big scenario, any intermediate sized objects we see today would be col-lisional fragments. Objects with D < 10 km accrete to D > 100 km objects,through unknown accretion processes that must be rapid to avoid the cre-ation of D < 100 km objects. Figure 1.7 shows, once D > 100 km objectsare formed from a born-big scenario, how they then collisionally evolve toproduce D < 100 km objects, forming a divot size distribution (discussed fur-ther below). An alternative explanation for the apparent lack of D < 100 km21objects could be that at this size there is a sudden change in the typicalalbedo of TNOs. However, the sudden lack of Neptunian Trojans seen bySheppard and Trujillo [2010b] would require a sudden change in the averagealbedo by a factor of 6 darker. Observational measurements of TNO albedos(by occultations and observations of binary TNOs) have revealed that theTNOs have a wide range of albedos and that the average albedo does notappear to be size-dependant [Grundy et al., 2005]. So while a size-dependantalbedo distribution could be used to explain the observed lack of objects, itappears less plausible than a transition in the size distribution.Around 1080 TNOs are known (with a > 30.07 and observations in mul-tiple years)5, however most of them were discovered in surveys that havenot been “well-characterised” (as defined in Jones et al. [2006]) or have nothad the characterisation published. Survey characterisations are importantin order to perform accurate modelling of the biases of the observations.CFEPS is one of the largest surveys, and has published characterisation [Pe-tit et al., 2011]. However, CFEPS had very little sensitivity to objects withD < 100 km. To improve upon the conclusions of CFEPS and to investi-gate the lack of D < 100 km objects described above, we performed a deepersurvey designed to constrain the faint objects beyond the sensitivity limit ofCFEPS.Several resonant populations may reach pericentre inside Neptune’s orbitwhile still being long-term stable. Such TNOs can therefore be observedwith even smaller diameters than those in the classical belt. Our surveywas specifically designed to probe the Plutinos and the Neptunian Trojans,as the Plutinos are very numerous and come to pericentres near the Trojanclouds, within Neptune’s orbit. We were able to probe to D < 100 kmwithin these populations. Our goal was to confirm or refute the results inSheppard and Trujillo [2010b], which finds a steep (α = 0.8) slope for bright5List of distant Solar System objects in the MPC database as of 2015 August 8: http://www.minorplanetcenter.net/iau/MPCORB/Distant.txt22Figure 1.7: Formation of a divot size distribution. This cartoonshows how a divot size distribution might form from a born-big scenarioas described in Morbidelli et al. [2009]. Here, an initial population of> 100 km objects (solid black line) evolve collisionally in a scenario morelikely to produce catastrophic disruption from a collision than a merger.Thus, with time, large objects are ground down by collisions, producinga horde of small objects that fill in the gap. After about 100 Myr apower-law still exists for large objects, a power-law with a collisional slop(q ≈ 3.5, α ≈ 0.5) for small objects, and a discontinuity at around D ≈100 km. If the planetesimal cloud continues to be collisionally active,the divot will eventually fill in to become a knee. However, if collisionalprocesses are halted prior to the divot being filled in (for example from aneccentric planet scattering the planetesimal cloud, reducing the densityand thus collision probability), the divot can remain for the age of theSolar System. (This cartoon was created with inspiration from Figure 6of Campo Bagatin and Benavidez [2012].)23objects, but then despite high sensitivity, detects nothing past Hr ∼ 8.6; theythus claimed a drastic drop in number density of Neptunian Trojans withD < 100 km We also hoped to investigate whether such a paucity could beconfirmed or rejected for the Plutinos. The Plutinos and Neptunian Trojansmay have had a similar origin [Levison et al., 2008, Lykawka et al., 2011],so the Plutinos, being vastly more populous, could provide even strongerevidence for or against such a paucity.Figure 1.8: Schematic of the different H-distribution models, as dif-ferential population versus absolute magnitude. A single exponential(left) has just one parameter, the slope (in a semi-logarithmic plot) ofthe curve, α. A knee (middle) has one exponential slope αb for ob-jects brighter than a transition magnitude Ht and another slope αf forfaint objects. A divot (right) has the same parameters as a knee, butalso features a discontinuity at the transition, parameterised by c, thecontrast between dN/dH immediately on each side of the transition.(This cartoon was created with inspiration from a similar cartoon by C.Shankman)Results published after this survey was begun [Shankman et al., 2013]suggested a similar sudden drop in the numbers of scattering objects withD . 100 km (Hg & 9.0), followed by a second power-law for faint objects,described as a “divot”, similar to the divot postulated by Fraser [2009]. This24result lends support to the idea that a drop in number might be present inall dynamically hot TNO populations. However, Fraser et al. [2014] proposesthat the dynamically hot TNOs all share a broken power-law size distribution(a “knee”) which is initially steep but then breaks to a much shallower,although still rising, distribution. We have thus investigated single power-law, knee and divot scenarios, and then explored which model parametersare compatible with our data.A visual representation of a single exponential, a knee and a divot is shownin Figure 1.8, also explaining the parameters of these models. The physicalexplanation for a knee or divot size distribution is not fully understood. Aknee in the size-distribution might imply a transition from a steep slopecreated from formation mechanics, to a shallower slope caused by collisionalgrinding (which would have had a larger impact on smaller objects, as thedensity of similar-sized objects increased towards smaller sizes). A divotmight imply a favoured size of planetesimals, such that planetesimals are“born big” [Morbidelli et al., 2009], in which case all objects just smallerthan the divot are collisional fragments.1.5.1 Discussion of albedo variationsThe albedos of TNOs are important if one wish to study the size distribution,rather than just the absolute magnitude distribution. TNOs with geometricalbedos from 3% to almost 90% are known [Stansberry et al., 2008, Vileniuset al., 2014], although albedos over ∼ 25% are only seen for the very largestobjects and the Haumea family. The diameter (or albedo) of TNOs is difficultto measure, and thus only 1756 of them have measured albedos (of a total¿1000). The three methods used for albedo measurements are: binaries, oc-cultations and thermal observations. Through observations of binary TNOs,the mass can be determined and through a density assumption the diameterand albedo can be estimated. The method used for most albedo estimates6http://www.johnstonsarchive.net/astro/tnoslist.html as of March 24 201525to date is to observe the TNO in far-infrared (mm) wavelengths, where theTNO’s emission dominates over reflected light; with a temperature model,the mm and optical observations can be combined to give a diameter andalbedo estimate. The most accurate method for determining the diameterand albedo of a TNO is through an occultation, in which the TNO passes infront of a background star, blocking out the star light; the shadow cast onthe Earth needs to be observed by several telescopes separated by up to thesize of the TNO, but from these observations, a very good estimate of theTNO’s size (and even shape) can be made. These occultation events are rareand require very high precision orbits in order to be predicted sufficiently inadvance to allow co-ordinating the many observatories required.Grundy et al. [2005] used various methods to determine albedos of 20TNOs, finding a mean of 14% with a 10% average deviation. Mommert et al.[2012] estimated albedos for 18 Plutinos using the Herschel infrared spacetelescope, finding a range of diameters from 150 to 730 km and albedos from4 to 28%, with a mean of 8±3%. Vilenius et al. [2014] used Herschel and theSpitzer infrared space telescope to estimate albedos for 18 classical TNOs.Combined with albedo measurements from literature, they use albedos of 44classical TNOs to find the mean albedo for dynamically hot and cold classicalTNOs to be 8.5+8.4−4.5% and 14+9−7%, respectively. A factor of 4 difference inalbedo corresponds to a factor of 2 in diameter for a fixed absolute magnitude,or a difference in absolute magnitude of 0.75 magnitudes for a fixed diameter.We therefore recommend that when no other information is known, an albedoof 16% is used in Equation 1.1, as the estimated diameter is thus at mostincorrect by a factor of 2 except in the most extreme cases.The range of TNO albedos means that any sharp features or transitions(such as a knee or a divot) in the size distribution will be smoothed out overa range of ∼ 0.8 magnitudes. The inverse is of course also true, but thereis less physical reason to believe there should be sharp transitions in theabsolute magnitude distribution than in the size distribution, as formation26processes would depend on the size and mass of objects and not on how muchlight they reflect. Thus it is unlikely that we will see a sharp transition inthe absolute magnitude distribution, however, adding a smooth transitionto a knee or divot model would add another parameter and is thereforeomitted, as our data will not have sufficient statistical constraint to warrantan additional parameter. Our planned observations will be sufficiently deepthat evidence of a knee or divot will still be apparent, despite the transitionpossibly being a bit blurred.The fact that different populations have differentalbedos [as shown by Vilenius et al., 2014] shows the importance of analysingthe absolute magnitude distribution (or luminosity function, as in some worksin the literature) for different populations separately.1.6 Data-sets used in this thesisThe majority of data used in this thesis was obtained from observationsperformed specifically for this thesis, using MegaCam of the 3.6-m Canada-France-Hawaii Telescope (CFHT). A few observations were also carried outusing Gemini Multi-Object Spectrograph (GMOS) on Gemini North.Additional data from CFEPS, specifically the publicly available list ofdetected objects and survey characterisation files required to perform sur-vey simulations, was also used. CFEPS was a large characterised survey,conducted from 2003 to 2007, which searched 321 sq.deg. of sky for TNOsat various ecliptic longitudes. The various different fields of the survey haddetection limits in the range mg ∼ 23.5− 24.4 [Gladman et al., 2011] (corre-sponding to mr ∼ 23.0− 23.9). The CFEPS observing strategy is describedin Jones et al. [2006], while the scientific results are presented in Gladmanet al. [2012], Kavelaars et al. [2009], Petit et al. [2011].Apart from the larger sky coverage and shallower depth, CFEPS and thesurvey conducted for this thesis are similar, as our new survey was designedwith the aid of the experience that members of our collaboration obtainedduring CFEPS. Both surveys were performed using MegaCam on CFHT.27While CFEPS used the g-band for most discovery observations, our newsurvey used r-band, as after the start of CFEPS it became clear that thetypical TNO colours combined with the quantum efficiency of the telescopeand camera gave a higher signal to noise ratio in the r-band, allowing asurvey in r-band to detect slightly smaller objects than a g-band survey withthe same exposure time.Combining the CFEPS data with our own is relatively straight forward,as both surveys were carefully characterised. We did not include data fromany other survey or observations, as no other survey (large/deep enough toconsider) has published the required characterisation information. As canbe seen in Figure 1.3 and 1.2, there are strong biases involved in discover-ing TNOs. First of all, there is the bias towards detecting high-eccentricityobjects, because they come much closer to the Sun than a low-eccentricity ob-ject with the same semi-major axis; the high-eccentricity object is thereforemuch brighter and more easily detectable during part of its orbit. Anotherbias is that many surveys for TNOs are carried out in or near the eclip-tic, where the object density is highest. Although observations carried outat the ecliptic (0◦ latitude) can potentially detect objects of any inclination(high-inclination objects pass through the ecliptic twice per orbit), high lat-itude observations are unable to discover TNOs with inclinations lower thanthe latitude of the observations. However, ecliptic fields are strongly biasedtowards detecting low-inclination objects, as high-inclination objects spendmost of their time at latitudes near the value of their inclination. The list ofall known TNOs, as curated by the MPC, is therefore full of biases; withoutcharacterisation information for all observations used, the exact extent of thebiases is unknown, and the full list of all known TNOs is thus of limited useJones et al. [2010], Petit et al. [2006].281.7 The CFEPS L7 modelAt the end of the survey analysis, CFEPS produced an absolutely calibratedmodel of all the populations of the trans-Neptunian belt, known as the L7model. One goal of CFEPS was to get an understanding of the gross featuresof the trans-Neptunian region and to get rough population estimates for allknown (and a few at the time unknown) populations. The classical belt wasmodelled in detail [Petit et al., 2011], and the resonant populations weremodelled in varying level of detail depending on the number of detections[Gladman et al., 2012]. Toy models were set up for resonances with onlyfew detected objects, with models and parameters based on resonances withmore detections and chosen simply such that the model was not rejected bythe data. This resulted in population estimates with large uncertainties forthe less populated resonances.The Plutinos were observed in sufficient numbers (24), that their orbitaland absolute magnitude distribution were modelled in more detail. The or-bital model had a parametric form determined empirically. The followingparameterisation used was:• Absolute magnitude distribution: A single exponential H-magnitudedistribution, of the differential formN(H) ∝ 10αH .α is often referred to as the slope of the absolute magnitude distribu-tion. CFEPS found that while different values of α are suitable fordifferent dynamical sub-populations, within a sub-population, a sin-gle value of α can be used to represent the entire range of availabledetections; for Plutinos this covered roughly 6.5 < Hr < 8.5.29• Eccentricity distribution: A Gaussian e distribution, withN(e) ∝ e−(e−ce)22w2e ,where ce is the eccentricity at the peak of the Gaussian and we is thewidth of the Gaussian. The value of e is restricted to be positive andwith an upper limit at e ≈ 0.44, as more eccentric orbits would beunstable due to Uranus encounters.• Inclination distribution: An inclination distribution of the functionalform first suggested by Brown [2001], withN(i) ∝ sin(i)e−i22w2i ,where wi is the width of the distribution.• Libration amplitude distribution: The intended (and documented) dis-tribution from CFEPS was supposed to have been a triangle, with theprobability distribution rising linearly from L32 = 20◦ to L32,peak, thenfalling linearly to L32 = 130◦, i.e.:N(L32) ∝0 L32 < 20◦L32−20◦Lpeak−20◦20 ≤ L32 < Lpeak130◦−L32130◦−LpeakLpeak ≤ L32 < 130◦0 L32 ≤ 130◦What was in fact implemented was a linear increase from 0◦ to L32,peak(so the term for the second range in the above piece-wise equation wasL32/Lpeak for 20 ≤ L32 < Lpeak), with libration amplitudes smaller than20◦ truncated off (so the first line in the above piece-wise equation stillapplied).• Semi-major axis distribution: The width of a resonance in semi-major30axis space depends on the eccentricity. However, as the width of theresonance is always < 1% of the semi-major axis at the centre, the ex-act semi-major axis is not important from a detectability standpoint,so a resonance shape was chosen that encompassed all detections androughly agreeing with the theoretical bounds set by Tiscareno and Mal-hotra [2009]. This accounts for the fact that the resonance narrows forsmall eccentricities, further limiting the low-eccentricity phase space.The CFEPS team determined the best values for these parameters to beα = 0.90 ± 0.25, ce = 0.18+0.03−0.04, we = 0.06+0.03−0.02, wi = 16◦+8◦−4◦, L32,peak = 95◦[Gladman et al., 2012], where uncertainties roughly correspond to the 95%confidence range. Additionally, CFEPS found that approximately 10% ofPlutinos are subject to the Lidov-Kozai mechanism [Gladman et al., 2012,Kozai, 1962, Lidov, 1962]. Objects stuck in this additional resonance un-dergo periodic exchange of angular momentum between eccentricity and in-clination, with the total,√(1− e2) cos i, being constant. This means thatan object oscillates between being on a highly inclined, low-eccentricity orbitand a low-inclination, highly eccentric orbit. This coupling causes the de-tection biases of detecting Lidov-Kozai Plutinos to be significantly differentthan for detecting other Plutinos [see Lawler and Gladman, 2013, for details].CFEPS determined that modelling 10% of Plutinos as being in Lidov-Kozairesonance is an appropriate fraction for modelling the Plutino population;we did not alter this fraction for this work.While this model is empirical and the parameterisation have no physi-cal foundation, the model is sufficient to get an understanding of the over-all features of the distribution, and a reasonable population estimate whencombined with the survey characterisation. The CFEPS Plutino populationestimate is 13000+6000−5000 (95% confidence) Plutinos with Hg < 9.16 [Gladmanet al., 2012] (corresponding to Hr < 8.66).311.8 The survey simulatorIn order to compare models of the intrinsic distribution of the outer SolarSystem with the highly biased samples obtained from observational surveys,the CFEPS team developed a survey simulator [Petit et al., 2011]. This sur-vey simulator produces a list of detections, simulating what a survey shoulddetect, based on an input model of the intrinsic population and the surveycharacteristics. It is possible to simulate multiple surveys at once, effectivelyadding to the CFEPS coverage, if the appropriate characterisation is avail-able. The survey simulator takes a synthetic object, and exposes it to theknown biases of the survey(s):• Field of view: an object cannot be discovered if its on-sky location isnot within the coverage of the survey. The survey simulator must knowthe date the observations were taken, in order to calculate the locationof the object on each day for which there are observations.• Filling factor: Most of the field of view is covered by the camera’sCharge-Coupled Device (CCD) chip(s), but some area will be occupiedby chip-gaps, permanent chip-defects, gaps between adjacent fields, etc.Moving objects cannot be detected in the areas that are not actuallyimaged. This is taken into account by the filling factor, the percentageof the field of view where objects are actually detectable.• Rate cuts: most TNO surveys are not interested in asteroids in theinner Solar System and the observations were therefore only searchedfor rates of motion corresponding to the outer Solar System. Similarly,there is also often a slow rate of motion cut, near the limit of detectablemotion (to prevent being overwhelmed with false detections causedby noise and atmospheric turbulence). These rate cuts must also beimposed on the simulated object.• Detection efficiency: A survey’s ability to detect moving objects de-32pend on the apparent magnitude of the object (and to some degreethe rate of motion). Bright objects are typically discovered with a80 − 100% efficiency, depending on the detection method, observingmethod, and star density (as a dense star field increases the likelihoodof a moving object being in front of a bright star, thus not being pickedout by some moving object detection methods). Fainter objects are de-tected at lower efficiency, partly because they are more prone to beingdrowned out even by faint stars in the case of background confusion,but primarily because as you go to fainter and fainter objects, the sig-nal to noise decreases and eventually the background noise dominatesand the detection efficiency is 0%. The newest version of the surveysimulator allows the detection efficiency to be defined for multiple ratesof motions, such that a possible drop in detectability of slow objectscan be simulated realistically.• Tracking fraction: Most surveys are unable to track all objects to well-determined orbits. The tracking fraction is the probability that anygiven object was successfully tracked. Depending on how the surveywas carried out, this might be magnitude dependant (if follow-up im-ages were not as deep as discovery images) or not (if the cadence orpointings for follow-up images were poor such that even bright objectswould not have been recovered).The synthetic objects passed through the survey simulator can be gener-ated in two ways: either the simulator draws objects from a list (generatedfrom another simulation or in another way), or it can be supplied with aparametric population model and one by one generate as many objects asare required. In the latter case, an orbit is generated from the orbital dis-tribution, and the object is given a random H-magnitude from the absolutemagnitude model distribution. When an object is within a survey field, thesimulator calculates the apparent magnitude in order to compare this tothe detection efficiency. The calculated apparent magnitude gets assigned a33noise-error and a ‘measured’ magnitude, according to magnitude-error pa-rameters from the survey characterisation7.For our work, we have used the combined characterisation of CFEPS andthe survey described in this thesis, as those are the only observations forwhich we have access to all the required characterisation needed to simulatethe surveys accurately. For the simulated objects, we use the same parame-terisation as used by CFEPS (described in Section 1.7), but explore a rangeof possible parameters. The goal of this work is to confirm the validity ofthe L7 orbital distribution model for the Plutinos, investigate the existenceof a transition in the absolute magnitude distribution of the Plutinos, andobtain population estimates for the Plutinos and Neptunian Trojans.7Here we indeed mean magnitude error as opposed to uncertainty; near the surveylimit, the survey preferentially detects objects that happen to be slightly brighter thantheir true brightness, causing a systematic error. This can be accounted for by recordingthe magnitude errors during survey characterisation (using planted objects of a knownmagnitude).34Chapter 2SurveyMeasure what is measurable,and make measurable what is not so.— Galileo GalileiAll truths are easy to understand once they are discovered;the point is to discover them.— Galileo GalileiFor this thesis project, we planned, programmed and obtained ≈ 90 hoursof observations in 2011 and 2012. This chapter describes the observationalportion of this thesis8.2.1 Survey designOur survey was designed to obtain 40 sq.deg. of high-cadence sky coveragenear Right Ascension (RA)= 2 hr, using the 1◦x1◦ field of view of Mega-Cam on the 3.6-m Canada-France-Hawaii Telescope (CFHT) on Mauna Kea,Hawaii. The pointings (on-sky positions) of the survey were chosen to benear Neptune’s L4 Lagrange point (where many Neptunian Trojans reside)8This chapter is based on work submitted for publication in Alexandersen et al. [2015].35and near the pericentre point of many n:2, n:3, and n:4 resonances [Glad-man et al., 2012]. The planned survey was to have one ‘low-lat’ block of20 sq.deg. (4◦x5◦) centred near the ecliptic plane and another ‘high-lat’ blockof 20 sq.deg. located ∼ 15◦ north (see Figure 2.1 for block geometry and Ta-ble 2.1 for exact co-ordinates). The reason for dividing our survey into twoblocks and locating half of our survey away from the ecliptic, where the skydensity would inevitably be lower, was to get a firm handle on the inclina-tion distribution of the populations. The Plutinos and Neptunian Trojansare dynamically hot populations, and thus their inclination distributions arewide, with characteristic widths around ∼ 15◦ [Gladman et al., 2012, Parker,2014]. High-inclination objects are strongly biased against being detected inecliptic surveys, as these objects spend most of their time well away fromthe ecliptic. Even with the drop in total number of detections, a survey athigher latitude should detect more high-inclination objects than an equiva-lent ecliptic survey, providing a better lever arm for statistical tests of theextent of the inclination distribution of the populations.The exact locations of the two blocks were selected such that all imagingover the two-year project would be in fairly low-density star-fields, particu-larly avoiding very bright stars. Crowded star fields and diffraction patternscan obscure the presence of moving objects, lowering our detection efficiency.Additionally, the location of the high-lat block was adjusted slightly (∼ 1◦)westward, such that it would be close to M33. In this way, our integratedr-band imaging and large contiguous field will provide a valuable additionalfilter, depth and coverage for the long-term exploitation of the large PAn-dAS program [A. McConnachie, private communication], which will use ourobservations to search for faint streams and structures in the vicinity of M33.We determined that our survey’s magnitude limit needed to reach mr ≈24.5 in order for our survey to be capable of detecting objects with Hr ≈ 9.7at 30 AU. This depth is more than a magnitude past the transition reported[Shankman et al., 2013, Sheppard and Trujillo, 2010b] near Hr ≈ 8.5 ± 0.2.36Our survey would detect 5-7 Neptunian Trojans if the sky density and D <100 km dearth reported in Sheppard and Trujillo [2010b] were true, or 30-40Neptunian Trojans if the lack of small objects in that work was a statisticalfluke and the steep (α ≈ 0.8) distribution continued.The discovery observations for each field were a set of three 320-secondexposures separated by ∼ 1 hour (“triplets”). MegaCam at CFHT has anoverhead time of 40 s/image, so we were able to acquire 10 exposures per hour.We designed our survey such that 10 fields were to be observed once, thenthe sequence was to be repeated twice more to get triplets of images spacedan hour apart. As a result, 10 fields would obtain triplets in three hours.The number of fields observed at once (and thus the time between images ina triplet) had to be sufficient that the objects had moved appreciably fromimage to image, but was also limited by the fact that we required good andsteady image quality for the duration of the triplets. If the weather turnedbad during a sequence, the whole sequence was lost (and would only be usefulas tracking images). Three hours is near the upper limit of duration thatthe observers can reasonably predict the weather to be stable, and near theupper limit of acceptable risk, if the weather turns bad. We were thereforelimited to an hour between each exposure of the triplets; however, withtypical image quality of ∼ 0.5 − 0.7′′, our lower limit of detectable rate ofmotion is ∼ 0.3′′/hour, equivalent to an object at ∼ 500 AU. At 500 AU anobject would have to be significantly larger than Pluto or Sedna in orderto be bright enough to be detectable in our survey. We therefore did notexpect the detection efficiency of the survey for objects within the maintrans-Neptunian belt to be strongly affected by the rate of motion.As both blocks were near RA= 2 hr, they come to opposition in October.Ideally, both blocks would therefore obtain discovery images in the Octoberdark-run of the 2011B semester9, as the rate of motion is highest at opposi-9At CFHT the A semester runs February to July inclusive, while the B semester runsAugust to January inclusive (so 2011B runs into 2012). Our survey only has observationfrom B semesters.37tion, and the objects are brightest (due to the Earth-object distance beingminimum). As a backup, we planned to observe triplets in August, whichcould be used for discovery if need be, and as a last resort, triplets could beobserved in December. Discovery images can be observed two months fromopposition, but not one month from opposition. One month off opposition,the range of rate of apparent motion of asteroids at ∼ 2 − 5 AU overlapswith the rate of motion for TNOs, causing confusion because it is impossibleto distinguish TNOs from nearer asteroids. Two months before or after op-position, nearby objects are moving prograde, while the TNOs are movingretrograde, albeit much slower than they do at opposition.As data delivery, data reduction, characterisation of the observations andidentification of objects takes several months, we would not be able to per-form targeted recoveries within the discovery year. However, as our blockswere large contiguous fields, shifting them across the sky at the mean TNOrate during the semester would allow us to successfully recover most objectsin several other dark runs, with very limited loss due to Keplerian shear.For follow-up observations in 2012B, we would then expand the blocks tobe 30 sq.deg. (5◦x6◦), roughly corresponding to the expected spread due toKeplerian shear.2.1.1 Why CFHT?CFHT is just one of several telescopes that we have access to either throughCanadian access or though collaborators. We chose to use CFHT for thisproject, because it has one of the largest fields of views on a large telescope.The MegaCam imaging camera has 36 CCD chips of each 2048x4612 pixelsarranged into a square, giving the camera a 1◦x1◦ field of view. This is morethan 100 times the area of Gemini’s GMOS imager, for example, so despiteCFHT not being the largest telescope we have access to, it provides the bestcompromise between deep observations and covering a large area of sky.We use r-band for this survey, despite the quantum efficiency of the tele-38Table 2.1: Field coverage of our survey. The width and height describea rectangle centred on the given RA and Dec. JD is the Julian date ofthe discovery observations. The fill factor is the fraction of the areainside the rectangle that was covered (less than 1 due to chip gaps andseparation of fields).Width Height RA Dec JD Fill factor(◦) (◦) (%)Low-latitude block4.930 0.995 01:51:08.00 +13:24:48.0 2455858.963195 90.94.914 0.995 01:51:08.00 +14:23:36.0 2455858.963195 91.24.896 0.995 01:51:08.00 +15:22:24.0 2455860.843750 91.54.878 0.995 01:51:08.00 +16:21:12.0 2455860.843750 91.6High-latitude block1.993 2.984 01:41:01.70 +28:10:00.0 2456220.904167 90.71.993 2.984 01:49:58.30 +28:10:00.0 2456221.947917 90.7scope being higher in g-band10. This is because TNOs are seen in reflectedsunlight, which is reddish; Sunlight has g−r = +0.45 [Holmberg et al., 2006].Furthermore, many TNOs have reddish surfaces, so that TNOs typically haveg−r = 0.4 to 0.9 [Sheppard, 2010, 2012]. TNOs are thus sufficiently brighterin red wavelengths that despite the slightly lower quantum efficiency, CFHTis more sensitive to small, faint TNOs in the r-band.2.2 Discovery observationsOur survey obtained 32 sq.deg. of high-cadence sky coverage using Mega-Cam on CFHT, near RA= 2 hr. Our “low-lat” block (5◦x4◦ fields centred on01:51:08 +14:53:00) was divided horizontally into two 10 sq.deg. sub-blocks,and observed as described in the survey design section above (Section 2.1).The southern half obtained discovery observations on 2011/10/24, the north-ern half on 2011/10/26, with image qualities mostly in the range 0.5′′− 0.6′′.10http://www.cfht.hawaii.edu/Instruments/Imaging/Megacam/specsinformation.html3920253035 RA (deg)15202530Dec (deg)Figure 2.1: Sky coverage of our survey. Black boxes are the 1 sq.deg.field of view of CFHT’s MegaCam, showing our discovery fields; crossesare the location of our detected objects at time of discovery; the inclinedblue line in the lower left is the ecliptic. The shaded grey areas are the“wallpaper” coverage described in Section 2.4.Unfortunately, poor weather and a short dark-run meant that our ”high-lat” block did not obtain any imaging in October 2011. The back-up tripletsobtained in August 2011 were unusable, as 14 of the 20 fields had image-quality poorer than our tolerable constraint (0.8′′). Had we used these tripletsfor discovery, our survey limit for the high-lat block would have been sig-nificantly brighter (and without images in September and October, linkageto follow-up observations would have been difficult). We therefore plannedto obtain high-lat triplets in December 2011; unfortunately, the dark-runstarted with cloudy weather and then the camera broke and was removedfrom the telescope for a month. The high-lat block therefore obtained discov-ery triplets in October of 2012. We had a reasonable amount of observationsfrom August and November 2011 and January 2012, allowing us to use these40as precovery images. Thus, rather than expand the 2012 high-lat block, wereduced it to 12 sq.deg. (4◦x3◦fields centred on 01:45:30 +28:10:00), such thatthe preemptive follow-up block was larger (as intended for follow-up obser-vations). This reduced our total field coverage from 40 sq.deg. to 32 sq.deg.,but allowed us to mostly complete this project in the planned two years.The October 2012 high-lat discovery block was divided vertically into two6 sq.deg. sub-blocks. The discovery triplets consisted of the 6 fields of thesub-block, interspersed with single images (“nailing” images) of fields fromthe other sub-block to extend the interval between subsequent images of thetriplets up to near an hour. The western half obtained discovery observationson 2012/10/20; the eastern half was observed on 2012/10/21.2.3 Tracking observationsObjects were tracked for up to 28 months of total arc. Single “nailing” imageswere taken at various times to extend the arc of the astrometry. At least onenailing image is required in the discovery dark run; two is preferable, in orderto prevent losing objects in chip-gaps. Another limitation in extending thearc of observations with single nailings is that you must have a comparisonimage in order to distinguish moving objects from stars, so most dark runshad two nailings of each field, taken on separate nights. In the discoveryyear (or both years for the high-lat block), nailings for tracking the objectsoutside the discovery dark-run were performed by simply shifting the entirecontiguous block across the sky at the mean TNO rate (see Figure 2.2).After most dark-runs, newly obtained images were searched manually (byeye) for our real objects (described in Section 2.6.1. This process was aidedby a software package developed by JJ. Kavelaars and significantly improvedfrom our use on this project. That software calculates the predicted locationfor the times of the new observations, and displays any images that coverthe predicted locations. The software also displays the estimated on-skyposition and uncertainty in form of a box around the predicted area (see41Figure 2.2: Illustration of our observing strategies for the low-lat(bottom) and high-lat (top) blocks. Each coloured rectangle shows thelocation and area of one CCD chip image that was observed. Here thelocation of observations from August 2011 (red), October 2011 (green),November 2011 (blue), January/February 2012 (cyan), August 2012(magenta) are plotted. Also shown is the grid of shallow “wallpaper”observations (yellow and orange) used to tie the stars in all of the imagestogether into one astrometric catalogue.42Figure 2.3). In cases where multiple nailings of a field were obtained withina dark-run, recovery was usually trivial. When only one image was available,a comparison image could be shown, if one exists, such that the operatorcould identify which point-source was not present in the comparison image.Once an object had been located in an image near the discovery night,the improved prediction (based on the supplemented astrometry) was usedto locate the object again on further images. Once the observed arc wasa few days long, the on-sky uncertainty was usually sufficiently small forthe object to be easily located on images from one month before or afterdiscovery. This iterative process continued until the object had been foundon all available images, or the remaining images have been deemed unusable.Figure 2.4 shows how the on-sky uncertainty evolves with time and withor without additional observations. It is clear that without observationsoutside the discovery dark-run, the object would have been impossible tolocate manually a year later; recovering it would have required several fieldswith triplets run through the moving object pipeline.By the end of the discovery year, most objects had several months of well-sampled arcs. The on-sky predicted locations for the following/previous yearswere therefore accurate to a few arcseconds (see Figure 2.4). For the low-latblock, pointed recoveries were trivial due to the small on-sky uncertainty ofmost objects (see Figure 2.5). Using pointed recoveries for the low-lat blockin 2012B, we saved ≈ 30% of time compared to a contiguous block, whilealso having a higher recovery rate as some objects had drifted so far thatthey would not have been on the planned 30 sq.deg. patch.2.3.1 Gemini observations15 objects received additional astrometry from images obtained with GMOSat Gemini North in 2013B. These images were obtained as part of a pho-tometric project to measure the g − r colours of all Plutinos in our low-latsample. This project will not form part of this thesis (as data-acquisition43Figure 2.3: Screenshots showing how objects are recovered. Thesoftware uses the current astrometry to predict the location of an objectat the times of follow-up images, then identifies any images that theobject might be seen in. The software displays those images (top) andplots the uncertainty (red box) of the predicted on-sky location. Acomparison image, i.e. an image of the same field from another night, canbe requested (centre) in order to identify the moving object. However,ideally another nailing image from a nearby night (bottom) exists, inwhich case the two nailings can serve as comparison for each other andthe moving object is easily identified. (To allow the reader to share theexperience of searching the images, the moving object in this image hasnot been highlighted/circled. Spoiler: the moving object is in roughlyat the centre of the left half of the error-box.)440 365 730 1095 1460 1825Days since discovery10-1100101102103104Positional uncertainty (")2011/10/242011/10/24-312011/10/24-11/262011/08/29-11/262011/08/29-2012/01/272011/08/29-2012/02/152011/08/29-2012/08/242011/08/29-2012/10/212011/08/29-2013/01/142011/08/29-2013/02/172011/08/29-2013/09/09MegaCam field of viewMegaCam CCD widthMegaCam pixel sizeFigure 2.4: Evolution of the ephemeris uncertainty of 2011 QF99with time and with added observations. The bold black line shows theevolution of the long-axis of on-sky uncertainty ellipse with time; suddenvertical drops occur when we added additional observations. The otherlines show how the uncertainty would have evolved, had we only hadthe labelled subsets of our total astrometry. It is clear that withoutmultiple recoveries the first year, recovery the following year would havebeen difficult and much more costly in telescope time. Note: this is theuncertainty in the predicted position of the TNO, which is largely dueto the uncertainty in the determined orbit; the orbit uncertainty doesnot increase with time.45Figure 2.5: Discovery field and first pointed recoveries of low-lat field.Left: Our discovery fields of the low-lat block from October 2011. Blackboxes are CFHT MegaCam fields. Coloured crosses are the locations ofour discovered objects that were brighter than our characterisation limit(the colours are purely to aid identifying the same object in the left andright panel). Right: Our first recovery observations of the low-lat blockin August 2012. The coloured crosses show the predicted locations of ourTNOs, based on only the astrometry from the August 2011 to February2012 opposition, including uncertainty ellipses (these are smaller thanthe crosses in all but one case). The black boxes shows the pointed fieldsthat we planned (and observed) based on these predictions. In grey arethe location of the original October 2011 fields, for reference.finished in December 2014). As a large part of this thesis concerns the dy-namics and distribution of the Plutinos, astrometry from the additional 2013images has been used to increase the arc of those objects to three oppositions,ensuring that we use the most accurately determined orbits possible.2.4 Accurate astrometryThe uncertainty in the heliocentric orbit of a TNO depends on the length ofthe observed arc, the number and temporal distribution of observations insidethe arc, and the accuracy of the measured astrometry. Larger astrometricuncertainty allows a wide range of acceptable orbital fits, and astrometry46affected by systematic errors push the determined orbit away from the trueparameters. Regular recovery over long arcs can help counter these prob-lems, but obtaining the best-possible orbit-determinations in a short amountof time requires the best possible astrometry. A particular concern with shortarcs is systematic errors, which can occur when an object moves from oneregion of the sky to another, where the astrometric reference catalogue canhave small systematic offsets, of order ∼ 0.2′′. To limit systematic errorsin our astrometry as much as possible, two large plate solutions and stel-lar catalogues were made (one for each block), which served as block-wideastrometric reference catalogues11.To make the unified astrometric catalogue, every image from 2011-2012was stitched together. The slow drift of fields across the sky produced consid-erable overlap, allowing the fields to be stitched together and the chip-gapsto be filled. To ensure that all portions of the sky coverage were connected(including outlying fields from pointed recoveries in 2012), we also shot twogrids of “wallpaper”; these were shallow, 20-second exposures in two offsetgrids covering all our other pointings. The two grids were offset by half a chipin the vertical direction and 1.5 chips in the horizontal direction to ensureoverlap between neighbouring fields and to allow removal of the distortion ofthe focal-plane. The coverage of the wallpaper, and thus the extent of ourastrometric catalogue, can be seen in Figure 2.1.The astrometry of each MegaCam image, both the science image and thewallpaper images, was individually calibrated using the 2MASS catalogue[Skrutskie et al., 2006] as a reference. Catalogues containing RA, Dec andmagnitude were created for each image. These catalogues were merged intoa single catalogue: sources found in multiple images were identified and theirpositions averaged together. This merged catalogue was used to calibrate the11Note: the task of creating the plate solutions, astrometric catalogues, calibrations andupdating image headers was performed by S. Gwyn and is described here to clarify whyour survey’s astrometry was of such high quality. M. Alexandersen measured the finalastrometry for our TNOs in the calibrated images.47plate solutions in the headers of all the images. The resulting astrometriccalibration has internal uncertainties/scatter on the order of 0.04 arcseconds.This was estimated by comparing the location of catalogue stars as measuredin individual images after they have been calibrated.This astrometric catalogue allowed us to measure accurate astrometryover a two year period leading to unprecedentedly small residuals in ourorbit-fits, with mean residuals of 0.13′′ and maximum residual of 0.45′′ (wherex and y residuals have been combined in quadrature). These residuals aredominated by measurement errors in the positions of the objects themselves(due to low signal to noise ratio, causing < 0.25′′ measurement uncertainty),while the uncertainties in the astrometric catalogue plays a smaller contribu-tion (∼ 0.1′′ catalogue systematic variations in the 2MASS catalogue [Gwyn,2014]). This is an improvement over the 0.25′′ RMS residuals of CFEPS (seeFigure 2.6), providing better orbit-determinations with a short arc. This im-provement allowed us to perform pointed recovery observations of almost alllow-lat objects in 2012B, with just six-month arcs in 2011B, and to securelyclassify most objects at the end of 2012B, see Section Characterisation methodsThe triplets from the discovery night were passed through the moving objectpipeline developed by the CFEPS team [Petit et al., 2004]. This pipelineemploys two different moving-object detection algorithms to search for lin-early moving sources. To minimise the number of false candidates (wherethe pipeline detects background noise or cosmic rays), only the overlappingset of detections shared by the two algorithms are considered “candidate”objects; these candidates were subsequently vetted by human inspection.In order to determine the characteristics of our survey (detection-efficiencyand false positive rate) we implanted artificial objects using methods out-lined in Petit et al. [2004] and Jones et al. [2006]. We here present a newimprovement to this process; we did not plant the artificial objects into the480.0 0.2 0.4 0.6 0.8 1.0Residual (") meanMA11 meanCFEPS residualsMA11 residualFigure 2.6: Comparison of orbit residuals for CFEPS and our survey.Histogram of all residuals for orbit-fits to astrometry from CFEPS (red)and this survey (MA11, blue), with the survey means shown with verticallines. Residuals are the x and y residuals combined in quadrature.“real” triplets that were searched for real objects. Instead we used a copyof the triplet images, in which we had first temporally scrambled the orderof the images. That is, we switched around the UT timestamps in the im-age headers so that images 1, 2, 3 were permutated to the sequence 2, 3,1. This scrambling of order meant that any detections by the pipeline inthese “fake” image triplets must either be artificially-implanted objects orbe false positives; nothing can be real objects, because no real object couldmove linearly in these out-of-order images (i.e. no real outer Solar System ob-ject can reverse apparent direction of sky motion within two hours). To our49knowledge, this approach, which allows us to measure the false positive rateexactly, has never been used before. This type of science (with short-termmonotonic time variability) is uniquely able to perform such false positivemeasurements, and this method should become standard for measuring falsepositive rates.Once the real and fake triplets had been passed through the automatedpipeline, a human operator inspected every candidate from both sets of im-ages. The candidate objects from the two sets of images were mixed together,such that the operator never knew whether the candidate being shown wasfrom a real or fake triplet. To ensure that the operator could not figure outwhen they were looking at a fake triplet, only a small part of the image (a“postage stamp”) was shown, rather than the full image (which, if planted,would contain dozens of moving objects, making it obvious that it was afake triplet). See Figure 2.7 for a screenshot of the inspection layout andFigure 2.8 to see a triplet showing a moving object. The blind inspectionensured that the fake and real objects were given the exact same treatment.During the human inspection (which for this project was done exclusivelyby M. Alexandersen), detections which were not believed to be valid (mostlycaused by noise and stellar diffraction spikes) were rejected by the opera-tor (see Figure 2.9 for examples of invalid candidate detections). Once theoperator decides that a detection is valid (real or planted; see Figure 2.8for an example), the inspection software performs photometry on the candi-date, and the operator is then tasked with inspecting the photometry andassigning flags to observations. Images are flagged when the astrometry orphotometry might not be trustworthy. This included images with stellar orcosmic interference in the object’s point spread function or bright sourcessignificantly polluting the photometric sky annulus (see Figure 2.10 for anexample of photometry that required rejections).Once all candidates had been inspected and either approved or rejected,the approved objects in the fake set of triplets reveals the efficiency function50Figure 2.7: Screenshot showing the layout of the inspection process.The software will load three postage stamps to the display (see Figure 2.8for all three postage stamps), marking the candidate with a green circle.The operator then blinks through the images to determine whether theyaccept the candidate (i.e.. operator believes the candidate to be real orplanted) or not (noise); the operators decision is recorded.and the false positive rate. The approved objects in the real set of tripletsare real objects (or false positives). These objects were then searched for intracking observations.2.5.1 Efficiency of searchThe detection efficiency was calculated as a function of magnitude; in orderto investigate whether our efficiency varied significantly with rate of motion,this was done for a few different magnitude ranges rather than all objectsat once. These efficiencies as a function of magnitude were found to be well51Figure 2.8: Triplet postage stamps of a candidate moving object. Thiscandidate is in fact the successful detection of a mr = 21.82 plantedobject moving at 2.52”/hr at an angle of −25.◦05 relative to the positivex-axis; but of course the operator was ignorant of the object’s nature atthe time.represented by a function described in Gladman et al. [2009c], of the form:f(mr) =f21 − k(mr − 21)21 + exp(mr−mew) , (2.1)where f21 is the efficiency at mr ' 21, k is a measure for the strength of aquadratic drop (caused by crowding; objects being obscured by backgroundstars is a problem that increases for faint objects), me is the characteristicmagnitude where the function transitions to behaving as an exponential tail(as signal to noise limits detection) and w is a measure for the width of thattail.The limiting magnitude of the survey for each rate range was chosen to bethe magnitude at which the efficiency was 40% of the maximum efficiency,rounded up to nearest 0.01 magnitude. The limiting magnitudes for theblocks are given in Table 2.2, as well as the parameters used to model theefficiency functions, which are shown graphically in Figure 2.11. All of ourobjects have rates between 2.5 and 4.4 ”/hr, except the two detected at r <28 AU. So the survey limit for main-belt TNOs is mr ' 24.6.52Figure 2.9: Examples of invalid candidates, which were not caused bya real or planted object. When image quality varies (even just slightly)between images in a triplet, invalid detections can be produced fromthree faint stars or galaxies (or just noise) fading in and out of view(top triplet). Images that contain bright, flared stars produce manyinvalid detections in the patterns caused by the diffraction and scatter-ing reflection of light from the bright star inside the telescope (bottomtriplet).We find that there is no significant dependence of the efficiency on therate of motion, except for the very fastest rates (≥ 6”/hr), which show lossesdue to trailing of the signal at the expected level of ∼ 0.2 mags. We alsoinvestigated whether the efficiency depended on the angle of the motion andfound no significant dependence. For objects in the 25 − 50 AU distancerange, detection therefore only depends on the apparent magnitude.53Figure 2.10: Screenshots of the inspection process, showing a triplet ofan object which is so close to a background star in the second image, thatits photometry (and possibly astrometry) is polluted as a result. Thephotometry of the second image was therefore rejected by the inspector,and the astrometry flagged such that once the object is recovered, poorastrometry can be ignored.54Table 2.2: Efficiency function parameters and characterisation limitsof our survey blocks in the searched rate ranges. The efficiency functionis parameterised in Equation 2.1. The limit is the magnitude at whichthe efficiency hits 40% of the maximum value. The functions are plottedin Fig. 2.11.Rate range (“/hr) f21 k me w LimitLow-latitude block0.50-2.06 0.96 0.0148 24.62 0.125 24.622.06-3.72 0.91 0.0118 24.60 0.133 24.613.72-5.38 0.90 0.0106 24.55 0.142 24.575.38-10.00 0.88 0.0136 24.48 0.130 24.49High-latitude block0.50-2.06 0.92 0.0194 24.64 0.118 24.602.06-6.00 0.90 0.0133 24.59 0.115 24.606.00-10.36 0.88 0.0139 24.49 0.124 24.4921.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0mr  magnitude0. efficiency0.50-2.06"/hr2.06-3.72"/hr3.72-5.38"/hr5.38-10.0"/hrUncertainty 2.06-3.72" 21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0mr  magnitude0.50-2.06"/hr2.06-6.00"/hr6.00-10.36"/hrUncertainty 2.06-6.00"Figure 2.11: Efficiency functions for our survey, low-lat block (left)and high-lat block (right). To prevent crowding, efficiency measurementsand associated statistical error-bars are shown only on the curve mostapplicable for TNOs at ∼ 40 AU.552.5.2 False positive rateThe fake triplets produced 22910 candidate detections, of which 1052 were re-jected and 21858 were accepted; of the accepted objects 21855 were planted,which means there were 3 false positives. In other words, the human oper-ator was willing to accept these three detections as real candidates near thenoise limit, which were actually coincidental noise. Having only three falsepositives despite the sheer number of candidates shows the ability of humansto successfully filter noise from valid low signal-to-noise sources. Moreover,these three false positives were all beyond our survey’s ∼ 40% characterisa-tion limit, so in fact the characterised portion of our survey had a zero falsepositive rate. This is the first time such a measurement of false positive ratehas been made, and we suggest that it should become standard. Our resultof zero false positives in the characterised portion of the survey should not beseen as evidence that surveys like this always have a zero/low false positiverate, but rather serves as evidence of the effort we put into the candidateinspection process. If a significant number of false positives brighter thanthe characterisation limit were found, it would call into question the validityof the efficiency function and thus our ability to de-bias the survey.In the real triplets, there were 1246 candidates; 1159 of these were re-jected, resulting in 87 candidate real objects: 70 in the low-lat block and 17in the high-lat block. Once the real objects were identified, we attempted totrack these objects in follow-up observations from 2011-2013.2.5.3 Tracking fraction and lost objectsWe were able to track nearly every object detected in our survey. Ten objectsdid not achieve two-opposition arcs. However, eight of these were fainterthan our characterisation limit and thus do not affect our scientific modelling(which only extends to the ∼ 40% characterization limit). The loss of the twoobjects brighter than our characterisation limit (with mags mr = 24.2 and24.5) was due to a combination of poor cadence and image quality of some56tracking observations, particularly the ones temporally closest to discovery.Serious efforts were put into recovering these two lost objects, however theseefforts were unsuccessfull. When the nailing images within the discoverydark-run are not sufficiently deep to see an object, it is very hard to identifythe object in images separated by several months, as the error-ellipse can bewider than the CCD chip, ∼ 7′. Because the characterization indicated thatwe have a zero false-positive rate in our characterised survey, we believe thatour two ‘lost’ objects are indeed real TNOs.To account for the (slightly) less than 100% tracking efficiency in ouranalysis in the coming sections, we modelled the tracking efficiency based onthe loss of these two objects, one in each block. We do not believe that ourtracking efficiency is dependant on the orbit of the object, as our observa-tions within the discovery year were always a large contiguous patch of sky,meaning tracking did not rely on assumed orbits. The low-lat untracked ob-ject, mal11nt = 2011UU412 (which had a one month arc but was then neverseen again) with mr = 24.5 ± 0.1, was near our characterisation limit andwas simply invisible in the tracking images, which were generally slightlyshallower than the discovery images (due to observing conditions being beston the night of discovery; all tracking images had the same exposure time asthe discovery images). We therefore modelled the tracking efficiency of thelow-lat block as 100% for mr < 24.5 followed by a linear falloff such thatintegrating over the tracking function gives the appropriate fraction of lostobjects (one object out of 62 characterised detections). In the high-lat block,mah11nt with mr = 24.2± 0.1 was not successfully tracked, being only seenin the discovery triplet (which is insufficient to be assigned a MPC designa-tion), despite being a fair bit brighter than our limit. The high-lat block lostmah11nt due to the poorer observing cadence and because mah11nt shearedoff the field during part of the discovery year; its lack of recovery is thus inde-pendent of the object’s brightness, so we modelled the tracking efficiency of57the high-lat block as a magnitude-independent 1415 ≈ 93.3%12. These trackingefficiencies were incorporated into our survey simulations.The on-sky uncertainties of mal11nt = 2011UU412 and mah11nt are nowso large that these objects would require survey-like observations to recoverthem. Such a survey would discover numerous other TNOs, and only oncethe new discoveries have been measured in significant tracking observations,the uncertainty would be small enough to make a confident linkage to ourlost objects.2.6 Orbit classificationAt the end of 2012B, our objects typically had 17 month arcs, from 15-30 observations in multiple months in both 2011 and 2012 (and Jan/Feb2013). The cadence of observations together with the superior astrometryensured that with just two oppositions, most of our objects have their orbitsdetermined to far better precision than most two-opposition objects in theMPC database. Two-opposition objects in the MPC database often havethe majority or entirety of their observations in the opposition month ofeach year, which is insufficient for secure classification. In order to classifyobjects with reasonable confidence, σa/a . 0.003 is needed, and in orderto constrain the libration amplitude of resonant objects, σa/a . 0.001 istypically required. 71 and 59 of our objects had σa/a smaller than thesevalues, respectively, by the end of 2012B (see Figure 2.12), as a result of ourhigh-precision astrometry and frequent observations.The objects were dynamically classified using the Solar System BeyondNeptune algorithm and nomenclature [Gladman et al., 2008]13. The nominalorbit, as well as the two orbits with the most extremal semi-major axisallowed by the astrometry, were integrated for 10 Myr. Instead of using a12There were 15 objects brighter than our survey characterisation limit in the high-latblock.13The classification process was carried out by B. Gladman.58Figure 2.12: Orbital element uncertainty after 1 and 2 years. Therelative semi-major axis uncertainty versus semi-major axis (left), ec-centricity uncertainty versus eccentricity (centre) and inclination uncer-tainty versus inclination (right) for all 77 objects from our characterisedsurvey. The uncertainties are shown for orbits derived with the astrom-etry from just the discovery opposition (top; August 2011 - February2012 for low-lat, August 2012 - February 2013 for high-lat) and fromtwo years of observations (bottom; August 2011 - February 2013). Thetwo lost objects are easily seen as the only objects that still have largeuncertainties after two years, despite efforts to recover these objects.Gaussian covariance matrix based on the Root-Mean-Squared (RMS) scatterof the observations, the two extremal orbits were determined from a slewof orbits identified by a Monte-Carlo process as being compatible with theastrometric residual pattern. If integrated particles with these three orbits asstarting conditions exhibit the same behaviour (resonant, classical, detached,scattering, etc.), the object is said to be securely classified. If all three clonesdo not experience similar behaviour, the object is insecurely classified; usually59the insecure classification will be based on the behaviour of two clones, ifonly one clone acts differently. Table 2.3 lists the resulting orbital classesand security of the classifications of the objects from our survey.2.6.1 Discovery summaryIn our discovery triplets, 87 objects were detected; 70 in the low-lat blockand 17 in the high-lat block. 77 of these 87 were above our characterisa-tion limit (62 in the low-lat block, 15 in the high-lat block), comprising ourcharacterised sample. The objects in the characterised sample were giveninternal designations consisting of the the letters “ma” denoting the sur-vey, “l” or “h” denoting low or high-lat block, and a number unique withinthat block; the objects were sorted such that objects closer to the Sun werenumbered with the lowest numbers. The uncharacterised sample was subse-quently labelled in the same manner, continuing the number sequence wherethe characterised sample ended. The internal designation would have a “u”prepended for uncharacterised objects, and would have “nt” appended forobjects that were not tracked successfully, despite our best efforts.Among the characterised sample, we found: one Uranian Trojan (1:1mean-motion resonance with Uranus, see Alexandersen et al. [2013a]), twoNeptunian Trojans (1:1 resonance with Neptune), two object in 4:3 reso-nance14, 18 Plutinos (3:2 resonance with Neptune), six 5:3 objects, three2:1 objects, two 5:2 objects, three 3:1 objects, one 4:1 object and a slew ofmain belt and detached objects. We present the entire sample, with orbitalinformation and classification, in Table 2.3. However, from here on we willonly use the characterised sample, focusing on the Plutinos in Chapter 3,co-orbitals in Chapter 4, and a few select other resonances in Chapter 5.14Unless explicitly stated otherwise (in the case of the Uranian and Neptunian co-orbitals), all resonances mentioned in this thesis are outer mean motion resonances withNeptune. We therefore do not explicitly state “outer mean motion resonance with Nep-tune” every time, as this is implicit.60Table 2.3: List of all objects detected in our survey. ID is the internal designation, “ma” denotingthe survey while “l” and “h” distinguish whether an object was discovered in the low-lat or high-latblock, respectively. MPC is the Minor Planet Center designation. The date of discovery (given as theJulian date−2 450 000) and heliocentric distance at discovery are Disc and r, respectively. N and t arethe total number of astrometric measurements and the total arc in years, respectively. The barycentricorbital elements, semi-major axis (a), eccentricity (e), inclination (i), longitude of ascending node(Ω), argument of pericentre (ω) and mean anomaly (M), are all given at the Epoch given as Juliandate−2 450 000. The discovery magnitude mr is the average r-band apparent magnitude as measuredby the moving object pipeline in the discovery triplet. Hr is the absolute magnitude calculated from mrusing appropriate phase-angle corrections. Column S denotes whether or not the classification is secure(S), insecure (I) or whether the object was never classified (N). Columns σ5 and σ10 show the long-axis(in arcseconds) of the on-sky uncertainty ellipse (generated using the Bernstein and Khushalani [2000]orbit-fitting code) of each object on 2016/10/25 and 2021/10/25, roughly 5 and 10 years after our firstobservations of the objects. Note that three objects were previously discovered objects that happenedto be in our fields; all information in this work, apart from the MPC designation, ignores the fact thatthese objects were previously discovered, and the information in this table is therefore derived solelyfrom this work. The MPC’s Minor Planet electronic Circulars containing the discovery astrometry isstated next to each class header. Our astrometry for the three previously known TNOs was publishedin Tomatic [2014a].ID MPC Disc r N t Epoch a e i Ω ω M mr Hr S σ5 σ10(AU) (yr) (AU) (◦) (◦) (◦) (◦) (”) (”)Temporary Uranian Trojan [Alexandersen et al., 2013b]mal01 2011 QF99 5858.9 20.296± 0.000 33 1.4722 5803.0 19.092± 0.001 0.17687± 0.00009 10.810± 0.000 222.499± 0.000 287.47± 0.02 258.64± 0.02 22.6± 0.1 9.57 S 12 100(Winnie)Stable Neptunian Trojan [Alexandersen et al., 2014o]mah02 2012 UV177 6220.9 29.578± 0.001 30 1.4614 5805.9 30.024± 0.004 0.0723± 0.0009 20.833± 0.000 265.668± 0.002 204.28± 0.10 285.94± 0.02 23.7± 0.1 8.93 S 6 30Continued on next page61Table 2.3 – Continued from previous pageID MPC Disc r N t Epoch a e i Ω ω M mr Hr S σ5 σ10(AU) (yr) (AU) (◦) (◦) (◦) (◦) (”) (”)Temporary Neptunian Trojan [Tomatic, 2014b]mah01 2012 UW177 6220.9 22.432± 0.001 17 1.0429 6158.0 30.072± 0.003 0.2591± 0.0002 53.886± 0.001 20.010± 0.001 34.4± 0.2 351.967± 0.004 24.2± 0.1 10.65 S 40 200Resonant 4:3 [Alexandersen et al., 2014n]mal09 2011 UZ412 5858.9 31.606± 0.001 24 1.4422 5803.0 36.433± 0.003 0.13295± 0.00012 6.812± 0.001 9.788± 0.003 27.4± 0.7 355.932± 0.001 24.0± 0.1 9.06 S 9 40mal10 2011 UA413 5860.8 31.794± 0.003 19 1.2866 5859.8 36.528± 0.009 0.1329± 0.0010 5.437± 0.000 292.353± 0.013 83± 2 11.076± 0.005 23.3± 0.1 8.29 S 20 110Resonant 3:2 [Alexandersen et al., 2014a]mal45 2011 UD411 5860.8 41.480± 0.001 28 2.2668 5804.0 39.21± 0.03 0.2884± 0.0011 15.700± 0.001 19.081± 0.001 131.88± 0.11 274.68± 0.09 23.1± 0.1 6.91 S 3 16mal04 2011 UA411 5860.8 29.828± 0.001 29 2.1003 5804.0 39.223± 0.004 0.24459± 0.00009 8.645± 0.000 237.962± 0.002 167.51± 0.09 351.158± 0.001 23.7± 0.1 9.01 S 2 13mal02 2011 UC411 5860.8 28.724± 0.000 43 2.0894 5804.0 39.270± 0.003 0.27179± 0.00009 4.652± 0.000 289.175± 0.005 88.35± 0.06 6.447± 0.001 22.3± 0.1 7.74 S 1.7 9mal03 2011 UU410 5858.9 28.997± 0.001 19 2.0893 5803.0 39.293± 0.004 0.26294± 0.00007 12.623± 0.001 19.330± 0.001 15.39± 0.08 356.454± 0.001 24.4± 0.1 9.85 S 2 13mah03 2012 UG177 6220.9 30.482± 0.002 18 1.2344 5889.9 39.304± 0.010 0.2248± 0.0003 21.876± 0.001 261.872± 0.002 130.6± 0.7 357.684± 0.001 24.4± 0.1 9.48 S 18 90mal30 2004 VT75 5858.9 37.958± 0.001 31 2.2717 5803.0 39.31± 0.02 0.2108± 0.0010 12.842± 0.001 26.610± 0.000 272.04± 0.03 68.66± 0.05 21.9± 0.1 6.11 S 3 13mal16 2011 UY410 5860.8 34.137± 0.002 24 1.4313 5804.0 39.312± 0.008 0.1364± 0.0007 8.590± 0.001 12.404± 0.002 2.9± 0.8 13.169± 0.005 24.4± 0.1 9.09 S 9 40mah15 2012 UH177 6221.9 45.464± 0.002 27 1.4559 5806.0 39.39± 0.02 0.1907± 0.0012 23.749± 0.001 255.149± 0.003 346.2± 0.6 137.39± 0.10 24.2± 0.2 7.57 S 4 20mal07 2011 UQ410 5858.9 30.974± 0.001 26 2.0071 5803.0 39.400± 0.008 0.2636± 0.0003 3.246± 0.001 10.821± 0.004 335.40± 0.06 26.946± 0.008 24.2± 0.1 9.36 S 1.9 10mal21 2011 UX410 5860.8 36.258± 0.001 28 2.1738 5804.0 39.40± 0.02 0.3279± 0.0007 17.479± 0.001 223.962± 0.000 263.515± 0.011 302.35± 0.04 23.7± 0.1 8.14 S 2 14mal06 2011 UR410 5858.9 30.875± 0.001 21 2.2693 5803.0 39.412± 0.009 0.2685± 0.0004 17.320± 0.001 27.050± 0.000 50.37± 0.09 332.881± 0.009 24.4± 0.1 9.59 S 3 18mal05 2011 UV410 5858.9 29.862± 0.001 26 2.1766 5803.0 39.421± 0.007 0.2591± 0.0003 4.308± 0.000 332.153± 0.006 30.14± 0.09 15.396± 0.004 22.5± 0.1 7.77 S 3 13mal17 2011 US410 5858.9 35.150± 0.001 27 2.1028 5803.0 39.510± 0.013 0.2554± 0.0005 11.876± 0.001 20.578± 0.001 290.41± 0.02 51.18± 0.03 23.7± 0.1 8.27 S 1.6 8mal35 2011 UT410 5858.9 39.355± 0.001 24 2.2420 5803.0 39.52± 0.02 0.1787± 0.0011 13.498± 0.001 19.796± 0.001 109.09± 0.04 281.59± 0.05 21.5± 0.1 5.58 S 3 15mal31 2011 UZ410 5860.8 38.402± 0.001 26 2.3184 5804.0 39.521± 0.013 0.1245± 0.0012 3.833± 0.000 298.461± 0.010 8.44± 0.04 69.87± 0.04 24.0± 0.1 8.17 S 3 14mal28 2011 UW410 5860.8 37.600± 0.001 30 2.1631 5804.0 39.578± 0.014 0.2175± 0.0007 10.752± 0.001 17.914± 0.001 285.455± 0.014 64.55± 0.03 23.8± 0.1 8.07 S 1.9 9mal32 2011 UB411 5860.8 38.548± 0.001 25 2.0862 5804.0 39.601± 0.006 0.0436± 0.0008 14.900± 0.001 13.374± 0.001 322.5± 0.6 50.471± 0.014 23.8± 0.1 7.92 I 2 11mal12 2011 UP410 5858.9 33.293± 0.001 32 2.2503 5810.0 39.602± 0.004 0.1749± 0.0003 2.733± 0.001 241.740± 0.006 179.2± 0.2 339.745± 0.004 23.4± 0.1 8.21 S 3 16Resonant 5:3 [Alexandersen et al., 2014e]mal60 2011 UM411 5858.9 48.408± 0.007 12 1.2891 5858.9 42.222± 0.014 0.1467± 0.0004 9.385± 0.003 11.445± 0.006 197± 4 183.8± 0.2 24.1± 0.1 7.24 S 14 60mal34 2011 UL411 5858.9 39.189± 0.002 16 1.4340 5803.0 42.224± 0.005 0.07190± 0.00009 1.974± 0.000 308.53± 0.04 82± 2 358.747± 0.000 24.3± 0.1 8.37 S 9 40mal14 2011 UO411 5860.8 33.891± 0.001 26 1.4395 5804.0 42.282± 0.011 0.2061± 0.0006 5.289± 0.000 310.100± 0.010 98.9± 0.5 347.583± 0.005 23.8± 0.1 8.51 S 9 40mal27 2011 UJ411 5858.9 37.500± 0.002 21 1.4338 5803.0 42.290± 0.009 0.1189± 0.0007 5.141± 0.001 21.604± 0.003 30.0± 1.0 344.336± 0.006 23.9± 0.1 8.20 S 8 40mal15 2011 UK411 5858.9 33.939± 0.001 27 1.4724 5803.0 42.32± 0.02 0.2328± 0.0011 13.424± 0.001 219.278± 0.001 210.3± 0.3 335.30± 0.02 21.9± 0.1 6.66 S 8 40mal13 2011 UN411 5860.8 33.447± 0.002 23 1.4395 5804.0 42.341± 0.007 0.2104± 0.0003 3.234± 0.000 326.45± 0.02 61.0± 0.7 2.553± 0.001 24.2± 0.2 8.99 S 12 60Resonant 19:10 [Alexandersen et al., 2014d]mal18 2011 UH411 5858.9 35.636± 0.002 18 1.4504 5803.0 46.14± 0.02 0.2386± 0.0009 27.041± 0.001 214.057± 0.000 198.3± 0.5 346.696± 0.009 24.2± 0.1 8.72 I 10 50Resonant 2:1 [Alexandersen et al., 2014c]mal22 2011 UE411 5858.9 36.574± 0.002 14 1.3792 5803.1 47.61± 0.09 0.320± 0.003 9.355± 0.001 17.099± 0.003 73.3± 0.3 329.11± 0.09 24.6± 0.1 9.00 S 13 70Continued on next page62Table 2.3 – Continued from previous pageID MPC Disc r N t Epoch a e i Ω ω M mr Hr S σ5 σ10(AU) (yr) (AU) (◦) (◦) (◦) (◦) (”) (”)mal47 2011 UG411 5860.8 41.990± 0.004 18 1.2784 5859.8 47.66± 0.04 0.139± 0.003 10.692± 0.001 238.833± 0.004 116± 2 27.11± 0.04 24.6± 0.2 8.35 S 14 60mal25 2011 UF411 5860.8 37.267± 0.002 31 1.4313 5804.0 47.73± 0.09 0.344± 0.002 5.872± 0.001 261.775± 0.010 195.8± 0.2 324.76± 0.10 24.1± 0.1 8.35 S 10 50Resonant 5:2 [Alexandersen et al., 2014h]mah04 2012 UJ177 6221.9 34.429± 0.001 25 1.4585 5805.0 55.20± 0.07 0.4333± 0.0013 15.632± 0.000 298.867± 0.003 46.9± 0.2 17.41± 0.04 24.1± 0.1 8.65 I 7 30mal61 2011 UT411 5860.8 51.239± 0.004 20 1.4338 5803.1 55.7± 0.3 0.406± 0.005 6.420± 0.002 4.876± 0.011 130.2± 0.2 304.2± 0.4 24.5± 0.1 7.46 I 7 40Resonant 3:1 [Alexandersen et al., 2014g]mal23 2011 UR411 5860.8 37.046± 0.003 15 1.2866 5859.9 62.1± 0.2 0.438± 0.003 26.581± 0.002 23.363± 0.001 45.8± 0.6 346.99± 0.07 24.0± 0.1 8.33 I 20 110mal08 2011 US411 5860.8 31.226± 0.002 21 1.4308 5810.1 62.431± 0.011 0.49983± 0.00008 22.040± 0.001 24.813± 0.000 9.4± 0.3 359.981± 0.000 24.0± 0.1 9.11 S 14 70mal62 2011 UQ411 5860.8 52.005± 0.002 25 1.4396 5804.0 62.4± 0.2 0.405± 0.004 40.400± 0.002 215.504± 0.000 263.65± 0.02 315.5± 0.2 23.9± 0.1 6.76 I 5 20Resonant 4:1 [Alexandersen et al., 2014f]mal33 2011 UP411 5860.8 38.569± 0.003 17 1.4228 5810.1 75.79± 0.02 0.49117± 0.00015 13.435± 0.001 231.961± 0.002 160.3± 0.5 359.450± 0.000 24.2± 0.1 8.35 S 13 60Resonant 16:3 [Alexandersen et al., 2014i]mah08 2012 UK177 6220.9 36.784± 0.001 25 1.4560 5805.9 92.3± 0.2 0.6173± 0.0009 24.852± 0.001 251.482± 0.002 162.5± 0.2 354.981± 0.014 24.1± 0.1 8.38 I 8 40Inner classical [Alexandersen et al., 2014k]mal29 2011 UO412 5860.8 37.892± 0.002 26 1.4146 5810.1 38.04± 0.02 0.129± 0.003 27.787± 0.001 21.128± 0.000 273.13± 0.12 80.85± 0.08 23.4± 0.1 7.62 I 8 40mal41 2011 UN412 5858.9 40.960± 0.004 13 1.2892 5858.9 38.96± 0.02 0.058± 0.003 20.411± 0.002 216.340± 0.001 326± 5 209.1± 0.2 24.5± 0.1 8.37 S 16 70Main classical [Alexandersen et al., 2014j]mah05 2012 UL177 6220.9 36.474± 0.001 30 1.4643 5805.9 40.666± 0.009 0.1166± 0.0008 19.236± 0.000 272.168± 0.003 147.6± 0.6 335.255± 0.009 22.9± 0.1 7.24 S 6 30mal24 2011 UK412 5860.8 37.131± 0.002 29 1.4476 5804.0 40.737± 0.007 0.0920± 0.0007 26.365± 0.001 218.923± 0.001 151.6± 1.3 14.430± 0.005 23.4± 0.1 7.74 S 7 40mah13 2012 UO177 6221.9 42.380± 0.002 25 1.4560 5805.9 41.562± 0.005 0.035± 0.002 16.446± 0.000 289.959± 0.006 336± 2 122.36± 0.06 24.4± 0.1 8.13 S 6 30mal50 2011 UJ412 5860.8 42.997± 0.002 17 1.4338 5803.1 42.031± 0.007 0.029± 0.002 29.757± 0.002 27.031± 0.000 148± 5 219.82± 0.10 24.3± 0.2 7.99 S 7 30mal37 2011 UX411 5858.9 39.653± 0.002 23 1.3793 5803.1 42.43± 0.02 0.098± 0.002 7.210± 0.001 24.163± 0.002 315.7± 1.0 44.02± 0.03 24.2± 0.2 8.27 I 8 40mal44 2011 UH412 5860.8 41.372± 0.002 22 1.4313 5804.0 42.60± 0.03 0.116± 0.003 15.118± 0.002 21.517± 0.001 94.4± 0.2 290.88± 0.07 24.2± 0.1 8.02 S 7 30mah09 2012 UP177 6221.9 39.922± 0.002 27 1.4585 5805.0 42.823± 0.014 0.099± 0.002 16.601± 0.001 280.170± 0.004 59.9± 0.8 42.49± 0.02 24.0± 0.1 7.97 S 6 30mal49 2011 UL412 5860.8 42.833± 0.002 21 1.4395 5804.0 43.03± 0.02 0.083± 0.003 10.371± 0.002 239.455± 0.005 60.14± 0.10 82.06± 0.07 23.7± 0.1 7.42 I 6 30mal40 1999 RU205 5858.9 40.875± 0.002 24 1.4339 5803.0 43.048± 0.012 0.071± 0.002 7.737± 0.001 14.293± 0.003 329.4± 1.4 41.43± 0.02 23.1± 0.1 7.04 S 6 30mah07 2012 UN177 6220.9 36.611± 0.002 29 1.4667 5805.0 43.181± 0.010 0.1589± 0.0006 20.094± 0.001 341.291± 0.002 71.2± 0.7 345.867± 0.006 22.9± 0.1 7.28 S 7 30mal43 2011 UV411 5858.9 41.355± 0.002 23 1.4340 5803.0 43.363± 0.013 0.067± 0.002 4.419± 0.001 20.174± 0.004 323± 2 43.37± 0.03 24.1± 0.1 7.93 S 7 30mal39 2011 UC412 5858.9 40.517± 0.002 26 1.4723 5803.0 43.408± 0.006 0.0705± 0.0008 3.132± 0.001 337.47± 0.02 33± 2 17.794± 0.007 23.8± 0.1 7.75 S 6 30mal59 2011 UE412 5858.9 48.245± 0.003 26 1.4339 5803.0 43.496± 0.005 0.1119± 0.0005 22.799± 0.002 218.117± 0.000 5.6± 1.4 165.93± 0.06 23.3± 0.1 6.45 S 4 20mal54 2011 UB412 5858.9 44.937± 0.005 17 1.2809 5858.9 43.50± 0.02 0.056± 0.005 15.311± 0.002 14.512± 0.002 246± 4 123.94± 0.14 24.3± 0.1 7.76 S 13 60mah12 2012 UM177 6220.9 41.687± 0.002 19 1.4586 5805.0 43.549± 0.011 0.058± 0.002 16.177± 0.000 287.738± 0.006 57± 2 40.52± 0.02 24.6± 0.2 8.33 S 7 30mal46 2011 UW411 5858.9 41.526± 0.002 20 1.4340 5803.0 43.84± 0.04 0.165± 0.003 4.529± 0.002 18.484± 0.005 94.7± 0.2 297.62± 0.08 24.5± 0.2 8.37 I 7 30mal52 2011 UU411 5858.9 43.301± 0.002 24 1.4723 5803.0 43.941± 0.008 0.031± 0.002 1.694± 0.001 357.00± 0.03 99± 2 299.72± 0.03 24.3± 0.1 7.96 S 6 30mal42 2001 RZ143 5858.9 41.288± 0.002 23 1.3795 5803.0 44.061± 0.005 0.0640± 0.0005 2.122± 0.002 8.32± 0.02 35± 3 350.411± 0.004 22.6± 0.1 6.44 S 7 30Continued on next page63Table 2.3 – Continued from previous pageID MPC Disc r N t Epoch a e i Ω ω M mr Hr S σ5 σ10(AU) (yr) (AU) (◦) (◦) (◦) (◦) (”) (”)mal51 2011 UY411 5858.9 43.191± 0.002 25 1.4423 5803.0 44.22± 0.02 0.064± 0.002 1.907± 0.000 309.43± 0.04 154.1± 0.7 294.69± 0.04 24.0± 0.1 7.64 S 6 30mal56 2011 UF412 5858.9 45.576± 0.004 12 1.1466 5803.1 45.07± 0.02 0.012± 0.003 2.146± 0.000 313.97± 0.07 278± 41 160.7± 0.9 24.4± 0.1 7.83 S 15 70mal38 2011 UM412 5860.8 40.339± 0.002 27 1.4309 5804.1 45.38± 0.02 0.132± 0.002 10.729± 0.001 231.692± 0.002 124.1± 1.0 28.50± 0.02 24.4± 0.2 8.36 I 8 40mal53 2011 UD412 5858.9 44.398± 0.001 22 2.0893 5803.0 45.419± 0.006 0.0535± 0.0011 5.496± 0.001 358.349± 0.007 101.0± 0.5 297.64± 0.01 23.2± 0.1 6.75 S 1.9 10mal55 2011 UZ411 5858.9 45.158± 0.002 22 1.4504 5803.0 45.535± 0.014 0.053± 0.003 1.214± 0.000 315.36± 0.07 352.2± 0.3 77.97± 0.04 24.3± 0.1 7.75 S 6 30mal36 2011 UA412 5858.9 39.507± 0.003 12 1.1468 5803.0 46.250± 0.011 0.1476± 0.0008 2.956± 0.001 252.56± 0.02 146± 2 352.267± 0.004 24.6± 0.2 8.66 I 16 80mal48 2011 UG412 5860.8 42.436± 0.002 33 1.4313 5804.0 46.45± 0.03 0.155± 0.002 2.496± 0.000 303.96± 0.03 24.9± 0.4 48.71± 0.06 23.8± 0.1 7.57 S 6 20Outer classical [Alexandersen et al., 2014m]mal58 2011 US412 5858.9 46.661± 0.004 16 1.4501 5803.1 47.88± 0.08 0.161± 0.005 2.606± 0.002 5.09± 0.02 116.38± 0.04 288.2± 0.2 24.4± 0.1 7.74 S 9 40mal57 2011 UT412 5860.8 46.124± 0.003 15 1.4395 5804.0 48.08± 0.07 0.183± 0.004 17.822± 0.002 224.314± 0.002 79.71± 0.05 66.9± 0.2 24.1± 0.1 7.50 I 6 30Detached [Alexandersen et al., 2014l]mal20 2011 UR412 5860.8 36.158± 0.002 20 1.4395 5804.0 50.07± 0.06 0.314± 0.002 17.873± 0.001 15.280± 0.001 54.2± 0.4 340.63± 0.03 24.2± 0.2 8.60 S 11 50mah10 2012 UQ177 6220.9 40.257± 0.002 26 1.4560 5805.9 51.97± 0.07 0.312± 0.002 19.603± 0.001 330.160± 0.003 120.1± 0.2 328.58± 0.07 23.8± 0.1 7.77 S 7 30mah06 2012 US177 6221.9 36.605± 0.002 25 1.4665 5805.1 56.09± 0.05 0.3665± 0.0011 17.214± 0.001 271.782± 0.003 92.5± 0.3 11.89± 0.02 24.2± 0.1 8.51 S 8 40mal26 2011 UP412 5858.9 37.450± 0.002 23 1.4422 5803.0 56.34± 0.03 0.3443± 0.0007 19.721± 0.001 23.081± 0.000 26.4± 0.4 351.390± 0.008 23.8± 0.1 8.09 S 8 40mal19 2011 UQ412 5858.9 35.912± 0.001 22 1.4311 5803.0 67.32± 0.08 0.4829± 0.0009 16.712± 0.001 219.944± 0.000 197.5± 0.2 352.205± 0.014 23.0± 0.1 7.51 S 9 50mah14 2012 UR177 6221.9 44.388± 0.003 26 1.4560 5805.9 73.8± 0.3 0.492± 0.003 16.353± 0.000 291.500± 0.006 158.2± 0.2 340.65± 0.13 24.2± 0.1 7.76 I 8 40Unclassified [Tomatic, 2014a]mah11nt 6221.9 38± 4 3 0.0002 6221.9 39± 21 0.0± 0.6 15± 4 316± 80 88± 1384 352± 7 24.2± 0.1 8.30 N 9 000 20 000mal11nt 2011 UU412 5860.8 32.8± 0.5 11 0.0986 5859.8 39± 10 0.2± 0.5 23± 2 25.5± 0.7 32± 248 343± 7 24.5± 0.1 9.40 N 8 000 30 000Uncharacterised & unclassified [Alexandersen et al., 2014b, Tomatic, 2014a]umal64nt 5858.9 36± 4 3 0.0002 5858.9 37± 20 0.0± 0.6 3± 3 291± 277 98± 1309 0.8± 0.7 24.7± 0.3 9.09 N 13 000 30 000umal70nt 2011 UV412 5858.9 47.85± 0.05 9 0.8354 5858.9 48± 5 0.3± 0.2 11.54± 0.03 219.29± 0.03 63± 11 74± 12 24.7± 0.1 7.89 N 200 1 000umal65nt 2011 UY412 5858.9 42± 2 4 0.0195 5858.9 43± 22 0.0± 0.6 8± 3 16± 6 20± 1486 357± 3 24.7± 0.1 8.52 N 9 000 20 000umah16nt 2012 UU177 6221.9 45± 2 7 0.0055 6220.0 46± 24 0.0± 0.6 16.1± 0.6 284± 5 119± 1789 352± 8 24.9± 0.3 8.30 N 6 000 15 000umal66nt 5858.9 46± 5 3 0.0002 5858.9 47± 24 0.0± 0.6 13± 25 24± 19 13± 1620 357± 3 24.6± 0.1 8.06 N 11 000 20 000umah17 2012 UT177 6221.9 52.65± 0.01 21 1.4614 5806.0 47.49± 0.08 0.127± 0.007 16.363± 0.001 320.497± 0.013 282± 6 144.9± 0.5 24.8± 0.1 7.49 N 14 70umal68nt 5858.9 47± 5 3 0.0002 5858.9 48± 25 0.0± 0.6 8± 24 10± 60 19± 1680 0.7± 0.7 24.8± 0.1 8.04 N 10 000 20 000umal69nt 5858.9 47± 5 3 0.0002 5858.9 48± 25 0.0± 0.6 3± 20 355± 304 40± 1708 358± 2 24.8± 0.1 8.10 N 9 000 19 000umal63 2011 UW412 5860.8 38.419± 0.003 18 1.4228 5810.1 78.2± 0.2 0.521± 0.002 12.976± 0.001 233.178± 0.002 134.1± 0.5 6.17± 0.03 24.7± 0.2 8.86 N 14 60umal67nt 2011 UX412 5858.9 46.9± 0.4 7 0.1010 5858.9 54± 25 0.4± 0.4 29± 2 216.9± 0.4 270± 16 311± 34 24.6± 0.2 7.94 N 2 000 6 00064Chapter 3PlutinosAn expert is a person who has found outby his own painful experienceall the mistakes that one can makein a very narrow field.— Niels BohrNever express yourself more clearlythan you are able to think.— Niels BohrIn this chapter15, we investigate the Plutinos (objects in 3:2 outer reso-nance with Neptune) in detail. The Plutinos are the most numerous popu-lation seen in our survey, apart from the classical belt (the large reservoir ofnon-resonant TNOs with 40 AU . a . 48 AU, the first (hence “classical”)predicted feature of the trans-Neptunian region). However, the classical beltis known to have a populous cold core that has a different absolute mag-nitude distribution from the hot populations that we are interested in. Wetherefore focused on the Plutinos, aiming to investigate what constraints ourPlutino sample could put on the absolute H-magnitude distribution, orbital15This chapter is based on work submitted for publication in Alexandersen et al. [2015].65distribution and population estimate of the Plutinos.Our Plutino sample consists of 18 objects, with semi-major axis, eccentric-ity, inclination and absolute magnitude range of 39.21 AU ≤ a ≤ 39.602 AU,0.0436 ≤ e ≤ 0.3279, 2.◦733 < i < 23.◦749 and 5.58 ≤ H ≤ 9.85. Detailsabout these 18 Plutinos are presented in Table 3.1 and their semi-major axesand eccentricities are plotted in Figure 3.1. All but one of these objectshave been found to be securely within the resonance. The lowest eccentricityobject, 2011 UB411, is the only insecurely classified object, despite the ec-centricity uncertainty being ±0.0008 and semi-major axis uncertainty being0.006 AU. This insecurity is due to the resonance being weaker and narrowerat low eccentricity. However, as the region with semi-major axis just largerthan the 3:2 resonance is unstable and quickly cleared by the ν8 secular res-onance [Duncan et al., 1995], we are confident that 2011 UB411 is indeed aPlutino.In this work, we began with only using our sample of 18 Plutinos, but asthe CFEPS survey was characterised in a nearly identical fashion, we choseto include the characterisation and detections from that survey to improvethe statistics. CFEPS detected 24 Plutinos, with 39.26 AU ≤ a ≤ 39.580 AU,0.0911 ≤ e ≤ 0.2968, 1.◦422 < i < 34.◦390 and 6.17 ≤ H ≤ 8.79 (a and eare plotted in Figure 3.1). Most of our analysis is thus based on a sampleof 42 Plutinos with semi-major axis, eccentricity, inclination and absolutemagnitude ranges of 39.21 AU ≤ a ≤ 39.602 AU, 0.0436 ≤ e ≤ 0.3279,1.◦422 < i < 34.◦390 and 5.58 ≤ H ≤ 9.85. As six of the Plutinos from oursurvey are fainter than any Plutinos from CFEPS, our survey helps constrainthe absolute magnitude distribution for faint objects.6639.1 39.2 39.3 39.4 39.5 39.6 39.7Semi−major axis [AU] boundariesAlexandersen boundariesCFEPS objectsAlexandersen objectsFigure 3.1: Scatter plot of the a-e elements of the 42 Plutinos fromCFEPS (black) and our survey (red), with error bars. Also shown indashed lines are the modelled resonance boundaries used when generat-ing objects in the survey simulator. As three of our Plutinos are outsidethe CFEPS bounds, but seen to be resonant, we were forced to changethe model distribution slightly. The CFEPS boundaries were modelledafter the diffusion map in Tiscareno and Malhotra [2009], whereas ournew bounds are inspired by the diffusion map in Morbidelli [1997] mod-ified to ensure that all of our real observed Plutinos are within thebounds.67Table 3.1: List of all Plutinos detected in our survey. ID is the internal designation, while MPCis the Minor Planet Center designation. The heliocentric distance at discovery is r. The barycentricorbital elements, semi-major axis (a), eccentricity (e), inclination (i), longitude of ascending node (Ω),argument of pericentre (ω) and mean anomaly (M) are all given. The discovery magnitude mr isthe average r-band apparent magnitude as measured by the moving object pipeline in the discoverytriplet. Hr is the absolute magnitude calculated from mr using appropriate phase-angle corrections.L32 is the libration amplitude of the resonant argument, centred on phi32 = 180◦. For objects thatare additionally in the Lidov-Kozai resonance [Kozai, 1962, Lidov, 1962], the libration centre CLKand libration amplitude LLK are given. Note that one of these Plutinos was discovered prior to oursurvey; all information in this work, apart from the MPC designation, ignores the fact that this objectswas previously discovered, and the information in this table is therefore derived solely from this work.Additional information on these objects is available in Table 2.3.ID MPC r a e i Ω ω M mr Hr L32 CLK LLK(AU) (AU) (◦) (◦) (◦) (◦) (◦) (◦) (◦)mal45 2011 UD411 41.480± 0.001 39.21± 0.03 0.2884± 0.0011 15.700± 0.001 19.081± 0.001 131.88± 0.11 274.68± 0.09 23.1± 0.1 6.91 130± 30mal04 2011 UA411 29.828± 0.001 39.223± 0.004 0.24459± 0.00009 8.645± 0.000 237.962± 0.002 167.51± 0.09 351.158± 0.001 23.7± 0.1 9.01 109± 2mal02 2011 UC411 28.724± 0.000 39.270± 0.003 0.27179± 0.00009 4.652± 0.000 289.175± 0.005 88.35± 0.06 6.447± 0.001 22.3± 0.1 7.74 94± 3mal03 2011 UU410 28.997± 0.001 39.293± 0.004 0.26294± 0.00007 12.623± 0.001 19.330± 0.001 15.39± 0.08 356.454± 0.001 24.4± 0.1 9.85 91± 3mah03 2012 UG177 30.482± 0.002 39.304± 0.010 0.2248± 0.0003 21.876± 0.001 261.872± 0.002 130.6± 0.7 357.684± 0.001 24.4± 0.1 9.48 86± 9 90 42± 2mal30 2004 VT75 37.958± 0.001 39.31± 0.02 0.2108± 0.0010 12.842± 0.001 26.610± 0.000 272.04± 0.03 68.66± 0.05 21.9± 0.1 6.11 66± 18 270 55± 2mal16 2011 UY410 34.137± 0.002 39.312± 0.008 0.1364± 0.0007 8.590± 0.001 12.404± 0.002 2.9± 0.8 13.169± 0.005 24.4± 0.1 9.09 79± 5mah15 2012 UH177 45.464± 0.002 39.39± 0.02 0.1907± 0.0012 23.749± 0.001 255.149± 0.003 346.2± 0.6 137.39± 0.10 24.2± 0.2 7.57 76± 4mal07 2011 UQ410 30.974± 0.001 39.400± 0.008 0.2636± 0.0003 3.246± 0.001 10.821± 0.004 335.40± 0.06 26.946± 0.008 24.2± 0.1 9.36 69± 1mal21 2011 UX410 36.258± 0.001 39.40± 0.02 0.3279± 0.0007 17.479± 0.001 223.962± 0.000 263.515± 0.011 302.35± 0.04 23.7± 0.1 8.14 50± 10 270 80± 10mal06 2011 UR410 30.875± 0.001 39.412± 0.009 0.2685± 0.0004 17.320± 0.001 27.050± 0.000 50.37± 0.09 332.881± 0.009 24.4± 0.1 9.59 54± 2 90 40± 1mal05 2011 UV410 29.862± 0.001 39.421± 0.007 0.2591± 0.0003 4.308± 0.000 332.153± 0.006 30.14± 0.09 15.396± 0.004 22.5± 0.1 7.77 73± 1mal17 2011 US410 35.150± 0.001 39.510± 0.013 0.2554± 0.0005 11.876± 0.001 20.578± 0.001 290.41± 0.02 51.18± 0.03 23.7± 0.1 8.27 90± 8 270 48± 1mal35 2011 UT410 39.355± 0.001 39.52± 0.02 0.1787± 0.0011 13.498± 0.001 19.796± 0.001 109.09± 0.04 281.59± 0.05 21.5± 0.1 5.58 114± 4mal31 2011 UZ410 38.402± 0.001 39.521± 0.013 0.1245± 0.0012 3.833± 0.000 298.461± 0.010 8.44± 0.04 69.87± 0.04 24.0± 0.1 8.17 75± 17mal28 2011 UW410 37.600± 0.001 39.578± 0.014 0.2175± 0.0007 10.752± 0.001 17.914± 0.001 285.455± 0.014 64.55± 0.03 23.8± 0.1 8.07 102± 18 270 70± 20mal32 2011 UB411 38.548± 0.001 39.601± 0.006 0.0436± 0.0008 14.900± 0.001 13.374± 0.001 322.5± 0.6 50.471± 0.014 23.8± 0.1 7.92 130± 50mal12 2011 UP410 33.293± 0.001 39.602± 0.004 0.1749± 0.0003 2.733± 0.001 241.740± 0.006 179.2± 0.2 339.745± 0.004 23.4± 0.1 8.21 115± 5683.1 StatisticsThroughout this chapter, except Section 3.3.3, 3.3.4 and, we haveused a modified version of the Anderson-Darling (AD) goodness of fit test[Anderson and Darling, 1954] in order to test the hypothesis that a sam-ple (either real observations or a small subset of simulated detections) hasbeen drawn from a larger population (in our case always simulated). Tra-ditionally, the AD test is used for one-dimensional data, as a test of therejectability of the hypothesis that a small data-set has been drawn from alarge set (or a model); the AD test is typically more sensitive to discrepan-cies near the tail of a distribution than, for example, the commonly-usedKolomogorov-Smirnov (KS) test. Because the calculated statistic (denotedAD) is a normalised unitless quantity, it is possible to create a pseudo-multi-dimensional test. In the past, some (for example, the CFEPS papers) havecreated a multi-dimensional AD test by testing various distributions indepen-dently and using the highest rejectability for the rejectability of the model asa whole. However, this fails to take into the “look elsewhere” effect, whichmeans that the more distributions you look at, the more likely it will be thatone of them will be rejectable at 95% confidence; the CFEPS team knew ofthis shortcoming but was unable to correct for it (the “look elsewhere” effectcan only be accurately accounted for when the distributions are independent,which is not the case for e, r, H). Others, (for example, Shankman et al.[2013]), attempted to correct for this; instead of testing the rejectability ofeach distribution independently and selecting the highest one, they picked theworst statistic (ADmax = max(ADdimension)) as representative and then boot-strapped this to find the rejectability of having such an ADmax. However, thiscan give a skewed view of the significance of the statistic, as a data-set thatgets a high statistic in all dimensions can have the same maximum statisticas a data-set that only has a high statistic in one dimension, even thoughthe one with multiple high statistics ought to get rejected more strongly. Analternate approach is to add the statistics from the various dimensions into69a summed statistic [Parker, 2014],∑AD =∑dimensionsADdimension. The sum isused, as∑AD will be small if the tested distributions are in good agreementin all dimensions (all ADdimension are low), large if one of the dimensions hasa poor match and even larger if several dimensions show large discrepanciesbetween the data and model [Parker, 2014]. Note that the summed statisticis thus not quite as sensitive to rejecting cases where a single distributiongives a large discrepancy, because the same summed statistic could be causedby several dimensions being moderately discrepant. Inversely, the summedstatistic will be slightly stronger at rejecting cases where multiple dimensionsare moderately discrepant than the maximum statistic method was. We se-lect that ideally a model should represent a data-set well in all dimensions,and choose the summed statistic as it rejects models that are discrepant withthe data in a broader sense.As the conversion table between AD and significance found in Andersonand Darling [1954] has asymptotic values (relevant for large sample sizes)and because∑AD has a wider distribution than individual AD, we cannot usethe Anderson and Darling [1954] conversion table. Therefore, the distribu-tion of∑AD for samples (equal in size to the real sample) actually drawnfrom the comparison model has to be calculated. This is done by calculating∑ADsim for 103−104 simulated data-sets of the same size as the real data-set,drawn from the comparison model, usually a large sample of 1×104−5×104simulated detections generated with our survey simulator (described in Sec-tion 1.8). The summed AD statistic for our real observations,∑ADreal is thencalculated and compared to the distribution of∑ADsim. The distribution ofsummed AD statistics for the simulated subsets reveal the probability of arandom subset being a worse match to the parent simulated detections thanthe real observations are, giving us the significance of the statistic of thereal sample; the rejectability of the hypothesis that the real detections were70drawn from the model isP (ADreal) =Nsim>realNsimwhere Nsim>real is the number of times ADsim > ADreal and Nsim is the totalnumber of simulated ADsim. This is analogous to the way that KS testsare usually interpreted. In this work, probabilities < 5% (rejection at >95% confidence) or < 1% (rejection at > 99% confidence) are consideredthresholds of significant or highly significant rejection, respectively.For our analysis of the Plutinos in this work, we used a 4-dimensionalAD test as described above, using the following four properties: eccentric-ity, inclination, heliocentric distance at discovery and absolute magnitude atdiscovery. These parameters were chosen because they are the most impor-tant parameters in revealing significant shortcomings of a combined orbit andmagnitude distribution model; for example, as all Plutinos have semi-majoraxes values within 1% of each other, detailed modelling of the a distributionis unimportant, as small semi-major axis variations in this range have nodiscernible effect on detectability, whereas changing the eccentricity modelor absolute magnitude model within reason changes the eccentricity, distanceand absolute magnitude distributions of detections and vice versa, makingthem highly important for model testing. The inclination model is less cou-pled than the e, H and r distributions are, so could be tested separately,but either way, it is important to constrain the inclination distribution, aschanging the inclination model changes the population estimate.In Section 3.3.3 we present an alternative statistical approach that focuseson a 1-magnitude range, testing the probability of getting the real number ofobserved objects in the second half, based on the number of real detectionsin the first half, survey sensitivity and assumed magnitude-model. In 3.3.4we present a likelihood analysis, in which we computed the likelihood of ourobservations based on a computed completeness factor and assumed models. compares our results from using the∑AD method described above71with results from repeating the same analysis with the CFEPS approach.3.2 Changes to the L7 model and surveysimulatorThere were two changes that we were forced to make to the previous orbitalmodel, as opposed to simply using the L7 model as is. As can be seen inFigure 3.1, three of the Plutinos detected in our survey have a combination ofa and e outside the model resonance boundary used by CFEPS. We thereforehad to adjust the resonance boundary slightly, as can be seen in Figure 3.1.Our new bounds are modelled after the diffusion map in Morbidelli [1997],although the bounds were also modified to ensure that all of the real observedPlutinos (objects found to be resonant in numerical integrations) are withinthe bounds. As all Plutinos have semi-major axis within ∼ 1% of each other,the slight change in semi-major axis range for e > 0.16 does not have anoticeable influence on detectability. However, having widened the e < 0.16part of the resonance, where the resonance narrows, means that there is nowmore phase space with e < 0.16 and thus the survey simulator will generateslightly more such objects; we investigated this and found it to have verylittle effect.The second change to the the model was in the form of correcting animplementation error in the libration-amplitude distribution. Rather thanhaving a triangular distribution from 20◦ to 130◦peaking at Lpeak as describedin Gladman et al. [2012], the old code had a triangle from 0◦ for which values< 20◦ were truncated off (thus making it a quadrilateral). While changingthis, we also changed the maximum end of the triangle to 135◦, as two of ourPlutinos have libration amplitudes of ∼ 130◦.These combined changes led to slightly fewer low-amplitude objects andslightly more low-eccentricity object being generated, however, this only in-creased the population estimate by ∼ 4%. This is insignificant in comparisonwith the changes caused by the changes in the survey simulator (Section 1.8)72itself; since Gladman et al. [2012], the survey simulator has had major up-dates, among which were a significant improvement to how it handles mag-nitude uncertainty. We found that in some situations, this change led tochanges of 10− 20% in the population estimate. The effects of the improve-ments in the survey simulator thus vastly outweigh the effects of our minorimprovements in the a-e and libration amplitude models.In this work, whenever we describe having done something with the L7model or the survey simulator, we refer to the corrected versions as describedin this section.3.3 Testing previously published modelsThe magnitude limits of most of the CFEPS sky blocks were more than amagnitude brighter than those for our observations, but CFEPS imaged amuch larger area, resulting in 24 Plutino detections. The CFEPS L7 model(as described in Section 1.7) of the Plutino H-magnitude and orbital distri-butions was constructed based on these 24 Plutinos [Gladman et al., 2012]with 6.1 < Hr < 8.8. As in this work, the primary motivation was to obtain apopulation estimate, rather than measure details of the orbital distribution.However, in order to calculate a population estimate, we must ensure thatwe use an acceptable model that cannot be rejected by the data.3.3.1 Comparison of our 18 Plutinos with publishedmodelsWe first investigated whether our independent Plutino sample was in agree-ment with the CFEPS L7 model, extrapolated to the fainter magnitudes thatour survey was sensitive to. This was done by running survey simulationsof just our survey, with the L7 model (with a single exponential absolutemagnitude distribution with α = 0.90 extrapolated out to Hr = 11, whereour survey’s sensitivity has fallen almost to zero) until it had found 250 00073tracked detections. The AD test as described in Section 3.1 was then runto compare our 18 Plutinos to these simulated detections; 104 subsets of 18detections were used to generate the probabilities. We found that our 18Plutinos reject an L7-like model extrapolated to Hr = 11, at 99.1% confi-dence. That is, only 0.9% of the time is such an extrapolated model a worsematch to 18 simulated detections than to the 18 real objects, thus rejectingthe model.As the CFEPS magnitude distribution was in fact only constrained byPlutinos in the detected range 6.17 < Hr < 8.8, it is not too troubling thatextrapolating the magnitude distribution to Hr = 11 is rejectable, since ourentire study was motivated by the prior arguments that a transition nearHr ≈ 8.5 would occur. Other works have found very different magnitudedistributions for the Plutinos or related populations. Elliot et al. [2005]favoured a single exponential with α = 0.52±0.08 for resonant TNOs, basedon 54 resonant objects from the Deep Ecliptic Survey (DES) with 2.2 ≤ Hr ≤9.0 (although only one object has Hr > 8.3 and only three have Hr > 7.6)16.Shankman et al. [2013] studied 11 Scattering TNOs with 6.6 ≤ Hr ≤ 9.5,finding that this population favours a divot magnitude distribution, whichhas αb = 0.8 for Hr < 8.5, at which point the number density drops by afactor of 6, followed by a shallower αf = 0.5 for faint objects. From analysisof dynamically hot TNOs from various surveys (no exact number or range ofmagnitudes given), Fraser et al. [2014] favours a broken exponential (a knee)which has a steep exponential with αb = 0.87+0.07−0.2 for Hr < 7.7+1.0−0.5 followedby a shallower αf = 0.2+0.1−0.6 for fainter objects. Most recently, a re-analysisof DES led Adams et al. [2014] to increase their estimate of slope to α =0.95 ± 0.16 for Plutinos (from 31 objects with magnitudes 5.5 < Hr < 7.3)and α = 1.04± 0.19 for Scattered objects (from 23 objects with magnitudes5.3 < Hr < 7.2).16We converted the DES magnitudes from V R-band using g − V R = 0.4 [B. Gladman& K. Volk, private communication] and g − r = 0.5 as assumed for Plutinos everywherein this work, combined as V R− r = 0.1.74As the Plutinos and Scattering Objects are both part of the dynamicallyhot population, all of which are thought to share a common origin, their mag-nitude distributions are expected to be similar. We therefore directly test thepostulated H-magnitude distributions from Elliot et al. [2005], Fraser et al.[2014], Shankman et al. [2013], extrapolated to Hr = 11, against our detec-tions17. For consistency, we assume the CFEPS orbital distribution (none ofthe other papers derived a proposed Plutino orbital parameterisation) andapply the mentioned proposed absolute magnitude distributions in the sur-vey simulator. From ∼ 250 000 simulated model detections in each case, wefind the hypotheses that our detections are drawn from a simulated popula-tion following the Elliot et al. [2005], Shankman et al. [2013] and Fraser et al.[2014] distributions (extrapolated to Hr = 11) are rejectable at 93.7%, 99.5%and 99.1% confidence; thus all models except the Elliot et al. [2005] modelare highly rejectable. It is peculiar that the Elliot et al. [2005] model doesnot get rejected, as the reanalysis of the same data (in Adams et al. [2014],by mostly authors of the original paper) indicated a far steeper slope, similarto the Gladman et al. [2012] slope. The cumulative distributions, as detectedby our real and simulated survey, can be seen in Figure 3.2; it is clear that allof the models produce a poor representation of the data. The inclinations ofthe real objects are generally much lower than those of the simulated objects(independent of the model used, as the observed inclinations are only weaklydependant on the intrinsic absolute magnitude distribution). The simulatedand real detections are also in poor agreement in the observed eccentricity,heliocentric distance and absolute magnitude distributions.Because we are extrapolating these model distributions far past the rangeof magnitudes for which they were determined, it is perhaps not surprisingthat these models are rejectable. When restricting the comparison rangein H-magnitude of the detections (real and simulated) to Hr < 9.0 (just17We omit testing the Adams et al. [2014] slope, as the CFEPS slope is within theuncertainty of the Adams et al. [2014] result.750.05 0.10 0.15 0.20 0.25 0.30Eccentricity0. fraction5 10 15 20Inclination [deg]30 32 34 36 38 40 42 44Heliocentric distance [AU] fractionObservedElliot et al. exponentialCFEPS exponentialShankman et al. divotFraser et al. knee6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5Absolute magnitude, HrFigure 3.2: Cumulative distributions of detections from our survey.The eccentricity, inclination, heliocentric distance at discovery and abso-lute magnitude at discovery of our real (black) and simulated detections,using our new survey and the full range of H-magnitudes. Simulateddetections from the magnitude distributions in Elliot et al. [2005], Glad-man et al. [2012], Shankman et al. [2013], Fraser et al. [2014] are plottedfor comparison. Notice that the inclination distribution of the real detec-tions differs significantly from the simulated ones in all cases, suggestingthat the underlying intrinsic inclination distribution is not as wide asthat used in the model. The reader is cautioned that the intrinsic distri-bution is not shown, only the biased detections after survey simulation.76fainter than most objects used in the cited works), our 12 remaining Plutinosreject the Elliot et al. [2005], Gladman et al. [2012] ,Shankman et al. [2013]and Fraser et al. [2014] distributions at 94.0%, 99.94%, 94.2% and 92.5%confidence, respectively; in other words, for Hr < 9.0, the divot and kneeare not rejectable and present an acceptable representation of the data. Itis interesting to note that our survey has no Plutino detections with 8.27 <Hr < 9.01, yet found six Plutinos (a third of the sample) with Hr ≥ 9.01.Truncating the detections at Hr = 8.3, our remaining 12 Plutinos reject thefour models (in same order) at 84.8%, 78.9%, 77.2% and 85.5% confidence.Thus, for Hr < 8.3, where the Shankman et al. [2013] divot is effectively justa single exponential with a slope slightly shallower than the Gladman et al.[2012] one, all of these models are non-rejectable when compared to our data.So we see that for Hr < 8.3, the Plutino absolute magnitude distribution iswell represented by a single exponential, but when extrapolating to Hr = 9.0,the steep single exponential becomes rejectable, while when extrapolating toHr = 11.0, all of the tested models become rejectable except for a shallowsingle exponential. In the next subsection, we combine our survey detectionsand characterisation with that of CFEPS and repeat the tests performed inthis subsection.3.3.2 Comparison using 42-Plutino combined sampleFrom this point onward, we will combine detections and characterisationsof our survey with that of CFEPS, for a total Plutino sample of 42 objectsin the range 5.58 ≤ Hr ≤ 9.85. As our survey was performed in r-bandand CFEPS was primarily performed in g-band, some assumptions had tobe made regarding g − r colours when combining the surveys. Most CFEPSPlutinos have had their g − r colours measured [Petit et al., 2011] allowingsimple conversion of the magnitude; the few that did not have measuredcolours had their magnitude converted using the average colour of the CFEPSPlutinos with measured colours (g − r = 0.5). In the Survey Simulator,770.05 0.10 0.15 0.20 0.25 0.30Eccentricity0. fraction5 10 15 20 25 30Inclination [deg]30 35 40 45Heliocentric distance [AU] fractionObservedElliot et al. exponentialCFEPS exponentialShankman et al. divotFraser et al. knee6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5Absolute magnitude, HrFigure 3.3: Cumulative distributions for combined Hr < 11.0 sample,similar to those in Figure 3.2, but for the combined surveys (our newsurvey combined with CFEPS), with the full range of H-magnitudes(Hr < 11). Plotted for comparison are simulated detections from themagnitude distributions in Elliot et al. [2005], Gladman et al. [2012],Shankman et al. [2013] and Fraser et al. [2014], extrapolated to Hr = 11;these models are rejectable at 99.8%, 98.7%, 99.2% and 99.4% confi-dence, respectively. Note how the extrapolated CFEPS model looks likea great match for the eccentricity and heliocentric distance, but is re-jectable due to the absolute magnitude, while the knee and divot havefairly good absolute magnitude distributions, but are rejectable due tothe eccentricity and heliocentric distance. All models predict more high-inclination objects than observed, suggesting that some of the modelsmight be made non-rejectable by adjusting the inclination model.780.05 0.10 0.15 0.20 0.25 0.30Eccentricity0. fraction5 10 15 20 25 30Inclination [deg]30 35 40 45Heliocentric distance [AU] fractionObservedElliot et al. exponentialCFEPS exponentialShankman et al. divotFraser et al. knee6.0 6.5 7.0 7.5 8.0 8.5Absolute magnitude, HrFigure 3.4: Cumulative distributions for combined Hr < 9.0 sample.This contains the Hr < 9.0 subsample of the objects that were seen inFigure 3.3. Here the Elliot et al. [2005], Gladman et al. [2012], Shankmanet al. [2013] and Fraser et al. [2014] models extrapolated to Hr = 9 arerejectable at 99.2%, 99.1%, 82.5% and 89.5% confidence, respectively.the two surveys were combined assuming a g − r colour distribution basedon the colour distribution of the CFEPS Plutinos, modelled as a Gaussiandistribution of width 0.2 centred on 0.5.With this combined, 42-Plutino data-set, we repeated the tests performedin Section 3.3.1. For the full sample, the Elliot et al. [2005], Gladman et al.[2012], Shankman et al. [2013] and Fraser et al. [2014] models extrapolatedto Hr = 11 are rejectable at 99.8%, 98.7%, 99.2% and 99.4% confidence, thusrejecting all of these extrapolated models; for a visual understanding of why790.05 0.10 0.15 0.20 0.25 0.30Eccentricity0. fraction5 10 15 20 25 30Inclination [deg]30 35 40 45Heliocentric distance [AU] fractionObservedElliot et al. exponentialCFEPS exponentialShankman et al. divotFraser et al. knee6.0 6.5 7.0 7.5 8.0Absolute magnitude, HrFigure 3.5: Cumulative distributions for combined Hr < 8.3 sample.This contains the Hr < 8.3 subsample of the objects that were seenin Figure 3.3, with only Hr < 8.3 detections. Here the Elliot et al.[2005], Gladman et al. [2012], Shankman et al. [2013] and Fraser et al.[2014] models are rejectable at 99.98%, 56.8%, 83.6% and 97.2% con-fidence, respectively. While the CFEPS exponential and the divot arenot rejectable, it appears that their representation of the data wouldbe improved by lowering the mean inclination and raising the mean ec-centricity of the orbital model (raising the mean eccentricity would alsoimprove agreement with the heliocentric distance distribution, in thelower left).80these models are rejectable, see the cumulative plots in Figure 3.3. With theHr < 9 cut (36 real Plutinos), these numbers become 99.2%, 99.1%, 82.5%and 89.5%, respectively (see Figure 3.4), and with the Hr < 8.3 cut (34 realPlutinos) they are 99.98%, 56.8%, 83.6% and 97.2% (see Figure 3.5). Thus,with the combination of our survey and CFEPS, we find that the Elliot et al.[2005] shallow slope is (as expected) rejectable to any of the limits tested,and that the Gladman et al. [2012] steep slope is (as expected) a very goodrepresentation of the data for Hr < 8.3, but is highly rejectable for highermagnitudes. The reasonably steep slope of the Shankman et al. [2013] divotis also not rejectable for Hr < 8.3 while the Fraser et al. [2014] knee isrejectable, suggesting that the steep α = 0.8 − 0.9 exponential does indeedextend to nearer to Hr = 8.3 than the transition in the Fraser et al. [2014]knee at Hr = 7.7. The fact that the steep exponential is non-rejectable forHr < 8.3 and highly rejectable for Hr < 9.0 and Hr < 11.0 suggests that themagnitude distribution indeed does not extend with that same exponentialform to faint magnitudes, with instead a drastic change in the magnitudedistribution around Hr = 8.3. This Hr ' 8.3 transition location is verysimilar to the drop in number density at HR ' 8.5 (Hr ' 8.6) found forthe Neptunian Trojans [Sheppard and Trujillo, 2010b] and the Hg ' 9.0(Hr ' 8.5± 0.2) transition concluded for the Scattering Objects [Shankmanet al., 2013]. The fact that all four models are rejectable when extrapolatedto Hr = 11.0 is unfortunate, and might mean that some of the parameters(either of the absolute magnitude distribution or orbital distribution) needto be adjusted in order to provide an acceptable solution; we will investigatethis possibility in Section 3.4. As discussed in Section 1.5.1, it is doubtfulthat the observed transition could be due to a sudden size-dependant albedofeature.813.3.3 Two-bin testIn 3.3.2 we confirmed that the Plutinos must have a transition in their ab-solute magnitude distribution, as extrapolating a single exponential is re-jectable (both for a shallow or a steep exponential). Because NeptunianTrojans [Sheppard and Trujillo, 2010b], scattering objects [Shankman et al.,2013] and hot populations in general [Fraser et al., 2014] were already pos-tulated to have a transition around Hr ∼ 7.7 − 8.6, we wanted a test todetermine whether such a transition is present in our data and also revealthe location of the transition. We here present such a test, designed toidentify whether and where a transition away from a single exponential H-magnitude distribution occurs. This test, which we refer to as the “two-bintest”, examines the absolute magnitude distribution within just two bins, inorder to determine whether a transition occurs between the two bins. Thistest investigates the rejectability of the hypothesis that the number of de-tections in the second bin is in agreement with the number expected giventhe number of detections in the first bin and an assumed absolute magnitudedistribution model. This test is envisioned as a simple, intuitive attempt toget a qualitative sense of the nature of the data. Focusing on only a smallmagnitude range will reveal the location and nature of a transition if present,without being potentially influenced by small number effects that might existat either end of the observed H-magnitude distribution.Figure 3.6a shows all of the 42 Plutinos from our combined surveys, as anumber density per 0.5 magnitudes at a 0.05 magnitude intervals. It is clearthat there is a roughly exponential rise in the number of detections fromHr = 5 to 8, but then there is a precipitous drop in number of detections justpast Hr = 8.0. Could that drop be simply due to a sudden drop in sensitivityto fainter objects? To investigate this, we determined the “visibility” of thePlutinos as a function of H-magnitude.We define the visibility as the fraction of objects with a givenH-magnitudewhich, assuming an orbital distribution, would be detected and tracked by82Hr  magnitude2468101214Plutinos/0.5 mag # PlutinosHr magnitude0. fractionPlutino visibilityPlausible range4 5 6 7 8 9 10Hr magnitude(Ht for c))10-810-710-610-510-410-310-210-1P(N≤N +|E +)P (N≤N+|Divot with αb =0.80, c=6, αf =0.50)P (N≤N+|Knee with αb =0.87, αf =0.20)P (N≤N+|Single Exponential with αb =0.90)Figure 3.6: Plutino histogram, visibility and two-bin test results. a)The number of Plutinos per 0.5 mag bin, at 0.05 mag intervals (to revealbinning-effects).b) The visibility of Plutinos to our combined surveys, assuming theCFEPS L7 orbital distribution (black curve), as a function of absolutemagnitude. Note that the visibility only includes objects detected andtracked by the survey simulator, and thus already takes the detectionefficiency and limiting magnitude into account; the visibility is thus re-liable even at small values. The green region shows the uncertainty invisibility caused by the uncertainty in the orbital distributions; the un-certainty is a small fraction of the visibility for Hr < 9.c) Illustration of the two-bin test, showing P (N ≤ N+|E+), the prob-ability of getting the observed number of detections or less in the nextbin, given the expectation based on the previous bin and labelled model.Low values thus shows that the number of detections is far fewer thanthe model predicts (so the model is overpredicting).83a given survey if the object was within the field of view; that is, the proba-bility that, if the object is in the field of view, it gets detected, tracked andis brighter than the survey characterisation limit18. An object’s visibilitydepends on its absolute magnitude, the object’s orbit and the detection effi-ciency. A low-H object will have a high visibility, as it is detectable even atapocentre; a moderate-H object would have a smaller visibility, as it wouldonly have a detectable apparent magnitude during part of its orbit; a verysmall (large-H) object in the field of view will not be detectable even atpericentre, thus having a visibility of 0. The visibility of Plutinos to ourcombined surveys, given the CFEPS L7 orbital model, was obtained by pass-ing a flat absolute magnitude distribution (i.e. same number of objects atevery H) through the survey simulator, thus obtaining a fraction of objectsdetected as a function of H-magnitude. This fraction was then normalisedas a visibility from 0 to 1 in order to only account for objects within thesurveyed fields and is shown in Figure 3.6b. Unsurprisingly, the visibilitycannot have sharp feature for a population with a wide eccentricity distri-bution like the Plutinos. Abrupt features in Figure 3.6a must thus be realvariations in the absolute magnitude distribution or statistical fluctuations.As we only used simulated objects with apparent magnitudes brighter thanour ∼ 40% survey characterisation limits to construct the visibility, we haveconfidence that the visibility is reliable even in the tail when it reaches smallvalues. To confirm this, we constructed the visibility for a range of orbitalmodels with parameters within the uncertainty of the L7 favoured parame-ters, to quantify the effect of slightly altering the orbital model. As can beseen in Figure 3.6b, none of the models can produce a sharp transition andall produce similar visibility curves.Having calculated the visibility, we perform the following algorithm, whichwe refer to as “the two-bin test”, to study the evidence for a slope change at18The visibility is thus analogous to the “completeness” in Section 3.3.4, only the visi-bility is normalised to the field of view as opposed to the whole sky.84a candidate transition Ht:1. Bin objects into two bins in H-magnitude space; B− covering [Ht −∆H , Ht) and B+ covering [Ht, Ht + ∆H). The number of objects inthese bins is N− and N+, respectively.2. Calculate E+, the “expectation value” for the number of objects in B+,given the known value of N−, the assumed H-model and the visibilityfraction in each bin.3. Calculate the Poisson probability P (N ≤ N+|E+) of observing in B+the detected number of objects N+ or less, given the expectation valueE+.P (N ≤ N+|E+) =N+∑N=0(E+)N exp(−E+)N !4. Shift Ht by δH and start over. It is important that δH is significantlysmaller than ∆H , in order to demonstrate that the choice of bin-centresand bin-widths are not crucial to any given result.For our work, we have used ∆H = 0.5 and δH = 0.05. This value for∆H was chosen as a compromise between having bins sufficiently large thatvery few bins had 0 objects in them, and the desire for smaller bins, inorder to be as sensitive as possible to the exact location of a transition. Wehave confirmed that changing our choice of ∆H and δH slightly did not havesignificant impacts on our results.Low values of P (N ≤ N+|E+) reject the model at that Ht, as a low prob-ability means that the model is overpredicting the number of detections (thatis, the expected number is significantly greater than the observed number).For a single exponential function, a low value at a given Ht means that thesingle exponential function cannot explain the number of detections in thebin after Ht. When testing a divot or knee function, we used the bright andfaint slope, αb and αf , as well as the contrast c in the case of the divot, from85the Shankman et al. [2013] divot and the Fraser et al. [2014] knee models.We let Ht freely vary as the division point between the two bins, thus testingevery value in the observed H-range; this way, the test gives an independentestimate of where a transition occurs.The probabilities, given different models, as a function of the varying Htmagnitude can be seen in Figure 3.6c. This plot shows a strong signature ofa drastic transition around Ht ∼ 8.3. However, how strong is this evidence?To check the efficacy of this test, we repeated the same test on simulated de-tections (rather than our real detections), created by passing various modelsthrough our survey simulator. This was done in such a way as to answer “Ifthe intrinsic distribution is a X distribution, how often would the detectionshave a distribution similar to that of our real detections?” where X will be theGladman et al. [2012] exponential, Fraser et al. [2014] knee and Shankmanet al. [2013] divot. Here simulated sets of detections were treated exactly asthe real detections have been, to reveal how often the simulated detectionsproduce a similar two-bin test result as the real detections.Simulated detections were generated from a single exponential model withα = 0.90, to see whether detections drawn from such a model could causesignatures (shapes and values) like those seen for the real detections in Fig-ure 3.6c. This was done with 1000 simulated sets of detections of 42 Plutinos,to get a range of possible outcomes, and the 1, 2 and 3σ boundaries werefound. When testing a single exponential against simulated detections gen-erated from a single exponential, the two-bin test should not give low prob-abilities; 1, 2 and 3σ outliers should have probabilities of ∼ 32%, ∼ 4.6%and 0.3%. Figure 3.7a confirms that this is indeed the case and that thesimulated detections never cause the test to give probabilities as small asour real detections do. The signature that we see at Hr ≈ 8.3 from the realdetections thus has insignificant probability of being caused by an underly-ing intrinsic single exponential magnitude distribution with α = 0.90, andsome transition must be causing this signature, confirming our result from86Hr magnitude10-710-610-510-410-310-210-1P(N≤N +|E +)P (N≤N+|Single Exponential with αb =0.90, real objects)Simulated from single exponential with αb =0.90 :1σ2σ3σHr magnitude10-710-610-510-410-310-210-1P(N≤N +|E +)P (N≤N+|Single Exponential with αb =0.90, real objects)Simulated from knee with αb =0.87,Ht =7.7, αf =0.20 :1σ2σ3σ4 5 6 7 8 9 10Hr magnitude10-710-610-510-410-310-210-1P(N≤N +|E +)P (N≤N+|Single Exponential with αb =0.90, real objects)Simulated from divot with αb =0.80, Ht =8.5, c=6, αf =0.50 :1σ2σ3σFigure 3.7: Demonstration of the efficacy of two-bin test. Each plotshows the 68.2 (dotted), 95.4 (dash-dotted) and 99.7% (dashed) boundsfrom testing a single exponential against 1000 sets of 42 simulated detec-tions using the two-bin test. The simulated detections were generatedfrom a) a single exponential magnitude distribution with αb = 0.90, b)a knee magnitude distribution with αb = 0.87, Ht = 7.7, αf = 0.2, c) adivot magnitude distribution with αb = 0.80, Ht = 8.5, c = 6, αf = 0.5.In all three plots, the solid red line is the same as in Figure 3.6c, for ref-erence. It is clear that in a) 99.7% of the simulated sets never generatesignatures as strong (i.e. probabilities as low) as the one seen for thereal data, making the hypothesis that “a single exponential magnitudedistribution could produce our real detections by chance” rejectable at> 3σ level. In b) and c), the signal from our real detections also dropsthrough the 3σ curve for the simulations, although far less. However,whereas our detections have their minimum near Hr 8.2−8.3, the simu-lated 3σ curve in c) does drop to even lower values near Hr = 8.5. Thissuggests that if c) was generated with Ht = 8.2−8.3 (in r-band) insteadof Ht = 8.5, the real detections would not reach values outside the 3σcurve. So although it would still be in the tail, the divot appears to bethe model most likely to create the observed data-set.87Section 3.3.1.The process was repeated in a similar way, generating a thousand sam-ples of 42 simulated detections using the Fraser et al. [2014] knee model, withαb = 0.87, Ht = 7.7, c = 1, αf = 0.2. The two-bin test was performed onthese simulated detections, in order to see whether a knee in the intrinsicdistribution can explain the signature seen in the real detections. Here weagain use the two-bin test of a single-exponential distribution against thesimulated detections; this is because we do not know the real intrinsic distri-bution, and it is important to treat simulated detections with the same levelof ignorance, in order to compare them fairly. We are thus always consideringwhether the simulated detections could explain the signature seen when thetwo-bin test ask the question “Is a single exponential a valid interpretation ofthe real detections over this magnitude range?”; in other words, if the intrin-sic population follows an X distribution (a knee in the current case), would itproduce data distributed as our real detections (interpreted by whether theyproduce a signature similar to that seen in Figure 3.6c). If not, then modelX is unlikely to be a valid representation of the intrinsic distribution. Thetest of a theoretical single exponential distribution against real or simulateddata is thus a constant, comparable metric.Although there is improvement when the simulated detections are drawnfrom a knee, as seen in Figure 3.7b, even this magnitude distribution cannotquite explain the signature that the real data gives us; the trough of the realsignature can happen, but is a & 3σ outlier. The transition is most likelymore drastic than the proposed knee.Lastly, we used the Shankman et al. [2013] divot magnitude distribution,with αb = 0.80, Ht = 8.5, c = 6, αf = 0.5, to generate simulated detectionsand repeat the procedure of the last two paragraphs. With the simulateddetections from the divot, the 1, 2 and 3σ curves from the two-bin test(Figure 3.7c) have a very deep dip at the transition magnitude, similar tothat seen for the real detections (although the trough is off-set slightly). The88Hr magnitude10-710-610-510-410-310-210-1P(N≤N +|E +)P (N≤N+|Single Exponential with αb =0.90, real objects)Simulated from knee with αb =0.87,Ht =7.9, αf =0.20 :1σ2σ3σHr magnitude10-710-610-510-410-310-210-1P(N≤N +|E +)P (N≤N+|Single Exponential with αb =0.90, real objects)Simulated from knee with αb =0.87,Ht =8.1, αf =0.20 :1σ2σ3σ4 5 6 7 8 9 10Hr magnitude10-710-610-510-410-310-210-1P(N≤N +|E +)P (N≤N+|Single Exponential with αb =0.90, real objects)Simulated from knee with αb =0.87,Ht =8.3, αf =0.20 :1σ2σ3σFigure 3.8: Tweaking the knee parameters. This plot is similar toFigure 3.7, although here we have used three slightly altered knee tran-sition magnitudes Ht to produce the intrinsic population. In all threeplots, the solid red line is the same as in Figure 3.6c, for reference. It isclear that in all three cases, 99.7% of the simulated sets never generatesignatures as strong (i.e. probabilities as low) as the one seen for thereal data, making the hypothesis that “a single exponential magnitudedistribution could produce our real detections by chance” rejectable at> 3σ level.test values for the simulations in fact dip to lower values than the test of thereal detections, but the minimum of the two is off-set, such that the signaturefrom the real detections still drops below the 3σ curve of the simulations atHr ∼ 8.2. While the agreement is not perfect, the shape of the signatureof the real and simulated detections are far more similar than any of theabove, and the 3σ curve for the simulations dips deeper than the real curve,89suggesting that a divot with a slightly smaller transition magnitude is themost likely model to produce a data-set that would generate the observedsignature.To further illustrate this point, we varied the parameters from the lit-erature models slightly to look for an improvement. Although this is not arigorous approach, it does give an impression of how tweaking the parame-ters affects the two-bin comparison. We investigated distributions (knees anddivots) with Ht = 7.7 to 8.5 at 0.1 magnitude intervals. Figure 3.8 clearlyshows that there is no way of tweaking the transition point such that an in-trinsic population with a knee has a non-negligible probability of producingthe observed detections (< 0.3% of cases have two-bin test probabilities aslow as our real detections). Figure 3.9, on the other hand, shows that tweak-ing the transition point to Ht = 8.2 does in fact allow a divot to produceobservations distributed in a similar manner to our real observations in 5%of cases. We take this as a reasonable argument that the Plutino magnitudedistribution is best represented by a divot with a transition in the magnituderange 8.1 < Ht < Maximum likelihood analysisThe boot-strapped AD routine used earlier in this chapter is a rejectabilitytest, not a maximum likelihood test. The least rejectable model is thusnot necessarily the most likely. We here implement a maximum likelihoodanalysis similar to that described in Heyl et al. [2015]. An additional benefitof this method, besides revealing which model is most likely to have producedthe observed data, is that the population estimate can easily be derived withvery little additional work. We implemented such a method on a simplifiedversion of our problem as a test case, fixing the orbital model and varyingonly the absolute magnitude distribution.The procedure described in Heyl et al. [2015] requires knowledge of afunction f(parameters), which describes the expected distribution of detec-90Hr magnitude10-710-610-510-410-310-210-1P(N≤N +|E +)P (N≤N+|Single Exponential with αb =0.90, real objects)Simulated from divot with αb =0.80, Ht =8.2, c=6, αf =0.50 :1σ2σ3σHr magnitude10-710-610-510-410-310-210-1P(N≤N +|E +)P (N≤N+|Single Exponential with αb =0.90, real objects)Simulated from divot with αb =0.80, Ht =8.3, c=6, αf =0.50 :1σ2σ3σ4 5 6 7 8 9 10Hr magnitude10-710-610-510-410-310-210-1P(N≤N +|E +)P (N≤N+|Single Exponential with αb =0.90, real objects)Simulated from divot with αb =0.80, Ht =8.4, c=6, αf =0.50 :1σ2σ3σFigure 3.9: Tweaking the divot parameters. This plot is similar toFigure 3.7, although here we have used three slightly altered divot Htvalues to produce the intrinsic population. In all three plots, the solidred line is the same as in Figure 3.6c, for reference. It is clear that inall three cases, the signature of the real detections (solid red line) donot drop below the 99.7% line, and in a) it only just reaches the 95%line. This means that for a divot with Ht = 8.2, 5% of the simulatedsets generate signatures stronger (i.e. lower probabilities) than the oneseen for the real data, making the hypothesis that “a single exponentialmagnitude distribution could produce our real detections by chance”rejectable at only ∼ 2σ level. Thus a divot is less rejectable than theknee and single exponential, meaning that a divot is more likely to haveproduced our observed sample.91tions as a function of various parameters; f is thus essentially the probabilitydistribution of detections19. This function can be constructed as a productof other functions:f(parameters) = C(parameters)D(parameters),where C is the completeness (or detectability) as a function of various para-meters, andD is the distribution of objects (including appropriate normalisa-tion, giving the population estimate). Here the completeness, C(parameters),is the fraction of all objects with those parameters that are detected; whenparameters = Hr, the completeness is thus analogous to the visibility usedin Section 3.3.3, except the completeness is normalised over the whole skyrather than just the survey field of view. D can furthermore be writtenas a product of the distributions of any independent variables, for exampleD(parameters) = D(Hr)D(i)D(e, a)D(other parameters) in the case of ourPlutino models.Unfortunately, due to complicated observational biases [see Gladmanet al., 2012, Jones et al., 2006], the completeness cannot be calculated ana-lytically and must be obtained using the survey simulator described earlier.The function C therefore must be binned. We chose a bin-size of 0.1 mag-nitudes, as we do not believe finer resolution is required and are limited byavailable processing power. We ran sets of survey simulations with the L7orbital model and just a single Hr magnitude until 10 000 detections werereached; comparing the number of detections to the number of drawn objectsgives the completeness at that Hr magnitude. 10 000 detections were chosen,in order to obtain 1% accuracy (Poisson-errors). The value of Hr (as thecentre of 0.1 mag wide bins) was incremented from 0.05 to 10.95 at 0.1 magincrements; these 110 jobs combined took ≈ 800 core-hours to complete (allreferences to core-time in this work refers to jobs run on Intel Xeon 3.0 GHz19f is thus unrelated to the true anomaly of an object, also typically assigned the letterf , as described in Section 1.1; the true anomaly makes no appearance in this section.920 2 4 6 8 10Hr10-710-610-510-410-310-210-1CompletenessCompletenessFigure 3.10: The Plutino completeness of our surveys (assuming theL7 orbital distribution), as a function of Hr. Note that this shows thatour combined surveys were able to detect and track ≈ 2% of all Plutinoswith Hr < 6.0 (assuming the L7 orbital model). Dashed lines corre-sponding to 10%, 1%, 0.1% and 0.01% of the maximum completenessare plotted for reference; the completeness drops below these values atHr = 8.5, 9.5, 10.0 and 10.4, respectively.processors). The completeness calculated can be seen in Figure 3.10.As a first iteration, we took the L7 orbital model of Gladman et al. [2012]as fixed, leaving only the magnitude distribution and population estimatevariable. For a single exponential distribution, the cumulative magnitudedistribution isN(Hr < Hr∗) = Nref10αb(Hr∗−Hr,ref ),93where Hr∗ is the r-band H-magnitude at which the cumulative populationis calculated and Nref is the normalisation such that when Hr∗ is equal to areference magnitude Hr,ref , N(Hr < Hr∗) = Nref . The differential form ofthis isdNdHr= Nrefαb ln(10)10α(Hr−Hr,ref ).The function f can be constructed by multiplying the differential magnitudedistribution with the completeness as a function of Hr. The function thusbecomesf(Hr) = C(Hr)Nrefαb ln(10)10α(Hr−Hr,ref ),for the case of a single exponential. For a knee and a divot, this becomesslightly more complicated; we will not go over the derivation here, but supplythe code used in Appendix D for the benefit of curious readers.Next, the logarithm of the likelihood was calculated by [Heyl et al., 2015]:logL =Nobs∑i=1log f(Hr,i)−Nbins∑j=1f(Hr,j)∆Hr (3.1)logL =Nobs∑i=1log f(Hr,i)−Npred (3.2)where the first sum is summed over the Nobs real observations (Hr,i beingthe Hr magnitude of the ith object), the second sum is summed over theNbins bins (Hr,j being the centre of the jth bin) and ∆Hr is the bin-size.The second term is the predicted (or expected) number of detections, andthus depends on the normalisation. Since the input model does not specifythe normalisation (population size), we set Npred = Nobs and instead applya normalisation such that∫ Hlimit0 f(Hr)dHr = Nobs, where Hlimit is the limitimposed on the H-range (8.3, 9.0 and 11.0 in the examples below). Thisensures that f is the probability distribution normalised such that it inte-grates to the same number of detections as the real sample. The form of f940 2 4 6 8 10Hr4321012logfElliotGladmanShankmanFraserObserved objectsFigure 3.11: The value of log f as a function of Hr for our survey(assuming the L7 orbital distribution), for various H-models and range0 < Hr < 11.0. The vertical dashed lines show the Hr of real detections.The value of logL is thus the sum of the values of log f at the intersectionof a curve with the dashed lines, minus the number of detections (42).for various models can be seen in Figure 3.11.The likelihood that the four different models produced our data for Hr <8.3 was calculated. We use Hr < 8.3 first, as this is the range for which theabsolute magnitude distribution is already best understood. Given that theShankman et al. [2013] divot transition does not occur until Hr = 8.5, inthis comparison it is really just an exponential slightly shallower than theGladman et al. [2012] one. The Elliot et al. [2005] shallow exponential, theGladman et al. [2012] steep exponential, the Shankman et al. [2013] divotand the Fraser et al. [2014] knee, have likelihoods of logL = 4.0, 6.9, 6.9and 6.6, respectively. Thus, the Elliot et al. [2005] distribution is clearly far95worse than the other models. To identify whether the models with the higherlikelihoods are not only most likely but also not rejectable, we drew 2500simulated samples of 34 Plutinos with Hr < 8.3 and calculated the likelihoodfor each of these samples. We find that for the four models logLreal is smallerthan logLsim of the simulated samples in 0.035%, 54%, 18%, 19% of cases.As logL is a two-tailed distribution, models are rejectable at 95% confidenceif they are in the 2.5% tail in either end; the Elliot et al. [2005] model is thusrejectable at 99.93% confidence, while none of the other models are rejectablefor the Hr < 8.3 sample. Integrating the absolute magnitude distribution(using the same normalisation as f), gives a Plutino population estimate.As we in this case had no other way of estimating the uncertainty of thepopulation estimate, we use the√Nobs/Nobs Poisson uncertainty; for Nobs =34, this is 17%. Thus our population estimates for Hr < 8.3 Plutinos giventhe four models are 5100±900, 7300±1300, 6800±1200, 6100±1100. CFEPSestimated the existence of 13 000+6000−5000 Plutinos with Hg < 9.16 [Gladmanet al., 2012], corresponding to Hr < 8.66 (here uncertainties indicate the95% confidence range). Scaling down, this CFEPS estimate corresponds to≈ 6000 ± 2500; thus, our population estimates are in good agreement withthe previously published value in this Hr < 8.3 range, where CFEPS wassensitive. Except for the rejectable very shallow Elliot et al. [2005] model,the other three models all give similar results.We now repeated the same procedure for Hr < 9.0 (36 real Plutinos) andHr < 11.0 (full set of 42 real Plutinos). With Hr < 9.0 we found likelihoodsfor the four models of logL = 1.9, 0.74, 3.8, 4.7, respectively, which weresmaller than the simulated values in 0.05%, 98.6%, 33% and 16% of cases(the four models are listed in the same order throughout). Thus now, anextrapolation of the Gladman et al. [2012] single exponential is rejectable at97.2% confidence, the Elliot et al. [2005] shallow slope is still highly rejectable,while the divot and knee remain acceptable. With Hr < 11.0 the four modelshad likelihoods of logL = 0.1, −2.6, 0.6, 2.0, respectively, which were smaller96than the simulated values in 6.9%, 99.5%, 92.6% and 82% of cases. Sothe full data-set rejects the Gladman et al. [2012] single exponential withα = 0.9 (extrapolated to Hr = 11) at 99.0% confidence, while the divot andknee are still acceptable. It is peculiar that the Elliot et al. [2005] slope ofα = 0.52, which was highly rejectable for Hr < 8.3 and Hr < 9.0, becomesnon-rejectable over the expanded range Hr < 11.0; however, the fact thatthis slope is highly rejectable on a (large) subset of the Hr range means thatthis model should still be considered rejected. For Hr < 11.0, our populationestimates are (105±16)×103, (770±120)×103, (45±7)×103, (38±6)×103 forthe extrapolated Elliot et al. [2005], Gladman et al. [2012], Shankman et al.[2013], Fraser et al. [2014] models, respectively, although only the last twoestimates can be considered reliable (as the first two models are rejectable).The transition magnitudes of both the Shankman et al. [2013] divot andthe Fraser et al. [2014] knee were stated to be uncertain at the 0.5 magnitudelevel. We therefore investigated whether we could improve the likelihood ofmodels by shifting Ht for both of them to Hr = 8.3, the location that ourtwo-bin test suggested as the location of a transition in Section 3.3.3. Thisincreased the likelihood of the divot to logL = 2.6, while decreasing thelikelihood of the knee to logL = 1.2; the divot and knee now have smallerlikelihoods than the simulations in 76% and 92% of cases, respectively. Thus,a divot at Hr = 8.3 instead of Hr = 8.5 (with all other parameters thesame) improves the match between model and data, while the knee appearsto slightly favour the original transition magnitude of Hr = 7.7. This isactually not surprising if one looks at Figure 3.11, which shows the functionlog f(Hr) for the four literature models, as well as the magnitude at whichour 42 real detections lie. This plot thus shows what values go into the firstterm of Equation 3.2 (the sum of log f(H) over the real detections). It is clearthat log f for the published knee peaks among the range of Hr most denselypopulated with detections, and is beginning to drop a bit before the suddenlack of detections; increasing the knee’s transition magnitude would mean97that more of the simulated detections are going to be in the Hr > 8.3 range,where there are very few real detections. On the other hand, the value oflog f for the Ht = 8.5 divot actually peaks past the range of dense detections,peaking in a range where the are no detections. Pushing the transition toHr = 8.3 clearly helps this model, as the value of log f will be highest inthe region of most detections, and lower for Hr > 8.3 where detections aresparse.The likelihood analysis and AD analysis (in Section 3.3.2) have similarresults, although not the exact same. For Hr < 11, the AD test rejectsall of the extrapolated models (with the Gladman et al. [2012] exponentialbeing the least rejectable), while the likelihood test of this section preferthe extrapolated Shankman et al. [2013] divot or Fraser et al. [2014] knee(although it also does not reject the Elliot et al. [2005] shallow exponential).For Hr < 9.0, the two methods agree that an extrapolated Elliot et al.[2005] shallow exponential and Gladman et al. [2012] steep exponential arerejectable at ≥ 99% confidence, and that extrapolating the Shankman et al.[2013] divot and Fraser et al. [2014] knee is non-rejectable. For Hr < 8.3,the two tests agree that the Elliot et al. [2005] shallow exponential is highlyrejectable, and that the Gladman et al. [2012] steep exponential and theShankman et al. [2013] divot is non-rejectable; they do disagree on whetheror not the Fraser et al. [2014] knee is rejectable.These differences could be caused by several factors. In this likelihoodanalysis, we have not incorporated a method for accounting for magnitudeerrors; the real data uses observed (uncertain) magnitudes, while the com-pleteness is a function of the intrinsic absolute magnitude. For a truly correctanalysis, the completeness should be a function of the surmised magnitude(the magnitude that would be measured, which be different, typically smaller(brighter), than the intrinsic magnitude). The magnitude error becomeslarger for smaller objects, which could explain why the two tests agree atHr < 9.0 and not for Hr < 11.0.98Another shortcoming of this likelihood analysis is the fact that we haveconstructed the completeness as a function of only Hr, with an assumedorbital distribution. The likelihood test is therefore only testing the Hrdistribution, while the AD test is testing both the orbital and magnitudedistribution, which could also cause the results to be slightly different if thedetected objects are in fact not drawn from the L7 orbital model as assumedfor the likelihood test. In order for the likelihood test to be truly useful com-parable to our AD test, the completeness (and f) should be constructed atthe very least as a function of e, i and H. We estimate that the completenesswould need to be calculated for a grid with 0 < e < 0.4420 in steps of 0.01,0◦< i < 90◦ in steps of 1◦ and 0 < Hr < 11 in steps of 0.1 magnitude. Thiswould therefore be a factor of ∼ 4000 more grid-points than in our aboveanalysis, many of which would take exceedingly long time to generate 10,000detections due to low completeness.We attempted to compute the completeness as a function of e, i and Hin this manner. For the full range of e and i stated above, we were able tocalculate the completeness for Hr < 8.3 using 47, 721 core-hours (7.5 core-years); as the time-requirements increase roughly as 1/C (where C is thecompleteness), the 7.8 < Hr < 8.3 grid-points accounted for ∼ 25% of thetotal time. The completeness, as a function of e, i and Hr can be seen incontour form in Figure 3.12; it is clear from this that at bright magnitudes,the combined surveys should have detected about 18% of all objects withi = 0◦− 1◦ (and any eccentricity), while the completeness drops to about0.7% for i = 80◦− 90◦ objects. As magnitudes increase past Hr = 6.0, thecompleteness first starts to drop for high-eccentricity objects; this is becauseat this magnitude, the Plutinos are no longer bright enough to be detectableat apocentre, while the low-eccentricity objects are still bright enough to bedetectable throughout their orbit. At about Hr = 7.4 this trend ends, and20Plutinos with q < 22 AU are not stable due to Uranian perturbations, so it is notnecessary to calculate the completeness beyond e ≈ 0.44990102030405060708090 Hr =0.00.095 0.0190.004 Hr =5.00.0950.0190.004Hr =5.50.0950.0190.004Hr =6.00.0950.0190.004010203040506070809 Hr =6.50.0950.0190.004Hr =7.00.095 0.0190.004Hr =7.40.0190.004Hr =7.50.0190.004010203040506070809Inclination (deg)Hr =7.60.0190.004Hr =7.70.0190.004Hr =7.80.0190.004Hr =7.90.0190.0040.0010.1 0.2 0.3 0.4010203040506070809Hr =8.00.0190.0040.0040.0010.1 0.2 0.3 0.4EccentricityHr =8.10.0190.0040.0010.1 0.2 0.3 0.4Hr =8.20.0190.0040.0010.1 0.2 0.3 0.4Hr =8.30.0190.0040.0010.0180.0280.0450.0710. (%)Figure 3.12: The Plutino completeness as a function of e, i and Hr, forour combined surveys. Other orbital parameters (libration amplitude,etc.) still assume the L7 distribution. Note that this shows that ourcombined surveys were able to detect and track ≈ 18% of all Plutinoswith Hr < 6.0 and i = 0. Black lines corresponding to 50%, 10%, 2%and 0.4% of the maximum completeness are plotted for reference.100for Hr > 7.4, the surveys have a higher completeness for high-eccentricityobjects; this is because the shallowest of the survey blocks are no longerable to detect objects on low-eccentricity orbits, while high-eccentricity stillallows objects to be detectable at pericentre. For Hr > 8.0, a peculiarpattern is evident where the completeness is noticeably higher for i ∼ 2◦− 6◦and i ∼ 15◦−18◦ than immediately below them; these are the latitudes of thesurvey blocks of this thesis. The fact that our high-lat block was ∼ 15◦ abovethe ecliptic means that the completeness of objects with i > 15◦ is boosted.The locations of our survey blocks are not apparent for brighter magnitudesbecause our survey is only a small addition when combined with the CFEPScoverage; however at faint magnitudes, where the CFEPS sensitivity hasdropped off, our survey dominates. At Hr = 8.3, CFEPS is insensitive tolow-eccentricity Plutinos even on i = 0◦ orbits, thus making the sensitivityof our survey (with most of its area 1◦− 5◦ above the ecliptic) noticeable.From this multi-dimensional completeness, we calculated the likelihoodof our data (using a modified version of the script in Appendix D) giventhe four models, the Elliot et al. [2005] shallow exponential, the Gladmanet al. [2012] steep exponential, the Shankman et al. [2013] divot and theFraser et al. [2014] knee, and find the data has likelihood smaller than thatof simulations in 1.1%, 45%, 22% and 19% of cases confidence. The Elliotet al. [2005] model is thus found to be rejectable at 97.8% confidence, whilethe other models are all acceptable to Hr = 8.3, in agreement with our 1Dlikelihood test earlier in this subsection.As mentioned earlier, the computing time goes roughly as the inverse ofthe completeness. Extending the computation of the completeness as a func-tion of e, i and Hr to Hr = 11 would therefore take a significant amount oftime. We estimate that it would require a factor of 100 more computationaltime, 750 core-years; even on 100 cores, this task is computationally pro-hibitive. As the AD-analysis is far less computationally heavy and appearsto give very similar (although not identical) results to the likelihood-method,101we will continue to use the AD-test for the rest of this work.3.3.5 Number of detections required for rejection.We have shown, repeatedly, that the single exponential distribution extrap-olated beyond Hr ' 8.3 is rejectable. However, we have found that theShankman et al. [2013] divot and the Fraser et al. [2014] knee are bothacceptable models with the current data, although the two-bin test in Sec-tion 3.3.3 suggests that a divot is slightly more likely than a knee. As futuresurveys (such as the Outer Solar System Origins Survey (OSSOS) and theLarge Synoptic Survey Telescope (LSST), see Chapter 8) search for Pluti-nos, how many detections will be required to uniquely identify the absolutemagnitude distribution as a knee or a divot? To answer this, we investigatedhow many detections CFEPS and our new survey would have had to detectin order to reject a knee or a divot using the summed AD method. Ideally,we would use the characterisation of future surveys to predict whether ornot they’ll detect enough Plutinos, but lacking that information (as they donot yet exist), the best approximation is to estimate how many detectionsour current surveys would have needed to detect (thus estimating how manytimes the coverage and depth of these surveys would need to be repeated).We generated 250 000 simulated detections from each of the Fraser et al.[2014] knee and Shankman et al. [2013] divot, as well as the CFEPS singleexponential, all extrapolated to Hr = 11. We then picked N objects from thesimulated detections, and calculated the rejectability (using the AD method)of each of the three models given this “observed” sample; this process wasrepeated 2500 times. This allowed us to answer the question “How often willa set of N detections from an intrinsic X distribution reject a Y distributionat > 95% confidence?” where X and Y are the specific single exponential,knee or divot from the literature, and N was investigated for a range of valuesfrom 42 to 500. The results can be seen graphically in Figure 3.13.When N = 42 (the number of Plutinos detected by the real surveys), an10202468 42 detections from an intrinsic SPL distribution.42 detections from an intrinsic Fraser distribution.42 detections from an intrinsic Shankman distribution.02468 100 detections from an intrinsic SPL distribution.100 detections from an intrinsic Fraser distribution.100 detections from an intrinsic Shankman distribution.02468Normalised frequency 150 detections from an intrinsic SPL distribution.150 detections from an intrinsic Fraser distribution.150 detections from an intrinsic Shankman distribution.02468 300 detections from an intrinsic SPL distribution.300 detections from an intrinsic Fraser distribution.300 detections from an intrinsic Shankman distribution.0 40 60 80 90 10002468 350 detections from an intrinsic SPL distribution.CFEPS 95%Fraser 95%Shankman 95%CFEPSFraserShankman 0 40 60 80 90 100Rejectability (%)350 detections from an intrinsic Fraser distribution.0 40 60 80 90 100350 detections from an intrinsic Shankman distribution.Figure 3.13: Number of detections needed to distinguish models. Thedistribution of rejectabilities of the CFEPS single exponential (red), theFraser et al. [2014] knee (blue) and Shankman et al. [2013] divot (green)by 2500 sets of N simulated detections (N given on subplots) drawn froman intrinsic population that has an absolute magnitude distribution inthe form of a single exponential with α = 0.9 (left), Fraser et al. [2014]knee (centre) and Shankman et al. [2013] divot (right), all extrapolatedto Hr = 11. The vertical lines (two of each colour, although often onlyone is visible) shows the rejectability that a model achieves less/morethan in 95% of cases. Note that the horizontal axis uses an exponentialprojection in order to better show the high-rejectability region (withoutcutting out the low region entirely).103intrinsic single exponential easily rejects the knee and divot, and vice versa,but an intrinsic divot or knee is rarely able to reject the each other. Wefound that, for N = 42 when the intrinsic population is in fact a single steepexponential distribution, the knee and divot model are both rejectable at> 95% confidence in over 99% of the cases. The single exponential itself was,obviously, only rejectable at > 95% confidence in 5% of cases. When theintrinsic distribution is a knee, the single exponential is rejectable at > 95%confidence in over 99% of cases, while the divot is only rejectable at thesame level in 18% of cases. When the intrinsic distribution is a divot, thesingle exponential is rejectable in over 99% of cases, while the knee is onlyrejectable in 15% of cases. It is therefore not surprising that our observationsare unable to reject either the divot or knee.If we increase N to 150 (similar to the expected number of detectionsfrom OSSOS plus the 42 from CFEPS and our survey), the knee is rejectablein ∼ 60% of cases when the intrinsic distribution is a divot, and vice versa.Although the odds are a lot better, it is thus not clear that OSSOS will be ableto definitively reject either model. This is especially true given that the realintrinsic distribution might not have the exact parameters of one of the twomodels used here. If the real intrinsic distribution lies somewhere betweenthe two models tested here (7.7 < Ht < 8.5, 1 < c < 6, 0.2 < αf < 0.5), thenthe two models would be rejectable less often. Of course, here we are simplyscaling up the number of detections from surveys identical to CFEPS and oursurvey. As the power to distinguish between a knee and a divot mostly comefrom the deeper observations of our survey, while CFEPS contributes littledue to its brighter limiting magnitude, we are most likely underestimatingthe abilities of OSSOS somewhat, as it will consist of deep observationswith depths more comparable to this thesis survey than CFEPS, weatherpermitting. The combined OSSOS+CFEPS+our survey should thereforelogically be able to distinguish (at 95% confidence) between a Shankmanet al. [2013] divot and a Fraser et al. [2014] knee in the Plutinos in more than10460% of possible data-sets.In order for our current surveys to be able to distinguish between thedivot and knee (reject one at > 95% confidence when the other is the trueintrinsic distribution) in ≈ 95% of possible data-sets, we find that N ' 350detections would have been required. However, as already discussed, manyof these detections would be attributed to the CFEPS fields which are notdeep enough to have a strong handle on the absolute magnitude distributionpast Hr > 8. Future surveys will almost certainly be able to distinguishthe distributions from fewer detections, as long as they probe deep enoughto detect the transition and have enough detections in the ±1 magnitudearound the transition to characterise its nature. As long as surveys havelimiting magnitudes mr ≈ 24.5, additional telescope time is better spent onsurveying a larger area of sky to that depth than on probing deeper on asmall field. OSSOS is thus optimally designed, given the available telescopetime, to scrutinise the Plutino absolute magnitude distribution.3.4 Investigating allowable parametersIn Section 3.3 we thoroughly showed that the single exponentialH-magnitudedistribution is not a valid representation for the Plutinos over the entire range5 < Hr < 11. Meanwhile, we found that the Shankman et al. [2013] divotand Fraser et al. [2014] knee could successfully be used to at least Hr = 9.Thus, we proceeded to explore the allowable parameters for the knee anddivot models. In addition to the allowable range of parameters for the H-magnitude distribution, we also identify the allowable range of some of theparameters of the orbital distribution. We do this partly because it is clearfrom Figure 3.2 that the 16◦width of the inclination distribution is too wide,and because the detected eccentricity and H-magnitude distributions arecorrelated.We have split the investigation of allowable parameters into two sections,in order to not be adjusting all parameters at once (which is computationally105expensive). As the orbital distribution should be independent of magnitude(at these sizes, there is no dynamical mechanism that would explain a size-dependent orbital distribution), we used the bright objects (Hr < 8.3), forwhich the absolute magnitude distribution is a simple exponential, to con-strain the orbital distribution and the bright slope αb. We then fixed theorbital distribution and αb at the least rejectable parameters, and used thefull range (Hr < 11) of detections in order to constrain the location of atransition, the slope for the faint objects and the contrast.3.4.1 Orbital distribution using Hr < 8.3 sampleWe here investigate the allowable range of orbital parameters, using the com-bined sample of Plutinos withHr < 8.3. We will also investigate the allowablerange of the slope parameter for the exponential H-magnitude distributionthat we have shown is valid for this range of magnitudes. We continued usingthe same parameterisation as in CFEPS 21 , in order to continue with a famil-iar and accepted parameterisation. The parameters we wished to investigatewere the width we and centre ce for the eccentricity distribution (e−(e−ce)22w2e cutsuch that q > 22 AU), the width wi of the inclination distribution (sin(i)e−i22w2i )and the exponent αb of the exponential H-magnitude distribution (10αbHr),where the b subscript denotes that we were only investigating the brightobjects (brighter than any transition). These model distributions do notcome from cosmogonic motivation, but CFEPS empirically found them tosatisfactorily represent the detections, once weighted by the detection biases[Gladman et al., 2012]. Given that these models, with the right parameters,continue to be highly non-rejectable (see Figure 3.14), their functional formsare still appropriate.We ran survey simulations for Plutinos with Hr < 8.3, using a four-dimensional grid of the ce, we, wi, αb model parameters. This Hr cut was21The few minor changes we made to the model were those described in Section 3.2.1060. e0.08 0.10 0.12 0.14 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.24 0.26Eccentricity centre ce0.400.500.600.700.800.850.900.951.−magindexα b0.02 0.04 0.05 0.06 0.07 0.08 0.10 0.12 0.14 0.16 0.18 0.20Eccentricity width we99.798.895.486.668.3ADrejectability(%)Leastrejectableparameters :ce =0.20we =0.07wi =14αb =0.95Figure 3.14: Rejectability in ce, we and αb space, for Hr ≤ 8.3. Thisshows the AD-rejectability of the hypothesis that our real detectionswith Hr ≤ 8.3 could be drawn from simulated detections from modelswith a grid of ce, we, wi and αb parameters run through the surveysimulator. This plot is equivalent to Fig. 4 in Gladman et al. [2012]and the patterns are in good agreement, with a peak only slightly offset,by values smaller than the uncertainties. Contour levels correspond to< 1σ (red), < 1.5σ (orange), < 2σ (yellow), < 2.5σ (green), < 3.0σ(cyan), > 3σ (blue). Contours are only shown for wi = 14◦, as the shapeof the contours are similar for all wi, but with shrinking size when goingaway from wi = 14◦, which produces the least rejectable values; this iswhy it is the value used for the plot. A four-dimensional grid of we,ce, wi and αf values was integrated. Black points indicate grid-points.The three plots are slices through the three-dimensional we, ce, αf space(with wi = 14◦), such that each plot is fixed to the least rejectable valuevalue of one parameter while the other two are varied.107chosen in order to only include objects brightwards of the transition, whilenot removing so many objects that it degrades our statistics. CFEPS andour new survey combined has 34 Plutinos with Hr < 8.3. From this analysis,we found that the L7 model parameters are near the centre of the allowablerange. Figure 3.14 shows contour plots showing the rejectability of variousparameters. It can be seen that for a given value of the eccentricity width,αb and ce have a clearly-defined allowable region. The eccentricity width we,however, is not well constrained, with non-rejectable values extending beyond0.20. It might seem odd that such a high eccentricity width is allowable,together with the high eccentricity centre (ce ∼ 0.2), but as our model hasa cut such q > 22 AU, the high-eccentricity tail that would make such acombination of we and ce rejectable does not exist. Values of αb up to 0.5 (asin Elliot et al. [2005]) can be seen to be rejectable for any combination of theother parameters, while values up to 1.50 are borderline acceptable in somecases. The eccentricity centre ce is the best constrained (smallest fractionalallowable range), as long as we is low; if we is increased, the allowable rangeof ce increases, primarily by smaller values of ce becoming non-rejectable.An important correlation remains that a lower eccentricity centre requires alarger width we to produce the copious numbers of e > 0.2 Plutinos.The allowable parameters found from this analysis are ce = 0.20+0.04−0.04,we = 0.07+0.15−0.03, wi = 14◦+7◦−3◦, αb = 0.95+0.45−0.35 (uncertainties represent the one-dimensional ranges for which the rejectability is < 95% when all other pa-rameters are fixed). The previous CFEPS best estimates were ce = 0.18+0.03−0.04,we = 0.06+0.03−0.02, wi = 16◦+8◦−4◦, αb = 0.90 ± 0.25. It is noteworthy that despitethis analysis using a factor ∼ 2 more objects, our constraint of the allowablerange of parameters is not improved. This is due to the difference in statisticused here, where our method is more rigorous but less constraining. Thesteep slope of αb = 0.95+0.45−0.35 (95% confidence), constrained using the range5.5 < Hr < 8.3, is nearly identical to recent results from the analyses of DES[Adams et al., 2014], which measured αb = 0.95 ± 0.16 (1σ) over the range1080.05 0.10 0.15 0.20 0.25 0.30Eccentricity0. 10 15 20 25 30Inclination [deg]30 35 40 45Heliocentric distance [AU] parametersLeast rejectableHigh we6.0 6.5 7.0 7.5 8.0Absolute magnitude, HrFigure 3.15: Cumulative distributions for different orbit distributions,for objects with Hr < 8.3. The distribution of real detections is shownin a black solid line. Simulated detections from the least rejectablemodel (with ce = 0.20, we = 0.07, wi = 14◦, αb = 0.95) are shown withgreen dashes, while simulated detections from the previous L7 model(with ce = 0.18, we = 0.06, wi = 16◦, αb = 0.90) is shown with ared dotted line. For comparison, a very different (although still notrejectable at 95% confidence) model is shown (magenta dash dottedline) with ce = 0.20, we = 0.20, wi = 14◦, αb = 0.95.1095.5 < Hr < 7.3. Figure 3.15 presents a visual representation of the simulatedand real detections illustrating differences between the CFEPS parametersand the least rejectable parameters found here. Comparison of new and old statistical methodIt is indeed peculiar that our analysis cannot constrain the we value morestrongly. Figure 3.15 shows, in a magenta dash-dotted line, the cumula-tive distribution of detections from a model with we = 0.2; it is clear tothe eye that this model produces an eccentricity distribution that is a verypoor match to the observed data. However, because the i, r and H distri-butions produced by that model are all very good matches to the data, thesummed AD statistic is the sum of one large and three small values. Thesummed statistic is not high enough to cause a rejection, because the boot-strapping frequently produces higher values (summed from four moderatevalues). This is a shortcoming of our summed AD method; while attemptingto not over-estimate our rejectability confidence, we have lost some abilityto reject models that should clearly be rejectable.For comparison, we repeated the analysis that went into producing Fig-ure 3.14, but with the statistical method used by CFEPS; that is, using a1D AD test on each of the four parameters (e, i, r and Hr) and selecting themost rejectable one. The results of this analysis can be seen in Figure 3.16This method clearly clearly far more constraining, rejecting we = 0.2 mod-els, although as mentioned in Section 3.1, it might be over-estimating therejectability confidence. Using the CFEPS method, we find the allowableranges for the parameters to be ce = 0.20+0.02−0.03, we = 0.07+0.05−0.03, wi = 14◦+6◦−2◦,αb = 0.95+0.30−0.25 (ranges for each are the range which has rejectability < 95%when the other parameters are held fixed). Our uncertainties are now similarto those of CFEPS, in some cases smaller. It is interesting to note that thetwo methods find the exact same parameters to be the least rejectable ones.This suggests that if one simply wishes to find the least rejectable model,1100. e0.08 0.10 0.12 0.14 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.24 0.26Eccentricity centre ce0.400.500.600.700.800.850.900.951.−magindexα b0.02 0.04 0.05 0.06 0.07 0.08 0.10 0.12 0.14 0.16 0.18 0.20Eccentricity width we99.798.895.486.668.3ADrejectability(%)Leastrejectableparameters :ce =0.20we =0.07wi =14αb =0.95Figure 3.16: Rejectability in ce, we and αb space using CFEPSmethod. This contour plots is similar to Figure 3.14, but uses the CFEPSstatistical method instead of the summed AD method. This shows theworst of the four values of the AD-rejectability of the hypotheses thate, i, r or Hr distribution of our real detections with Hr ≤ 8.3 couldbe drawn from the same distributions of the simulated detections. Thisplot is equivalent to Fig. 4 in Gladman et al. [2012] and the patterns arein good agreement, with a peak only slightly offset, by values smallerthan the uncertainties. Contour levels correspond to < 1σ (red), < 1.5σ(orange), < 2σ (yellow), < 2.5σ (green), < 3.0σ (cyan), > 3σ (blue).111the two methods are equal. The shape of the contours in Figure 3.14 and3.16 are also similar, so the only difference is the magnitude of the rejectabil-ity values given. We have discussed above (and in Section 3.1) that theCFEPS approach of using the most rejectable distribution over-estimatesthe rejectability. Figure 3.14 and 3.15 suggest that out summed AD ap-proach might underestimate the rejectability confidence. It is quite possiblethat a maximum likelihood analysis would produce a more correct measure-ment of the rejectability of various parameter combinations; unfortunately,conducting this whole analysis with a maximum likelihood analysis is com-putationally unfeasible (see Section 3.3.4). Erring on the side of caution inorder to not over-state our certainties, we will continue to use our summedAD approach in the following section.3.4.2 Magnitude distribution for Hr < 11Making the reasonable starting assumption that the orbital distribution isnot absolute magnitude dependent, we fixed the Plutino orbital distributionto the least rejectable parameters from Section 3.4.1. This allowed us toexplore the remaining H-magnitude space, to determine what it could re-veal about the absolute magnitude distribution. We implemented the fourparameter divot H-magnitude distribution as parameterised by Shankmanet al. [2013], which turns into a knee when c = 1 and further becomes asingle exponential when c = 1 and αb = αf . We fixed αb = 0.95 (from theleast rejectable parameter-set in Section 3.4.1 and investigated the remainingthree-dimensional parameter space (Ht, c, αf ) for Plutinos with Hr < 11,thus comparing to the full set of 42 detected Plutinos.Figure 3.17 shows contour plots of the rejectability of the explored Ht,c and αf combinations. For all values of Ht a correlation exists, for whichincreasing the contrast requires an increase in αf . This is because our surveyfound six Plutinos with Hr > 9, so the distribution must have sufficientHr > 9 objects to explain these detections; if there is a deep contrast, the1120. =7.3 Ht =7.5 Ht = =7.9 Ht =8.1 Ht = =8.4 Ht =8.5 Ht =8.71 2 3 5 9 17contrast c0.α fHt =8.92 3 5 9 17Ht =9.12 3 5 9 17Ht =9.3 99.798.895.486.668.3ADrejectability(%)Figure 3.17: Rejectability in Ht, c, αf space, with αb = 0.95 and theleast rejectable orbital parameters from Section 3.4.1. Each subplot hasthe same axes and is a slice through the three-dimensional space at aparticular value of Ht (labelled on each subplot). Colours are the sameas in Figure 3.14; models which are not rejectable at 95% confidenceare shown in red, orange and yellow. With Ht > 9, all solutions arerejectable, because the steep αb cannot continue to Hr = 9; there mustbe a transition at a brighter magnitude. For Ht < 8.8, knees (solutionswith c = 1) are not rejectable at 95% confidence. Note that knees toαf ≥ 0.8 are always rejectable at 95% confidence; for Ht > 8.2, kneesto αf ≥ 0.6 are also rejectable. The least rejectable set of parametersis marked with a green star (Ht = 8.4 panel). Also marked are theknee parameters favoured by Fraser et al. [2014] for hot TNOs (magentastar; Ht = 7.7 panel), the divot parameters favoured by Shankman et al.[2013] for Scattering TNOs (cyan star; Ht = 8.5 panel) and the singleexponential model (red star on every plot, as varying Ht when c = 1and αf = αb changes nothing).113post-divot slope must be steep in order to generate the Hr > 9 objects. Ata given Ht, knees thus have the smallest allowable αf and the maximumallowable αf is also lowest for knees. For the brightest possible transitionsmagnitude we investigated (Ht = 7.3), a range of divot and knee parametersare non-rejectable; a knee requires a faint slope in the range 0.2 ≤ αf ≤ 0.8,while as the transition contrast c increases, the range of allowable αf valuesgets narrower and shifts to higher values. As Ht increases, the smallest non-rejectable αf decreases (or inversely, a deeper contrast c becomes allowablefor a given αf ). Deep contrasts are allowable, as long as the post-divot slopeis steep; this is because of the dearth of detections with 8.3 < Hr < 9.0 inour sample combined with the six detections with Hr > 9.0. Eventually, themaximum allowed αf falls with increasing Ht, especially for the knee; thefaintest location for a knee to be rejectable at less than 95% confidence isbetween Ht = 8.7− 8.9; for Ht > 8.9, a knee is always rejectable. With thefull set of data, we find that the least rejectable set of parameters are fora moderately deep divot (c = 6) at Ht = 8.4 to a moderately steep slope(αf = 0.8 for Hr = 8.4 to 11.0), near the Shankman et al. [2013] parameters,while the least rejectable knee is, with Ht = 7.7 and αf = 0.4, very near to theFraser et al. [2014] parameters. Figure 3.18 shows the cumulative distributionof real and simulated detections from various models; to the eye, the divotprovides the best match to the observations, (although the improvement overa knee is not statistically significant). This is not surprising, given that oursurvey detected no Plutinos with 8.27 < Hr < 9.01, yet found six Plutinoswith Hr ≥ 9.01.A steep (αf > 0.6) post-transition slope cannot continue, because if sucha slope is extrapolated indefinitely, the total mass of the trans-Neptunian re-gion would diverge [Chiang and Brown, 1999, Gladman et al., 2001]. So, anαf ' 0.8 post-divot slope can only be suitable over a limited H-magnituderange; there would have to be yet another transition at smaller sizes to agentler slope. The main asteroid belt is seen to have multiple gentle tran-114sitions in the known size distribution; it is quite possible that TNOs wouldhave a similar rolling weave pattern. Alternately, Figure 3.17 makes it clearthat many αf < 0.6 divot models are not rejectable, including the original(Ht = 8.5, c = 6, α = 0.5) proposal from Shankman et al. [2013]; it is thusentirely possible that the Plutinos feature an αf < 0.6 slope starting from aHr ∼ 8.4 transition.3.4.3 Population estimateCFEPS predicted 13000+6000−5000 Plutinos with Hg < 9.16 [Gladman et al., 2012],corresponding to Hr < 8.66, where uncertainties indicate the 95% confidencerange. Our survey combined with CFEPS has 34 Plutinos with Hr < 8.66,so we ran 10 000 survey simulations of the combined surveys until each sim-ulation detected 34 synthetic objects with Hr < 8.66. We counted the totalnumber of generated objects required to provide these detections for each runin order to estimate the total Plutino population size and uncertainty. Thiswas done for two absolute magnitude distributions; the least rejectable kneeand the least rejectable divot found above. With this boot-strap approach,we estimate 9000 ± 3000 Plutinos with Hr < 8.66, regardless of whetherwe use the divot magnitude distribution or the knee distribution, showing a∼ 1√2improvement in relative uncertainty as expected from roughly doublingthe number of detections. This revised population estimate is in agreementwith the CFEPS prediction, although this is not too surprising due to thepartial overlap of the sample.Figure 1 of Adams et al. [2014] shows that they estimate ∼ 3000 Plutinoswith HV R ≤ 8.1 (Hr = 8.0, as V R − r = 0.1, see footnote 16 on page16). With the divot magnitude distribution, which for Hr < 8.4 is simplya single exponential with α = 0.95, we estimate 8000+3000−2000 Plutinos withHr < 8.4. Scaling this down, our estimate corresponds to 3300+1300−800 Plutinoswith Hr < 8.0. The observations of Adams et al. [2014] are thus in goodagreement with our own population estimates and those of CFEPS.1150.05 0.10 0.15 0.20 0.25 0.30Eccentricity0. 10 15 20 25 30Inclination [deg]30 35 40 45Heliocentric distance [AU] modelSingle exp, α=0.95Least rejectable kneeLeast rejectable6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5Absolute magnitude, HrFigure 3.18: Cumulative distributions of the least rejectable models,for the full combined Plutino sample. Our simulations generated objectsdown to Hr = 11, well beyond our faintest real detection of 9.85, toprevent artificially cutting the simulated detections. The least rejectabledivot (blue dash double-dotted line) had Ht = 8.4, c = 6, αf = 0.80,while the least rejectable knee (magenta dash dotted line) had Ht = 7.7,c = 1, αf = 0.40. It is clear that both of these solutions provide abetter representation of the observations than the single exponential ofsame αb (green dashed line) and the L7 model (with original orbital andmagnitude parameters; red dotted line) over this full magnitude range(compare to Figure 3.18).116Table 3.2: Population estimates for the Plutinos, based on our work.Population estimates are medians given with 95% confidence ranges(2.5% in each tail). Estimates are given for two different absolute magni-tude distribution models, the least rejectable knee (αb = 0.95, Ht = 7.7and αf = 0.40) and divot (αb = 0.95, Ht = 8.4 and αf = 0.80) fromour analysis. Note that for N(Hr ≤ 8.4), the divot is equivalent to asingle exponential with α = 0.9 (as the transition occurs at Hr = 8.4).Population estimates for all populations described in this work can befound in Table B.1 for easy comparison.H-model N(Hr ≤ 8.4) N(Hr ≤ 8.66) N(Hr ≤ 9.1) N(Hr ≤ 10.0)PlutinosKnee 7000+3000−2000 9000± 3000 14 000+5000−4000 35 000+12 000−10 000Divot 8000+3000−2000 9000± 3000 12 000+4000−3000 37 000+12 000−10 000The addition of the new, deeper survey block described in this thesissurvey allows us to estimate the population of even fainter objects (Table 3.2).We estimate the existence of 37000+12000−10000 Plutinos with Hr < 10.0 assumingthe divot distribution, and 35000+12000−10000 if the knee is assumed. It may seemsurprising that these two models produce such similar population estimates,as they have such different values of αf , but this is because the proposedknee occurs at a brighter magnitude than the divot, so the divot has more7.7 < Hr < 8.4 objects than the knee, while the knee has more objectsjust past Hr = 8.4, thus roughly equalising the total population estimate.If one could probe several magnitudes past the transition, the difference inpopulation would become clearer. These population estimates show thatdespite having transition away from a single exponential absolute magnitudedistribution, the Plutinos continue to significantly outnumber the inner SolarSystem’s asteroid belt, which has 592 objects22 with Hr < 10.22The main asteroid belt (2.06 AU < a < 3.65 AU) is completely known for H <15.75 [Ryan et al., 2015]. We have thus simply counted the number of objects with2.06 AU < a < 3.65 AU and Hr < 10 in the MPC database file MPCORB.DAT fromhttp://www.minorplanetcenter.net/iau/MPCORB.html. By comparing the magnitudesof the objects from our survey, we found that MPCORB.DAT had a H−Hr ≈ 0.20 offset,which we accounted for.117Chapter 4Co-orbitalsWe have a genetic kinship with all life on earth,an atomic kinship to all matter in the cosmos.So when I look at the universe, I feel large,because I remind myselfthat not only are we living in this universe,the universe is living within us.— Neil deGrasse Tyson (2009)Co-orbitals are objects in the 1:1 mean-motion resonance with a planet(see Section 1.3 for details)23. There are two types of co-orbital objects :stable co-orbitals are on long term stable orbits that keep them co-orbitalfor the age of the Solar System; temporary co-orbitals leave the co-orbitalmotion on timescales significantly shorter than the age of the Solar System.Temporary co-orbitals are almost certainly scattering objects temporar-ily captured into co-orbital motion. They therefore likely have completelydifferent histories and origins from the stable co-orbitals, besides obviouslyhaving a different orbital distribution. We have therefore treated the stableand temporary co-orbitals as two separate populations. Our survey found23This chapter is based on work published in Alexandersen et al. [2013a] and submittedfor publication in Alexandersen et al. [2015].118one temporary Uranian co-orbital and two Neptunian co-orbitals (one stableTrojan and one temporary Trojan), while CFEPS found one Neptunian co-orbital (a temporary Trojan). Normally, population estimates based on sofew detections are highly uncertain and depend strongly on the choice of toymodel orbit distribution used to make the population estimate. However,in this case we have access to independent estimates of the orbital distribu-tions of the temporary Uranian and Neptunian co-orbitals [theoretical workin Alexandersen et al., 2013a] and the stable Neptunian Trojans [statisticalanalysis of all known Neptunian Trojans in Parker, 2014]. The analysis isdetailed below. Population estimates of the three populations are presentedtogether in Table 4.1, assuming magnitude-distributions from the literature.4.1 Stable TrojansOur survey discovered one long-term stable Trojan of Neptune; CFEPS foundnone. With our one detection, combined with the characterisation of our sur-veys, we derive order-of-magnitude population estimates using orbital andsize distributions derived elsewhere in literature. This population estimateshould not be considered unreliable simply because we only have one detec-tion, as the non-detection of objects also factor into the population estimate.While the detection of only one object means that the lower limit of the pop-ulation estimate will make it essentially consistent with 0, the non-detectionof more objects sets a strong upper-limit on the population estimate.119Table 4.1: Population estimates for various co-orbital populations studied in this work. Populationestimates are either 95% confidence upper limits (in cases of no real detections) or medians given with95% confidence ranges (2.5% in each tail). Estimates are given for two different absolute magnitudedistribution models, the Fraser et al. [2014] knee with αb = 0.87, Ht = 7.7 and αf = 0.20, and theShankman et al. [2013] divot with αb = 0.8, Ht = 8.5, c = 6 and αf = 0.50. Note that for N(Hr ≤ 8.4),the divot is equivalent to a single exponential with α = 0.8 (as the transition occurs at Hr = 8.5).Population estimates for all populations described in this work can be found in Table B.1 for easycomparison.H-model N(Hr ≤ 8.4) N(Hr ≤ 8.66) N(Hr ≤ 9.1) N(Hr ≤ 10.0) N(Hr ≤ 11.0)Temporary Knee ≤ 300 ≤ 300 ≤ 300 110+400−100 190+800−180Uranianco-orbitals Divot ≤ 300 ≤ 300 ≤ 300 110+500−100 270+1200−260Temporary Knee 70+300−60 70+300−60 90+400−80 200+900−190 1200+3000−1000Neptunianco-orbitals Divot 70+300−60 80+300−70 90+400−80 210+900−200 2500+6000−2100Stable Knee ≤ 250 ≤ 250 80+300−70 140+600−130NeptunianTrojans Divot ≤ 250 ≤ 260 80+300−70 150+600−1401204.1.1 About 2012 UV177The stable Neptunian Trojan, 2012 UV177, with Hr = 8.93, was discoveredin the high-lat block of our new survey. Via 4 Gyr numerical integrations,using the range of orbits established by the orbital classification algorithm ofGladman et al. [2008], we find this to be a secure stable 1:1 Trojan resonator,with a libration amplitude around the L4 point of 13◦ (see Figure 4.1. Thefact that 2012 UV177 has an inclination of 21◦and the fact that we discoveredno stable Neptunian Trojans in our low-lat block continues to strengthenthe assertion that the stable Neptunian Trojans have a very dynamicallyhot inclination distribution [Sheppard and Trujillo, 2006, 2010a]. Recentstudies indicate that the standard 15◦ width (of other hot populations) isstill plausible, although at the low end of the allowable range [Parker, 2014].4.1.2 Population estimateWe generated population estimates (Table 4.1) or upper limits for variousHr-cuts using different absolute magnitude distributions. As we cannot con-strain the absolute magnitude distribution of the Trojans, we use distribu-tions as found for other related populations in the literature (the Shankmanet al. [2013] divot and Fraser et al. [2014] knee); we did not use a steep singleexponential, as that has already been rejected for the Trojans [Sheppard andTrujillo, 2010b]. For this, we used the favoured orbital distribution derivedby Parker [2014] for the stable Neptunian Trojans. The visibility of Nep-tunian Trojans (both L4 and L5 combined), given this orbital distribution,can be seen in Figure 4.2a, showing that our new survey was as sensitiveto Neptunian Trojans as CFEPS was, despite our smaller sky-coverage, be-cause our fields were chosen to optimise Neptunian Trojan detection. Ournew survey also pushes deeper, meaning that overall we are far more sensitiveto Neptunian Trojans than CFEPS was.Sheppard and Trujillo [2010b] estimated about 400 Neptunian Trojans(stable + temporary) with D > 100 km, which corresponds to Hr < 8.6121050100150200250300350λ−λ N0. 2 4 6 8 10time (Myr)29.829.930. (AU)Figure 4.1: The future evolution of 2012 UV177. Shown is the numer-ical integration of the best-fit orbit (black) as well as the smallest andlargest semi-major axis orbits compatible with the astrometry (green andred, respectively). All three clones remain as L4 Trojans for the 10 Myrintegration used in classification. Subsequent integration showed thatorbit is stable for the age of the Solar System.122given their assumptions and appropriate colour conversion. CFEPS [Glad-man et al., 2012] put an upper limit of 300 (95% confidence) stable NeptunianTrojans with Hg < 9.16 (chosen to correspond with D > 100 km), which (as-suming colours similar to the Plutinos) correspond to Hr < 8.66. Our 95%confidence upper limit, < 250 Trojans (L4+L5) with Hr < 8.66, is even lowerthan previous estimates and disagrees with the Sheppard and Trujillo [2010b]prediction of 400. However, part of this difference is due the fact that onlyseven of the nine objects on which Sheppard and Trujillo [2010b] based theirpopulation estimate have in fact turned out to be Trojans; additionally, theydid not consider the stable and temporary Trojans separately.Our estimated cumulative populations of stable Trojans for Hr < 9.1 and< 10.0 are 80+300−70 and 150+600−140 respectively (see Table 4.1 for all numbers).Even at these higher magnitude-cuts, our population estimates are lower thanthe Sheppard and Trujillo [2010b] prediction. Stable Neptunian Trojans arethus not nearly as numerous as the initial estimates, and almost certainlyless numerous than the main asteroid belt, which has 592 asteroids24 withHr < Temporary co-orbitalsThere were three temporary co-orbitals in our sample, one from CFEPS andtwo in our new survey, including the first known Uranian Trojan.In our survey, 2011 QF99 and 2012 UW177 were the only objects with asemi-major axis within the planetary region (defined here as a < 34 AU toinclude Neptune co-orbitals but exclude the exterior stable trans-Neptunianpopulations); both of these objects are temporary co-orbitals, one with Uranus,one with Neptune. CFEPS detected two25 a < 34 AU objects; one of these24See footnote 22 on page 117.25Erratum: Alexandersen et al. [2013a] states three; this error was caused by the in-clusion of an object discovered by the CFEPS-team in a later survey [to be published inPetit et al., 2015]. However, as the characterisation of that later survey was not used inour survey-simulations, it should not be included here.123Hr  magnitude0. fractionStable Neptunian Trojan visibilityFraction from our 2011-2012 survey alone5 6 7 8 9 10 11Hr  magnitude0. fractionTemporary Neptunian Co-orbital visibilityFraction from our 2011-2012 survey aloneFigure 4.2: The visibility of Neptunian co-orbitals. a) The visibil-ity of stable Neptunian Trojans (black curve) to the combined surveys,assuming the Parker [2014] intrinsic orbital distribution, as a functionof absolute magnitude. The blue dashed curve shows the visibility con-tributed by our new survey alone, which, due to being pointed very closeto the centre of the Trojan cloud, was as sensitive to bright Trojans asthe entirety of CFEPS was, despite the latter’s larger areal coverage.Our survey’s greater depth is also evident.b) The visibility of temporary Neptunian co-orbitals (black curve) tothe combined surveys, assuming the Alexandersen et al. [2013a] orbitaldistribution for temporary Neptunian co-orbitals, as a function of abso-lute magnitude. Our new survey contributes a much smaller fractionshere (blue dashed curve) compared to the larger coverage of CFEPS,because temporary co-orbitals have larger libration amplitudes and in-cludes horse-shoe orbits, and can thus be found at all longitudes. Ournew survey still probes fainter objects, supplying the majority of thesensitivity in the tail beginning at Hr ≈ 9.0.124two objects, 2004 KV18 is a temporary co-orbital of Neptune. The Interna-tional Astronomical Union (IAU) MPC database contains 247 objects with6 AU < a < 34 AU as of 2013-Jul-09; here the additional a > 6 AU constraintis imposed to exclude Jovian Trojans and main belt asteroids, which neitherour survey nor CFEPS are sensitive to.Noticing how large the fraction of temporary co-orbitals was of the 6 AU <a < 34 AU sample, we investigated their possible origin from the scattering/-Centaur population. One might be surprised that ∼ 2% of the known objectsin this a-range are exhibiting temporary co-orbital motion with Uranus orNeptune, since being trapped into this state is presumably a rare occur-rence. We thus seek to determine if (to factor of two) it is reasonable that, ina model of Centaur supply from the Scattering Disk, a large-enough fractionshould be in resonance at any time to explain the observed discovery rate oftemporary co-orbitals. To address this question, we estimated the fractionof Centaurs in temporary co-orbital states with Uranus and Neptune usingnumerical simulations of orbital dynamics, similar to what has been donefor the Earth [Morais and Morbidelli, 2002] and Venus [Morais and Mor-bidelli, 2006]. Here, “scattering objects” are those [Gladman et al., 2008,Petit et al., 2011] that experience ∆a > 1.5 AU in 10 Myr; scattering objectswith a < 30 AU are called “Centaurs”, whereas those with a > 30 AU arethe “Scattering Disk”.We constructed a Centaur model26, supplied from the trans-Neptunianregion, to estimate temporary co-orbital capture frequency and duration (tofactor of two accuracy), finding that at any time 0.4% and 2.8% of the popu-lation will be Uranian and Neptunian co-orbitals, respectively. Even thoughthe scattering population is depleting with time, the co-orbital fraction doesnot (see Section A.1). We subsequently found, through survey simulations,that there are no strong biases towards or against discovering Neptunian26The numerical integrations and co-orbital identification was primarily carried out bySarah Greenstreet, and is thus only mentioned lightly in the main text here. For moredetails, see Appendix A and Greenstreet [2015].125or Uranian co-orbitals as opposed to Centaurs in general. This is largelydue to the fact that CFEPS surveyed areas at many different latitudes to asimilar depth, so that the observational biases are cancelled out. Similarly,the MPCs database contains objects found in dozens of surveys, which whencombined has observations at almost every latitude to similar depth; we thusdo not expect the MPCs database of Centaurs to contain any significant(latitudinal) biases. The co-orbital fraction (∼ 2.4%) among Centaurs in theIAU MPC database is thus as expected under trans-Neptunian supply.4.2.1 About 2011 QF99We discovered the first known Uranian Trojan [Alexandersen et al., 2013a,b].We detected 2011 QF99 at a heliocentric distance of 20.3 AU, where its ap-parent magnitude mr = 22.6 ± 0.1 sets its absolute magnitude at Hr = 9.6.This magnitude indicates that 2011 QF99 is ∼ 75 km in diameter, assuminga 5% albedo (or ∼ 40 km assuming a 16% albedo).As more observations constrained the orbit, it became clear that 2011QF99 was not simply a Centaur that happened to be near the distance ofUranus. Our current astrometry, consisting of 36 measurements from 10dark runs with total arc of 742 days, indicates the following orbital elements:a = 19.0919 ± 0.0009 AU, e = 0.17687 ± 0.00009, i = 10.◦8104 ± 0.◦0003,Ω = 222.◦4986 ± 0.◦0003, ω = 287.◦47 ± 0.◦02, Tperi = 2 464 382 ± 2 JD. Herea, e, i, Ω, ω, Tperi are the osculating J2000 barycentric semi-major axis,eccentricity, inclination, longitude of ascending node, argument of pericentreand Julian day of pericentre. The low eccentricity along with a semi-majoraxis similar to that of Uranus (aU ≈ 19.2 AU) indicated already after a yearof observations that 2011 QF99 might be a Uranian co-orbital.A short (50 kyr) numerical integration showed that 2011 QF99 is libratingaround the leading (L4) Lagrange point (Figure 4.3). Could 2011 QF99 be aprimordial Trojan? Jupiter hosts a large population of Trojan asteroids sta-ble for 5 Gyr. The recently-detected stable population of Neptunian Trojans126Figure 4.3: The short-term motion of 2011 QF99. Shown here is thebest fit trajectory of 2011 QF99 (thin black line), from its current posi-tion (red square) to 10 libration periods (59 kyr) into the future. Theco-ordinate frame co-rotates with Uranus (on right, black circle) anddotted circles show the semi-major axis of the giant planets. Green dia-monds denote the L4 (upper) and L5 (lower) Lagrange points. The ovaloscillations occur over one heliocentric orbit, while the angular extentaround the Sun is the slower libration around L4.127was believed to outnumber Jovians for objects with diameter > 100 km [Chi-ang and Lithwick, 2005, Sheppard and Trujillo, 2010b], although our updatedpopulation estimate in Table 4.1 shows a smaller population. In contrast,the Trojan regions of Saturn and Uranus are believed to be mostly unstabledue to overlapping resonances with the neighbouring planets [Dvorak et al.,2010, Nesvorny´ and Dones, 2002, demonstrated by long term numerical in-tegrations]. Saturn and Uranus are thus unlikely to host long-lived Trojans(Nesvorny´ and Dones [2002] demonstrate that their population must be de-pleted by at least a factor of 100 from the primordial population), eventhough a few stable niches exist [Dvorak et al., 2010]; it is unclear how mi-gration affects the likelihood of these niches being populated [Kortenkampand Joseph, 2011, Kortenkamp et al., 2004].In longer integrations, using the 10 Myr classification method described inSection 2.6, both the nominal orbit of 2011 QF99 and all other orbits withinthe (already small) orbital uncertainties librate around the L4 Lagrange pointfor at least the next 70 kyr (Figure 4.4). On time scales of 100 kyr to 1 Myr,all integrated orbits transition out of the L4 Trojan region, either escapingdirectly to scattering behaviour (that is, become Centaurs) or transitioningto other co-orbital behaviour before escaping and scattering away within3 Myr. 2011 QF99 is therefore not classified as a stable resonant object; it isinstead only temporarily (by astronomical standards) in the co-orbital state.Figure 4.4 shows the evolution of the three orbits (the best fit and the twoextremes) for the next 1 Myr into the future, showing that 2011 QF99 alwaysexecutes 10 or more angular librations of the φ11 resonant argument aroundthe L4 point, over the next 70 kyr or more27. Due to the chaotic nature of theresonance dynamics, by this point the orbits have sufficiently separated thatthey show different evolutions as to when they exit the L4 libration. Thisexit is usually into another form of co-orbital behaviour (transition to anL5 librator or a horseshoe orbit). We numerically estimate that the object’s27This is also true in all other experiments we have conducted with different time steps.128Figure 4.4: The future evolution of 2011 QF99. Shown is the numericalintegration of the best-fit orbit (black) as well as the smallest and largestsemi-major axis orbits compatible with the astrometry (green and red,respectively). The detailed evolution is highly chaotic, but all threeclones remain as L4 Trojans for at least 70 kyr and remain co-orbital forat least 450 kyr. The nominal orbit (black) switches back and forth fromL4 to L5, before a horseshoe period at t ≈ +800 kyr and then escapefrom the resonance at t ≈ +880 kyr. The small-a clone (green) remainsaround L4 the longest, until breaking loose around t ≈ +450 kyr andnever getting recaptured. The large-a clone (red) is the most stable andremains in or near the co-orbital state for the entire 1 Myr shown here,before leaving soon after. (This figure was produced by B. Gladman forAlexandersen et al. [2013a]). 129


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