Ph.D Thesis

You can download my Ph.D thesis by clicking on the following link

Variational methods and Hamiltonian perturbation theory applied to the N-body problem
A theoretical and computational approach

Publications

1. M. F., A. Carbognani: 2024. Long-term orbital evolution of Dimorphos boulders and implications on the origin of meteorites, Monthly Notices of the Royal Astronomical Society, Volume 528, Issue 4, Pages 6660–6665 (arXiv). See also related news on Media INAF, National Geographic, The Times, and IFL Science. Abstract ▿▵

   By using recent observations of the Dydimos-Dimorphos system from the Hubble Space Telescope, 37 boulders with a size of 4 to 7 meters ejected from the system during the impact with the DART spacecraft were identified. In this work, we studied the orbital evolution of a swarm of boulders with a similar size to that of the detected ones. By using recent estimates for the ejection velocity of the boulders, we numerically propagated the dynamics of the swarm for 20 kyr in the future. We found that the ejection velocities and the non-gravitational effects are not strong enough to change the secular evolution significantly. The minimum orbit intersection distance (MOID) with the Earth will be reached in about 2.5 kyr, but it will not fall below 0.02 au. On the contrary, the Mars MOID will be very small in four instances, two near 6 kyr and the other two near 15 kyr. Therefore, there may be a chance for them to impact Mars in the future. Given the rarefaction of the Martian atmosphere, we expect the boulders to arrive intact on the ground and excavate a small impact crater. The results presented here provide a further indication that some meteorites found on Earth originated in collisions of ∼100 m near-Earth asteroids with projectiles of ∼1 m in size.

   
   

2. M. F., M. Micheli, F. Gianotto, L. Faggioli, D. Oliviero, A. Porru, R. Rudawska, J. L. Cano, L. Conversi, R. Moissl: 2024. An automated procedure for the detection of the Yarkovsky effect and results from the ESA NEO Coordination Centre, Astronomy and Astrophysics 682, A29 (arXiv). Abstract ▿▵

   Context. The measurement of the Yarkovsky effect on near-Earth asteroids (NEAs) is common practice in orbit determination today, and the number of detections will increase with the developments of new and more accurate telescopic surveys. However, the process of finding new detections and identifying spurious ones is not yet automated, and it often relies on personal judgment.
   Aims. We aim to introduce a more automated procedure that can search for NEA candidates to measure the Yarkovsky effect, and that can identify spurious detections.
   Methods. The expected semi-major axis drift on an NEA caused by the Yarkovsky effect was computed with a Monte Carlo method on a statistical model of the physical parameters of the asteroid that relies on the most recent NEA population models and data. The expected drift was used to select candidates in which the Yarkovsky effect might be detected, according to the current knowledge of their orbit and the length of their observational arc. Then, a nongravitational acceleration along the transverse direction was estimated through orbit determination for each candidate. If the detected acceleration was statistically significant, we performed a statistical test to determine whether it was compatible with the Yarkovsky effect model. Finally, we determined the dependence on an isolated tracklet.
   Results. Among the known NEAs, our procedure automatically found 348 detections of the Yarkovsky effect that were accepted. The results are overall compatible with the predicted trend with the the inverse of the diameter, and the procedure appears to be efficient in identifying and rejecting spurious detections. This algorithm is now adopted by the ESA NEO Coordination Centre to periodically update the catalogue of NEAs with a measurable Yarkovsky effect, and the results are automatically posted on the web portal.

   
   

3. B. Novaković, M. F., D. Marčeta, D. Pavela: 2024. ASTERIA - Asteroid Thermal Inertia Analyzer, Planetary Science Journal, Vol. 5, Num. 11. (arXiv). The software developed for this paper can be downloaded from my GitHub page: ASTERIA. Abstract ▿▵

   Thermal inertia estimates are available for a limited number of a few hundred objects, and the results are practically solely based on thermophysical modeling (TPM). We present a novel thermal inertia estimation method, Asteroid Thermal Inertia Analyzer (ASTERIA). The core of the ASTERIA model is the Monte Carlo approach, based on the Yarkovsky drift detection. We validate our model on asteroid Bennu plus ten well-characterized near-Earth asteroids (NEAs) for which a good estimation of the thermal inertia from the TPM exists. The tests show that the ASTERIA provides reliable results consistent with the literature values. The new method is independent from the TPM, allowing an independent verification of the results. As the Yarkovsky effect is more pronounced in small asteroids, the noteworthy advantage of the ASTERIA compared to the TPM is the ability to work with smaller asteroids for which TPM typically lacks the input data. We used the ASTERIA to estimate the thermal inertia of 38 NEAs, with 31 of them being sub-km asteroids. Twenty-nine objects in our sample are characterized as Potentially Hazardous Asteroids. On the limitation side, the ASTERIA is somewhat less accurate than the TPM. The applicability of our model is limited to NEAs, as the Yarkovsky effect is yet to be detected in main-belt asteroids. However, we can expect a significant increase in high-quality measurements of the input parameters relevant to the ASTERIA with upcoming surveys. This will surely increase the reliability of the results generated by the ASTERIA and widen the model’s applicability.

   
   

4. A. Carbognani, M. F.: 2023. Identifying parent bodies of meteorites among near-Earth asteroids, Monthly Notices of the Royal Astronomical Society, Volume 525, Issue 2, 1705–1725 (arXiv). See also related news on Media INAF and Universe Today. Abstract ▿▵

   Meteorites provide an important source of information about the formation and composition of asteroids, because the level of accuracy of studies and analyses performed in a laboratory cannot be achieved by any ground or space based observation. To better understand what asteroid types a meteorite represents, it is crucial to identify the body they originated from. In this paper, we aim to determine possible parent bodies for the known meteorite falls among the known population of near-Earth asteroids (NEAs). By using the similarity criterion 𝐷𝑁, based on geocentric quantities, we found 20 possible NEA-meteorite pairs. By performing additional numerical simulations of the backward dynamics, we found that 12 of these pairs may be associated with a possible separation event from the progenitor NEA, while the remaining 8 pairs appear to be ambiguous or random associations. The most interesting are the Pribram and Neuschwanstein meteorites, which are dynamically associated with (482488) 2012SW20 with a common separation age dating back to about 20−30 kyr ago, and the Motopi Pan meteorite, that has three candidate parent bodies: (454100) 2013BO73, 2017MC3, and 2009FZ4. The average time of separation between our meteorite list and the progenitor body appears to be about 10 kyr, a time consistent with what is expected from the collision frequency of small NEAs. Based on our results, we suggest that about 20−30 per cent of meteorites do not originate in the main belt, but mainly from little collision events happening between NEAs in the inner Solar System.

   
   

5. M. F., B. Novaković, D. Marčeta: 2023. The low surface thermal inertia of the rapidly rotating near-Earth asteroid 2016 GE1, Astronomy and Astrophysics 675, A134 (arXiv). Abstract ▿▵

   Context. Asteroids smaller than about 100 meters are observed to rotate very fast, with periods often much shorter than the critical spin limit of 2.2 h. Some of these super-fast rotators can also achieve a very large semi-major axis drift induced by the Yarkovsky effect, that in turn, is determined by internal and surface physical properties.
   Aims. We consider here a small super-fast rotating near-Earth asteroid, designated as 2016 GE1. This object rotates in just about 34 seconds, and a large Yarkovsky effect has been determined from astrometry. By using those results, we aim to constrain the thermal inertia of the surface of this extreme object.
   Methods. We used a recently developed statistical method to determine the thermal properties of near-Earth asteroids. The method is based on the comparison between the observed and the modelled Yarkovsky effect, and the thermal conductivity (inertia) is determined by a Monte Carlo approach. Parameters of the Yarkovsky effect model are either fixed if their uncertainty is negligible, modelled with a Gaussian distribution of the errors if they are measured, or deduced from general properties of the population of near-Earth asteroids when they are unknown.
   Results. Using a well-established orbit determination procedure, we determined the Yarkovsky effect on 2016 GE1, and verified a significant semi-major axis drift rate. Using a statistical method, we showed that this semi-major axis drift rate could be explained only by low thermal inertia values below 100 J m^-2 K^-1 s^-1/2. We benchmarked our statistical method using the well-characterised asteroid Bennu and found that the sole knowledge of semi-major axis drift rate and rotation period is generally insufficient to determine the thermal inertia. However, when the statistical method is applied to super-fast rotators, we found that the measured Yarkovsky effect can be achieved only for very low values of thermal inertia: namely, 90\% of the probability density function of the model outcomes is contained at values smaller than 100 J m^-2 K^-1 s^-1/2.
   Conclusion. We propose two possible interpretations for the extremely low thermal inertia of 2016 GE1: a high porosity or a cracked surface, or a thin layer of fine regolith on the surface. Though this seems somewhat unexpected in either case, it opens up the possibility of a subclass of low inertia, super-fast rotating asteroids.

   
   

6. M. F., G. F. Gronchi, B. Novaković: 2023. Maps of secular resonances in the NEO region, Astronomy and Astrophysics 672, A39 (arXiv). Abstract ▿▵

   Context. From numerical simulations, it is known that some secular resonances may affect the motion of near-Earth objects (NEOs). However, the specific location of the secular resonance inside the NEO region is not fully known, because the methods previously used to predict their location can not be used for highly eccentric orbits and when the NEOs cross the orbits of the planets.
   Aims. In this paper, we aim to map the secular resonances with the planets from Venus to Saturn in the NEO region, even for high values of the eccentricity.
   Methods. We used an averaged semi-analytical model that can deal with orbit crossing singularities for the computation of the secular dynamics of NEOs, from which we can obtain suitable proper elements and proper frequencies. Then, we computed the proper frequencies over a uniform grid in the proper elements space. Secular resonances are thus located by the level curves corresponding to the proper frequencies of the planets.
   Results. We determined the location of the secular resonances with the planets from Venus to Saturn, showing that they appear well inside the NEO region. By using full numerical N-body simulations we also showed that the location predicted by our method is fairly accurate. Finally, we provided some indications about possible dynamical paths inside the NEO region, due to the presence of secular resonances.

   
   

7. Z. Zhong, J. Yan, S. Chen, L. Liu, M. F., Q. Wen: 2022. The Likely Thermal Evolution of the Irregularly Shaped S-Type Astraea Asteroid, Remote Sensing 14, 6320. Abstract ▿▵

   The thermal evolution of asteroids provides information on the thermal processes of the protoplanetary disk. Since irregular bodies have a large surface subject to fast heat loss, we used the finite element method (FEM) to explore the likely thermal pathways of one of these bodies. To test our FEM approach, we compared the FEM to another algorithm, the finite difference method (FDM). The results show that the two methods calculated a similar temperature magnitude at the same evolutionary time, especially at the stage when the models had temperatures around 800 K. Furthermore, this investigation revealed a slight difference between the methods that commences with a declining temperature, particularly around the center of the model. The difference is associated with the tiny thickness of the boundary used in the FDM, whereas the FEM does not consider the thickness of the boundary due to its self-adapting grid. Using the shape data provided by DAMIT, we further explored the likely thermal evolution pathway of the S-type asteroid Astraea by considering the radionuclide 26Al. Since we only focused on the thermal pathways of conduction, we considered that the accretion lasts 2.5 Ma (1 Ma = 1,000,000 years) by assuming that Astraea has not experienced iron melting. The results show a high interior temperature area with a shape similar to the shape of Astraea, indicating the influence of the irregular shape on thermal evolution. The interior of Astraea achieved the highest temperature after 4.925 Ma from the accretion of planetesimals. After that time of high temperature, Astraea gradually cooled and existed more than 50 Ma before its heat balanced approximately to its external space. We did not find signs of apparent fast cooling along the shortest z-axis as in previous studies, which could be due to the hidden differences in the distances along the axes. The methodology developed in this paper performs effectively and can be applied to study the thermal pathways of other asteroids with irregular shapes.

   
   

8. M. F., G. F. Gronchi, M. Saillenfest: 2022. Proper elements for resonant planet-crossing asteroids, Celestial Mechanics and Dynamical Astronomy 134 (arXiv). Abstract ▿▵

   Proper elements are quasi-integrals of motion, meaning that they can be considered constant over a certain timespan, and they permit to describe the long-term evolution with a few parameters. Near-Earth objects (NEOs) generally have a large eccentricity and therefore they can cross the orbits of the planets. Moreover, some of them are known to be currently in a mean-motion resonance with a planet. Thus, the methods previously used for the computation of main-belt asteroid proper elements are not appropriate for such objects. In this paper, we introduce a technique for the computation of proper elements of planet- crossing asteroids that are in a mean motion resonance with a planet. First, we numerically average the Hamiltonian over the fast angles while keeping all the resonant terms, and we describe how to continue a solution beyond orbit crossing singularities. Proper elements are then extracted from a frequency analysis of the averaged orbit-crossing solutions. We give proper elements of some known resonant NEOs, and provide comparisons with non- resonant models. These examples show that it is necessary to take into account the effect of the resonance for the computation of accurate proper elements.

   
   

9. M. F., B. Novaković: 2022. Mercury and OrbFit packages for the numerical integration of planetary systems: implementation of the Yarkovsky and YORP effects, Serbian Astronomical Journal 204, pp 51-63 (arXiv). The software developed for this paper can be downloaded from my GitHub page: OrbFit, mercury. Abstract ▿▵

   For studies of the long-term evolution of small Solar System objects, it is fundamental to add the Yarkovsky and Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effects in the dynamical model. Still, the implementation of these effects in publicly available N-body codes is often lacking, or the effects are implemented using very simplified models. In this paper, we present an implementation of the coupled Yarkovsky/YORP effects in the mercury and OrbFit N-body codes. Along with these two effects, to properly model the asteroid spin state we have also included the effects of collisions and breakups. Given the stochastic nature of many effects included, the software is suitable for statistical dynamical studies. Here we explained the scientific aspect of the implementation, while technical details will be made freely available along with the source codes.

   
   

10. L. Asselle, M. F., A. Portaluri: 2022. Bifurcations of balanced configurations for the Newtonian n-body problem in R⁴, Journal of Fixed Point Theory and Applications 24, 22 (Symplectic geometry - A Festschrift in honour of Claude Viterbo’s 60th birthday) (arXiv). Additional material for this paper can be found here. Abstract ▿▵

    For the gravitational n-body problem, the simplest motions are provided by those rigid motions in which each body moves along a Keplerian orbit and the shape of the system is a constant (up to rotations and scalings) configuration featuring suitable properties. While in dimension d≤3 the configuration must be central, in dimension d≥4 new possibilities arise due to the complexity of the orthogonal group, and indeed there is a wider class of S-balanced configurations, containing central ones, which yield simple solutions of the n-body problem. Starting from recent results of the first and third authors, we study the existence of continua of bifurcations branching from a trivial branch of collinear S-balanced configurations and provide an estimate from below on the number of bifurcation instants. In the last part of the paper, by using the continuation method, we explicitly display the bifurcation branches in the case of the three body problem for different choices of the masses.

    
    

11. M. F.: 2022. Local minimality properties of circular motions in 1/r^α potentials and of the figure-eight solution of the 3-body problem, Partial Differential Equations and Applications, Volume 3, 10 (arXiv). Abstract ▿▵

    We first take into account variational problems with periodic boundary conditions, and briefly recall some sufficient conditions for a periodic solution of the Euler-Lagrange equation to be either a directional, a weak, or a strong local minimizer. We then apply the theory to circular orbits of the Kepler problem with potentials of type 1/r^α, α > 0. By using numerical computations, we show that circular solutions are strong local minimizers for α > 1, while they are saddle points for α ∈ (0,1). Moreover, we show that for α ∈ (1,2) the global minimizer of the action over periodic curves with degree 2 with respect to the origin could be achieved on non-collision and non-circular solutions. After, we take into account the figure-eight solution of the 3-body problem, and we show that it is a strong local minimizer over a particular set of symmetric periodic loops.

    
    

12. M. F., B. Novaković: 2021. The role of the Yarkovsky effect in the long-term dynamics of asteroid (469219) Kamo'oalewa, The Astronomical Journal, Volume 162, Number 6 (arXiv). Abstract ▿▵

    Near-Earth asteroid (469219) Kamo'oalewa (aka 2016 HO3) is an Earth co-orbital and a potential space mission target. Its short-term dynamics is characterized by a periodic switching between quasi-satellite and horseshoe configurations. Due to its small diameter of only about 36 meters, the Yarkovsky effect may play a significant role in the long-term dynamics. In this work, we addressed this issue by studying the changes in the long-term motion of Kamo'oalewa caused by the Yarkovsky effect. We used an estimation of the magnitude of the Yarkovsky effect assuming different surface compositions and introduced the semi-major axis drift by propagating orbits of test particles representing the clones of the nominal orbit. Our simulations showed that the Yarkovsky effect may cause Kamo'oalewa to exit from the Earth co-orbital region a bit faster when compared to a purely gravitational model. Nevertheless, it still could remain an Earth companion for at least 0.5 My in the future. Our results imply that Kamo'oalewa is the most stable Earth's co-orbital object known so far, not only from a short-term perspective but also on long time scales.

    
    

13. M. F., B. Novaković, D. Vokrouhlický, R. J. Weryk: 2021. Low thermal conductivity of the super-fast rotator (499998) 2011 PT, Astronomy and Astrophysics 647, A61 (arXiv). Abstract ▿▵

    Context. Asteroids with a diameter of up to a few dozen meters may spin very fast and complete an entire rotation within a few minutes. These small and fast-rotating bodies are thought to be monolithic objects because the gravitational force due to their small size is not strong enough to counteract the strong centripetal force caused by the fast rotation. This argument means that the rubble-pile structure is not feasible for these objects. Additionally, it is not clear whether the fast spin prevents dust and small particles (regolith) from being kept on their surface.
    Aims. We develop a model for constraining the thermal conductivity of the surface of the small, fast-rotating near-Earth asteroids. This model may suggest whether regolith is likely present on these objects.
    Methods. Our approach is based on the comparison of the measured Yarkovsky drift and a predicted value using a theoretical model that depends on the orbital, physical and thermal parameters of the object. The necessary parameters are either deduced from statistical distribution derived for near-Earth asteroids population or determined from observations with associated uncertainty. With this information, we performed Monte Carlo simulations and produced a probability density distribution for the thermal conductivity.
    Results. Applying our model to the superfast rotator asteroid (499998) 2011 PT, we find that the measured Yarkovsky drift can only be achieved when the thermal conductivity K of the surface is low. The resulting probability density function for the conductivity is bimodal, with two most likely values being around 0.0001 and 0.005 W m⁻¹ K⁻¹. Based on this, we find that the probability that K is lower than 0.1 W m⁻¹ K⁻¹ is at least 95%. This low thermal conductivity might indicate that the surface of 2011 PT is covered with a thermal insulating layer, composed of a regolith-like material similar to lunar dust.

    
    

14. M. F., G. F. Gronchi: 2021. Symmetric constellations of satellites moving around a central body of large mass, Journal of Dynamics and Differential Equations 35, pp. 1511–1559 (arXiv). Abstract ▿▵

    We consider a (1+N)-body problem in which one particle has mass m₀≫1 and the remaining N have unitary mass. We can assume that the body with larger mass (central body) is at rest at the origin, coinciding with the center of mass of the N bodies with smaller masses (satellites). The interaction force between two particles is defined through a potential of the form U ∼ 1/r^α, where α ∈ [1,2) and r is the distance between the particles. Imposing symmetry and topological constraints, we search for periodic orbits of this system by variational methods. Moreover, we use Gamma-convergence theory to study the asymptotic behaviour of these orbits, as the mass of the central body increases. It turns out that the Lagrangian action functional Gamma-converges to the action functional of a Kepler problem, defined on a suitable set of loops. In some cases, minimizers of the Gamma-limit problem can be easily found, and they are useful to understand the motion of the satellites for large values of m₀≫1. We discuss some examples, where the symmetry is defined by an action of the groups Z4, Z2 x Z2, and the rotation groups of Platonic polyhedra on the set of loops.

    
    

15. G. Lari, M. Saillenfest, M. F.: 2020. Long-term evolution of the Galileian satellites: the capture of Callisto into resonance , Astronomy and Astrophysics 639, A40 (arXiv). Abstract ▿▵

    Context. The strong tidal dissipation in the couple Jupiter-Io is spread to all the moons involved in the Laplace resonance (Io, Europa, and Ganymede), leading to a migration of their orbits.
    Aims. We aim to characterize the future behavior of the Galilean satellites over the Solar System lifetime and to quantify the stability of the Laplace resonance. Since tidal dissipation makes possible the exit from the current resonances or capture into new ones, we investigate the capture of Callisto into resonance.
    Methods. We perform hundreds of propagations using an improved version of a recent semi-analytical model. As Ganymede moves outwards, it approaches the 2:1 resonance with Callisto, inducing a temporary chaotic motion in the system. For this reason, we draw a statistical picture of the outcome of the resonant encounter.
    Results. The system can settle into two distinct outcomes: A) a chain of three 2:1 two-body resonances (Io-Europa, Europa-Ganymede and Ganymede-Callisto), or B) a resonant chain involving the 2:1 two-body resonance Io-Europa plus at least one pure 4:2:1 three-body resonance, most frequently between Europa, Ganymede and Callisto. In case A (56\% of the simulations), the Laplace resonance is always preserved and the eccentricities remain confined to small values below 0.01. In case B (44\% of the simulations), the Laplace resonance is generally disrupted and the eccentricities of Ganymede and Callisto can increase up to about 0.1, making this configuration unstable and driving the system into new resonances.
    Conclusion. From our results, the capture of Callisto into resonance appears to be extremely likely (100\% of our simulations). Assuming the most recent estimate of the dissipation between Io and Jupiter, the resonant encounter happens at about 1.5 Gyrs from now. Therefore, the stability of the Laplace resonance is guaranteed at least up to about 1.5 Gyrs.

    
    

16. M. F., A. Jorba: 2019. Braids with the symmetries of Platonic polyhedra in the Coulomb (N+1)-body problem , Communications in Nonlinear Science and Numerical Simulation 83, 105105 (arXiv). Abstract ▿▵

    We take into account the Coulomb (N+1)-body problem with N = 12, 24, 60. One of the particles has positive charge Q > 0, and the remaining N have all the same negative charge q < 0. These particles move under the Coulomb force, and the positive charge is assumed to be at rest at the center of mass. Imposing a symmetry constraint, given by the symmetry group of the Platonic polyhedra, we were able to compute periodic orbits, using a shooting method and continuation with respect to the value Q of the positive charge.
    In the setting of the classical N-body problem, the existence of such orbits is proved with Calculus of Variation techniques, by minimizing the action functional. Here this approach does not seem to work, and numerical computations show that the orbits we compute are not minimizers of the action.

    
    

17. M. F., G. F. Gronchi: 2018. On the stability of periodic N-body motions with the symmetry of Platonic polyhedra, Nonlinearity, Vol. 31, Num. 11, pp 4935-4954 (arXiv). Abstract ▿▵

    In (Fusco et. al., 2011) several periodic orbits of the Newtonian N-body problem have been found as minimizers of the Lagrangian action in suitable sets of T-periodic loops, for a given T>0. Each of them share the symmetry of one Platonic polyhedron. In this paper we first present an algorithm to enumerate all the orbits that can be found following the proof in (Fusco et. al., 2011). Then we describe a procedure aimed to compute them and study their stability. Our computations suggest that all these periodic orbits are unstable. For some cases we produce a computer-assisted proof of their instability using multiple precision interval arithmetic.

    
    

Visits and Seminars

• From 15th April to 15th July 2022 I was a visiting researcher at the European Space Research and Technology Centre (ESTEC) of the European Space Agency (ESA), under the supervision of Prof. Detlef Koschny.

• From 11th January to 11th April 2021 I was a visiting researcher at the University of Pisa within the Stardust-R project, under the supervision of Prof. G. F. Gronchi.

• On 8th July 2020 I held a seminar titled "Numerical methods for the computation of symmetric periodic orbits of the N-body problem", in a serie organized by the I-CELMECH project.

• In the period from 23rd October 2019 to 25th October 2019 I visited the Departement of Mathematics "Giuseppe Peano" of the University of Turin (Italy). During this week I gave a presentation with title "Symmetric constellations of satellites around a massive body: a Gamma-convergence approach".

• In the period from 1st October 2018 to 31st January 2019 I visited the University of Barcelona (Spain), under the supervision of Prof. Á. Jorba.