**Organizers**: Marco Sansottera and Giulio Baù.

## Upcoming Talks

**Thursday 10 June****2021, 4.30 pm (Rome local time)****Sylvio Ferraz Mello**(University of Padua)*Exoplanets: Resonant chains and Apsidal Corotation resonances*### Show abstract

Updated lists of the discovered extrasolar planets a.k.a. exoplanets, record, presently, 4759 planets, 783 of which belonging to multi planet systems. They present an extraordinary diversity and many of these systems may be examples of naturally engendered solutions of the gravitational N-body problem. In this lecture we will consider the so-called resonant chains showing several examples that are solution candidates of the 1+3, 1+4 and 1+5 problems. In the more simple 1+2 case we will discuss some known resonant periodic solutions of the planetary 3-body problem, found among the known planets.

## Previous Talks

**Thursday 27 May****2021, 4.30 pm (Rome local time)****Christos Efthymiopoulos**(University of Padua)*Closed-form perturbation theory with Lie series*

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Closed form perturbation theory provides a framework for series calculations in perturbed Keplerian problems without expansions in powers of the bodies’ orbital eccentricities. The talk will review the subject starting from the well known (since Brouwer’s and Deprit’s works) case of Delaunay normalization. We will then present possible extensions of the method in the case of the three-body problem, as well as in the case of satellite motions around a planet when passage to a rotating frame is required. We will finally discuss applications of the method for a semi-analytical theory of the Earth’s or Moon’s satellite orbits, as well as of small bodies orbiting within the Sun-Jupiter system. In the latter case, we will discuss a variant of the closed-form method that avoids the use of relegation.

**Thursday 13 May****2021, 4.30 pm (Rome local time)****Irene Cavallari**(Università di Pisa)*On the Sun-shadow dynamics*

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The patched conics approximation is one of the most classical methods adopted to design spacecraft trajectories in multi-body environments. The subject of the presentation concerns the results that we have obtained by considering the planar motion of a mass particle in a force field defined by patching Kepler’s and Stark’s dynamics. This model is called “Sun-shadow dynamics”, referring to the motion of an Earth satellite perturbed by the solar radiation pressure taking into account the Earth shadow effect. The existence of periodic orbits of “brake” type is proved, and the Sun-shadow dynamics is investigated by means of a Poincaré-like map defined by a quantity that is not conserved along the flow. The results of numerical investigations on the map are also presented.

This is a joint work with Giulio Baù and Giovanni Federico Gronchi.

**Thursday 29 April****2021, 4.30 pm (Rome local time)****Federica Spoto**(Harvard & Smithsonian Center for Astrophysics)*High precision asteroid astrometry: challenges and results*

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**Abstract**

The ESA Gaia mission, launched in December 2013 and currently surveying the sky from the L2 Lagrangian point, has shown for the first time that it’s possible to obtain high precision astrometry for asteroids (Spoto et al. 2018, Gaia Data Release 2) even wiout the use of radar observations. This opens new perspectives in the study of the asteroid populations in the main belt, in particular for what concerns asteroid families.

But on the other side, the second Gaia Data Release (April 2018) has also proved that the available orbit determination algorithms had visible limitations when dealing with such high precision astrometry. An important step that we made consisted in the creation of a completed new weighting scheme that allow us to perform orbit determination without the needed of revise the existent old astrometry.

I will present the latest results on the ESA Gaia mission, the new weighting scheme we have developed and some recent applications to asteroid occultations and to the measurement of the Yarkovsky effect. I will also show how this last point is extremely important in the study of the chronology of the collisional events in the Solar System.

**Thursday 15 April****2021, 4.30 pm (Rome local time)**(IMCCE, Observatoire de Paris)**Philippe Robutel****TOI-178: from a hypothetical co-orbital system to planets in a chain of Laplace resonances**

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**Abstract**

The star TOI-178-a has been the subject of particular attention since the first TESS (Transiting Exoplanet Survey Satellite) observations, in August 2018, hinted at the possible presence of three planets near a 2:3:3 resonant chain. On the basis of these data, TOI-178 was identified as being able to harbor a co-orbital system with two planets oscillating around the same period of about 10 days. After CHEOPS (CHaracterising ExOPlanet Satellite) mission visits the system in August 2020, this planetary system appeared under a new light: no more co-orbital system, but at least six planets, five of which are trapped in a Laplace resonance chain. Although our first hypothesis is not the right one, we will take the opportunity to discuss co-orbital dynamics, and show how it has allowed us to define CHEOPS observation strategy for this system. We will then talk about the dynamics in Laplace resonance chain as observed by CHEOPS mission.

**Thursday 1 April****2021, 4.30 pm (Rome local time**)**Melaine Saillenfest**(IMCCE, Observatoire de Paris)*The large obliquity of Saturn explained by the fast migration of Titan*### Show abstract

**Abstract**

The large tilt of Saturn’s spin axis (27°) as compared to Jupiter’s (3°) requires a dynamical explanation. For decades, it was thought that Saturn had tilted during the late planetary migration, more than four gigayears ago, because of a capture in secular spin-orbit resonance. However, recent studies show that this traditional picture is at odds with the observed orbital expansion of Saturn’s biggest moon, Titan. Instead, the resonance has probably been encountered recently, and Saturn may still be tilting today. In this talk, I will explain how a migrating satellite can tilt its host planet, with particular focus on the most likely evolution pathway followed by Saturn and Titan. I will also discuss the implications of these findings for other bodies, including Jupiter and exoplanets.

**Thursday 18 March****2021, 4.30 pm (Rome local time)**(Università di Pisa)**Daniele Serra***Approximate symmetries in the BepiColombo radio science experiment: a numerical approach*

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**Abstract**

The problem of the determination of the orbit of a celestial body given a set of observations was very efficiently solved in the XIX century by C.F. Gauss on the occasion of the discovery of Ceres. As a matter of fact, the ideas and the main techniques introduced by the German mathematician are still used nowadays in the data analysis of interplanetary space missions. Although the mathematics of orbit determination is relatively simple – mainly consisting of linear algebra and iterative procedures for the computation of the roots of a function – there are a number of problems of easy formulation that do not always find an easy solution. One of these is the problem of identifying possible symmetries in the space of the parameters which would make the whole orbit determination underdetermined, thus impossible to achieve in practice. Exact symmetries are only found in theoretical situations: in real applications there are usually some perturbations that break the exact symmetry, leaving an approximate one. Nevertheless, such approximate symmetry can be detrimental to the determination of the parameters of interest, thus it is interesting to identify it and study possible ways to prevent its disruptive effects. In this talk we will focus on reviewing the approximate symmetries of the BepiColombo radio science experiment, and the exact symmetries of the three-body problem they derive from. In an attempt to assess the impact on the exact symmetry of the various sources of perturbations, we will present a toy model which approximates the real experiment and enables us to quantify numerically the degree of approximation of these symmetries. This is still a work in progress, in collaboration with Giulia Schettino and Giacomo Tommei.

**Thursday 18 February****2021, 4.30 pm (Rome local time)****Tudor Vartolomei**(Università degli Studi di Roma “Tor Vergata”)**Proper Elements for Space Debris – Computational Methods and Applications**

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**Abstract**

Proper elements are quasi-invariants of a Hamiltonian system, obtained through a normalization procedure. Proper elements have been successfully used to identify families of asteroids, sharing the same dynamical properties. We show that proper elements can also be used within space debris dynamics to identify groups of fragments associated to the same break-up event. The proposed method allows to reconstruct the evolutionary history and possibly to associate the fragments to a parent body. At the end of the procedure, the results will consist in a set of new elements that are quasi-invariants of motion, namely they are almost constant for a long period of time. The concentration of proper elements in specific regions in the phase space allows to identify families of fragments associated to the parent body.

**Thursday 4 February****2021, 4.30 pm (Rome local time)****Benedetto Scoppola**(Università degli Studi di Roma “Tor Vergata”)**Tides and dumbbell dynamics**### Show abstract

**Abstract**

We discuss a model describing the effects of tidal dissipation on satellite’s orbits. Tidal bulges are described in terms of a dumbbell, coupled to the rotation by a dissipative interaction. The assumptions on this dissipative coupling turns out to be crucial in the evolution of the system.

Joint work with Alessio Troiani and Matteo Veglianti.

**Thursday 17 December****2020, 4.30 pm (Rome local time)****Giuseppe Pucacco**(Università degli Studi di Roma “Tor Vergata”)**Normal forms for the Laplace resonance**

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**Abstract**

We describe a comprehensive model for systems locked in the Laplace resonance. The framework is based on the simplest possible dynamical structure provided by the Keplerian problem perturbed by the resonant coupling truncated at first order in the eccentricities. The reduced Hamiltonian, constructed by a transformation to resonant coordinates, is then submitted to a suitable ordering of the terms and to the study of its equilibria. Henceforth, resonant normal forms are computed. The main result is the identification of two different classes of equilibria. In the first class, only one kind of stable equilibrium is present: the paradigmatic case is that of the Galilean system. In the second class, three kinds of stable equilibria are possible and, at least one of them, is characterised by a high value of the forced eccentricity for the ‘first planet’: here the paradigmatic case is the exo-planetary system GJ- 876. The normal form obtained by averaging with respect to the free eccentricity oscillations, describes the libration of the Laplace argument for arbitrary amplitudes and allows us to determine the libration width of the resonance.

**Thursday 3 December****2020, 4.30 pm (Rome local time)**(Università degli Studi di Roma “Tor Vergata”, Dipartimento di Matematica)**Joan Gimeno****Invariant KAM tori, breakdown, and estimates for the dissipative spin-orbit problem**

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**Abstract**

In this talk I will present a recent project concerning the construction, breakdown and estimates for the existence of invariant tori in the spin-orbit problem in Celestial Mechanics with a time-dependent tidal torque. This model is conformally symplectic, namely it is a dissipative system enjoying the property that the it transports the symplectic form into a multiple of itself. We will see an efficient Newton method allowing us to compute an invariant attractor for a given Diophantine frequency. After that, we will see how the torus changes with the continuation of a perturbative parameter bringing the torus to its breakdown. Finally, we will see how computing certain quantities verifying KAM estimates leads to the proof of these tori near the breakdown.

This is a joint work with R. Calleja, A. Celletti, and R. de la Llave.

**Thursday 19 November****2020, 4.30 pm (Rome local time)****Bhanu Kumar**(Georgia Institute of Technology)**Rapid and accurate computation of invariant tori, manifolds, and connections near mean motion resonances in periodically perturbed planar circular restricted 3-body problem models**

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**Abstract**

When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, most unstable resonant periodic orbits become invariant tori. In this study, we 1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable directions; 2) implement continuation by both perturbation parameter as well as rotation numbers; 3) compute Fourier-Taylor series parameterizations of the stable and unstable manifolds; 4) globalize these manifolds; 5) rapidly search for homoclinic and heteroclinic connections in the 4D phase space by taking advantage of the manifolds? internal dynamics, with computational assistance from a GPU; and 6) refine the approximate connections found. Our methodology improves on efficiency and accuracy compared to prior studies, runs quickly on a standard laptop, and applies to a variety of periodic perturbations. We demonstrate the tools on the planar elliptic RTBP.

**Thursday 5 November****2020, 4.30 pm (Rome local time)****Adrian Perez Bustamante**(Georgia Institute of Technology)**Gevrey estimates and domains of analyticity for asymptotic expansions of tori in weakly dissipative systems**

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**Abstract**

We consider the problem of following quasi-periodic tori in perturbations of some Hamiltonian systems which involve friction and external forcing. In a first goal, we use different numerical methods (Pade approximants, Newton continuation till boundary) to obtain numerically the domain of convergence. We also study the properties of the asymptotic series of the solution. In a second goal we study rigorously the (divergent) series of formal expansions of the torus obtained using Lindstedt method. We show that, for some systems in the literature, the series is Gevrey. We hope that the method can be of independent interest: we develop KAM estimates for the divergent series. In contrast with the regular KAM method, we lose control of all the domains, so that there is no convergence, but we can generate enough control to show that the series is Gevrey.

This is joint work with R. Calleja and R. de la Llave.

**Monday 20 July 2020, 2.30 pm (GMT + 2)****Giacomo Lari**(University of Pisa, Department of Mathematics)**On the stability of the Laplace resonance under tidal effects**

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**Abstract**

The strong tidal dissipation between Jupiter and Io leads to significant migration of all the moons involved in the so-called Laplace resonance (Io, Europa, and Ganymede). Using an accurate averaged dynamical model of the Galilean moons developed to investigate their future long-term behavior, we show that the Laplace resonance is preserved despite the tidal effects, at least until Ganymede approaches the 2:1 resonant region with Callisto (about 1.5 billion years from now, assuming the current estimation of the tidal dissipation in the system). The resonant encounter can then result 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 and a pure 4:2:1 three-body resonance between Europa, Ganymede and Callisto. In case A, the Laplace resonance is always preserved and the eccentricities remain confined to small values below 0.01. In case B, 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. This is a joint work with M. Saillenfest and M. Fenucci.

**Monday 6 July 2020, 2:30 pm (GTM + 2)****Chiara Caracciolo**(University of Rome “Tor Vergata”, Department of Mathematics)*Librational KAM tori in the secular dynamics of the Upsilon-Andromedae planetary system*

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**Abstract**

Abstract: We investigate the stability of the secular motion of the two most massive planets in the Upsilon-Andromedae planetary system. Such a system has some remarkable characteristics: both the major planets move in rather eccentric orbits (e ~ 0.3) with a relevant mutual inclination (~30**°**); moreover, since their masses are about ten times larger than the Jupiter one and the mutual distance is relatively short (being the semi-major axes ~0.8AU and ~2.5AU, resp.), there is a strong interaction between them. We study the secular approximation at order 2 in the masses of the planetary three-body problem. In this framework, using a normal form approach, we show the existence of librational KAM tori, whose configuration is such that the pericenters of the two orbits are in libration around the anti-alignment, namely in a state of apsidal locking. In practice, we start from the construction of a suitable elliptic lower dimensional torus, i.e. a solution where the pericenters stay completely anti-aligned; then, we construct tori of maximal dimension, by focusing on the one which corresponds to initial conditions compatible with the observations.

This work is made in joint collaboration with U. Locatelli, M. Sansottera and M. Volpi.

**Monday 22 June 2020, 2:30 pm (GTM + 2)****Veronica Danesi**(University of Milan, Department of Mathematics)*Variation on Kolmogorov’s theorem: KAM with knobs*

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**Abstract**

In this talk I will reconsider the proof of the Kolmogorov’s theorem, introducing a variation on the handling of the frequencies and avoiding the so-called translation step. The motivation behind the development of this approach has its origin in the problem of persistence of lower dimensional elliptic invariant tori under sufficiently small perturbation.

This is a joint work with M. Sansottera.

**Monday 8 June 2020, 2:30 pm (GTM + 2)****Marco Fenucci**(University of Belgrade, Faculty of Mathematics)*Numerical methods for the computation of symmetric periodic orbits of the N-body problem*

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**Abstract**

In the last two decades, several periodic solutions of the N-body problem with equal masses have been found as minimizer of the Lagrangian action. The first and the most famous periodic orbit of this kind is the figure-eight solution of the three-body problem, whose existence has been proved by A. Chenciner and R. Montgomery in 2000. Here three equal masses follow an eight shaped curve, with the same time law and a constant shift in phase. Orbits with this property are called choreographies. Motivated by the above example, in this talk we first describe some numerical methods for the computation of symmetric periodic orbits, then we see how their stability can be studied, using also rigorous numerical techniques. Moreover, since some orbits are found as minimizers of the Lagrangian action, we give an idea of how local minimality properties can be studied with numerical computations. After, we show how to apply the numerical methods to the computation of periodic orbits of the*N*-body problem and the Coulomb (*N*+1)-body problem with the symmetry of Platonic polyhedra, and see how they provided clues for rigorous proofs of existence.

**Monday 25 May 2020, 2:30 pm (GTM + 2)****Joan Gimeno**(Università degli Studi di Roma “Tor Vergata”, Dipartimento di Matematica)*Delay perturbations of an ODE*

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**Abstract**

In this talk we will start introducing some basics results on Delay Differential Equations (DDEs) in order to focus our attention on the state-dependent DDEs (SDDEs). To bypass the phase space notion for SDDEs, we will consider a very singular perturbation of an ODE to provide an a-posteriori theorem of the existence of a parametrization of a subfamily of solution of the infinite stable invariant manifold of the new system. We will also discuss a numerical implementation stressing the hardest part in the coding of the algorithm. In particular, the results admit a straightforward extension for advanced or even mixed differential equations as well as smooth dependence on parameters. Similar techniques allow us to prove the persistence of periodic solutions under this kind of singular perturbation with only mild assumptions on the original ODE.

This is a joint work with J. Yang and R. de la Llave.

**Monday 11 May 2020, 2:30 pm (GTM + 2)****Rocío Isabel Páez**(Università degli Studi di Padova, Dipartimento di Matematica “Tullio Levi-Civita”)*The Levi-Civita Hamiltonian normalization, the analytical construction of large Lyapunov orbits and their manifolds*

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**Abstract**

In this seminar, I will introduce and describe at full extent the construction of the normalized Levi-Civita Hamiltonian, a fundamental tool for the study of the invariant manifolds emanating from planar Lyapunov orbits, in particular for large values of the energy. We will start from the Hamiltonian of the CR3BP in Cartesian coordinates, whose radius of convergence after being expanded at $L_1$ or $L_2$ is limited in amplitude by $|1 – \mu – x L_1|$, and therefore it is inadequate for dealing with large amplitudes. We will introduce regularized variables, so as to construct the Levi-Civita Hamiltonian, and we will investigate if regularizations prior to the normalization scheme allow us to overcome this limit. Finally, we will discuss the results obtained from the study of the normalized Hamiltonian, on which we notice variations in the structure of the tubes manifolds, allowing circulations in either sense and even collisions.

This research has been done in collaboration with M. Guzzo.

**Monday 4 May 2020, 2:30 pm (GTM + 2)****Sara Di Ruzza**(Università degli Studi di Padova, Dipartimento di Matematica “Tullio Levi-Civita”)*Symbolic dynamics in a binary asteroid system*

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**Abstract**

We consider a system of three point masses undergoing Newtonian attraction. Two of them (“binary asteroids”) have equal mass, while the third one (“planet”) is much heavier. The three masses are constrained on a plane. The two lighter particles orbit one to the other, while the trajectory of the third particle is external to the two, and far. We look at the secular motions of this system, meaning that, from a reference frame centred with one of the asteroids, we average out the position of the other asteroid, and look at the movements of the eccentricity and the pericenter of its instantaneous ellipse, as determined by the attraction by the planet. In the limit when the planet describes a circular trajectory with infinite radius, the asteroidal ellipse periodically squeezes to a segment while the pericenter oscillates about an equilibrium. The question we ask concerns the onset of chaos, once the planet exercises its attraction at a large, but finite distance from the asteroids. Our analysis is purely numerical. Based on covering relations as in a recent paper by A. Gierzkiewicz and P. Zgliczynski (2019), a topological horseshoe is highlighted, indicating symbolic dynamics.

Joint work with Jerome Daquin and Gabriella Pinzari.