Analytical methods in Celestial Mechanics: satellites' stability and galactic billiards

Abstract

In this paper, two models of interest for Celestial Mechanics are presented and analysed, using both analytic and numerical techniques, from the point of view of the possible presence of regular and/or chaotic motion, as well as the stability of the considered orbits. The first model, presented in a Hamiltonian formalism, can be used to describe the motion of a satellite around the Earth, taking into account both the non-spherical shape of our planet and the third-body gravitational influence of Sun and Moon. Using semi-analytical techniques coming from Normal Form and Nekhoroshev theories it is possible to provide stability estimates for the orbital elements of its geocentric motion. The second dynamical system presented can be used as a simplified model to describe the motion of a particle in an elliptic galaxy having a central massive core, and is constructed as a refraction billiard where an inner dynamics, induced by a Keplerian potential, is coupled with an external one, where a harmonic oscillator-type potential is considered. The investigation of the dynamics is carried on by using tools of ODEs' theory and is focused on studying the trajectories' properties in terms of periodicity, stability and, possibly, chaoticity.

Figures (13)

• Figure 1.1: Examples of orbits of refraction galactic billiards. The orbit goes inside and outside the domain, being deflected at every passage through the interface. Left: three-periodic trajectory. Right: quasi-periodic trajectory (figure taken from deblasiterraciniOnSome).
• Figure 2.1: Stability times computed for different values of semimajor axis, eccentricity and inclination using the nonresonant version of Nekhoroshev theorem. The color scale refers to the computed stability times (in years), while the white region correspond to the values of $(e,i)$ where Theorem \ref{['thm:nekh']} can not be applied with the present algorithm. The red lines are in correspondence of the inclinations of the secular geolunisolar resonances. Data taken from cellettiDeBlasiEft2023nekhoroshev.
• Figure 2.2: $LogLog$ plot of the points $\{|h_1|_{}A,r_0,s_0, K\}$ for $a_*=13\ 000\ km$, $e_*=0.2$ and $i_*\in[0^\circ,90^\circ]$. Data taken from cellettiDeBlasiEft2023nekhoroshev.
• Figure 3.1: Left: Snell's refraction law. The angles $\alpha_E$ and $\alpha_I$ are the angles respectively of the outer and the inner arc with respect to the normal direction to $\partial D$ in $z$. The two angles are connected by the relation \ref{['eq:snell']}. Right: concatenations from $p_0$ to $p_1$ with an outer and inner arc, for different positions of the transition point $p$. The left figure is taken from barutello2023chaotic.
• Figure 3.2: First return map: starting from initial conditions $(p_0, v_0)$, determined by the one-dimensional parameters $(\xi_0,\alpha_0)$, the trajectory is follow through an outer arc, a refraction from outside to inside, an inner arc and a refraction from inside to outside to find the final conditions $(p_1,v_1)$, defined by $(\xi_1,\alpha_1)$.

Theorems & Definitions (11)

• Theorem 2.1
• Proposition 3.1
• Proposition 3.2
• Theorem 3.3
• Definition 3.4
• Proposition 3.5
• Proposition 3.6
• Theorem 3.7
• Definition 3.8
• Theorem 3.9