Accurate modelling of the low-order secondary resonances in the spin-orbit problem

Abstract

We provide an analytical approximation to the dynamics in each of the three most important low order secondary resonances (1:1, 2:1, and 3:1) bifurcating from the synchronous primary resonance in the gravitational spin-orbit problem. To this end we extend the perturbative approach introduced in [10], based on normal form series computations. This allows to recover analytically all non-trivial features of the phase space topology and bifurcations associated with these resonances. Applications include the characterization of spin states of irregular planetary satellites or double systems of minor bodies with irregular shapes. The key ingredients of our method are: i) the use of a detuning parameter measuring the distance from the exact resonance, and ii) an efficient scheme to 'book-keep' the series terms, which allows to simultaneously treat all small parameters entering the problem. Explicit formulas are provided for each secondary resonance, yielding i) the time evolution of the spin state, ii) the form of phase portraits, iii) initial conditions and stability for periodic solutions, and iv) bifurcation diagrams associated with the periodic orbits. We give also error estimates of the method, based on analyzing the asymptotic behavior of the remainder of the normal form series.Keywords. Normal form -primary and secondary resonances -spin-orbit problem. IntroductionThe study of resonant configurations is of primary importance in many astronomical problems. One of the most frequently observed commensurabilities in our Solar system is that between the orbital and the rotational period of natural satellites. Our Moon, for example, is locked in a synchronous (1:1) spin-orbit resonance and this is probably the case also for all large planetary satellites. In a simple spin-orbit coupling model, the dynamics about the synchronous resonance can be described with a pendulum approximation. The phase-space is separated by a separatrix into a rotation and a libration domain. The frequency of the librations is determined to a first-order approximation by the shape of the satellite. For particular values of the asphericity parameter, used to measure the divergence of the real shape from a sphere, this frequency can become resonant with the orbital frequency. This situation, known as a secondary resonance,

Dynamical Structures Associated with High-Order and Secondary Resonances in the Spin–Orbit Problem

“…The normalized Hamiltonian, governing the evolution of satellite's spin axis, can be written as (Goldreich & Peale 1966;Celletti 1990aCelletti , 1990bFlynn & Saha 2005;Gkolias et al 2016Gkolias et al , 2019…”

“…In this respect, Wisdom (2004) developed a perturbative model to study the secondary 3:1 spin-orbit resonance of Enceladus and discussed the rate of tidal dissipation for Enceladus inside this secondary resonance. Gkolias et al (2016Gkolias et al ( , 2019 took advantage of Lie-series transformation theory to formulate analytical approximations for those low-order secondary resonances (1:1, 2:1 and 3:1) bifurcating from the synchronous primary resonance. For the bifurcation diagram of secondary 1:1 resonance, the analytical (normal form) solution can agree with the numerical result in the range of α ä [0, 1.2], but starts to diverge when α is greater than 1.2 (Gkolias et al 2019).…”

“…It is not difficult to see that the Hamiltonian model based on Lie-series transformation adopted by Flynn & Saha (2005) can describe high-order primary resonances outside the primary island but cannot describe secondary resonances inside the primary island, while the Hamiltonian model based on Lieseries method adopted by Gkolias et al (2016Gkolias et al ( , 2019 can be used to study secondary resonances but cannot be used to study those high-order spin-orbit resonances. Thus, a unified resonant Hamiltonian model for describing high-order and secondary resonances in the spin-orbit problem is required to explain phase-space structures.…”

The Excited Spin State of Dimorphos Resulting from the DART Impact

“…where is the mean motion, and , , and are the secondary's three principal moments of inertia (which correspond to the axis lengths ≥ ≥ ). For certain combinations of the three moments of inertia, the free libration frequency can become resonant with the forced libration frequency (i.e., the mean motion) and a secondary resonance can occur (Melnikov, 2001;Gkolias et al, 2019). This can lead to an intricate dynamical environment, which only becomes more…”

Lie-series transformations and applications to construction of analytical solutions

“…Perturbation theories based on Lie-series transformations have been widely applied to varieties of subjects in celestial mechanics (Beaugé et al, 2006;Ceccaroni et al, 2016;Deprit and Rom, 1970;Deprit et al, 1969;Galgani, 1978, 1985;Giorgilli et al, 1989;Gkolias et al, 2019;Hou and Xin, 2017;Jorba and Masdemont, 1999;Lara, 2022;Lara et al, 2018;Lei, 2021;Meyer and Schmidt, 1986;Milani and Knežević, 1990;Nayfeh and Kamel, 1970;Riaguas et al, 2001). There are three classical versions of Lie-series transformations, developed by Hori (1966), Deprit (1969) and Dragt and Finn (1976).…”