Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Fladeland, Logan
Master of Science - MSc
University of British Columbia
Humanity is on course to return to the surface of the Moon within the next decade, with plans and aspirations of permanently crewed stations that could eventually become self-sustained through advancements of in-situ resource utilization (ISRU). During NASA’s Apollo era, which saw the first and only humans step foot on the Moon, the issue of lunar dust rose to prominence as it was found to be quite abrasive and became coated on practically every type of material. Natural mechanisms have been discovered that can levitate lunar dust and soil above the surface, at times creating a faint haze around the Moon. Increasing lunar surface activity, including potential construction and mining operations, as well as a significant increase in landings and departures, could see large amounts of material artificially jettisoned, if proper mitigation techniques are not introduced. N-body simulations of 10,000 particles ejected from the Moon are performed using REBOUND to examine some of the different dynamical pathways available to material escaping the low lunar gravity. Particles that do not collide with the Moon or the Earth generally end up in orbit around the Earth or escape onto heliocentric orbits, with very few left orbiting the Moon after one year. A constant delivery of particles to the L1 and L2 Earth- Moon Lagrange points is seen, with sporadic activity at the three other Lagrange points. Moreover, the possibility of long-lived dust in the cis-lunar environment raises questions about how lunar activities will impact the night sky. To that end, the particles are projected onto the sky for an observer on Earth and the associated brightness is calculated.
Investigations of the Dynamical Evolution of Lunar EjectabyLogan FladelandB.Sc., Combined Honours Physics and Astronomy, The University of BritishColumbia, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Astronomy)The University of British Columbia(Vancouver)October 2022© Logan Fladeland, 2022The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Investigations of the Dynamical Evolution of Lunar Ejectasubmitted by Logan Fladeland in partial fulfillment of the requirements for thedegree of Master of Science in Astronomy.Examining Committee:Aaron Boley, UBC Physics and AstronomySupervisorJ.J. Kavelaars, NRC Canada and UBC Physics and AstronomySupervisory Committee MemberiiAbstractHumanity is on course to return to the surface of the Moon within the next decade,with plans and aspirations of permanently crewed stations that could eventually be-come self-sustained through advancements of in-situ resource utilization (ISRU).During NASA’s Apollo era, which saw the first and only humans step foot on theMoon, the issue of lunar dust rose to prominence as it was found to be quite abra-sive and became coated on practically every type of material. Natural mechanismshave been discovered that can levitate lunar dust and soil above the surface, at timescreating a faint haze around the Moon. Increasing lunar surface activity, includingpotential construction and mining operations, as well as a significant increase inlandings and departures, could see large amounts of material artificially jettisoned,if proper mitigation techniques are not introduced. N-body simulations of 10,000particles ejected from the Moon are performed using REBOUND to examine someof the different dynamical pathways available to material escaping the low lunargravity. Particles that do not collide with the Moon or the Earth generally end up inorbit around the Earth or escape onto heliocentric orbits, with very few left orbitingthe Moon after one year. A constant delivery of particles to the L1 and L2 Earth-Moon Lagrange points is seen, with sporadic activity at the three other Lagrangepoints. Moreover, the possibility of long-lived dust in the cis-lunar environmentraises questions about how lunar activities will impact the night sky. To that end,the particles are projected onto the sky for an observer on Earth and the associatedbrightness is calculated.iiiLay SummaryMultiple ongoing efforts to return humans to the Moon and establish long-term op-erations present a need to better understand the transport and evolution of dust andmeteoroids throughout the Earth-Moon environment. This study investigates suchevolution by running dynamics simulations of ejected particles from the Moon’ssurface to analyze the short and long-term behavior of such material. The accumu-lation of material in locations of potential future lunar operations is studied, alongwith statistics of collisions with the Earth or Moon, as well as the number of par-ticles escaping onto orbits about the Sun. Projections of the night sky from Earthallow for estimations of the brightness of reflected sunlight arriving from these par-ticles. Further applications of this work can help determine potential limits on theamount of material allowed to escape the lunar surface in order to avoid impactingspacecraft operations or increasing the brightness of the night sky.ivPrefaceThe content of this thesis is original unpublished work by the author. The idea forthis study came about through discussions with the author’s supervisor, Dr. AaronBoley. All necessary python scripts, as well as the resulting plots and data werecreated by the author. Individual techniques and methods used but not developedby the author have been cited where appropriate.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1 Dynamical Simulations . . . . . . . . . . . . . . . . . . . . . . . 142.2 Coordinate Transformations . . . . . . . . . . . . . . . . . . . . 172.2.1 Lunar Rotating Frame . . . . . . . . . . . . . . . . . . . 182.2.2 Sky Projection . . . . . . . . . . . . . . . . . . . . . . . 182.3 Time Series Analysis . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Kernel Density Estimator . . . . . . . . . . . . . . . . . . 202.3.2 Lagrange Points . . . . . . . . . . . . . . . . . . . . . . 212.3.3 Sky Brightness Estimates . . . . . . . . . . . . . . . . . . 223 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1 Dynamical Behavior of Particles . . . . . . . . . . . . . . . . . . 23vi3.2 Night Sky Projections . . . . . . . . . . . . . . . . . . . . . . . . 344 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.1 Particle Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 Radiation Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Sky Brightness . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51viiList of FiguresFigure 1.1 Surveyor 6 Lunar Sunset Haze . . . . . . . . . . . . . . . . . 4Figure 1.2 Apollo 8 Earth Rise . . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.3 NRHO Example Orbits . . . . . . . . . . . . . . . . . . . . . 10Figure 2.1 Example of Polar Projection Plot . . . . . . . . . . . . . . . . 19Figure 3.1 10 Day XY Moon’s Rotating Frame . . . . . . . . . . . . . . 24Figure 3.2 10 Day L1 and L2 Kernel Density Estimations . . . . . . . . 26Figure 3.3 30 Day XY Moon’s Rotating Frame . . . . . . . . . . . . . . 27Figure 3.4 30 Day XZ Moon’s Rotating Frame . . . . . . . . . . . . . . 28Figure 3.5 30 Day L1 and L2 Kernel Density Estimations . . . . . . . . 28Figure 3.6 One Year XY Moon’s Rotating Frame Zoomed . . . . . . . . 29Figure 3.7 One Year XY Moon’s Rotating Frame . . . . . . . . . . . . . 30Figure 3.8 One Year Kernel Density Estimations . . . . . . . . . . . . . 31Figure 3.9 One Year XY Heliocentric Frame . . . . . . . . . . . . . . . 32Figure 3.10 One Year Z-Dispersion of Particles . . . . . . . . . . . . . . 33Figure 3.11 31.4 Day Extended Polar Projection . . . . . . . . . . . . . . 34Figure 3.12 31.4 Day Flux Projection . . . . . . . . . . . . . . . . . . . . 35Figure 3.13 331.2 Day Flux Projection . . . . . . . . . . . . . . . . . . . 37Figure 4.1 10 Day XY Moon’s Rotating Frame Beta Comparison . . . . 40Figure 4.2 30 Day XY Moon’s Rotating Frame Beta Comparison . . . . 41Figure 4.3 30 Day XZ Moon’s Rotating Frame Beta Comparison . . . . . 41Figure 4.4 One Year Z-Dispersion Beta Comparison . . . . . . . . . . . 42Figure 4.5 One Year XY Heliocentric Beta Comparison . . . . . . . . . 44viiiFigure 4.6 One Year Kernel Beta Comparison . . . . . . . . . . . . . . . 45ixGlossaryILRS International Lunar Research StationNASA National Aeronautics and Space AdministrationLEO Low Earth OrbitCSM Command and Service ModuleLM Lunar ModuleEVA Extravehicular ActivityISS International Space StationESA European Space AgencySLS Space Launch SystemNRHO Near-Rectilinear Halo OrbitCR3BP Circular Restricted Three Body ProblemISRU In-Situ Resoure UtilizationxChapter 1IntroductionHumanity is renewing its effort to return to the Moon with plans to develop long-term human habitation on the Moon’s surface and in cis-lunar space. Moreover,a larger range of actors are seeking to take part in possible lunar development,including national space agencies and private enterprises. Uses of the Moon areenvisaged to include scientific sampling, lunar-based observations, space resourceprocessing for exploration and energy needs, and even tourism [9, 28]. Whilemany benefits are expected to be derived from this expansion into space, we mustbe mindful of any unintended and potentially harmful consequences of our actions.As we begin to create a long-term presence on the Moon, it’s important we do soin a way that is as sustainable as possible, to allow future generations to continueto explore and enjoy the wonders and beauty of our celestial companion.Two major multinational groups are emerging in this return to the Moon, withthe United States led Artemis program forming concurrently as China and Rus-sia develop plans for their own jointly led international lunar base. The ArtemisAccords, signed by 22 countries and territories1, seeks to build upon the limitedinternational framework surrounding the uses of outer space, while supporting andpromoting peaceful activities and best practices for space exploration. Several sig-natories of the Accords, working together with private companies, are seeking toreturn to the surface of the Moon by the mid 2020s, with plans of eventually build-1As of July 14, 2022. The Accords can be accessed at: https://www.nasa.gov/specials/artemis-accords/img/Artemis-Accords-signed-13Oct2020.pdf1ing a long-term base previously referred to as the Artemis Base Camp [28]. Mean-while, China and Russia are developing plans to build the International Lunar Re-search Station (ILRS), with construction expected to begin before the end of the2020s, however it is currently not expected to be crewed until about a decade afterconstruction begins [9]. While the two groups are promoting peace and coopera-tion, with both open to new nations joining them, it does not appear that the twogroups are looking to work together any time soon. The co-emergence of the twogroups may push each other to exciting new advancements, but with the prospectsof emerging economic opportunities, combined with the competitiveness of humannature, we must be cautious as we proceed. As such, it may be worth examiningthe last time the US and Russia (formerly USSR) engaged in such a race.During the late 1950s and early 1960s, amid the tense Cold War, the SovietUnion and the United States become the first nations to achieve spaceflight, withtheir rivalry pushing each other to greater and greater heights. In 1957, the USSRsuccessfully launched the first artificial satellite (Sputnik 1) into orbit around theEarth, signifying the beginning of the space race. Just two years later, the Sovietswere already conducting several lunar missions, including the first fly-by of theMoon (Luna 1), the first spacecraft to impact the surface of the Moon (Luna 2), andsuccessfully capturing and sending back to Earth the first images of the far side ofthe Moon (Luna 3). While the US were also making significant strides matching theachievements of the Soviets, it seemed that the USSR was often just a few monthsahead, with several notable firsts, including the first animal in orbit (Sputnik 2)2.By 1961, just four years after Sputnik 1 was launched, both the USSR and theUS would successfully complete human spaceflight missions, though the Sovietswould again just beat the Americans, this time by less than a month. Followingthese crewed missions, former U.S. President John F. Kennedy delivered his nowfamous address to Congress, calling for America to land a man on the Moon andreturn him to Earth by the end of the 1960s. Although the former President didnot explicitly say in this speech that the US needed to land a man on the Moon2This was a stray dog named Laika. While Sputnik 2 was able to successfully make it to orbit,Laika ultimately passed away reportedly due to overheating caused by a broken air conditioning.Additionally, prior to launch it was fully understood she would not survive the entire mission [6],so, personally, I find it difficult to characterize the mission as a success. Nevertheless, she providedvaluable information about living organisms in spaceflight.2before the Soviets, it was quite clear that doing so would ultimately determine ifthe mission was a complete success or not. This address set the foundations forNASA’s Apollo program, which would attempt to deliver on JFK’s bold ambitionsand, as a result, win the Space Race.As the two competing space superpowers charted their journeys to the Moon,neither had much of an idea about what they were getting into, as very little wasknown about the Moon prior to the Apollo era. Regardless of the sheer magni-tude of technological innovations that would be required to make it there, it wasunknown exactly what the surface would be like and how extreme the radiationwould be. In 1955, Gold [15] theorized that impacts and the intense lunar radiationenvironment would eventually erode the lunar surface into a fine dust. While thistheory would help explain why the Moon had such a surprisingly low albedo, Goldwas unable to estimate the thickness of the dust layer as the speed of this erosionwas dependent on factors still uncertain at the time [4]. This created some fearsthat the Moon could have a deep layer of dust covering the surface that would causeheavy spacecraft to sink, potentially preventing any safe landing attempts.Several more missions in the mid 1960s by both countries would give the twovaluable insight and help answer some of the unknowns about the Moon at the time,while also highlighting the perils of the task ahead. The USSR would continueits record of significant milestones, including the first successful soft landing ofa spacecraft on the Moon with its Luna 9 mission in 1966. Importantly, Luna 9confirmed that the surface of the Moon could support a lander and that it wouldnot sink into the ground, although Gold [15] was correct as the surface was indeedcovered in dust [37]. Just as before though, the US was not far behind as fourmonths later, the American Surveyor 1 was also able to safely land on the surfaceof the Moon. Impressively however, the Surveyor 1 mission was successful on thefirst soft landing attempted by the US, while the Soviet Union had several failedattempts in the years prior, including Luna 4, 5, 6, 7, and 8. Following a multi-yeartest campaign of the new Saturn family of rockets and the success of Surveyor 1,the US would look to attempt its first crewed flight of the Apollo era in February of1967 with Apollo 1. Unfortunately, the crew members would never get the chanceto lift-off, as all three tragically died in a fire during testing on the launch pad lessthan one month prior to their expected launch. This would ultimately lead to a320-month hiatus in attempted crewed flights for the Apollo program.Following this tragic start to Apollo, NASA continued their uncrewed opera-tions as the rest of 1967 saw them launch 8 lunar missions, including the Surveyor5 lander, which provided an unanticipated discovery. Images taken from the sur-face of the Moon as the Sun was setting below the horizon revealed a haze extend-ing away from the Moon. This haze indicated that the sunlight was being scattered,similar to what happens on Earth due to its atmosphere, but the Moon’s atmosphereis practically non-existent. What Surveyor 5 had discovered was lunar dust float-ing above the surface that is capable of creating a similar effect, which was furtherconfirmed by follow up images from Surveyor 6 and 7, shown in Figure 1.1 [31].Although unexpected, the idea had been brought forward before by Gold [15], whooriginally theorized the lunar surface was covered in a layer of dust, as he predictedFigure 1.1: Surveyor 6 image of a sunset seen from the surface of the Moon[3]. A haze can be seen, indicating the light is getting scattered. Credit:NASA4the same mechanism that forms the dust could be capable of dispersing it up offthe surface [4]. At the time, it was still unclear entirely what the source of the dustwas, but the amount observed was not enough to give pause to any future missions.Several more significant missions followed that would give the two nationsconfidence that a crewed lunar landing could be done, and potentially still withinthis decade. In September 1968, the Soviets Zond 5 spacecraft completed the firstmission to fly around the moon with animals onboard and return them to Earthsafely, and less than 1 month later, the Apollo crewed program was ready to make areturn. A successful launch saw the Apollo 7 crew spend 10 days and complete over160 orbits in low Earth orbit (LEO)3. The crewed Apollo 8 would launch just twomonths later, surpassing the achievements of the Soviets Zond 5 by successfullyachieving the first humans in orbit about the Moon, capturing the iconic ‘Earthrise’photo in the process, shown in Figure 1.2. With JFK’s deadline to land on theMoon before the end of the decade quickly approaching, 1969 would see a numberof historic missions launched for the Apollo program. In accordance with theirnewly condensed launch timeline, Apollo 9 was ready to launch in March, justthree months after Apollo 8. It was the first mission to contain and successfullydemonstrate the capabilities of both the Command and Service Module (CSM) andthe Lunar Module (LM). Apollo 10 would launch just two months later, serving asa dress rehearsal for an actual landing attempt, and following this successful test,NASA was confident it could attempt a crewed Moon landing.Launching atop a Saturn V rocket on July 16th, 1969, two months after themost recent Apollo mission, 29 months after the fatal Apollo 1 disaster, and eightyears after JFK’s proclamation, Apollo 11 set out on course to the Moon to at-tempt the first crewed lunar landing. Four days into the mission, Neil Armstrongand Buzz Aldrin piloted their LM ‘Eagle’ to a successful landing on the Moon.During their stay, they collected soil and rock samples, took photos and videos,and deployed instruments including a retroreflector for lunar ranging and a seismicdevice for measuring moonquakes [43]. After almost a full day on the surface,including over two hours of extravehicular activities (EVAs), Eagle ascended to re-join the CSM ‘Columbia’ and set out back for Earth with over 20 kg of lunar rocks3LEO is often defined as orbits contained within 2000 km of the Earth’s surface, but generallyare just a few hundred kilometers in altitude. For reference, the Moon is almost 400,000 km away.5Figure 1.2: Image of the Earth taken by Apollo 8 astronauts as it rose frombehind the Moon while the crew was in orbit about the Moon. Credit:NASAand soil [46]. The Apollo 11 crew safely returned to Earth after being in spacefor over eight days, achieving JFK’s goal just five months before the end of thedecade, and in doing so, beating the Soviets to the Moon, and effectively declaringthemselves as the winners of the space race.NASA would go on to launch six additional crewed missions under the Apolloprogram, with the final, Apollo 17, launching in December of 1972. The entireApollo program would see 12 different men plant their feet on the Moon, spendingover one week combined on the surface, with almost half the time spent duringEVAs and almost 400 kg (840 lbs) of lunar rocks and soil returned to Earth [46].Apollo 17’s Lunar Ejecta and Meteorites Experiment also provided additional ev-idence of the lunar dust floating above the surface of the Moon that was first ob-served by the Surveyor probes. Designed to detect lunar impacts and the subse-quent ejecta, the resulting data was overwhelmed by dust from the Moon that was6instead highly charged and slow moving, not what is expected for impact ejecta[1]. While these missions inspired generations and provided us with incredibleknowledge and perspectives, the extreme dangers and costs of the program werehard to ignore.Although initially denied publicly, the Soviets had simultaneously been de-veloping a crewed lunar program of their own, attempting to mirror the Apolloprogram, but they would never attempt a crewed lunar landing. Their first crewedmission of the program (Soyuz 1), which launched just two months after the Apollo1 tragedy, would again show the perils of spaceflight. After successfully launchingto orbit, the solo-crewed mission quickly encountered a variety of issues, resultingin the spacecraft returning to Earth early. The lone cosmonaut would unfortunatelynot survive reentry, with this becoming the first in-flight fatality of a space mis-sion. The Soviets were also developing a massive new rocket to take their crewsto the moon, but four failed launch attempts of the N1 would see the rocket’s de-velopment suspended, and ultimately canceled. These difficulties, paired with thesuccess of the Apollo missions, would eventually dissolve Soviet ambitions of acrewed landing attempt [7]. While the Soviets would still beat the Americans togetting the first rover onto the surface of the Moon with the Lunokhod 1 mission, itwould prove to be one of their last major space achievements regarding the Moon.The 1975 joint Apollo-Soyuz mission saw a historic first as the two ‘rivals’ dockedtheir crewed spacecraft together in orbit, with a symbolic handshake arguably end-ing the space race and marking a new era of cooperation in space. At the time ofthis writing, the US remains the only country to have safely landed a crew on theMoon and returned them to Earth, with almost 50 years passing since the final tripwas made.During the Apollo Moon landings, one unforeseen issue that plagued the crewswas the dust on the surface. Although the presence of the lunar dust was becomingwidely accepted prior to the first missions, the number of different problems it cre-ated was not predicted [2, 12]. The troubles began immediately during their visit,as large amounts of material were kicked up during the landings, greatly decreas-ing visibility for the pilots [12, 42]. Once on the surface, the problems continued asthe dust was found to stick to everything, covering their spacesuits and equipment,and when the astronauts tried to wipe it off their suits, they found it got rubbed7deeper into the fabrics [8, 12]. Lunar regolith is created by the shattering of lunarbedrock from billions of years of impacts, and without the weathering that occurson Earth, these particles can remain quite sharp and jagged [13]. This became abigger problem still once the astronauts would return inside their lunar module, asthe dust remained on their suits, with some getting kicked up into the air and be-coming a lung and eye irritant [12, 42]. The full extent of the toxicological of lunardust is still not fully understood to this day, and while no long-term problems forthe astronauts were attributed to the dust, prolonged exposure could have adverseeffects on astronauts’ lungs, eyes, and skin [13, 18].Despite rising concerns about how dust could impact crewed lunar missions,NASA was not deterred and performed several studies in the 1970s about the hab-itability of the Moon. During the early part of the decade, while still conductingApollo missions, NASA was already studying lunar orbital stations and locationsfor surface bases [16, 17, 23]. One such proposal was a 24-person lunar surfacebase called MOONLAB. It would require 15 years and 37 Saturn V launches tobuild the base, and would have contained a farm for growing food, doubling as aCO2 removal system [18]. Additionally, to support long-term lunar missions andhabitats, several studies investigated the feasibility of mining the Moon. A varietyof methods were proposed included using nuclear explosives [19] and drilling gaswells [35]. Methods of mining water on the Moon were also studied, at a timewhen it was still unknown if there was even water available [34]. Unfortunatelyfor the lunar enthusiasts, while the issue of dust remained a constant throughoutthese studies, the political ambitions funding the lunar program were not. A re-cently elected US president no longer saw the risk of lunar missions necessary, ashe thought a disaster could impact his re-election chances [22]. Gradually, the pro-gram was brought to an end as three planned Apollo missions were cancelled andfunds got diverted towards NASA’s new poster child, the Space Shuttle.The decades following the Apollo era would see a relatively consistent humanpresence in space, although the crewed missions would remain within LEO, as lu-nar activity faded despite a strong interest in the Moon remaining. Several spacestations were launched to LEO, notably the International Space Station (ISS), pro-viding a safer, more accessible option than traveling to the Moon. While the 1980swould see no lunar missions launched by either the US or the Soviet Union, the81990s would start with a new country launching a mission to the Moon. TheJapanese Hiten spacecraft, launched in 1990, experienced some initial setbacksbut was eventually able to successfully enter lunar orbit. The US would launch justtwo lunar missions that decade, with Clementine launched in 1994, and the Lu-nar Prospector in 1998. However, the 2000s would see a resurgence in activity andplans for lunar missions, with several new agencies joining in. The European SpaceAgency (ESA) would launch their first lunar orbiter, SMART-1, in 2003. This wasfollowed shortly after by the Chinese Chang’e 1 in 2007 and India’s Chandrayaan-1 the year after. As new countries were setting out to explore the Moon for thefirst time, America was finally making plans for a crewed return to the surface. In2005, the NASA Constellation program was announced, with part of its goal beinga crewed return to the Moon by 2020. This goal was not met, as since the programwas announced, its priorities have shifted with the changing of presidents. Theprogram was rebranded as Artemis in 2017, and the first crewed landing is nowscheduled for 2025. Although its uncertain if this deadline will be met, it appearsto be a question of when, rather than if, the mission will launch.The Artemis program, as well as the agreements in place between China andRussia, are a sign that we are about to enter a new era of lunar exploration, thistime with the intent to stay, permanently. Part of the objectives of the Artemisprogram is to build an outpost on the surface of the Moon that could be crewedfor long-duration missions [28]. The ILRS, which is currently expected to beginconstruction around 2026, will seek to do the same [9]. The first test flight of theSpace Launch System (SLS) rocket, which is being built to launch the Artemiscrew, is scheduled for later this year (2022). Regardless of when the first launch ofSLS takes place, the Artemis era has begun, as the first support mission has alreadybeen launched. CAPSTONE was launched in June of 2022 and is currently on itsway4 into a special lunar orbit called a near-rectilinear halo orbit (NRHO).Figure 1.3 demonstrates what some NRHOs look like, seen from a side-onview of the Moon’s rotational frame about the Earth. The NRHO family of orbitsare a result of the circular restricted three body problem (CR3BP), which assumesone body is on a circular orbit around a more massive object, while the third body4At the time of writing.9is required to have a negligible mass compared to the other two. This is a safeassumption here as the Earth and the Moon are both much more massive than anyspacecraft and the Moon’s orbit is approximately circular about the Earth for thepurposes of this discussion. A consequence of the CR3BP is the five equilibriumpositions in the rotating frame, known as the Lagrange points. Three of thesepoints, known as the collinear points, lie along the x-axis of the rotating frame,while the other two, L4 and L5, are about 60◦ ahead and behind the Moon in orbitabout the Earth. The two collinear points closest to the Moon are L1 and L2, withL1 located between the Earth and the Moon, and L2 beyond the far side of theMoon, shown in Figure 1.3. L3 is the final collinear point, located about one lunardistance away from the Earth in the opposite direction of the Moon. Halo orbits areFigure 1.3: Examples of the northern and southern NRHOs about L1 and L2.The figure is shown from a side-on perspective of the Moon’s rotationalframe about the Earth. The Earth-Moon L1 and L2 Lagrange points areshown for reference. Credit: Davis et al. [11] (Used with Permission)10stable orbits that can form about the three collinear points. NRHOs are a subset ofthese halo orbits, and although the CR3BP is not exact for the Earth-Moon system,they are still relatively stable, so not a lot of energy would be required to keep aship in such an orbit [21, 47].CAPSTONE was launched to verify the stability of NRHOs, as NASA envis-ages using the same orbits eventually for the Lunar Gateway, which is planned tobe the first space station to orbit the Moon. One benefit of these orbits is that thespacecraft will be able to maintain almost constant communication with the Earth,as it is designed to limit the Moon passing in between the spacecraft and the Earth[21, 47]. Another feature of a NRHO is that it has a very close approach to theMoon at periselene and such a distant aposelene that it sets an easy transfer pointfor Earth-Moon connections. Gateway will act as a transfer station in the south-ern L2 NRHO for missions to the lunar surface and beyond. Once the technologyfor in-situ resource utilization (ISRU)5 becomes available, we could see Gatewayserve as a fuel depot, enabling lunar landers to extend their mission lifetimes andrefueling spacecraft bound for deep-space [21].The new space race seeks to build human lunar habitats and increase surfaceactivity on the Moon in general, with ISRU at its core, but we must be mindful toproceed in a sustainable manner if we want it to last. One issue we must be carefulof as we return to the Moon is orbital debris, including discarded rocket stages anddefunct satellites, as well as disturbed lunar soil that can get lofted into cis-lunarspace. It should be clear by now that we cannot be allowed to underestimate ourcollective ability to pollute seemingly enormous bodies, as we’ve already provenwith the Earth’s oceans and waterways, atmosphere, and now orbital environment.Although nations have previously been extremely careful to avoid contaminatingthe lunar surface, common practice has also seen these same nations discard rocketstages by crashing them into the Moon (both intentionally and unintentionally).In addition to ejecta from natural and artificial impactors, landing and takeoff se-quences on the lunar surface will eject a large amount of material, as experiencedby the Apollo missions. Some particles will be able to reach escape velocities and5ISRU is another way of saying space mining. Water ice, discovered in craters near the lunarsouth pole, is likely to be the target of the first mining operations, as water is not only an importantresource for human survival, it can also be broken down and used as rocket fuel [32].11enter orbit, while buildings or equipment in the vicinity will also be prone to dam-age [20, 44]. Once on the surface of the Moon, the disturbances will only continue.The construction of infrastructure on the surface is likely to disrupt lots of soil, inaddition to the inevitable emergence of ISRU and the mining of the lunar south polefor water ice [32]. Additionally, while private companies are currently focused onsupporting national agencies, in a not-too-distant future, we could see a shift to-wards the commercialization of the Moon, with lunar hotels and resorts becominga reality. If we are to move forward in a sustainable manner, it is important that wetry to avoid making mistakes we’ve previously made while learning from our past.Since humans last stepped foot on the Moon, we now know much more aboutour celestial companion and technology has vastly improved, but problems suchas the lunar dust could still limit any progress. The dust first discovered extend-ing beyond the lunar surface by the Surveyor program is now better understood.Electrostatic charges on the particles interacting with lunar surface charges causedby solar radiation and the lunar plasma environment can launch particles off thesurface and allow them to levitate [10, 36, 40]. The majority of the charged parti-cles that get lofted above the surface are on ballistic trajectories that will eventuallyreturn to the surface [40]. In addition to natural mechanisms, lunar soil can alsoget dispersed from the surface through anthropogenic processes. Some of this ma-terial could exceed escape speeds and ultimately get ejected out into the cis-lunarorbital environment. Escaping material could potentially build up in regions ofhigh spacecraft activity, including the Earth-Moon Lagrange points. Understand-ing the behaviour of these particles, both those on suborbital ballistic trajectories,as well as the high-speed orbital ejecta, is important for our anticipated return tothe Moon. Yang et al. [45] recently studied the distribution of dust that is ejectedfrom the Moon, integrating a variety of particle sizes released from plausible ejec-tion models for 100 years and analyzed the end state. Their simulations showedmost particles escaped the Earth-Moon system by the end of the 100 years, withsome particles found to collide with the Earth or the Moon.One final consideration we should keep in mind when examining our possibleimpacts on the cis-lunar environment is the ability to alter the brightness of thenight sky. Ejected lunar material can reflect sunlight, and a sufficient spatial densityof material in orbit could produce brightness levels that exceed the background sky.12This idea is not necessarily new, as particles have been theorized to gather in theL4 and L5 Earth-Moon Lagrange points, referred to as the Kordylewski clouds6.These clouds have been imaged several times but are not bright enough to see withthe naked eye [39]. The night sky is arguably common heritage of humankind,and as such, it should be protected. Human-caused changes to the sky could havea number of unintended consequences, including for science, as well as naturaland cultural heritage. While studies on the impacts of lunar dust on the night skybrightness could not be found, one study discussed the negative impacts it couldhave on future astronomical observations attempted from the Moon [27].As we proceed to return to the Moon, we should attempt to limit our impact onthe natural lunar environment to the best of our ability. This includes preventingthe buildup of any long-lasting orbital debris, including lunar soil disturbed fromthe surface. To address some of these issues, this study seeks to gain a better un-derstanding of the dynamical behavior of lunar ejecta. Particles ejected from theMoon are simulated and tracked for one year to determine the short and longer-term behavior of the particles in the cis-lunar environment. The accumulation ofmaterial near the Earth-Moon Lagrange points is studied, as well as statistics re-garding particle collisions with the Earth or Moon and the amount which remaingravitationally bound to the two. Additionally, the ability of this material to alterthe Earth’s night sky is examined. Particles are projected onto the night sky forEarth based observers. When combined with estimations for the flux due to re-flected sunlight, the dust distribution is used to predict local sky brightness. Themethods used to run the simulations, as well as the analysis required for particledensity estimations and flux calculations and projections are described in Chap-ter 2. The results of the simulations and the subsequent analysis are shown inChapter 3. Discussion of all the results, including additional simulations with ra-diation forces, as well as potential future work are presented in Chapter 4, whileconclusions are provided in Chapter 5.6These are likely interplanetary dust getting trapped, rather than material from the Moon.13Chapter 2MethodsThe dynamical behaviour of material in the cis-lunar environment is explored byrunning computational N-body simulations of particles ejected from the lunar sur-face. In total, 10,000 particles are released at random locations around the Moonwith a velocity distribution that allows the particles to just escape the influenceof the lunar gravity. The short and long-term results of the simulations are ana-lyzed to understand where this material is transported, including if accumulationor over-densities are seen in any of the Lagrange point regions. The particles arealso projected onto the sky for an observer on Earth to examine how the brightnessof the night sky could be impacted by the reflection of sunlight by this material. Itis important to point out that these simulated particles can be thought of as tracerparticles. Each simulated particle may represent a large number of actual particles,all comparable in size and released with similar trajectories. The exact numberdepends on the mass that is assumed to be released. The analysis and results pre-sented in this study are only for the simulated particles, but a number of the resultscan be scaled accordingly, as will be discussed in Chapter 4.2.1 Dynamical SimulationsSimulations in this study are performed using REBOUND, a N-body simulatorpackage for python which has its main routines written in C [29]. The defaultintegrator for REBOUND, IAS15, was used for these simulations. IAS15 is a14high order, non-symplectic integrator with an adaptive time-step that allows for in-creased precision during close approaches while minimizing the overall run time[30]. REBOUNDx, a package designed to incorporate additional effects into RE-BOUND, is included for simulations where radiation forces are taken into account[41]. REBOUNDx models the radiation forces based on Burns et al. [5], usinga≈ β GM⊙r2[(1− rˆ ·v/c) rˆ−v/c] . (2.1)Here, a is the acceleration of the particle, r is the particle’s heliocentric positionvector, v is the heliocentric velocity, and c is the speed of light. The parameter β isthe ratio of the solar radiation pressure force to the gravitational force experiencedby a particle in orbit, given byβ ≡ Fr/Fg = 5.7×10−5Qpr/ρs, (2.2)where s and ρ are the particle radius and density in cgs units, respectively, andQpr is the radiation coefficient. The parameter β is thus large for very small par-ticles and becomes negligible as the particle size increases. The primary resultspresented in Chapter 3 are for simulations of large particles where beta is assumedto be zero. However, separate simulations were run with a high beta (0.1), repre-senting particles that are approximately a few to ten microns in diameter. Thesesimulations were completed after a majority of this thesis was written, and thus theresults are included in Section 4.2.Individual simulations containing only one massless particle, along with theSun, Earth, and Moon, are run in parallel for a total of 10,000 simulations. Asthe particles are massless, there are no interactions between the particles duringthe simulations. Therefore, provided the simulations have identical starting times,they can be run separately and subsequently combined for analysis to keep thecomputational times manageable. This parallelization is accomplished using themultiprocessing package in python. All simulations are performed on a 2020 Mac-Book Pro with an Apple M1 chip and 8 GB of ram, using 6 of the 8 cores avail-able. The starting time for each simulation is set to be August 22, 2007, 01:14:00TDB (2454334.551388889 JD), with the initial conditions for the Sun, Earth, and15Moon in the heliocentric ecliptic coordinate system obtained using NASA’s JPLHORIZONS Web-Interface.7 While initial conditions are defined in the heliocen-tric frame, the simulations are run in the center of mass frame, with REBOUNDcontaining built-in functions to handle these transformations. Simulations are runfor varying lengths of time with different archiving time-steps to allow for analysisof both the long-term behavior and higher temporally resolved short-term behavior.Particle collisions with the Sun, Earth, and Moon are tracked and recorded duringthe simulations within REBOUND using its built-in “line” collision search, whichchecks if the trajectory of two objects overlap during the time-step, assuming theyare traveling straight. If a particle collides with one of the three large bodies, theparticle is removed from the simulation. Particle on particle collisions are ignoredfor this study.Rather than attempting to model specific mechanisms or processes that couldeject material from the lunar surface, a simplified approach is taken. Particles arereleased in a random uniform spatial distribution spread evenly across the Moon,at an altitude of 100 km, using a method based on Shao and Badler [33]. This isdone by first using python’s random module to generate two random variables:a= Uniform[0,1],b= Uniform[0,1].(2.3)These random numbers, a and b, are then converted into angles:θ = arccos(2b−1),φ = 2πa,(2.4)where θ can range [0,π] radians measured from the pole and φ is the azimuthal an-gle, ranging [0,2π] radians. This sampling results in a uniform distribution acrossthe Moon, assuming it is a perfect sphere. The particles are chosen to be initially100 km above the lunar surface to ensure there are no strange interactions or over-laps between the Moon and the particles when the simulations begin.7Orbital data retrieved from https://ssd.jpl.nasa.gov/?horizons on November 1, 2021. The arbi-trary starting time was chosen for preliminary work that is not included here and is kept purely forconvenience.16A velocity distribution is assigned to the particles, vkick, relative to the Moon’sorbital velocity about the Sun, with a magnitude calculated as follows:vkick =√vmin2+ vrandom2, (2.5)where vmin is the escape velocity of the Moon from an altitude of 100 km andvrandom is a random velocity drawn from a Rayleigh distribution with a mode of1 m/s. The Rayleigh distribution is chosen for convenience and to generate a dis-persion among the velocity magnitudes, while the choice of 1 m/s is to give theparticles a speed just above the escape speed, on average. The particles’ initial ve-locity vector is not directly perpendicular to the lunar surface, but rather, a randomdirection within a cone is used. The launch angle is tilted in a uniform randomdirection with a uniform random angle from zenith between [0,π4 ] radians, with 0corresponding to a perpendicular ejection.2.2 Coordinate TransformationsOne of the first steps for analyzing the simulation results is transforming the parti-cles, along with the 3 larger bodies, into different frames of reference. Examplesinclude the geocentric and selenocentric frames, which allow for observing the tra-jectory and behavior of the particles relative to the Earth and lunar orbital planes.Different combinations of rotations, combined with translations, can allow the sim-ulations to be viewed from a variety of different frames of reference. Two rotationmatrices are required to accomplish the transformations in this study:Rx(Θ) =1 0 00 cosΘ −sinΘ0 sinΘ cosΘ ,Rz(Φ) =cosΦ −sinΦ 0sinΦ cosΦ 00 0 1 .(2.6)Here, Rx(Θ) and Rz(Φ) represent rotation matrices for rotations about the x-axisand z-axis, respectively, for the given angles, Θ and Φ.172.2.1 Lunar Rotating FrameA majority of the analysis is performed in the Moon’s rotating frame about theEarth, where the xy-plane is aligned with the lunar orbital plane, such that the Earthand Moon are along the x-axis and the Earth-Moon barycenter is at the origin. Totransform the simulations into the Moon’s rotating frame, they are first transformedinto the Earth’s rotating frame about the Sun. Beginning in the heliocentric frame,the simulations are rotated about the z-axis by the longitude of Earth’s ascendingnode, effectively aligning the Earth’s orbital nodes along the x-axis. Next a ro-tation is performed about the x-axis equal to the Earth’s inclination, placing theEarth’s orbital plane in the xy-coordinate plane. The Earth can then be rotated tothe x-axis by applying another z-axis rotation, this time with an angle equal to theEarth’s true anomaly plus the Earth’s argument of pericenter. Finally, a translationin the negative x-direction, equal to the Earth’s instantaneous heliocentric distance,results in an Earth-centered rotational reference frame, with the Sun always on thenegative x-axis.Applying similar rotations to this new Earth-centered, or geocentric, frame, butwith respect to the Moon, will result in the simulations being transformed into theMoon’s rotational frame. The lunar orbital elements used here are calculated inthe geocentric frame. Beginning from the geocentric frame, a z-axis rotation isapplied, equal to the longitude of the Moon’s ascending node. With the Moon’snodes now along the x-axis, the simulations can be rotated about the x-axis by theMoon’s inclination to place the lunar orbital plane in the xy-plane. A final z-axisrotation equal to the Moon’s true anomaly plus the Moon’s argument of pericenterplaces the simulations into the Moon’s rotating frame, with the Earth at the originand the Moon along the x-axis in the xy-plane. The lunar rotating frame’s origin isthen translated to the Earth-Moon’s barycenter, as is common convention.2.2.2 Sky ProjectionA different series of translations and rotations are performed to place an Earthbased observer at the origin of a new frame, which allows for easier computationof the positions of the Sun, Moon, and particles across the sky. For this study,the observer is assumed to be located near Vancouver, British Columbia (49◦ N,18123◦ W). Beginning in the simulation’s heliocentric frame, the coordinates aretranslated to have the origin centered on the Earth. A rotation about the x-axis ofthis translated frame, by the Earth’s obliquity relative to the ecliptic plane, placesthe Earth’s north pole along the z-axis. This can be done as the simulations areoriginally in the ecliptic coordinate system. At this point, a rotation about thez-axis based on the time of the simulation and the given observer’s longitude isperformed to place the observer along the y-axis. A final x-axis rotation basedon the observer’s latitude is then used to put the observer on the x-axis as well.Subtracting the z-values by the Earth’s approximate radius (6471 km) places theobserver at the origin. In this frame, the observer’s zenith is along the z-axis, northis in the positive y-direction, and east is in the positive x-direction.Figure 2.1: An example of a polar projection of the Sun, Moon, and particlesfrom an observer’s perspective on Earth. The inner grey rings representsteps of 15◦ from the zenith, with the last inner ring representing thevisible horizon. In this example, the Sun is just beginning to set alongthe western horizon, while the Moon is rising out of the south-east.19A polar projection is used to visualize the position, and ultimately brightness,of the particles from the observer’s perspective, with an example shown in Fig-ure 2.1. This projection represents what an observer who is lying on their backwith their head pointed north would see. To accomplish this, the signs of the x-coordinates are flipped to have west pointed in the positive x-direction. The radiusof the polar projection plot is the angle of the object relative to the observer’s zenith,with objects at a radius greater than 90◦ being considered below the horizon. Thehorizon is indicated by the final grey inner ring in the example above, with eachring representing a step of 15◦. The outer black ring marks 180◦, which is actuallyjust a point directly below the observer, representing anti-zenith. These projectionswere tested by verifying the position of the Sun and the Moon at different times.2.3 Time Series AnalysisWhen running the simulations, snapshots are archived at constant time intervalsfor analysis. Simulations that are run for 30 days are archived four times per sim-ulated day, while yearlong simulations have snapshots saved just once a day. Thespatial distribution of the particles is plotted and analyzed from a variety of refer-ence frames at different times during the simulations. Special attention is paid toanalyzing particle densities around the Earth-Moon Lagrange points and the corre-sponding density evolution. The brightness of these particles is of interest as well,so their projections onto the sky of an Earth based observer are also analyzed atdifferent times. In the following subsections, the methods for obtaining particlevolume densities, determining Lagrange points, and estimating sky brightness areall discussed.2.3.1 Kernel Density EstimatorA cubic spline kernel density estimator, based on Monaghan [26], is used to deter-mine the spatial densities of particles at any given location. Here, the density at a20given point j is approximated byρ j =N∑i1πh31− 32(ri jh )2+ 34(ri jh )3 if 0≤ ri jh ≤ 1;14(2−ri jh )3 if 1≤ ri jh ≤ 2;0 otherwise,(2.7)where N is the number of interpolants, ri j is the distance to particle i from the pointj, and h is the length scale. The length scale is defined as half the radius of a spherethat is centered at point j and contains the closest 32 particles. A kd-tree is used toquery the 32 closest particles, using an algorithm in SciPy’s spatial algorithms anddata structures package, described by Maneewongvatana and Mount [24].To avoid the kernel from including too distant of particles, a maximum distancelimit is placed on the length scale, which is 30,000 km for this study. Due tothe nature of kernel, this has the effect of excluding any particle that is greaterthan 60,000 km away from the point of interest. This value is chosen as it is theapproximate distance of L1 and L2 from the Moon.2.3.2 Lagrange PointsThe three collinear Earth-Moon Lagrange points (L1, L2, and L3) are calculated as-suming the circular restricted three body problem holds. This is not strictly correctdue to the Moon’s eccentricity of about 0.05 but is sufficient for the calculationshere. Within the Moon’s rotating frame, the origin is assumed to be the Earth-Moon center of mass with the Earth at x=−µ and the Moon at x= 1−µ in unitsof lunar distances, where µ = m2m1+m2 , with m2 the smaller of the two masses, whichis the Moon in this case. Numerically solving the equation of motion0 = x− (1−µ)(x+µ)|x+µ|3 −µ(x−1+µ)|x−1+µ|3 , (2.8)results in the three real roots corresponding to the x-positions of the three collinearLagrange points.The L4 and L5 Lagrange points are approximated to be ±60◦ in front of andbehind the Moon, assuming again an instantaneous circular orbit. These two pointscan also be determined analytically from the CR3BP setup, but the approximation21used here is sufficient for the depth of this study.As noted, the CR3BP is assumed for these calculations, which is not exact dueto the Moon’s eccentricity. To accommodate this variation, including the changein the Earth-Moon distance with time, the positions of the Lagrange points areuniquely calculated for each time-step analyzed. As a result, the Lagrange pointsoscillate radially, with respect to the Earth in the Moon’s rotating frame, at thesame rate as the Moon.2.3.3 Sky Brightness EstimatesThe flux of sunlight that reaches the Earth after reflecting off a particle is approxi-mated asfp = SIP(ω)ApSAp/(πdEarth2). (2.9)Here, SI is the solar irradiance value, set to be 1373 W/m2 for this study, represent-ing the total amount of solar energy per square meter reaching Earth every second,given here for the full solar spectrum. This is kept constant for all the particlesin these simulations, as they are all approximately 1 au from the Sun. The phasefunction of the particles, P(ω), represents how much light is reflected relative tothe angle of the Sun, particle, and observer. Additionally, Ap is the Bond albedoof a particle ‘p’, SAp is the particle’s cross-sectional surface area, and dEarth is thedistance from the particle to Earth. For this study, several key assumptions are re-quired. A linear phase function is assumed, such that P(ω) = 1 for particles 180◦away from the Sun relative to the observer, while P(ω) = 0 for particles with aphase angle of zero. Particles are assumed to have an albedo similar to the Moon(Ap = 0.14) [25] and a radius of one millimeter for these flux calculations. Thecross-sectional surface area is calculated assuming the particles are spheres.22Chapter 3ResultsThe short and long-term behavior of the simulated particles, including density es-timates at the Lagrange points and collision statistics, are presented in Section 3.1.Polar projections of the particles onto the night sky for an Earth based observer areshown in Section 3.2, including 2d binned projections of the expected solar fluxscattered off the particles. Comparisons of simulations with and without radiationforces (β = 0.1 vs β = 0.0) are discussed in Section 4.2. However, the value of β= 0.1 is an extreme case and should be taken as an approximate upper limit. Thecase of β = 0.0 will apply to a wide range of particle sizes and has the greaterpotential to be threatening to cis-lunar operations. As will be shown in Section 4.2,such particles will also remain in the cis-lunar environment longer than very highbeta particles, and thus, could contribute more to sky brightness in the long term.For these reasons, all results presented in this chapter are for simulations with noradiation forces on the particles (β = 0.0).3.1 Dynamical Behavior of ParticlesFigure 3.1 shows the release and subsequent spread of 10,000 particles in the xy-plane of the Moon’s rotating frame at six different times during the first 10 daysof the simulations8. The origin has been translated to be centered on the Moonto examine the initial trajectory of the particles relative to the Moon. The Earth8The number of particles shown decreases over time due to collisions.23XY Particle Distribution in the Moon’s Rotating Frame over 10 DaysFigure 3.1: The xy-positions of the debris particles, Earth, Moon, and fourof the five Earth-Moon Lagrange points, in the Moon’s rotating frame,with the origin centered on the Moon, over the course of 10 days.24and the four nearby Lagrange points have been included for reference. The axesare labeled in units of lunar distance [1 LD ≈ 3.844e5 km ≈ 0.0025 au], which isdefined as the average distance between the Earth and the Moon.During the first few days after release, the particles can be seen expandingisotropically away from the Moon, reaching just beyond the L1 and L2 Lagrangepoints before the particles begin to shear. Two trails of particles continue awayfrom the Moon in opposite directions, through the vicinity of L1 and L2. Someparticles in these trails escape the lunar gravity well and becoming increasinglyinfluenced by the gravity of the Earth and the Sun. Not all particles escape, withsome falling back and colliding with the Moon after several days in the lunar orbitalenvironment, while others enter lunar orbits.Ten days after release, some particles have traveled beyond a lunar distance,stretching past the Earth, while over 40% have already crashed back into the surfaceof the Moon. The earliest, as well as the majority, of the impacts occur on daysfour and five, with almost 3600 lunar collisions recorded on just these two days.Although no particles travel very close to L4 or L5 during the first 10 days, thesame cannot be said about the two nearby collinear Lagrange points. L1 and L2both see a steady stream of particles flowing directly through them during the first10 days. This can be further seen through the particle density evolution at theLagrange points.Figure 3.2 shoes the kernel density estimations for L1 and L2 during the first10 days of the simulations. The densities for both points first peak about two daysinto the simulations, which corresponds to when the particles are first expandingthrough the two points, as seen in Figure 3.1. Between days four and six there is adecrease seen in the density estimations, which is supported by the xy-positions ofthe particles at this time. The following days show an increase in particle densitiesat L1 and L2, although never exceeding the initial peak values from around daytwo. While the densities at L1 and L2 differ at times, they follow a similar patternand reach comparable peak values. The remaining three Lagrange points do notproduce any density estimates during this time as they do not have any particlesclose enough to be included within the maximum smoothing length.The distribution of the particles during the first 30 days of the simulations isshown in the xy-plane in Figure 3.3, as well as in the xz-plane in Figure 3.4. For25Figure 3.2: The kernel density estimations for the L1 and L2 Earth-MoonLagrange points during the first 10 days of the simulations. The densityis given in number of simulated particles per cubic km.both figures, the origin is centered on the Earth-Moon center of mass. The xy-view shows the two previously seen trails further extending, creating two differentpopulations. The trail that originally extended through L1 is now in orbit aroundthe Earth, staying within L3, L4, and L5, and wrapping completely around back toL1 within about three weeks. The other trail of particles continues to drift furtheraway from the Earth-Moon system, appearing to orbit the pair from outside theLagrange points. All the particles seem to stay close to the lunar orbital planeduring the first few weeks of the simulations, but by the end of the first month, theouter most particles extend to almost half a lunar distance in both directions.The kernel estimations of the particle densities at the Lagrange points over 30days is shown in Figure 3.5. During the first month of the simulations, the L3,L4, and L5 Lagrange points appear to stay clear of particles, while the L1 and L2points seem to exhibit a slowly decreasing amount of material. Large variationsexist within the overall trend, including an order of magnitude spike in the L1density around day 14, which is due to a dense cloud of particles in lunar orbitpassing through L1. The remaining three Lagrange points are not included for this26XY Particle Distribution in the Moon’s Rotating Frame over 30 DaysFigure 3.3: The xy-positions of the debris particles, Earth, Moon, and thefive Earth-Moon Lagrange points, in the Moon’s rotating frame, withthe origin at the Earth-Moon center of mass, over the course of 30 days.27XZ Particle Distribution in the Moon’s Rotating Frame over 30 DaysFigure 3.4: The xz-positions of the debris particles, Earth, Moon, and thethree collinear Earth-Moon Lagrange points, in the Moon’s rotatingframe, with the origin at the Earth-Moon center of mass, over 30 days.Figure 3.5: The kernel density estimations for the L1 and L2 Earth-MoonLagrange points during the first 30 days of the simulations.28plot once again as there are still no particles within the maximum scale length ofthe kernel for each point during this time. After 30 days, almost half the particles(48.9%) have collided with the Moon, while none is found to have yet collidedwith the Earth (or Sun).As noted in Section 2.3, the simulations are further propagated up to one year,however, the output is less frequent. This nonetheless allows one to get a betterunderstanding of the longer-term behavior of the particles. Figure 3.6 shows the xy-XY Particle Distribution in the Moon’s Rotating Frame over One YearFigure 3.6: The xy-positions of the debris particles, Earth, Moon, and thefour nearby Earth-Moon Lagrange points, in the Moon’s rotating frame,with the origin centered on the Moon, over the course of a year.29positions of the particles relative to the center of the Moon, while Figure 3.7 showsthe xy-positions of the particles at the same times, but with respect to the Earth-Moon center of mass. In the Moon centered frame, particles are found to remain inlunar orbits for the full year, although there are considerably fewer particles after ayear than at 60 days. After 60 days, there are about 800 particles within the Moon’shill sphere, while after a year there are only 26. Figures 3.6 and 3.7 both show thatXY Particle Distribution in the Moon’s Rotating Frame over One YearFigure 3.7: The xy-positions of the debris particles, Earth, Moon, and thefive Earth-Moon Lagrange points, in the Moon’s rotating frame, withthe origin at the Earth-Moon center of mass, over the course of a year.30large numbers of particles remain in orbit around the Earth during the full first year,with less of a decrease seen in comparison to particles in lunar orbit. Particles arenot bound to either orbit, as several are often seen transferring between differentorbits about the Earth and the Moon.While the majority of the particles orbiting the Earth appear to be within theLagrange points, there remains a scattered population of particles extending be-yond the Lagrange points after one year. Once again, the kernel can be used toestimate the particle densities found at the five Earth-Moon Lagrange points to de-termine if particles are crossing through any of the points. Figure 3.8 shows thedensity estimations for the five Lagrange points during one year of the simulations.After about 200 days, the particle density around L1 and L2 becomes roughly con-stant on average, but with order of magnitude variations over weekly timescales.L1 retains a slightly higher density compared with L2, more noticeably during thesecond half of the year. After about two months, the other three Lagrange pointsFigure 3.8: The kernel density estimations for all five of the Earth-Moon La-grange points during one year of the simulations.31Figure 3.9: The xy-positions of the debris particles, the Earth, and the Sun, inthe simulation’s original heliocentric frame, one year after the particleswere released. The Earth’s orbit is shown with the green line to givereference for the radial spread of the particles.begin to sporadically encounter particles, at times reaching levels that are compa-rable to the L1 and L2 densities.By the end of the year, only 25% of the initial particles remain, with a largenumber of those appearing to escape onto heliocentric orbits, shown in Figure 3.9.Most of the particles that remain gravitationally bound to the Earth-Moon systemduring the simulations eventually get swept up by the two bodies. In total, 21particles collide into the Earth, while 7479 collided with the Moon. Only about1000, or 40%, of the remaining particles are within the Earth-Moon hill sphere afterone year. The particles that do escape are slowly shearing out in both directionsand will eventually spread throughout the Earth’s entire orbit in approximately 5-10 more years.32Figure 3.10: The maximum z-distance, normal to the Earth’s orbital plane,between any two particles during one year of the simulations. Theblue line represents the z-distance between the two furthest particles.The orange line is the z-distance between the two furthest particlesif 5% of the particles at either extreme are excluded, while the greenexcludes 25% of the particles at either end.Figure 3.10 shows the maximum z-distance between any two particles, normalto the Earth’s orbital plane, during the simulated year. The blue line is the truemaximum between any of the particles, while the orange and the green lines seek toexclude any outliers, as well as demonstrate the spread of the bulk of the material.The orange line excludes the furthest 5% of the particles at either end, while thegreen excludes 25% of both ends. The largest z-distance between two particlesduring the first year was almost 3.5 lunar distances. This is likely due to a strayparticle kicked out of the ecliptic plane after a close encounter with the Earth orthe Moon. The orange lines shows that the majority (≥ 90%) of the particles aredispersed less than one lunar distance, normal to the ecliptic, for the entire year.333.2 Night Sky ProjectionsAt various points throughout the simulations, the particles are projected onto po-lar maps of the sky to determine if and where on the sky the particles would bevisible at night for an Earth based observer. Particles are generally visible (abovethe horizon) for most nights, apart from a few of the first nights of the simulationsduring the new moon phase, when the particles are not yet spread beyond the lu-nar hill sphere. After the streams of particles become sheared out during the firstfew months, any given night will have roughly half the particles visible above thehorizon, usually oscillating between 30% and 70%.Figure 3.11: The polar projection of the particles, the Sun, and the Moon, atnight for a northern hemisphere observer, approximately one monthinto the simulations. The inner grey rings are at 15◦ steps from thezenith, with the outer grey ring representing the visible horizon. TheMoon is covered in this plot by the particles just below the south-westhorizon. Particles are shaded relative to their solar phase angle.34Occasionally, the particles create a large stream across the night sky, with anexample of this shown in Figure 3.11. As a reminder, this and the subsequent plotsin this section are for an observer near Vancouver, Canada (49◦ N, 123◦ W). Here,the particles are plotted with an alpha corresponding to their phase angle, such thatparticles that would appear close to the Sun in the sky are more transparent andlight grey in color. Particles that are on the opposite side of the Earth from the Sun,and thus would reflect a large amount of light towards the observer, are shadedthe darkest. The local time of this projection for the observer near Vancouver isapproximately 4:00 am in September, so the Sun, which will rise above the easternhorizon in a couple hours, is where it is expected to be. The Moon is hidden behindthe particles in this plot, just below the horizon in the south-west.Figure 3.12 shows a polar projection of the bolometric flux of reflected sunlightFigure 3.12: The flux of reflected sunlight from particles visible above thehorizon approximately one month into the simulations. The plot isdivided into bins 1◦ by 1◦ in size, with the color mapping normalizedon a log scale. The Sun and the Moon are both below the horizon.Note this is the bolometric flux.35that would reach the Earth from these particles, calculated using Equation 2.9, andfiltered to just show the visible sky above the horizon of the observer. The plothas been divided into a 2d grid, with each bin one square degree in size. As theflux values are calculated in units of W/m2, the units of each bin in the figure aretherefore W/m2 per square degree. The solar irradiance value listed in Section 2.3.3accounts for the full solar spectrum and not just visible light. Therefore, these arebolometric flux values, which can ultimately be scaled to visible flux values, asdiscussed further in Section 4.3. The color mapping of the bins is normalized on alog scale, with the minimum and maximum non-zero bin values set as the limits.At this time in the simulation, the Sun and the Moon are both below the horizon,so the sky would be dark, and nothing would be concealing the particles.A wide range of values are seen in the previous figure, driven largely by thevarying distances to the particles from the Earth, with the minimum and maximumflux separated by almost three orders of magnitude. Further particles will have agreater distance for the reflected light to disperse and therefore will be less bright.Different phase angles will also add to this variation, although not to the sameextent. The differing phase angles accounts for the gradient of yellow and greenseen across Figure 3.11, while the darker blue patch extending from the easternhorizon is a different population of particles further away.While particles are found to be visible in the sky almost every night of thesimulation, they do not always appear as a streak across the sky, and at times theyappear to wrap around the horizon, as seen in Figure 3.13. Here, an even greaterdifference in flux values is seen, with values both higher and lower than the pre-vious plot. This is likely due to the particles having more time to disperse, andtherefore the particles are both closer and further to the Earth than earlier in thesimulations. The dark blue streak seen above the south-west horizon is some ofthe particles that are seen escaping into heliocentric orbit in Figure 3.9, with closerparticles orbiting the Earth passing in between. The Sun and the Moon are againbelow the horizon for this plot.36Figure 3.13: The bolometric flux of reflected sunlight from particles visibleabove the horizon, almost one year into the simulations. The plot isdivided into bins 1◦ by 1◦ in size, with the color mapping normalizedon a log scale. The Sun and the Moon are both below the horizon.37Chapter 4Discussion4.1 Particle BehaviorThe simulations presented in Chapter 3 for particles with no radiation forces in-cluded (β = 0.0) showed that three quarters of the initial particles eventually col-lided with the Moon or the Earth. A large number of particles fall back towards theMoon during the first week, with days four and five recording almost half of thetotal collisions that occur in the first year. The collisions will likely occur at dif-ferent locations on the Moon than where the particles are released from, impactingat speeds comparable to their ejection velocities. Only 0.2% of the total particlesreleased are found to collide with the Earth during the first year.Particles that avoid collisions can have a number of different pathways. Thesimulated particles are observed to expand away from the Moon in all directionsfor about two days before they begin to spread. Two streams of particles extendaway from the Moon in opposite directions during the first several weeks due toshear, while another population of particles appears to remain in orbit about theMoon. One stream enters into orbit around the Earth inside of the Earth-MoonLagrange points, with some particles occasionally transferring back and forth intolunar orbits. The other stream wraps into a larger Earth orbit, remaining outsidethe Lagrange points, and eventually scattering into the greater cis-lunar environ-ment and beyond. About 15% of the initially released particles are able escape theEarth’s hill sphere, entering into heliocentric orbits similar to the Earth’s. After one38year, roughly 1000 of the initial 10,000 particles remain in orbit about the Earth,while only a few dozen still orbit the Moon. Most particles remain within a lunardistance from the ecliptic in the z-direction, with the largest dispersion reachingjust under 3.5 lunar distances.Once the initial expansion of particles reaches the L1 and L2 Lagrange pointsafter just a couple of days, a continuous presence of material is seen at the twopoints. Both reach their peak densities during this initial expansion at comparablelevels before trending down. After several months, L1 appears to retain higherdensities, likely a result of the larger population of particles in orbit about the Earththan the Moon. Particles eventually start to cross through L3, L4, and L5 after acouple of months, with scattered peaks seen for all three, some briefly reachingvalues similar to L1 and L2. No particles seem to remain in any of these threepoints, as all the density measurements are spikes, nor is residency of particles atthose locations expected based on energetics, at least from the point of view of theCR3BP.As mentioned briefly at the start of Chapter 2, the simulated particles discussedhere can also be thought of as tracer particles representing a larger number of actualparticles. With this assumption, the reported kernel density values could be scaleddirectly by the number of actual particles represented by one simulated particle. Itis also important to note that the results here assume the particles are large enoughto neglect the radiation forces acting on them. We will now see in the next section,smaller particles with non-negligible beta values can exhibit differing behavior.4.2 Radiation ForcesIdentical simulations are run with the sole difference being that the particles arenow subjected to radiation forces throughout the simulations. The particles aregiven a large beta value (β = 0.1), corresponding to particles that are about 10microns in diameter, assuming a density of 1.4 g/cc and Qpr ≈ 1. The Earth and theMoon are both still assumed to have no radiation forces acting on them (β = 0.0).Figure 4.1 shows that the high beta (β = 0.1) particles which initially escapethe lunar gravity do not behave like the escaping particles with no radiation forces(β = 0.0). The influence of the radiation forces is seen immediately, with the par-39XY Particle Distribution in the Moon’s Rotating Frame at 10 DaysFigure 4.1: The xy-positions of the debris particles, Earth, Moon, and four ofthe five Earth-Moon Lagrange points in the Moon’s rotating frame, withthe origin centered on the Moon, at 10 days, for two different β s.ticles stretching away from the Moon in one direction, rather than in two oppositedirections as seen before. In fact, these particles are no longer bound to the Earth-Moon system and begin to expand away from the Sun. Due to the sudden exposureto radiation forces upon leaving the lunar surface, the particles’ heliocentric semi-major axes are instantaneously increased. Thirty days into the simulations, thecis-lunar environment looks noticeably different. Figure 4.2 shows the Earth is un-able to capture the same population of particles found to orbit within the Lagrangepoints in the original (β = 0.0) simulations.The z-distribution of the particles in the lunar orbital plane is seen to increasefor the particles exposed to the radiation forces, shown in Figure 4.3. This trendalso appears for the dispersion of the particles, normal to the Earth’s orbital plane,throughout the simulations as seen in Figure 4.4. Not only are the maximum z-distances of β = 0.1 particles much greater than the β = 0.0 particles, radiationforces disperse the bulk of the particles as well.One year after the particles have been released, the simulation states are quitedifferent. About 20% more of the initial particles subjected to radiation forcesavoid collisions with the Earth and the Moon. Interestingly, almost 700 particles40XY Particle Distribution in the Moon’s Rotating Frame at 30 DaysFigure 4.2: The xy-positions of the debris particles, Earth, Moon, and thefive Earth-Moon Lagrange points in the Moon’s rotating frame, with theorigin at the Earth-Moon center of mass, at 30 days, for two differentβ s.XZ Particle Distribution in the Moon’s Rotating Frame at 30 DaysFigure 4.3: The xz-positions of the debris particles, Earth, Moon, and thefive Earth-Moon Lagrange points in the Moon’s rotating frame, with theorigin at the Earth-Moon center of mass, at 30 days, for two differentβ s.41The Dispersion of Particles in the Z-Direction for One YearFigure 4.4: The maximum z-distance, normal to the Earth’s orbital plane, be-tween any two particles during one year of the simulations, for two dif-ferent β s. As before, the blue line represents the z-distance between thetwo furthest particles, the orange line is the z-distance between the twofurthest particles if 5% of the particles at either extreme are excluded,and the green excludes 25% of the particles at either end.42in total collide with the Earth, far more than the 21 in the original simulations.Almost no β = 0.1 particles remain orbiting the Earth or the Moon one year in, withjust 62 found within the Earth-Moon hill sphere. Instead, these smaller particlesare on heliocentric orbits, as seen in Figure 4.5, trailing far beyond the Earth andspreading radially outward. It should be noted that although the semi-major axis ofthese particles is changed, their orbits will still return through their release pointsand therefore this stream will still be intersecting the Earth’s heliocentric orbit.As expected from these results, the particle kernel density estimations calcu-lated at the Earth-Moon Lagrange points are also impacted by the introduction ofradiation forces. Figure 4.6 shows the L2 Lagrange point initially peaks higherthan L1 for the β = 0.1 particles. This is supported by Figure 4.1, as the particlesare initially drifting away from the Sun, rather than getting captured by the Earth.However, this is likely just due to the orientation of the Earth-Moon system withrespect to the Sun at the time of release, and the opposite could be seen for differentrelease times. Due to the radiation forces quickly clearing out the cis-lunar envi-ronment, more significant decreases are seen over time for the L1 and L2 points.The remaining three Lagrange points actually see more activity for the β = 0.1particles, with peaks seen more frequently and sooner. By the end of one year, all5 points seem to reach comparable density values, much lower than what L1 andL2 experience after one year with no radiation forces.As we can see, radiation forces can significantly alter the short and long-termbehavior of particles released from the Moon. These results do not support findingsfrom Slı´z-Balogh et al. [38], in which they presented results that showed gravita-tional forces dominate for particles greater than 1 µm. They concluded that radi-ation forces do not need to be included for short-term simulations, which we findto be incorrect. Nonetheless, it must be recalled that both the β = 0.0 and β = 0.1simulations reflect real behaviour, as larger particles, which would be ejected dur-ing some lunar operations (as well as natural cratering events), would have aneffective beta of nearly zero, at least for the purposes of these calculations. Futurerelated work, such as those discussed in Section 4.4, is advised to include radiationforces representative of a range of dust/meteoroid sizes and properties.43XY Particle Distribution in the Heliocentric Frame after One YearFigure 4.5: The xy-positions of the debris particles, the Earth, and the Sun,in the heliocentric frame, one year after the particles were released, fortwo different β s.44All 5 Lagrange Point Kernel Density Estimations for One YearFigure 4.6: The kernel density estimations for all five of the Earth-Moon La-grange points during one year of the simulations, for two different β s.454.3 Sky BrightnessPolar projections of the particles across the sky of an Earth based observer showthat at least some particles would be visible above the horizon every night. Largedifferences are seen in the distribution of the particles on the sky, as well as theincoming flux. Differences of over 6 orders of magnitude are seen at times, withthese differences largely driven by the varying distances from the Earth to the par-ticles. A large streak of particles can occasionally be seen stretching from one edgeof the sky to another. Depending on the time of year and latitude of the observer,these particles can also appear just above the horizon along most of the sky. Duringthese times, the particles may be less visible due to any neighbouring geography orcity light pollutionSimilar to the kernel density estimations, the brightness values calculated herecan be scaled by the number of actual particles assumed to be represented byone simulation particle. The flux would scale linearly as it depends on the cross-sectional surface area of the actual particles, which is just the cross-section of oneparticle multiplied by the number of particles. As a reminder, the size of the sim-ulated particles is arbitrarily assumed for the flux calculations in this study. How-ever, as shown in the previous sections, future studies should be performed withappropriate radiations levels. As a result, given a particle’s beta value, one canthen estimate the actual particle size, for a given particle mass density.Additionally, the flux values presented in this study are bolometric estimations,or the flux of sunlight across the full solar spectrum. Although the solar spectrumpeaks in the visible wavelength, radiation is still produced in the ultraviolet andinfra-red portions of the spectrum. As such, more detailed calculations should beconverted from the bolometric flux to visible flux estimations, in order to under-stand the potential brightness values for the human eye. In addition, atmosphericcorrections may be necessary.4.4 Future WorkCertain aspects of this study were intentionally simplistic by design to allow fora more generalized approach, and several changes could be made to investigatemore specific topics. For example, the particle release model could be adjusted46to simulate individual events such as spaceship landings and takeoffs, as well asmeteoroid, cometary, or anthropogenic impact events. As further plans for theeventual sustained return to the Moon become more clear, surface disruptions suchas mining or construction operations could also be studied in greater detail.Using realistic release models will also allow for improvements in makingpredictions for actual particle densities in cis-lunar space and corresponding skybrightness. Depending on the type of material and how it is released, estimationsof the particle size distributions, as well as the albedo and phase functions, couldalso be determined and used to create better estimations of the brightness of the re-flected sunlight. As shown previously, radiation forces impact the results of thesesimulations, so along with the particle sizes, estimations of the particle densitycould allow for radiation forces to be accurately included. Additionally, if the sizeand density of the particles are known, the kernel could be used to estimate themass density of particles at locations of interest. With these more realistic assump-tions, values from the brightness and density estimates could be used to determinethe amount of material that could create disturbances. Examples of this include theamount of material that would generate a certain brightness, or what density/fluxvalues become dangerous to orbiting spacecraft.While the accuracy of these simulations was sufficient for the depth of analysisattempted here, it must be understood that these simulations are still incompletemodels and greater detail could always be included. Currently the large bodiesare modeled to be point sources, but asymmetric mass distributions for the Earthand the Moon create non-trivial perturbations on objects in cis-lunar space. The J2and J4 perturbations can be included with REBOUND and REBOUNDx, but likelywould only be necessary for highly detailed, longer-term simulations. Additionally,electrostatic surface phenomena that can transport material above the surface of theMoon could impact the trajectory of particles.The kernel used to study the density of material at the Lagrange points couldalso be used to study a variety of different orbits and positions. One type of orbitthat would be of great interest to look at is the NRHO family of orbits that will beoccupied by NASA’s CAPSTONE and Gateway missions. The kernel could calcu-late the estimated density or flux of material at any given point in the orbits. Thiswould provide insight about if debris could ever cross trajectories with objects in47these orbits, and if so, what the risk to the spacecraft or station might be. Otherorbits that would be worth investigating include high-Earth orbits such as geo-stationary (GEO) and the orbits of important space telescopes beyond GEO likeTESS and Chandra. The Earth-Sun L2 Lagrange point, which is home to the re-cently launched JWST, is likely safe from this material based on visual observationof Figure 3.9, although it could be worth including just for a safety check.48Chapter 5ConclusionDebris particles ejected from the lunar surface with enough velocity to escape theMoon’s hill sphere can remain in orbit for at least a year, although most are foundin orbit about the Earth rather than the Moon. A large number of particles are ableescape the gravity of both, extending out into heliocentric orbits. The majorityof the initially released particles do not survive a full year, with three out of fourcolliding with the Moon (and occasionally the Earth). Little dispersion is observednormal to the Earth’s orbital plane, as most particles remain within a lunar distance.The L1 and L2 Earth-Moon Lagrange points see a large influx of material withinthe first few days of the simulations, and although the number density of the par-ticles falls off, a constant presence remains throughout. After several months, L1appears to have slightly higher density values recorded than L2, while L3, L4, andL5 begin to see sporadic particles passing through. Occasionally these three pointsreach comparable densities to the other two, but a sustained presence is never seen,indicating no mechanisms discovered for feeding these points, in particular the Ko-rdylewski clouds located at L4 and L5. Additionally, radiation forces can have asignificant impact on the behavior of the ejected particles.The particles that avoid collisions are visible above the horizon for an Earthbased observer almost every night of the simulations. At times these particles cancreate a large stream extending across the sky, while other times a ring will formaround the horizon, potentially obscured from the observer. While the necessaryassumptions that were made for this study strongly influence the density and bright-49ness values reported, insight can be gathered from the results, and the techniquescan be applied in the future to a number of specific problems.As we return to the Moon, we must be careful to act sustainably in order toavoid any potential harmful consequences to our actions. 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Thesis/Dissertation
eng
Astronomy
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Boley, Aaron
Investigations of the dynamical evolution of lunar ejecta
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