## Periodic Reporting for period 4 - StableChaoticPlanetM (Stable and Chaotic Motions in the Planetary Problem)

Reporting period: 2020-03-01 to 2021-08-31

The planetary problem consists in determining the motions of n planets, interacting among themselves and with a sun, via gravity only. Its deep comprehension has relevant consequences in Mathematics, Physics, Astronomy and Astrophysics. The problem is perturbative, but perturbation does not apply directly to the problem, because of its degeneracies. The goal of this research is to develop techniques in order to prove long-time stability or diffusive motions.

The PI of the project, Gabriella Pinzari, has been in charge in the direction of the team and advance the project’s scientific purposes.

In the publication [Perihelia reduction and global Kolmogorov tori in the planetary problem, Mem. Amer. Math. Soc. 255 (2018), no. 1218], extending previous results in [Arnold, 1963], [Laskar & Robutel, 1996], [Fejoz 2004], [Chierchia & Pinzari, 2011], G. Pinzari proved a stability results for a planetary system. In the publication [On the co--existence of maximal and whiskered tori for the planetary three--body problem, J. Math. Phys. 59 (2018), no. 5, 052701] she discussed the existence of a zone, in the three-body problem, where chaos and stability are possible simultaneously. She also produced the following preprints, which have been submitted to scientific journals: the preprint [A first integral to the partially averaged Newtonian potential of the three-body problem, https://arxiv.org/abs/1607.03056] where G. Pinzari proved the existence of a first integral to a suitable average of the Newtonian potential; the preprint [Exponential stability of Euler integral in the three--body problem, https://arxiv.org/abs/1808.07633] where she discussed that, as consequence of this, the same first integral that determines the integrability remains constant over exponentially long times and an application of this to predict collisions.

Alexandre Pousse joined the team in November 2016, as a post--doctoral fellow. He has been awarded of two yearly post-doc positions (November 2016-October 2017 and December 2017-November 2018), funded by the project. His research focuses on the dynamics of “quasi-satellites”. The results of his research appeared in the publication [Pousse, A., Robutel, P. & Vienne, A. Celest Mech Dyn Astr (2017) 128: 383]. He also produced the preprint [L. Niederman, A. Pousse, P. Robutel, 2018; https://arxiv.org/abs/1806.07262] and the popularized preprint [L. Niederman, A. Pousse, P. Robutel, https://arxiv.org/abs/1807.10220].

Santiago Barbieri joined the team in October 2017, as a three-year PhD Student. His research focuses on Nekhorossev Theory, applied to planetary systems. Barbieri elaborated on a careful evaluation of the times for "quasi--convex" systems, in the joint work [S. Barbieri, L. Niederman. Sharp Nekhoroshev estimates for the three body problem around periodic orbits. https://arxiv.org/abs/1810.05987].

Sara Di Ruzza joined the team in March 2018, as a three-year senior researcher (“Ricercatore a tempo determinato sub. lett. a) L 240-2010”; RTDA), funded by the project. She is currently studying, in collaboration with the PI, the existence of unstable orbits in the three body problem.

Rocio Paez joined the team in July 2018, as a yearly post-doctoral fellow, funded by the project. Her research is focused on orbital dynamics. She is currently studying, in collaboration with M. Guzzo, fast transitions in phase space, associated to close encounters between gravitationally interacting bodies, including both natural and artificial hyperbolic dynamics.

Jerome Daquin joined the team in July 2018, as a yearly post-doctoral fellow, funded by the project. His research attains the long time dynamics of a Earth-Satellite-Moon-Sun system, including oblateness of one of the major bodies. They found a lack of correlation between local hyperbolicty and global instabilty in the phase-space. Their result appeared in the publication [Daquin et al., Drift and its mediation in terrestrial orbits, 2018, https://www.frontiersin.org/articles/10.3389/fams.2018.00035/full].

Massimiliano Guzzo is a professor of the University of Padova who started collaborating with the project on March, 1 2017. He studies the dynamics of Hamiltonian systems with the methods of perturbation theory (Nekhoroshev theorem, Hamilton-Jacobi equations, symplectic integrations) and hyperbolic dynamics (stable and center manifolds theorems, regularizations, chaos indicators). The recent outcomes of these researches are: the proof of a result about the integrability of close encounters in the spatial restricted 3-body problem [Cardin Guzzo 2018], [Cardin Guzzo preprint2018]; a semi-analytic model predicting the speed of Arnold diffusion along the resonances of multiplicity 1 of a priori stable systems [Eftymiopoulos Cardin Guzzo 2018, in preparation].

In the publication [Perihelia reduction and global Kolmogorov tori in the planetary problem, Mem. Amer. Math. Soc. 255 (2018), no. 1218], extending previous results in [Arnold, 1963], [Laskar & Robutel, 1996], [Fejoz 2004], [Chierchia & Pinzari, 2011], G. Pinzari proved a stability results for a planetary system. In the publication [On the co--existence of maximal and whiskered tori for the planetary three--body problem, J. Math. Phys. 59 (2018), no. 5, 052701] she discussed the existence of a zone, in the three-body problem, where chaos and stability are possible simultaneously. She also produced the following preprints, which have been submitted to scientific journals: the preprint [A first integral to the partially averaged Newtonian potential of the three-body problem, https://arxiv.org/abs/1607.03056] where G. Pinzari proved the existence of a first integral to a suitable average of the Newtonian potential; the preprint [Exponential stability of Euler integral in the three--body problem, https://arxiv.org/abs/1808.07633] where she discussed that, as consequence of this, the same first integral that determines the integrability remains constant over exponentially long times and an application of this to predict collisions.

Alexandre Pousse joined the team in November 2016, as a post--doctoral fellow. He has been awarded of two yearly post-doc positions (November 2016-October 2017 and December 2017-November 2018), funded by the project. His research focuses on the dynamics of “quasi-satellites”. The results of his research appeared in the publication [Pousse, A., Robutel, P. & Vienne, A. Celest Mech Dyn Astr (2017) 128: 383]. He also produced the preprint [L. Niederman, A. Pousse, P. Robutel, 2018; https://arxiv.org/abs/1806.07262] and the popularized preprint [L. Niederman, A. Pousse, P. Robutel, https://arxiv.org/abs/1807.10220].

Santiago Barbieri joined the team in October 2017, as a three-year PhD Student. His research focuses on Nekhorossev Theory, applied to planetary systems. Barbieri elaborated on a careful evaluation of the times for "quasi--convex" systems, in the joint work [S. Barbieri, L. Niederman. Sharp Nekhoroshev estimates for the three body problem around periodic orbits. https://arxiv.org/abs/1810.05987].

Sara Di Ruzza joined the team in March 2018, as a three-year senior researcher (“Ricercatore a tempo determinato sub. lett. a) L 240-2010”; RTDA), funded by the project. She is currently studying, in collaboration with the PI, the existence of unstable orbits in the three body problem.

Rocio Paez joined the team in July 2018, as a yearly post-doctoral fellow, funded by the project. Her research is focused on orbital dynamics. She is currently studying, in collaboration with M. Guzzo, fast transitions in phase space, associated to close encounters between gravitationally interacting bodies, including both natural and artificial hyperbolic dynamics.

Jerome Daquin joined the team in July 2018, as a yearly post-doctoral fellow, funded by the project. His research attains the long time dynamics of a Earth-Satellite-Moon-Sun system, including oblateness of one of the major bodies. They found a lack of correlation between local hyperbolicty and global instabilty in the phase-space. Their result appeared in the publication [Daquin et al., Drift and its mediation in terrestrial orbits, 2018, https://www.frontiersin.org/articles/10.3389/fams.2018.00035/full].

Massimiliano Guzzo is a professor of the University of Padova who started collaborating with the project on March, 1 2017. He studies the dynamics of Hamiltonian systems with the methods of perturbation theory (Nekhoroshev theorem, Hamilton-Jacobi equations, symplectic integrations) and hyperbolic dynamics (stable and center manifolds theorems, regularizations, chaos indicators). The recent outcomes of these researches are: the proof of a result about the integrability of close encounters in the spatial restricted 3-body problem [Cardin Guzzo 2018], [Cardin Guzzo preprint2018]; a semi-analytic model predicting the speed of Arnold diffusion along the resonances of multiplicity 1 of a priori stable systems [Eftymiopoulos Cardin Guzzo 2018, in preparation].

The research pursued by the team includes the following new methodologies.

- The results of the papers [G. Pinzari. Perihelia reduction and global Kolmogorov tori in the planetary problem. Mem.Amer.Math.Soc.2018] and [G. Pinzari. On the co--existence of maximal and whiskered tori for the planetary three--body problem. Journ. Math. Phys. 2018] are based two versions of a system of canonical coordinates for planetary systems, named "perihelia reduction", realizing a full reduction of SO(3) symmetry and simultaneously integrating the Keplerian part of the Hamiltonian, that were non known before, but have been introduced for the first time by the author.

- In the preprint [G.Pinzari. A first integral to the partially averaged Newtonian potential of the three-body problem. arXiv:1607.03056v1] it is shown that the partially averaged Newtonian potential of three-body problem is integrable, which is a result not previously known. The papers [G. Pinzari. Exponential stability of Euler integral in the three--body problem.arXiv:1808.07633] [G. Pinzari. An analysis of the Sun-Earth-Asteroid systems based on the two-centre problem, arXiv 1702.03680] are based on that result. In addition, the latter paper uses a new "weak" Nekhorossev-type normal form result, which holds beyond the Nekhorossev regime, because it does not use "small divisors" assumptions, so it can be applied (as it is, in the mentioned paper) regardless steepness assumptions.

- In the preprint [F. Cardin, M. Guzzo, 'Integrability of the spatial restricted three-body problem near collisions',

arXiv:1809.01257v1] a proof of the integrability of the spatial circular restricted three-body problem in a neighbourhood of its collision singularity is given. Team member R. Paez plans to analyse thoroughly the properties of close encounters and the consequences of such integrability, through both analytical and numerical means, that will provide a valuable insight for both natural and artificial hyperbolic dynamics.

- Some of numerical experiments by Daquin and collaborators have revealed that, in the regime of strong chaos, the time evolution of the second moment (variance) of some actions variables may depart significantly from normal diffusion. This abnormal diffusion raises the issue of the derivation of a reliable diffusive index.

- Pousse and collaborators developed a new analytical formalism that replaces the classical expansions in power and/or trigonometric series with an expansion in Laurent series.

Indeed, with a suitable choice of function that measures the distance from a singularity (the quasi-circular collision), a Laurent series expansion allows to reach the quasi-satellite domain, even for high eccentricities. In particular, this result opens some perspectives in order to prove the existence of stable quasi-satellite fixed points in the restricted and also planetary problem.

Until the end of the project, we expect to obtain concrete results concerning dynamics of gravitational systems especially for what concerns the existence of extremely unstable orbits, as well as extremely stable ones.

- The results of the papers [G. Pinzari. Perihelia reduction and global Kolmogorov tori in the planetary problem. Mem.Amer.Math.Soc.2018] and [G. Pinzari. On the co--existence of maximal and whiskered tori for the planetary three--body problem. Journ. Math. Phys. 2018] are based two versions of a system of canonical coordinates for planetary systems, named "perihelia reduction", realizing a full reduction of SO(3) symmetry and simultaneously integrating the Keplerian part of the Hamiltonian, that were non known before, but have been introduced for the first time by the author.

- In the preprint [G.Pinzari. A first integral to the partially averaged Newtonian potential of the three-body problem. arXiv:1607.03056v1] it is shown that the partially averaged Newtonian potential of three-body problem is integrable, which is a result not previously known. The papers [G. Pinzari. Exponential stability of Euler integral in the three--body problem.arXiv:1808.07633] [G. Pinzari. An analysis of the Sun-Earth-Asteroid systems based on the two-centre problem, arXiv 1702.03680] are based on that result. In addition, the latter paper uses a new "weak" Nekhorossev-type normal form result, which holds beyond the Nekhorossev regime, because it does not use "small divisors" assumptions, so it can be applied (as it is, in the mentioned paper) regardless steepness assumptions.

- In the preprint [F. Cardin, M. Guzzo, 'Integrability of the spatial restricted three-body problem near collisions',

arXiv:1809.01257v1] a proof of the integrability of the spatial circular restricted three-body problem in a neighbourhood of its collision singularity is given. Team member R. Paez plans to analyse thoroughly the properties of close encounters and the consequences of such integrability, through both analytical and numerical means, that will provide a valuable insight for both natural and artificial hyperbolic dynamics.

- Some of numerical experiments by Daquin and collaborators have revealed that, in the regime of strong chaos, the time evolution of the second moment (variance) of some actions variables may depart significantly from normal diffusion. This abnormal diffusion raises the issue of the derivation of a reliable diffusive index.

- Pousse and collaborators developed a new analytical formalism that replaces the classical expansions in power and/or trigonometric series with an expansion in Laurent series.

Indeed, with a suitable choice of function that measures the distance from a singularity (the quasi-circular collision), a Laurent series expansion allows to reach the quasi-satellite domain, even for high eccentricities. In particular, this result opens some perspectives in order to prove the existence of stable quasi-satellite fixed points in the restricted and also planetary problem.

Until the end of the project, we expect to obtain concrete results concerning dynamics of gravitational systems especially for what concerns the existence of extremely unstable orbits, as well as extremely stable ones.