The dynamics of the spin-spin problem in Celestial Mechanics

Abstract

This work investigates different models of rotational dynamics of two rigid bodies with the shape of an ellipsoid, moving under their gravitational influence. The focus of this study is on their behavior, their linear stability, and numerical investigation of the main resonances. We assume that the spin axes of the two bodies are perpendicular to the orbital plane and coinciding with the direction of their shortest physical axis. In the basic approximation, we assume that the orbits of the centers of mass are Keplerian and we retain the lowest order of the potential, according to which the rotational motions of the two bodies are decoupled, the so-called spin-orbit problem. When considering highest order approximation of the potential, the rotational motions become coupled giving rise to the so-called spin-spin problem. Finally, we release the assumption that the orbit is Keplerian, which implies that the rotational dynamics is coupled to the variation of the orbital elements. The resulting system is called the full spin-spin problem. We also consider the above models under the assumption that one or both bodies are non rigid; the dissipative effect is modeled by a linear function of the rotational velocity, depending on some dissipative and drift coefficients. We consider three main resonances, namely the (1:1,1:1), (3:2,3:2), (1:1,3:2) resonances and we start by analyzing the linear stability of the equilibria in the conservative and dissipative settings (after averaging and keeping only the resonant angle), showing that the stability depends on the value of the orbital eccentricity. By a numerical integration of the equations of motion, we compare the spin-orbit and spin-spin problems to highlight the influence of the coupling term of the potential in the conservative and dissipative case...

Figures (4)

• Figure 1: Keplerian spin-spin problem, maximum of the real part of the eigenvalues on a color scale for the conservative case, $(0,\pi)$ (left panel), single-dissipative averaged case, $(0,0)$ with $\delta_1=10^{-3}$ (middle panel), double-dissipative averaged case, $(0,0)$ with $\delta_1=10^{-3}$ and $\delta_2=2\cdot 10^{-3}$ (right panel).
• Figure 2: Keplerian spin-spin problem, maximum of the real part of the eigenvalues on a color scale for the conservative case, $(0,\pi)$: (3:2,3:2) resonance (left panel), (1:1,3:2) resonance (right panel).
• Figure 3: Poincaré sections projected on $(\theta_1,p_1)$, $(\theta_2,p_2)$ for Patroclus-Menoetius for the Keplerian spin-orbit problem (upper panels) and the Keplerian spin-spin problem (second row). Mixed case with $\bar{\gamma}_1=10^{-3}$ (third row). Dissipative case with $\bar{\gamma}_1=10^{-4}$ and $\bar{\gamma}_1=10^{-4}$ (last row).
• Figure 4: Comparison between conservative (1st and 2nd row) and dissipative (3rd and 4th row) dynamics in the full spin-spin problem for Patroclus-Menoetius. 1st row: Poincaré sections projected on $(\theta_1,p_1)$, $(\theta_2,p_2)$ in the conservative case for the same initial conditions and parameters as shown in the second row in Fig. 3. 2nd row: evolution of semi-major axis $a$ and eccentricity $e$ for the initial condition $(p_1,\theta_1)=(0.6,0)$, $(p_2,\theta_2)=(0.4,0)$ (close to the centers of the librational islands). 3rd row: long-term integration for two specific initial conditions, $(p_1,\theta_1)=(0.2,0)$, $(p_2,\theta_2)=(0.4,0)$ (blue) and $(p_1,\theta_1)=(1,0)$, $(p_2,\theta_2)=(0.4,0)$ (yellow) in the dissipative case with $\bar{\gamma}_1=6\times10^{-6}$ and $\bar{\gamma}_2=4\times10^{-6}$; the black dots indicate the state of the orbit at the end of the integration time. 4th row: shows the corresponding evolution of $a$ and $e$ for the orbits shown in row 3.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Remark 3
• Proposition 4
• proof
• Proposition 5
• proof
• Proposition 6
• proof
• Remark 7