On the stability of periodic N-body motions with the symmetry of Platonic polyhedra

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

On the linear stability of some periodic orbits of the N-body problem with the simmetry of Platonic polyhedra

In the last few years, many interesting periodic motions of the classical Newtonian N-body problem have been discovered as minimizers of the Lagrangian action functional on a particular subset of T-periodic loops. The interest in this classical problem was revived by the numerical discovery of the now famous figure eight solution of the three-body problem, by C. Moore in 1993. In 2000 A. Chenciner and R. Montgomery rediscover this particular orbit, giving a formal proof of its existence that uses the direct method of Calculus of Variations. The figure eight is a first example of a N-body choreography, that is a solution of the classical N-body problem in which N equal masses chase each other around a fixed closed curve, equally spaced in phase along the curve: since 2000 many other choreographies that present a strong symmetrical structure have been found. Moreover, in 2007 T. Kapela and C. Simò proved the linear stability of the figure eight: this fact is quite surprising, since that no other stable choreographies are known. In this thesis we prove the existence of a number of periodic motions of the classical N-body problem which, up to relabeling of the N particles, are invariant under the rotation group R of one of the five Platonic polyhedra. The number N coincides with the order of the rotation group R and the particles have all the same mass. We use again variational techniques to minimize the Lagrangian action A on a suitable subset K of the H^1 T-periodic maps. The set K is a cone and is determined by imposing on $u$ both topological and symmetry constraints which are defined in terms of the rotation group R. For a certain number of cones K, using level estimates and local perturbations, we show that minimizers are free of collisions and therefore they are classical solutions on the N-body problem. A natural question that comes out in presence of a periodic orbit is whether is it stable or not. To perform a study of the linear stability we use numerical methods, since our problem is not integrable. In fact we know only that periodic orbits with the previous symmetries exist, but we do not have their analytic expression. These particular solutions were found numerically, using a method described by C. Moore and called relaxation dynamics. The numerical implementation of this method boils down to a gradient search of the minima in some finite-dimensional approximation of the path space: in short, it is a numerical implementation of a direct method of Calculus of Variations. Starting from these numerical solutions, we can propagate numerically the variational equation in order to produce an approximation of the monodromy matrix, from which we can determine the linear stability studying its spectral properties: this is a first method that we develop. However, because of the convergence of the gradient search is slow, especially when the orbit presents some close approaches, this method could result inefficient. An alternative approach is to find an initial condition of the periodic orbit and then propagate numerically the equation of motion and the variational equation coupled together. A classical method to find initial conditions is the well known multiple shooting method. This method has been successfully used by T. Kapela and C. Simò to find initial conditions for the figure eight and some other non-symmetric choreographies. However, since this is an iterative method too, it could fail to converge and this typically happens when the orbit passes close to a collisions. Therefore, it is clear that the problem of close approaches must be treated with more care. The thesis is structured as follows: Chapter 1. It contains results on the existence of periodic orbits of the classical N-body problem with the symmetries of Platonic polyhedra. Chapter 2. In this chapter we try to develop an automatic procedure in order to find all the periodic orbits described in Chapter 1. Chapter 3. We present the classical theory of linear stability for periodic solutions of autonomous systems. In particular, we introduce here the monodromy matrix, the Floquet multipliers and the Poincaré map. Chapter 4. It is the heart of the work, in which we develop the two different numerical methods to study the linear stability of periodic orbits found in Chapters 1 and 2. Tests of the software written are reported at the end of the chapter. Chapter 5. In this chapter we list all the results obtained with our software, from which we can get some conclusions. At the end we suggest some improvements of the methods of Chapter 4 and of the software, which could represent a continuation of the present work, in order to produce a true computer assisted proof of stability or instability of these orbits

The low surface thermal inertia of the rapidly rotating near-Earth asteroid 2016 GE1

Asteroids smaller than about 100 meters are observed to rotate very fast, with periods often much shorter than the critical 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. We consider the small super-fast rotating near-Earth asteroid 2016 GE1. This object rotates in just 34 seconds, and a large Yarkovsky effect has been determined from astrometry. Here we aim to constrain the thermal inertia of the surface of this extreme object. 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. 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$^{-1/2}$: namely, 90\% of the probability density function of the model outcomes is contained at values smaller than 100 J m$^{-2}$ K$^{-1}$ s$^{-1/2}$. We propose two possible interpretations for the extremely low values: a high porosity or a cracked surface, or a thin layer of fine regolith on the surface. Though this seems unexpected in either case, it opens up the possibility of a subclass of low thermal inertia, super-fast rotating asteroids.Comment: Accepted for publication on A&

Low thermal conductivity of the superfast rotator (499998) 2011 PT

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. 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.Comment: Accepted for publication in A&A. 13 pages, 7 figures, 2 table

Assessment of flyby methods as applied to close encounters among asteroids

Orbital flybys have been extensively studied for spacecraft missions, resulting in effective mathematical and physical models. However, these models’ applicability to natural encounters involving asteroids has not been explored. This paper examines the applicability of two such theories, patched conics (PC) and the Keplerian map (KM), to asteroid encounters. A review of the two methods will be provided, highlighting their assumptions and range of applicability. Simulations of asteroid–asteroid encounters will then be performed to evaluate their effectiveness in these scenarios. The simulation parameters are set by collecting data on actual asteroid–asteroid encounters, hereby presented, generally characterised by high close approach distances and small masses of the perturbing bodies, if compared to those used to build the flyby theories. Results show that the PC theory’s effectiveness diminishes with increasing approach distances, aligning with its assumptions. Moreover, the prediction of the model is better in the geometric configurations where the flyby has major effects on the orbital energy change. The KM theory has shown good effectiveness for encounters occurring outside the sphere of influence of the perturbing body, even for very high distances. This research investigates flyby models’ strengths and weaknesses in asteroid encounters, offering practical insights and future directions.European Space Agency (ESA) Open Space Innovation Platform (OSIP) campaign and by Cranfield University (ESA Contract no. 4000134762/21/NL/MH/hm-Asteroid Collisions).Aerospac

An automated procedure for the detection of the Yarkovsky effect and results from the ESA NEO Coordination Centre

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.Comment: Accepted for publication on A&

Fusing Acoustic Ranges and Inertial Measurements in AUV Navigation: the Typhoon AUV at CommsNet13 Sea Trial

The paper presents some experimental results of autonomous underwater navigation, based on the fusion of acoustic and inertial measurements. The work is in the framework of the Thesaurus project, funded by the Tuscany Region, aiming at developing techniques for systematic exploration of marine areas of archaeological interest through a team of Autonomous Underwater Vehicles (AUVs). The test was carried out with one Typhoon vehicle, a 300m depth rated AUV with acoustic communication capabilities, during the CommsNet13 experiment, organized and scientifically coordinated by the NATO S&T Org. Ctr. for Maritime Research and Experimentation (CMRE, formerly NURC), with the participation of several research institutions. The fusion algorithm is formally casted into an optimal stochastic filtering problem, where the rough estimation of the vehicle position, velocity and attitude, are refined by using the depth measurement, the relative measurements available on the acoustic channel and the vehicle surge speed