# Long-term Dynamical Evolution of Pallene (Saturn XXXIII) and its Diffuse,
Dusty Ring

Marco A. Muñoz-Gutiérrez,1 A. P. Granados Contreras,1 Gustavo Madeira,2,3
Joseph A. A’Hearn,4 and Silvia Giuliatti Winter2
1Institute of Astronomy and Astrophysics, Academia Sinica, 11F of AS/NTU
Astronomy-Mathematics Building, No.1, Sec. 4,
Roosevelt Rd, Taipei 10617, Taiwan, R.O.C.
2Grupo de Dinâmica Orbital e Planetologia, São Paulo State University (UNESP),
333 Av. Dr. Ariberto Pereira da Cunha, Guaratinguetá-SP, 12516-410, Brazil
3Université de Paris, Institut de Physique du Globe de Paris, CNRS, F-75005
Paris, France
4Department of Physics, University of Idaho, 875 Perimeter Drive, Moscow,
Idaho 83844, USA E-mail<EMAIL_ADDRESS>(MAM)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The distinctive set of Saturnian small satellites, Aegaeon, Methone, Anthe,
and Pallene, constitutes an excellent laboratory to understand the evolution
of systems immersed in co-orbital dusty rings/arcs, subjected to perturbations
from larger satellites and non-gravitational forces. In this work, we carried
out a comprehensive numerical exploration of the long-term evolution of
Pallene and its ring. Through frequency map analysis, we characterised the
current dynamical state around Pallene. A simple tidal evolution model serves
to set a time frame for the current orbital configuration of the system. With
detailed short and long-term N-body simulations we determine whether Pallene
is currently in resonance with one or more of six of Saturn’s major moons. We
analysed a myriad of resonant arguments extracted from the direct and indirect
parts of the disturbing function, finding that Pallene is not in mean motion
resonance from the present up to 5 Myr into the future; nonetheless, some
resonant arguments exhibit intervals of libration and circulation at different
timescales and moon pairings. We studied the dynamical evolution of
micrometric particles forming the ring, considering gravitational and non-
gravitational forces. Non-gravitational forces are responsible for particles
vertical excursions and outward migration. By estimating the satellite’s mass
production rate, we find that Pallene could be responsible for keeping its
ring in steady-state only if it is mainly composed of large micrometre-sized
particles. If mainly composed of particles with a few micrometres for which
Pallene is the only source, the ring will spread out, both radially and
vertically, until it finally disappears.

###### keywords:

planets and satellites: individual: Pallene – methods: numerical – planets and
satellites: dynamical evolution and stability – planets and satellites: rings

††pubyear: 2021††pagerange: Long-term Dynamical Evolution of Pallene (Saturn
XXXIII) and its Diffuse, Dusty Ring–Long-term Dynamical Evolution of Pallene
(Saturn XXXIII) and its Diffuse, Dusty Ring

##  1 Introduction

Pallene (Saturn XXXIII) is a satellite of only 2.23 km in radius (Thomas et
al., 2013), orbiting Saturn at an average distance of $\sim 212\,283$ km, with
an eccentricity of $\sim 0.004$, and a relatively large inclination of $\sim
0.18^{\circ}$ (Spitale et al., 2006; Jacobson et al., 2008).

This small Saturnian moon was first observed in a single photograph of the
_Voyager 2_ spacecraft. It was reported, together with a preliminary orbital
and physical characterisation, by Synnott (1986). Pallene was then
rediscovered in 2005 by the _Cassini_ Imaging Science team (Porco et al.,
2005) and positively identified as the S/1981 S14 object from _Voyager 2_.

Pallene is one of three small moons located between the orbits of Mimas and
Enceladus, called collectively as the Alkyonides. Despite the presence of a
vast number of resonances in the region, accentuated by the commensurabilities
of Mimas with Tethys and Enceladus with Dione (see e.g. Sinclair, 1972;
Greenberg, 1973; Peale, 1976, 1999), Pallene doesn’t seem to be in a mean
motion resonance (MMR), unlike Methone and Anthe, trapped in 14:15 and 10:11
corotation eccentricity resonances with Mimas, respectively (Cooper et al.,
2008; Hedman et al., 2009; El Moutamid et al., 2014). Nonetheless, Pallene is
migrating away from Saturn via tidal evolution, though at a slower rate than
Mimas. Thus the orbits of Pallene and Mimas are converging, which at some
point either in the past or in the future should have resulted or should
result in Pallene being captured into resonance with Mimas. A simple tidal
evolution model (Murray & Dermott, 1999) suggests that the most recent first-
order resonance that Pallene might have escaped from, perhaps around 40 Myr
ago, is the 4:5 resonance with Mimas. Pallene’s current eccentricity or
inclination could be signs of this or another past resonance.

After Pallene’s rediscovery, Spitale et al. (2006) determined the orbital
parameters of Pallene with high accuracy using images from _Cassini_ and
_Voyager 2_. Spitale et al. suggested that Pallene could be in an inner third-
order resonance with Enceladus (i.e.,
$\phi=19\lambda_{\mathrm{Enc}}-16\lambda_{\mathrm{Pal}}-\varpi_{\mathrm{Pal}}-2\Omega_{\mathrm{Pal}}$).
However, recent short-term numerical integrations have shown that this
resonance’s libration angle actually circulates (e.g. Fig. 1 in Muñoz-
Gutiérrez & Giuliatti Winter, 2017).

Furthermore, synchronous periodic eccentricity and inclination oscillations
were found while exploring a longer-term dynamical evolution of Pallene (of up
to $10^{5}$ yr, Callegari & Yokoyama, 2010; Muñoz-Gutiérrez & Giuliatti
Winter, 2017), which could be the result of a resonant perturbation produced
by either Mimas or Enceladus. Moreover, Callegari & Yokoyama (2010) identifies
a possible argument for the proposed quasi-resonance involving the apsidal and
nodal longitudes of Pallene and Mimas, given by
$\phi=\varpi_{\mathrm{Pal}}-\varpi_{\mathrm{Mim}}+\Omega_{\mathrm{Pal}}-\Omega_{\mathrm{Mim}}$.
Nonetheless, Muñoz-Gutiérrez & Giuliatti Winter (2017) found that this
argument also circulates with a period of $\sim 4762.2$ yr, though,
interestingly, with the same period of the observed oscillations of Pallene’s
eccentricity and inclination.

Pallene shares its orbit with a diffuse ring of micrometre-sized dust, first
reported by Hedman et al. (2009). The constant resupply of ring material is
expected to come from impact debris, expelled from the satellite’s surface by
collisions between interplanetary dust particles (IDPs) and the moon. A
similar mechanism has been proposed and explored in order to explain the
existence of Aegaeon’s ring arc inside the G ring (Hedman et al., 2010;
Madeira et al., 2018), the ring arcs of Methone and Anthe (Sun et al., 2017;
Madeira & Giuliatti Winter, 2020), as well as the Neptunian rings and arcs
(Gaslac Gallardo et al., 2020; Giuliatti Winter et al., 2020).

In this work we carry out a comprehensive study of the long-term dynamics of
Pallene, as well as of the possible origin and dynamical evolution of its
diffuse dusty ring, formed by micrometre-sized particles subject to
gravitational and non-gravitational forces. We organise this paper as follows:
in Section 2, we describe the different set-ups of our numerical simulations,
performed to address various aspects of our study; we characterise the current
dynamical environment of Pallene and its ring through frequency map analysis
in Section 3. In Section 4, we first estimate the time span in which the
current orbital configuration of the Saturnian system would remain
approximately unchanged by using a simple tidal evolution model; then, with
detailed short- and long-term simulations, we re-evaluate at different
timescales all possible libration angles between Pallene and the six major
Saturnian satellites considered in our study. Finally, a characterisation of
the evolution of Pallene’s ring is carried out in Section 5, where all the
relevant non-gravitational forces that affect small particles are considered.
We summarise our work and present our main conclusions in Section 6.

##  2 Methods and Simulations

Table 1: Saturn’s physical parameters.

Parameter | Value | Reference
---|---|---
$R_{\mathrm{S}}$ [km] | $60\,330$ | Kliore et al. (1980)
$GM_{\mathrm{S}}$ [km3 s-2] | 3.793120749865220E+07 |  gm_de431.tpc _a_
$J_{2}$ | 1.6290573E-02 | Iess et al. (2019)
$J_{4}$ | -9.35314E-04 | Iess et al. (2019)
$J_{6}$ | 8.6340E-05 | Iess et al. (2019)
$\Omega_{\mathrm{S}}$ [rad s-1] | 1.65269E-04 | Helled et al. (2015)

  * _a_

Available at
https://naif.jpl.nasa.gov/pub/naif/generic_kernels/pck/gm_de431.tpc

Table 2: Summary of physical parameters of the six large moons in our system.

Name |  $GM_{m}$_a_ | $\rho_{m}$ |  $R_{m}$_b_
---|---|---|---
| [km3 s-2] | [g cm-3] | [km]
Mimas | 2.503522884661795E+00 | 1.152 | 198.2
Enceladus | 7.211292085479989E+00 | 1.606 | 252.6
Tethys | 4.121117207701302E+01 | 0.956 | 537.5
Dione | 7.311635322923193E+01 | 1.469 | 561.4
Rhea | 1.539422045545342E+02 | 1.233 | 763.8
Titan | 8.978138845307376E+03 | 1.880 | 2574.7

  * _a_

$GM_{m}$ values are taken from the planetary constant kernel gm_de431.tpc.

  * _b_

Radius values, $R_{m}$, are taken from the planetary constant kernel
pck00010.tpc (available at
https://naif.jpl.nasa.gov/pub/naif/generic_kernels/pck/pck00010.tpc, Archinal
et al., 2011).

We carried out extensive and detailed numerical simulations of the evolution
of the dynamical system formed by Pallene and six major Saturnian satellites,
those gravitationally relevant in our region of interest, namely: Mimas,
Enceladus, Tethys, Dione, Rhea, and Titan. Throughout this work, we consider
Saturn’s oblateness and take into account zonal harmonic terms up to $J_{6}$
in all simulations. Our numerical integrations cover several time spans, in
order to study different aspects of the dynamics of Pallene, its phase-space
surroundings, as well as the evolution of its dust-ring. Our shortest
simulation lasts 18 yr, while the longest simulation is $5\times 10^{6}$ yr
long.

Unless otherwise stated, the physical parameters of Saturn and the six major
moons used throughout this work are summarised in Tables 1 and 2. We use a
rotation rate for Saturn $\Omega_{\mathrm{S}}=1.65269\times 10^{-4}$ rad/s
from Helled et al. (2015). As initial conditions for the major Saturnian
moons, we use the satellite’s Saturn-centric state vectors taken from the JPL
Horizons ephemerides service111https://ssd.jpl.nasa.gov/horizons.cgi on
$JD=2459305.5$, corresponding to April 1, 2021. We scale the satellite semi-
major axes and masses to Pallene’s semi-major axis and Saturn’s mass,
respectively. The system’s gravitational constant is scaled accordingly, for
which we use Pallene’s average semi-major axis
$\bar{a}_{\mathrm{Pal}}=2.1228335\times 10^{5}$ km (as found in Muñoz-
Gutiérrez & Giuliatti Winter, 2017) and the $GM_{\mathrm{S}}$ parameter given
in Table 1. Consequently, our gravitational constant for this system is
$G=29.59895344398\;\bar{a}_{\mathrm{Pal}}^{3}\,M_{\mathrm{S}}^{-1}\,d^{-2}$.

Pallene’s mass is derived from its size, which has been measured with small
uncertainty, i.e., $R_{\mathrm{Pal}}=2.23\pm 0.07$ km, as well as from its
reported range of bulk density, i.e. $0.19\leq\rho_{\mathrm{Pal}}\leq 0.34$
g/cm3 (Thomas et al., 2013). We explore three different density values to
cover the uncertainty reported by Thomas et al., i.e., $\rho_{\mathrm{Pal}}=$
0.19, 0.25, and 0.34 g/cm3. This means that for each simulation suite
described in the following paragraphs, we run three versions, each with
Pallene’s gravitational parameter given by $GM_{\mathrm{Pal}}=$
5.89064055531E-07, 7.75084283594E-07, and 1.05411462568E-06 km3/s2,
corresponding to the selected density values. At the end of each of our
integrations, we convert the state vectors ($\vec{r}$ and $\vec{v}$) to
geometric orbital elements (Renner & Sicardy, 2006), which reduces the short-
term oscillations of the osculating elements due to the oblateness of the
central mass.

In order to place Pallene within its current dynamical context, our first
objective is to characterise the dynamics of a broad region of the geometric
semi-major axis-eccentricity ($a$-$e$) phase-space plane around Pallene. With
this in mind, we performed two numerical simulations (lasting 18 and $10^{4}$
yr, respectively), including a total of $13\,025$ test particles covering a
grid of the geometric $a$-$e$ plane in the vicinity of Pallene. For these
integrations, we used the Bulirsch-Stoer integrator from the Mercury6 package
(Chambers, 1999), with a toleration accuracy parameter of $10^{-12}$ and an
initial time-step of 0.1 days.

Secondly, to examine the big picture of Pallene’s tidal evolution, we use a
simple model based on Murray & Dermott (1999), which assumes a linear tidal
dissipation mechanism and a constant $Q$, independent of frequency. We only
examine the tidal evolution of Pallene and the large moons in its vicinity,
Mimas and Enceladus, in order to look at resonances that may have been crossed
in the recent past, as well as to establish a time limit of the validity of
the current orbital configuration of the system for the longer-term
simulations.

Next, in order to determine the possible resonant behaviour of Pallene, we
performed a set of N-body simulations, spanning from 50 up to $5\times 10^{6}$
yr of integration time. In this instance, the test particles are not included,
and there are only seven bodies orbiting Saturn. The N-body simulations are
performed with our implementation of the Implicit integrator with Adaptive
time-Stepping of 15th-order (IAS15, Rein & Spiegel, 2015) taking into account
Saturn’s gravitational moments (Table 1). Subsequently, we integrate the
satellite system for 50, $5\times 10^{3}$, $5\times 10^{4}$, $5\times 10^{5}$,
and $5\times 10^{6}$ yr. We use the geometric orbital elements to calculate
several libration angle combinations, among all satellites in Table 2 and
Pallene.

Finally, we study the evolution of the diffuse ring through two distinct
scenarios: (a) particles initially co-orbital to the satellite and (b) by
considering the temporal evolution of particles launched from Pallene’s
surface. The study is performed considering the system’s gravitational effects
and also non-gravitational forces acting in the region, such as solar
radiation force, plasma drag, and the electromagnetic force. Using an adapted
version of the Mercury6 package which includes the effects of these forces and
Saturn’s gravitational moments, we integrated the system formed by Pallene,
the six large moons, and a set of 5,000 test particles until all the particles
were removed from the simulation.

##  3 Pallene’s Current Dynamical Context

###  3.1 Characterisation Through Frequency Map Analysis

Figure 1: Diffusion map for a wide region of the geometric semi-major axis -
eccentricity phase-space plane around Pallene. A colour scale indicates the
stability of orbits, where bluer regions represent the more stable, and redder
ones the more unstable. Locations where particles were ejected or collided
with Pallene before the end of the simulation are coloured in white. The solid
black lines stand for the constant pericentre and apocentre distances of
Pallene, delimiting the collision region of the small moon. All the MMR ratios
which were explored for libration in the simulations of Section 4.2, going
from first to fourth order, are labelled at the top of the figure. Colours
correspond to MMRs with Mimas (blue), Enceladus (red), Tethys (orange), Dione
(brown), and Rhea (green). The final conditions of a longer simulation
($10^{4}$ yr), of the same particles used to create the diffusion map, are
over-plotted on the map (black dots) to highlight the predictive power of the
frequency analysis technique for the characterisation of the dynamical
stability of wide regions of phase space.

To gain a better understanding of the dynamical behaviour and future stability
of Pallene, as well as of the micrometric dust particles in its vicinity, we
carried out a frequency map analysis (FMA, Laskar, 1990; Laskar et al., 1992;
Robutel & Laskar, 2001) of a broad region of the geometric $a$–$e$ phase space
plane, surrounding Pallene. We performed a short-term numerical integration
(of $\sim 18$ yr, or approximately $5\,700$ Pallene orbital periods and
$2\,000$ orbital periods of the most external particle in the map). We used
this time span since at least $2\,000$ revolutions of each particle are
required to confidently recover the main orbital frequencies.

We included $13\,025$ test particles distributed in a homogeneous grid
covering the $a$–$e$ geometric plane, with the following conditions: $a$ is
sampled from 0.95 to 1.05 $D_{\mathrm{Pal}}$ (where $D_{\mathrm{Pal}}$ is the
normalised average geometric semi-major axis of Pallene) in steps of size
$\Delta a=5\times 10^{-5}$. In $e$ we sampled from 0 to 0.02 in steps of
$\Delta e=2\times 10^{-3}$. The remaining orbital elements are all set to zero
for simplicity, namely, inclination $I$, longitude of pericentre $\varpi$,
longitude of the ascending node $\Omega$, and mean anomaly $M$. We recall that
test particles are subject to the gravitational perturbations of an oblate
Saturn, Pallene, and the six gravitationally dominant moons in our region of
interest.

A frequency analysis for each test particle in the grid was performed, using
the algorithm of Šidlichovský & Nesvorný (1996), over the dynamical variable:

$\xi(t)=a(t)\exp(i\lambda(t)),$ (1)

where $a(t)$ and $\lambda(t)$ are the semi-major axis and mean longitude of
each particle, respectively. The variable $\xi(t)$ expresses a combination
closely related to a formal combination of the action and angle variables
($J_{i}$,$\eta_{i}$), of each orbit, expressed as
$\xi^{\prime}_{i}(t)=J_{i}\exp{\eta_{i}}$. Though it is clear that $\xi(t)$
and $\xi^{\prime}(t)$ are not equal, they are still related as
$\xi(t)=f(\xi^{\prime}_{1},\xi^{\prime}_{2},...,\xi^{\prime}_{n})$, being $f$
is a function close to unity (Laskar, 1993).

When we perform a frequency analysis of $\xi(t)$, we obtain a decomposition of
the form

$\xi(t)=\alpha_{0}\exp(i\beta_{0})+\sum_{k=1}^{N}\alpha_{k}\exp(i\beta_{k}).$
(2)

For a Keplerian orbit, the decomposition of $\xi(t)$ would have only one term,
i.e. $\alpha_{0}=a$ and $\beta_{0}=n$, where $\beta_{0}$ is what we call the
“mean frequency”, while $a$ and $n$ are the semi-major axis and mean motion of
the particle, respectively. For non-Keplerian orbits, the decomposition given
in Eq. 2 contains many periodic terms. Nonetheless, frequency analysis ensures
that if a particle remains in a stable orbit, the conditions expressed by the
approximations $\alpha_{0}\approx a$ and $\beta_{0}\approx n$ will prevail;
also, for stable orbits $\alpha_{0}\gg\alpha_{k}$. These conditions do not
hold for particles following unstable orbits, for which $\beta_{0}$ will
change dramatically from one time interval to the next, since the evolution of
chaotic orbits does not remain on the surface of KAM tori.

To compute the change of the main frequencies, we perform a frequency analysis
of $\xi(t)$ in two adjacent time intervals of length $T$, equal to half the
total integration time. We call $\beta_{01}$ and $\beta_{02}$ the main
frequencies obtained in each interval, respectively. Finally, we define a
diffusion parameter, $D$, which provides a measure of the stability of the
orbits. Following Correia et al. (2005); Muñoz-Gutiérrez & Giuliatti Winter
(2017) we have

$D=\frac{\left|\beta_{01}-\beta_{02}\right|}{T}.$ (3)

It can be seen that small values of $D$ will be obtained for stable
trajectories, while larger values of $D$ are the result of unstable orbital
evolution.

###  3.2 Diffusion Map of Pallene’s Neighbourhood

A diffusion map for the region around Pallene, shown in Fig. 1, was obtained
after applying the above procedure to all the grid particles covering the
geometric $a$–$e$ plane. A coloured rectangle is plotted for each particle
according to its initial location in the plane, where colour is scaled
according to the value of the logarithm of $D$. Redder colours indicate more
unstable orbits, while bluer colours represent the more stable trajectories.
Particles that are lost from the simulation before it finished, mainly due to
collisions with Pallene, are coloured white. Solid black lines delimit
Pallene’s collision region, i.e. the region in which, at their apocentric or
pericentric excursions, particles will cross Pallene’s orbit, thus having a
higher probability of colliding with the small moon.

The diffusion map provides a quick method to globally characterise the
dynamical state of a vast region of phase-space, at a low computational cost,
i.e. using only short-term numerical simulations. Unstable regions are
immediately highlighted by the colour contrast. We can quickly identify MMRs,
as well as their relative strength. The semi-major axis parameter space from
0.98 to 1.02 $D_{\mathrm{Pal}}$ encompasses completely both Pallene’s orbit
and the co-orbital dusty ring. In this region the strongest MMRs are due to
first-order commensurabilities with either Mimas or Enceladus, however,
higher-order MMRs with Dione, Tethys, and Rhea can also be observed. The
location of all the existing commensurabilities with the six major moons (up
to order 4 and degree 30) are indicated at the top of Fig. 1; outside this
interval we only indicate the location of first-order MMRs with Mimas and
Enceladus. The stronger resonances are characterised by thin vertical
structures of homogeneous yellow to orange colour, such as the 4:5, 5:6, 6:7,
and 7:8 MMRs with Mimas (blue labels), as well as the 5:4, 6:5, 7:6, 8:7, 9:8,
and 10:9 with Enceladus (red labels). Second-, third-, and fourth-order MMR
bands are thinner than first-order resonances. Furthermore, MMR chords are
less stable than the broader, non-resonant, blue bands, regardless of
eccentricity. Aside from possible exceptions at MMRs, lower eccentricity
orbits are far more stable in general throughout the map.

From Fig. 1, it is apparent that Pallene, whose location is indicated by the
large black circle, is not currently trapped inside any strong MMR, despite
the very close proximity of three resonances: the 9:11 with Mimas and the
19:16 and 25:21 with Enceladus.

Moreover, two interesting regions stand out from the map, corresponding to the
clustering of several MMRs with various moons. The first of such regions,
$b_{1}$, is located at $\sim 0.986$ $D_{\mathrm{Pal}}$, where the 5:6 MMR with
Mimas, the 17:14 and 23:19 MMRs with Enceladus, the 5:3 MMR with Tethys
(orange label), and the 4:1 MMR with Rhea (green label) lie in close proximity
to each other. The second region, $b_{2}$, is located around $\sim 1.014$
$D_{\mathrm{Pal}}$; in this region two first-order resonances, the 4:5 with
Mimas and the 7:6 with Enceladus, are in close proximity to the 8:5 MMR with
Tethys (orange label), and the 7:3 MMR with Dione. It is apparent that the
interaction of several low-order resonances results in especially unstable
regions at these locations. A similar case occurs at $\sim 0.966$
$D_{\mathrm{Pal}}$, where the two first-order resonances, 6:7 with Mimas and
5:4 with Enceladus, produce a particularly wide unstable region.

To reassess the predictive power of the frequency analysis technique, we
integrated up to $10\,000$ yr the same set of $13\,025$ particles of the grid
covering the geometric $a$–$e$ phase-space plane. The final conditions of this
simulation were over-plotted on the diffusion map of Fig. 1 with black dots.

To the left of the collision region of Pallene, the largest perturbations in
eccentricity are observed for particles located in the bands of MMRs, as
expected. The most unstable region, however, is the one located in the top
left corner of the map, roughly above 0.015 in eccentricity and to the left of
the 11:13 MMR with Mimas; here the direct influence of Mimas is stronger and
particles are removed faster. To the right of the collision region, all the
particles remain nearly unperturbed, except for the $b_{2}$ band where several
resonances converge, as well as at the locations of other first-order MMRs
with Mimas and Enceladus.

Notably, inside the collision region of Pallene, only three particles survive
after 10 kyr, one of them is a co-orbital with the small moon; a second lies
at the location of the 19:16 MMR with Enceladus, and the last one lies inside
the band of the 11:9 MMR with Enceladus, which overlaps with the 21:25 MMR
with Mimas.

Both the map and the long-term simulation of the particles serve as an
indication of the future evolution of large dust particles, with radii larger
than $\sim 30$ $\mu$m, i.e. those unaffected by non-gravitational forces known
to act in this region. Towards the internal zone of the Pallene collision
region, even this kind of large particles would be removed (though at times
greater than 10 kyr) due to perturbations from Mimas. Exterior to the Pallene
collision region, large particles could in principle survive for very long
times. This indicates that the Pallene ring would find greater stability
towards semi-major axes larger than that of the small moon, increasing the
eccentricity of its conforming particles as they find MMR regions with
Enceladus. On the other hand, ring-forming particles within the Pallene
collision region could survive mainly as co-orbitals; however, with only one
co-orbital and two apparently resonant particles surviving in this region in
the 10 kyr simulation, we cannot provide quantifiable predictions for the
behaviour of the ring, based exclusively on the diffusion map. For a more in-
depth analysis of the possible origin of the ring and its future evolution, we
performed a large set of detailed simulations, presented in Sections 5.2 to
5.4 of this paper.

##  4 Dynamical evolution of Pallene in different timescales

###  4.1 Tidal Evolution

To gain an appropriate perspective on the timescales of Pallene’s dynamical
evolution, we first look at Pallene’s tidal evolution in between Mimas and
Enceladus. Although more complex analyses of tidal evolution in the Saturn
system have recently been done (e.g. Fuller et al., 2016; Lainey et al.,
2020), here we employ a simpler model to gain a general understanding of the
context in which Pallene may have evolved. Using Equation 4.213 from Murray &
Dermott (1999), we can calculate previous semi-major axes

$a=a_{0}\left[1-\left(\frac{k_{2}}{Q}\frac{39M_{m}R_{S}^{5}}{2a_{0}^{13/2}}\sqrt{\frac{G}{M_{S}}}t\right)\right]^{2/13},$
(4)

assuming that the tidal dissipation mechanism is linear and that $Q$ is
frequency-independent.

For our tidal evolution calculations, we take our value for Saturn’s Love
number, $k_{2}=0.390$, from Lainey et al. (2017). We estimate a quality factor
$Q=2000$ also based on Lainey et al. (2017) and similar to what is used in Ćuk
et al. (2016), which was based on the earlier work of Lainey et al. (2012),
though there is less agreement on this value and it is meant to apply only
near the semi-major axes roughly around Mimas and Enceladus. Previous
estimates of $Q$ an order of magnitude higher were due to the assumption that
Mimas was primordial (Murray & Dermott, 1999; Meyer & Wisdom, 2008). However,
recent studies that argue Saturn’s rings and the mid-sized moons are probably
young, use a $Q$ value in the range we have assumed (Ćuk et al., 2016; Fuller
et al., 2016; Lainey et al., 2017; Neveu & Rhoden, 2019; Hesselbrock & Minton,
2019). Other values for this calculation are given in Tables 1 and 2.

Using these values, we measured the change in semi-major axis with respect to
today’s semi-major axis value $\frac{\Delta a}{a}$ over the past five million
years for Mimas, Pallene, Enceladus, Tethys, and Dione. Out of these
measurements, Mimas has $\frac{\Delta a}{a}=0.0017$, which is the largest
among these moons. Because this change in semi-major axis due to tidal
evolution is small, we expect our long-term simulations of 5 Myr without the
inclusion of tidal evolution to be accurate enough.

From the semi-major axis calculations, if Pallene is old enough, it may have
recently escaped the 4:5 resonance with Mimas (40 Myr ago with $Q=2000$).
Prior to escape, Pallene could have migrated with Mimas for a substantial
period of time. For this reason, it becomes difficult to project Pallene’s
previous tidal evolution with much certainty. If Pallene was not captured in
any resonance with Mimas for a significant period of time, which is unlikely
because their orbits are converging, then further in the past Pallene’s orbit
may have crossed that of Enceladus (400 Myr ago with $Q=2000$), suggesting
that Pallene could be a fragment from Enceladus, similar to the way Showalter
et al. (2019) propose that Hippocamp could have fragmented off of Proteus,
possibly from a cometary impact.

Hippocamp is close to the orbit that is synchronous with Neptune’s rotation,
which, together with the fact that it is the least massive of Neptune’s moons,
implies that the rest of Neptune’s moons are diverging from Hippocamp. In
contrast, Pallene’s orbit is converging with Mimas’s orbit. For this reason,
Pallene is expected to have been captured into resonance with Mimas at each
resonance crossing, but it is difficult to determine the duration of the
capture in each resonance.

Proteus and Hippocamp have mean radii of 203.8 km and 17.4 km (Showalter et
al., 2019), while Enceladus and Pallene have mean radii of 252 km and 2.23 km
(Roatsch et al., 2009; Thomas et al., 2013). Using these mean radii and masses
of $1.08\times 10^{20}$ kg for Enceladus (Jacobson et al., 2006) and
$4.4\times 10^{19}$ kg for Proteus (multiplying the volume from Stooke (1994)
by an assumed density of 1.3 g/cm3), the escape velocity
$v_{\mathrm{esc}}=\sqrt{2GM_{m}/R_{m}}$ from the surface of Enceladus is 240
m/s, while for Proteus it is 170 m/s. Pallene has a smaller size ratio to
Enceladus than Hippocamp has to Proteus, but perhaps Pallene is evidence of
the proposed impactor in the south polar terrain of Enceladus (Roberts &
Stickle, 2017).

Not too long in the past, however, is the Mimas-Enceladus 3:2 resonance
crossing (115 Myr ago with $Q=2000$). Meyer & Wisdom (2008) studied a triplet
of Mimas-Enceladus 3:2 resonances and found that Mimas’s eccentricity can be
explained either by passage through the 3:2 $e$-Mimas resonance or the 6:4
$ee^{\prime}$-mixed resonance (but not the 3:2 $e$-Enceladus resonance), and
found dynamical escape to be possible for both of these resonances. Ćuk et al.
(2016) proposed that Tethys, Dione, and Rhea all formed in one event about 100
Myr ago, and suggests that Mimas and Enceladus could have formed during the
same epoch or could be even younger. Neveu & Rhoden (2019), however, have
suggested that Mimas could be significantly younger than Enceladus. This last
scenario allows for the possibility of Pallene migrating away from Enceladus
after an impact before the formation of Mimas.

Thus, given a constant $Q$ tidal model, it looks like Pallene has crossed some
resonances, which, especially if it had been trapped in any of them for some
period of time, could have affected its eccentricity and inclination. However,
the new tidal models indicate the evolution of the satellites could be more
complex than previously thought (Fuller et al., 2016; Lainey et al., 2020).
Still, small moons such as Pallene are likely sensitive probes of this tidal
evolution (see, for example, El Moutamid et al., 2017) and so should be
considered in those contexts.

###  4.2 Resonance Analysis

In view of the rich dynamical structure of the phase-space close to Pallene,
where many resonances are in close proximity to each other, we seek to
determine whether any particular resonance between Pallene and one or more of
the major Saturnian moons drives the evolution of Pallene, or could be a
possible mechanism to confine the particles of the dusty ring. Hence, we ran
five sets of numerical N-body simulations with different integration times,
i.e. 50 (or approximately $15\,766$ Pallene’s orbits), $5\times 10^{3}$,
$5\times 10^{4}$, $5\times 10^{5}$, and $5\times 10^{6}$ yr. The output
interval in each integration is always a multiple of Pallene’s orbital period,
$P\approx 1.2$ d, so that in each output file there are a total of $15\,220$
data points. For each integration, several libration angles from the direct
and indirect arguments of the disturbing function were explored, up to fourth-
order (Murray & Dermott, 1999). Due to the uncertainties in Pallene’s density
and therefore its mass, three different densities were considered as described
in Section 2, which means that in total 15 realisations were performed, three
per each integration time; we designate these as density-sets per integration
time.

We referred to the resonant arguments of the disturbing function for two
reasons: (1) the number of possible arguments is constrained and (2) in the
case that one of these arguments librates, then the corresponding argument
would facilitate its use in future secular theory calculations of this system.
The libration angle among an outer (with primed orbital elements) and an inner
satellite (un-primed elements), is expressed as

$\phi=j\lambda^{\prime}+(k-j)\lambda+\gamma(\varpi^{\prime},\varpi,\Omega^{\prime},\Omega),$
(5)

where $k$ is the order, $j$ the degree, and $\gamma$ is a linear combination
of $\varpi^{\prime},\,\varpi,\,\Omega^{\prime}$, and $\Omega$. The examined
libration angles range in order $k$ from 1 to 4, while the degree $j$
corresponds to possible resonances within 0.98 and 1.02 $D_{\mathrm{Pal}}$.
The linear combination $\gamma(\varpi^{\prime},\varpi,\Omega^{\prime},\Omega)$
in Eq. 5 is determined from the direct and indirect arguments of the
disturbing function described in Murray & Dermott (1999), which have the form
$\gamma=k_{1}\varpi^{\prime}+k_{2}\varpi+k_{3}\Omega^{\prime}+k_{4}\Omega$,
where $k_{1}+k_{2}+k_{3}+k_{4}=k$.

In the rest of this section, we denote the libration angles of a given moon
with Pallene by their capitalised initials, e.g., $\phi_{\mathrm{PM}}$ for the
Pallene-Mimas libration angle, except for Tethys which will be denoted by a
“t” to distinguish it from Titan. For the semi-major axis interval considered
above, the possible resonant combinations are summarised in Table 3. The
majority of explored direct arguments involve either Pallene and Mimas, or
Pallene and Enceladus. In contrast, the combination between Pallene and Titan
lacks possible resonant combinations in this semi-major axis interval. For
completeness, additional zeroth-order resonances were also evaluated for
degrees $j=$ 0 to 15.

Table 3: Order $k$ and degree $j$ explored for libration angles with Pallene Moon | $k$ | $j$
---|---|---
Mimas | 1 | 5, 6
2 | 10 – 12
3 | 15 – 19
4 | 20 – 25
Enceladus | 1 | 6, 7
2 | 11 – 15
3 | 17 – 22
4 | 22 – 30
Tethys | 2 | 5
3 | 8
4 | 10
Dione | 4 | 7
Rhea | 3 | 4

We inspected 75 indirect arguments per moon pair, i.e., 450 in total,
denominated as $\psi$ to distinguish them from the direct arguments, $\phi$.
Most of the indirect arguments explore all the angular range in every
timescale. Only two fourth-order indirect arguments show interesting
behaviour: the Dione-Pallene argument
$\psi_{\mathrm{DP}}=\lambda^{\prime}+3\lambda-2\varpi^{\prime}-2\varpi$
displays temporal libration (Fig. 2a) for about 30 kyr; while the Titan-
Pallene argument
$\psi_{\mathrm{TP}}=3\lambda^{\prime}+\lambda-2\varpi^{\prime}-2\Omega$ (Fig.
2b) presents a long circulation period of 494 yr.

(a) $\psi_{\mathrm{DP}}=\lambda^{\prime}+3\lambda-2\varpi^{\prime}-2\varpi$

(b) $\psi_{\mathrm{TP}}=3\lambda^{\prime}+\lambda-2\varpi^{\prime}-2\Omega$

Figure 2:  Unique indirect arguments in our search with either librating
properties or long period circulation. The remaining 448 arguments displayed
short period circulation.

In contrast, the direct arguments displayed a broader variety of phenomena
depending on the timescale of the integration: circulation, alternating
intervals of circulation, libration, or overall circulation with ‘steps’ of
near constant value. In Sections 4.3 to 4.4, we only present angles that show
resonant-like features and that coincide within a given density-set; we
display the evolution of $\rho_{\mathrm{Pal}}=0.25$ g/cm3 integrations only.
Nevertheless, when the resonant-like libration angles are compared within
density-sets, we find that for integrations longer than $5\times 10^{3}$ yr
the angles evolve similarly within $5\times 10^{4}$ yr but differ after this
threshold. Consequently, the effect of Pallene’s mass in its dynamical
evolution is small and only noticeable after $10^{4}$ yr or $\sim 10^{6}$
Pallene orbits.

We divide our analysis into short (50 yr) and long-term ($t\geq 5\times
10^{3}$ yr), demonstrating that Pallene has different resonant behaviour with
one or more Saturnian satellites depending on the timescale, some emerging
just in either short- or long-term simulations.

###  4.3 Short-term evolution of direct arguments

The intention of the 50 yr simulations was to re-examine the suggested third-
order resonance between Pallene and Enceladus (Spitale et al., 2006). We
probed for libration all ten possible direct arguments with
$19\lambda^{\prime}-16\lambda$, finding one additional combination with
interesting behaviour in this interval. Figure 3 shows a comparison between
the resonant angle suggested by Spitale et al. (2006) (Fig. 3a) and our
finding (Fig. 3b). The angle
$\phi_{\mathrm{EP}}=19\lambda^{\prime}-16\lambda-\varpi-2\Omega$, circulates
with a period of $10.6$ yr. Similarly, Muñoz-Gutiérrez & Giuliatti Winter
(2017) found this angle to circulate but with a period 1.8 times shorter. The
angle
$\phi_{\mathrm{EP}}=19\lambda^{\prime}-16\lambda-\varpi^{\prime}-2\Omega^{\prime}$
differs from that suggested in Spitale et al., in that the longitudes of
ascending node and pericentre belong to the outer satellite instead of the
inner one. The evolution of this argument exhibits a softer negative slope
that circulates with a period of $\sim 30$ yr.

(a) $\phi_{\mathrm{EP}}=19\lambda^{\prime}-16\lambda-\varpi-2\Omega$

(b)
$\phi_{\mathrm{EP}}=19\lambda^{\prime}-16\lambda-\varpi^{\prime}-2\Omega^{\prime}$

Figure 3:  Two different 19:16 libration angles between Enceladus and Pallene
over 50 yr. The top panel corresponds to the resonant angle suggested by
Spitale et al. (2006) and the bottom panel corresponds to our finding.
Although both libration angles circulate, the argument in Fig. 3b has a
circulation period 3 times longer than the one in Fig. 3a.

In contrast to other small moons in the region, clearly trapped in first-order
MMRs with Mimas, such as Aegaeon in the 7:6 (Hedman et al., 2010; Madeira et
al., 2018), Methone in the 14:15 (Spitale et al., 2006; Hedman et al., 2009;
Callegari et al., 2021), and Anthe in the 10:11 (Cooper et al., 2008;
Callegari & Yokoyama, 2020), our short-term (and long-term) simulations
indicate that Pallene’s evolution is not characterised uniquely by any MMR,
either with Mimas or Enceladus. Although some of the 19:16 libration angles
between Pallene and Enceladus present features associated with a near-
resonance, they all clearly circulate in longer timescales. It is likely that
Pallene is just outside the parameter space that characterises the 19:16 MMR
with Enceladus. Similarly, several of the libration angles shown outside of
the Mimas-Anthe 10:11 MMR (Fig. 8 Callegari & Yokoyama, 2020) resemble the
evolution of some of the direct arguments we studied in this work. The latter
suggest that an analysis of the “individual dynamic power spectra” (IPS in
Callegari & Yokoyama, 2020) of the 19:16 MMR between Pallene and Enceladus
could disclose the nature of the current resonant state of Pallene (Fig. 3),
however, we consider such analysis beyond the scope of the current work.

####  4.3.1 Simultaneous zeroth-order direct argument among all moons

While examining the zeroth-order direct arguments of the 50 yr simulations, a
simultaneous resonant libration angle was detected between Pallene and four
other moons: Mimas, Tethys, Dione, and Titan. Here ‘simultaneous’ means that
more than one pair of satellites (Pallene and another large Saturnian moon)
displays apparent resonant properties for the same libration angle expression.
In this case, this simultaneity emerged for
$\Phi\equiv\varpi^{\prime}-\varpi+\Omega^{\prime}-\Omega$ as presented in Fig.
4. In this time interval, $\Phi$ appears to be constant with small
oscillations, except for the pairs Enceladus-Pallene and Rhea-Pallene, which
circulate with a period of 12 and 36 yr, respectively. Nonetheless, Enceladus
displays a semi-resonant behaviour due to the step-like oscillation of
$\Phi_{\mathrm{EP}}$. Each “step” has a semi-constant value that changes in
each full circulation. For example, in the first step (from 1 to 4 yr) the
nearly-constant value is $90^{\circ}$ while on the fourth step (from 14 to 18
yr) the corresponding value is $60^{\circ}$, therefore, there are $\sim 4$ yr
intervals where this angle librates followed by a shift of $\sim 130^{\circ}$
during $\sim 1.5$ yr to another semi-constant step.

Figure 4:  50 yr evolution of the libration angle
$\Phi=\varpi^{\prime}-\varpi+\Omega^{\prime}-\Omega$ between each moon in
Table 2 and Pallene. The known libration angle between Pallene and Mimas
(Callegari & Yokoyama, 2010, top panel), also presents resonant behaviour
between Pallene and three other moons: Tethys, Dione, and Titan (3rd, 4th, and
6th panels from top to bottom). In contrast, $\Phi_{\mathrm{EP}}$ exhibits
circulation with semi-constant ‘steps’, whereas $\Phi_{\mathrm{RP}}$ (5th
panel) circulates.

Callegari & Yokoyama (2010) suggested this quasi-resonant relationship between
Mimas and Pallene ($\Phi_{\mathrm{PM}}$) and demonstrated that it has a long
circulation period ($\sim 5000$ yr, later confirmed by Muñoz-Gutiérrez &
Giuliatti Winter, 2017)). In order to explore possible circulation of $\Phi$
for Tethys, Dione, and Titan, we looked for circulation of this direct
argument in our $5\times 10^{3}$ yr integrations and, if circulation existed,
determined the corresponding period using Fourier frequency analysis. Table 4
lists the circulation periods of $\Phi$ for each moon pair, including our
estimate for $\Phi_{\mathrm{PM}}=4708$ yr. The measured circulation periods
for Tethys-Pallene (tP), Dione-Pallene (DP) and Titan-Pallene (TP), are 872
yr, 844 yr, and 794 yr, respectively. Even though these angles are not
resonant, their long circulation relative to Pallene’s orbital period might
significantly affect the dynamics of Pallene in the short-term.

Table 4: Period of libration angle $\phi=\varpi^{\prime}-\varpi+\Omega^{\prime}-\Omega$ for the moon pairs in Fig. 4.  Moon pair | $P_{\mathrm{circ}}$ | No. orbits
---|---|---
| [yr] | [$10^{3}$]
PM | 4708 | 1495.5
EP | 12 | 3.9
tP | 872 | 277.0
DP | 844 | 268.2
RP | 36 | 11.3
TP | 794 | 252.4

The possible existence of a quasi-resonance with the same combination of
angles that excludes the mean longitudes suggests an alignment of the lines of
nodes and apses of Pallene, Mimas, Tethys, and Dione, most likely with Titan.
In other terms, a combination of the eccentricity and inclination vectors of
these satellites may be aligned to some extent to Titan’s. This is not
entirely unexpected, since secular resonances could lead to apsidal
alignments; in the Saturnian system an example of this has long been known to
occur between Rhea and Titan (see Greenberg, 1975, and references therein).
The well known example of Tethys-Mimas 2:4 MMR
($\phi_{\mathrm{tM}}=4\lambda^{\prime}-2\lambda-\Omega^{\prime}-\Omega$), for
which the variation in inclination drives the resonance (Greenberg, 1973;
Allan, 1969), is another important example of node alignment. Moreover,
alignment of the nodes has been discussed in several works involving the
dynamics of compact extrasolar systems (e.g., Kaib et al., 2011; Boué &
Fabrycky, 2014; Granados Contreras & Boley, 2018); the later works refer to
this alignment as the interaction of an outer massive planet/companion with an
inner compact system (of planets) which affects the inner system as if it were
a rigid body. In the case of the Saturnian moon system, a study of the
compactness of the orbits interior to Titan could reveal whether this
phenomenon also occurs in this system. Nonetheless, a detailed study of this
scenario is currently beyond the scope of this paper focused on Pallene
dynamics; thus we consider this idea for future work.

###  4.4 Long-term evolution of direct terms

We performed four long-term simulations, lasting $5\times 10^{3}$, $5\times
10^{4}$, $5\times 10^{5}$, and $5\times 10^{6}$ yr. In these simulations, most
of the explored arguments circulate. Although a handful of angles display
resonant characteristics during definite time intervals, there is not a single
case in which the libration angle has a constant value for the total length of
the simulations.

In Sections 4.4.1 to 4.4.5, we present libration angles of interest separated
by order, at least one per order, from first to fourth-order finishing with
zeroth-order. The second-order arguments in Section 4.4.2 failed to produce
similar behaviour among the density-set in all timescales. However, we include
the results of two Tethys-Pallene arguments (each with distinct densities for
Pallene) displaying temporal libration to exemplify the long-term effect of
Pallene’s mass in determining its resonant state.

####  4.4.1 First-order arguments

Only one first-order argument presenting unusual features was recovered from
our simulations (Fig. 5). Although it circulates at all times in the $5\times
10^{5}$ yr integration, this 7:6 argument between Enceladus and Pallene shows
a change in circulation frequency that slows down and holds for more than
$2\times 10^{5}$ yr, a considerable interval in terms of Pallene’s orbital
period. The exact resonance is located at 1.012$D_{\mathrm{Pal}}$ and is one
of the strongest resonances in the region considered in this work (see map of
Fig. 1). However, due to its semi-major axis being far from the 7:6 MMR
location, it is unlikely that Pallene would be trapped or suffer strong
perturbations from Enceladus through this resonance.

Figure 5:  First-order argument
$\phi_{\mathrm{EP}}=7\lambda^{\prime}-6\lambda+\varpi-2\Omega$ between
Enceladus and Pallene over $5\times 10^{5}$ yr. A change in the circulation
frequency is observed between 100 to 300 kyr.

####  4.4.2 Second-order arguments

Fig. 6 presents the evolution of two second-order libration angles with the
same degree, 5:3, over $5\times 10^{5}$ yr. The argument involving the
longitudes of ascending nodes of Tethys and Pallene (6a), corresponding to the
simulation with $\rho_{\mathrm{Pal}}=0.19$ g/cm3, exhibits two librating
intervals, one between 350 and 400 kyr and another extending from 450 to 500
kyr, with a slow circulation period enclosed by both intervals. On the other
hand, the second argument (6b) is an outcome of the $\rho_{\mathrm{Pal}}=0.34$
g/cm3 simulation and involves both the nodal and apsidal longitudes. This
argument briefly librates at different intervals of the simulation, the most
notable of which covers the 400 to 450 kyr interval.

(a) $\phi_{\mathrm{tP}}=5\lambda^{\prime}-3\lambda+\Omega^{\prime}-3\Omega$

(b)
$\phi_{\mathrm{tP}}=5\lambda^{\prime}-3\lambda+\varpi^{\prime}-\varpi+\Omega^{\prime}-\Omega$

Figure 6:  Second-order 5:3 MMR between Tethys and Pallene. Both arguments
present temporal libration at different times that last for thousands of
years.

####  4.4.3 Third-order arguments

(a) $\phi_{\mathrm{EP}}=22\lambda^{\prime}-19\lambda-\varpi^{\prime}-2\varpi$

(b) $\phi_{\mathrm{tP}}=8\lambda^{\prime}-5\lambda-\varpi^{\prime}-2\varpi$

(c) $\phi_{\mathrm{EP}}=19\lambda^{\prime}-16\lambda-3\varpi$

(d) $\phi_{\mathrm{tP}}=8\lambda^{\prime}-5\lambda-\varpi^{\prime}-2\varpi$

Figure 7:  Third-order direct arguments between Enceladus and Pallene and
between Tethys and Pallene on different timescales. From all the arguments
with librating properties, the argument
$\phi_{\mathrm{tP}}=8\lambda^{\prime}-5\lambda-\varpi^{\prime}-2\varpi$ (Figs.
7b and 7d) librates for about 10 kyr, the longest time amongst our findings in
Section 4.

In total, three different third-order arguments were found (Fig. 7),
associated with resonances with Enceladus and with Tethys. On the $5\times
10^{3}$ yr timescale, the direct argument
$\phi_{\mathrm{EP}}=22\lambda^{\prime}-19\lambda-\varpi^{\prime}-2\varpi$
between Enceladus and Pallene (Fig. 7a) circulates for most of the integration
yet displays intervals of libration which last about 400 yr, and, similar to
the argument in Fig. 5, shifts the constant value at which it librates, e.g.,
in the first 400 yr it librates close to 0∘ and then shifts to librate close
to 90∘ from the 1200 to 1600 yr interval. The argument associated with the 8:5
MMR between Tethys and Pallene,
$\phi_{\mathrm{tP}}=8\lambda^{\prime}-5\lambda-\varpi^{\prime}-2\varpi$,
exhibits a clear ample libration for the duration of the integration. Figure
7b is the clearest example of libration found in our exhaustive exploration of
resonant arguments between Pallene and a major Saturnian moon.

On the $5\times 10^{4}$ yr realisations (Figs. 7c and 7d), we recover a
Tethys-Pallene 8:5 argument and find an additional 19:16 direct argument
between Enceladus and Pallene. The latter argument (Fig. 7c),
$\phi_{\mathrm{EP}}=19\lambda^{\prime}-16\lambda-3\varpi$, presents a distinct
libration interval between $3.8$ and $4\times 10^{4}$ yr around 90∘. Finally,
the Tethys-Pallene 8:5 argument is displayed in Fig. 7d. We observe that the
soft slope visible in the shorter timescale (Fig. 7b) is maintained in this
scale, which then steepens after $\sim 1.2\times 10^{4}$ yr until the argument
initiates an erratic behaviour, followed by alternating circulation and
libration intervals. Similar behaviour occurs in the $5\times 10^{5}$
realisation, but not in our longest integrations ($5\times 10^{6}$ yr) where
the 8:5 argument no longer exhibits signs of libration, just circulation.

####  4.4.4 Fourth-order arguments

We identified three direct arguments of fourth-order with temporal libration
with Dione and with Enceladus. Figure 8a illustrates the argument
$\phi_{\mathrm{DP}}=7\lambda^{\prime}-3\lambda-4\Omega^{\prime}$ between Dione
and Pallene; this inclination-type resonance involves only the longitude of
the ascending node of Dione; it was recovered in the timescale of $5\times
10^{3}$ yr only. Despite the general circulation of this argument, some
libration intervals with large amplitude about 180∘ are observed.

(a) $\phi_{\mathrm{DP}}=7\lambda^{\prime}-3\lambda-4\Omega^{\prime}$

(b) $\phi_{\mathrm{EP}}=22\lambda^{\prime}-18\lambda-\Omega^{\prime}-3\Omega$

(c)
$\phi_{\mathrm{EP}}=25\lambda^{\prime}-21\lambda-\varpi^{\prime}-\varpi-2\Omega^{\prime}$

Figure 8:  Evolution of fourth-order direct arguments (of Pallene) with Dione
and with Enceladus on different timescales.

The remaining arguments are between Enceladus and Pallene (Figs. 8b and 8c)
over $5\times 10^{4}$ yr which have the particularity that they coincide in
the location of their temporal libration and have similar width in Fig. 1. The
argument
$\phi_{\mathrm{EP}}=22\lambda^{\prime}-18\lambda-\Omega^{\prime}-3\Omega$
corresponds to the Enceladus-Pallene 11:9 MMR location in Fig. 1, while the
argument in Fig. 8c, i.e.,
$\phi_{\mathrm{EP}}=25\lambda^{\prime}-21\lambda-\varpi^{\prime}-\varpi-2\Omega^{\prime}$,
is situated at 0.9983 $D_{\mathrm{Pal}}$ almost overlapping the Mimas-Pallene
9:11 MMR.

####  4.4.5 Zeroth-Order arguments

We recovered several zeroth-order arguments with various degrees of Pallene
with Dione, Rhea, and Titan from the $5\times 10^{4}$ yr and $5\times 10^{6}$
yr integrations. The clearest libration occurs in the argument
$\phi_{\mathrm{TP}}=5\lambda^{\prime}-5\lambda+\varpi^{\prime}+\varpi-\Omega^{\prime}-\Omega$
between Titan and Pallene (Fig. 9c) which coincides with the libration
intervals of
$\phi_{\mathrm{RP}}=13\lambda^{\prime}-13\lambda+\varpi^{\prime}-\varpi-\Omega^{\prime}+\Omega$
(Fig. 9b) and the reversal of circulation of
$\phi_{\mathrm{DP}}=3\lambda^{\prime}-3\lambda+\varpi^{\prime}+\varpi-2\Omega^{\prime}$
(Fig. 9a). A different argument involving Titan and Pallene also with degree 5
is shown in Fig. 9d. It displays a slow circulation with intervals of faster
circulation coincident with the libration period of the argument in Fig. 9c.

(a)
$\phi_{\mathrm{DP}}=3\lambda^{\prime}-3\lambda+\varpi^{\prime}+\varpi-2\Omega^{\prime}$

(b)
$\phi_{\mathrm{RP}}=13\lambda^{\prime}-13\lambda+\varpi^{\prime}-\varpi-\Omega^{\prime}+\Omega$

(c)
$\phi_{\mathrm{TP}}=5\lambda^{\prime}-5\lambda+\varpi^{\prime}+\varpi-\Omega^{\prime}-\Omega$

(d) $\phi_{\mathrm{TP}}=5\lambda^{\prime}-5\lambda+2\varpi-2\Omega^{\prime}$

(e) $\phi_{\mathrm{DP}}=\varpi^{\prime}-\varpi$

(f) $\phi_{\mathrm{RP}}=2\varpi-\Omega^{\prime}-\Omega$

Figure 9:  Several zeroth-order arguments over timescales of $10^{4}$ and
$10^{6}$ yr. In these timescales, none of the zeroth-order arguments repeat as
in Section 4.3.1. Furthermore, two Titan-Pallene direct arguments of degree 5
were recovered (Figs. 9c and 9d); the latter displays a clearer temporal
libration about 0∘ at $3.75\times 10^{4}$ yr.

Finally, the bottom two panels in Fig. 9 only involve the apsidal and nodal
longitudes. The longer and most evident circulation found in our simulations
occurs between the longitudes of pericentres of Dione and Pallene (Fig. 9e),
while a reversal in the circulation of the argument
$\phi_{\mathrm{RP}}=2\varpi-\Omega^{\prime}-\Omega$ between Rhea and Pallene
(Fig. 9f) takes place around 2 and 3 Myr, producing a temporary libration in
this interval.

###  4.5 What all these arguments mean

In our exhaustive search for resonant behaviour, we did not find any clear
libration for either first- or second-order resonances among any of the
Pallene pairings with the six large moons considered in this work. This means
that the proposed 19:16 MMR between Enceladus and Pallene does not exist.

The quasi-resonant zeroth-order argument suggested by Callegari & Yokoyama
(2010) between Pallene and Mimas is also present with other moons. Taking into
account the values of $e$ and $I$, we consider that the most important
contribution of this combination to the disturbing function would be the one
arising from the Mimas-Pallene pair, followed by the Titan-Pallene pair. The
small discrepancy in the circulation period found in this paper with respect
to the value found in Muñoz-Gutiérrez & Giuliatti Winter (2017) may be due to
the updated values of both $GM_{m}$ and Saturn’s zonal harmonics.

The clearest librations, observed for arguments of third- and fourth-order
resonances, would, however, have only a slight contribution to the disturbing
function, given the small eccentricities and inclinations of both Pallene and
the other moons. For the same reason, we do not expect that any resonance of
major order, or with the same order but of a larger degree, would result in
any significant contribution to the evolution of Pallene.

Based on our analysis, we can conclude that Pallene is not currently trapped
in any two-body MMR of any order or degree. This does not exclude the
possibility of the existence of a more complex, three-body resonance,
involving Pallene and some of the other moons, not exclusive to Mimas and
Enceladus. Although a preliminary analysis of this possibility does not show
any clear signs for the existence of such a configuration, an in-depth
analysis of three-body resonances is left for future work.

We find no significant variations in the overall results of simulations
shorter than $\sim 2\times 10^{4}$ yr as a function of density (this includes
all the simulations referring to the evolution of Pallene’s ring). For longer
simulations, the accumulation of numerical errors, resulting from differences
in the $GM_{m}$ values of order $10^{-7}$ \- $10^{-8}$, and the weak chaotic
nature of the N-body problem, lead to a loss of coherence among different
simulations; nonetheless, statistically, all the longer-term simulations are
equivalent to each other up to our longest integration time of $5\times
10^{6}$ yr. Despite the shift in angular phases, the main orbital elements,
$(a,e,I)$, remain confined and evolve regularly up to 5 Myr.

##  5 Origin and Dynamical Evolution of the Pallene Ring

Pallene shares its orbit with a complete dusty ringlet (Hedman et al., 2009,
2010) seen by _Cassini_ images in high phase angle, while a concentration of
large particles ($\gtrsim 100~{}\mu m$) was detected in other phase angle
images (Hedman et al., 2009). These data indicate that the ring is composed of
micrometre-sized particles and denser bodies. Hedman et al. (2009) found that
the ring has a radial full-width of $\sim$2500 km and a vertical profile with
a full-width at half-maximum (FWHM) of $\sim$50 km, that is, the ring is
vertically thin. More recently, Spahn et al. (2019) measured the FWHM of the
Gaussian vertical profile as $\sim$270 km while obtaining the same radial
full-width as Hedman et al. (2009). Spahn et al. (2019) also found that the
radial mean position of the ring is shifted radially outwards by $\sim$1100
km.

###  5.1 Pallene’s Mass Production by Impacts

In theory, satellites of a few kilometres in radius are efficient sources of
debris for rings and arcs due to their reasonably large cross-section and low
escape velocity (Poppe, 2016). However, Madeira et al. (2018, hereafter M18)
and Madeira & Giuliatti Winter (2020, hereafter M20) found that Saturn’s three
smallest moons (Aegaeon, Anthe, and Methone) do not replenish the material
lost by their associated arcs due to non-gravitational forces. It raises the
question of whether Pallene can maintain its diffuse ring in a steady state,
as proposed by Hedman et al. (2009). In this section, we compute the amount of
debris ejected from Pallene and analyse the fate of the ejecta in Section 5.4.

The production of material by Pallene is the result of energetic collisions
between the surface of the satellite and fluxes and interplanetary dust
projectiles (IDPs) (Grun et al., 1985; Divine, 1993). Typically, IDPs are
supplied by families of comets (Jupiter-family, Halley-type, and Oort-Cloud
comets, Dikarev et al., 2005; Nesvorný et al., 2010; Poppe et al., 2011) and
by the Edgeworth-Kuiper Belt (EKB, Landgraf et al., 2002). Data obtained by
the Student Dust Counter (SDC) on board the New Horizons spacecraft indicate
that the Saturn neighbourhood is dominated by EKB dust (Piquette et al., 2019;
Poppe et al., 2019) corresponding to the population that reaches the orbits of
Saturn’s satellites.

In addition to the impacts with IDPs, Pallene may produce material due to
impacts with the E ring particles (ERPs). The icy-dust emission from
Enceladus’s volcanism is the principal source of the E ring (Spahn et al.,
2006; Kempf et al., 2010), producing a dense debris that impacts the surface
of satellites immersed in the E ring (Spahn et al., 2006). The mass production
rate by Pallene (or any other satellite) is given by (Krivov et al., 2003):

$M^{+}=\pi R_{m}^{2}(F_{\rm IDP}Y_{\rm IDP}+F_{\rm ERP}Y_{\rm ERP})$ (6)

where $R_{m}$ is the satellite radius, $F_{\rm IDP}$ and $F_{\rm ERP}$ are the
mass flux of impactors due to IDPs and ERPs, respectively, and $Y_{\rm IDP}$
and $Y_{\rm ERP}$ are the ejecta yields associated to each projectile-type.

The ejecta yield is the ratio between the mass produced during the impact and
the impactor’s mass. This quantity is calculated using the empirical
prescription obtained by Koschny & Grün (2001) for pure-ice satellites:

$Y=\frac{6.69\times 10^{-8}}{2^{1.23}~{}{\rm
kg/m^{3}}}\left(\frac{1}{927~{}{\rm kg/m^{3}}}\right)^{-1}\left(\frac{m_{\rm
imp}}{\rm kg}\right)^{0.23}~{}\left(\frac{v_{\rm imp}}{\rm m/s}\right)^{2.46}$
(7)

where $m_{\rm imp}$ and $v_{\rm imp}$ are the mass and velocity of the
impactor.

Pallene, Aegaeon, Anthe, and Methone are likely porous satellites (Hedman et
al., 2020), due to their bulk densities, $\rho_{m}$, being lower than the
density of ice ($\rho_{\rm ice}$=927 kg/m3). Since an impact on a porous body
is expected to generate more material than an impact on a non-porous surface,
we artificially modified Equation 7 by introducing a porosity ratio of
${\rm\alpha_{p}=\rho_{m}/\rho_{ice}}$:

$Y_{p}=\frac{(6.69\times 10^{-8})^{\alpha_{p}}}{2^{1.23}~{}{\rm
kg/m^{3}}}\left(\frac{\alpha_{p}}{927~{}{\rm
kg/m^{3}}}\right)^{-1}\left(\frac{m_{\rm imp}}{\rm
kg}\right)^{0.23}~{}\left(\frac{v_{\rm imp}}{\rm m/s}\right)^{2.46}.$ (8)

We must point out that Equation 8 is theoretical, and there is no experimental
evidence that it actually rules the yield for a porous body. In this work, we
will use Equation 8 only as an artifice to demonstrate the uncertainties
related to the collision yield. The parameters assumed for the two projectile
populations are presented below.

####  5.1.1 Interplanetary Dust Projectiles

In Saturn’s vicinity, the (unfocused) IDP mass flux is estimated to be $F_{\rm
IDP}^{(\infty)}=10^{-16}$ kgm-2s-1 (Altobelli et al., 2018; Piquette, 2019).
We assume the IDPs’ velocity near Saturn as the median speed of EKB grains,
$v_{\rm imp}^{(\infty)}=3.1$ km/s (Poppe, 2016) and the mass of the impactors
as $m_{\rm imp}=10^{-8}$ kg. When IDPs enter Saturn’s Hill sphere, the
planet’s gravitational force is responsible for enhancing the flux and
velocity of the projectiles (Krivov et al., 2003). Respectively, the mass flux
and velocity of IDPs at an orbital radius $r$ are (Colombo et al., 1966;
Krivov et al., 2003):

$\frac{F_{\rm imp}}{F_{\rm imp}^{(\infty)}}=\frac{1}{2}\left(\frac{v_{\rm
imp}}{v_{\rm imp}^{(\infty)}}\right)^{2}+\frac{1}{2}\frac{v_{\rm imp}}{v_{\rm
imp}^{(\infty)}}\left[\left(\frac{v_{\rm imp}}{v_{\rm
imp}^{(\infty)}}\right)^{2}\right.\\\
\left.-\left(\frac{R_{\mathrm{S}}}{r}\right)^{2}\left(1+\frac{2GM_{S}}{R_{\mathrm{S}}(v_{\rm
imp}^{(\infty)})^{2}}\right)\right]^{1/2},$ (9)

and

$\frac{v_{\rm imp}}{v_{\rm
imp}^{(\infty)}}=\sqrt{1+\frac{2GM_{S}}{r\left(v_{\rm
imp}^{(\infty)}\right)^{2}}}.$ (10)

####  5.1.2 E Ring Impactors

We assume the E ring is composed of sub-micrometric ejecta from Enceladus onto
highly eccentric orbits (Nicholson et al., 1996; Kempf et al., 2008; Postberg
et al., 2008; Ye et al., 2014a). The average mass of impactors is assumed to
be $m_{\rm imp}=2.3\times 10^{-15}$ kg ($0.65~{}\mu$m, Spahn et al., 2006) and
the impact velocity is given by (Hamilton & Burns, 1994; Spahn et al., 2006):

$v_{\rm imp}=\frac{1}{2}\sqrt{\frac{GM_{S}}{r}}$ (11)

The flux of impactors on the equator plane is assumed to be $F_{\rm
ERP}=m_{\rm imp}v_{\rm imp}N_{\rm ERP}$, where $N_{\rm ERP}$ is the particle
number density in the E ring, extracted from the Cosmic Dust Analyser data
(Kempf et al., 2008):

$N_{\rm
ERP}(r)=N_{0}\exp\left(-\frac{z_{0}(r)^{2}}{2\sigma(r)^{2}}\right)\left\\{\begin{array}[]{ll}\left(\frac{r}{3.98~{}R_{\mathrm{S}}}\right)^{50}&\textrm{for}~{}r\leq
3.98~{}R_{\mathrm{S}}\\\
\left(\frac{r}{3.98~{}R_{\mathrm{S}}}\right)^{-20}&\textrm{for}~{}r>3.98~{}R_{\mathrm{S}},\end{array}\right.$
(12)

with

$\sigma(r)=1826~{}{\rm
km}+(r-3.98~{}R_{\mathrm{S}})\left\\{\begin{array}[]{ll}-\frac{467~{}{\rm
km}}{0.82~{}R_{\mathrm{S}}}&\textrm{for}~{}r\leq 3.98~{}R_{\mathrm{S}}\\\
\frac{510~{}{\rm
km}}{0.77~{}R_{\mathrm{S}}}&\textrm{for}~{}r>3.98~{}R_{\mathrm{S}},\end{array}\right.$
(13)

and,

$z_{0}(r)=\left\\{\begin{array}[]{ll}-1220\left(\frac{r-3.98~{}R_{\mathrm{S}}}{0.82~{}R_{\mathrm{S}}}\right)~{}{\rm
km}&\textrm{for}~{}r\leq 3.98~{}R_{\mathrm{S}}\\\
0&\textrm{for}~{}r>3.98~{}R_{\mathrm{S}},\end{array}\right.,$ (14)

where $N_{0}$ is the maximum particle number density – near Enceladus’ radius
– set as $N_{0}=1~{}$m-3 (Ye et al., 2014b).

####  5.1.3 Mass Production Rate of Aegaeon, Anthe, Methone, and Pallene

Following the prescription described in Sections 5.1.1 and 5.1.2 and using Eq.
7, we estimate the mass production rate of Pallene as

$M^{+}\sim 7.4\times 10^{-4}~{}{\rm kg/s}.$ (15)

In order to determine whether Pallene can maintain the ring, we need to
estimate the mass of the structure and compare it with the lifetime of the
ejected material, which is obtained by N-body numerical simulations in Section
5.4. If the time $\mathcal{T}$ for Pallene to produce the amount of mass
observed in the ring is shorter than the particles’ lifetime, then the
satellite is an efficient source for the ring and the structure will be in a
steady state. On the other hand, if $\mathcal{T}$ is longer than the lifetime
of the particles, the ring will disappear unless another source keeps it in a
steady-state.

The time for the satellite to produce the observed mass of the ring is (M20)

$\mathcal{T}=M_{\mathrm{Ring}}/M^{+},$ (16)

if $M_{\mathrm{Ring}}$ is the mass of a ring (or arc), as given by (Sfair &
Giuliatti Winter, 2012):

$M_{\mathrm{Ring}}=A\left(\frac{4}{3}\pi\rho_{\rm ice}\right)\int_{0.1~{}\mu
m}^{100~{}\mu m}C\pi s^{3-q}ds,$ (17)

where $s$ is the physical radius of the particles, $C$ is a constant, and $q$
is the slope of the size distribution of the particles. The surface area is
$A=r\Delta\theta\Delta r/2$ (M20), where $\Delta\theta$ is the angular width
of the ring/arc in radians and $\Delta r$ is the radial width. The constant
$C$ can be obtained from the observed optical depth $\tau$ (Sfair & Giuliatti
Winter, 2012)

$\tau=\int_{0.1~{}\mu m}^{100~{}\mu m}C\pi s^{2-q}ds.$ (18)

The distribution of particles in Pallene’s ringlet is not constrained by
observational data. However, the data regarding the size distribution of the E
ring provides us with a range of possible slopes $q$ for the ringlet, with
values ranging from 1.9 to 5 (Horányi et al., 2008; Kempf et al., 2008; Ye et
al., 2014a; Srama et al., 2020). For instance, Horányi et al. (2008) estimated
from numerical simulations that the grain density in the E ring follows a
power law distribution with $q=2.5$, while Kempf et al. (2008) obtained slopes
between 4 and 5 for $s>0.9~{}\mu$m from Cassini data. The slopes reported by
Ye et al. (2014a) vary between 3 and 4 for $s>10~{}\mu$m. To cover all
possible values of $q$, we assume slopes between 1 and 6.

Figure 10:  Estimated time $\mathcal{T}$ for Aegaeon, Methone, Anthe, and
Pallene to produce the mass of their associated arc/ring as a function of the
slope $q$ of the particle radius distribution. The solid and dashed lines
correspond to the time calculated following the prescription given in Section
5.1. The solid (dash-dotted) black line corresponds to Pallene’s system
assuming a non-porous (porous) satellite and the grey area gives the error in
the calculation of $\mathcal{T}$ due to the uncertainties in Pallene’s bulk
density. The coloured red, blue, and green lines correspond to the arcs of
Aegaeon, Methone, and Anthe, respectively. The arc lifetime is given by
different coloured dashed lines. The red star gives $\mathcal{T}$ obtained for
Aegaeon by M18 and the triangles the times obtained for Methone (blue) and
Anthe (green) by M20.

Figure 10 shows the time $\mathcal{T}$ for Pallene to produce the ringlet mass
(solid black line) for slopes between 1 and 6, assuming a non-porous satellite
(Eq. 7). The figure also shows the time for the moons Aegaeon, Methone, and
Anthe to produce the material of their associated arcs (solid coloured lines).
Meanwhile, the dash-dotted lines provide the estimated production time
$\mathcal{T}$ assuming that the satellites are porous. For Aegaeon, Anthe, and
Methone, we assume a bulk density of 500 kg/m3, while for Pallene this value
is 250 kg/m3. The filled region surrounding the dashed black line gives the
$\mathcal{T}$ calculated using the minimum and maximum bulk densities
estimated for Pallene ($\rho_{\mathrm{Pal}}$=190-340 kg/m3). The mass
production rate depends only on the cross-section of the satellite, so if we
assume a non-porous Pallene, the uncertainties regarding its bulk density do
not affect the mass production, since the physical radius of the satellite is
constrained by observational data (Hedman et al., 2009).

M18 and M20 estimated $\mathcal{T}$ following a simple prescription assuming
production due to IDP impacts of cometary origin (with lower focused fluxes
and velocities than the EKB grains), and assumed a single slope, $q=3.5$. The
prescription here presented goes a step further in relation to their model
because it incorporates recent data and the production due to ERP impacts. The
time $\mathcal{T}$ obtained in M18 for the arc of Aegaeon is shown by the red
star in Fig. 10 and the times obtained in M20 for the arcs of Methone and
Anthe are the triangles with matching colours. The dashed lines correspond to
the lifetime of ${\rm 10~{}\mu}$m-sized particles, obtained by M18 and M20.

Our times are shorter than those estimated in previous works. M18 obtained
that Aegaeon’s arc will most likely disappear if it is composed exclusively of
micrometre-sized grains. Here, we also obtained that a non-porous Aegaeon
cannot replenish the arc material when we disregard other sources in the
arcs,222We do not compute production due to ERPs because Aegaeon is immersed
in the G ring. since $\mathcal{T}$ is at least an order of magnitude higher
than the lifespan of the particles. However, if we mimic the effect of
porosity on the yield, the satellite can maintain the arc for $q\gtrsim 4$.
Unlike M20, Methone can replenish the arc material for $q>3.3$ regardless of
its porosity. Although the lifetime of the particles in Anthe’s arc is shorter
than our $\mathcal{T}$ for the non-porous case, the radial width of the arc is
unknown 333We assume the same radial width as Methone’s arc due to the
proximity of the systems and the similar evolution of the particles under the
effects of the 14:15 and 10:11 corotation resonance. and we cannot be sure if
the satellite can produce by itself the amount of material necessary to keep
the arc in a steady-state or not. Assuming a porous limit, the Anthe arc seems
to be in a steady-state for $q\gtrsim 4$.

Table 5: Radial width ($\Delta r$), angular width ($\Delta\theta$), and optical depth ($\tau$) assumed for the systems of Aegaeon, Methone, Anthe, and Pallene (Hedman et al., 2009, 2010, 2020; Sun et al., 2017; Spahn et al., 2019). The table shows the fractions of yield $Y$, flux $F$, and mass rate $M^{+}$ between the IDP and ERP, and the total mass rate production in kg/s. | Aegaeon | Methone | Anthe | Pallene
---|---|---|---|---
$\Delta r$ [km] | 250 | 1000 | 1000 | 2500
$\Delta\theta$ [∘] | 60 | 10 | 20 | 360
$\tau$ | $10^{-5}$ | $10^{-6}$ | $10^{-6}$ | $10^{-6}$
$Y_{\rm IDP}/Y_{\rm ERP}$ | – | ${\rm 447}$ | ${\rm 448}$ | ${\rm 449}$
$F_{\rm IDP}/F_{\rm ERP}$ | – | ${\rm 10}$ | ${\rm 4}$ | ${\rm 10^{-1}}$
$M^{+}_{\rm IDP}/M^{+}_{\rm ERP}$ | – | ${\rm 4\times 10^{3}}$ | ${\rm 2\times 10^{3}}$ | ${\rm 50}$
$M^{+}$[kg/s] | ${\rm 2.6\times 10^{-5}}$ | ${\rm 3.7\times 10^{-4}}$ | ${\rm 4.2\times 10^{-5}}$ | ${\rm 7.4\times 10^{-4}}$

Table 5 summarises the initial ring (arc) parameters and the estimated
fraction of yield, flux, and mass production between the IDP and ERP
populations. We also include the total mass production for Aegaeon, Methone,
Anthe, and Pallene for the non-porous case. Ejecta production due to IDP
impacts is the most efficient for all systems. For the arcs of Aegaeon,
Methone, and Anthe, production due to ERPs can be disregarded because the
$M^{+}$ due to IDP impacts is more than 1000 times higher than for ERPs. The
production due to ERPs corresponds to 2% of the total amount produced by
Pallene.

###  5.2 Dynamical Model

We study the evolution and fate of Pallene’s ringlet by analysing the temporal
evolution of two distinct sets of particles: i) particles initially co-orbital
to the satellite (Section 5.3) and ii) particles ejected from Pallene’s
surface (Section 5.4). The first set corresponds to a scenario in which the
ringlet, and perhaps Pallene, would have formed by a disruption of an ancient
satellite; while the second, mimics the evolution of the material produced by
impacts into the satellite (Section 5.1).

The numerical simulations were performed using Mercury6 (Chambers, 1999) with
the Bulirsch-Stoer algorithm. We used 5,000 particles with micrometric sizes
ranging from 0.1 $\mu$m to 100 $\mu$m, and integrated the system until either
all particles collide with Mimas, Pallene, or Enceladus or migrate outwards
beyond the orbit of Enceladus. We adopted the collision detection treatment
between particles and satellites as implemented in Mercury6 (for details, see
Chambers, 1999; Liu et al., 2016).

Micrometre-sized particles are affected by non-gravitational forces that
decrease their lifetimes. Thus it is necessary to include these effects in the
system. In our simulations, the particles are under the effect of a total
force,

$\vec{\rm F}=\vec{\rm F}_{\rm SR}+\vec{\rm F}_{\rm PD}+\vec{\rm F}_{\rm
EM}+\vec{\rm F}_{\rm G},$ (19)

where $\vec{\rm F}_{\rm SR}$ is the solar radiation force, $\vec{\rm F}_{\rm
PD}$ is the plasma drag force, $\vec{\rm F}_{\rm EM}$ is the electromagnetic
force, and $\vec{\rm F}_{\rm G}$ corresponds to the sum of the gravitational
forces of the system: Saturn (including its gravitational coefficients),
Mimas, Enceladus, Tethys, Dione, Rhea, Titan, and Pallene.

####  5.2.1 Non-Gravitational Forces

The solar radiation force ($\vec{\rm F}_{\rm SR}$) includes two components
(Burns et al., 1979; Mignard, 1984): the radiation pressure (RP) caused by
collisions of solar radiation on the dust grain,

$\vec{\rm F}_{\rm RP}=\frac{\Phi\pi
s^{2}}{c}Q_{pr}\frac{\vec{r}_{sp}}{r_{sp}},$ (20)

and the Poynting-Robertson drag (PR), caused by the re-emission of the solar
radiation absorbed by the particles,

$\vec{\rm F}_{\rm PR}=-\frac{\Phi\pi
s^{2}}{c}Q_{pr}\left\\{\frac{\vec{V}_{P}+\vec{V}}{c}+\left[\left(\frac{\vec{V}_{P}}{c}+\frac{\vec{V}}{c}\right)\cdot\frac{\vec{r}_{sp}}{r_{sp}}\right]\frac{\vec{r}_{sp}}{r_{sp}}\right\\},$
(21)

where $c$ is the speed of light, $\Phi$ is the solar flux (Burns et al.,
1979), and $\vec{V}$ is the velocity vector of the particle relative to the
planet. The solar radiation pressure efficiency $Q_{pr}$ (in Eqs. 20 and 21)
depends on the radius of the particle and is computed from Mie theory (Irvine,
1965; Mishchenko et al., 1999, 2002) assuming spherical ice grains. The
particle is in a circumplanetary orbit $\vec{r}$ ($r=|\vec{r}|$), and the
planet in a circular heliocentric orbit. The heliocentric position of Saturn
$\vec{r}_{sp}$ ($r_{sp}=|\vec{r}_{sp}|$) and the magnitude of the planet’s
velocity $\vec{V}_{P}$ are considered constants. We also assume that Saturn
shields particles from solar radiation when the planet eclipses the Sun from
the particle’s perspective, i.e., the solar radiation force is neglected when
the particle is in the planet’s shadow, which happens when
$\vec{r}\cdot\vec{r}_{sp}<0$ and
$(r^{2}-R_{\mathrm{S}}^{2})r_{sp}-|\vec{r}\cdot\vec{r}_{sp}|^{2}<0$ (Liu et
al., 2016).

The principal source of plasma for Saturn’s magnetosphere in the E ring region
is the ionisation of neutrals provided by the Enceladus plume. The E ring
region is dominated by water group ions, i.e., O+, OH+, H2O+, and H3O+, the O+
ion being the most abundant (Cassidy & Johnson, 2010; Tseng et al., 2010;
Tseng & Ip, 2011; Sittler & Johnson, 2015). Direct collision of the plasma
with the ring particles is responsible for a drag force ($\vec{\rm F}_{\rm
PD}$) (Morfill & Gruen, 1979; Morfill et al., 1993; Horányi et al., 2008),
given by

$\vec{\rm F}_{\mathrm{PD}}=\pi
s^{2}m_{i}N_{i}a^{2}(n-\Omega_{\mathrm{S}})^{2}\hat{u}_{t},$ (22)

where $n$ is the mean motion of the particle, $m_{i}$ and $N_{i}$ are the mass
and number density of the plasma ions, respectively, and $\hat{u}_{t}$ is the
unit vector in the tangential direction to the osculating orbit of the
particle.

Cassini measurements have shown seasonal variations in ion densities ranging
from $N_{i}\sim 40~{}{\rm cm}^{-3}$ to $N_{i}\sim 120~{}{\rm cm}^{-3}$ in
Pallene’s vicinity (Elrod et al., 2014; Persoon et al., 2015; Persoon et al.,
2020). For simplicity, we assume the plasma in the Pallene region is only
composed of O+ ions (molecular mass of 16 a.m.u.) with constant number density
$N_{i}=65.9~{}{\rm cm}^{-3}$ (Persoon et al., 2015). Moreover, we neglect the
indirect Coulomb interaction between charged ring particles and the plasma
material, since this effect is at least two orders of magnitude weaker than
the direct collisions (Northrop & Birmingham, 1982; Grun et al., 1984; Sun et
al., 2015).

The ring particles are also influenced by Saturn’s magnetosphere due to the
charging of the particles by the ambient plasma and electrons photoemission
(solar UV). Therefore, the electromagnetic force ($\vec{F}_{\rm EM}$)
(Northrop & Birmingham, 1982; Burns et al., 1985), is included in our
simulations as

$\vec{\rm
F}_{\mathrm{EM}}=\frac{4\pi\epsilon_{0}sV}{c}\left\\{\left[\vec{V}-\Omega_{\mathrm{S}}(\hat{u}_{n}\times\vec{r})\right]\times\vec{B}\right\\},$
(23)

where $\epsilon_{0}=8.8542\times 10^{-12}$ F/m is the vacuum permittivity
(Chapman & Bartels, 1940), $V$ is the electric potential, $\vec{B}$ is the
magnetic field vector, and $\hat{u}_{n}$ is the unit vector perpendicular to
the planet’s equatorial plane. We adopt an equilibrium potential of $V=-3$ V
for the Pallene region, as determined by Hsu et al. (2011) in their
investigation of the dynamics of the Saturnian stream particles.

We assumed the Saturnian magnetic field as a composition of an aligned dipole
and a quadrupole (Chapman & Bartels, 1940; Hamilton, 1993):

$\vec{B}=g_{1.0}R_{\mathrm{S}}^{3}\vec{\nabla}\left(\frac{\cos{\zeta}}{r^{2}}\right)+\frac{g_{2.0}}{2}R_{\mathrm{S}}^{4}\vec{\nabla}\left(\frac{3\cos^{2}{\zeta}-1}{r^{3}}\right)$
(24)

where $g_{1.0}=0.21$ G is the Saturnian dipole momentum and $g_{2.0}=0.02$ G,
the quadrupole momentum (Hamilton, 1993; Belenkaya et al., 2006); $\zeta$ is
the angle between $\hat{u}_{n}$ and $\vec{r}$.

####  5.2.2 Orbital Elements Of One Representative Particle

The non-gravitational forces are responsible for variations in the shape and
orientation of the orbits, affecting the temporal evolution of the particles.
The mean temporal variations of the osculating orbital elements of a particle
with mass $m$ are (Mignard, 1984; Hamilton, 1993; Madeira & Giuliatti Winter,
2020)

$\dot{a}=-\frac{2na^{2}\alpha_{\rm
r}}{c}\frac{5+\cos^{2}{I}}{6}+\frac{2|\vec{F}_{\mathrm{PD}}|}{mn}\sqrt{1-e^{2}},$
(25) $\dot{e}=\alpha_{\rm
r}\sqrt{1-e^{2}}(\cos{\Omega}\sin{\omega}+\sin{\Omega}\cos{\omega}\cos{I})\\\
-\frac{3}{2}\frac{e|\vec{F}_{\mathrm{PD}}|}{mna}\sqrt{1-e^{2}}-\frac{qg_{1.0}R_{\mathrm{S}}^{3}\Omega_{\mathrm{S}}}{4mcna^{3}}e\sqrt{1-e^{2}}\sin^{2}{I}\sin{2\omega},$
(26) $\dot{I}=\frac{\alpha_{\rm
r}e}{\sqrt{1-e^{2}}}\sin{\Omega}\cos{\omega}\sin{I}+\frac{3}{2}\frac{|\vec{F}_{\mathrm{PD}}|}{mna}\sqrt{1-e^{2}}\sin{I}\\\
+\frac{qg_{1.0}R_{\mathrm{S}}^{3}\Omega_{\mathrm{S}}}{8mcna^{3}}\frac{e^{2}}{\sqrt{1-e^{2}}}\sin{2I}\sin{2\omega},$
(27) $\dot{\Omega}=-\dot{\Omega}_{\rm obl}+\frac{\alpha_{\rm
r}e}{\sqrt{1-e^{2}}}\sin{\Omega}\sin{\omega}-(2-e)\frac{|\vec{F}_{\mathrm{PD}}|}{mna}\cos{I}\sqrt{1-e^{2}}\\\
+\frac{qg_{1.0}R_{\mathrm{S}}^{3}\Omega_{\mathrm{S}}}{mcna^{3}}\frac{1}{\sqrt{1-e^{2}}}\left[\cos{I}-\frac{1}{(1-e^{2})}\left(\frac{n}{\Omega_{\mathrm{S}}}\right)\right],$
(28)

and

$\dot{\varpi}=\dot{\varpi}_{\rm obl}+\frac{\alpha_{\rm
r}\sqrt{1-e^{2}}}{e}(\cos{\Omega}\cos{\omega}-\sin{\Omega}\sin{\omega}\cos{I})\\\
+(2-e)\frac{|\vec{F}_{\mathrm{PD}}|}{mna}\sqrt{1-e^{2}}+\frac{qg_{1.0}R_{\mathrm{S}}^{3}\Omega_{\mathrm{S}}}{mcna^{3}}\frac{2\cos{I}}{(1-e^{2})^{3/2}}\left(\frac{n}{\Omega_{\mathrm{S}}}\right),$
(29)

where

$\alpha_{\rm r}=\frac{3\Phi\pi s^{2}}{2mcna}Q_{pr}.$ (30)

$\dot{\Omega}_{\mathrm{obl}}$ and $\dot{\varpi}_{\mathrm{obl}}$ are the
temporal variation of longitude of ascending node and argument of pericentre,
respectively, due to the non-sphericity of Saturn (see Renner & Sicardy,
2006).

Figure 11: From top to bottom: Geometric semi-major axis, eccentricity,
inclination, longitude of ascending node, and argument of pericentre of a
${\rm 10~{}\mu}$m-sized particle co-orbital to Pallene with displacement in
the mean anomaly of 180∘ in relation to the satellite. The top row of each
panel shows the orbital elements when only gravitational effect is included.
The following rows display the evolution of the particle when different non-
gravitational forces are included (i.e., solar radiation force,
electromagnetic force, and plasma drag). Finally, the bottom row of each panel
shows the effect of all forces.

Figure 11 illustrates the variation of geometric orbital elements ($a$, $e$,
$I$, $\Omega$ and $\varpi$) of one representative ${\rm 10~{}\mu}$m particle
due to each non-gravitational force and the total force (Eq. 19). The particle
is initially co-orbital to Pallene with
$\lambda=\lambda_{\mathrm{Pal}}+180^{\circ}$, where $\lambda$ and
$\lambda_{\mathrm{Pal}}$ are the mean longitude of the particle and Pallene,
respectively. As one can see in the top panel of Fig. 11 (Eq. 25), the semi-
major axis is affected secularly by two distinct drag effects: the Poynting-
Robertson component that produces an inward migration, and the plasma drag,
which increases the semi-major axis of the particle. We find that the plasma
drag is at least one order of magnitude stronger than the Poynting-Robertson
component for all particle sizes. While the electromagnetic force only induces
short-term variations in the semi-major axis, the net outcome is that grains
migrate outward when all the effects are included.

In the eccentricities, we have that the electromagnetic and solar radiation
forces produce oscillations with constant period and amplitude for the same
particle size (Hamilton, 1993; Madeira et al., 2018; Gaslac Gallardo et al.,
2020). As we can see in Eq. 26, the intensity of these effects depends on the
radius of the particles, with $\dot{e}\propto s^{-3}$ for the electromagnetic
force and $\dot{e}\propto s^{-1}$ for solar radiation. Thus, the effect of the
electromagnetic force dominates over the solar radiation for smaller
particles, while for larger sizes the electromagnetic force can be disregarded
in relation to the solar radiation.

Plasma drag, on the other hand, produces only short-term variations in the
eccentricities (M20). The jumps of this element, seen in Fig. 11, result from
the crossing of the particle with resonances with Enceladus, as will be shown
in Section 5.3. For Pallene ringlet particles, the electromagnetic force
dominates for ${\rm s\leq 5~{}\mu}$m, while the solar radiation force is the
most important effect on the eccentricity of ${\rm s>5~{}\mu}$m particles. We
obtain that the non-perturbative forces produce only small variations in the
inclination ($I\sim 10^{-3}$ deg) for the time intervals considered by us in
this section.

The longitude of ascending node and argument of pericentre are mainly affected
by the plasma drag, which is responsible for the precession of the elements in
relation to Pallene. Fig. 12 displays snapshots of the osculating orbit (solid
lines) of a representative particle (coloured dots) and Pallene (black dot).
We rotate the systems on each snapshot to keep Pallene in the fixed position
$x=1$ ${\rm D_{Pal}}$. We show particles with radius of ${\rm 20~{}\mu m}$,
${\rm 50~{}\mu m}$, ${\rm 100~{}\mu m}$, as well as with radius of
centimetres, which corresponds to the case with only gravitational forces.

Figure 12: Snapshots of the osculating orbit (solid lines) and spatial
position (dots) of Pallene (in black) and of a co-orbital particle with
$\lambda=\lambda_{P}+90^{\circ}$. The colour indicates the body, as labelled.
We assume the single particle has a radius of either ${\rm 20~{}\mu}$m, ${\rm
50~{}\mu}$m, or ${\rm 100~{}\mu}$m. Displayed in red, we include the case
solely with gravitational forces (“cms”). The orbits are provided in the
rotating frame in which Pallene is stationary at $x=1$ ${\rm D_{Pal}}$. An
animation of this figure is included in the electronic version; it requires
Adobe Reader version $\geq$9 or similar.

As we can see in Fig. 12, without non-gravitational forces, the particle
remains in the same orbit as Pallene and lacks vertical variation in relation
to the satellite’s orbital plane. When the non-gravitational forces are
included, the orbit precesses, exhibiting vertical excursions in relation to
Pallene’s orbital plane. This phenomenon could be responsible for the observed
vertical width of $\sim 10^{2}~{}$ km of the ring (Hedman et al., 2009; Spahn
et al., 2019) indicating that the ringlet may evolve into a torus, as observed
in the gossamer rings of Jupiter (Burns et al., 1999). The formation of the
torus occurs when the precession of the pericentre acts long enough to
completely randomise the orientation of the particles’ orbits. These results
will be discussed in detail in Section 5.3.

The osculating semi-major axis and eccentricity of a representative particle
under the effects of the non-gravitational forces are presented in Fig. 13.
The lines correspond to numerical simulations where the physical radius of the
single particle is modified (${\rm 0.1~{}\mu}$m, ${\rm 0.2~{}\mu}$m, ${\rm
0.5~{}\mu}$m, ${\rm 1~{}\mu}$m, ${\rm 2~{}\mu}$m, ${\rm 5~{}\mu}$m, ${\rm
10~{}\mu}$m, ${\rm 20~{}\mu}$m, ${\rm 50~{}\mu}$m, and ${\rm 100~{}\mu}$m).
The solid and dotted horizontal lines indicate the orbits of Pallene and
Enceladus, respectively. In this work, we consider a particle to be removed
from the ringlet if it collides with a satellite or migrates outside the
generous limit of $a_{\mathrm{Pal}}+1100$ km (${\rm\sim
1.05~{}D_{\mathrm{Pal}}}$). The latter can be seen in the figure by the
horizontal dot-dashed line.

Figure 13: Osculating semi-major axis and eccentricity of representative
particles co-orbiting Pallene. The particles have a size of ${\rm
0.1~{}\mu}$m, ${\rm 0.2~{}\mu}$m, ${\rm 0.5~{}\mu}$m, ${\rm 1~{}\mu}$m, ${\rm
2~{}\mu}$m, ${\rm 5~{}\mu}$m, ${\rm 10~{}\mu}$m, ${\rm 20~{}\mu}$m, ${\rm
50~{}\mu}$m, and ${\rm 100~{}\mu}$m (coloured lines). The horizontal dotted
line indicates Enceladus’s semi-major axis, while the horizontal dot-dashed
line is the maximum semi-major axis of the particle to be considered as a
ringlet particle. The particles are under the effects of the solar radiation
force, plasma drag, and electromagnetic force.

Particles with ${\rm s\leq 2~{}\mu}$m migrate beyond the orbit of Enceladus
(horizontal dotted line) in less than 100 yr and reach $e>10^{-2}$. In the
case shown in Fig. 13, the particles of $0.1~{}\mu$m and $1~{}\mu$m are
ejected from the Saturnian system ($e>1$) while the particles of $0.2~{}\mu$m
and $0.5~{}\mu$m collide with a satellite outside the orbit of Enceladus. The
${\rm 2~{}\mu}$m-sized particle collides with Enceladus in about 80 yr.

The effects of the non-gravitational forces are weaker for larger grains and
particles with $s>{\rm 5~{}\mu}$m remain with eccentricities of the order of
$10^{-3}$. These particles migrate outwards but still are considered ringlet
particles according to our definition. These results roughly demonstrate that
the permanence of the particles in the ring is strongly affected by non-
gravitational forces and only particles with a radius of tens of micrometres
or greater should have significantly long lifetimes in the ringlet (several
hundreds of years). In the next sections, we perform full N-body simulations
of the ring particles evolution.

###  5.3 Particles co-orbital to Pallene

In this section, we analyse Pallene’s ringlet as formed by a set of 5,000
particles co-orbital to the satellite. We assume particles with the same
orbital elements as Pallene, except for the mean anomaly that was randomly
selected from a uniform distribution between 0∘ and 360∘. The ring composed of
co-orbital particles corresponds, e.g., to a scenario where the structure
could be formed by the disruption of a proto-Pallene. In this scenario, the
ring would also be composed of centimetre-sized or even larger particles.
Nevertheless, we do not perform simulations for this size range since the
effects of non-gravitational forces can be neglected. The orbital evolution of
the centimetre-sized particles would correspond to the analysis in Section 3.2
which demonstrated that most of the particles initially located inside the
Pallene collision region would eventually collide with the satellite, reducing
the survival rate of co-orbital particles.

As a general outcome, particles with $s\leq 10~{}\mu$m present a dynamical
evolution similar to those shown in Fig. 11. The particles migrate towards
Enceladus and show an increase in eccentricity. However, we obtain a more
complex dynamical evolution for particles with $s\geq 20~{}\mu$m caused by
capture in resonances with Enceladus. Roughly speaking, a migrating particle
is captured at a given resonance with a satellite if the migration timescale
is shorter than the libration period of the resonance (Batygin, 2015). In our
case, this condition is achieved for the largest particles ($20~{}\mu$m,
$50~{}\mu$m, and $100~{}\mu$m) which are captured, even for a short period of
time, in the 7:6, 8:7, 9:8, and 10:9 $e$-type MMRs with Enceladus.

Figure 14:  Snapshots showing the percentage of particles as a function of the
geometric semi-major axis (at left) and the geometric eccentricity vs.
geometric semi-major axis (at right). From top to bottom, we show the data for
0, 200, 750, 5000, and 8000 yr. The ${\rm 20~{}\mu}$m, ${\rm 50~{}\mu}$m, and
${\rm 100~{}\mu}$m sized particles are shown in different colours, as
indicated. Pallene is represented by a black filled-circle. The locations of
MMRs with Enceladus are indicated by dashed vertical lines. Similarly to Fig.
12, an animation of this figure is provided in the electronic version.

Figure 14 shows the evolution of the fraction of particles with $s>20~{}\mu$m
(left column), as well as their geometric eccentricity (right column), as a
function of the geometric semi-major axis. Initially, all particles have the
same semi-major axis and eccentricity as Pallene (black dot). As the particles
migrate outward, they cross resonances with Enceladus, increasing their
eccentricities. After 200 yr, a fraction of $20~{}\mu$m-sized particles is
trapped in the 7:6 and 8:7 MMRs, while most of the set is located between the
8:7 and 9:8 MMRs. Particles in the 7:6 MMR are confined for a longer period of
time, reaching the highest eccentricity values ($\approx$0.05). The ${\rm
20~{}\mu}$m-sized particles that are not in MMRs at 200 yr had their
eccentricity increased during the passage through the two innermost
resonances, reaching values $\sim 0.01$. Particles with radius of $50~{}\mu$m
and $100~{}\mu$m have not yet crossed any resonances and remain with the same
initial eccentricity.

At 750 yr, the ${\rm 100~{}\mu}$m-sized particles have crossed the 7:6 MMR,
and the ${\rm 50~{}\mu}$m-sized particles have crossed all four resonances.
Most of the ${\rm 20~{}\mu}$m-sized particles migrated outside the limit of
${\rm\approx 1.05~{}D_{Pal}}$, leaving only the particles confined in MMRs. A
similar result is seen for 5,000 yr, when only ${\rm 100~{}\mu}$m-sized
particles in MMRs remain in the ring, indicating that capture in resonances
increases their longevity. Therefore, the vicinity of MMRs would correspond to
brighter regions of the ring, as will be shown later. Finally, after 8000 yr,
the ring is completely depleted of $\mu$m-sized particles.

(a)  (b)  (c)

Figure 15: a) The half-life (in blue) and the lifetime (in red) of the ring as
a function of the physical radius of the co-orbital particles. b) The fraction
of the particles that collides with the satellites Mimas (in red), Pallene (in
black), and Enceladus (in blue), and the fraction of particles that migrates
out of the orbit of Enceladus (in green). c) The time $\mathcal{T}$ for the
satellite to produce the mass of the ring, assuming a non-porous (black solid
line) and a porous (black dot-dashed line) Pallene. The red and blue lines
give the ring’s lifetime and half-life, respectively, as a function of the
slope $q$.

Figure 15a shows two different timescales as a function of particle radius: in
blue, the time required for 50% of particles to collide with a satellite or
migrate outside the limit of ${\rm\sim 1.05~{}D_{Pal}}$ – hereafter referred
to as the ring’s half-lifetime – and in red the time required for all
particles to be lost – referred as the ring’s lifetime. The ring is completely
depleted of sub-micrometric particles in less than a decade, while particles
of radius of $1-10~{}\mu$m have lifetimes of the order of ${\rm 10^{2}}$ yr.
Particles that last longer are those with $s\geq 20~{}\mu$m, with lifetimes of
${\rm\sim 10^{3}}$ yr – same order of the time $\mathcal{T}$ for Pallene to
produce the mass of the ring (see Fig. 10).

Particle sinks are shown in Fig. 15b. Due to the intense migration caused by
the plasma drag, almost all the sub-micrometric particles migrate beyond the
orbit of Enceladus and collide with an external satellite or are ejected from
the system. By increasing the radius of the particles, the slower rate of
migration increases the period that the particles interact gravitationally
with Enceladus in the vicinity of the satellite. Consequently, the number of
collisions with Enceladus increases, as seen in Fig. 15b. Also due to
migration, the number of particles that collide with Pallene is less than 5%
for all sizes; this rules out Pallene as an efficient secondary source of
material, produced by subsequent impact with these particles.

Figure 15c shows in black lines the same curves shown in Fig. 10: the solid
line is the time for Pallene to produce the ring mass in the non-porous case,
while the dot-dashed line is the same for the porous case. The red and blue
lines indicate the ring’s lifetime and half-lifetime, respectively, obtained
by a time-weighted average:

$\bar{T}=\frac{\sum_{s}m_{s}\left(\frac{s~{}{\rm(\mu m)}}{\rm 100~{}\mu
m}\right)^{-q}T_{s}}{\sum_{s}m_{s}\left(\frac{s~{}{\rm(\mu m)}}{\rm 100~{}\mu
m}\right)^{-q}}$ (31)

where $m_{s}$ is the mass of a particle with radius $s$ and $T_{s}$ is the
(half)-lifetime of the particles.

Focusing on the red curve in Fig. 15c, we verify that the ring would not be in
a steady-state, assuming ejection by Pallene as the only source of material.
However, given the uncertainties in the yield calculation and the proximity of
the values between the black and red solid curves, towards the lower values of
$q$, we can conclude that Pallene might be able to maintain its ring if the
particle distribution is given by ${\rm q\lesssim 3}$. Lower slope values mean
that the ring has higher concentrations of larger particles, which seems to be
the case of the ringlet of Pallene – given that larger particles can be
captured in MMRs with Enceladus, while smaller ones have lifetimes of only a
few years. If the particle distribution in the ring is given by slopes ${\rm
q\gtrsim 4}$, Pallene by itself certainly cannot maintain the ring, since the
lifetime is lower than $\mathcal{T}$ even for the porous limit.

Figure 16: Animations showing the normalised optical depth ${\rm\tau_{norm}}$
in the $\theta$-$r$ (top panels) and $r$-$z$ (bottom panels) planes in the
rotating frame for co-orbital particles. The green dot gives Pallene’s
position and the dashed lines indicate the MMRs with Enceladus. The upper
limit of the radius in the panels corresponds to the limit ${\rm
1.05~{}D_{Pal}}$. Adobe Reader version $\geq$9 or similar is required.

Figure 16 shows animations of the co-orbital particle profiles in the planes
$\theta$-$r$ (top panels) and $r$-$z$ (bottom panels). The colour of each
pixel gives the normalised optical depth of that pixel, assuming a particle
distribution with slope $q=2.5$. The particles are initially distributed along
the orbit of Pallene. In 10 yr, we can identify ring-like structures in the
$r$-$z$ plane, produced by the precession of the longitude of pericentre (Fig.
12), where each structure is composed of particles with different radii. After
100 yr, the ring shows an asymmetrical profile, with the brightest part close
to Pallene’s orbit, and structures with lower brightness outside the
satellite’s orbit. We do not see any bright regions inside the orbit of
Pallene, since outward migration is dominant for all particles.

At 400 yr, the torus structure is completely formed, and the ring has an
asymmetric structure. The brightest part of the ring is in the region of the
7:6 MMR with Enceladus, but we see dimmer structures inside and outside this
location, as an effect of the increased eccentricity of resonant particles.
After 1000 yr, the complete structure of the ring has moved outward and the
brightest region is located in the 8:7 MMR. After 4000 yr, the structure has
moved further away and only a few particles have remained in the ring region.

###  5.4 Particles Ejected from Pallene

In the numerical simulations presented in this section, 5000 particles were
randomly and uniformly distributed in a spherical shell within the Hill radius
of Pallene. Particles are ejected radially with random velocities that follow
the normalised distribution (Hartmann, 1985; Krivov et al., 2003; Sun et al.,
2017):

$f_{v}=\frac{1}{v_{0}}\left(\frac{v}{v_{0}}\right)^{-2}\Theta[v-v_{0}],$ (32)

where $\Theta(x)$ denotes the Heaviside function. The minimum ejecta speed,
$v_{0}$, is obtained from the transcendental equation (Krüger et al., 2000)

$\frac{K_{e}}{K_{i}}=Y\left(\frac{v_{0}}{v_{\rm
imp}}\right)^{2}\left[\left(\frac{v_{0}}{v_{\rm max}}\right)^{-1}-1\right],$
(33)

where $v_{\rm max}$ is the maximum ejecta speed and $K_{e}/K_{i}$ is the ratio
between the kinetic energy partitioned to the ejecta and the impactor’s
kinetic energy, assumed as $K_{e}/K_{i}=0.1$ (Sun et al., 2017).

Figure 17:  Normalised optical depth ${\rm\tau_{norm}}$ for the ejected
particles. Similarly to Fig. 16, we present a cut in the $\theta$-$r$ and
$r$-$z$ planes in the rotating frame. The green dot gives Pallene’s position
and the vertical dashed lines are MMRs with Enceladus. Adobe Reader version
$\geq$9 or similar is required.

Figure 17 is similar to Fig. 16 but for the ejected particles. The temporal
evolution of the ejected particles is similar to the co-orbital particles
scenario. The same is true for the ring profiles, with greater distinctions
only in the first years of the simulation, due to the different initial
conditions. Figure 18 shows the half-lifetime and lifetime of the ring (top
panel), the particle sinks (middle panel), the times required for Pallene to
produce the ring material, as well as the lifetimes as a function of the slope
of the size distribution (bottom panel). Our results are similar to those
discussed in Section 5.3. In both scenarios, Pallene could produce the
material to keep the ring in a steady-state if the distribution of the
particles in the ring is given by $q\lesssim 3$.

(a)  (b)  (c)

Figure 18:  a) The solid lines in blue and red show the time for 50% and 100%
of the ejected particles to be removed from Pallene ring, respectively. b) The
coloured lines show the fraction of particles that collide with Mimas (in
red), Pallene (in black), and Enceladus (in blue), and the fraction that
migrates outside the orbit of Enceladus (in green). c) The time for Pallene to
produce the ring material is given by the black lines, in the non-porous
(solid) and porous (dot-dashed) cases, while the ring lifetime and half-life
are given by the red and blue lines, respectively.

###  5.5 Comments on ring sources

Similar to Madeira et al. (2018) and Madeira & Giuliatti Winter (2020), we
only computed the production due to external projectile impacts with the
immersed moon. Therefore, we are analysing whether the satellite can produce
the amount of material needed to keep the systems in steady-state, not whether
they are in steady-state. In fact, the most likely case is that all the
mentioned dusty arcs/rings are in a quasi-steady state, demonstrating that
more sophisticated models are needed to understand their stability.

As we pointed out in this section, satellite porosity can be a factor
influencing material production; however, the systems also have other sources.
For example, ring particles are also impacted by external projectiles and
therefore also produce material. However, following the prescription given in
Dikarev et al. (2005), we obtained that such source is at least three orders
of magnitude less efficient than the satellite for the systems analysed here.

The mentioned arcs/rings have the similarity of having a population of larger
particles ($\sim$ cm-m, Hedman et al., 2009, 2010; Spahn et al., 2019), which
lead us to speculate whether the mutual collision of these objects or their
impacts with the moon would be the main source of these systems (Colwell &
Esposito, 1990a, b). Just as a proof of concept, we will assume that in the
Pallene ring is immersed a family of moonlets with radii ranging from $1$ m to
$100$ m, following a size distribution $N\sim s^{-3.5}$ and total optical
depth $\tau_{\rm mlets}=10^{-8}$. Production due to impacts between the
moonlets can be roughly estimated as (Sun et al., 2015)

$\dot{M}_{\rm mlets}=3\tau_{\rm mlets}NM_{\rm col}$ (34)

where $M_{\rm col}$ is the amount of dust released per collision, assumed as
$0.12M_{\rm mlet}$ (Canup & Esposito, 1995), and $M_{\rm mlet}$ is the total
mass of the moonlet population.

As a result, we get $\dot{M}_{\rm mlets}\sim 10^{-2}~{}{\rm kg/s}$
corresponding to a value more than one order of magnitude higher than the
production due to the non-porous Pallene. This shows that impacts between
larger particles are an appealing possibility to keep the arcs/rings in
steady-state. However, production due to impacts between centimetric-metric
bodies is a very intricate problem, and is beyond the scope of this work.

##  6 Summary and Conclusions

In this work, we performed an exhaustive numerical exploration of the
evolution of the small Saturnian moon Pallene, as well as of the diffuse dusty
ring sharing its orbit. We used both short- and long-term numerical
simulations, spanning a wide range of timescales to cover in detail the
evolution of Pallene and its ring.

By using the frequency map analysis technique, we produced a diffusion map to
characterise the current dynamical state of a wide region of phase-space
surrounding Pallene. We identified all the MMRs of relevance in the region,
among Pallene and any of the six major moons considered in this study, up to
fourth order. We used a simple tidal evolution calculation for Mimas, Pallene,
and Enceladus in order to set the context for our longer-term simulations. We
made note that the most recent resonance Pallene may have escaped from is the
4:5 resonance with Mimas. Pallene’s current eccentricity or inclination could
be signs of this or another past resonance crossing.

From the short- and long-term N-body simulations, we analysed all the direct
and indirect arguments of the disturbing function identified in the diffusion
map in the vicinity of Pallene. These arguments included zeroth-order
arguments, with degrees $j\leq$ 15, and first- to fourth-order arguments with
degrees $j\leq 30$. In brief, we found that some arguments displayed
interesting behaviour by temporally librating at various timescales. In
particular, the direct argument
$\phi_{\mathrm{tP}}=8\lambda^{\prime}-5\lambda-\varpi^{\prime}-2\varpi$ of
Pallene with Tethys that librates for $\sim 10$ kyr and the zeroth-order
argument $\Phi=\varpi^{\prime}-\varpi+\Omega^{\prime}-\Omega$ of Pallene with
Tethys, Dione and Titan, which coincides with the angle combination suggested
for Pallene with Mimas by Callegari & Yokoyama (2010). The recurrence of this
zeroth-order combination suggests a possible secular alignment of the lines of
apsides and nodes among Pallene, Dione, Rhea, and Titan in timescales $\sim
800$ yr.

Furthermore, after a thorough search of possible (two-body) resonant arguments
for Pallene, we conclude that the small moon is not currently in resonance
with either Mimas, Enceladus, Tethys, Dione, Rhea, or Titan. It is unlikely
that Pallene would be in a higher-order MMR, i.e., $\geq$ 5th order, with any
of these satellites, due to their small eccentricity/inclination, and the
corresponding $e$-$I$ coefficients of the disturbing function. Nevertheless,
the lack of two-body MMRs for Pallene does not exclude the hypothesis that
Pallene might be part of a three-body resonance. Moreover, under the present
considerations and without accounting for Saturn’s tidal forces in the
numerical simulations, we cannot dismiss either the past escape of Pallene
from a resonance or its future trapping, particularly at times longer than 5
Myr.

We analysed the dynamical evolution of the Pallene ring assuming a scenario
where particles are ejected from the satellite’s surface, as well as a
scenario where the material is originally co-orbital to Pallene. We found that
non-gravitational forces dynamically dominate the system and the material
experiences a similar dynamical evolution in both scenarios.

The outward migration due to plasma drag causes the loss of particles with
radius of a few micrometres in just tens of years, while larger particles
($\gtrsim 10~{}\mu$m) can survive for a few hundred years in the ring. Spahn
et al. (2019) measured the radial mean position of the ring to be more than
$1000$ km beyond the satellite’s orbit; this is likely caused by plasma drag.
Our ring profiles clearly show the formation of particle clusters beyond
Pallene’s orbit. Furthermore, the profiles show that the ring evolves into
structures that are radially asymmetrical in relation to the satellite’s
orbit.

The precession of the longitude of pericentre due to non-gravitational forces
produces vertical excursions of the particles in relation to Pallene’s orbital
plane. This could be the mechanism responsible for vertical excursions
discussed in Hedman et al. (2009).

_Cassini_ data indicate a concentration of larger particles around Pallene’s
orbit, which is in line with the significantly longer lifetime of the larger
particles that we found. In fact, when calculating the mass production rate
due to IDPs and ERPs, we find that Pallene can keep the ring in a steady-state
only if it is predominantly composed of larger micrometre-sized particles
($q\lesssim 3$).

If we assume Pallene as the only source of material for the rings, we conclude
that the ring would spread for $q\lesssim 4$. This corresponds to the slope
range given by Kempf et al. (2008); Ye et al. (2014a) for the E ring, in which
Pallene is immersed. In this scenario, our profiles show that the ring will
evolve into a toroidal structure similar to the gossamer rings of Jupiter, and
then it will continuously spread out, both radially and vertically, until it
finally disappears. From our numerical results, we cannot constrain whether
the ring originated from the material ejected from the satellite or from the
disruption of an ancient proto-Pallene.

We must point out that our dynamical model is not complete; if the ring has a
high concentration of larger particles, additional effects such as collisions
between the particles, self-gravity, and local viscosity may be significant to
the system. However, even in this case, plasma drag may dominate, and our main
results would still hold valid.

## Acknowledgements

We thank the anonymous referee for a detailed and careful report that helped
to greatly improve the quality of this paper. G. Madeira thanks FAPESP for
financial support via grant 2018/23568-6. J. A’Hearn thanks M. Hedman, M.
Tiscareno, and M. Showalter for useful discussions; and also thanks NASA for
partial support through the Cassini Data Analysis and Participating Scientist
Program grant NNX15AQ67G. S. M. Giuliatti Winter thanks FAPESP (2016/24561-0),
CNPq (313043/2020-5) and Capes for the financial support.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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