UNIVERSIDADE ESTADUAL PAULISTA "JÚLIO DE MESQUITA FILHO" CAMPUS DE GUARATINGUETÁ GIULIA VALVANO DO PRADO RIBEIRO Formation and stability around irregular asteroids Guaratinguetá/Brazil 2024 GIULIA VALVANO DO PRADO RIBEIRO Formation and stability around irregular asteroids Doctoral Thesis presented to the Graduate Program in Physics and Astronomy of the São Paulo State University "Júlio de Mesquita Filho" as a partial requirement to obtain the degree of Doctor in Physics and Astronomy Universidade Estadual Paulista "Júlio de Mesquita Filho" Graduate Program in Physics and Astronomy Supervisor: Prof. Dr. Othon Cabo Winter Co-supervisor: Prof. Dr. Rafael Sfair Guaratinguetá/Brazil 2024 R484f Ribeiro, Giulia Valvano do Prado Formation and stability around irregular asteroids / Giulia Valvano do Prado - Guaratinguetá, 2024. 158 f : il. Bibliografia: f. 131-144 Tese (Doutorado) – Universidade Estadual Paulista, Faculdade de Engenharia e Ciências de Guaratinguetá, 2024. Orientador: Prof. Dr. Othon Cabo Winter Coorientador: Prof. Dr. Rafael Sfair 1. Asteroides - Órbitas. 2. Astronomia. 3. Sistema solar. I. Título. CDU 523.44(043) Luciana Máximo Bibliotecária/CRB-8/3595 unesp UNIVERSIDADE ESTADUAL PAULISTA CAMPUS DE GUARATINGUETÁ GIULIA VALVANO DO PRADO RIBEIRO ESTA TESE FOI JULGADA ADEQUADA PARA A OBTENÇÃO DO TÍTULO DE “DOUTORA EM FÍSICA” I dedicate this thesis to myself and the academic community. For you, for all of us. CURRICULAR DATA GIULIA VALVANO DO PRADO RIBEIRO BIRTH 20/01/1997 - Taubaté FILIATION Edson Francisco Ribeiro Juliene de Fátima do Prado Ribeiro 2015 - 2019 Bachelor of Science in Mathematics Education Faculdade de Engenharia e Ciências de Guaratinguetá Universidade Estadual Paulista "Júlio de Mesquita Filho" 2019 - 2024 Doctorate in Physics and Astronomy Faculdade de Engenharia e Ciências de Guaratinguetá Universidade Estadual Paulista "Júlio de Mesquita Filho" Acknowledgements I would like to express my deepest gratitude to my parents, Juliene and Edson, and my godparents, Eliete and Antonio, for their support, encouragement, and sacrifices throughout my academic journey. I appreciate the partnership of my sister, Luisa Valvano, for making my life more challenging. I also would like to express my eternal gratitude to my best friends Toddy and Cacau for helping me overcome all the problems and challenges of my quotidian life. A heartfelt appreciation goes to my esteemed advisors, Othon Winter and Rafael Sfair, for their mentorship and intellectual insights. Their dedication to advising me has been instrumental in shaping my academic and professional growth. I am privileged to have had the opportunity to learn from such marvellous scholars. I extend my thanks to my colleagues and friends at São Paulo State University, who have been sources of inspiration and collaboration. In special a huge thank you to Tamires Moura, André Amarante and Raí Machado. Finally, I would like to express my gratitude to everyone who has played a role in the completion of this doctoral thesis. Your collective support has been invaluable, and I am deeply grateful to have such an amazing network of individuals in my life. This study was financially supported by: • Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001 • Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) - Process: 2016/24561-0, 2017/26855-3, 2019/23963-5 and 2022/01678-0 • Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) - Process: 305210/2018-1 "I am burdened with glorious purpose" Loki Laufeyson Abstract Asteroids are rocky or metallic bodies leftovers from the early Solar System. Their sizes range from a few meters to hundreds of kilometers and have irregular formats. Some of them cross the orbit of Earth being called Near Earth Asteroids (NEAs). A set of NEAs also represent a potential threat to the Earth and, for this reason, are called potentially hazardous asteroids (PHAs). Due to their potential threat to the Earth or unique characteristics, some asteroids are targets of space missions such as the asteroids Bennu, Apophis and the triple system 2001 SN263. For this reason, this work aims to better understand the dynamic of these asteroids, studying their topography, the stability of their neighbourhood and even their formation. Since Apophis will have its closest encounter with Earth in 2029, we study its geometric and geopotential topography to comprehend the dynamics on its surface and also the stability in its nearby. Then, we study how the Earth will affect its surface and neighbourhood through the slope angle and numerical simulation with massless particles. We extended the studies to track the destination of the particles that were ejected during the encounter with Earth. May they cause a meteor activity on Earth? Focusing on the ASTER mission, we also study the topography and the stability of the neighbourhood of the triple system 2001 SN263. In the first publication, we make a detailed analysis of their geometric and geopotential topography, and initial stability studies around the largest satellite of the system and numerically integrate a cloud of particles to comprehend the preferential regions where particles may fall. In a second work, we decided to extend the stability studies and make a detailed analysis. We consider the three asteroids and the particles in the simulations. We also assumed the irregular shape of them and the perturbation of the solar radiation pressure. During these studies, we were excited with the discovery of the first quadruple asteroid system. Thus, we also analysed their dynamical stability in its nearby since it was reported uncertainties on the orbital elements of the third moonlet discovered. Thrilled by the studies including asteroids with spinning-top shapes as the main body of the triple system 2001 SN263, we decided to investigate the possible scenarios and conditions for their formation. An initial study using Smooth Particle Hydrodynamics and Finite Element Method was made to analyse their formation and the structure failure of their interiors. However, these works are still under development. The findings of all these studies contributed to a comprehensive understanding of the dynamics and physical properties of asteroids and multiple asteroid systems. Our results may contribute to future space missions and a better understanding of the asteroid dynamics. Keywords: NEAs. Dynamical stability. Numerical simulations. Asteroids. Multiple as- teroid systems. Spinning-top asteroids. Smooth Particle Hydrodynamics. Finite Element Method. Resumo Asteroides são corpos rochosos ou metálicos remanescentes do início do Sistema Solar. Suas dimensões variam de alguns metros a centenas de quilômetros e possuem formatos irregulares. Alguns deles cruzam a órbita da Terra, sendo chamados de Near Earth Asteroids (NEAs). Alguns NEAs representam uma ameaça potencial à Terra e, por esse motivo, são chamados de asteroides potencialmente perigosos (PHAs, do inglês Potentially Hazardous Asteroids). Por esse motivo e por suas características únicas, alguns asteroides são alvos de missões espaciais, como os asteroides Bennu, Apophis e o sistema triplo 2001 SN263. Por esse motivo, este trabalho tem como objetivo entender melhor a dinâmica desses asteroides, estudando sua topografia, a estabilidade de seu entorno e até mesmo sua formação. Como Apophis terá seu encontro próximo com a Terra em 2029, estudamos sua topografia geométrica e geopotencial para compreender a dinâmica em sua superfície e também a estabilidade em suas proximidades. Em seguida, estudamos como a Terra afetará sua superfície e sua vizinhança por meio de estudos de sua topografia e simulação numérica com partículas sem massa. Estendemos os estudos para rastrear o destino das partículas que foram ejetadas durante o encontro com a Terra. Elas poderiam causar uma atividade meteorítica na Terra? Focando na missão ASTER, também estudamos a topografia e a estabilidade do entorno do sistema triplo 2001 SN263. Na primeira publicação, fazemos uma análise detalhada de sua topografia geométrica e geopotencial. Também realizamos estudos iniciais de estabilidade ao redor do maior satélite do sistema. Para compreender as regiões preferenciais onde as partículas podem cair nos corpos do sistema, integramos numericamente uma nuvem de partículas em cada um deles. Em um segundo trabalho, decidimos estender os estudos de estabilidade e fazer uma análise detalhada. Consideramos os três asteroides e as partículas nas simulações. Também assumimos a forma irregular deles e a perturbação da pressão de radiação solar. Durante esses estudos, ficamos extasiados com a descoberta do primeiro sistema de asteroides quádruplo. Assim, também analisamos sua estabilidade dinâmica em suas proximidades, já que foram relatadas incertezas nos elementos orbitais do terceiro satélite descoberto. Empolgados com os estudos que incluem asteroides com formas de "pião" como corpo principal do sistema triplo 2001 SN263, decidimos investigar os cenários e condições possíveis para a formação de corpos com esse formato. Um estudo inicial usando Hidrodinâmica de Partículas Suaves (SPH, do inglês Smooth Particle Hydrodynamics) e o Método dos Elementos Finitos foi realizado para analisar sua formação e a falha estrutural de seus interiores. Os resultados de todos esses estudos contribuíram para uma compreensão abrangente da dinâmica e propriedades físicas de asteroides e sistemas de asteroides múltiplos. Nossos resultados podem contribuir para futuras missões espaciais e uma melhor compreensão da dinâmica dos asteroides. Palavras-chave: NEAs. Estabilidade. Simulações Numéricas. Asteroides. Asteroides com formato "pião". Smooth Particle Hydrodynamics. Elementos Finitos. List of Figures Figure 1 – Representation of the NEAs orbital categories: Amor, Atira, Apollo and Aten adapted from Araújo (2011). The yellow, blue and black dots represent, respectively, the Sun, the Earth and an asteroid. . . . . . . . 17 Figure 2 – A representation of an asteroid region with the total acceleration (a1 and a2) and normal (n1 and n2) vectors on each asteroid decline. Their respective slope angles are also shown. A representation of the gravi- tational (Fg) and centrifugal (Fw) perturbations are shown, as well as, the total applied force (Fr). (a) represents the asteroid region in a time t1, while (b) in a time t2. . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Figure 3 – Slope angle considering the Earth’s perturbation at the moment of maximum encounter in 2029 mapped over the Apophis surface under different views and for the small and large densities. . . . . . . . . . . . 25 Figure 4 – Map of the gravitational and rotational contribution, and the geopoten- tial across the surface of Alpha, Beta, and Gamma, respectively. . . . . 59 Figure 5 – The periodogram for the orbital periods of (a) Beta and (b) Gamma. . 63 Figure 6 – A ring of particles coloured according to their lifetime in the xoy plane. Alpha and Gamma are represented by their irregular shape in grey. . . 63 Figure 7 – Representation of six spinning-top-shaped asteroids in scaled sizes. . . . 109 Figure 8 – Schematic of stress-strain diagram adapted from Teo et al. (2014). The green region represents the elastic region, while the blue one the plastic region. Young’s modulus represents the slope of this linear region defined by Hooke’s law. Yield strength is the point where the deformation starts to be plastic. Ultimate Strength is the maximum value of stress that a material can achieve. The fracture point represents the break point of the material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Figure 9 – Schematic of stress-strain diagram for glass, steel and rubber materials. Adapted from Engineer (2023). . . . . . . . . . . . . . . . . . . . . . . 111 Figure 10 – Representation of the domain of particle a with kernel function W . The distance between the particle of interest a and other particle b is r. Figure adapted from Duckworth et al. (2021) . . . . . . . . . . . . . . 113 Figure 11 – Representation of the tree method showing the regions divided into octants. The red dot represents the particle of interest, the green ones represent far particles and the pink, the nearest ones. The red particle will interact with the centre of mass of each octant. . . . . . . . . . . . 115 Figure 12 – Exemplification of a sphere of 2 km created by the SEAGen algorithm. 116 Figure 13 – Snapshots from different simulations showing the initial condition and outcome for a body after the period of simulation. The colours represent the velocity magnitude. The simulations were made considering the Hydro case. The snapshots are taken at the beginning and the end of the simulation. The xyz orientation is shown in each snapshot. . . . . . 117 Figure 14 – Time evolution of the semi-axes a : b : c of a body simulated by the Hydro case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Figure 15 – Snapshots showing different periods of simulation for each simulated case. The colours represent the velocity magnitude. The simulations were made considering the Solid case. The xyz orientation is shown in each snapshot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Figure 16 – Time evolution of the semi-axes a : b : c of a body simulated by the Solid case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Figure 17 – The representation of three bodies shape, being them: (a) 2001 SN263 Alpha, (b) our outcome considering a spin period of 3.0 hours and (c) 2.5 hours. The shape of Alpha was smoothed to not evidence of the faces originating from the polyhedron model. The view is set to −y direction similar to Fig. 15. . . . . . . . . . . . . . . . . . . . . . . . . 122 Figure 18 – Representation of some types of elements in FEM. Figure adapted from Engineer (2023). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Figure 19 – Representation of a two-dimensional beam and its degrees of freedom. . 124 Figure 20 – FEM mesh for the asteroid Itokawa considering 4247 nodes and 2347 elements. The mesh is shown on (a) the asteroid’s surface and (b) its interior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Figure 21 – Process of applying constraints (coloured as blue) and loads (coloured as blue as red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 List of Tables Table 1 – The physical properties of some spinning-top-shaped asteroids organized in a descending way of size. . . . . . . . . . . . . . . . . . . . . . . . . . 108 List of abbreviations and acronyms au Astronomical Units FEM Finite Element Method MOID Minimum Orbit Intersection Distance NEAs Near Earth Asteroids PHA Potentially Hazardous Asteroid SPH Smooth Particle Hydrodynamics Contents 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 THE ASTEROID APOPHIS . . . . . . . . . . . . . . . . . . . . . . 21 2.1 Contextualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 APOPHIS – effects of the 2029 Earth’s encounter on the surface and nearby dynamics: The Making-of . . . . . . . . . . . . . . . . . . 22 2.2.1 The Making-of: APOPHIS - May a meteor activity happen on Earth after the 2029 closest approach? . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.2 APOPHIS – An Extension . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3 APOPHIS – effects of the 2029 Earth’s encounter on the surface and nearby dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4 Erratum: APOPHIS – effects of the 2029 Earth’s encounter on the surface and nearby dynamics . . . . . . . . . . . . . . . . . . . . . . . 46 2.5 APOPHIS - May a meteor activity happen on Earth after the 2029 closest approach? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3 THE TRIPLE SYSTEM 2001 SN263 . . . . . . . . . . . . . . . . . 56 3.1 Contextualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2 2001 SN263 - a topography study and an initial stability analysis: The Making-of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3 2001 SN263 - the contribution of their irregular shapes on the neighborhood dynamics: The Making-of . . . . . . . . . . . . . . . . 60 3.3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4 Asteroid triple-system 2001 SN263: surface characteristics and dy- namical environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.5 2001 SN263 - the contribution of their irregular shapes on the neighborhood dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4 ELEKTRA - THE FIRST QUADRUPLE SYSTEM . . . . . . . . . . 95 4.1 Contextualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2 Elektra Delta - on the stability of the new third moonlet: The Making-of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.3 (130) Elektra Delta - on the stability of the new third moonlet . . . 98 5 SPINNING-TOP SHAPE ASTEROIDS: UNDER DEVELOPMENT 105 5.1 Basics concepts of solid body mechanics . . . . . . . . . . . . . . . . 108 5.2 Smooth Particle Hydrodynamics . . . . . . . . . . . . . . . . . . . . . 111 5.2.1 Hydrodynamic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.2.2 SPH test simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.3 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.3.1 Ansys test simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6 FINAL REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 A THE DYNAMICAL STRUCTURE OF A HYPOTHETI- CAL DISC OF PARTICLES AROUND THE ASTEROID 99942 APOPHIS . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 17 1 Introduction Asteroids are rocky or metallic bodies ranging in size from a few meters to hundreds of kilometers in diameter. They orbit the Sun and can provide valuable insights into the formation and evolution of our Solar System. Asteroids preserve information about the Solar System’s past and can assist scientists in understanding the evolution of planets and other celestial bodies. Most asteroids are located in the main asteroid belt, between the orbits of Mars and Jupiter. However, some asteroids migrate closer to the Sun and consequently start crossing the orbits of Mars, Earth, Venus, and Mercury. This migration occurs due to gravitational perturbations in the Solar System, for example, the perturbations caused by the orbits of Jupiter and Saturn. The asteroids with a perihelion that passes within 1.3 au of the Sun and an aphelion largest or equal to 0.983 au are classified as NEAs (Near Earth Asteroids)1. Figure 1 – Representation of the NEAs orbital categories: Amor, Atira, Apollo and Aten adapted from Araújo (2011). The yellow, blue and black dots represent, respec- tively, the Sun, the Earth and an asteroid. There are subgroups of NEAs based on their orbital characteristics. Figure 1 shows the orbital representation of these subgroups. Amor asteroids have orbits entirely outside the Earth’s orbit and they cross the orbit of Mars, while Atira has orbits within the Earth’s orbit and does not cross the orbit of Mars. The Apollo and Aten asteroids have orbits crossing the Earth’s orbit. However, the orbits of the Apollo group have larger semi-major axis than the Earth’s orbit and Aten, smaller. The first discovered NEA was the Amor asteroid 433 Eros in 1898 (Berberich, 1898). Eros also was the target of the mission NEAR-Shoemaker (Veverka et al., 2000; Prockter et al., 2002), the first one to orbit and land on an asteroid. Discovered in 1998 (MPEC 1998-S45), the Apollo asteroid Itokawa was the target of the Hayabusa spacecraft, the first mission that collected samples from the asteroid’s surface (Fujiwara et al., 2006; Abe et al., 2006; Tsuchiyama et al., 2011; Yoshikawa et al., 2015). In 2019, the Hayabusa 2 spacecraft performed a touchdown on the Apollo asteroid (162173) Ryugu, samples were 1 https://www.iau.org/public/themes/neo/ visited on 20 November 2023 https://minorplanetcenter.net//mpec/J98/J98S45.html https://www.iau.org/public/themes/neo/ Chapter 1. Introduction 18 collected and reached Earth in December 2020 (Kawaguchi et al., 2008; Müller et al., 2017; Watanabe et al., 2017; Sawada et al., 2017; Yada et al., 2022). The OSIRIS-REx mission also collected a sample from an asteroid, the Apollo asteroid Bennu (Lauretta et al., 2015, 2017; Beshore et al., 2015). The sample reached the Earth on Sept. 24, 2023, and it is now being studied (Polit et al., 2023; Lauretta et al., 2023). The mission did not stop after collecting samples from Bennu, it will continue on as OSIRIS-APEX. The goal of OSIRIS-APEX is to study physical changes to the Aten asteroid Apophis after its 2029 close encounter with Earth (DellaGiustina et al., 2022, 2023). The Main-belt asteroid 243 Ida was the first asteroid found to have its own natural satellite (Disasters, 1994; Kerr, 1994; Chapman et al., 1995). Years later, the 87 Sylvia system was discovered to be the first triple system detected (Marchis et al., 2005). In 2021, the (130) Elektra was reported to be the first observed quadruple asteroid system (Berdeu et al., 2022). The studies of asteroid systems may provide valuable perspectives on the comprehension of our Solar System. Their formation can occur through collisions, accretion, disruption, or various other processes (Walsh and Richardson, 2006; Walsh et al., 2008; Vokrouhlickỳ and Nesvornỳ, 2008; Jacobson and Scheeres, 2011; Walsh et al., 2012; Jacobson et al., 2013; Margot et al., 2015). Understanding the formation of multiple asteroid systems may increase our knowledge about their dynamics, compositions, interior structure, bulk density, and the mechanics involved in the evolutionary processes of asteroids and our Solar System. The studies of asteroids are also important due to their potential threat to the Earth (Azadmanesh et al., 2023). Asteroid collisions with our planet constitute a real danger. Thus, studying them closely is crucial for developing strategies for protection against hazardous impacts. Within the group of NEAs, there are asteroids considered potentially hazardous, or PHAs. In particular, any asteroid with an Earth Minimum Orbit Intersection Distance (MOID) of 0.05 au or less is classified as PHA2. The asteroid Apophis is a PHA that has been continuously tracked ever since a significant potential threat to the Earth was reported (Larsen and Descour, 2004; Chesley, 2005; Giorgini et al., 2008). Using radar observation data from the Arecibo radio telescope obtained in mid-2005-2006, Giorgini et al. (2008) predicted closer approaches of Apophis with the Earth in the following years. The closest encounter will occur on 13 April 2029 and Apophis will approach the Earth at a distance of about 6 Earth radii (Giorgini et al., 2008). Considering the potential threats of asteroids similar to Apophis, the first planetary defense mission was planned3. The Double Asteroid Redirection Test Mission (DART) has as its goal the on-orbit demonstration of asteroid deflection (Cheng et al., 2018). The target of the mission was the PHA binary system Didymos and its moon Dimorphos. The impact’s effects on Dimorphos’ orbit will be measured and the Hera mission4 will further 2 https://cneos.jpl.nasa.gov/about/neo_groups.html visited on 20 November 2023 3 https://dart.jhuapl.edu/Mission/Planetary-Defense.php visited on 20 November 2023 4 https://www.esa.int/Space_Safety/Hera/Hera visited on 20 November 2023 https://cneos.jpl.nasa.gov/about/neo_groups.html https://dart.jhuapl.edu/Mission/Planetary-Defense.php https://www.esa.int/Space_Safety/Hera/Hera Chapter 1. Introduction 19 assess the orbital changes in Dimorphos after the impact. A multiple system of asteroids offers a range of observational opportunities and exploration into their environment. Thus, the PHA system (153591) 2001 SN263 was chosen as a target of the ASTER mission (Sukhanov et al., 2010). The ASTER mission will be the first Brazilian deep space mission. One of its aims is to investigate the triple asteroid system’s origin since it is formed from two bodies with similar densities (the central body and the larger moon) and a third one with a different and larger density (smaller moon). This difference in densities may be a clue to the internal structure of the asteroids. For example, a parent body with a density similar to the central body and the larger moon may have been impacted by an impactor with a density of the smaller moon. Certainly, there are multiple factors and parameters to analyze. However, this difference may indicate the formation of (153591) 2001 SN263 by an impacting agent with a density different from the parent body. Thus, the parent body’s region of the core that was exposed by the collision with the impactor may be observed and, consequently, information about its internal structure may be obtained. The composition of an asteroid can be estimated by the combination of observational techniques, remote sensing, and sometimes direct sample return missions (Zellner et al., 1977; Bowell et al., 1978; Degewij et al., 1979; Tholen, 1989; Gaffey et al., 1993b; Nelson et al., 1993; Pravec et al., 2012; Kitazato et al., 2019). Different materials absorb and reflect light at characteristic wavelengths, allowing researchers to infer the presence of certain compounds. Asteroids also exhibit an extensive range of albedos and their variation indicates differences in their surface properties and compositions. The asteroid’s albedo refers to the ability to reflect sunlight, determining how bright or dark its surface appears. An albedo near zero represents a completely black (non-reflective) surface, while near one represents a perfectly reflective surface that reflects all incident light5. The albedo of an asteroid is influenced by factors such as its mineral composition, surface texture, and the presence of reflective or absorbing materials. The majority of asteroids can be classified into three main categories: C-type, S-type, and M-type (Bowell et al., 1978; Bottke Jr et al., 2002). The carbonaceous (C- type) represents about 75% of known asteroids. They are characterized by their very dark appearance with an albedo ranging from 0.03 to 0.09. The C-type asteroids are rich in carbon and organic compounds and found in the outer regions of the main asteroid belt(Vernazza et al., 2017; Yada et al., 2022). The silicate asteroids (S-type) constitute about 17% of known asteroids and an albedo between 0.10 and 0.22. These asteroids are composed of metallic iron mixed with iron- and magnesium-silicates and they are predominant in the inner regions of the asteroid belt (Gaffey et al., 1993a; Chapman, 1996; Nakamura et al., 2011). The metallic (M-type) asteroids are relatively bright, with an 5 https://nssdc.gsfc.nasa.gov/planetary/text/asteroids.txt visited on 20 November 2023 https://nssdc.gsfc.nasa.gov/planetary/text/asteroids.txt Chapter 1. Introduction 20 albedo ranging from 0.10 to 0.18. Their composition is primarily dominated by metallic iron and they are located in the middle region of the main asteroid belt (Lupishko and Belskaya, 1989; Fornasier et al., 2010; Ockert-Bell et al., 2010). Despite the variety of compositions, asteroids also have a diversity of irregular shapes, reflecting their diverse origins and physical processes that have shaped them over time. The study of their shapes is fundamental for understanding their formation processes, internal structures, and the dynamics of the Solar System. Some asteroids have been observed to exhibit a spinning-top or rotating shape, with pronounced bulges around their equators. The following asteroids have a spinning-top shape: the central bodies of the systems 2001 SN263 (Becker et al., 2015), 1994 CC (Brozović et al., 2011), 1999 KW4 (Ostro et al., 2006), 2000 DP107 (Naidu et al., 2015), Didymos (Naidu et al., 2020), 2004 DC (Taylor et al., 2008) and the asteroids 1950 DA (Busch et al., 2007), Ryugu (Watanabe et al., 2019), Bennu (Scheeres et al., 2016), 2008 EV5 (Busch et al., 2011), Phaeton (Marshall et al., 2019), 1991 VH and 1996 FG3 (Scheeres et al., 2021). Considering the importance of asteroid studies, our thesis aims to explore the nearby environment stability of asteroids and multiple asteroid systems. Our thesis compiles a set of studies with interesting results. We show research papers and projects that were developed during our doctorate and other ones that are under development. Each chapter of our thesis presents independent subjects from each other. A critical analysis of the results was made for the published papers. Hence, this work is structured as follows: • In Chapter 2, a study of Apophis topography, stability, and how the 2029 encounter with the Earth would affect its topography and the stability of its nearby is presented. The fate of the ejected particles during the 2029 encounter with Earth is investigated. The idea is to analyse if a meteor activity could happen on Earth. Two papers are also presented in this chapter. • In Chapter 3, the analysis of the topography on the triple system 2001 SN263 surface and the stability around its nearby is presented. The two published papers about these studies are also presented. • Chapter 4 presents the stability analysis of the first quadruple system of asteroids and the published paper about it. • Chapter 5 shows a contextualization of the spinning-top-shaped asteroids, basic concepts of solid mechanics, Smooth Particle Hydrodynamics, and Finite Element Method. These studies are still under development and some preliminary results are presented. • Finally, Chapter 6 discusses the main findings and important points of this thesis. 21 2 The asteroid Apophis 2.1 Contextualization The potential threat of an asteroid impact on Earth has always been a concern for humanity. The Earth has been impacted by asteroids many times since its formation and some of these impacts have changed the Earth forever1. The Cretaceous extinction occurred about 66 million years ago and is one of the most famous catastrophic events in the history of Earth. Paleontologists claimed indications that an asteroid of about 5 km in radius hit the Earth causing the dinosaur extinction (Brusatte et al., 2015; Chiarenza et al., 2020). The impact left a crater of about 180 km wide in Mexico. The 1908 Tunguska event2 caused a considerable change in its vicinity, knocking down about 80 million trees with an explosion more powerful than Hiroshima’s atomic bomb. However, it is not clear if the impactor was a 40-meter-radius asteroid or a comet (Gladysheva, 2020). A recent event occurred in 2013 in Russia, the Chelyabinsk event was considered the most airburst on Earth since the Tunguska event (Popova, 2021). It was caused by an asteroid of about 20 m in diameter and, luckily, its explosion was in the air avoiding a larger disaster. It is not a secret that the Earth will be hit again by a sizeable asteroid, the big problem is when. The asteroids with orbits that can come near the Earth are called Potentially Hazardous Asteroids. According to the Minor Planet Center3, there are about 2,300 PHAs detected at the moment of this thesis writing. The list of PHAs is daily updated. PHAs need to be constantly searched and monitored to ensure any of them will not impact and devastate the Earth. There is a risk impact list that summarizes asteroids with potential threats to the Earth in the future4. For this reason, the discussion about planetary defense, searching and tracking of asteroids that may pose an impact on Earth is important. The asteroid Apophis was discovered in 2004 (Garradd, 2004; Smalley et al., 2004) and classified as PHA. Considering the initial data from its observation, a probability larger than 2% of collision with Earth in 2029 was estimated, but it was diminished after some studies (Larsen and Descour, 2004; Chesley, 2005). Apophis orbits continued to be observed, and with additional data, the orbital uncertainties were reduced (Giorgini et al., 2008). After observations in 2021, Apophis was removed from the risk list after remaining in it for more than 15 years. Despite that, the encounter with the Earth on April 2029 1 https://www.planetary.org/notable-asteroid-impacts-in-earths-history visited on 24 November 2023 2 https://www.nasa.gov/history/115-years-ago-the-tunguska-asteroid-impact-event/ visited on 24 No- vember 2023 3 https://www.minorplanetcenter.net/iau/Dangerous.html visited on 24 November 2023 4 https://cneos.jpl.nasa.gov/sentry/ visited on 24 November 2023 https://www.planetary.org/notable-asteroid-impacts-in-earths-history https://www.nasa.gov/history/115-years-ago-the-tunguska-asteroid-impact-event/ https://www.minorplanetcenter.net/iau/Dangerous.html https://cneos.jpl.nasa.gov/sentry/ Chapter 2. The asteroid Apophis 22 continued to be the closest one for the next years. There are some studies analyzing what the 2029 encounter could cause on Apophis. Apophis spin state could change drastically according to Scheeres et al. (2005). However, Souchay et al. (2018) reported that these changes actually could be small and the larger changes may be on Apophis’ obliquity and precession in longitude. On its surface, landslides and migration of material may happen (Binzel et al., 2010) but the tidal effects may only trigger small landslides (Yu et al., 2014). Hence, this research aims to analyse how the Earth will affect Apophis and its nearby in 2029. The work entitled APOPHIS – effects of the 2029 Earth’s encounter on the surface and nearby dynamics by Valvano et al. (2022b) published in Monthly Notices of the Royal Astronomical Society (MNRAS) presents a study about Apophis’ surface and its nearby environment (Section 2.3). We consider a scenario without the perturbation of the Earth and the 2029 encounter scenario. The goal is the compare both scenarios and analyse how the Earth and the 2029 trajectory may affect Apophis’ surface and nearby. An erratum to this paper was produced as a consequence of a misimplementation of a loop of our program. This program calculates the energy of the particles to analyse if a particle was ejected. The results of the ejected particles were underestimated and the corrected results of a set of simulations were published on Erratum: APOPHIS – effects of the 2029 Earth’s encounter on the surface and nearby dynamics"also in MNRAS (Section 2.4). Since the majority of the particles were ejected in the hypothetical disc during the 2029 encounter, we wondered "What could happen with these ejected particles from Apophis gravitational field? Could they collide with the Earth? Could they generate a meteor activity on the Earth or even the Moon?". Thus, considering these questions, we analyse the fate of these ejected particles and we produce the paper entitled APOPHIS - May a meteor activity happen on Earth after the 2029 closest approach? (Section 2.5). The paper was submitted to the Monthly Notices of the Royal Astronomical Society and we are awaiting reviewer reports. 2.2 APOPHIS – effects of the 2029 Earth’s encounter on the surface and nearby dynamics: The Making-of Considering that Apophis will have one of the closest encounters between an asteroid and the Earth, we decided to study it. First, we searched for previous studies about Apophis, its discovery, physical parameters, shape models and possible effects that Earth may cause on its nearby environment and surface. During the search for detailed information about Apophis, we noticed that its size and density were not well-defined, consequently, its mass was not well known. Cellino Chapter 2. The asteroid Apophis 23 et al. (2006) estimated Apophis size to be around 270 m using data from an observing campaign from 2006. Müller et al. (2014) estimated a size of 375 m from thermal infrared observations, while Licandro et al. (2015) with the same approach, estimated a size of 370-393 m. Considering radar observations from Goldstone and Arecibo observatories, Brozović et al. (2018) evaluated Apophis to be about 340 m in size. Since Apophis has a spectral type similar to LL ordinary chondrite and a porosity of about 4% to 62% (Binzel et al., 2009), Apophis’ density is reported to range from 1.29 g·cm−3 to 3.46 g·cm−3. To study more realistically the surface and the nearby environment around Apophis, we need a polyhedral shape model of it. From a photometric observational campaign, Pravec et al. (2014) constructed an unscaled polyhedron shape model for Apophis with 2024 triangular faces and 1014 vertices. They also reported that Apophis is a tumbling asteroid having a well-known precession period of 263±6 h and a retrograde rotational period of 27.38±0.07 h. Brozović et al. (2018) improved the late Apophis shape model using their new radar data and created a more precise polyhedric shape model with 3996 triangular faces and 2000 vertices. We decided to assume the Pravec et al. (2014) shape model and Brozović et al. (2018) size for Apophis. It is interesting to point out that both the density and the shape model of Brozović et al. (2018) were the most recent that we had at the paper writing time. However, we used the Pravec et al. (2014) shape model since the Brozović et al. (2018) one was not available. Thus, all of our results are valid for the Pravec et al. (2014) format. We take three density values into account in our work: 1.29 g·cm−3, 2.2 g·cm−3, and 3.5 g·cm−3. The geometric altitude computes the distance between the geometric central point of the body and the barycenter of each triangular face of the polyhedron. The smaller altitude computed was about 131 m and it is called "sea-level"(Scheeres et al., 2016). Apophis has an asymmetric ellipsoidal format with the largest altitude on the equatorial region extremities (Fig. 1 of Valvano et al. (2022b)). The sea levels are preferentially located at the poles. The north pole presents a restricted and smaller region of sea levels, while the south pole has a larger region of sea levels. Since the volume of Apophis was preserved for the estimation of its mass, the geometric altitude is valid for the three density models. The slope is the angle that shows how inclined the region is concerning the local acceleration vector. It helps us understand how the migration of free particles may be on the asteroid’s surface. The minimum slope is larger than 0.5◦ (Fig. 3 of Valvano et al. (2022b)), these minimum values are distributed along all the asteroids’ surfaces. When a surface has a zero slope, it means that there is an equilibrium on the surface. The maximum slopes are located in the extremities of the equatorial regions and are smaller than 37◦. Since the slope value is smaller than 90◦ the behaviour of the slope angle could Chapter 2. The asteroid Apophis 24 be associated with the repose angle. The angle of repose represents the maximum inclination at which the material can be piled without undergoing collapse. For geological materials, the repose angle is about 35◦–40◦ (Lambe and Whitman, 1969; Mitchell et al., 1974; Al-Hashemi and Al-Amoudi, 2018). Hence, considering the slope distribution, we can estimate where landslides may happen and where the free material may accumulate. The regions where the slope is larger than the repose angle may be susceptible to landslides. Thus, considering a repose angle of 35◦, the small regions in the equatorial extremities may generate smaller landslides. However, if the angle of 40◦ was assumed, the landslides will not happen. If landslides happen, where the free material may go? To discuss the migration of free material, we need to analyse the direction of the tangential acceleration on the asteroids’ surface. Figure 4 of Valvano et al. (2022b) shows the map of slope distribution but with the tangential acceleration vectors for the density of 2.2 g·cm−3. Since there is a small difference in the slope behaviour of the three models of densities considered, we decided to show only one map on the paper. Note that the tangential acceleration vectors point from the larger slope to the smaller slopes. Thus, the free particles could migrate from the regions with slopes larger than the repose angle to the near-equilibrium regions (slope near zero). What may happen with Apophis’ surface during its closest approach with Earth? May the Earth’s perturbation trigger a landslide? Let us consider an applied total force (Fr) in a random asteroid region (Fig. 2). During a time t1, the region suffers the perturbation of the centrifugal (Fw) and gravitational (Fg) forces. The slope angles in two different regions are 150◦ and 114◦ (Fig. 2(a)). However, in a time t2 the gravitational perturbation (Fg) deceased due to an external perturbation such as Earth’s tide. Then, the slope in a region increases to 159◦, while in another region, it decreases to 105◦ (Fig. 2(b)). Thus, an external perturbation could be able to increase and decrease slopes on the asteroid’s surface. Considering this, we decided to compute the perturbations caused by the Earth into the total acceleration of each face on Apophis’ surface. Hence, we would have a slope with the tides caused by Earth on the 2029 encounter. In the paper, we presented the ∆slope map, and here we extend this analysis by presenting the slope map considering the Earth’s perturbation at the moment of maximum encounter in 2029 (Fig. 3) for the smaller and larger densities. This map is important to show that the Earth causes a small variation in the slope range. Thus, the slope pattern during the encounter looks similar to the one without the Earth’s perturbation (Fig. 3 of Valvano et al. (2022b)). The Earth’s perturbation may intensify the landslides on the extremities of the equatorial region of Apophis. However, this phenomenon will continue to be small, without drastic changes. To compute the slope variation caused by the Earth, we consider the difference Chapter 2. The asteroid Apophis 25 (a) (b) Figure 2 – A representation of an asteroid region with the total acceleration (a1 and a2) and normal (n1 and n2) vectors on each asteroid decline. Their respective slope angles are also shown. A representation of the gravitational (Fg) and centrifugal (Fw) perturbations are shown, as well as, the total applied force (Fr). (a) represents the asteroid region in a time t1, while (b) in a time t2. (a) ρ = 1.29 g·cm−3 (b) ρ = 3.5 g·cm−3 Figure 3 – Slope angle considering the Earth’s perturbation at the moment of maximum encounter in 2029 mapped over the Apophis surface under different views and for the small and large densities. Chapter 2. The asteroid Apophis 26 between the acceleration with the presence and absence of Earth’s gravity, resulting in a slope referred to as a ∆slope. The ∆slope illustrates the changes in slope on the asteroids’ surface. The variation of the ∆slope is about 4◦ for the model of Apophis with the smaller density and about 2◦ for the larger density. It is a smaller variation but it could be sufficient to trigger landslides in some regions where the slope angles may exceed the value of the angle of repose. Simulations have shown that a variation of about 2◦ may trigger erosion in regions with larger slopes (Ballouz et al., 2019). Thus, the perturbation of the Earth during the 2029 encounter may trigger landslides but it will not change drastically its surface and shape. It could cause only local landslides (Yu et al., 2014). Once the surface dynamic was studied, we investigated the instability that the Earth may cause on Apophis’ nearby environment during the trajectory of the 2029 encounter. We restrict our analysis to 12 hours after and before the encounter with Earth in 2029 totalling 24 hours of trajectory. The used trajectory is shown in Fig. 2 of Valvano et al. (2022b). However, before considering the trajectory of the 2029 encounter, we analysed the stability of the neighbourhood of Apophis without the external perturbation of the Earth in 2029. We only considered the irregular gravitational field of Apophis. For the first 24 hours, almost all of the particles survived, only particles near the equatorial extremities of the body collided with Apophis (blue dots of Fig. 9 of Valvano et al. (2022b)). The irregularities of Apophis’ format in these regions contributed to the collisions of the particles. Since solar radiation pressure plays an important role in the nearby dynamic of asteroids, we decided to investigate how solar radiation pressure perturbs the Apophis neighborhood. We simulated the same distribution of particles previously studied but with the addition of solar radiation pressure. We simulated micrometric-sized particles but none of them survived. Thus, we increase to cm-sized particles. Since the gravitational field of Apophis with 1.29 g·cm−3 of density is small, only particles larger than 15 cm of radius survived for the entire simulation (30 years), while for the densities of 2.2 g·cm−3 and 3.5 g·cm−3, particles of at least 5 cm survived (Fig. 11 of Valvano et al. (2022b)). Almost the entire set of particles collided with Apophis and only a small portion survived. For both models of 1.29 g·cm−3 and 2.2 g·cm−3of density, the region of surviving has a pattern similar to a moustache, while for the larger density, we have a small disc of surviving. When we simulated Apophis during its close encounter, the results for the three density models are similar. A large ring of particles survived for the entire simulation (24 hours), similarly, a large ring of particles was ejected. The small region of collision near the body’s equatorial extremities became a small disc of collisional particles. A set of particles located in the region of the odd equilibrium points also formed a region of collisional particles. We expanded our disc of particles to a disc external to the equilibrium points to 10 Chapter 2. The asteroid Apophis 27 times the distance of the equilibrium points from Apophis. A large set of particles survived this external condition, in the same way a massive quantity of particles were ejected. Only a small portion of particles collided with Apophis. After the paper APOPHIS – effects of the 2029 Earth’s encounter on the surface and nearby dynamics was published, we noticed a misimplementation in the loop of one of our programs. This program was responsible for calculating the energy of the particles to analyse if a particle was ejected. The outcome of a set of particles was underestimated. Section 2.4 presents the correct results of the instability caused by the 2029 close encounter with Earth. The patterned behaviour remains the same, however, the set of survived particles decreased and the ejected particles increased. We concluded from this work that despite the 2029 encounter being the closest, the Earth’s perturbation may not influence the Apophis surface to trigger a drastic reshaping of its format. Only isolated regions will be significantly affected by the encounter perturbation. Different from the surface results, the numerical simulations of a hypothetical disc around Apophis show that a massive set of particles will be ejected. However, a considerable portion of particles survive the 2029 approach. The big question now is "may the ejected particles hit or cause a meteor activity on Earth?" In the paper described in this section, I was the leading author and led the writing process. I conducted all the simulations and talked about the results with the other authors. I also implemented and tested the solar radiation pressure to our N-body integrator. 2.2.1 The Making-of: APOPHIS - May a meteor activity happen on Earth after the 2029 closest approach? After the publication of the paper of section 2.3, we started to think about the fate of the massive amount of ejected particles during the 2029 encounter with the Earth. Could they collide with the Earth? or even generate a meteor activity? Guided by these questions, we decided to study what may happen with the ejected particles after the encounter with the Earth. We found interesting works involving meteor activities (Kováčová et al., 2020; Wiegert, 2020; Carbognani et al., 2022) and a good way of studying the fate of the particles was by calculating the closest distance between the orbits of two celestial bodies, also called Minimum Orbit Intersection Distance (MOID). We find that MOID is the closest distance between two celestial bodies’ orbits and can assist in the determination of the possibility of a collision (Bowell et al., 2002). Lowe presents a set of calculated MOIDs in his website. We started trying to reproduce Lowe’s results but we did not get the same values as him. We contacted him to discuss the differences and he identified that we are using the ephemerides with respect to http://andrew-lowe.ca/ALL_MOID.TXT Chapter 2. The asteroid Apophis 28 the center of the Earth, not the Earth/Moon barycenter. He also suggests that we increase our numerical step for the MOID calculation. The changes suggested by him were made and we obtained satisfactory results similar to his. Once our program for calculating the MOID was completely functional, we began the particles’ fate studies. We take into account all the particles that were ejected from the 24-hour simulations during the 2029 encounter with Earth. The state vector of the particles was considered at the end of the simulation. Then, we integrated them for 200 years in a system considering the Sun, Earth, Moon, Venus, Mars, Jupiter and Apophis. Due to our previous study, for the particles to survive to Apophis gravitational field and the influence of solar radiation pressure, the particles need to be cm-radius-sized as bolides. After the integration, we calculate the Apophis’ MOID and the particles’ with Earth. During the first 38 years, the MOID of Apophis and the particles have approximately the same values. Apophis experiences a significant encounter with the Earth in 38 years into the integration. This encounter provokes an increase in Apophis MOID magnitude. Since the dispersion of the particles from Apophis is smaller, their MOID follows Apophis’ during the first 73 years into the integration. This encounter triggers an increase in the particle’ and Apophis’ MOID and, after that, their MOID starts to diverge. Between the period of 87 to 98 years, the particles spread quickly and the stream is fully distributed along the Apophis’ orbit. The particles’ MOID and dispersion continue to grow until the end of the integration (200 years). Once the particles are fully distributed along the Apophis’ orbit the, probability of a meteor activity on Earth is larger. However, we have to consider that the particles MOID have to be as small as possible and they need to be near the Earth’s orbital plane. However, this scenario did not happen. Note that there are two regions (Figs. 2(c) and 2(d) of the paper presented in section 2.5)on the particle stream with smaller z amplitude, meaning the particles in this region are near the Earth’s orbital plane. However, none of these two regions are near enough to the Earth for a meteor activity to happen. Since meteor activity on Earth has a lower possibility, may debris impact happen on the Moon? In the upper panel of Fig. 1 of the paper presented in section 2.5, we can see blue lines, the full one represents the Moon’s distance and the dashed one the Lagrangian points L1 and L2 of the Moon-Earth system. During the first years, the particles’ MOID with respect to Earth have smaller values than the Moon’s distance from Earth. Thus impact debris on the Moon could happen. However, the particles are not completely encircled and have a smaller azimuthal dispersion. For this reason, impact debris in this period is improbable. In the period of 87 to 98 years into integration the particles are azimuthally spreading, achieving the complete encircling in 98 years into the integration. Thus, during this period debris might impact the Moon or/and the Lagrangian points L1 and L2 of the Chapter 2. The asteroid Apophis 29 Moon-Earth system but the possibility is deeply small. We thank the paper’s referee for suggesting the analysis of activity on the Moon. Analysing the MOID, we note that the first 75 years after the closest approach could be the most promissory period for a meteor activity considering only the MOID magnitude. However, the particles are too close to Apophis and they are not completely encircled. Thus, in about 75 years after 2029 encounter an activity on the Earth and the Moon will not happen. After this period the particles will encircle, completing it in 98 years into the integration. During the period of quickly encircling (87 to 98 years), there is a pretty small possibility of activity on the Moon. However, for the Earth is not probable. As the first author of the paper described in this section, I took charge of writing it and performed all the simulations. I also led the implementation of the MOID calculations. Additionally, I engaged in discussions about the results with the assistance of all co-authors. 2.2.2 APOPHIS – An Extension During the process of publication of the paper "APOPHIS – effects of the 2029 Earth’s encounter on the surface and nearby dynamics", we decided to investigate the patterns of outcomes from Fig. 10(a) of Valvano et al. (2022b), mainly explaining the formation of the collisional ring. To analyse in a more detailed way, we subdivided each region of the disc with a different colour despite their final fate (Fig. 3(a) of the paper presented in section A). The analysis shows that all the regions are significantly influenced by the elongated shape of Apophis and some resonances. Note that the regions C2, S1 and S2 have an elliptical aspect similar to Apophis’ shape. The C1 group is also directly influenced by Apophis’ shape, since the particles are removed in the first hours of simulation due to the tapered ends. Since Apophis has an irregular shape model, the osculating elements may not accurately describe the particle’s orbit. Thus, the resonant angles are complex to obtain. We need to look at the particle trajectories searching for patterns similar to those described on Murray and Dermott (2000). We identify a set of spin-orbit resonances among all the disc and they also play an important role in the formation of the disc’s pattern. The first author led the paper writing and all the analysis of the paper. The paper was published in the special topics of The European Physical Journal. As a co-author, I write some sections of the work including the sections of introduction, a discussion about the previous paper (section 2.3) and the physical properties of the asteroid. I participated in all discussions of the paper and the process of writing these discussions. I also provided the initial data of the simulations originating from the paper of the section 2.3. The paper is presented in the appendix section A. Chapter 2. The asteroid Apophis 30 2.2.3 Methodology Throughout this section, the methodology used will be briefly introduced. For a more detailed explanation of the methodology used in these works see the manuscripts presented in sections 2.3 and 2.5. Since asteroids have an irregular shape, their format needs to be carefully considered in asteroid studies. For Apophis studies, its format is modelled by a polyhedron with 1014 vertices and 2024 triangular faces (Pravec et al., 2014). This polyhedron shape is modelled through observational methods such as, for example, radar observations. Once a polyhedron models the format of the asteroid, its surface is covered by triangles which are used to compute different quantities such as altitude, slope, acceleration, geopotential and many others (Scheeres et al., 2016). Usually, the barycentre of each triangular face is considered to calculate these quantities. To analyse the stability around Apophis, particles were integrated around it using an N-body integrator package called N-BoM (Moura et al., 2020; Winter et al., 2020). N-BoM uses the Bulirsch–Stoer algorithm and the mass concentration (MASCONS) model to manipulate the gravitational potential of irregular bodies (Geissler et al., 1996; Borderes-Motta and Winter, 2018). The solar radiation pressure is implemented in N-BoM according to Scheeres and Marzari (2002). The 2029 closest approach trajectory with the Earth was modelled using JPL’s HORIZONS ephemerides. Then, the interpolation of this trajectory with the Earth was added as an additional external force to N-BoM. The system was made by Apophis modelled as MASCONS and a disc of massless particles. Three scenarios were assumed, being them: the simulation considering the gravitational field of Apophis, the gravitational field of Apophis plus the solar radiation pressure and the gravitational field of Apophis plus the 2029 encounter as an additional external force. The ejected particles that were ejected from the 2029 closest encounter with Earth were also analysed to comprehend if a meteor activity happened on Earth after this encounter. To perform these numerical simulations the Rebound package (Rein and Liu, 2012) using the IAS15 integrator (Rein and Spiegel, 2015). The particles were considered massless bodies and also Apophis since the particles had escaped from Apophis’s gravitational field and were not gravitationally bound to it. The Earth, Moon, Venus, Mars, Jupiter and the Sun were assumed to be point masses. We integrated the system for 200 years after the closest 2029 encounter. MNRAS 510, 95–109 (2022) https://doi.org/10.1093/mnras/stab3299 Advance Access publication 2021 No v ember 13 APOPHIS – effects of the 2029 Earth’s encounter on the surface and nearby dynamics G. Valvano , 1 ‹ O. C. Winter , 1 R. Sfair , 1 R. Machado Oliveira , 1 G. Borderes-Motta 2 and T. S. Moura 1 1 Grupo de Din ̂ amica Orbital e Planetologia, S ̃ ao Paulo State University, UNESP, Guaratinguet ́a, CEP 12516-410 S ̃ ao Paulo, Brazil 2 Bioengineering and Aerospace Engineering Department, Universidad Carlos III de Madrid, Legan ́es, E-28911 Madrid, Spain Accepted 2021 No v ember 9. Received 2021 November 8; in original form 2021 October 19 A B S T R A C T The 99942 Apophis close encounter with Earth in 2029 may provide information about asteroid’s physical characteristics and measurements of Earth’s effects on the asteroid surface. In this work, we analysed the surface and the nearby dynamics of Apophis. The possible effects of its 2029 encounter on the surface and environment vicinity are also analysed. We consider a 340 m polyhedron with a uniform density (1.29, 2.2, and 3.5 g cm −3 ). The slope angles are computed, as well their variation that arises during the close approach. Such variation reaches 4 ◦ when low densities are used in our simulations and reaches 2 ◦ when the density is high. The zero-velocity curves, the equilibrium points, and their topological classification are obtained. We found four external equilibrium points and two of them are linearly stable. We also perform numerical simulations of bodies orbiting the asteroid, taking into account the irregular gravitational field of Apophis and two extra scenarios of perturbations: the solar radiation pressure and the Earth’s perturbation during the close approach. The radiation pressure plays an important role in the vicinity of the asteroid, only cm-sized particles survived for the time of integration. For densities of 2.2 and 3.5 g cm −3 , a region of 5 cm radius particles survived for 30 yr of the simulation, and for 1.29 g cm −3 , only particles with 15 cm of radius survived. The ejections and collisions are about 30–50 times larger when the close encounter effect is added but around 56–59 per cent of particles still survive the encounter. Key words: celestial mechanics – minor planets, asteroids: general – planets and satellites: dynamical evolution and stability – methods: numerical. 1 I N T RO D U C T I O N Upon its disco v ery on 2004 June at Kitt Peak by R.A. Tucker, D.J. Tholen, and F. Bernardi (Garradd 2004 ; Smalley et al. 2004 ; Tucker, Tholen & Bernardi 2004 ), 99942 Apophis (originally designated as 2004 MN4) has its orbit constantly monitored since it was reported a high probability of collision with Earth of more than 2 per cent (Chesley 2005 ). Although this possibility was later discharged (Larsen & Descour 2004 ), other potential impact or approaches with Earth were reported in the following years, and Apophis has been considered as a potentially hazardous asteroid (PHA). The trajectory and the future close encounter parameters were impro v ed with ephemeris derived from radar observations in 2005–2006 from Arecibo radiotelescope (Giorgini et al. 2008 ). With this data, they were able to reduce orbital uncertainties and infer a nominal distance for the 2029 approach of about 6 Earth radii, predict a distance of 0.34 au for the 2036 encounter, and find that an impact probability still remains (Giorgini et al. 2008 ). The next approach will be the closest encounter with the Earth and will occur on 2029 April 13. Apophis will pass at a distance near to the geostationary orbit and a distance about one-tenth the distance between the Earth and Moon. � E-mail: giulia.v alv ano@unesp.br The predictions for the pre-2029 orbit turn to be well defined, ho we ver, for the post-2029 orbit, the predictions are not well determined due to the uncertainties caused by perturbations. The Yarko vsk y effect is a considerable source for the orbital uncertainties of Apophis’ orbit (Giorgini et al. 2008 ; Farnocchia et al. 2013 ; Vokrouhlick ̀y et al. 2015 ). Thus considering the Yarko vsk y effect and using astrometric data from 2004 to 2008, Farnocchia et al. ( 2013 ) presented a new impact risk for Apophis. They infer an impact probability of about ∼10 −9 for the 2036 encounter and ∼10 −6 for 2068. Observations from the 2021 encounter provided measurements that impro v ed the fit for the orbit of Apophis and eliminate the possibility of impact for the next 100 yr. It led Apophis to be remo v ed from the ESA’s asteroid Risk List after remaining on this list for almost 17 yr. 1 The 2029 flyby may provide an opportunity to impro v e the 3D shape model, investigate possible changes on the spin state, reshaping and effects of the encounter on the surface. Scheeres et al. ( 2005 ) predicted that the terrestrial torques caused by the encounter will change Apophis’ spin state drastically, and consequently the Yarko vsk y accelerations. Conv ersely, Souchay et al. ( 2018 ) showed that the changes in the spin rate may be small, and the larger 1 https:// cneos.jpl.nasa.gov/sentry/ removed.html © 2021 The Author(s) Published by Oxford University Press on behalf of Royal Astronomical Society D ow nloaded from https://academ ic.oup.com /m nras/article/510/1/95/6427373 by U niversitaet Tuebingen user on 17 D ecem ber 2021 Chapter 2. The asteroid Apophis 31 2.3 APOPHIS – effects of the 2029 Earth’s encounter on the surface and nearby dynamics 96 G. Valvano et al. effects may occur in the obliquity and precession in longitude. The effects of a close approach with the Earth also could cause material landslides and migration as is shown in Binzel et al. ( 2010 ). Ho we ver, the numerical simulations made for Yu et al. ( 2014 ) using soft-sphere code implementation predicted that the effects of the tidal pertubations on the Apophis’ surface may be small but could produce small landslides. Therefore, the closest approach of 2029 with the Earth may provide measures through observations before, during, and after the encounter. The observational data from these observations may impro v e some physical characteristics, the understanding of the effects of the closest encounters with the Earth, validate models about the material, and other possibilities. In this work, we used the conv e x shape model from Pravec et al. ( 2014 ) to analyse the surface and the nearby dynamics of the asteroid Apophis considering the effects caused by the 2029 closest approach with the Earth. The paper is composed of the following sections. In the next section, we present the asteroid model used in this work, discussing its poly- hedral shape model, general characteristics, and the 2029 encounter configuration. Section 3 introduces the gravitational potential and geopotential considering the polyhedra method. We discuss physical features using the slope angle and its variation due to the Earth perturbation. In Section 4, we explore the nearby environment of Apophis by calculating the zero-velocity curves and equilibrium points. We also present a set of numerical simulations of a disc of particles around Apophis considering the gravitational field and two additional scenarios of perturbations: the solar radiation pressure and, in Section 5, the Earth perturbation on the 2029 trajectory . Finally , in the last section we provide our final comments. 2 T H E ASTEROID M O D E L Apophis was observed by the VLT (Very Large Telescope) and the data obtained from a polarimetric observing campaign in 2006 resulted in an approximate diameter of 270 m for the asteroid (Cellino, Delb ̀o & Tedesco 2006 ). In 2012 and 2013, Pravec et al. ( 2014 ) made a photometric observational campaign and disco v ered that Apophis is a tumbling asteroid. They computed the Apophis’ tumbling spin state as a retrograde rotational period of 27.38 ± 0.07 h and a precession period of 263 ± 6 h. The 2012–2013 Apophis’ apparition also allowed the construction of a conv e x shape model. The Apophis’ shape model is represented as an unscaled polyhedron with 1014 vertices and 2024 triangular faces (Prav ec et al. 2014 ). Brozo vic et al. ( 2018 ) also pro vided a shape model for the asteroid Apophis but they used radar observations from Goldstone and Arecibo to impro v e the previous model. This impro v ed shape model is a 340 m polyhedron with 2000 vertices and 3996 faces. From thermal infrared observations, M ̈uller et al. ( 2014 ) reported an approximated size of 375 m for (99942) Apophis. Licandro et al. ( 2015 ) also provided an estimated size for Apophis using this technique, ho we v er, the y combined their data with previous thermal observations. The resulting size for Apophis ranges from 380 to 393 m. We adopted the Apophis’ diameter presented by Brozovic et al. ( 2018 ), 340 m, and the shape model provided by Pravec et al. ( 2014 ), hence the model by Brozovic et al. ( 2018 ) is not publicly available. We used the rotational period of 27.38 h with an obliquity of 180 ◦ to reproduce the Apophis’ retrograde rotation (Pravec et al. 2014 ) and do not consider the precession period as it is 10 times larger than the rotational period. Figure 1. Geometric altitude map computed across the surface of Apophis under dif ferent vie ws. The sea-le vel (the smaller distance between the geometric centre of the body and the barycentre among all triangular faces) is 130.54 m. The model for Apophis has an ellipsoid elongated format and tapered ends in the equatorial regions, as shown in its geometric map (Fig. 1 ). To measure the geometric altitude, we determine the distance between the geometric centre of the body and the barycentre of each triangular face of the polyhedron. Then, we identify the smaller distance between these points and set it as ‘sea-level’ (Scheeres et al. 2016 ). The sea-level is 130.54 m and it is almost half the value of the maximum distance calculated between the geometric centre and the barycentre among all the faces. Analysing the geometric map, we notice that the maximum values of the geometric altitude are on the equatorial regions, while the minimum values are on the poles, which means that these regions are at the sea-level. The top and bottom views in Fig. 1 show the poles of the asteroid, and they show that the north pole (top view) has a concentrated region of the smaller altitudes on its middle. Ho we ver, the south pole (bottom view) presents a band of a larger distribution of small altitudes. Apophis has a grain density in the range 3.4–3.6 g cm −3 and a total porosity with range of 4–62 per cent because of its similar spectral characteristics with LL ordinary chondrite (Binzel et al. 2009 ). This results in an approximate bulk density of 1.29–3.46 g cm −3 . As the composition and some similarities suggest, Apophis could have a total porosity similar to the asteroid (25143) Itokawa. So Binzel et al. ( 2009 ) adopted Apophis’ radius as 135 m, density of 3.2 g cm −3 and Itokawa’s total porosity of 40 per cent to estimate Apophis’ mass as 2.0 × 10 10 kg. Dachwald, Kahle & Wie ( 2007 ) assumed a spherical radius of 160 m and a bulk density of 2.72 g cm −3 to determine the Apophis’ mass as 4.67 × 10 10 kg. M ̈uller et al. ( 2014 ) also estimated a mass for Apophis, ho we v er, the y used a radius of 187.5 m, density of 3.2 g cm −3 , and total porosity of 30–50 per cent, resulting in a total mass of 4.4–6.2 × 10 10 kg. Therefore, for this work, we assumed three bulk densities of 1.29, 2.2, and 3.5 g cm −3 . Considering the density values adopted and the Apophis’ size provided from Brozovic et al. ( 2018 ), we constructed three models for the asteroid, one for each density. As the mass is not a known parameter, we set the volume of the asteroid as the same of a sphere with an equi v alent radius of 170 m. So, we preserved the volume and the size and estimate the mass for each model according to the density. MNRAS 510, 95–109 (2022) D ow nloaded from https://academ ic.oup.com /m nras/article/510/1/95/6427373 by U niversitaet Tuebingen user on 17 D ecem ber 2021 Chapter 2. The asteroid Apophis 32 APOPHIS – effects of the 2029 Earth’s encounter 97 The volume adopted was 0.0205 km 3 and the three calculated masses are 2.64 × 10 10 , 4.50 × 10 10 , and 7.16 × 10 10 kg for the densities of 1.29, 2.2, and 3.5 g cm −3 , respectively. With the three models, we defined a coordinate system with the origin at the object centre of mass and align the system axes to the axes of the principal moment of inertia. The x , y , and z axes correspond, respectively, to the smallest, intermediate, and largest moments of inertia. This process was made using the algorithm presented by Mirtich ( 1996 ) and it was assumed that Apophis has a constant density and a uniform rotation about the largest moment of inertia. The values of the principal moment of inertia normalized by the mass are I xx /M = 559 . 23 m 2 , (1) I yy /M = 897 . 72 m 2 , (2) I zz /M = 969 . 39 m 2 . (3) From the principal moment of inertia, we calculated the second- order degree terms C 20 and C 22 of the gravity expansion. The coefficient J 2 ( −C 20 ) represents how oblate Apophis is and C 22 how elongated. The values are (MacMillan 1958 ; Hu & Scheeres 2004 ) C 20 = − 1 2 R 2 n (2 I zz − I xx − I yy ) = −0 . 00837, (4) C 22 = 1 4 R 2 n ( I yy − I xx ) = 0 . 00294 . (5) where the equi v alent radius, R equi v alent = 170 m, was used for normalization. Note that the coefficient C 20 is of the same order of magnitude as the coefficient C 22 but the absolute value of C 22 is smaller than J 2 . We also modelled Apophis’ shape as an ellipsoid with the parameters a , b , and c representing the ellipsoid semi-axes of the asteroid. The equi v alent ellipsoid parameters for Apophis are a = 228 . 966631 m , (6) b = 159 . 026432 m , (7) c = 139 . 797960 m . (8) Observe that a is about 64 per cent larger than the parameter c , evidencing the elongated shape at the equator and flattened at the poles. So, considering the harmonic coefficients and the equivalent ellipsoid parameters, the shape of Apophis has a proportion of 5:3.5:3 between the semi-axes a : b : c , respectively. 2.1 2029 Earth’s encounter On 2029 April 13, Apophis will have the closest approach with the Earth at an approximate distance of ∼38 000 km according to the JPL’s HORIZONS ephemerides (Giorgini et al. 1996 ). The trajectory is shown in Fig. 2 , where the black dot represents the Earth, the green circle and line illustrate the equatorial plane and Moon’s orbit, respectively. The filled and the dashed blue lines represent the trajectory of the object abo v e and below the equatorial plane, respecti vely, while the arro w illustrates the direction of the mo v ement of Apophis. Note that Apophis approaches the Earth from left to right and it takes 34.30 h to enter and leave the Moon’s orbit. Apophis traverses the equatorial plane from the bottom-up and just crosses the equatorial plane near to the closest approach. Figure 2. Apophis’ trajectory of the encounter with Earth in 2029 in the xoy plane. The black dot represents the Earth and the green line and circle represent the Moon’s orbit and the equatorial plane, respectively. The blue line represents the trajectory abo v e the equatorial plane and the blue dashed line represents the trajectory below the equatorial plane. The arrow represents the mo v ement direction of Apophis. Since we do not know the exact orientation at the encounter, we inv estigate sev eral hypothetical orientations for the asteroid Apophis at the moment of the encounter with the Earth in 2029 and see if this approach could change the surface characteristics of the body (Sec- tion 3). We also used the 2029 approach trajectory (Fig. 2 ) to compute the effects on the dynamical nearby environment (Section 4). This analysis may provide insights to the 2029 observational campaign, and eventual space missions designed to study the asteroid. 3 EFFECTS O N T H E SURFAC E The geopotential is an ef fecti ve way to measure the relative energy on the body surface since it considers the gravitational and rotational potential (Scheeres 2012 ; Scheeres et al. 2016 ). It is possible to relate the geopotential energy with the motion of a cohesionless particle by associating the quantity of energy necessary to mo v e this particle across the body surface. Assuming a constant angular v elocity v ector ω ω ω , the reference frame centred at the centre of mass and the axes aligned to the principal inertia axes for the asteroid Apophis, the expression for the geopotential is (Scheeres et al. 2016 ) V ( r r r ) = −1 2 ω 2 ( x 2 + y 2 ) − U ( r r r ) , (9) where r r r is the position vector of a massless particle in the body-fixed frame and U ( r r r ) is the gravitation potential that is given by the method of polyhedra. To model the object, we used the polyhedra method that computes the gravitational potential energy of an irregular body modelled by a uniform density polyhedron with a given number of faces and vertices. Then, the expression for the gravitational potential is MNRAS 510, 95–109 (2022) D ow nloaded from https://academ ic.oup.com /m nras/article/510/1/95/6427373 by U niversitaet Tuebingen user on 17 D ecem ber 2021 Chapter 2. The asteroid Apophis 33 98 G. Valvano et al. (Werner & Scheeres 1996 ) U = Gρ 2 ∑ e∈ edges r r r e ·E E E e · r r r e · L e − Gρ 2 ∑ f ∈ faces r r r f ·F F F f · r r r f · ω f , (10) where ρ is the density of Apophis, G is the gravitational constant; r r r f and r r r e represent, respectively, the position vectors from a point in the gravitational field to any point in the face f and edge e planes; F F F f and E E E e are the faces and edges tensors; ω f and L e are the signed angle viewed from the field point and the integration factor, respectively. From the geopotential, Scheeres et al. ( 2016 ) derive the slope angle using its gradient (see Section 3.1), this measure considers the effects on the surface of the body, but we also can derive an approach for a particle in the nearby environment of the asteroid. Since it is considered that Apophis has a constant angular velocity ( ω ω ω ) about its axis of maximum moment of inertia, the mo v ement of a particle orbiting the nearby environment is described by (Scheeres et al. 2016 ) r̈ r r + 2 ω ω ω × ṙ r r = −∂V ∂ r r r , (11) where r r r is the position vector of the particle, and ṙ r r and r̈ r r are its velocity and acceleration vectors, respectively. Equation (11) is time-invariant since the Apophis’ spin-rate is assumed to be a constant value. Therefore, this equation can be associated with a conserved quantity, C j , called ‘Jacobi constant’. This constant is defined by (Scheeres et al. 2016 ) C j = 1 2 v 2 + V ( r r r ) , (12) where the magnitude of the v elocity v ector relativ e to the rotating frame of the Apophis is denoted by v. 3.1 Slope The slope angle is the supplement between the normal and total acceleration vectors (Scheeres 2012 ; Scheeres et al. 2016 ). The slope is an angle that quantifies how inclined is a region on the Apophis’ surface with respect to its local acceleration vector. Physically, slope assists in understanding the mo v ement of free particles on the Apophis surface. The slope angle distribution across the Apophis’ surface for the larger and smaller densities is shown in Fig. 3 . The variation between the models is small and the slope distribution is almost identical. The slope variation amplitude goes from 35.81 ◦ for the smaller density, up to 36.16 ◦ for the larger density, a difference of only 0.35 ◦. For all three cases, the maximum slope angles are smaller than 37 ◦, and the minimum angles are larger than 0.5 ◦. The maximum values of slope occur just in the small regions on the equatorial extremities of Apophis (see front, left, and bottom views in Fig. 3 ). The minimum values of the slope are distributed preferentially at the central region of the body as we can see in the bottom view in Fig. 3 , the south pole has a strip of minimum values passing through the middle of the surface. By definition, slope angles varies between 0 ◦ and 180 ◦. When it is larger than 90 ◦, the centrifugal force is larger than the gravitational force, causing cohesionless particles to escape from the body’s surface. If the slope angle is smaller than 90 ◦, the mo v ement of the cohesionless particles can be associated with the repose angle of a given material. The angle of repose for geological material is about 35–40 ◦ (Lambe & Whitman 1969 ; Mitchell et al. 1974 ; Al-Hashemi & Al-Amoudi 2018 ). Figure 3. Slope angle maps across the surface of Apophis under different views. The panels (a) and (b) represent, respectively, the slope distribution considering the densities of 1.29 and 3.5 g cm −3 . As already referred, the maximum value of the slope considering the three values of density is about 37 ◦. So we can see two regimes, one for the lower limit value of the angle of repose, 35 ◦, and the other for the upper limit value, 40 ◦. For the lo wer limit, the flo w of the cohesionless particles may occur, since we did not find a slope value higher than this limit. To understand the possible flow of free particles on Apophis’ surface, we compute the directions of the tangential component of the local acceleration vector (Scheeres 2012 ; Scheeres et al. 2016 ). Fig. 4 shows the tangential acceleration vectors considering a density of 2.2 g cm −3 . As expected, the vectors are pointing to the regions where the slope is smaller. The flow of loose material might occur from the small regions on the equatorial extremities to the middle of the Apophis’ surface. The dark regions in Fig. 4 represent the near-zero slopes, MNRAS 510, 95–109 (2022) D ow nloaded from https://academ ic.oup.com /m nras/article/510/1/95/6427373 by U niversitaet Tuebingen user on 17 D ecem ber 2021 Chapter 2. The asteroid Apophis 34 APOPHIS – effects of the 2029 Earth’s encounter 99 Figure 4. Slope angle map computed across the surface of Apophis with the directions of the local acceleration tangent vectors and under dif ferent vie ws for the density of 2.2 g cm −3 . thus are stable resting areas. Those regions are propitious regions to accumulate cohesionless particles. Apophis will have a close approach with the Earth in 2029 and will be affected by time-varying forces. Thus, to evaluate the strongest possible effect on its surface due to the Earth‘s gravity, we calculated a � slope that computes the slope variation caused due to the encounter at the closest distance of the whole approach. In order to identify the most extreme condition that may occur in 2029, we assume a hypothetical configuration for the Apophis–Earth system. We set the Earth on the equatorial plane of Apophis and the distance between the centre of mass of Apophis and the Earth as the closest distance during the whole encounter, and we calculate the slope on the Apophis’ surface considering its geopotential including the Earth’s gravitational force on the closest approach. Then, we rotate the shape model of Apophis all along the 360 ◦, so that each Apophis’ orientation experiences the extreme approach condition. The difference between the total acceleration vectors, with and without the Earth’s gravity, leads to a slope that we call a delta slope. The delta slope is the variation of the slope angle produced by the Earth’s gravitational perturbation on the surface of Apophis at the extreme approach condition. The � slope map was also made over the rotation of 360 ◦ of the Apophis’ shape model, in order to compute the � slope on each Apophis’ orientation. Fig. 5 shows the � slope for the smaller and larger density of Apophis (see the complete animation of the � slope angle maps in the videos of the complementary material). The top and bottom views are not shown since they are not directly pointed to the Earth at the closest approach, as we defined before. For the 1.29 g cm −3 density model, the slope variation was smaller than 4 ◦, and for 3.5 g cm −3 density model was about 2 ◦. This variation represents about 11 per cent of the regular slope angle for the smaller density model and 5.5 per cent for the larger density model. With the perturbation caused by the Earth, some slope angle values may exceed the angle of repose. Ballouz et al. ( 2019 ) showed that a variation around 2 ◦ may be sufficient to start a slow erosion process in regions with high slope that experiment larger perturbations. Ho we ver, the perturbations due to the 2029 encounter will not trigger drastic reshaping on Apophis’ surface and shape. The numerical Figure 5. The � slope angle maps across the surface of Apophis under dif ferent vie ws and considering the Earth perturbation in 2029 encounter. The panels (a) and (b) represent, respectively, the slope distribution considering the densities of 1.29 and 3.5 g cm −3 . MNRAS 510, 95–109 (2022) D ow nloaded from https://academ ic.oup.com /m nras/article/510/1/95/6427373 by U niversitaet Tuebingen user on 17 D ecem ber 2021 Chapter 2. The asteroid Apophis 35 100 G. Valvano et al. simulations provided by Yu et al. ( 2014 ) revealed that the effects caused by the Earth are small and just can cause local landslides. 4 EFFECTS O N T H E N E A R B Y STABILITY In this section, we present an investigation of the environment around Apophis. The analysis of the variation of the Jacobi constant, through zero-v elocity curv es, allows us to identify where the mo v ement is permitted or not. These curves delimit the external equilibrium points of the body, the location, and the topological classification of these points are presented. A set of numerical simulations with a disc of particles is also presented in order to analyse the stability around the asteroid considering the perturbation of its own gravitational field and two additional perturbations: the solar radiation pressure and the 2029 flyby scenario. 4.1 Equilibrium points and zero-velocity curves To understand the stability in the vicinity of Apophis, we calculated the zero-v elocity curv es and equilibrium points. The zero-v elocity curves limit the movement of a particle, delimiting where its mo v ement is allowed or not. This delimited mo v ement depends on the Jacobi constant, C j , and it arises from the inequality C j − V ( r r r ) ≥ 0 , (13) since the velocity term, 1 2 v 2 , from equation (12) shall be al w ays positive. So, when C j < V ( r r r ) there will be forbidden regions to the mo v ement of the particle, considering that the inequality is infringed. In the case of the regions where C j > V ( r r r ), there will be no a priori limitation to the mo v ement of the particle, since the inequality is satisfied. When C j = V ( r r r ) we have the zero-v elocity curv es that separate the regions between the permitted and prohibited movement. Fig. 6 illustrates the zero-velocity curve countourplots in the xoy plane for the model with the mean density. The colour of the zero- v elocity curv es changes as the Jacobi constant value changes. At a certain value of C j , the curves could form confined regions that will encompass an equilibrium point, similar to a ‘banana’ shape in Fig. 6 . The region between these confined curves also will include an equilibrium point. The equilibrium points are critical points of geopotential where there is a balance between the gravitational and centrifugal forces. Since the resulting force is null at this points, the equilibrium points of the asteroid Apophis are calculated by the solution of ∇V = 0 , (14) where the ∇V means the gradient of the geopotential. In general, there is no fixed number of solutions for equation (14), since the solution depends on the body shape model, its density, and rotational period. Ho we ver, we could estimate the number of equilibrium points just by looking at the zero-v elocity curv es of the body. The zero-velocity curves of Apophis show it has at least four equilibrium points, two of them inside the curves with a ‘banana’ shape and two between the larger ‘banana’ curves near to the y = 0 line (Fig. 6 ). The location of the equilibrium points also depends on the rotational period, density, and shape of the asteroid. Once defined, the shape, model, and the spin, if we change its density, the location of the equilibrium points will change as well. If the density decreases, the location of the equilibrium points will be closer to the body due to the reduction of the gravitational force. Conversely, if the density is increased, the equilibrium points will be more distant from the body. Figure 6. Zero-v elocity curv e contourplots in the xoy plane considering the mean density of 2.2 g cm −3 . The colourbox represents the value of the Jacobi constant. Considering the polyhedral model derived from Pravec et al. ( 2014 ) and the densities of 1.29, 2.2, and 3.5 g cm −3 , we found five equilibrium points for each density, being one in the centre of the body. The position of the equilibrium points are shown in T able 1 . W e note they are close to the equatorial plane, so Fig. 7 shows the projection of the asteroid Apophis and the equilibrium points in the xoy plane. Note that the number of equilibrium points does not change, differently from other systems (Jiang & Baoyin 2018 ). Observe that although the model with a larger density is almost three times larger than the model with a smaller density, the difference in the radial distance between the equilibrium points of the model with larger and smaller densities is about 300 m. Applying the linearization method to the equations of motion, we analysed the six eigenvalues of the characteristic equation for each point in order to identify their topological classification (Jiang et al. 2014 ). Table 1 shows the coordinates and the topological classification of each equilibrium point for each density of Apophis. The odd inde x ed points, E 1 and E 3 , are classified as a Saddle–Centre– Centre, while the even points ( E 2 and E 4 ) are Centre–Centre–Centre, implying that they are linearly stable points. According to Jiang et al. ( 2014 ) and Yu & Baoyin ( 2012 ), when analysing the eigenvalues of the characteristic equation, we identified the existence of three families of periodic orbits in the tangent space of the equilibrium points E 2 and E 4 , which have characteristic times or oscillation periods of approximately 27.3, 28.9, and 88.4 h for E 2 , and 27.3, 28.8, and 90.9 h for E 4 , when we consider the density of 1.29 g cm −3 . Only the period value of the third periodic orbit family underwent a significant change, with an increase of about 21 per cent and 43 per cent, respectively, for the Apophis intermediate and upper MNRAS 510, 95–109 (2022) D ow nloaded from https://academ ic.oup.com /m nras/article/510/1/95/6427373 by U niversitaet Tuebingen user on 17 D ecem ber 2021 Chapter 2. The asteroid Apophis 36 APOPHIS – effects of the 2029 Earth’s encounter 101 Table 1. Coordinates and characteristic time of the equilibrium points of 99942 Apophis considering the three densities (1.29, 2.2, and 3.5 g cm −3 ) and their topological structures. Point x (km) y (km) z (km) Topological classification Characteristic time (h) ρ = 1.29 g cm −3 E 1 0 .764033 − 0 .014211 − 0 .001470 Saddle–Centre–Centre 26.83192 26.97251 E 2 − 0 .038260 − 0 .752882 0 .000598 Centre–Centre–Centre 27.28828 28.90402 88.39612 E 3 − 0 .765131 − 0 .007504 − 0 .001544 Saddle–Centre–Centre 26.57328 26.90200 E 4 − 0 .041584 0 .752713 0 .000675 Centre–Centre–Centre 27.30720 28.79664 90.94898 ρ = 2.2 g cm −3 E 1 0 .910268 − 0 .013482 − 0 .001033 Saddle–Centre–Centre 26.98514 27.08051 E 2 − 0 .038718 − 0 .900849 0 .000424 Centre–Centre–Centre 27.31676 28.38482 107.46649 E 3 − 0 .911036 − 0 .007877 − 0 .001078 Saddle–Centre–Centre 26.82109 27.04486 E 4 − 0 .041482 0 .900733 0 .000470 Centre–Centre–Centre 27.32790 28.32686 110.03305 ρ = 3.5 g cm −3 E 1 1 .060747 − 0 .012966 − 0 .000760 Saddle–Centre–Centre 27.08273 27.15419 E 2 − 0 .039031 − 1 .052625 0 .000314 Centre–Centre–Centre 27.33415 28.09159 126.74930 E 3 − 1 .061310 − 0 .008169 − 0 .000789 Saddle–Centre–Centre 26.97445 27.13409 E 4 − 0 .041388 1 .052541 0 .000343 Centre–Centre–Centre 27.34118 28.05680 129.32771 densities, when we analysed the E 2 equilibrium point eigenvalues. The same conclusion can be applied to point E 4 . While for points E 1 and E 3 , there are two families of periodic orbits, which have periods of oscillation around 26.8 and 26.9 h for E 1 , and 26.6 and 26.9 h for E 3 , and these values undergo small changes, less than 30 min, for other Apophis densities (Table 1 ). Note that despite changing the density of Apophis, the topological classification of the equilibrium points E 2 and E 4 remains as linearly stable. Thus, we performed numerical simulations of a disc of particles encompassing the equilibrium points regions in order to identify possible stable zones around them. 4.2 Stability regions There have been some studies (Aljbaae et al. 2020 , 2021 ; Lang 2021 ) investigating suitable orbits around Apophis for a spacecraft taking into account the irregular gravitational field of the asteroid and the solar radiation pressure for an area to mass ratio similar to the OSIRES-REX spacecraft. Some stable orbits were found, but during the 2029 close encounter, the majority of the orbits for a spacecraft suffer a collision or ejection. The small particles around Apophis may be subject to a plethora of forces besides the gravitational potential of the main body: the oblateness coefficient add an extra gravitational pull and there are the tides by the Sun and the Earth at the close encounter. If the grains are small they can also experience the disturbance due to the solar radiation force. To estimate which perturbation may be rele v ant, we computed dimensional parameters that allow us to analyse the relative strength of each force (for a detailed definition of the parameters, see Hamilton & Krivov 1996 ; Moura et al. 2020 ). Fig. 8 shows the parameter strengths according to the distance from Apophis: the solar tide (green), Earth’s gravity at the closest approach distance (black), the solar radiation force for grains with radius of 1 cm (pink) and 15 cm MNRAS 510, 95–109 (2022) D ow nloaded from https://academ ic.oup.com /m nras/article/510/1/95/6427373 by U niversitaet Tuebingen user on 17 D ecem ber 2021 Chapter 2. The asteroid Apophis 37 102 G. Valvano et al. Figure 7. Location of the equilibrium points in the xoy plane for each density. The point’s colour represents the density of Apophis. Figure 8. Variation of the parameter oblateness (blue), Earth and Sun tide (black and green, respectively), the solar radiation pressure for a particle with a radius of 1 cm (pink) and 15 cm (red), and the Apophis gravity (yellow) as a function of the distance from Apophis. The shaded region indicates the variation of these parameters between the pericentre and apocentre. The vertical lines represent the limits of the radial location of the equilibrium points for the density model of 1.29 and 3.5 g cm −3 . The equi v alent radius, R equi v alent , of Apophis is 170 m. (red); it also shows the oblateness effect (blue) and the Apophis gravity (yellow). For each parameter, the lines are calculated for the nominal density (2.2 g cm −3 ) and distance equal to the semimajor axis, while the shaded region corresponds to the variation of these parameters between the pericentre and apocentre. In Fig. 8 , the vertical lines indicate the limits of the radial location of the equilibrium points for the density model of 1.29 and 3.5 g cm −3 . The Apophis gravity has the major magnitude among the other perturbations, about seven orders of magnitude larger than the Sun tide and five orders larger than the Earth tide at the closest approach distance. The oblateness of Apophis as the Sun tide reaches the same magnitude at a distance of about 9 radii of Apophis. The Earth tide is also lower than the oblateness near the asteroid and they both are comparable in the region of the equilibrium points. Beyond this distance the Earth tide dominates the dynamic o v er the oblateness. So, the Sun tide will not contribute with major perturbations to the system. The solar radiation pressure for a particle with a 1 cm of radius gets a larger order of magnitude than the oblateness at about 3 radii of Apophis, which means that this particle will suffer a major influence of the solar radiation pressure. Ho we ver, for a particle with a 15 cm of radius, the solar radiation pressure is about one order of magnitude smaller, and it is equi v alent to the oblateness near the region of the equilibrium points. Thus, a particle of 15 cm is more likely to survive in the nearby environment. In this work, we are concerned with natural objects, such as dust, boulders, fragments, or even larger bodies that might be orbiting around Apophis. Therefore, aiming to identify the size and location of possible stable regions around Apophis and/or around the equilibrium points, we performed sets of numerical simulations of particles around these regions. We numerically integrated a disc with 15 000 massless particles initially for 24 h (in order to compare with the close encounter with the Earth in 2029 shown in Fig. 2 ) and subsequently for 30 yr ( ∼10 000 times the rotation period of the asteroid). The particles were distributed at an initial distance of 300 m from the asteroid centre of mass and different widths for each density model, since the position of the equilibrium points changes as the density changes. For the 1.29, 2.2, and 3.5 g cm −3 densities, the final distances were 1.0, 1.15, and 1.3 km, respectively. All the particles were assumed to be initially with Keplerian angular velocity in the equatorial plane and circular orbits. To perform these simulations, we used an N -body integrator package called N-BOM (Moura et al. 2020 ; Winter et al. 2020 ). It considers the gravitational potential of an irregular body as a mass concentration model, MASCONS (Geissler et al. 1996 ). To reproduce the Apophis’ gravitational potential field, we calculate the sum of the gravitational potential of all the masses points as (Borderes-Motta & Winter 2018 ) U ( x , y , z) = N ∑ i= 1 Gm r i , (15) where N is the number of mass points (which is 20 457 in our model), r i is the distance mascon-particle, and m is the mass of each mascon. As the sum of all mascons is equal to the Apophis’ total mass, M Apophis = Nm , the mass of each mascon is different according to the density model. We considered a particle as ejected if it reaches a distance larger than 10 times the distance between the Apophis’ centre of mass and the equilibrium point, this is almost equi v alent to half of the Hill radius (Hamilton & Burns 1992 ) of Apophis with respect to the Sun. For the 24 h simulations, an additional criterion of ejection was considered since some particles may not have sufficient time to exceed the ejection distance due to the short time of the simulation. So we also considered as ejected these particles with positive energy. In this way, a region is called stable where the particles remain at the final integration time, obviously, without collision with the body or be ejected. Fig. 9 shows the initial conditions of the 15 000 particles, coloured according to their final outcome at the end of the 24 h of integration. The blue dots represent the particles that collided with Apophis, while the black dots indicate the particles that have survived the entire simulation. We note that the distribution of the particles that MNRAS 510, 95–109 (2022) D ow nloaded from https://academ ic.oup.com /m nras/article/510/1/95/6427373 by U niversitaet Tuebingen user on 17 D ecem ber 2021 Chapter 2. The asteroid Apophis 38 APOPHIS – effects of the 2029 Earth’s encounter 103 Figure 9. Initial conditions of the 15 000 particles around the asteroid Apophis in the xoy plane. The green dots represent the equilibrium points. The black dots represent the particles that survived after the 24 h of integration and the blue dots the particles that collide with the asteroid. The panels (a)–(c) represent, respectively, the densities of 1.29, 2.2, and 3.5 g cm −3 . collided with the surface is similar among the three density models, and they are located in two preferential regions closer to the surface of Apophis and near to the equatorial extremities of the body, so the gravitational field of the equatorial extremities is relate