# A Hamiltonian for 1:1 Rotational Secondary Resonances, and Application to
Small Satellites of Saturn and Jupiter

N. Callegari Jr<EMAIL_ADDRESS>Instituto de Geociências e Ciências
Exatas, Unesp - Univ Estadual Paulista,  Departamento de Estatística,
Matemática Aplicada e Computação.  Av. 24-A, Rio Claro/SP/Brazil, CEP
13506-900.

###### Abstract

In this work, we study the dynamics of rotation of the small satellites
Methone and Aegaeon and revisit previous works on the rotation of Prometheus,
Metis, and Amalthea. In all cases, the surfaces of section computed with the
standard spin-orbit model reveal that the synchronous regime shares another
large domain in the rotation phase space. We reproduce and apply the
hamiltonian theory given in Wisdom (2004) to analytically characterize the
detected structure as being a secondary resonance where the period of
oscillations around the synchronism is similar to the orbital period. Being
that the current rotational states of this sort of satellite should be
synchronous (Thomas and Helfenstein 2020), our results can be taken into
account in evolutionary studies of their rotation.

###### keywords:

Aegaeon; Amalthea; Metis; Methone; Prometheus; Saturnian and Jupiter small
satellites; Spin-Orbit resonance.

##  1 Introduction

Consider the rotation of a rigid and homogeneous satellite orbiting a punctual
planet in the case the satellite rotates around a single axis corresponding to
its largest moment of inertia. The mutual perturbations between the bodies
belonging to multiple systems or due to the non-sphericity of the secondary
are neglected in this work, such that the motion of the system is governed by
the laws of the two-body problem (Goldreich and Peale 1966).

A hamiltonian for the sort of problem is given by:

$H(\theta,p,t)=\frac{p^{2}}{2C}-\frac{\epsilon^{2}n^{2}C}{4}\left[\frac{\textsf{a}}{r(t)}\right]^{3}\cos{2[\theta-f(t)]},$
(1)

where $\theta$ is an angle of rotation of the satellite measured from an
inertial line; $p=C\dot{\theta}$ is the angular momentum of rotation around
the $z-$axis assumed to be perpendicular to the orbital plane, and
$\dot{\theta}$ is the angular velocity of rotation. $A<B<C$ are the moments of
inertia around the principal axis of the satellite around the $x$, $y$, $z$,
respectively; a, $e$, $f(t)$, $r(t)=\frac{\textsf{a}(1-e^{2})}{1+e\cos f(t)}$,
$n$ are the semi-major axis, orbital eccentricity, true anomaly, planet-
satellite distance, and the mean-motion of the satellite, respectively.

It is well known that the dynamics of rotation of mid-sized satellites of the
outer planets and the Moon are mainly characterized by synchronous orbit-
rotation resonances (see Peale 1977, Melnikov and Shevchenko 2022). The exact
synchronism occurs when the angular velocity of rotation of the satellite
equals its mean motion, $n$. In practice, the synchronous resonance of the
satellites is characterized by the physical libration of the angle
$\Psi(t)=\theta(t)-f(t)$. $\Psi$ contains all main components of the spin-
orbit perturbations, namely, the optical, free, and forced librations (see
Callegari and Ribeiro 2015, and references therein). In the case of a small
amplitude of libration of $\Psi$ _and_ small orbital eccentricity of the
satellite, the physics of the synchronous resonance can be interpreted by an
analog of the forced harmonic oscillator (see Murray and Dermott 1999).
Suppose also to model the shape of the satellite with a Roche ellipsoid with
semi-axes $a$, $b$, and $c$ in the $x$, $y$, $z$ directions, respectively; in
this case, $a>b>c$ (see Callegari and Rodríguez 2013, and references therein).
Denote the free frequency by $\omega$ (which corresponds to the frequency of
the oscillator in the harmonic approximation). The average linear
theory111W.r.t. mean anomaly and valid for small eccentricity. of synchronous
resonance predicts that the ratio of the free and orbital frequencies
$\frac{\omega}{n}$ is given by:

$\displaystyle\epsilon\equiv\frac{\omega}{n}=\left(3\frac{B-A}{C}\right)^{1/2}=\left[3\left(\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\right)\right]^{1/2}.$
(2)

Thus, when $\epsilon$ is a rational number, an inner resonance can rise due to
the commensurability between $n$ and $\omega$. We have therefore a definition
for secondary resonance within the synchronous regime (in the case of a
synchronous rotating satellite). The approximation of a simple oscillator is
no longer enough to describe the rotational dynamics in this situation, and
additional analyses (e.g. perturbation theory) are necessary to describe the
dynamics of rotation in more detail.

The case of the Saturnian satellite Enceladus studied in Wisdom (2004) is a
good example to illustrate the dynamics of secondary resonances. Under the
hypothesis of a homogeneous body, and based on _Voyager_ spacecraft data, the
dimensions of Enceladus are $a=256.3\pm 0.3$ km, $b=247.3\pm 0.3$ km,
$c=244.6\pm 0.3$ km (Dermott and Thomas 1994). From Equation (2) we obtain
$\epsilon\sim 0.327$, a value close to $\frac{1}{3}$, so that Wisdom
conjectured if Enceladus would be currently rotating close to the
$\frac{\omega}{n}=\frac{1}{3}$ secondary resonance222After _Cassini_ images,
the amplitude of the physical libration of Enceladus has been detected ($\sim
0.12$ degree; Thomas et al. 2016). This value is too high and therefore is not
consistent with a homogeneous satellite. The best fitting given in Thomas et
al. (2016) corresponds to a multi-layered satellite including a global
subsurface ocean..

While it is certain that current rotational states of _all_ mid-sized
satellites of the giant planets are characterized by synchronous resonance,
the same is not necessarily true for _all_ other smaller secondary companions
having mean-radius of the order of a few dozens kilometer. Several of them are
irregularly shaped, and due to this, $\epsilon$ is large. Let us consider the
case of Amalthea (J15), a member of the inner group of Jupiter satellites
composed of Metis, Adrastea, Amalthea, and Thebe. Amalthea is significantly
out of round with dimensions $a\times b\times c$ of the order of $\sim$
$125\times\sim 73\times\sim 64$ such that $\epsilon\sim 1.214$. These values
are given after analyses of _Galileo_ tour in Jupiter system; Thomas et al.
1998 (see also Pashkevich et al. 2021, and references therein). The phase
space of rotation of Amalthea shows the existence of two resonant modes within
the synchronism. They denote these modes by “$\alpha$-resonance” and
“$\beta$-resonance”, and named this property of the synchronous regime by the
“Amalthea effect”. The existence of such kind of “bifurcation” inside the
synchronous domain leads to questioning and discussions on topics related to
the evolution of rotation and the true current equilibrium configuration of
the satellite (see discussions in Melnikov and Shevchenko 2002, and references
therein). Another body with similar rotational properties of the phase space
is the Prometheus (Melnikov and Shevchenko 2008), from Saturn, a close-in with
a mean radius of 42.8 km (see Table 1).

Table 1: Selected physical and orbital parameters of the satellites studied in
this work. In the case of Saturn,
the values of the orbital eccentricities correspond to the maximum geometric
values (Callegari et al. 2021;
Ceccatto et al. 2021, Callegari and Rodríguez 2023).

| Methonea | Aegaeona | Prometheusa | Amaltheab | Metisb
---|---|---|---|---|---
semi-major axis (km) | 194,707 | 168,031 | 140,029 | 181,400 | 128,000
eccentricity | 0.0014 | 0.0005 | 0.00226 | 0.0031 | 0.0002
dimensions (km) | $1.94\times$ | $0.7\times$ | $68.5\times$ | $125\times$ | $30\times$
| $1.29\times$ | $0.25\times$ | $40.5\times$ | $73\times$ | $20\times$
| 1.21 | 0.20 | 28.1 | 64 | 17
$\epsilon$ | 1.077 | 1.5236 | 1.1717 | 1.2140 | 1.074

$(^{a})$ Thomas and Helfenstein 2020; $(^{b})$ Thomas et al. 1998; Pashkevich
et al. 2021

The objects we study in this project belong to another class of even smaller
inner satellites of the Jovian planets having mean diameters of a few
kilometers or even less. We consider in this work the Saturnian satellites
Methone (S/2004 S 1) and Aegaeon (S/2008 S 1) discovered by the _Cassini_
spacecraft in 2004 (Porco 2004) and 2008 (Porco 2009), respectively. Similarly
to Amalthea and Prometheus, their phase spaces also display the co-existence
of the $\alpha$ and $\beta$-“resonances”. Methone, jointly with Aegaeon and
Pallene is classified as ellipsoidal-like satellites333The other denominations
are irregular (e.g. Prometeus, Pandora, Epimetheus, Janus, Telesto, Calypso,
and Helene), and irregular with equatorial ridges (e.g. Atlas, Pan, Daphnis)
(Thomas and Helfenstein 2020).. We will focus next and throughout the paper on
the case of Methone, and at the end of the paper, similar results will be
shown and discussed for Aegaeon and Prometheus. The cases of Amalthea and
Metis will also be revisited given the theory developed here.

To investigate the rotation of Methone, we apply the standard model of spin-
orbit resonance. We utilize the Everhart (1985) algorithm to solve numerically
the rotation differential equations derived from hamiltonian (1). The full
equation is non-autonomous such that, at first, we compute the surfaces of
section (hereafter denoted by SOS), a practical procedure often applied in
studies of rotation after Wisdom et al. (1984). In this technique, the pair of
generalized variables $(\theta,\dot{\theta}/n)$ is plotted every time the
satellite passes through the pericenter of its orbit. Figure 1a) shows the SOS
for Methone in the vicinity of the synchronous domain. The fixed points of the
$\alpha$ and $\beta$-“resonances” are indicated by vertical arrows. Note that
the regimes are separated by a new separatrix (red curve in Fig. 1a)), while
they are encompassed by a thin chaotic layer associated to the synchronous
domain.

The main goal of this work is to investigate the nature of the $\alpha$ and
the $\beta$-regimes in the case of the satellite Methone. Both share the
synchronous domain, so it is worth giving a cinematic description of the
rotation of clones of Methone departing from initial conditions very close to
their fixed points to identify their differences. Figs. 1(b,c) show the time
variations of $\frac{\dot{\theta}}{n}$ and $\Psi=\theta-f$ corresponding to
the $\alpha$ and $\beta$-regimes, respectively. The initial conditions are
$\theta_{0}=0$ in both cases and $\frac{\dot{\theta}(0)}{n}=1$,
$\frac{\dot{\theta}(0)}{n}=0.45$ in b) and c), respectively.

In the case of the $\beta$-regime (Fig. 1b)), $\Psi$ oscillates around zero
with amplitude of $\sim 0.04$ radian (or $\sim 2$ degree) and
$\frac{\dot{\theta}}{n}$ oscillates with amplitude of $\sim 0.036$. In both
variables there is a “long-term” mode resulting from close values of the
fundamental frequencies $n$ and $\omega$. In fact, from Equation (2) we have
$\omega=n\epsilon$ and $\epsilon\sim 1.077$. Denoting by $T$, $P$,
$T_{\omega}$ the periods of the beating, the orbital period, and the period of
the free libration, we obtain $T\sim 13.2$ day from the relation
$\frac{1}{T}=\frac{1}{P}-\frac{1}{T_{\omega}}$ considering $P\sim 1.014$ day
and $T_{\omega}=P/\epsilon$.

In the case of the $\alpha$-regime (Fig. 1c)), $\Psi$ oscillates around zero
with amplitude of $\sim 0.5$ radian (or $\sim 29$ degree).
$\frac{\dot{\theta}}{n}$ oscillates around the unit with amplitude of $\sim
0.5$ such that at the minima (indicated by black points in Fig. 1(b)), the
satellite is always located at the pericenter of its orbit (open circles in
Fig. 1(b)). Thus, due to this relatively large amplitude of oscillation of
$\frac{\dot{\theta}}{n}$, the SOS displays $\frac{\dot{\theta}}{n}\sim 0.45$,
the same as that of the initial value.

Figure 1: (a) Surfaces of section of solutions of the model (1) applied to the
satellite Methone. Each orbit has been integrated for 500 orbital periods of
the satellite. The sections (projections) are made at each orbital revolution,
and all the time the true anomaly of the satellite reaches $f=0$. The $x$ and
$y$-axes are the (projected) rotation angle ($\theta$) around the axis of the
largest moment of inertia and the ratio of the angular velocity of rotation
and the mean motion, $\frac{\dot{\theta}}{n}$. The vertical arrows indicate
the fixed points associated with the $\alpha$ and $\beta$ regimes of rotation.
The numerical values of the parameters of the model adopted in this work which
is necessary in the numerical solution are: $\textsf{a}=194,707.6$ km,
$e=0.0014$, $a=1.94$ km, $b=1.29$ km (see Section 1 for definitions). (b,c)
Time variations of $\frac{\dot{\theta}(t)}{n}$, $\Psi(t)=\theta(t)-f(t)$ for
two trajectories within the domains of the $\alpha$ and $\beta$-regimes,
respectively. $\theta_{0}=0$ in both cases and $\frac{\dot{\theta}(0)}{n}=1$,
$\frac{\dot{\theta}(0)}{n}=0.45$ in b) and c), respectively. The horizontal
dashed lines plotted in (c) guide the solutions close to the fixed points at
the pericenter passage, while the horizontal line in (a) gives the straight
$\frac{\dot{\theta}(t)}{n}=1$.

Since the Methone’s phase space of rotation displays additional structures
within the synchronous regime, all discussion given above leads us to
investigate the existence of secondary resonances. Accordin to the most recent
analyses of data taken from _Cassini_ measurements, Thomas and Helfenstein
(2020) determined that the dimensions of Methone are $a=1.94\pm 0.02$ km,
$b=1.29\pm 0.04$ km, $c=1.21\pm 0.02$ km so that $\epsilon\sim 1.077$. This
value of $\epsilon$ is close to the unit so that we conjecture if a 1:1
secondary resonance of the type $\frac{\omega}{n}\sim\frac{1}{1}$ would
explain the existence of $\alpha$ and $\beta$ structures within the
synchronous domain. To prove the conjecture we return to the Wisdom’s (2004)
paper. He developed _a simple_ analytical model for spin-orbit problem in
hamiltonian formalism aiming to explain the phase space of Enceladus _in the
close vicinity_ of the synchronous regime. Having in hand an expanded version
of hamiltonian of Equation (1), he selected the correct terms and explained
analytically the properties of the secondary $\frac{1}{3}$ resonance. We
follow his steps and reproduce its main results, and we choose those terms in
hamiltonian which are proportional to the arguments containing the
$\frac{1}{1}$ secondary resonance. By adopting a similar methodology given in
Wisdom (2004), we successfully explain the co-existence of the $\alpha$ and
$\beta$-regimes in synchronous resonance by comparing the SOS with level
curves of the constructed hamiltonian444It is worth noting that the mapping of
the 1:1 secondary resonance within the synchronous regime have been done in
many works where very sophisticated models have been adopted (see Gkolias et
al. 2016, Gkolias et al. 2019, Lei 2023.) .

The presentation of the results of this paper is divided as follows. In
Section 2, we deduce the expanded hamiltonian developed in Wisdom (2004) in
detail, showing the terms of the hamiltonian explicitly (that ones related to
equation 20 of the referred paper); some extensions of Wisdom’s model are also
given. In Sections 3.1 and 3.2, we show exactly how the perturbed hamiltonian
can give rise to the secondary resonances and write the hamiltonian for the
$\frac{1}{1}$ secondary resonance. The level curves applied in the case of
Methone are shown in Section 3.3. Analytical estimative of the bifurcations of
the level curves are also provided. In Section 4 we analyze the cases of
Aegaeon and Prometheus, and Section 5 is devoted to conclusions and more
general discussion involving other satellites. Since the phase space of
rotation of Methone, Aegaeon, Prometheus, Amalthea, Metis, and others share
similar structures and properties within the synchronous regime, we are led to
question their current states of rotation.

##  2 Expansion of the hamiltonian (1)

###  2.1 An integrable hamiltonian

The hamiltonian (1) is analogous to that one of a simple _non-autonomous_
pendulum. An expanded version of the hamiltonian can be obtained resulting in
a sum of autonomous-like pendulums. This can easily achieved after
substituting in hamiltonian (1) the developed forms of
$\left[\frac{\textsf{a}}{r(t)}\right]^{3}$, $\cos{2f(t)}$, $\sin{2f(t)}$:

$\displaystyle\left[\frac{\textsf{a}}{r(t)}\right]^{3}\approx
1+3e\cos{nt}+\frac{3}{2}e^{2}(1+3\cos{2nt}),$
$\displaystyle\cos{2f(t)}\approx\cos{nt}+e(\cos{2nt}-1)+\frac{9}{8}e^{2}(\cos{3nt}-\cos{nt}),$
(3)
$\displaystyle\sin{2f(t)}\approx\sin{nt}+e\sin{2nt}+\frac{1}{8}e^{2}(9\sin{3nt}-7\sin{nt}).$

We have:

$\displaystyle H(\theta,p,t)=\frac{p^{2}}{2C}$ $\displaystyle-$
$\displaystyle\frac{\epsilon^{2}n^{2}C}{4}\left(+\frac{17}{2}e^{2}\right)\cos{(2\theta-4nt)}$
(4) $\displaystyle-$
$\displaystyle\frac{\epsilon^{2}n^{2}C}{4}\left(+\frac{7}{2}e\right)\cos{(2\theta-3nt)}$
$\displaystyle-$
$\displaystyle\frac{\epsilon^{2}n^{2}C}{4}\left(1-\frac{5}{2}e^{2}\right)\cos{2(\theta-
nt)}$ $\displaystyle-$
$\displaystyle\frac{\epsilon^{2}n^{2}C}{4}\left(-\frac{1}{2}e\right)\cos{(2\theta-
nt)}+\ldots.$

Each term in the right side of (4) proportional to $\epsilon^{2}$ can be
considered a distinct disturbing resonance associated to the
commensurabilities $\dot{\theta}=2n$, $\dot{\theta}=\frac{3}{2}n$,
$\dot{\theta}=n$ and $\dot{\theta}=\frac{n}{2}$, defining therefore the 2:1,
3:2, 1:1 (synchronous), and the 1:2 spin-orbit resonant states.

We can isolate a single resonance after applying the average principle (see
footnote 1) and we can note that, whatever resonance, the resulting
hamiltonian is always analogous to a simple _non-autonomous_ pendulum.
Canonical transformations can make each of the resonances given in Equation
(4) locally integrable. For instance, consider the synchronous resonance and
define $H_{0}$ by:

$\displaystyle
H_{0}(\theta,p,t)=\frac{p^{2}}{2C}-\frac{\epsilon^{2}n^{2}C}{4}\left(1-\frac{5}{2}e^{2}\right)\cos{2(\theta-
nt)}.$ (5)

Given the canonical transformation:

$\displaystyle\phi$ $\displaystyle=$ $\displaystyle\theta-nt$ (6)
$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle p-nC,$

the new $H^{\prime}_{0}$ is given by:

$\displaystyle H^{\prime}_{0}(\phi,\Phi,-)$ $\displaystyle=$
$\displaystyle-n\Phi+\frac{(\Phi+nC)^{2}}{2C}-\frac{\epsilon^{2}n^{2}C}{4}\cos{2\phi}\Rightarrow$
(7) $\displaystyle=$
$\displaystyle\frac{\Phi^{2}}{2C}-\frac{\epsilon^{2}n^{2}C}{4}\cos{2\phi},$

where the term $-n\Phi$ comes from phase space extension (see Sussman and
Wisdom 2001, Ferraz-Mello 2007). We have also neglected the term in $e^{2}$
since the orbital eccentricity of Methone is of the order of $10^{-3}$.
Moreover, an irrelevant additive constant ($n^{2}C/2$) is not included in (7).

Thus, we have an _integrable_ hamiltonian for the synchronous spin-orbit
resonance. Since the hamiltonian (7) is analogous to the simple autonomous
pendulum, we can use the equations of transformation to the action-angle
variables developed to the simple pendulum which are given in several books.
In this work, we consider only the libratory regime in the case of small
oscillations.

The action variable $J$ is defined as being the area on the plane
$(\phi,\Phi)$ divided by $2\pi$. The conjugated action $\psi$ varies linearly
in time. The new hamiltonian depends only on the action and can be written as
follows:

$\displaystyle H^{{}^{\prime\prime}}_{0}(-,\tilde{J})$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{2}n^{2}C}{4}\left(-1+\tilde{J}-\frac{1}{16}\tilde{J}^{2}-\frac{1}{256}\tilde{J}^{3}+\ldots\right),$
(8)

where

$\displaystyle\tilde{J}=\frac{4J}{\epsilon nC},$ (9)

such that

$\displaystyle
H^{{}^{\prime\prime}}_{0}(-,J)=-\frac{\epsilon^{2}n^{2}C}{4}+\epsilon
nJ-\frac{1}{4C}J^{2}-\frac{1}{16\epsilon nC^{2}}J^{3}+\ldots.$ (10)

The algebraic expression between the new variables $(\psi,J)$ and the old ones
$(\phi,\Phi)$ is:

$\displaystyle\sin{2\phi}\approx
2\phi=F_{1}\sin{\psi}+F_{3}\sin{3\psi}+\ldots,$ (11)

where

$\displaystyle F_{1}$ $\displaystyle=$
$\displaystyle\left(2\tilde{J}+\frac{1}{4}\tilde{J}^{2}+\frac{27}{512}\tilde{J}^{3}+\ldots\right)^{1/2}\approx\sqrt{\tilde{J}}+\frac{1}{16}(\sqrt{2\tilde{J}})^{3},$
(12) $\displaystyle F_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{192}\left[(2\tilde{J})^{3}\left(1+\frac{9}{16}\tilde{J}+\ldots\right)\right]^{1/2}\approx\frac{1}{192}(\sqrt{2\tilde{J}})^{3}+\ldots.$
(13)

Equations (8)-(13) have been taken from Wisdom (2004). A proof of them can be
obtained after applying the pendulum model to the hamiltonian (7) in the case
of small oscillation approximation (see for instance appendix B.3 in Ferraz-
Mello 2007).

###  2.2 Perturbation of the integrable hamiltonian

Let us consider now the time-dependent perturbations on the integrable part of
the synchronous resonance of the hamiltonian such that

$\displaystyle H^{\prime}(\phi,\Phi,t)=H^{\prime}_{0}+H^{\prime}_{1},$ (14)

where $H^{\prime}_{0}$ is given in (7) and $H^{\prime}_{1}$ is the
perturbation. We will include in $H^{\prime}_{1}$ the two neighboring
resonances of the synchronous resonances, namely, 3:2 and 1:2 (see Equation
(4)). In terms of old variables $(\phi,\Phi)$ we can obtain from (4) that:

$\displaystyle
H^{\prime}_{1}(\phi,\Phi,t)=-\frac{\epsilon^{2}n^{2}C}{4}\left[\frac{7e}{2}\cos{(2\phi-
nt)}-\frac{e}{2}\cos{(2\phi+nt)}\right].$ (15)

To write $H^{\prime}_{1}$ in terms of the action-angle variables we must
consider the transformation equation (11). The algebraic manipulations can be
much more simple considering the limit of small oscillations (Wisdom 2004).
Thus, first write (13) as follows:

$\displaystyle H^{\prime}_{1}(\phi,\Phi,t)$ $\displaystyle=$
$\displaystyle-\frac{\epsilon^{2}n^{2}C}{4}\left[\frac{7e}{2}(C_{2}\cos{nt}+S_{2}\sin{nt})-\frac{e}{2}(C_{2}\cos{nt}-S_{2}\sin{nt})\right],$
(16) $\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle\cos{2\phi}\approx
1-\frac{1}{2}(2\phi)^{2},S_{2}=\sin{2\phi}\approx
2\phi-\frac{1}{6}(2\phi)^{3}.$ (17)

After substituting the expression (11) in (17) and the result in (16), we can
obtain the perturbation $H^{\prime\prime}_{1}(\psi,J,t)$ as follows:

$\displaystyle H^{\prime\prime}_{1}(\psi,J,t)$ $\displaystyle=$
$\displaystyle\epsilon^{2}n^{2}Ce\times$ (18) $\displaystyle+$
$\displaystyle(3/16)F_{1}^{2}\cos{nt}+(3/16)F_{3}^{2}\cos{nt}-(3/4)\cos{nt}$
$\displaystyle-$ $\displaystyle(1/2)F_{1}\cos{(nt-\psi)}$ $\displaystyle+$
$\displaystyle(1/16)F_{1}^{3}\cos{(nt-\psi)}$ $\displaystyle-$
$\displaystyle(1/16)F_{1}^{2}F_{3}\cos{(nt-\psi)}$ $\displaystyle+$
$\displaystyle(1/8)F_{1}F_{3}^{2}\cos{(nt-\psi)}$ $\displaystyle-$
$\displaystyle(3/32)F_{1}^{2}\cos{(nt-2\psi)}$ $\displaystyle+$
$\displaystyle(3/16)F_{1}F_{3}\cos{(nt-2\psi)}$ $\displaystyle+$
$\displaystyle(1/16)F_{3}^{3}\cos{(nt-3\psi)}$ $\displaystyle-$
$\displaystyle(1/48)F_{1}^{3}\cos{(nt-3\psi)}$ $\displaystyle+$
$\displaystyle(1/8)F_{1}^{2}F_{3}\cos{(nt-3\psi)}$ $\displaystyle-$
$\displaystyle(1/2)F_{3}\cos{(nt-3\psi)}$ $\displaystyle-$
$\displaystyle(3/16)F_{1}F_{3}\cos{(nt-4\psi)}$ $\displaystyle+$
$\displaystyle(1/16)F_{1}F_{3}^{2}\cos{(nt-5\psi)}$ $\displaystyle-$
$\displaystyle(1/16)F_{1}^{2}F_{3}\cos{(nt-5\psi)}$ $\displaystyle-$
$\displaystyle(3/32)F_{3}^{2}\cos{(nt-6\psi)}$ $\displaystyle-$
$\displaystyle(1/16)F_{1}F_{3}^{2}\cos{(nt-7\psi)}$ $\displaystyle-$
$\displaystyle(1/48)F_{3}^{3}\cos{(nt-9\psi)}$ $\displaystyle-$
$\displaystyle(1/16)F_{1}^{3}\cos{(nt+\psi)}$ $\displaystyle+$
$\displaystyle(1/2)F_{1}\cos{(nt+\psi)}$ $\displaystyle-$
$\displaystyle(1/8)F_{1}F_{3}^{2}\cos{(nt+\psi)}$ $\displaystyle+$
$\displaystyle(1/16)F_{1}^{2}F_{3}\cos{(nt+\psi)}$ $\displaystyle+$
$\displaystyle(3/16)F_{1}F_{3}\cos{(nt+2\psi)}$ $\displaystyle-$
$\displaystyle(3/32)F_{1}^{2}\cos{(nt+2\psi)}$ $\displaystyle-$
$\displaystyle(1/16)F_{3}^{3}\cos{(nt+3\psi)}$ $\displaystyle+$
$\displaystyle(1/48)F_{1}^{3}\cos{(nt+3\psi)}$ $\displaystyle+$
$\displaystyle(1/2)F_{3}\cos{(nt+3\psi)}$ $\displaystyle-$
$\displaystyle(1/8)F_{1}^{2}F_{3}\cos{(nt+3\psi)}$ $\displaystyle-$
$\displaystyle(3/16)F_{1}F_{3}\cos{(nt+4\psi)}$ $\displaystyle+$
$\displaystyle(1/16)F_{1}^{2}F_{3}\cos{(nt+5\psi)}$ $\displaystyle-$
$\displaystyle(1/16)F_{1}F_{3}^{2}\cos{(nt+5\psi)}$ $\displaystyle-$
$\displaystyle(3/32)F_{3}^{2}\cos{(nt+6\psi)}$ $\displaystyle+$
$\displaystyle(1/16)F_{1}F_{3}^{2}\cos{(nt+7\psi)}$ $\displaystyle+$
$\displaystyle(1/48)F_{3}^{3}\cos{(nt+9\psi)}.$

Note that at this step we are not utilizing the expanded versions of $F_{1}$
and $F_{3}$ given in the right side of Equations (12) and (13).

##  3 Perturbed hamiltonian and the rise of secondary resonances

###  3.1 The 1:1 secondary resonance

The transformation variable given in Equation (6), $\phi=\theta-nt$, consists
of an approximation of the physical libration $\Psi$ valid for small
eccentricity.

For small amplitude of oscillation, the frequency of the angle variable $\psi$
is the frequency of $\phi$ divided by $2\pi$. Thus, Equation (18) can be
utilized to study several secondary resonances. Inspection of (18) shows the
terms in arguments of the cosines of the form $nt-k\psi$, where
$k=1,2,\ldots$. Therefore, when $k$ is a multiple of $n$, we should have a
secondary resonance.

For instance, after collecting the terms proportional to $\cos{(nt-3\psi)}$,

$H^{\prime\prime}_{1}(\psi,J,t)=-\frac{\epsilon^{2}n^{2}Ce}{4}\left(\frac{1}{12}F_{1}^{3}+2F_{3}-\frac{1}{4}F_{3}^{3}-\frac{1}{2}F_{1}^{2}F_{3}\right)\cos{(3\psi-
nt)},$ (19)

we can obtain the disturbing part of the hamiltonian associated with the 3:1
secondary resonance utilized in the theory developed by Wisdom (2004) to study
the rotation of Enceladus (see Section 1).

The problem we are considering in this work is related to the nature of the
$\alpha$ and $\beta$ regimes of motion in the phase space of rotation of
Methone and, as pointed out in Section 1, we conjectured that the coexistence
of the regimes is explained by a 1:1 secondary resonance. We will prove our
conjecture in the next two subsections after considering the terms factored by
$\cos{(\psi-nt)}$ in (18) so that we have the following disturbing part of the
hamiltonian:

$H^{\prime\prime}_{1}(\psi,J,t)=-\frac{\epsilon^{2}n^{2}Ce}{4}\left(-\frac{1}{4}F_{1}^{2}F_{3}-2F_{1}+\frac{1}{4}F_{1}^{3}+\frac{1}{2}F_{1}F_{3}^{3}\right)\cos{(\psi-
nt)}.$ (20)

###  3.2 The second extension of the phase space and the non-singular
variables

Consider the canonical transformation

$\displaystyle\psi^{\prime}$ $\displaystyle=$ $\displaystyle\psi-nt,$ (21)
$\displaystyle J^{\prime}$ $\displaystyle=$ $\displaystyle J,$

and define

$\displaystyle\hat{J}$ $\displaystyle=$ $\displaystyle\frac{J^{\prime}}{nC},$
$\displaystyle\delta$ $\displaystyle=$ $\displaystyle\epsilon-1.$ (22)

where $\hat{J}$ is a dimensionless quantity (Wisdom 2004), and $\delta$ can be
interpreted as being the distance of the exact resonance $\frac{1}{1}$
secondary resonance.

After collecting the terms in (20) up to order $J^{\prime 2}$, neglecting the
constant terms, and factoring the parameter $n^{2}C$, we have the final form
in Angle-Action variables of the hamiltonian (12) for 1:1 secondary resonance:

$H^{\prime\prime\prime}_{1:1}(\psi^{\prime},\hat{J},t)=n^{2}C\left[\delta\hat{J}-\frac{\hat{J}^{2}}{4}+\frac{3}{8}e\sqrt{\epsilon}\left(\sqrt{2\hat{J}}\right)^{3}-e\sqrt{\epsilon^{3}}\sqrt{2\hat{J}}\right]\cos{\psi^{\prime}}.$
(23)

Define

$H\equiv\frac{H^{\prime\prime\prime}_{1:1}}{n^{2}C}.$ (24)

It is straightforward to prove that $H$ is dimensionless so that the $H$
depends only on two parameters: $\epsilon$ and the orbital eccentricity $e$.

Let us consider the canonical transformation:

$\displaystyle x$ $\displaystyle=$
$\displaystyle\sqrt{2\hat{J}}\cos{\psi^{\prime}}$ $\displaystyle y$
$\displaystyle=$ $\displaystyle\sqrt{2\hat{J}}\sin{\psi^{\prime}}.$ (25)

Thus, the hamiltonian (24) becomes:

$H(x,y)=\frac{\delta}{2}(x^{2}+y^{2})-\frac{1}{16}(x^{2}+y^{2})^{2}+\frac{3}{8}e\sqrt{\epsilon}(x^{2}+y^{2})x-e\sqrt{\epsilon^{3}}x.$
(26)

###  3.3 Atlas of the phase space

Keeping in hand a time-independent one-degree-of-freedom hamiltonian for our
problem, we can now explore the rotation phase space by computing the level
curves of Equation (26). We also aim to compare the level curves with the
surfaces of sections and a suitable plane of variables which can be the same
utilized in Fig. (1), $(\theta,\frac{\dot{\theta}}{n})$. To relate them to the
variables $(x,y)$ (Equation (25)), we first recall that in the linear
approximation of harmonic oscillator the relation between variables
$(\phi,\Phi)$ (Equation (6)) and the Angle-Action $(\psi,J)$ are given by the
canonical transformation

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\left(\frac{2J}{\alpha}\right)^{1/2}\sin{\psi},$
$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle(2J\alpha)^{1/2}\cos{\psi},$ (27) $\displaystyle\alpha$
$\displaystyle=$ $\displaystyle\epsilon nC.$

By fixing a instant of time ($t=0$ for instance), manipulating the Equations
(6), (25), (27), and by noting that at $t=0$, $\psi^{\prime}=\psi$ (Equation
(21)), we can show that,

$\displaystyle\theta$ $\displaystyle=$
$\displaystyle\frac{y}{\sqrt{\epsilon}},$
$\displaystyle\frac{\dot{\theta}}{n}$ $\displaystyle=$ $\displaystyle
x\sqrt{\epsilon}+1.$ (28)

Fig. 2 shows the level curves of the hamiltonian (26) for six values of
$\epsilon$. Let $\epsilon_{c}$ be a critical value of $\epsilon$. For
$\epsilon<\epsilon_{c}$, only the equilibrium point associated with the
$\alpha$-regime exists. The rising of the $\beta$-regime occurs at values of
$\epsilon$ slightly larger than $\epsilon_{c}$, and now three equilibria
centers share the $y$-axis: two of them are stable points, namely those
associated to $\alpha$ and $\beta$-regimes, and the third one is an unstable
equilibrium point associated to the separatrix of the $\beta$-regime. The plot
in bottom-middle panel in Fig. 2 reproduces with good agreement the surface of
section given in Fig. 1. Next we will study in detail the bifurcation of the
regimes in the phase space and determine $\epsilon_{c}$ numerically.

Figure 2: Contour plots of the hamiltonian (26) for several values of the non-
sphericity parameter $\epsilon$. The orbital eccentricity is $0.0014$ and the
range of values of $\epsilon$ are compatible with the values of Methone
($\epsilon\sim 1.077$ (the same utilized in Fig. 1). Green and blue lines
correspond to level curves within the domains of the $\alpha$ and
$\beta$-regimes, respectively. Red curves are the separatrix of the
$\beta$-regime. Right-bottom plot is a detailed close of the fixed point
associated with the $\beta$-regime. The dashed line indicates the forced
component.

###  3.4 The forcing of the $\beta$-regime and the quasi-synchronous regime.

The loci of the equilibria centers (fixed points) of the $\alpha$ and
$\beta$-regimes in the phase space can be obtained analytically from equations
of motion:

$\displaystyle\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle+\frac{\partial
H}{\partial y}=-\frac{1}{4}(x^{2}+y^{2})y+\delta y+2\gamma_{1}yx$ (29)
$\displaystyle\frac{dy}{dt}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial x}=+\frac{1}{4}(x^{2}+y^{2})x-\delta
x-3\gamma_{1}x^{2}-\gamma_{1}y^{2}+\gamma_{2},$ (30)

where $\gamma_{1}=\frac{3}{8}e\sqrt{\epsilon}$,
$\gamma_{2}=e\sqrt{\epsilon^{3}}$.

The equilibria conditions $(\frac{dx}{dt},\frac{dy}{dt})=(0,0)$ lead to a
system of non-linear algebraic equations, and equilibria solutions in the
x-axis where $y=0$ must satisfy the equation:

$\frac{x^{3}}{4}-3\gamma_{1}x^{2}-\delta x+\gamma_{2}=0.$ (31)

The three solutions of Equation (31) are only real for values of $\epsilon$
larger than $\epsilon_{c}$. For $\epsilon<\epsilon_{c}$, only one root is real
and there is only one fixed point in the phase space, namely, that one related
to the $\alpha$-regime. At $\epsilon=\epsilon_{c}$ occurs the rising of the
$\beta$-regime and the unstable point. Considering the current
$\epsilon=1.077$ of Methone, the solution of Equation (31) is
$(x_{1},x_{2},x_{3})=(-0.5616685,0.02034028,0.5478663)$. Substituting these
roots into Equation (28) we obtain $\frac{\dot{\theta}}{n}=0.4171$ ($x_{1}$:
$\alpha$-regime); $\frac{\dot{\theta}}{n}=1.0211$ ($x_{2}$: $\beta$-regime);
and $\frac{\dot{\theta}}{n}=1.5686$ ($x_{3}$: separatrix). The three values
agree with those obtained after inspection of the level curves given Fig. 2.
Fig. 3 shows the real solutions calculated in this way in the range of
$0.96<\epsilon<1.08$. We can see that the bifurcation occurs at
$\epsilon_{c}\sim 1.0149$. Note also the quasi-symmetry of the roots $x_{1}$
and $x_{3}$.

The fixed point of the $\beta$-regime is not centered at the origin as it is
indicated by the dashed line in Fig. (2), bottom-right. The forced component
calculated above deviates from the unit such that
$\frac{\dot{\theta}}{n}=1.0211$. More generally note that in the case of non-
circular orbits, $x=0$ is never a solution of Equation (31), so that from
Equation (28) the forcing is never null in this case (recall the definition of
$\gamma_{1}$).

The forced component in the SOS can also be explained by the classical linear
theory of the average solution of the spin-orbit dynamics within the
synchronous resonance. The forced harmonic oscillator-analogue model gives
$\frac{\dot{\theta}}{n}=1+A$ where the amplitude of the forced component $A$
is given by:

$\displaystyle
A=\frac{2\omega^{2}e}{\omega^{2}-n^{2}}=\frac{2e}{1-\epsilon^{-2}},$ (32)

(Callegari and Rodríguez 2013; see also equation 5.123 in Murray and Dermott
1999). From Equation (32) we obtain $\frac{\dot{\theta}}{n}=1.0203$, showing
good agreement with the value obtained above.

Figure 3: The real roots of Equation (31) as function of $\epsilon$. Red, blue
and green curves are the loci of the fixed points of the separatrix ($s$), the
$\beta$, and the $\alpha$-regimes, respectively. Dashed horizontal line,
dashed-dot-vertical line, and dashed-dot-dot vertical line indicate the
straight $\frac{\dot{\theta}}{n}=1$, the critical value $\epsilon_{c}=1.0149$
where the $\beta$-regime rises, and the current value for Methone
($\epsilon=1.077$), respectively.

Figure 4: The surfaces of section (SOS; first column), and level curves of the
hamiltonian (29) (second column), and (33) (third column) in the cases of
Aegaeon and Prometheus (top and bottom, respectively). The separatrix of the
$\alpha$-regime is shown in red. Dashed-dot-horizontal lines indicate the
fixed points obtained from hamiltonians and compared to the SOS corresponding
values.

##  4 Additional application: Aegaeon and Prometheus.

Fig. (4) shows the rotational phase space around the synchronous regime in the
cases of Aegaeon ($\epsilon\sim 1.5236$), and Prometheus ($\epsilon\sim
1.1727$). As pointed out in Section 1, both rotational phase spaces display
the $\alpha$ and $\beta$-regimes. Inspection of the left and middle panels
shows us that the position of the three fixed points of the $\alpha$ and
$\beta$-regimes obtained from level curves (middle) agree with surfaces of
section in the case of Prometheus, but the same is not true in the case of
Aegaeon555Note that our model does not allow to obtain the loci of the
separatrix of the synchronous regime since it is valid, by construction, only
in the interior of the resonance..

To improve the model we include terms of order higher than $J^{2}$ in the
actions so that the hamiltonian (26) becomes:

$\displaystyle
H(x,y)=\frac{\delta}{2}(x^{2}+y^{2})-\frac{1}{16}(x^{2}+y^{2})^{2}+\frac{3}{8}e\sqrt{\epsilon}(x^{2}+y^{2})x-e\sqrt{\epsilon^{3}}x$
$\displaystyle-\frac{1}{128\epsilon}(x^{2}+y^{2})^{3}+\frac{17}{96\sqrt{\epsilon}}e(x^{2}+y^{2})^{5/2},$
(33)

where terms up to $J^{3}$ have been included (corresponding to the last two
terms in Equation (33). A significant improvement can be seen, mainly in the
case of Aaegeon. As discussed in Lei (2023), and first shown in Gkolias et al.
(2019), in the specific case of the 1:1 secondary resonance, the analytical
and numerical mapping of the associated fixed points diverge for values of
$\epsilon>1.2$.

##  5 Conclusions

This work shows the results of the dynamics of rotation of out-of-round close-
in small Saturnian satellite Methone and Aegaeon discovered by the Cassini
spacecraft. Their shapes and physical parameters have been updated recently by
Thomas and Helfenstein (2020). As pointed out in Section 1, the main problem
we are considering in this work is related to the nature of the $\alpha$ and
$\beta$-regimes of motion in the phase space of rotation of these satellites
already detected for Amalthea, Metis, and Prometheus (and probably others -
see Melnikov and Shevchenko 2022, and references therein).

We first conduct our investigation with the well-known model of rotation
developed by Goldreich and Peale (1966) and applied by many authors in
distinct sorts of problems through decades (e.g. Wisdom et al. 1984, Wisdom
2004, Callegari and Rodriguez 2013). The exact equations of the 1.5 degree-of-
freedom model are solved numerically Everhart (1985) code, given very accurate
rotational trajectories able to identify the main properties of the phase
space with the surface of section technique. The $\beta$-regime is located
close to the origin and corresponds to the “classical” fixed point associated
with the synchronous rotation motion where the amplitude of oscillation of the
physical angle $\Psi=\theta-f$ is null at the equilibrium. In the case of
Methone, its center is slightly forced by an amount $\sim 0.02$ such that
$\frac{\dot{\theta}}{n}\sim 1.02$, a value which can be confirmed by linear
theory (Wisdom et al. 1984; Callegari and Rodriguez 2013). The $\alpha$-regime
engulfs the $\beta$-regime and it is located far from origin at
$\frac{\dot{\theta}}{n}\sim 0.45$ such that $\frac{\dot{\theta}}{n}$
oscillates around 1. Also interestingly the amplitude of variation of $\Psi$
is not null in the equilibrium point of the $\alpha$-regime keeping an
amplitude of the order of 30 degrees.

To understand the complexity of such rotation phase space of Methone and
Aegaeon, we apply the model developed by Wisdom (2004) to study the rotation
of Enceladus. Thus, we follow exactly his steps and reproduce his results;
having completed this task, we could apply the methodology to our case. Our
hamiltonian (18) generalizes Wisdom’s theory showing all terms at that order
of expansion (second order in action). Hamiltonian (33) includes also terms up
to $J^{3}$ in the actions. The terms in the expansion involving those which
are proportional to 1:1 resonance between the frequency of physical libration
$\Psi$ and the mean motion $n$ are responsible for the co-existence of the
$\alpha$ and $\beta$ regimes in the rotation of the current Methone and
Aegaeon’s phase spaces. Therefore, we have shown the existence of a secondary
resonance within the synchronism, completing our initial goals.

Figures (2) and (3) show the dependence of the domains of the $\alpha$ and
$\beta$ regimes with the shape parameter of the Methone. The $\beta$-regime
occurs at $\epsilon=\epsilon_{c}\sim 1.0149$ such that for
$\epsilon<\epsilon_{c}$ only the $\alpha$-regimes exists. While there is good
agreement between numerical and analytical calculations in the case of
Methone, the same is not true for Aegaeon, where the model is useful only when
higher order terms are included in the hamiltonian since $\epsilon>1.2$, a
superior limit for convergence of analytical estimative (Gkolias et al. 2019).

We revisit the previous works on the rotation of Prometheus, Amalthea, and
Metis given the current theory. The case of Metis is similar to that Methone
since $\epsilon\sim 1.0742$ (see Table 1). The case of Amalthea is similar to
that of Prometheus since $\epsilon\sim 1.214$ (see Table 1).

As we pointed out above, trajectories within this regime of motion are forced
and suffer “long-term” oscillations even in the case of initial conditions
located over the fixed point. Being that the current rotational states of
Saturnian close-in small satellites are probably synchronous (Thomas and
Helfenstein 2020), our results can be taken into account in evolutionary
studies of the rotation of these bodies. Tidal models (e.g. Ferraz-Mello et
al. 2008, Ferraz-Mello 2013) can be applied to estimate the final destiny of
the rotation of such small bodies and also the role of the secondary
resonances on its thermal emission (Wisdom 2004).

ACKNOWLEDGEMENTS.

The São Paulo Research Foundation (FAPESP) (process 2020/06807-7).

Prof. Adrián Rodríguez (Valongo Observatory, UFRJ/Brazil).

Part of this work has been presented at the ‘XXI Brazilian Colloquium on
Orbital Dynamics’ (INPE, from December 12 to 16, 2022); and at the ‘New
Frontiers of Celestial Mechanics: theory and applications’ (Department of
Mathematics Tullio Levi-Civita, University of Padua, from February 15 to 17,
2023).

## References

  * [1] Callegari Jr., N., Rodríguez, A.. Dynamics of rotation of super-Earths. Celest Mech. Dyn. Astr. 116, 389–416 (2013).
  * [2] Callegari Jr., N., Ribeiro, F. B.. Computational and Applied Mathematics 34, 423-435 (2015).
  * [3] Callegari Jr., N., Rodríguez, A., Ceccatto, D. T.. The current orbit of Methone (S/2004 S 1). Celest. Mech. Dyn. Astr. 133:49 (2021).

  * [4] Callegari Jr., N., Rodríguez, A..The orbit of Aegaeon and the 7:6 Mimas-Aegaeon resonance. Celest. Mech. Dyn. Astr. 135:21 (2023).

  * [5] Ceccatto, D. T.; Callegari, N., Jr.; Rodríguez, Adrián (2021). The current orbit of Atlas (SXV). Proceedings of the International Astronomical Union, IAU Symposium, 15, 120 - 127.
  * [6] Dermott, S. F., Thomas, P. C.. The Determination of the Mass and Mean Density of Enceladus from its Observed Shape. Icarus 109, 241- (1994).
  * [7] Everhart, E.. An efficient integrator that uses Gauss-Radau spacings. In: IAU Coloquium 83, 185-202 (1985).
  * [8] Ferraz-Mello, S.. Canonical Perturbation Theories - Degenerate Systems and Resonance. Series: Astrophysics and Space Science Library, Vol. 345. Springer US (2007).
  * [9] Ferraz-Mello, S., Rodríguez, A., Hussmann, H.. Tidal friction in close-in satellites and exoplanets: The Darwin theory re-visited. Celest. Mech. Dyn. Astr. 101, 171-201 (2008).
  * [10] Ferraz-Mello, S.. Tidal synchronization of close-in satellites and exoplanets. A rheophysical approach. Celest. Mech. Dyn. Astr. 116, 109-140 (2013).
  * [11] Gkolias, I., Celletti, A., Efthymiopoulos, C., Pucacco, G.. The theory of secondary resonances in the spin–orbit problem. MNRAS, 459, 1327 (2016).
  * [12] Gkolias, I., Efthymiopoulos, C., Celletti, A., Pucacco, G.. Accurate Modelling of the low-order resonances in the spin-orbit problem. Communications in Nonlinear Science and Numerical Simulation, 77, 181 (2019).
  * [13] Goldreich, P., Peale, S.. Spin-orbit coupling in the solar system. The Astronomical Journal, 71, 425-437 (1966).
  * [14] Lei, H.. Dynamical structures associated with high-order and secondary resonances in the spin-orbit problem. https://arxiv.org/pdf/2312.14413.pdf (2023).
  * [15] Melnikov, A. V., Shevchenko. On the rotational dynamics of Prometeus and Pandora. Celest Mech. Dyn. Astr. 101, 31-47 (2008).
  * [16] Melnikov, A. V., Shevchenko. Rotational Dynamics and Evolution of Planetary Satellites in the Solar and Exoplanetary Systems. Solar System Research, 56, No. 1, pp. 1–22 (2022).
  * [17] Murray, C. D, Dermott, S. F.. Solar System Dynamics, Cambridge University Press (1999).
  * [18] Pashkevich, V. V. ; Vershkov, A. N. ; Mel’nikov, A. V.. Rotational Dynamics of the Inner Satellites of Jupiter. Solar System Research, 55, Issue 1, p.47-60 (2021).
  * [19] Peale, S. J.. Rotation Histories of the Natural Satellites. In: Planetary Satellites. Proceedings of IAU Colloq. 28, held in Ithaca, NY, August 1974. Edited by J. A. Burns. University of Arizona Press, 1977, p.87.
  * [20] Porco, C. C.. S/2004 S 1 and S/2004 S 2. IAU Circ. 8401 (2004 August 16) (2004).
  * [21] Porco, C. C. S/2008 S 1. IAU Circ. 9023 (2009 March 3) (2009).
  * [22] Press, W. H., Teukolsky, S. A., Vetterling, W. T., B. P. Flannery: ‘Numerical Recipes in Fortran 77’. Cambridge University Press (1996).
  * [23] Sussman, G. J., Wisdom, J. with Mayer, M. E.: Structure and Interpretation of Classical Mechanics. The MIT Press. Cambridge, Massachusetts (2001).
  * [24] Thomas, P.C., J.A. Burns, L. Rossier, D. Simonelli, J. Veverka, C.R. Chapman, K. Klaasen, T.V. Johnson, M.J.S. Belton, Galileo Solid State Imaging Team. The Small Inner Satellites of Jupiter. Icarus, 135, 360-371 (1998).
  * [25] Thomas, P. C.; Helfenstein, P.. The small inner satellites of Saturn: Shapes, structures ans some implications. Icarus, 344, 113355 (2020).
  * [26] Wisdom, J., Peale, S. J., Mignard, F.. The chaotic rotation of Hyperion. Icarus, 58, 137-152 (1984).
  * [27] Wisdom, J.. Spin-Orbit Secondary Resonance Dynamics of Enceladus. The Astronomical Journal 128, 484-491 (2004).