Long-term Evolution and Stability of Saturnian Small Satellites: Aegaeon, Methone, Anthe, and Pallene

TL;DR

This study uses numerical simulations and frequency analysis to investigate the long-term stability and dynamical behavior of four small Saturnian moons, revealing their stability over hundreds of thousands of years and the influence of resonances.

Contribution

It provides the first detailed long-term dynamical analysis of these moons, identifying their stability and resonance interactions over extended timescales.

Findings

Aegaeon, Methone, and Anthe are stable for at least 0.5 million years.

Pallene remains stable for at least 64 million years.

Resonance interactions influence the orbital variations of these moons.

Abstract

Aegaeon, Methone, Anthe, and Pallene are four Saturnian small moons, discovered by the Cassini spacecraft. Although their orbital characterization has been carried on by a number of authors, their long-term evolution has not been studied in detail so far. In this work, we numerically explore the long-term evolution, up to $10^5$ yr, of the small moons in a system formed by an oblate Saturn and the five largest moons close to the region: Janus, Epimetheus, Mimas, Enceladus, and Tethys. By using frequency analysis we determined the stability of the small moons and characterize, through diffusion maps, the dynamical behavior of a wide region of geometric phase space, $a$ vs $e$, surrounding them. Those maps could shed light on the possible initial number of small bodies close to Mimas, and help to better understand the dynamical origin of the small satellites. We found that the four small moons are long-term stable and no mark of chaos is found for them. Aegaeon, Methone, and Anthe could remain unaltered for at least $\sim0.5$Myr, given the current configuration of the system. They remain well-trapped in the corotation eccentricity resonances with Mimas in which they currently librate. However, perturbations from nearby resonances, such as Lindblad eccentricity resonances with Mimas, seem responsible for largest variations observed for Methone and Anthe. Pallene remains in a non-resonant orbit and it is the more stable, at least for 64 Myr. Nonetheless, it is affected by a quasi-resonance with Mimas, which induces long-term orbital oscillations of its eccentricity and inclination.

Tables

Table 1. Table 1: Summary of physical parameters of the moons.

Name	GM (km3 s-2)	$\rho$ (g cm-3)	$R_m$ (km)	Reference
Janus	1.2656324971531704E-01	0.63	89.2	Thomas et al. (2013)
Epimetheus	3.512421991952764E-02	0.64	58.2	Thomas et al. (2013)
Aegaeon	1.2606299971854121E-09	0.54	0.33	Thomas et al. (2013)
Mimas	2.502784093954375E+00	1.15	198.2	Cooper et al. (2015)
Methone	6.261297469640338E-08	0.31	1.45	Thomas et al. (2013)
Anthe	2.334499994563023E-08	0.35	0.5	Thomas et al. (2013)
Pallene	1.7980185560068351E-07	0.25	2.23	Thomas et al. (2013)
Enceladus	7.211597878640501E+00	1.6	252.6	Cooper et al. (2015)
Tethys	4.121706150116760E+01	0.956	537.5	Cooper et al. (2015)

Table 2. Table 2: Saturn parameters.

Constant	Value	Reference
GM (km3 s-2)	3.793120706585872E+07	Cooper et al. (2015)
$R_S$ (km)	60,330	Cooper et al. (2015)
$J_2$	1629.054382E-05	Hedman et al. (2010)
$J_4$	-93.6700366E-05	Hedman et al. (2010)
$J_6$	8.6623065E-05	Hedman et al. (2010)

Table 3. Table 3: Initial conditions in geometric elements for the large moons.

Parameter	Janus	Epimetheus	Mimas	Enceladus	Tethys
$a$ (DMim)	0.816224821915373	0.816481357153813	1.00001997053844	1.282926463121003	1.588211135154469
$a$ (km)	$\mathrm{151\hspace{0.17em}441.5372}$	$\mathrm{151\hspace{0.17em}489.1345}$	$\mathrm{185\hspace{0.17em}542.7053}$	$\mathrm{238\hspace{0.17em}032.8930}$	$\mathrm{294\hspace{0.17em}675.1058}$
$e$	0.006783798913451	0.009648290023497	0.01935393988488	0.004717750162711	0.000318152308649
$i$ (∘)	0.163561569028246	0.351528831169069	1.56783494576081	0.004011975196723	1.091284647139937
$\overline{\omega }$ (∘)	6.132967855641644	116.348018800562642	88.06783652593947	57.40490801677754	29.38263145675355
$\Omega$ (∘)	6.369960248480534	44.525088430924164	239.785329147143301	337.68709939428453	141.711912205570002
$\lambda$ (∘)	129.813146812610796	263.577236051360842	318.616779106079946	217.90585212208282	271.834917459084238

Table 4. Table 4: Initial conditions in geometric elements for the small moons.

Parameter	Aegaeon	Methone	Anthe	Pallene
$a$ (DMim)	0.90274553988027	1.046749724679189	1.065312819759956	1.144147730931239
$a$ (km)	$\mathrm{167\hspace{0.17em}494.5047}$	$\mathrm{194\hspace{0.17em}212.8971}$	$\mathrm{197\hspace{0.17em}657.0752}$	$\mathrm{212\hspace{0.17em}284.0258}$
$e$	0.000312629006216	0.000781255131993	0.000980088315003	0.003986601194199
$i$ (∘)	0.002368099675087	0.010191700663056	0.018728031233752	0.181964040454351
$\overline{\omega }$ (∘)	240.784600532626655	101.055122930558355	196.379681976898155	320.254677876382459
$\Omega$ (∘)	246.490855111672857	150.999808094857514	115.172628200709838	131.420435093503784
$\lambda$ (∘)	325.21506542212154	205.403404370634803	159.569919065624163	293.49347042844056

Table 5. Table 5: Summary of mean values for the evolution of geometrical orbital elements and frequency analysis.

Satellite	${\overline{a}}_{Geo}$ (DMim)	${\overline{a}}_{Geo}$ (km)	${\overline{e}}_{Geo}\times {10}^{-3}$	${\overline{i}}_{Geo}\times {10}^{-2}$ (∘)	$\log \left(D\right)$	$t_{Diff}$ (Myr)
Aegaeon	$0.90272626\pm 3.472\times {10}^{-5}$	$\mathrm{167\hspace{0.17em}490.93}\pm 6.44$	$0.27542\pm 0.23335$	$0.16156\pm 0.12165$	-3.0138	$0.466565$
Methone	$1.0468417\pm 1.531\times {10}^{-4}$	$\mathrm{194\hspace{0.17em}229.97}\pm 28.40$	$0.85436\pm 0.61407$	$1.5441\pm 1.2678$	-3.1172	$0.473853$
Anthe	$1.0652882\pm 1.514\times {10}^{-4}$	$\mathrm{197\hspace{0.17em}652.51}\pm 28.09$	$1.2091\pm 0.8610$	$1.3255\pm 0.8787$	-3.1291	$0.474469$
Pallene	$1.1441441\pm 1.760\times {10}^{-5}$	$\mathrm{212\hspace{0.17em}283.35}\pm 3.26$	$3.9240\pm 0.1677$	$17.6289\pm 1.0308$	-5.3085	$64.41534$

Equations

$$\dot{{\theta }_j}=\frac{\partial H_0(J_j)}{\partial J_j}={\nu }_j(J),$$

$$z^{\prime }(t)={\alpha }_0\exp (i{\nu }_{0}t)+\sum _j^N{\alpha }_j\exp (i{\nu }_{j}t),$$

$$D=\frac{\left|{\nu }_1-{\nu }_2\right|}{T}.$$

Full text

††pubyear: 2016††pagerange: Long-term Evolution and Stability of Saturnian Small Satellites: Aegaeon, Methone, Anthe, and Pallene.–Long-term Evolution and Stability of Saturnian Small Satellites: Aegaeon, Methone, Anthe, and Pallene.

1 Introduction

The Cassini space mission has been a highly successful endeavor that helped to expand significantly our knowledge and understanding of Saturn and its environment, both physically and dynamically. Among other important results, several new moons around Saturn were discovered thanks to images taken by the spacecraft. Four of such small moons, now called Aegaeon, Methone, Anthe, and Pallene, lie between the position of the co-orbital duet formed by Janus and Epimetheus, at $\sim 152\,000$ km from Saturn centre, and roughly the inner edge of the E ring, where the moon Enceladus orbits, at $\sim 238\,000$ km from Saturn centre. Those small bodies remain a challenge to be successfully characterized due to the complex dynamical environment they inhabit, nonetheless, such dynamical environment has started to be explored. In the present work, we review our current knowledge of the four small moons, and explore their dynamical environment and evolution on a wider and longer basis than current studies did.

Aegaeon was discovered and reported by Hedman et al. (2010). It is a small moon of roughly 0.33 km in diameter (Thomas et al., 2013) that lies in the middle of a prominent arc structure, located inside and close to the inner edge of the G ring. The G ring is a faint and thin ring formed by $\mu$m to cm sized particles (Hedman et al., 2007), which extends from $\sim 165\,000$ to $\sim 175\,000$ km from Saturn centre (Horányi et al., 2009). Some particles forming the G ring presumably emanate from collisions of micrometeoroids with the largest objects of the region; like Aegaeon itself, however, nowadays the origin of the ring remains uncertain. As already mentioned, the G ring possesses a unique bright arc located close to its inner edge, at $\sim 167\,500$ km from Saturn centre, which covers $60^{\rm o}$ in longitudinal extend, while has just $\sim$250 km of radial width. The arc is located at the 7:6 corotation eccentricity resonance (CER) with Mimas. It has been assumed that this resonance confines the longitudinal extent of the arc (Hedman et al., 2009).

Being the most prominent member of the arc of the G ring, Aegaeon orbits trapped at the 7:6 CER with Mimas, with a resonant argument given by $\varphi_{CER}=7\lambda_{Mimas}-6\lambda_{Aegaeon}-\bar{\omega}_{Mimas}$, where $\lambda$ and $\bar{\omega}$ are the mean longitude and longitude of pericenter, respectively. This resonant argument librates with amplitude $\sim 10^{\circ}$ around $\sim 180^{\circ}$. However, Aegaeon’s longitude deviates from the expected position, if one only considers its $\sim 10^{\circ}$ libration due to the CER. This small deviation has a period of $\sim$70 yr, similar to the period of the Mimas longitude libration due to a 4:2 mean motion resonance (MMR) with Tethys (Hedman et al., 2010), therefore Tethys perturbs the orbit of Aegaeon through its interaction with Mimas.

Furthermore, variations in Aegaeon’s eccentricity and inclination, not related to the CER, indicate a possible influence of other resonances, particularly the 7:6 Inner Lindblad Resonance (ILR), which resonant argument is given by $\varphi_{ILR}=7\lambda_{Mimas}-6\lambda_{Aegaeon}-\bar{\omega}_{Aegaeon}$. This resonant argument librates, although not so tightly as the argument of the CER, with an amplitude of $\sim 90^{\circ}$ around zero (Hedman et al., 2010); therefore the ILR could be inducing second or higher order perturbations, not well characterized so far.

Methone is a small moon of $\sim$1.45 km in diameter (Thomas et al., 2013). First reported by Porco et al. (2005), it is located at $\sim 194\,200$ km from Saturn centre. It inhabits inside an arc of dust for which most probably it is the source (Sun et al., 2017). The arc’s longitudinal extend is $\sim 10^{\circ}$, roughly centered at the small moon. Both Methone and its arc orbit near the 15:14 CER and the 15:14 outer Lindblad resonance (OLR) with Mimas (both resonances are separated by just 4 km, El Moutamid et al., 2014). Both resonant arguments of the 15:14 CER and OLR librate with amplitude $\sim 90^{\circ}$ around zero. However, the arc confinement is consistent with the maximum extension permitted by the 15:14 CER with Mimas alone (Hedman et al., 2009).

Anthe is a tiny moon of an estimated $\sim$0.5 km in diameter; a better define measurement has not been carried out to date, due to the lack of resolution in Cassini images (Thomas et al., 2013). Anthe is located at $\sim 197\,600$ km from Saturn centre. It was first reported by Cooper et al. (2008), whom characterized its short-term dynamics. They found that Anthe orbits close to an 11:10 outer resonance with Mimas, with two librating arguments, one of which is associated with the 11:10 CER with Mimas, $\varphi_{CER}=11\lambda_{Anthe}-10\lambda_{Mimas}-\bar{\omega}_{Mimas}$, while the second one is given by $\varphi_{2}=11\lambda_{Anthe}-10\lambda_{Mimas}-\bar{\omega}_{Anthe}-\Omega_{Anthe}+\Omega_{Mimas}$. Both periods of libration for these arguments are 761 days (Cooper et al., 2008) and librate with amplitude $\sim 90^{\circ}$ around zero. Likewise Aegaeon and Methone, Anthe lies at the centre of an arc of dusty material that extends for $\sim 20^{\circ}$, consistent with the confinement produced by the 11:10 CER with Mimas (Hedman et al., 2009).

Both arcs of Methone and Anthe have been deeply studied recently by Sun et al. (2017). They explored the evolution of $\mu$m sized particles which origin is the ejecta produced by collisions of interplanetary dust particles (IDP) on the surfaces of the small moons. They then explored the dynamical evolution of the ejected particles considering an oblate Saturn, the gravitational perturbations of Mimas, Enceladus, and Methone or Anthe, depending on the system, plus a number of non-gravitational effects, such as solar radiation pressure, Lorentz force, and plasma drag. The authors are able to reproduce some of the properties of both arcs, such as their longitudinal extend and optical depth, although dust density in the simulations is an order of magnitude smaller compared to estimations derived from observations. According to Sun et al. (2017), particles smaller than 5 $\mu$m are quickly removed from the arcs due mainly to the action of plasma drag, while larger particles could survive up to 100 yr in the arcs before recolliding with the parent small moon or leaving the arc.

Regarding Pallene, also first reported by Porco et al. (2005), it is a small moon of diameter $\sim$2.23 km (Thomas et al., 2013), orbiting at $\sim 212\,300$ km from Saturn centre. It shares its orbit with a faint full ring of dusty material. This ring is only visible at extremely high phase angles, implying that it is formed by $\mu$m sized particles. The Pallene ring is very thin, with a vertical width of just $\sim$50 km. For comparison, Pallene achieves vertical excursions of $1\,300$ km above Saturn’s equatorial plane, due to its relatively large inclination. The ring is therefore tightly constrained to the orbital plane of Pallene, sharing the same inclination of the small moon (Hedman et al., 2009). Regarding their dynamics, Spitale et al. (2006) suggested that both the moon and its ring were probably in a 19:16 inner mean longitude resonance with Enceladus, with resonant argument given by $\theta=19\lambda_{Enc}-16\lambda_{Pall}-\bar{\omega}_{Pall}-2\Omega_{Pall}$. Meanwhile, Callegari & Yokoyama (2010) argued for a secular resonance relating Pallene and Mimas, with resonant argument given by $\theta=\bar{\omega}_{Pal}-\bar{\omega}_{Mim}-\Omega_{Mim}+\Omega_{Pal}$. Nonetheless, both arguments circulate, with different periods, suggesting that Pallene is most probably evolving out of any strong resonance. We provide some details in sections 3 and 5.

The short-term dynamics of the four small moons have been explored in a number of works, mainly those concerning their discovery and orbit determination from Cassini raw images. As we have recalled, Aegaeon, Methone and Anthe orbit near CER with Mimas, while Methone and Anthe also lie close to Lindblad resonances with Mimas. CER and Lindblad resonances are separated by just some km and are very close to the mean semimajor axis of Methone and Anthe. Motivated by this, El Moutamid et al. (2014) proposed a simplified analytical model, named CoraLin model, to explore the dynamics of test particles close to both CER and Lindblad eccentricity resonances (LER), which can be applied, under some assumptions, to these small moons. The CoraLin model is a simplification of the Hamiltonian for a restricted three body problem, where the large perturbing moon is eccentric and the test particle orbits near a MMR of the form $m+1:m$, where $m$ is an integer. The model depends upon just two free parameters. The first one, $D_{C}$, is related to the separation distance between the positions of the CER and LER, while the second parameter, $\epsilon_{L}$, is related to the mass and eccentricity of the perturbing satellite. $D_{C}=0$ means that both CER and LER lie at the same position, and the Hamiltonian is integrable in this case. For $D_{C}>2$ both resonances are separated enough as to not interfere with each other and the Hamiltonian is quasi integrable. Intermediate cases, $0<D_{C}<2$ leads to strong coupling between the resonances, which in turn leads to a chaotic evolution of the orbits.

The results from applying the CoraLin model to the small moons, by inspecting the corresponding Poincaré surfaces of section, suggest that Anthe is closer to the chaotic region than any of the other small moons. Methone is farther from the chaotic region than Anthe, while Aegaeon is the more stable, for being far enough from the chaotic region. Those small moons form a distinctively family of objects which dynamics is dominated by CER and LER coupling resonances (Cooper et al., 2015).

The only long-term exploration so far regarding the dynamics of the small moons (except for Aegaeon), was carried out by Callegari & Yokoyama (2010). They performed $\sim 60\,000$ yr numerical integrations of the orbits of Methone, Anthe and Pallene under the influence of Saturn’s oblateness until $J_{4}$, and the gravitational perturbations from Mimas, Enceladus, Tethys, Dione, Rhea, and Titan. They found a high sensitivity on initial conditions for both Methone and Anthe, suggesting that their evolution could be chaotic, while Pallene seems to be very stable. The authors suggest a possible resonance involving the longitudes of pericenter and ascending nodes of Pallene and Mimas, however, as previously noted, such resonant argument does not librate, but circulates with a long period, as we will verify in sections 3 and 5.

Regarding their physical characteristics, the Cassini images have revealed smooth surfaces for two moons, Methone and Pallene, for which there are enough resolution in the images. It is expected that both Aegaeon and Anthe share the same feature, i.e. a smooth surface; this because of their proximity, size and likely same formation process. Moreover, since the discovery of the arc/ring structures associated with the small moons, the source of the micrometric material forming such structures has been assumed to be the escaped debris particles produced when IDP collided on the surface of the small moons. Due to their small size, their escape velocity is very low, letting almost all impact ejecta freely evolve out of the surface of the small moon into the arcs/ring. However, a considerable proportion of those dusty particles end up colliding again with the moons, smoothly depositing themselves on the surface of the small moons, thus being likely responsibly for the even surfaces observed. Numerical evidence for such scenario can be found in Sun et al. (2017).

In this paper we explore the long-term evolution of the four small moons, exceeding any previous integration of their orbits. We also explore, with thousands of test particles, a wide region of the geometric phase space of semimajor axis, $a$, vs eccentricity, $e$, surrounding the small moons, to gain a global understanding of the dynamical behavior and evolution of such regions where arcs/ring structures coexist.

This paper is organized as follows: in section 2 we describe the frequency analysis technique and the simulations used in this work. Section 3 is devoted to the short-term evolution of the small moons, while in section 4 we present several diffusion maps, based on frequency analysis, which characterize the dynamical behavior of all the regions of interest. In section 5 we present our main results regarding the long-term evolution of the small moons. Finally, in section 6 we enumerate our main conclusions.

2 Methods and Numerical Simulations

2.1 Frequency Analysis

Frequency analysis is a powerful technique to quantify the weak chaotic behavior of a dynamical system with arbitrary degrees of freedom. Since its original formulation by Laskar (1990, 1993), who develop it in order to prove the chaotic nature of the solar system’s secular evolution, it has been successfully applied to a wide variety of dynamical problems, going from planetary sciences (Nesvorný & Morbidelli, 1998; Robutel & Laskar, 2001) to galactic dynamics (Papaphilippou & Laskar, 1998; Valluri & Merritt, 1998), and even through fundamental particle physics (Nadolski & Laskar, 2003; Papaphilippou, 2014).

The frequency analysis algorithm developed by Laskar, look for the fundamental frequencies of the system resulting from a numerical integration. A brief sketch of the method and, at the same time, a justification for its use, can be stated as follows (Laskar et al., 1992): consider the Hamiltonian of an integrable dynamical system, $H(J,\theta)$; once the system is reduced to action angle variables, $(J_{j},\theta_{j})$, the Hamiltonian will depend only on the actions, $H(J,\theta)=H_{0}(J_{j})$, where $j=1,2,...,n$ for a system of $n$ degrees of freedom. We know that actions are constants of the motion, $\dot{J}_{j}=0$, while angles evolve according to:

[TABLE]

where the $\nu_{j}(J)$ are the fundamental frequencies of the motion.

In numerical experiments, we will rarely work on action angle variables, nonetheless, we can still consider some close related dynamical variables, say $z^{\prime}_{j}$, of the form: $z^{\prime}_{j}=J^{\prime}_{j}\,\exp(i\theta^{\prime}_{j})=f(z_{1},z_{2},...,z_{n})$, where $z_{j}=J_{j}\,\exp(i\theta_{j})$, this is, any variable will be a certain function, $f$, close to unity, that depends on the actual actions and angles of the system. Although $(J^{\prime}_{j},\theta^{\prime}_{j})$ are not the action angle variables, the analysis of $z^{\prime}_{j}(t)$ will still give us the fundamental frequencies, $\nu_{j}$, since for periodic and quasi periodic motions, these frequencies will remain constant.

If we recover the fundamental frequencies in different intervals of a numerical integration, we will be able to determine the existence of periodic motion, if frequencies remain almost constant, or on the contrary, if largely variable frequencies are found, the implication is an irregular, maybe chaotic motion. This is because, by inverting Eq. 1, we can state that $J=F(\nu_{1},\nu_{2},...,\nu_{n})$, therefore variable frequencies imply variable actions.

We are not interested in recovering the analytical representation of particles orbits in our region of interest. On the contrary, we are interested in the variations of their motions due to different perturbations, thus we focus on the global variations of the frequencies and not in their specific values. In this work we calculate the main frequency, $\nu$, of the quantity $z^{\prime}_{j}=a(t)\exp(i\lambda(t))$, from a time series dataset resulting from a numerical integration. To obtain the frequencies we use the frequency modified Fourier transform (FMFT) algorithm, described in Šidlichovský & Nesvorný (1996), made publicly available by the author 111https://www.boulder.swri.edu/~davidn/fmft/fmft.html.

According to Robutel & Laskar (2001), the main frequency of $z^{\prime}(t)$ will be related to the mean motion, $n$, of the orbit. A full representation of $z^{\prime}(t)$ through frequency analysis would be:

[TABLE]

where $\nu_{0}=n$ in the case of pure Keplerian motion. As this is not the case, both $|\alpha_{0}|$ and $\nu_{0}$ are close, but not equal, to the initial semimajor axis and mean motion, respectively. Nonetheless, the amplitudes of the following terms are 100 to $1\,000$ times smaller for quasi-periodic trajectories, therefore, variations in the main frequency, $\nu_{0}$, will be enough to provide an estimation of the orbit’s stability.

2.2 Numerical simulations

In order to obtain a global dynamical perspective of the region inhabited by Aegaeon, Methone, Anthe, and Pallene, we performed short-term ($\sim$18 yr) numerical simulations of thousands of test particles that initially cover the entire $a$ vs $e$ geometric phase-space, from the orbit of Janus-Epimetheus, to beyond the orbit of Enceladus. The choice of our integration time relies on a compromise, between the large volume of data produced and the requirement of enough orbital periods to accurately calculate the main frequencies for our analysis. In $\sim$18 yr Mimas performs $\sim 6\,950$ orbital revolutions around Saturn, while Enceladus, close to the exterior limit of our maps, performs $4\,780$ revolutions, i.e. we obtain enough orbital revolutions for all particles for an accurate recovery of their frequencies.

Test particles are subject to gravitational perturbations from the oblateness of Saturn up to $J_{6}$ in zonal harmonics, plus the five largest moons of the region, namely Janus, Epimetheus, Mimas, Enceladus, and Tethys, in a first set. In later simulations we consider the gravitational perturbations from Saturn and the large five moons, as well as the gravitational perturbations from the four small moons. After the global exploration, we zoomed in towards those regions closer to each small moon, performing simulations that include thousands of test particles covering a smaller patch of the geometric $a$ vs $e$ phase-space, but in finer steps. We provide the details for each simulation along with the results in the next section.

In section 5, the results for a single long-term ($1\times 10^{5}$ yr) numerical integration are reported, where only the interaction between the five large moons and the four small moons, orbiting an oblate Saturn, is considered.

All of the numerical simulations presented in this work were performed using the Bülirsch-Stöer integrator from the MERCURY package (Chambers, 1999), where a toleration accuracy parameter was set to $10^{-12}$ and an initial time-step of $0.1$ day was used. Most of the simulations were performed at the Saturn Cluster belonging to the Group of Planetology and Orbital Dynamics of the Mathematics Department of the São Paulo State University (UNESP).

The physical parameters of all the nine different moons used throughout this work are shown in Table 1, where GM is the gravitational parameter, $\rho$ is the bulk density of the body, and $R_{m}$ the mean radius. In Table 2 we show the parameters of Saturn.

3 Short-term Dynamics

To test the robustness of the initial conditions used in all of our simulations, we performed a detailed short-term integration, lasting for 24 yr, in order to reproduce the resonant dynamics of Aegaeon, Methone, and Anthe, as has been found by previous authors. This simulation includes the nine moons presented in Table 1, plus Saturn and its gravity coefficients given in Table 2. The initial conditions for all the bodies were taken from the Horizons website for the Julian Day $2\,457\,601.5$, which corresponds to the date August 01, 2016. The geometric elements used as initial conditions for the nine objects which constitute the n-body part of the simulations, are shown in Tables 3 (for the five large moons) and 4 (for the four small moons), where all variables have their usual meaning of: geometric semimajor axis, $a$, geometric eccentricity, $e$, geometric inclination, $i$, longitude of pericenter, $\bar{\omega}$, longitude of the ascending node, $\Omega$, and mean longitude, $\lambda$. In Tables 3 and 4, DMim stands for the normalized geometric semimajor axis of Mimas, D${}_{Mim}=1$, used as a reference distance unit throughout this work. In physical units D${}_{Mim}=1.85539\times 10^{5}$ km.

The most important constraint for the dynamics of the small moons is the resonant dynamics associated with several resonances, among them the CER and LER with Mimas. In Fig. 1 we show the short-term evolution of the resonant arguments for Aegaeon, Methone, and Anthe resonances. In addition, we plotted the evolution of the two arguments associated with the quasi-resonances of Pallene with Enceladus and Mimas, as suggested by Spitale et al. (2006) and Callegari & Yokoyama (2010), respectively.

Hedman et al. (2010) found that Aegaeon is trapped in the 7:6 CER with Mimas, with a libration argument $\phi_{Aeg}=7\lambda_{Mim}-6\lambda_{Aeg}-\bar{\omega}_{Mim}$, a period of libration around 180∘ of $\sim 1\,264$ days, and amplitude of $\sim 10^{\circ}$. We found (top panel of Fig. 1) a libration period of $\sim 1\,252.5$ days, a difference of only 0.9% with the previous result, and the same libration amplitude of 10∘ around 180∘.

For Methone, both Spitale et al. (2006) and Jacobson et al. (2006) found a libration argument $\phi_{Met}^{LER}=15\lambda_{Met}-14\lambda_{Mim}-\bar{\omega}_{Met}$, a period of libration of $\sim$450 days, and an amplitude of the residual longitude of $\sim 5^{\circ}$. This argument corresponds to the 15:14 LER with Mimas. Nonetheless, Hedman et al. (2009) also found that the argument associated to the 15:14 CER with Mimas, $\phi_{Met}^{CER}=15\lambda_{Met}-14\lambda_{Mim}-\bar{\omega}_{Mim}$, also librates with the same period and amplitude as the one associated with the LER. This means than Methone is perturbed by both the CER and the LER with Mimas, which are separated by just 4 km. We found that effectively both arguments for the CER (blue curve in second panel of Fig. 1) and the LER (red curve in same panel) of Methone, librate, and we have normalized the two arguments for them to librate around zero, in order to compare their amplitudes, which is around 90∘. Their period of libration is of $\sim$515.7 days, a difference of $\sim$14% with previous results. Such differences could be the result of the updated orbital data and masses we have used. An evidence in this sense is the recent result found by Sun et al. (2017) for the Methone libration period of the CER. They found this period to be $\sim 520$ days and an amplitude of the residual longitude of $\sim 5^{\circ}$. The result of Sun et al. (2017) is in better agreement with ours, since they used updated data from the Horizons system of the Jet Propulsion Laboratory, as we did.

In the case of Anthe, Cooper et al. (2008) found two resonant arguments that librate, one is the associated with the 11:10 CER with Mimas, $\phi_{Ant}=11\lambda_{Ant}-10\lambda_{Mim}-\bar{\omega}_{Mim}$, and the second argument is given by $\phi_{Ant_{2}}=11\lambda_{Ant}-10\lambda_{Mim}-\bar{\omega}_{Ant}-\Omega_{Ant}+\Omega_{Mim}$. For both arguments, Cooper et al. (2008) found a period of libration of $\sim$761 days and an amplitude of 78∘. We find for both arguments (blue and red curves in third panel of Fig. 1) a period of $\sim 731$ days, a difference of just 4% with the previous result, with an amplitude of libration of $\sim 90^{\circ}$. Hedman et al. (2009) and Sun et al. (2017) argue for the 11:10 CER of Anthe with Mimas to be responsible for the confinement of the Anthe’s arc, being then the most relevant for the evolution of the moon and its environment.

Finally, although Pallene is not in resonance neither with Mimas or Enceladus, both Spitale et al. (2006) and Callegari & Yokoyama (2010) suggest two different arguments that circulate and could indicate a quasi-resonance of Pallene with any of those large moons. First, Spitale et al. (2006) suggest the argument: $\theta_{Pall}^{Enc}=19\lambda_{Enc}-16\lambda_{Pall}-\bar{\omega}_{Pall}-2\Omega_{Pall}$, a 19:16 inner mean longitude resonance between Pallene and Enceladus, as responsible for the long-term variations in the orbital elements of Pallene. On the other hand, Callegari & Yokoyama (2010) argue for an argument given by $\theta_{Pall}^{Mim}=\bar{\omega}_{Pal}-\bar{\omega}_{Mim}-\Omega_{Mim}+\Omega_{Pal}$, which implies a relation between Pallene and Mimas, based on the high stability found by the authors in their long-term simulations. They obtain a period of circulation of $\sim 4\,400$ yr. We find that in fact both cited arguments circulate, where the one involving Pallene and Enceladus has a period of circulation of $\sim 2\,192$ days (blue curve in bottom panel of Fig. 1), while the second argument, involving Mimas and Pallene, although at first glance could seem as librating (see red curve in Pallene’s panel), it actually circulates but with a much larger period, not visible in this short-time scale plot. We show the circulation of this second argument in section 5, for which we find a period of $\sim 4\,762.21$ yr, a difference of $\sim$7.6% compared with Callegari & Yokoyama (2010) work.

4 A Global Dynamical Perspective Through Frequency Analysis

Maps

4.1 The Full Region Around Mimas

The aim of this section is to delineate the lookalike of a wide dynamical region around Mimas, going from 0.7 to 1.3 $D_{Mim}$, to better understand the influences that can drive the evolution of the small moons and their ring/arcs. In Fig. 2 we plot the location of all first and second order MMRs with Janus, Mimas, Enceladus, and Tethys, present in this region of the Saturnian system. All this resonances are somewhat responsible for the shaping of structures, cleaning or maintaining of the particles in the region, which are subject to gravitational forces only. Although it is well-known that all Aegaeon, Methone, Anthe, and their associated arcs lie in CER with Mimas, here we provided a global framework for further detailed explorations near any specific MMR of interest, where other kinds of dynamics could better fit to the particular case, such as corotation or Lindblad resonances.

The global dynamical picture of the full region was obtained from the detailed numerical integration during $\sim$18 yr of $11\,000$ test particles distributed in a rectangular grid, covering the geometric phase-space of $a$ vs $e$. In $a$, initial conditions are taken from 0.7 to 1.3 DMim in steps of $6.0\times 10^{-4}$. In $e$ we cover from 0 to 0.04 in steps of $4\times 10^{-3}$. In all this work we explore the evolution of particles initially in the Saturn’s equatorial plane (thus all test particles have initially zero inclination). As for the remaining orbital elements of each particle, $\bar{\omega}$, $\Omega$, and $\lambda$, they are set to zero, following the approach of Robutel & Laskar (2001) in their global exploration of the dynamics of the solar system. Indeed, the variations of the angular elements ($\bar{\omega}$, $\Omega$, and $\lambda$) do not change significantly the results for the diffusion of the particles outside resonances (which constitutes most of the dynamical space in all our experiments). Such variations lead only to different capture rates for particles which are initially close to MMRs and therefore are subject to get trapped. In this sense, angles’ values are important mainly because they determine the width of the libration regions of MMRs. Nonetheless, in our case, as we only explored very low eccentricities, the width of MMRs is, as we will see, very thin, thus the contribution from variations of the angles is negligible.

All test particles are subject to the non-spherical potential of Saturn up to an order $J_{6}$ in zonal harmonics, plus the gravitational perturbations of the five large moons close to the region, namely Janus, Epimetheus, Mimas, Enceladus, and Tethys, in the first set. For the second set we also added the gravitational perturbations from Aegaeon, Methone, Anthe, and Pallene. To account for the rapid orbital oscillations due to the oblateness of Saturn, the state vectors resulting from the integrations are everywhere transformed to geometric elements following Renner & Sicardy (2006).

In order to construct the frequency analysis map we explore the evolution of the quantity $z^{\prime}(t)=a(t)\exp(i\lambda)$, for each particle, where $a$ is the geometric semimajor axis and $\lambda$ is the mean longitude of the orbit. We have $T\sim 9$ yr, equivalent to half the total integration time. We applied the FMFT (Šidlichovský & Nesvorný, 1996) over $z^{\prime}(t)$ on the adjacent time intervals, $[0,T]$ and $[T,2T]$, in order to compare the main frequencies of $z^{\prime}(t)$, $\nu_{1}$ and $\nu_{2}$, from each time interval.

Since we are interested in the global dynamical structure of the region, we focus on the diffusion parameter, $D$, for each particle, defined following Robutel & Laskar (2001) and Correia et al. (2005), as:

[TABLE]

The $D$ parameter provides a measure of the particle’s orbit stability. Some particles do not survive the whole simulation and we do not compute $D$ for them. For unstable particles, that nonetheless survive until the end of the simulation, the difference $\left|\nu_{1}-\nu_{2}\right|$ will be significant, leading to large values of $D$. On the contrary, for stable particles the same difference will be small, leading to very small values of $D$. An estimation of a diffusion time-scale, $t_{Diff}$, this is, an estimation of the time required until an appreciable change in the $n$ of the particle is observed, or the time required for a diffusion of the orbit in the radial direction, can be crudely estimated as $t_{Diff}=(DP)^{-1}$ yr, where $P$ is the period of the orbit in yr.

The FMFT algorithm let us to measure with confidence the variations of $\nu_{1}$ and $\nu_{2}$, in the very short-time integrations performed. In order to characterize the dynamical region of interest, we have plotted, in a color scale, the $\log(D)$ for each particle, according to its initial position in the geometric $a$ vs $e$ phase-space plane. The resulting map of the entire region is shown in Fig. 3.

To highlight the relevance of the small moons over their environment, an analogous map to that of Fig. 3 is shown in Fig. 4. This map was obtained from the integration of the same set of test particles subject to the gravitational perturbations from the five large moons, plus the four small moons: Aegaeon, Methone, Anthe, Pallene, and an oblate Saturn.

The maps presented in Figs. 3 and 4, show the general dynamical characteristics of the region. In both figures, black lines delimit the position of constant pericenter and apocenter of test particles, at the positions of the apocenter and pericenter of each major body, respectively. In Fig. 3, the positions of the four large moons are pointed by filled black circles. Near the coorbital moons Janus-Epimetheus, the constant pericenter and apocenter lines are those corresponding to the position of Janus’ apocenter and pericenter, respectively, as it is the more massive moon of the two coorbitals. For Mimas, both curves of constant apocenter and pericenter are plotted. For Enceladus, only the line delimiting the constant apocenter of particles at the position of Enceladus’ pericenter is shown. The redder colors in both maps are enclosed by such black lines. Particles that enter, or that are initially located inside such regions, are strongly perturbed by the major bodies, resulting in a major probability for them to collide with the corresponding moon, thus being lost from the simulation. In the map of Fig. 3 most particles inside the Enceladus region are removed before the end of the simulation, while much less particles are removed inside the regions of Mimas and Janus-Epimetheus. However, some of the particles inside the Enceladus region survived, remarkably those that are coorbital with the large moon. A small patch of stability is also seen for particles of small eccentricity, coorbital with Mimas, while no evident coorbital stability region is seen inside the Janus-Epimetheus region. Outside of the regions delimited by black lines, the more stable particles are those of small eccentricity, expected since they do not cross the orbit of any large moon.

Regarding particle collisions with the large moons we see that, for the simulation shown in Fig. 3, due to the Mimas larger eccentricity, compared to that of Enceladus or Janus and Epimetheus, it takes longer time for Mimas to clean up its region despite its large size. From the $1\,454$ particles initially inside the Mimas region, only $\sim$12% of them collided with the large moon. The surviving particles have an average $Log(D)=-1.4091149$. This implies a diffusion time of $t_{Diff}=9.946\times 10^{3}$ yr, considering the orbital period of Mimas. For Enceladus we found 885 particles initially inside its region, out of which almost 70% collided, and the surviving particles are mainly coorbitals with the large moon. The surviving particles have an average $Log(D)=-3.2606267$, which implies a $t_{Diff}=4.857\times 10^{5}$ yr, considering the period of Enceladus. Finally, in the region dominated by Janus-Epimetheus, out of the 806 particles initially inside the corresponding region, around 36% collided with any of the large moons. The average $Log(D)=-1.5613890$ of the surviving particles leads to a diffusion time of $t_{Diff}=1.915\times 10^{4}$ yr, by considering the period of Janus.

In the simulation shown in Fig. 4, Aegaeon, Methone, Anthe, and Pallene were included as n-bodies. The positions of the four small moons are indicated by brown circles in the map. All lines for constant pericenter and apocenter of test particles at the positions of apocenters and pericenters of the large and small moons are shown in black solid lines. In this case, a greater number of particles is removed from the collision regions (those enclosed by solid black lines) leading to the appearance of big white zones. As in the previous case, inside these collision regions, some coorbital particles survive, mainly with Enceladus and with Mimas to a lesser extent.

The results from the map of Fig. 4 suggest that the small eccentricity of Aegaeon, Methone, and Anthe, let them to better clean up their orbits in a short period of time ($\sim$18 yr), when compared to the much more massive Janus, Epimetheus, or Mimas. Nonetheless, inclinations likely play also an important role for the clearing ability of the moons. To test this, we perform a simulation with the same conditions of that of Fig. 4, but this time with the initial inclinations of all the nine moons set to zero. The result is shown in the map of Fig. 5. Clearly, when the moon lies in the orbital plane, it better clear its path, this is manifest in the large white regions inside the black lines limiting the regions of Janus-Epimetheus and Mimas. Moreover, the coorbital stability region is clearly visible now for Mimas, and two high eccentricity coorbitals with Janus-Epimetheus also survive. In the case of Aegaeon, Methone, and Anthe, the changes are insignificant as all they had from the beggining a small inclination ($<0.02^{\circ}$). The same lack of change is observed for Enceladus, which has an $i\approx 0.004^{\circ}$.

Therefore, moons of small eccentricity and inclination clean their orbits efficiently, regardless their mass (see Aegaeon, Methone, Anthe, and Enceladus), and coorbital particles with the lowly inclined moons are more likely to survive. On the other hand, moons with larger eccentricities and inclinations require longer times for the clearing of their orbits despite their large masses (see Janus-Epimetheus, Mimas, and Pallene), and coorbital particles are not likely to coexist with them due to a large inclination. Interestingly for Pallene, from the previous exercises we have gained an important hindsight about the high stability of the region surrounding this small moon. The combination of a small mass and a large inclination ($\sim 0.182^{\circ}$) make Pallene a weak perturber and inefficient clearer of its region.

In order to explore in detail the dynamical environment of the small moons, we mapped the geometric phase-space around them, zooming from the general view into three finer maps, one around Aegaeon, a second around Methone and Anthe, and one more around Pallene. Although the simulations last the same amount of time as the ones describe earlier, the coverage of $a$ and $e$ is far better. We describe the initial conditions used for each map in the next subsections.

4.2 The Dynamical Environment of Aegaeon

To get a better understanding of the dynamical evolution of Aegaeon and particles nearby, like the ones forming the G ring, we perform the numerical integration of the orbits of $11\,000$ test particles, covering a small patch of the geometric $a$ vs $e$ phase-space around the small moon. In $a$ we cover from 0.85 to 0.95 DMim, in steps of $1.0\times 10^{-4}$. In $e$ we cover from 0.0 to 0.02 in steps of $2\times 10^{-3}$. Initial inclination and the angles $\bar{\omega}$, $\Omega$, and $\lambda$, are all set to zero. Test particles are subject to gravitational perturbations from the oblateness of Saturn up to $J_{6}$, the five large moons of the global case described earlier, and Aegaeon, included as a massive body of $1.89\times 10^{13}$ g, assuming a bulk density of $0.54$ g/cm3 (Thomas et al., 2013). The resulting diffusion map is showed in Fig. 6.

The average location of Aegaeon is shown in the map of Fig. 6 by a filled brown circle. This location is close but not equal to the location of the 7:6 MMR, as Aegaeon librates in the 7:6 CER with Mimas. The long-term dynamics of Aegaeon in this resonance will be further analyzed in section 5. In this map, black solid lines indicate the constant apocenter and pericenter distances of particles, at the position of Aegaeon’s pericenter and apocenter, respectively. Most of the particles inside this region end up colliding with the small moon before the end of the simulation, which results in a wide white triangle feature. Initially, $2\,042$ particles were inside the region enclosed by solid black lines, while just 303 of them survived, i.e. almost 85% of the particles collided with Aegaeon in just $\sim$18 yr. In average, surviving particles have $Log(D)=-4.6857625$, which implies a $t_{Diff}=2.192\times 10^{7}$ yr.

As we can see from Fig. 2, a great number of first and second order MMRs, mainly with Janus and Mimas, populate the region around Aegaeon. The evenly colored vertical features in the map of Fig. 6, both interior and exterior to the orbit of Aegaeon, are related to any of these MMRs. To identify the strongest resonances, and therefore particles librating inside them, we make use of the frequency curve (Robutel & Laskar, 2001). In Fig 7 we show the frequency curve, i.e. the ratio of the particle’s mean motion over Mimas mean motion, $n/n_{Mim}$, vs the initial particle geometric $a$. MMRs are characterized for the flatness of the frequency curve, due to a constant $n/n_{Mim}$ ration of nearby particles of different $a$; this facilitates the identification of librating particles. In the curve of Fig. 7, we point out (with arrows in the figure) to the presence of the next first order MMRs: 6:5, 7:6, 8:7, 9:8, 10:9, 11:10, 11:12, and 13:12; although the flatness of the frequency curve associated to the mentioned resonances is barely visible, some particles cause the curve to flatten, making possible their identification. To exemplify the way how frequency curve flats in resonances, in Fig. 7 we zoomed into a region very close to the 7:6 MMR with Mimas, where a number of particles maintains a constant ratio $n/n_{Mim}=7/6$. We note that the small maximum eccentricity covered by our grid of initial conditions accounts for the width of the resonant region to be very small, thus making difficult for such regions to stand out from the Keplerian trend of the rest of the frequency curve.

It is in this sense that the resonant regions are better recognized directly from the diffusion map of Fig. 6. In the map we have plotted as black rectangles the resonant particles with Mimas identified from the frequency curve, i.e. the particles for which the ratio $n/n_{Mim}$ is equal to the corresponding MMR ratio $p/q$. Nonetheless, apart from this first order MMRs with Mimas, other homogenous vertical regions are clearly recognizable, and other resonances can be related to them. For this, we have indicated at the top of the Figure the corresponding ratios at the position of first and second order MMRs with Janus, in orange, and with Mimas, in blue. Such resonances could not be as strong as first order MMRs with Mimas, however, their influence is imprint in the homogeneous $D$ parameter they produced over particles of different eccentricities.

We highlight the importance of MMRs on the evolution of test particles, even as there are not known structures or particles in such regions in the real system, i.e. outside the edges of the G ring. If ever a population of particles inhabited some of the stable regions currently empty, then it needs to be an explanation, other than the pure gravitational evolution, for the lost of such particles. Non-gravitational forces are a likely explanation, given that particles inhabiting such regions are expected to be small and therefore subject to drift forces like solar radiation pressure and plasma drag (see for example Sun et al., 2017, Madeira et al., in prep.).

Regarding the G ring, we know it is formed by $\mu$m-sized particles and extends from $\sim 165\,000$ to $175\,000$ km from Saturn centre, or from $\sim$0.8893 to 0.9432 DMim (Horányi et al., 2009). In Fig. 6 the location and size of the G ring are indicated by the double-headed black arrow. Also showed in the figure, in dashed lines, are the constant pericenter and apocenter of particles at the position of the inner and outer edges of the G ring, respectively. It is assumed that particles inside this region can form part of the ring, since particles outside of it (some of them with the same $a$ but larger $e$) would have pericentric and apocentric excursions farther away from the observed edges of the ring. Initially $3\,913$ particles form part of the ring, while $2\,937$ survive at the end of the simulation. This is, around 25% of the G ring particles are lost due to collisions with Aegaeon. Besides, surviving particles have in average a $Log(D)=-4.3779971$, which implies a $t_{Diff}=1.079\times 10^{7}$ yr. Such particles are dynamically heated, i.e. by increasing their inclination, from 0 to 0.00425∘ in average, mainly due to the effect of MMRs with Mimas. This average inclination implies that the G ring is vertically widened in a short time period, by $\sim 12.6$ km, considering an average $a\approx 170\,000$ km for the vertical excursions of particles ($a\times i$).

4.3 The Region Around Methone and Anthe

For Methone and Anthe, we study the region around both moons in the same map, due to their proximity. The simulation performed for this aim covers from 1.02 to 1.10 DMim in geometric $a$ and from 0 to 0.02 in geometric $e$. The steps in $a$ and $e$ are, as in the previous case, $1.0\times 10^{-4}$ and $2\times 10^{-3}$, respectively. This makes a total of $8\,800$ test particles. Inclination is set to 0 as well as $\bar{\omega}$, $\Omega$, and $\lambda$. Particles are subject to the oblateness of Saturn, gravitational perturbations from the five large moons, plus Methone and Anthe, whose masses were considered to be: $9.38\times 10^{14}$ g for Methone and $3.5\times 10^{14}$ g for Anthe. Regarding densities, we assumed values of 0.31 g/cm3 for Methone, following Thomas et al. (2013), while for Anthe we used a value of 0.35 g/cm3, by considering an average of the densities of the other three small moons.

The diffusion map for the region of Methone and Anthe is shown in Fig. 8. Brown filled circles mark the average position of the small moons. Black lines indicate the pericentric and apocentric distances of particles at the positions of the apocenters and pericenters of Methone and Anthe, respectively. The blue line mark the pericentric distance of particles at the apocenter of Mimas.

An important fraction of this map is unstable, mainly due to the overlapping of the collision regions of Mimas, Methone, and Anthe. Nonetheless, some particles in the region interior to the orbit of Methone, lie in MMRs with Mimas, which are tightly packaged the closer they are to the large moon. Another stability zone exists, located between the Methone and Anthe locations, while the predominantly stable region can be found beyond the orbit of Anthe, where the perturbative influence of Mimas is weaker. Inside the combined collision regions of Methone and Anthe, there are initially $4\,204$ particles, out of which $2\,254$ survive the whole simulation, therefore, around 46% of the particles end up colliding with any of the moons. For the surviving particles an average $Log(D)=-2.9265259$ is found, which implies a $t_{Diff}=3.023\times 10^{5}$ yr. This is a considerable shorter stability time than that found for the surviving particles inside the Aegaeon region; this since although there are few of them, several of the Aegaeon surviving particles are those of smaller eccentricity, while others remain in resonance, avoiding larger perturbations from Mimas and collision with Aegaeon. In this case, however, many particles that survive the simulation are part of the high eccentricity and largely perturbed region of the map, for which it is to be expected shorter diffusion time-scales.

Fig. 9 shows the frequency curve for the Methone/Anthe region. Interestingly, the unstable region close to Mimas is very easy to identify due to the scatter of the points (or $n/n_{Mim}$ values) away of the Keplerian trend. In the figure we point out the existence of the Mimas MMRs: 7:8, 8:9, 9:10, 10:11, 11:12, 12:13, 13:14, and 14:15. The particles with the constant ratios responsible for the plateaus are colored in black in the map of Fig. 8.

4.4 The Dynamical Environment of Pallene

The region around Pallene was studied through a simulation of test particles covering a patch of geometric $a$ vs $e$ phase-space, going from 1.1 to 1.2 DMim in $a$, in steps of $1.0\times 10^{-4}$. In $e$ the values run from 0.0 to 0.02, in steps of $2\times 10^{-3}$, totalizing $11\,000$ test particles. They are subject to the gravitational perturbations from an oblate Saturn, the five large moons of the region plus Pallene. The mass of Pallene is taken to be $2.69\times 10^{15}$ g, with a density of 0.25 g/cm3 (Thomas et al., 2013).

The diffusion map for the Pallene region is shown in Fig. 10. Same as in previous cases, the average location of Pallene is mark by the filled brown circle. Solid black lines delimit the region where particles have the highest probability for colliding with Pallene. Dashed black lines delimit the apocentric and pericentric distances of particles at the position of the inner and outer edges of the Pallene ring, respectively. The width of the ring is accounted for with the double-headed arrow. The ring is about $2\,500$ km in width and centered at the small moon (Hedman et al., 2009). Interestingly, the radial extend of the ring is constrained by the largest pericentric and apocentric excursions of Pallene itself.

We can see from Fig. 10 that Pallene does not lie close to any first order MMR, neither with Mimas or Enceladus, suggesting the non-resonant dynamics of the small moon and its ring. If by chance, the second order MMRs 9:11 with Mimas and 13:11 with Enceladus, lie close to the position of Pallene, these weak resonances seems unable to affect the dynamics of the small moon whatsoever. Nonetheless, a rich structure of resonances shapes this intermediate region between the major satellites Mimas and Enceladus, as can be seen from Fig. 2. In the map of Fig. 10 we plot in black the resonant particles in first order MMRs either with Mimas or Enceladus. At the top of the figure we indicate the ratios of the resonances (in blue for Mimas, in red for Enceladus). It is clear that some of the vertical features visible on the map are not related to first order MMRs, therefore we also indicate, by labeling the ratio at the top of the figure, the location of second order MMRs with Mimas and with Enceladus, with the same color code. Despite the abundance of such first and second order MMRs, all the region is globally very stable and only a minor fraction of particles are lost in our short time integration.

Inside the collision region of Pallene, there are initially $3\,522$ particles, while $3\,375$ of them survive at the end of the simulation. This is, less than $\sim$5% of the particles collide with Pallene in $\sim$18 yr. In average, for the surviving particles we have a $Log(D)=-5.4219$, which implies a stability time of $t_{Diff}=8.365\times 10^{7}$ yr, the longest for any of the regions studied so far. Regarding the Pallene ring, there are initially inside the region delimited by the dashed lines a total of $266$ particles, out of which only one collided with Pallene. A $Log(D)=-6.0716074$ is found in average for the ring surviving particles, leading to $t_{Diff}=3.733\times 10^{8}$ yr. The average inclination of the surviving ring particles is 0.001023∘. This inclination implies vertical excursions of just $\sim 3.8$ km, considering an average $a\approx 212\,000$ km. This is an order of magnitude smaller than the vertical width of around 50 km, determined by Hedman et al. (2009) for the Pallene ring. Nonetheless, we expect a larger value of the final average inclination in longer simulations. Such explorations are planned to be presented in a future work.

It is worth to note the high stability of the orbits inside the region of the maximum orbital excursions of Pallene. Even more, particles of small eccentricity laying close to the small moon, at least below $e\sim 0.006$, are those actually forming part of the ring and are expected to survive in a long-term basis. If the ring itself is a stable feature, it should then be formed by this low $e$ particles, leading in its way to a ring of small eccentricity also. Due to the limited number of Cassini images in which the Pallene ring is visible, it has not been possible to accurately determine the eccentricity of the ring, among other properties such as its mass. Nonetheless, the ring seems to follow closely the orbital path of the small moon. This fact implies that most likely the ring shares the same orbital properties of the parent moon. A further detailed study about the ring is planned to be presented in a future paper.

5 Long-term evolution of Aegaeon, Methone, Anthe, and Pallene

In this section we present the results of a single long-term simulation, lasting for $10^{5}$ yr, which included the nine large bodies of Table 1, and the oblate Saturn with physical parameters given in Table 2. In Fig. 11 we plot the evolution of the geometric orbital elements of the four small moons of Table 4. For them, we calculate the mean and maximum variations of the geometric parameters. Also, we apply the frequency analysis to the small moons in order to estimate a diffusion parameter and an associated diffusion time-scale for them in the current configuration of the saturnian system. Our results are summarize in Table 5.

Second and third columns of Table 5 show the mean and maximum variations of the geometric semimajor axis, $\bar{a}_{Geo}$, in units of DMim (second column) and km (third column) for an easier interpretation. The largest variations in this parameter are those of Methone and Anthe. We can see that all the orbits of the small moons are highly stable over the $10^{5}$ yr simulation. The largest ones, for Methone and Anthe, are of only $\sim 28$ km. It has been suggested that those two moons are closer to a chaotic zone produced by the overlapping of the CER and LER resonances with Mimas (El Moutamid et al., 2014). The variations, well above the 4 km separation of the CER and LER may lead to Methone and Anthe to be influenced by both resonances at different times, nonetheless, the combined perturbations are weak enough as being ineffective in leading to a chaotic evolution of their orbits in $10^{5}$ yr.

In the case of Aegaeon, the variations $<6.5$ km implies that the moon is well trapped inside the 7:6 CER with Mimas. For Pallene, the variations of only $\sim$3 km turns it into the more stable of the small moons. Pallene is far from any first order MMR, this allows it to avoid any strong perturbation from Mimas or Enceladus.

Regarding geometric $e$ and $i$, Anthe turns out to be the more perturbed in $\bar{e}_{Geo}$, probably as a result of perturbations produced by the overlapping the CER and LER with Mimas, being that, as El Moutamid et al. (2014) have pointed out, Anthe is closer to the chaotic zone than any of the other small moons, although variations in this parameter for Methone are comparable. Regarding inclinations, for Aegaeon, Methone, and Anthe, they are also small, so the small moons remain very close to Saturn’s equatorial plane; Aegaeon reachs maximum excursions above the plane of only $\sim 4.7$ km, while for Methone and Anthe vertical excursions are of $\sim 52$ and $\sim 46$ km, respectively. For Pallene such excursions reach $\sim 653$ km.

The orbital behavior of Pallene is interesting, showing that variations in $e$ and $i$ are periodic and clearly correlated. The period of such oscillations is $4\,762.21$ yr, for both $e$ and $i$, and although its mean values of $e$ and $i$ are the largest among the small moons, it presents the more stable orbit. We can confirm the last statement by calculating the diffusion parameter and the associated diffusion time for the orbits of the small moons, as shown in the last two columns of Table 5. The value of $D$ is determined from the evolution of the same quantity $z^{\prime}(t)=a(t)exp(i\lambda)$ as for the test particles of the diffusion maps. Here, $t_{Diff}$ gives us an estimation of the time required for an appreciable change in the orbital evolution of the small moons to be observed. As shown in the last column of Table 5, Pallene could remain stable for as long as $\sim$64 Myr in its current orbit, while Aegaeon, Methone, and Anthe could do the same for only $\sim 0.45$ Myr. This short-time period results from a crude estimation and could result from the large variations of the three small moons produced by the resonances they are trapped in. Nonetheless, for Pallene it seems to represent a confident value for the stability of the moon in a long term basis.

Regarding the resonant nature of the small moons, we show in Fig. 12 the long term evolution of the same resonant arguments previously shown if Fig. 1, where we explored their short-term evolution. Now we show a period of $10\,000$ yr out of the $10^{5}$ yr of our simulation, as the behavior remains unchanged along the whole run. From Fig. 12 we can conclude that the three small moons currently trapped in CERs with Mimas, will remain there for a long-term basis, as is shown by blue curves in the three first panels of Fig. 12. Such arguments remain librating for the whole time of the simulation. Aegaeon, Methone, and Anthe are strongly affected by CER resonances and they could hardly escape from them, probably explaining the existence of their associated arcs. Indeed, if the small moons remain for a long time trapped in the resonances, the slow impact-ejecta processes that are believed to suply the arcs with dusty material, could have plenty of time to act and originate such structures, even if the tiny grains are constantly removed due to the action of gravitational and non-gravitational forces, such as collisions, solar radiation, and plasma drag.

For Methone, the argument associated to the LER with Mimas also remains librating for the whole simulation, therefore, this resonance also affects the dynamics of the small moon; this could explain the large variations in $a$ seen for Methone. In the case of Anthe, the argument $\varphi_{2}=11\lambda_{Ant}-10\lambda_{Mim}-\bar{\omega}_{Ant}-\Omega_{Ant}+\Omega_{Mim}$, shown in Sec. 3, starts librating but then it changes and starts to circulate, then librates again repeating the cycle with a constant period and sequence. Although the erratic regime changes from libration to circulation are a signature of chaotic behavior, in this case, the regularity in the behavior of this argument seems not to lead to the chaotic evolution of Anthe.

In the case of Pallene, we show the two arguments previously presented in Sec. 3; now it is clear that both arguments circulate. The resonant argument associated with Mimas, $\theta=\bar{\omega}_{Pal}-\bar{\omega}_{Mim}-\Omega_{Mim}+\Omega_{Pal}$, circulates with a period of $\sim 4\,762.21$ yr; this is the same period as the one found for the long-term oscillations of $e$ and $i$. Such correlation implies that Mimas induces those variations on the orbit of the small moon, despite leaving it in a highly stable orbit.

6 Conclusions

In this work we have explored the dynamical evolution of four Saturnian small moons: Aegaeon, Methone, Anthe, and Pallene, as well as characterized their dynamical environment by using thousands of test particles surrounding them, by means of short and long term simulations, going up to $10^{5}$ yr, this is, longer than any previous numerical exploration. We have considered the current configuration of the Saturnian system, including the oblateness of Saturn up to $J_{6}$ and the five largest moons close to the region of interest, namely Janus, Epimetheus, Mimas, Enceladus, and Tethys. Through frequency analysis and the long term evolution of the moons we have analyzed their stability and that of the region surrounding them. The frequency analysis has allowed us to obtain a global dynamical perspective of the region inhabited by the small moons by using detailed but short time integrations. With it we gain an estimation of the stability times for particles in such regions and it also let us characterize the resonances involved in shaping or perturbing such regions. We have also found that:

• •

all the four small moons are stable in a long-term basis. If the current configuration of the system would remain unchanged, this is, if the migration of the largest moons, mainly Mimas, is slow enough, then the stability time for the small moons would be at least $\sim 0.4$ Myr for Aegaeon, Methone, and Anthe, and up to 64 Myr for Pallene.

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Aegaeon remains trapped in the 7:6 CER with Mimas, with maximum variations of only $\sim 6.4$ km in $a$. Its $e$ remains small, and variations in $i$ implies maximum excursions above Saturn’s equatorial plane of 4.7 km.

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Methone and Anthe share similar stability times, as well as variations of the same order in their orbital parameters. Methone librates in both the 14:15 CER and LER with Mimas, while Anthe remains in the 10:11 CER with Mimas but it is also close to the chaotic region produced by the overlapping of the CER and LER. The influence of two resonances on those small moons likely lead to largest variations than those observed for Aegaeon. For both Methone and Anthe, variations in $a$ are of $\sim$28 km, while their $i$ imply vertical excursions of 52 and 46 km, respectively.

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Pallene is the most stable of the small moons analyzed. It remains as non-resonant but, nonetheless, it suffers long-term perturbations from Mimas, through the quasi-resonance found by Callegari & Yokoyama (2010). Such perturbations only affect the evolution of $e$ and $i$, inducing long-term oscillations, while its $a$ changes by only $\sim$3.2 km.

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Regarding the G ring, 25% of ring particles collide with Aegaeon, while the remaining are stirred in their $i$ up to $0.00425^{\circ}$ in average. This implies a vertical widening of 12.6 km in just $\sim 18$ yr, of an initially flat distribution of particles.

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The Pallene ring may be formed of particles of small eccentricity, below $\sim$0.006. Despite the larger size of Pallene compared to Aegaeon, Methone or Anthe, the small moon is unable to efficiently clean up its orbit, due to its anyway small size and large $i$. The average final inclination of ring particles is $0.001023^{\circ}$, which leads to a vertical width of $\sim$3.8 km.

All the regions explored in this work have some stable zones where no particles are found in the real system. If ever some particles existed in such regions, then some mechanism should be responsible for their removal. Such mechanisms are most likely the non-gravitational forces, such as solar radiation force and plasma drag, as it is expected that particles close to the small moons were originated when micrometeoroid IDPs hit the surfaces of the small moons, leaving ejecta that is also micrometrical in size.

The influence of non-gravitational forces have been studied for the regions of Aegaeon (for example in Madeira et al., in prep.), Methone and Anthe (Sun et al., 2017). The work exploring such shaping forces for the Pallene ring is under investigation.

Acknowledgements

We acknowledge an anonymous referee for insightful comments that help to improve the present work. We thank the financial support from FAPESP (Proc. No 2016/01467-8 and 2011/08171-3). SGW also thanks CNPq project number 309254/2012-4. We acknowledge the use of the Saturn Cluster of the Grupo de Dinâmica Orbital e Planetologia at UNESP, campus of Guaratinguetá.