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

https://doi.org/10.1016/j.cnsns.2019.04.015Get rights and content

Highlights

  • Accurate analytical modelling of the 1:1, 2:1 and 3:1 secondary resonances of the synchronous state of the spin-orbit gravitational problem.

  • General normal form method based on two main ingredients: the introduction of a detuning term, measuring the distance from the exact resonance, and the ordering different terms through a book-keeping parameter.

  • 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.

  • Error analysis and study of the asymptotic behavior analysis of the proposed non-standard normal form construction.

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.

Introduction

The 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 libration 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, creates a non-trivial topology in the synchronous resonance librational domain which has to be studied further.

In astronomical literature, examples of the study of secondary resonances around the synchronous primary resonance are motivated by possible connections to the problem of tidal evolution of systems such as a satellite with aspherical shape around a planet, or a double configuration of minor bodies (e.g., asteroids) where one or both bodies have irregular shapes. An example of the former case is Enceladus: it was originally conjectured ([29]) that the asphericity ratio of this satellite would make possible a past temporary trapping into the 3:1 secondary resonance located within the synchronous spin-orbit resonance with Saturn. Such a scenario would justify an amount of tidal heating substantially larger than far from the secondary resonance. The efficiency of this scenario was questioned as Cassini’s observations reduced Enceladus’ estimated asphericity closer to ε ≈ 1/4 rather than 1/3 ([24]; see the review by [20]). On the other hand, the overall role that secondary resonances could have played for the tidal evolution of planetary satellites towards their final synchronous state is a largely open problem. As regards minor planetary satellites or double minor bodies (e.g., double asteroids), exploration of the subject is still bounded by the scarcity of observations (see e.g., [25]). A question of central interest regards predicting changes in the stability character of a certain spin ‘mode’ (or periodic orbit) associated with a resonance, as the main parameters of the problem (eccentricity, asphericity) are varied. Varying the parameters leads to bifurcations of new periodic orbits, accompanied by a change of stability of their parent orbits. For secondary resonances l: k of order |l|+|k|>4, such bifurcations are described by a general theory (see, for example, [1]). Instead, for low order resonances (2|l|+|k|4) such transitions are case-dependent, and they lead to important changes in the topology of the phase portrait in the neighbourhood of one resonance. Besides theoretical interest in modelling these cases, the determination of stability of the various resonant modes can be useful to the interpretation of observations. An additional motivation stems from the need for precise models of spin-orbit motion in connection with future planned missions to double minor body systems.

With the above applications in mind, in the present paper we discuss the implementation of our method recently introduced in [10] with the aim to provide an analytical modelling allowing to fully reproduce the dynamics of the 3:1, 2:1 and 1:1 secondary resonances around the synchronous primary spin orbit resonance. Besides demonstrating the ability to analytically deal with all peculiarities encountered in the phase space features and bifurcation properties of these secondary resonances, the provision of analytical formulas with high precision is of practical utility, as it can substitute expensive numerical treatments with practically no loss of accuracy. In fact, we make an analysis of the error introduced in our approximation, based on well known methods used in asymptotic analysis of series expansions in classical perturbation theory. More precisely, after computing a Hamiltonian normal form for the secondary resonance, we measure the goodness of the approximation by the estimate of the remainder function, whose size is determined by two principal factors: i) The way we ‘book-keep’ the series terms including the detuning as a small parameter in the series (see Section 2 below), and ii) the accumulation of small divisors in the series terms. The typical behavior of the size of the remainder is that it decreases up to a certain order and then it increases. The order at which the size of the remainder attains its minimum is called the optimal order of the normal form. In this work we outline a procedure to estimate the optimal order, and hence obtain explicit estimates of the error of our analytical approximation. In fact, a key result is that our normal form construction, albeit non-standard in the way we ‘book-keep’ the Hamiltonian terms, still exhibits the desired asymptotic behavior of more conventional constructions, as, e.g., multivariate series in powers of more than one small parameters (see for example [26]).

The paper is organized as follows. The general problem is introduced in Section 2. The normalization process is discussed in a general setting in Section 3, along with a demonstration of how estimates of the errors follow from an asymptotic analysis of the normal form’s remainder. The specific application to the description of secondary resonances in the spin-orbit problem is given in Section 4, with concrete applications to the 1:1, 2:1 and 3:1 secondary resonances. Error analysis for each secondary resonance is discussed in Section 5. Finally, our results are summarised in Section 6. Explicit formulas for use in analytic computations are provided in the Appendix A.

Section snippets

Hamiltonian of the spin-orbit problem

The Hamiltonian describing the orbital and rotational coupling of a satellite in a Keplerian orbit, rotating about one of its primary axes of inertia, which is assumed perpendicular to the orbital plane, is given by [4], [11], [22]:H(pθ,θ,t)=pθ22ν2ε24a3r3(t)cos(2θ2f(t)),where θ is the angle formed by the largest physical axis of the satellite and the orbit apsis line, a is the orbit’s semi-major axis, ν is the orbital frequency, f the true anomaly, r the distance between the two bodies and ε

General normal form theory

In this section we discuss our proposed canonical normalization procedure and generalise our method for the study of an arbitrary order secondary resonance appearing in the vicinity of a primary resonance that can be described locally by a pendulum approximation. First we assume a Hamiltonian model which has the form (10). Then, we introduce the main ingredients that will be used to compute the normal form: the introduction of a detuning term, measuring the distance from the exact resonance,

Application to the secondary resonances of the synchronous resonance in the spin-orbit problem

The general method described in the previous section is now applied to the particular cases of the secondary resonances of the 1:1 primary resonance in the spin-orbit problem. More specifically, we study the three lowest order secondary resonances: 1:1, 2:1 and 3:1. For each case we construct a high-order normal form and provide a series of analytical computations. First, we compare the analytical Poincaré surfaces of section with the numerical ones, and confirm that our integrable

Series asymptotic behavior and error analysis

In this section we apply the error analysis estimates introduced in Section (3.8), based on the asymptotic behavior of the remainder function associated with the normal forms computed in the previous sections. The basic quantity of interest is R(n,N)(cα,δ,ξ), introduced in Eq. (36). Given particular parameter values e, δ, the first step in the analysis is to check that the successive normalizations keep our transformed Hamiltonian convergent within the domain |Ji| < ξ, for a value of ξ

Conclusions

The normal form theory can be used to study a wide variety of astronomical systems. The study of resonances, primary and secondary, can give us very important results in understanding and exploiting the natural dynamics of the system. In this work, we further generalise the method presented in [10] for the study of secondary resonances. The spin-orbit model still serves as our test problem to apply the proposed techniques and study their efficiency. The result for the 2:1 secondary resonance

Acknowledgements

AC was supported by GNFM-INdAM and acknowledges MIUR Excellence Department Project awarded to the Department of Mathematics of the University of Rome Tor Vergata (CUP E83C18000100006). GP was supported by GNFM-INdAM.

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