# The dynamics of self-gravity wakes in the Mimas 5:3 bending wave: modifying
the linear theory

Daniel D. Sega<EMAIL_ADDRESS>Laboratory for Atmospheric and Space
Physics, University of Colorado at Boulder, Boulder, CO, 80309, USA  Glen
Stewart  Laboratory for Atmospheric and Space Physics, University of Colorado
at Boulder, Boulder, CO, 80309, USA  Joshua E. Colwell  Department of Physics,
University of Central Florida, Orlando, FL, 32816, USA  Girish M. Duvvuri
Laboratory for Atmospheric and Space Physics, University of Colorado at
Boulder, Boulder, CO, 80309, USA  Center for Astrophysics and Space Astronomy,
University of Colorado at Boulder, Boulder, CO, 80309, USA  Richard Jerousek
Department of Physics, University of Central Florida, Orlando, FL, 32816, USA
Larry Esposito  Laboratory for Atmospheric and Space Physics, University of
Colorado at Boulder, Boulder, CO, 80309, USA

###### Abstract

The satellite Mimas launches a bending wave —a warping of the rings that
propagates radially through self-gravity— at the 5:3 inner vertical resonance
with Saturn’s rings. We present a modification of the linear bending wave
theory (Shu et al., 1983) which includes the effects of satellite self-gravity
wakes on the particles in the wave. We show that, when treated as rigid, these
wakes generate an extra layer of particles whose number density is
proportional to the magnitude of the slope of the warped ring. Using a ray-
tracing code we compare our predictions with those of Shu et al. (1983) and
with 60 stellar occultations observed by the Cassini Ultraviolet Imaging
Spectrograph (UVIS) and find that the extra layer of particles of our
perturbed bending wave model has a considerable explanatory power for the UVIS
dataset. Our best model explains the most discrepant and surprising features
of the Mimas 5:3 bending wave; the enhancement of the signal for the cases of
occultations with high ring opening angle and the bigger-than-expected
viscosity, $\nu=576\,\mathrm{cm^{2}/s}$, which is more than double the
viscosity computed from density waves (Tiscareno et al., 2007). This shows
that self-gravity wakes can be effective at transporting angular momentum in a
vertically perturbed disk. Relative to neighboring density waves (Tiscareno et
al., 2007), we find a lower-than-expected value for the surface mass density,
$\sigma=36.7\,\mathrm{g/cm^{2}}$, which suggests that the enhanced viscous
interactions may be transporting material into the surrounding regions.

Saturn, Planetary Rings, Celestial Mechanics, Stellar Occultation, Saturn
Satellites

##  1 Introduction

A bending wave (BW) is a warping of a disk (a circular and thin mass plane)
caused by perturbations normal to the disk that propagates due to self-
gravity. Warped disks are ubiquitous in astrophysics and can be found around
young stars (Epstein-Martin et al., 2022), black holes (Thomas et al., 2021),
and around planets (Shu et al., 1983), for which we have a vast amount of
observations available. While warps are only understood to a limited extent
(for instance, see Nelson & Tremaine 1995), they are known to occur when out-
of-the-plane forces perturb a disk that orbits a central body. In the case of
the rings of Saturn, the out-of-the-plane force is produced when satellites
with inclined orbits perturb the ring particles at a frequency that coincides
with a rational multiple of these particles’ vertical motion at some radial
distance from Saturn (Murray & Dermott, 2000). Many of these resonances occur
in the A ring of Saturn, where we focus on the 5:3 vertical resonance due to
the moon Mimas at $131902\,\mathrm{km}$ from Saturn’s center. A radial cut
through an idealized bending wave is shown on the top panel of Figure 1.

Figure 1: Theoretical predictions for the vertical displacement (top panel)
and apparent optical depth (bottom panel) according to SCL. The red rays in
the top panel represent light rays that are being transmitted through the
ring, and the corresponding optical depth probed for each light ray is marked
by the red dashed lines in the bottom panel. The angle between the mean ring
plane and the incident light rays is $B=26^{\circ}$ (note that the axes do not
have the same scale), and the angle between the radial direction and the light
rays projected onto the plane is $\phi=0^{\circ}$. We can see how the light
ray goes through more material when it aligns with the ring’s slope (which
corresponds to a peak in optical depth), and through less material when it
travels perpendicular to the slope (which corresponds to a trough in optical
depth).

Shu, Cuzzi, & Lissauer (1983) (henceforth SCL) were the first to confirm the
presence of bending waves in the rings by adapting the theory of warped
galaxies developed by Bertin & Mark (1980) and analyzing images from Voyager
I. While there was general agreement with the SCL theory for weak bending
waves, the Mimas 5:3 BW proved difficult to fit. Later, Lissauer et al. (1984)
focused on this specific wave and obtained a value for the viscosity by using
similar methods. A general study of waves using occultations from Voyager’s
photopolarimeter was then done by Esposito et al. (1983), while Gresh et al.
(1986) did an extensive study focusing on bending waves using low opening
angle radio occultations. Neither managed to explain the shape of the Mimas
5:3 BW, and Gresh et al. (1986) suggested that the damping mechanism for this
wave needs revision. On the theoretical side, Chakrabarti (1988, 1989)
considered radial shear of particles as a damping mechanism for the Mimas 5:3
BW but it yielded an inconsistent result for the viscosity. All of these
efforts were based on the Voyager observations; however, Cassini data has yet
to be used to improve the theory of bending waves.

Figure 1 shows how a BW can influence the optical depth of the ring measured
by stellar occultations (which are photometric measurements of a star as it is
occulted by the rings) assuming the ring behaves as predicted by the SCL
theory. The red dashed lines in the bottom panel show the optical depths
corresponding to the light rays in the top panel: the optical depth maxima
(minima) occur when the light rays are the most parallel (perpendicular) to
the local warped ring. From looking at Cassini occultations (Figures 2 and 3),
and by comparing to Gresh et al. (1986), we have identified 3 problems that
appear when comparing SCL with the data:

(1) Wave profile.
The poor prediction of the wave’s morphology can be observed in Figure 2,
where we show a plot of the optical depth of an occultation of the star
$\gamma$-Pegasus measured by UVIS (red line) compared to SCL theory (black
line). The effective opening angle of an occultation is defined as
$B_{\mathrm{eff}}=\tan^{-1}(\tan B/\cos\phi)$ where $B$ is the angle between
the mean ring plane and the incident light rays, and $\phi$ the angle between
the radial direction and the light rays, projected onto the plane (see Gresh
et al. 1986); the lower $B_{\mathrm{eff}}$ is, the higher the variations in
the optical depth. As shown in Figure 1, the peaks and troughs of this optical
depth depend on the inclination, or the slope, of the ring relative to the
line-of-sight to the star. The overprediction of the optical depth peak
interior to $-80\,\mathrm{km}$ seen in Figure 2 suggests that the SCL theory
overestimates this inclination.

SCL also predicts that all variation of the optical depth of the disk relative
to the flat ring is caused only by the inclination of the ring; therefore, in
occultations where the line-of-sight is normal to the rings, no variation of
optical depth is expected. In the left panel of Figure 3, however, the data
shows otherwise. An approximately normal (to the rings) occultation still
shows a symmetric variation of the optical depth, centered about
$80\,\mathrm{km}$ inside the resonance location, which suggests an extra layer
of material being generated within the BW that may be caused by the
fragmentation of particles in the ring (Gresh et al., 1986). The right panel
of Figure 3 shows that the position of the peak and the overall shape of this
optical depth enhancement is correlated with the theoretical prediction for
the maximum slope of the wave (solid black line) instead of the maximum
elevation (dashed blue line), which further suggests that this fragmentation
is higher where the slope of the ring is steeper. In this work, we suggest a
plausible physical mechanism that explains this extra layer of particles and
its apparent correlation with the slope of the wave.

This suggested extra layer of particles could be described as a haze, as a
comparison between the Visible and Infrared Mapping Spectrometer (VIMS) and
UVIS data (see Table 1) shows the presence of a small amount of sub-mm size
particles which diffract the infrared light out of the detector increasing the
infrared optical depth (Jerousek et al., 2016). This is an uncommon sight in
the A-ring as sub-mm particles normally are rapidly accreted into bigger
particles (Bodrova et al., 2012; Harbison et al., 2013), further motivating a
modification of SCL theory that accounts for the disruption of material in the
BW region.

| $\lambda\;[\mathrm{nm}]$ | $\frac{\bar{\tau}_{\mathrm{wave}}}{\bar{\tau}_{\mathrm{bg}}}\;\mathrm{[-]}$
---|---|---
VIMS | $2886-2977$ | $1.520+0.026\,/-0.025$
UVIS | $110-190$ | $1.352+0.028\,/-0.027$
Table 1: Simultaneous Infrared and Ultraviolet occultations of
$\alpha$-Scorpius taken by the VIMS and UVIS instruments. VIMS optical depth
is larger by $13\%$ compared to UVIS. For $\tau_{\mathrm{bg}}$ we averaged
data between $131600$ and $131700\,\mathrm{km}$ and for $\tau_{\mathrm{wave}}$
we averaged data between $131862$ and $131762\,\mathrm{km}$ Figure 2: We
compare an occultation of $\gamma$-Pegasus (rev 32I, $B=20.3^{\circ}$,
$B_{\mathrm{eff}}=26^{\circ}$) seen by Cassini (red) to the standard bending
wave theory (black). The resonance occurs at a radius of $131902\,\mathrm{km}$
and the wave propagates towards Saturn (left of the plot). We see three major
discrepancies: (1) The SCL theory overpredicts the peaks of the optical depth
and underpredicts the troughs. (2) The wavelengths do not match for the first
three cycles. (3) The theory underpredicts how fast the wave damps, as peaks
are still predicted to appear around $-150\,\mathrm{km}$, but the observed
wave gets buried in the noise before that point. Using the viscosity of
$\nu=260\,\mathrm{cm^{2}/s}$ from Lissauer et al. (1984)’s analysis of this
wave and Tiscareno et al. (2007)’s analysis of the Prometheus 11:12 density
wave.

(2) Wavelength near resonance.
Figure 2 indicates that, while the theoretical wavelength works fairly well at
later wave crests (this wave propagates to the left in the plot), it differs
from observed values near resonance, a result seen in Gresh et al. (1986). The
extra layer of particles suggested by Figure 3 and Table 1 may produce a
radial change in surface density—which affects the local wavelength (SCL;
Lehmann et al. 2019)—or modify the shape of the first optical depth peaks.
These effects may aid in explaining the observed wavelength mismatch.

(3) Viscosity.
The ring’s viscosity controls the shape and amplitude of the peaks of the
optical depth profile seen in Figure 2. Viscous interactions reduce the
vertical displacement of ring particles; which, in turn, changes the maximum
extinction at the point where the ring’s slope and the line-of-sight coincide.
In other words, the peak amplitude of the optical depth pattern depends on the
viscosity. In Figure 2, the SCL model (black line) predicts a highly peaked
optical depth profile and a wave that propagates beyond the observed
variations seen in the data (red line). This suggests that the fluid shear
description of the damping is inadequate, and/or the values suggested by
previous Mimas 5:3 BW studies (Lissauer et al., 1984; Gresh et al., 1986), and
Cassini density waves studies (Tiscareno et al., 2007), are too low to
describe this wave. Regardless, the observed peaks of the optical depth and
how fast the wave damps indicate that the current theory underpredicts the
amount of energy and momentum diffused in the wave region. This missing
diffusion may be caused by the same mechanism that produces the extra layer of
particles suggested by Figure 3.

All three issues can be explored in depth using Cassini ring occultation data.
Cassini observed the Mimas 5:3 BW in $217$ occultations with the UVIS High
Speed Photometer (HSP) (Esposito et al., 2004) out of which $130$ have a
signal-to-noise high enough to detect the variations in the data due to the
bending wave. Moreover, using Cassini data, Colwell et al. (2006) confirmed
the existence of self-gravity wakes, which are elongated clumps of particles
that periodically disaggregate, and are $10$ to $100$ times bigger than a
typical A ring particle ($\sim 1\,\mathrm{m}$) (Jerousek et al., 2016; Salo et
al., 2018). The consequences of adding self-gravity wakes to the SCL theory
are derived in section §2 via the assumption that they react to the relevant
torques as rigid bodies; here we present a mechanism that explains the extra
optical depth signal seen in normal occultations. In section §3 we describe
the transmission model, our selection criteria for the $60$ UVIS occultations
used, and how the data were reduced. In section §4 we describe the ray-tracing
code. In section §5 we compare our bending wave model to the data followed by
a discussion in section §6, where we consider how our model addresses the
above problems. In section §7 we summarize our conclusions.

Figure 3: Left panel: we compare an occultation of $\beta$-Centauri (rev85I,
$B=66.7^{\circ}$, $B_{\mathrm{eff}}=87^{\circ}$) (solid red), to the SCL model
(solid back). The error bars are displayed as a light red filled curve in the
background of the plot. The SCL theory predicts a uniform optical depth;
however, the data shows a symmetric rise in the optical depth centered at
$-80\,\mathrm{km}$ from resonance. Right panel: the same optical depth profile
of $\beta$-Centauri (solid red) is now compared to the theoretical (SCL
theory) maximum slope of the wave (solid black) and the amplitude of the wave
(dashed blue) for a viscosity of $566\,\mathrm{cm^{2}/s}$ and a ring surface
density of $\sigma=36.3\,\mathrm{g/cm^{2}}$; these values are the best-fit
wave parameters of the model presented in this paper.

##  2 Self-gravity wakes and the Bending wave

Self-gravity wakes are clumps of particles with an elongated structure at an
angle $\theta_{w}$ from the azimuthal direction (see Figure 4), which are held
together by self-gravity against the Keplerian shear, and whose constituent
particles present a resistance to changing their relative distances, $\Delta
r_{ij}\approx 0$ (Daisaka & Ida, 1999; Lu et al., 2018). In other words,
particles within the wakes show coherent motion, causing the wake to keep a
consistent shape and orientation in timescales of one orbital period
(Karjalainen & Salo, 2004). Self-gravity wakes were first hypothesized in the
galactic context (Toomre, 1964) as non-axisymmetric disturbances generated by
the interplay of the self-gravity of a rotating disk and its shear. They were
first proposed to exist in the rings by Colombo et al. (1976), and Dones &
Porco (1989) and Dunn et al. (2004) modeled and corroborated their effects on
Voyager images and Very Large Array observations, respectively. They were
first numerically modeled for planetary rings by Salo (1992) and were detected
in Saturn’s rings in Colwell et al. (2006) as an azimuthal asymmetry in the
optical depth of UVIS occultations.

The SCL linear bending wave theory takes the rings to be smooth and of uniform
density, so the presence of these structures was not accounted for in the
dynamics. To study the effects of self-gravity wakes we assume that they
remain rigid at least during the time it takes the wave to advance one
wavelength in the frame co-rotating with the wake, which is about one orbital
period (see Eqn. 5 below). At least some self-gravity wakes are expected to
remain together for that long (Karjalainen & Salo, 2004): simulations even
suggesting possible lifetimes of many orbital periods (Michikoshi et al.,
2015).

Figure 4: Snapshot of self-gravity wakes in a shear-box simulation produced
with the shear-sheet REBOUND example (Rein & Liu, 2012), for visualization
purposes only. Circled in red we can see an oriented self-gravity wake; we can
also see they tend to point at an angle $\theta_{w}$ with the azimuthal.
Figure 5: Bending wave (to scale) and the two frames of reference used in this
work. The frame on the right is centered at resonance ($x=0$ at resonance),
co-rotating and defined by the unprimed axis $(\hat{x},\hat{y},\hat{z})$ while
the Eulerian frame on the wakes (top left) is centered on the center of mass
of a self-gravity wake at a distance $r$ from resonance and it is defined by
the primed coordinates $(\hat{x^{\prime}},\hat{y^{\prime}},\hat{z^{\prime}})$.
We also represent here the angles
$\theta_{q^{\prime}}=\int\omega_{q^{\prime}}dt$ where $\omega_{q^{\prime}}$ is
the angular velocity of a rotation about the $q^{\prime}$-axis where
$q^{\prime}$ can be $x^{\prime},y^{\prime},$ or $z^{\prime}$ (the principal
axes of the wake). The angles are $0$ when the prime and unprimed axis align.

Within the BW the wakes are subjected to different torques that affect their
orientation ($\theta_{q^{\prime}}$, see Figure 5) as they oscillate in
elevation ($z$) with the wave; these torques and their effects on the
rotational motion of the wakes are computed in section §2.1. We will see that,
as the wakes change their orientation, the wake particles at some distance
from the center of the wake will gain a difference in vertical speed with
respect to the surrounding ring particles (which are assumed to follow the
expected SCL vertical motion), resulting in collisions of order
$1\,\mathrm{cm/s}$ in relative velocities which release material residing on
the wake’s surface. The predicted properties of the resultant haze are
described in section §2.2.

###  2.1 The torques that couple the wake’s rotation and elevation

Assuming the wakes remain rigid, we can compute their rotation by solving
Euler’s equations of rigid motion:

$\dot{\omega}_{x^{\prime}}=\frac{\tau_{x^{\prime}}}{I_{x^{\prime}}}+\frac{I_{y^{\prime}}-I_{z^{\prime}}}{I_{x^{\prime}}}\omega_{y^{\prime}}\omega_{z^{\prime}}$
(1)

$\dot{\omega}_{y^{\prime}}=\frac{\tau_{y^{\prime}}}{I_{y^{\prime}}}+\frac{I_{z^{\prime}}-I_{x^{\prime}}}{I_{y^{\prime}}}\omega_{z^{\prime}}\omega_{x^{\prime}}$
(2)

$\dot{\omega}_{z^{\prime}}=\frac{\tau_{z^{\prime}}}{I_{z^{\prime}}}+\frac{I_{x^{\prime}}-I_{y^{\prime}}}{I_{z^{\prime}}}\omega_{x^{\prime}}\omega_{y^{\prime}}$
(3)

where $I_{q^{\prime}}$, $\omega_{q^{\prime}}$, and $\tau_{q^{\prime}}$ are the
moment of inertia, angular velocity, and torques, respectively, about the
principal axis $q^{\prime}$ (where $q^{\prime}=x^{\prime}$, $y^{\prime}$, or
$z^{\prime}$).

To solve these equations we need to compute the torques about each principal
axis ($\tau_{x^{\prime}}$, $\tau_{y^{\prime}}$, $\tau_{z^{\prime}}$), which
will depend, among other things, in the elevation of the center of mass (CM)
of the wake, effectively coupling the rotation of the wake with its vertical
position. To compute these torques we set the reference frame in the center of
mass of a wake a distance $x$ from resonance (see Figure 5), and take the time
dependence of the vertical coordinate, $z$, of the CM to act harmonically in
time with a frequency Doppler-shifted by the orbit of the wake:

$\omega^{\prime}=m(\Omega_{M}-\Omega)+\mu_{M}$ (4)

where $\Omega$ is the mean motion frequency of the wake’s CM, $\Omega_{M}$ and
$\mu_{M}$ are the mean motion and vertical frequencies of Mimas, and $m$ is
the azimuthal wave number ($m=4$ for our case). This is in accord with the SCL
linear bending wave theory translated to a co-rotating frame represented by
the unprimed frame $(x,y,z)$ in Figure 5, where $x$ and $y$ are the radial and
azimuthal coordinates respectively. We also introduce angle coordinates which
represent rotations about the principal axis of rotation
$(x^{\prime},y^{\prime},z^{\prime})$ of the wake and for concreteness consider
the wake to be a parallelopiped (or a ‘granola bar’) with sides
$(W,L,H)=(18\,\mathrm{m},232\,\mathrm{m},4\,\mathrm{m})$ accordingly. We
extract these dimensions from transmission models of the A ring (Hedman et
al., 2007; Jerousek et al., 2016), and numerical simulations (Salo et al.,
2018) where $H/W\sim 0.2$, $L\sim 4\lambda_{T}$ and $W\sim\lambda_{T}/3$;
$\lambda_{T}\equiv 4\pi^{2}G\sigma/\Omega^{2}\sim 58\,\mathrm{m}$ is the
Toomre critical wavelength (Toomre, 1964), where $\sigma$ is the surface mass
density and $\Omega$ the mean motion frequency.

The motion of a wake particle at a point $(x^{\prime},y^{\prime},z^{\prime})$
with respect to the unprimed co-rotating frame will then be given by adding
the CM velocity (whose motion is governed by the SCL theory) to the velocity
due to the wake’s rotation, obtained from Eqns. (1), (2), and (3). The
equation for the z-coordinate for the CM of a wake a position $x$ from
resonance as a function of time $t$ is given by:

$z(x,t)=\mathrm{Re}\big{[}h(x)e^{i(\omega^{\prime}t)}\big{]},$ (5)

$h(x)=\mathfrak{H}A_{V}e^{-(\frac{x}{\chi_{D}(\nu)})^{3}}$

where $\mathfrak{H}$ is a complex Fresnel integral, $A_{V}$ is the wave
amplitude (see Eqn. 27 in SCL), $\chi_{D}$ is the damping length, which
depends on the kinematic viscosity $\nu$ (see Eqn. 98 in Shu 1984), and where
$\omega^{\prime}$ is the Doppler-shifted forcing frequency. While for $kx>>1$,
the wave profile $h(x)$ takes a WKBJ form $h(x)\sim A(r)e^{\int ikdx}$, we
will use the exact expressions for the amplitude ($\sqrt{hh^{*}}$) and slope’s
magnitude ($\sqrt{\frac{dh}{dr}(\frac{dh}{dr})^{*}}$). For the explicit form
of $h(x)$ see Sega (2023), Eqn. (C.54).

Note that we have eliminated $z$’s usual explicit dependence on azimuthal
position ($\Theta$) by using the Doppler-shifted frequency $\omega^{\prime}$,
which amounts to making the substitution $\Theta$ = $\Omega t$ in Eqn. (26) of
SCL. In other words, we are moving from an Eulerian to a co-rotating
Lagrangian description of the rings. Thus, the Lagrangian derivative used in
SCL, $D/Dt=\partial/\partial t+\Omega\partial/\partial\Theta$, becomes the
time derivative $\partial/\partial t$ in the co-rotating frame.

Knowing the motion of the CM of a wake at a distance $x$ from resonance, we
proceed to compute the relevant torques on the wake. We first consider the
torques caused by tidal forces and the local environment of the BW, while
ignoring the gravitational effects of other wakes. These amount to: the tidal
torque due to Saturn ($\tau_{\mathrm{tidal}}$, §2.1.1), the torque due to the
Keplerian shear ($\tau_{\mathrm{Kep}}$, §2.1.2), the acceleration torque due
to the BW ($\tau_{\mathrm{BWacc}}$, §2.1.3), and the torque due to the
vertical shear of the BW ($\tau_{\mathrm{BWsh}}$, §2.1.4). These torques cause
the most important aspects of the wake’s rotation, which we analyze in the
idealized case of an isolated wake (§2.1.5). Later, we derive the torque due
to the wake-to-wake gravitational interaction ($\tau_{\mathrm{wake}}$, §2.1.6)
and study its effect (§2.1.7).

####  2.1.1 Tidal torque

We begin by deriving the torques about the $z^{\prime}$-axis, which are also
present outside the BW region. We assume that the wake’s inclination outside
the ring’s mean plane remains small, so that
$\theta_{x^{\prime}},\theta_{y^{\prime}}<15^{\circ}$ and $\theta_{z^{\prime}}$
represents the pitch-angle of the self-gravity wake (in Appendix A we show the
general expression for an arbitrary wake orientation). The torques about the
$z^{\prime}$-axis are the tidal torque—which tends to align the wake’s long-
axis with the radial direction—and the Keplerian shear torque that aligns the
wake with the azimuthal. Later, we will see how the BW torques change this
picture.

We start with the tidal force on a piece of the wave with mass $dm$ (Murray &
Dermott, 2000)

$F_{\mathrm{tidal}}=3\frac{GM_{S}}{a^{3}}y^{\prime}*\sin\theta_{z^{\prime}}\cos\theta_{z^{\prime}}dm$

where $a$ is the distance from Saturn’s center, $G$ the gravitational
constant, $M_{s}$ the mass of Saturn and $y^{\prime}$ the coordinate along the
long-axis of the wake. Using $dm=HW\rho_{w}dy^{\prime}$ for the differential
mass of the wake (where $\rho_{w}$ is the wake’s density) and integrating with
respect to $y^{\prime}$ we arrive at the tidal torque:

$\tau_{\mathrm{tidal};z^{\prime}}=2\frac{GM_{S}}{a^{3}}\sin{\theta_{z^{\prime}}}\cos{\theta_{z^{\prime}}}\left(\frac{L}{2}\right)^{3}*\rho_{\mathrm{Roche}}HW$
(6)

For the density of the wake ($\rho_{w}$) we will use $\rho_{\mathrm{Roche}}$,
defined as the bulk-density of an object that fills its Hill sphere (Tiscareno
et al., 2013a). Aggregates within the rings, such as moonlets, have been shown
to have this density within a $20\%$ margin (Porco et al., 2007), and
numerical simulations have shown similar results for self-gravity wakes (Salo
et al., 2018). Notice that this torque is positive when
$0<\theta_{z^{\prime}}<90^{\circ}$ and becomes negative when
$\theta_{z^{\prime}}>90^{\circ}$, causing the pitch-angle to orient radially
(i.e $\theta_{z}^{\prime}=90^{\circ}$).

####  2.1.2 Keplerian shear torque: explaining the pitch-angle

While real wakes are transient agglomerations of particles in a shearing disk,
we can reproduce the orientation of these structures by introducing the torque
on a rigid bar caused by collisions with particles moving past the wake due to
the Keplerian shear. To compute this torque, we have to consider the 3-Body
Hill’s problem. As done in Morishima & Salo (2004) and Yasui et al. (2014) for
the case of embedded moonlets, we integrate the Hill equations in 3D to find
the colliding trajectories of the ring particles:

$\begin{split}\ddot{x}-2\dot{y}=\frac{\partial U_{H}}{\partial x}\\\
\ddot{y}+2\dot{x}=\frac{\partial U_{H}}{\partial x}\\\ \ddot{z}=\frac{\partial
U_{H}}{\partial z}\end{split}$ (7)

The Hill potential ($U_{H}$) consists of the tidal potential
($\frac{3}{2}x^{2}$) plus the potential of a line of mass of finite size,
modified to allow for arbitrary wake orientations. The zero-velocity curves
for this potential are shown in Figure 6; once the collisions change the
orientation of the wake, the trajectories are recalculated with the updated
potential. The particles that are radially inward orbit faster than the CM of
the wake, and hence they will tend to align the wake with the azimuthal
direction; the radially outward particles have the same effect. To find the
colliding trajectories we integrate the Hill equations to then compute the
rate of momentum transfer into the self-gravity wake:

Figure 6: Zero-velocity curves of the Hill potential of a line of mass
oriented $45^{\circ}$ from the azimuthal. The self-gravity wake’s potential is
modeled as a uniform line of mass of length $L$.

$\tau_{\mathrm{Kep};z^{\prime}}=2(1+\epsilon)\int
y^{\prime}\frac{dp_{\perp}}{dt}=2(1+\epsilon)\int^{z_{\mathrm{max}}}_{z_{\mathrm{min}}}\int^{\frac{L}{2}}_{0}\rho_{s}(z)|v_{\perp}|v_{\perp}y^{\prime}dy^{\prime}dz^{\prime}$
(8)

where $\rho_{s}$ is the space density of the colliding particles and
$p_{\perp}$ and $v_{\perp}$ are the momentum and velocity perpendicular to the
long axis of the wake, $\epsilon$ is the coefficient of restitution. To
numerically compute this integral we create a grid of particles for the
incoming impact parameter $b$ and the vertical coordinate $z$, where $\Delta
b=\Delta z=0.01r_{H}$ where $r_{H}$ is the Hill radius that was taken to be
$\frac{W}{2}=9\,\mathrm{m}$ (which is consistent with our value for
$\rho_{\mathrm{Roche}}$). The highest and lowest initial $z$ values of
particles that collide with the wake are $z_{\mathrm{max}}$ and
$z_{\mathrm{min}}$, respectively. We use the vertical density profile
$\rho_{s}(z)=e^{-(\frac{z}{z_{0}})^{2}}$ where $z_{0}$ is the rings half
thickness which was taken to be $5\,\mathrm{m}$ and a temporal resolution of
$dt=0.01T$ where $T$ is the orbital period at resonance.

We proceed to study the angle at which the torques equilibrate in the case of
a single self-gravity wake as a way to test our framework. The total torque in
the z-axis can be written as:

$\tau_{z^{\prime}}=\tau_{\mathrm{tidal}}(\theta_{z^{\prime}})+\tau_{\mathrm{Kep};z^{\prime}}(\theta_{z^{\prime}},\rho_{s},\epsilon)$
(9)

We find that these torques equilibrate stably at
$\theta_{z^{\prime}}=25^{\circ}$ when $\rho_{s}\approx
425,\,500\,\mathrm{kg/m^{3}}$ for $\epsilon=1,\,0.5$, respectively. This
equilibrium angle is a boundary condition for our problem and any pair
($\epsilon$, $\rho_{s}$) that achieves it yields the same wake motion inside
the wave region. Figure 7 show how the pitch-angle derived by equilibrating
the torques matches the one suggested by UVIS occultations (Colwell et al.,
2006; Jerousek et al., 2016), and aligns with the pitch-angles apparent in
shear-box simulations conducted with different shear rates (Michikoshi et al.,
2015; Salo et al., 2018). As a reference for these space density ($\rho_{s}$)
values, consider that a preliminary multiwavelenght analysis suggests an
average space density of $85\,\mathrm{kg/m^{3}}$ in between wakes (Jerousek,
2018); with this density the torques equilibrate at
$\theta_{z^{\prime}}=50^{\circ}$ for $\epsilon=1$. The higher value of
$425\,\mathrm{kg/m^{3}}$ required by the $25^{\circ}$ pitch-angle suggests
that wake-to-wake collisions—which are not included in our model—can
significantly elevate the average amount of material interacting with a wake
within an orbital period. Note that $\rho_{s}$ is overestimated by our
parallelepiped model for the wake: the lesser inertia of a more realistic
ellipsoidal wake would increase the angular acceleration due to collisions
while not affecting the tidal angular acceleration. This decreases the
required $\rho_{s}$ by the ratio of the inertia coefficients between the two
models, $3/5$.

Figure 7: Relationship between the pitch-angle $\theta_{w}$ (see Figure 4) and
the shear rate $q=-\frac{d\ln{\Omega}}{d\ln{x}}$ where $\Omega$ is the mean
motion frequency and $x$ the radial coordinate from Saturn. We calibrate our
model to equilibrate at $\theta_{w}=25^{\circ}$ for $q=1.5$ which is the case
for the Mimas 5:3 BW. The blue triangles correspond to equilibrium
configurations extracted from a shear-box simulation presented in Figure 16.23
of Salo et al. (2018). The black square represents the values reported in
Jerousek et al. (2016) at the region near the Mimas 5:3 BW (at a Saturncentric
distance of $13.2$ Mm) with the error bars representing the range of the
values presented in Figure 7 of that work; the upper error bar given by the
pitch-angle derived from UVIS occultations. Our rigid-body model matches the
shear-box simulations’ equilibrium angle within $4^{\circ}$.

The computationally intensive nature of integrating the Hill equations self-
consistently forces us to resort to a two-body approximation of the three-body
problem. Specifically, a gravitational focusing approximation common in planet
formation (Greenberg et al., 1991; Rafikov, 2004; Armitage, 2020), where the
incoming speed of the scattering event can either be the dispersion velocity
or the Keplerian velocity depending on which one dominates in the wave region.
Cuzzi et al. (1979) already argued that the dispersion velocity dominates over
the extent of a particle, but we find that both velocities are similar over
the extent of the Hill radius of a wake. Thus, the safest approach for our
case is to include 3-body corrections in our two-body approximation. From
Greenberg et al. (1991) and Armitage (2020) we can get an approximate
collision rate for our regime:

$\frac{dM}{dt}=q\Omega\cdot\sigma\cdot f\cdot b\cdot db$ (10)

where $f$ is the fraction of particles that enter the Hill sphere and collide,
$q$ is the shear rate $q=-d\ln{\Omega}/d\ln{x}$ ($q=1.5$ for a Keplerian
disk), $\Omega$ is the mean motion frequency of the particle, $b$ is the
impact parameter and $\sigma$ is the surface mass density of the disk. To get
a torque we use two-body dynamics to find the speed and angle of incidence for
the incoming particles at the moment of impact.

$\tau_{\mathrm{Kep};z^{\prime}}=-2(1+\epsilon)\int^{b_{\mathrm{max}}}_{b_{\mathrm{min}}}\frac{dM}{dt}v_{\theta}(b,\theta_{z^{\prime}})r_{2B}(b,\theta_{z^{\prime}})$
(11)

where $v_{\theta}(b)=b^{2}*q\Omega/r_{2B}$ and

$r_{2B}(b,\theta_{z^{\prime}})=\frac{\frac{(b^{2}q\Omega)^{2}}{G\rho_{\mathrm{Roche}}WLH}}{1+e\cos(\theta_{z^{\prime}}-\varpi)}$

is the two-body orbit equation, where $\varpi$ is the argument of periapsis
taken from the azimuthal direction. We have set the true anomaly to the
angular position at which the wake is oriented, which is the point in the
orbit where the collision takes place (valid for the case of small
$\theta_{x^{\prime}}$ and $\theta_{y^{\prime}}$). Here we are taking the
angular momentum to be conserved about the CM of the wake, so that for a given
impact parameter $b$ we can determine a collisional speed $v_{\theta}$ at a
distance $r_{2B}$ from the CM. Under this approximation the trajectories of
the colliding particles are hyperbolas and the three-body effects are
accounted for by using Hill’s equations to determine the limit of the integral
$b_{\mathrm{min}}$. This and the length of the long axis ($L$) relative to the
Hill radius, which guarantees that particles entering the Hill sphere collide
with the wake, allow us to do the following simplification of the Greenberg et
al. (1991) formulation:

  * •

$f$, the fraction of particles that collide after entering the Hill Sphere, is
1.

  * •

Tadpole and horseshoe orbits are excluded from the colliding particles by
using the three-body code to determine $b_{\mathrm{min}}$. We find that
$b_{\mathrm{min}}$ varies from $35\,\mathrm{m}$ to $43\,\mathrm{m}$ with the
different orientations of the wake (from radially aligned to the azimuthally
aligned), with a mean value of $b_{\mathrm{min}}=38\,\mathrm{m}$, which we use
as a constant $b_{\mathrm{min}}$ for the two-body simulation.

  * •

We define $b_{\mathrm{max}}$ as the solution of
$r_{2B}(b_{\mathrm{max}},\theta_{z^{\prime}})=\frac{L}{2}$

The collisional torque then becomes:

$\tau_{\mathrm{Kep};z^{\prime}}=-2(1+\epsilon)\int^{b_{\mathrm{max}}}_{b_{\mathrm{min}}}\left|v_{\theta}+\omega_{z^{\prime}}r_{2B}\right|(v_{\theta}+\omega_{z^{\prime}}r_{2B})r_{2B}\int^{\frac{H}{2}}_{-\frac{H}{2}}\rho_{s}(z)dzdb$
(12)

where we have replaced the incoming speed in Eqn. (10) ($q\Omega b$) with
$|v_{\theta}|$ and used $\int^{H/2}_{-H/2}\rho_{s}(z)dz$ for $\sigma$.
Additionally, by adding $\omega_{z^{\prime}}r_{2B}$, we allow for the rotation
of the wake to affect the incoming relative speed of the particles. This
integral is solved numerically since we have to determine $b_{\mathrm{max}}$
every time the pitch angle $\theta_{z^{\prime}}$ changes. For a Keplerian
shear rate, we find that this formulation equilibrates stably with the tidal
force (Eqn. 6) at $\theta_{z^{\prime}}=25^{\circ}$ for
$\rho_{s}=500\,\mathrm{kg/m^{3}}$ and $\epsilon=1$. Therefore Eqn. (9), with
the Keplerian shear torque given by Eqn. (12), meets the boundary condition
for the orientation of the wake outside of the wave. We proceed to explore the
torques and the motion of the wake’s inside the BW region.

####  2.1.3 BW acceleration torque

The main torque that causes the wakes’ rotation in the BW region stems from
the difference between the acceleration of the CM and that of a wake particle
a radial distance $\Delta x$ from the CM; we write this as
$\ddot{z}_{\mathrm{CM}}-\ddot{z}(x)$. Given that all the torques are derived
in the frame of the CM of the wake, differences in acceleration will appear as
a force. To compute this force we take two time derivatives of Eqn. (5), which
denotes the vertical position of a ring particle as governed by the
interactions in SCL theory; namely, the self-gravity of the bent ring, the
gravitational force of Saturn, and a viscous force. Note that the time
dependence of Eqn. (5) lies entirely in the exponential factor
$e^{i\omega^{\prime}t}$, and, given the $\sim 10\,\mathrm{km}$ wavelength and
the tightly-wound nature of the wave, the change of $z(x,\Theta)$ over a wake
can be approximated linearly on $\Delta x$. The difference in vertical
acceleration can then be expressed as:

$\ddot{z}_{\mathrm{CM}}-\ddot{z}=-\omega^{\prime
2}({z}_{\mathrm{CM}}-{z})\approx-\omega^{\prime
2}\left.\frac{dz}{dx}\right|_{\mathrm{CM}}\Delta x$ (13)

Let’s consider the rotation about the long axis (the $y^{\prime}$ axis). We
need to find the directional derivative of $\ddot{z}(x)$ in the directions
perpendicular to $y^{\prime}$. Given the flattened nature of self-gravity
wakes (Colwell et al., 2006; Hedman et al., 2007; Jerousek et al., 2016),
variation of forces across the vertical axis, $H$, will be ignored and only
variations along $x^{\prime}$ and $y^{\prime}$ will be considered. Then for
the torque about $y^{\prime}$ we only need the variation of $\ddot{z}$ along
the $x^{\prime}$ axis. The force then becomes:

$F_{z}(x^{\prime})=-\omega^{\prime
2}\left.\frac{dz}{dx}\right|_{\mathrm{CM}}(\hat{x}\cdot\hat{x^{\prime}})x^{\prime}dm$
(14)

where $dm$ is a differential mass a distance $x^{\prime}$ from the CM.
Projecting it onto $\hat{z}^{\prime}$ and crossing it with the lever arm we
get:

$\vec{\tau}_{\mathrm{BWacc},y^{\prime}}=\vec{x}^{\prime}\times\vec{F_{z^{\prime}}}=-2\omega^{\prime
2}\left.\frac{\partial z}{\partial
x^{\prime}}\right|_{CM}(\hat{z}\cdot\hat{z}^{\prime})(\hat{x}^{\prime}\times\hat{z}^{\prime})\int^{\frac{W}{2}}_{0}\rho_{\mathrm{Roche}}HLx^{\prime
2}dx^{\prime}$ (15)

where we substituted $dm=\rho_{\mathrm{Roche}}H*L*dx^{\prime}$ and the
directional derivative along $x^{\prime}$ defined as
$\frac{dz}{dx}\Bigr{|}_{\mathrm{CM}}(\hat{x}\cdot\hat{x}^{\prime})\equiv\frac{\partial
z}{\partial x^{\prime}}\Bigr{|}_{CM}$.

####  2.1.4 BW shear torque

The wake’s CM vertical speed will differ from that of the neighboring
particles and hence there will be a collision which will generate a torque.
This is similar to the Keplerian torque, and, for the general case, the
vertical and horizontal speeds are added to compute a total collisional torque
(see Appendix A). If $\theta_{x^{\prime}}$ and $\theta_{y^{\prime}}$ are
small, however, these two torques are independent. To model the BW shear
torque we consider the shear $\frac{d\dot{z}}{dr}$ and note that the particles
a distance $y^{\prime}$ along the long axis will have a relative vertical
velocity of $\frac{d\dot{z}}{dr}(\hat{x}\cdot\hat{y^{\prime}})$ with respect
to the CM. If these particles have a space density $\rho_{s}$ then the amount
of mass colliding with the wake at a distance $y^{\prime}$ in a time $dt$ will
be:

$M(r)=\rho_{s}\left|\frac{\partial\dot{z}}{\partial
y^{\prime}}\Bigr{|}_{CM}y^{\prime}dt\right|Wdy^{\prime}$

which causes a momentum transfer of:

$dp_{z^{\prime}}(y^{\prime})=[M]*[\Delta
v_{z^{\prime}}]=[\rho_{s}\left|\frac{\partial\dot{z}}{\partial
y^{\prime}}\Bigr{|}_{CM}y^{\prime}\right|dtWdy^{\prime}]*[(1+\epsilon)\frac{\partial\dot{z}}{\partial
y^{\prime}}\Bigr{|}_{CM}y^{\prime}]$

This momentum transfer corresponds to a force of:

$F_{\mathrm{BWsh};x^{\prime}}=\frac{dp_{z^{\prime}}}{dt}=(1+\epsilon)\rho_{s}\left|\left(\frac{\partial\dot{z}}{\partial
y^{\prime}}\Bigr{|}_{CM}-\omega_{x^{\prime}}\right)y^{\prime}\right|\left(\frac{\
\partial\dot{z}}{\partial
y^{\prime}}\Bigr{|}_{CM}-\omega_{x^{\prime}}\right)Wy^{\prime}dy^{\prime}$
(16)

where we have included $\omega_{x^{\prime}}$, the angular velocity about the
$x^{\prime}$-axis, to account for the change in the relative incoming speed
that occurs when the wake rotates. The torque associated with this momentum
transfer is:

$\tau_{\mathrm{BWsh};x^{\prime}}=2\int^{\frac{L}{2}}_{0}{y^{\prime}\frac{dp_{z^{\prime}}}{dt}}=\frac{1+\epsilon}{2}\rho_{s}W\left|\frac{\partial\dot{z}}{\partial
y^{\prime}}\Bigr{|}_{CM}-\omega_{x^{\prime}}\right|\left(\frac{\partial\dot{z}}{\partial
y^{\prime}}\Bigr{|}_{CM}-\omega_{x^{\prime}}\right)\left(\frac{L}{2}\right)^{4}$
(17)

where we have taken the wake to be much more massive than the incoming ring
particles so that the CM of the collision is the CM of the wake. Doing the
same computation for $\tau_{\mathrm{vert};y^{\prime}}$, we can arrive at the
total torques about the $x^{\prime}$ and $y^{\prime}$ axis, now written as a
function the angle $\theta_{z^{\prime}}$ and radial distance $x$. For small
angles $\theta_{x^{\prime}}$ and $\theta_{y^{\prime}}$ we have:

$\tau_{x^{\prime}}\approx-\frac{2}{3}\rho_{\mathrm{Roche}}HW\frac{d\ddot{z}}{dx}\Biggr{|}_{CM}\sin{\theta_{z^{\prime}}}\left(\frac{L}{2}\right)^{3}+\rho_{s}W\frac{1+\epsilon}{2}\left|\frac{d\dot{z}}{dx}\Biggr{|}_{CM}\sin{\theta_{z^{\prime}}}-\omega_{x^{\prime}}\right|\left(\frac{d\dot{z}}{dx}\Biggr{|}_{CM}\sin{\theta_{z^{\prime}}}-\omega_{x^{\prime}}\right)\left(\frac{L}{2}\right)^{4}$
(18)

$\tau_{y^{\prime}}\approx\frac{2}{3}\rho_{\mathrm{Roche}}HL\frac{d\ddot{z}}{dx}\Biggr{|}_{CM}\cos\theta_{z^{\prime}}\left(\frac{W}{2}\right)^{3}+\rho_{s}L\frac{1+\epsilon}{2}\left|\frac{d\dot{z}}{dx}\Biggr{|}_{CM}\cos{\theta_{z^{\prime}}}+\omega_{y^{\prime}}\right|\left(\frac{d\dot{z}}{dx}\Biggr{|}_{CM}\cos{\theta_{z^{\prime}}}+\omega_{y^{\prime}}\right)\left(\frac{W}{2}\right)^{4}$
(19)

where, in this approximation, $\theta_{z^{\prime}}$ is the pitch-angle of the
wake.

####  2.1.5 Motion of an isolated self-gravity wake

Real wakes consist of many transient objects that can gravitationally
interact. However, to understand the underlying dynamics within a BW, it is
worth considering the simpler motion of an isolated wake subjected to the
torques computed thus far; later, we will include the effects of surrounding
wakes. Unlike the case of the flat ring, inside the BW the wakes are
continuously perturbed via out-of-the-plane torques; thus, there is no torque
equilibrium and we must integrate the equations of rigid motion (Eqns. 1, 2,
and 3) These equations are numerically integrated with a midpoint method
(Press et al., 2007) (differential rotation is commutative, so the order of
rotation does not affect the integration).

Note that the inertial terms

$\tau_{\mathrm{inertia}_{k}}=(I_{i}-I_{j})\omega_{i}\omega_{j}$ (20)

(where $i\neq j\neq k$ and $i,j,k=x^{\prime},y^{\prime},z^{\prime}$) will
prove important in understanding the wake’s rotation since they couple the
rotations about the different axis.

In Figure 8 we describe the rotation of the wakes in the BW, for a maximum
slope of $|dz/dx|_{\mathrm{max}}=14^{\circ}$. The top panel shows the change
in the wake’s orientation, the middle panel shows the magnitude of the
torques, and the bottom panel shows the vertical impact speeds. To calculate
Figure 8 we relaxed the approximation that $\theta_{x^{\prime}}$ and
$\theta_{y^{\prime}}$ are small, which amounts to computing all 4 torques
symmetrically for the 3 axes. The full form of the torques can be found in
Appendix A.

In order to describe the orientation of the wake in inertial space, we have
introduced the angles $(\theta_{x},\theta_{y},\theta_{z})$ (upper panel),
which correspond to the wakes’ ‘pitch’, ‘roll’, and ‘yaw’, respectively.
$\theta_{x}=\cos^{-1}{(\hat{x}^{\prime}\cdot\hat{z})}$ and
$\theta_{y}=\cos^{-1}{(\hat{y}^{\prime}\cdot\hat{z})}$ increase as the
$x^{\prime}$ and $y^{\prime}$ axes, respectively, point towards the vertical
direction.
$\theta_{z}=\tan^{-1}\left(-\hat{y}^{\prime}\cdot\hat{x}/\hat{y}^{\prime}\cdot\hat{y}\right)$
is $0$ when the $y^{\prime}$-axis, projected onto the ring’s plane, points at
the azimuthal and it increases counterclockwise. In the specific
configurations of the wakes in Figure 5, these angles are equivalent to their
corresponding primed counterparts. Moreover, if $\theta_{x^{\prime}}$ and
$\theta_{y^{\prime}}$ are small, then $\theta_{z^{\prime}}\approx\theta_{z}$.

In the top panel of Figure 8 we show the motion of a wake when the slope of
the wave is constant and zero (black line). We find that the wake equilibrates
stably at an angle $\theta_{z}=25^{\circ}$. However, when the slope is
changing, the orientation of the wake becomes significantly more complex
(colored lines in the top panel). The sole difference causing this divergence
in behavior is the introduction of a slope via the BW acceleration and shear
torques—even if this slope is relatively small
($|dz/dx|_{\mathrm{max}}=\tan{14^{\circ}}$). The dependence on the small slope
and the forcing frequency ($\omega^{\prime}$) makes the BW torques (on
average) two orders of magnitude smaller than the Keplerian and the tidal
torques (middle panel), but the inclinations they cause provoke a non-linear
behavior in the angular variables
$(\theta_{x^{\prime}}$,$\theta_{y^{\prime}}$, $\theta_{z^{\prime}})$ which are
strongly coupled. The coupling mainly comes from the inertial term.

Since the BW torques are small, the inclination of the wake comes from the
action of this inertial term Eqn. 20), but the inertial term is non-zero only
due to the presence of the BW torques. Therefore, the smallest torques in
magnitude, namely the ones introduced by the BW, have the biggest effect on
the dynamics. The wakes no longer stay on the plane of the ring simply
oscillating about an equilibrium orientation; rather, they spin full circles
and reach inclinations of $80^{\circ}$ from the equatorial plane for the case
of a maximum slope of $14^{\circ}$ (this occurs at the vertical dashed line).

The bottom panel of Figure 8 shows the relative vertical speed between the
wake and the ring particles. These are computed as

$v_{\mathrm{rel};z^{\prime}}=[\frac{L}{2}(-\omega_{x^{\prime}}+\frac{d\dot{z}}{dx})+\frac{W}{2}*(\omega_{y^{\prime}}+\frac{d{\dot{z}}{dx}}{)}](\hat{z^{\prime}}\cdot\hat{z})+\frac{L}{2}\omega_{x^{\prime}}(\hat{x}^{\prime}\cdot\hat{z})$
(21)

where the dominant terms are the projections of the rotational speeds onto the
$z$-axis. The relative vertical velocities within the wake and the ring
particles associated with this motion are in the $\mathrm{cm/s}$ range (bottom
panel of Figure 8). Note, however, that we are still to account for the
gravitational interactions between self-gravity wakes, which, as we will see,
can have a stabilizing effect on the pitch-angle of the wakes.

Figure 8: Angular motion (top panel), torques (middle panel), and speed
relative to the ring (bottom panel) of a uniform density self-gravity wake
satellite evolved integrating the equations of rigid motion (Eqns. 1, 2 and 3)
and the 3D Hill equations (Eqn. 7).$\theta_{z}=0$ when the long axis
($y^{\prime}$), projected onto the plane, points to the azimuthal,
$\theta_{x}=0$ and $\theta_{y}=0$ when the $x^{\prime}$ and $y^{\prime}$ axes,
respectively, lie in the rings’ plane and $\theta_{x},\theta_{y}90^{\circ}$
when they point in the vertical direction. The wake has uniform density and
dimensions $W=18\,\mathrm{m}$, $L=232\,\mathrm{m}$ and $H=4\,\mathrm{m}$, and
the coefficient of restitution $\epsilon$ and the space density $\rho_{s}$ of
the ring are set to balance the torques outside of the wave at a pitch-angle
is $25^{\circ}$ (see Figure 4 and related discussion). The maximum slope is
set to $14^{\circ}$ and the slope is $0$ at $t=0$, and varies harmonically
with the forcing frequency $\omega^{\prime}$ (see Eqn. 4). For comparison, we
also show the time evolution of the orientation of a wake on a flat ring
(black dashed line). Top panel: the wake inclines back and forth as the wave
passes through, and its long axis spins around in the plane until it reaches a
steady state where the wake oscillates about the azimuthal direction. Marked
with a dashed gray line is the point where the motion becomes too extreme,
given that $\theta_{x}=80^{\circ}$, and given that the wake points completely
in the radial direction. Middle panel: the torques associated with the
displayed motion. The inertial torque becomes important, and the BW
acceleration torque is on average the smallest one by two orders of magnitude.
Bottom panel: z-component of the relative velocities between the wake and a
particle colliding with the edge of the wake.

####  2.1.6 Torque due to self-gravity wakes

For the gravitational force that self-gravity wakes exert on each other we
consider the case of a wake with variable orientation embedded on a field
self-gravity wakes oriented at a constant $\theta_{z^{\prime}}=\theta_{w}$.
Taking this field as a set of density plane waves, we can use Pringle and
Lyden-Bell equations for the torque of a spiral arm on a galaxy (Lynden-Bell &
Pringle, 1974), which have been applied to self-gravity wakes in proto-lunar
disks in Takeda & Ida (2001). The potential of the spiral arms is given in
Binney & Tremaine (2008) in terms of Saturncentric distance $r$ and longitude
$\Theta$ as:

$\Phi^{\prime}=\frac{2\pi G}{|k|}\sigma_{w}Re[e^{i(m_{w}\Theta+k_{r}r)}]$ (22)

where $k=\frac{2\pi}{\lambda_{T}}$ and $m_{w}$ and $k_{r}$ are the azimuthal
and radial wavenumbers, and $\sigma_{w}$ is the wake surface density minus the
mean surface density of the ring.

Following the appendix in Takeda & Ida (2001), we relate the wavenumbers by
$m_{w}=kr\sin\theta_{w}$ and $k_{r}=k\cos\theta_{w}$, where
$\theta_{w}=25^{\circ}$ is the pitch-angle. To compute $\sigma_{w}$ we will
approximate the wake surface density to be
$\rho_{\mathrm{Roche}}\int^{\infty}_{-\infty}e^{-(\frac{z}{H/2})^{2}}dz\rho_{\mathrm{Roche}}\frac{\sqrt{\pi}}{2}H$,
noting that the mean surface density $\sigma$ for the A ring region is about
$400\,\mathrm{kg/m^{2}}$ (Tiscareno et al., 2007) and has a negligible effect
on $\sigma_{w}$. We will now expand the potential (Eqn. 22) about the center
of a wake located at $(r_{0},\Theta_{0}=y_{0}/r_{0})$ where $y_{0}=0$, similar
to what is done in Cook & Franklin (1964). The potential then becomes:

$\Phi^{\prime}=\frac{2\pi
G}{|k|}(\rho_{\mathrm{Roche}}\frac{\sqrt{\pi}}{2}H)(1-k^{2}(\cos{\theta_{w}}(r-r_{0})+\sin{\theta_{w}}y)^{2})$
(23)

To write the potential in terms of the orientation of the wake
$\theta_{z^{\prime}}$ (in the approximation where $\theta_{x^{\prime}}$ and
$\theta_{y^{\prime}}$ are small), we consider the distance about the long
axis, $y^{\prime}$, and write $r-r_{0}=-y^{\prime}\sin(\theta_{z^{\prime}})$
and $y=y^{\prime}\cos(\theta_{z^{\prime}}$)

This yields the potential

$\Phi^{\prime}=\frac{2\pi
G}{|k|}(\rho_{\mathrm{Roche}}\frac{\sqrt{\pi}}{2}H)[1-k^{2}y^{\prime
2}\cos^{2}{(\theta_{z^{\prime}}-\theta_{w})}]$

We now apply the
$-\frac{1}{y^{\prime}}\frac{\partial}{\partial\theta_{z^{\prime}}}$ operator
to the potential and multiply by the differential mass times the lever arm
($dm\cdot y^{\prime}=\rho_{\mathrm{Roche}}HWdy^{\prime}\cdot y^{\prime}$) to
get the torque at distance $y^{\prime}$ from the center of the wake; we then
proceed to integrate over $y^{\prime}$ and get:

$\tau_{\mathrm{wake};z^{\prime}}=\frac{4\pi^{2}}{\lambda_{T}}G\rho_{\mathrm{Roche}}\frac{\sqrt{\pi}}{2}H^{2}W\frac{L^{3}}{24}*\sin{2(\theta_{w}-\theta_{z^{\prime}})}$
(24)

which changes the equation of the torque about the z-axis to:

$\tau_{z^{\prime}}=\tau_{\mathrm{tidal};z^{\prime}}(\theta_{z^{\prime}})+\tau_{\mathrm{Kep};z^{\prime}}(\theta_{z^{\prime}},\rho_{s})+\tau_{\mathrm{wake};z}(\theta_{z^{\prime}})$
(25)

The torques in the z-axis now equilibrate at an angle of
$\theta_{z^{\prime}}=25^{\circ}$ for $\rho_{s}=390\,\mathrm{kg/m^{3}}$ for the
full Hill equations computation and $\epsilon=1$. The equilibrium orientation
of the wake is now less sensitive to the surrounding space density since the
wake potential creates a minimum when wakes align (note that Eqn. 24 vanishes
when $\theta_{z^{\prime}}=\theta_{w}$). This may explain why the pitch-angle
of the wakes is nearly uniform over the rings (Jerousek et al., 2016) even if
the density and the shear rate vary between the A and the B rings. We are
assuming the coherent structure of $\theta_{w}=25^{\circ}$ is kept within the
wave, but the wakes may not have a coherent orientation or well-defined
wavenumbers, which is suggested by the reduction of the number of coherent gap
structure in UVIS occultations to $25\%$ the outside value (Rehnberg et al.,
2017) and by our equations of motion which predict that the wakes can spin up
or oscillate about the azimuthal; in which case Eqn. (24) would be modified by
the wakes that are not part of the plane-wave field. While we lack an
expression for the case of neighboring wakes with multiple orientations, we
will heuristically extend the case of a single misaligned wake embedded in a
wake field with orientation $\theta_{w}$, to the case of a misaligned wake in
the BW region embedded in the oriented wake field of the A-ring. The total
wake potential acting on a wake is then given by Eqn. (22).

####  2.1.7 Motion of an embedded self-gravity wake

The relative velocity and orientation of a self-gravity wake embedded on a
field of self-gravity wakes with fixed orientation $\theta_{w}$, evolved under
Eqns. (1), (2), and (3), is plotted in Figure 9.

Figure 9: Time evolution of the average velocities of particle collisions with
self-gravity wakes within the bending wave; motion evolved as in Figure 8 with
the addition of the wake potential. The maximum slope is set to $14^{\circ}$
and $t=0$ when the slope is $0$; the slope varies harmonically with the
vertical frequency. Adding the wake potential increases the maximum relative
velocity between the wakes and ring particles. For the case of
$L=232\,\mathrm{m}$ we see vertical velocities well within the $\mathrm{cm/s}$
range, while the $L=77\,\mathrm{m}$ case peaks at about $1\,\mathrm{cm/s}$.

We see a similar non-linear behavior as the isolated wake (Figure 8) resulting
in $\mathrm{cm/s}$ collisional speeds (Figure 9), but the values of
$\theta_{z}$ are more conservative—oscillating about $\theta_{z}=0$ and
staying below $90^{\circ}$ for most of the motion. The restoring force to the
equilibrium angle has now increased, and this makes the vertical collisional
speeds — which are coupled to $\theta_{z^{\prime}}$ by the inertial term (Eqn.
20) — increase as the wake swings back into position.

The extreme values for the orientation angles, particularly $\theta_{z}$ which
at times has the long axis pointing in the radial direction, make us skeptical
that the rigidity assumption holds throughout all of the motion. To remedy
this we also computed the motion of a wake with long axis $L=77\,\mathrm{m}$,
a third of the usual length. Note that we are not suggesting that self-gravity
wakes will shed two thirds of their mass (that would create a much bigger haze
signal than the one observed), rather we are verifying that even if a wake
were to shed the regolith near its edges while rotating, it will still
maintain collisional speeds in the $\mathrm{cm/s}$ range. Hence, the process
is not self-limiting, nor does it necessitate a wake of long axis
$L=232\,\mathrm{m}$.

The rapid change of $\theta_{z}$ for the $L=232\,\mathrm{m}$ case also makes
us consider the possibility of the release of material due to rotation (e.i
rotational disruption). Note that in this scenario, the launched regolith
itself will exit the wave at $\mathrm{cm/s}$ speeds, hence contributing to the
haze. In the next section we will show that both, the rotational disruption
and collisional ejecta scenarios, create a haze with the same observable
features.

###  2.2 Haze predicted properties

The relative velocity curves in Figures 8 and 9 are of the order of
$1\,\mathrm{cm/s}$. Microgravity experiments have shown that at
$\mathrm{cm/s}$ velocities we begin to see ejecta in particle collisions with
simulant regolith aggregates (Brisset et al., 2018). The $\mathrm{cm/s}$ range
is also important because the characteristic vertical speeds of the Mimas 5:3
BW are in the $\mathrm{cm/s}$ range: a $\mathrm{cm/s}$ change in velocity is
necessary to have particles reset their motion into trajectories that are no
longer in phase with the wave.

If these collisional speeds increase with slope amplitude, the amount of
released material would change with radial position. Figure 10 shows precisely
this: the peaks of the relative speed over a period have a monotonically
increasing trend with the amplitude of the slope of the wave. To determine how
the amount of material changes due to this increase in the velocities, we
apply a square-root regression to Figure 10 and find that:

Figure 10: Average maximum relative speed of self-gravity wakes for different
slopes. The average is taken over a range of initial conditions to show that
the increase of collisional speed with the slope is robust to initial
conditions. At each value of the slope considered, we start the wake out-of-
phase (at an angle with the ring) by $0$, $\pm 10$, $\pm 20$, and $\pm 30$
degrees at $15$ equidistant points in the wave’s vertical motion and run the
simulation for 1 period. Therefore, each point is an ensemble average of $105$
runs, where the error bars represent the $1\sigma$ of the distributions. We
see a clear trend: the bigger the slope amplitude of the BW the bigger the
velocity of the collisions between the wakes and the ring particles. We model
this increase with a square-root law.

$v_{\mathrm{rel};z}=a\sqrt{\left|\frac{dz}{dx}\right|_{\mathrm{max}}}+w$ (26)

where $a=2.42\,\mathrm{cm/s}$ and $w=0.47\,\mathrm{cm/s}.$

Microgravity experiments also show the existence of a speed threshold before
which the collisions do not lift ejecta (Brisset et al., 2018). Once this
minimum threshold has been passed, numerical simulation of collisions in free
space (Stewart & Leinhardt, 2009) and drop tower experiments (Colwell et al.,
2008) suggest that the amount of ejected regolith increases linearly with the
specific kinetic energy of the impactor; we represent this by the equation:

$N_{\mathrm{haze}}\propto\mathcal{H}(v_{\mathrm{impact}}-g)\cdot(v_{\mathrm{impact}}-g)^{2}$
(27)

where $N_{\mathrm{haze}}$ is the number of haze particles liberated per wake,
$g$ is the impact speed threshold, and $\mathcal{H}$ is the Heaviside function
whose value is $0$ if the argument is negative, $1$ otherwise.

We do not know where this threshold is for the case of wakes in the rings, but
it must be higher than the dispersion velocity ($\sim\Omega
z_{0}=0.13\,\mathrm{cm/s}$), since this does not generate sub-mm ejecta
outside of the BW (Table 1). From now on we heuristically assume that $g=w$ so
that $N$ is proportional to the slope amplitude. Misestimating the value for
$g$ by a factor of $2$ would change the quantity of haze produced at low
slopes ($<$$0.03$) by a factor of $\sim 4$; however, more than $90\%$ of the
wave lies in the high-speed regime where the square-root trend works well.
Given this value of $g$, and assuming a set number of wakes per area, the
amount of haze particles per area ($\Sigma_{\mathrm{haze}}$) has the
proportionality of:

$\Sigma_{\mathrm{haze}}\propto\left|\frac{dz}{dx}\right|_{\mathrm{max}}$ (28)

Note that the number density of the haze, $n_{\mathrm{haze}}$, is given by
$\frac{\Sigma_{\mathrm{haze}}(r)}{D(r)}$ where $D(r)$ is the vertical
thickness of the haze. Combining Eqns. (26) and (28) and the equation for the
differential optical depth
$d\tau_{\mathrm{haze}}(r)=n_{\mathrm{haze}}\bar{\sigma}_{\mathrm{haze}}dl$,
where $\bar{\sigma}_{\mathrm{haze}}$ is the typical cross-section of particles
in the haze and $dl$ is a differential path-length through the haze, we arrive
at the relation:

$d\tau_{\mathrm{haze}}(r)=\frac{\beta}{D}\left|\frac{dz}{dx}\right|_{\mathrm{max}}dl$
(29)

where $\beta=\Sigma_{0}\bar{\sigma}_{\mathrm{haze}}$ is the normal optical
depth of the haze when the slope is unity, and a free parameter in our model;
$\Sigma_{0}$ is the surface density generated by a slope of unity.

For the case of rotational disruption, we take the number of released haze
particles per wake to be proportional to the centrifugal force due to the
rotation: $N_{\mathrm{haze}}\propto\omega_{x^{\prime}}^{2}r_{p}$ where
$\omega_{x^{\prime}}$ is the angular speed and $r_{p}$ is the radial distance
from the CM. Note that the angular speed and $v_{\mathrm{rel};z}$ are related
by $\omega_{x^{\prime}}\approx v_{\mathrm{rel};z}/r_{p}$; combining this with
Eqn. (26), we again arrive at Eqn. (29). Therefore, given our rigid model for
self-gravity wakes, we predict that in the BW region the extra optical depth
must be proportional to the maximum slope of the wave. Note that $\beta$ is a
dimensionless absorption per slope coefficient related to the ejecta-
generating efficiency of rotation or collisions.

To construct the geometry of the haze we consider the trajectories of the
particles after the collision with the self-gravity wake. Given that in Figure
9 $>$$1\,\mathrm{cm/s}$ collisional speeds are achieved at many points during
a period, we consider that the collisions with the self-gravity wakes are
equally likely to occur at any point in the vertical motion of the particles.
Moreover, we consider the imparted velocity to be in equal proportions radial
and vertical with a magnitude of $a\sqrt{|\frac{dz}{dx}|_{\mathrm{max}}}+w$
(where $a$ and $w$ are defined in Eqn. 26). The subsequent trajectories can be
easily computed by taking the CM of the collision to be the center of the
self-gravity wake. Let $V_{\mathrm{CM}}$ be the velocity of the CM, then the
outgoing speed will be given by

$v_{0}=V_{\mathrm{CM}}(1+\epsilon)\pm\epsilon(a\sqrt{\left|\frac{dz}{dx}\right|_{\mathrm{max}}}+w)$
(30)

the case of $\epsilon=1$ representing the outgoing speed for the rotational
disruption scenario. Relative to a particle in the wave the ejected particles
move vertically with a Doppler shifted frequency of $\mu_{V}\pm
k_{b}v_{0_{x}}$, where $\mu_{V}$ is the vertical frequency of the particle and
$k_{b}$ is the wavenumber of the BW. The subsequent trajectories are shown in
the left panel of Figure 11. In the right panel of Figure 11 we show the
envelope these trajectories form over the radial extent of the wave; note that
the thickness of the haze varies radially and is greater at the peaks and
troughs than at the equatorial plane. Moreover, the haze’s thickness varies
with $\epsilon$, and we find it to range from ${90-190}\,\mathrm{m}$ for
$\epsilon={0.1-1}$.

While there’s phase dependence in the haze’s geometry (Figure 11), the
dynamics predict the same number of particles at any given phase for a given
radial location $x$. Hence, the change in the optical depth profile of a
normal occultation can only come from the change of scattered particles in the
collisions, which is related to the change in the slope amplitude with $x$. We
then predict that the phase should not appear in the haze signal, which is a
desired feature of the model, given that the observed signal of
$B_{\mathrm{eff}}\cong 90^{\circ}$ occultations show no correlation with the
phase of the wave.

Figure 11: Left panel: trajectories in time of the vertical position of haze
particles (blue) and the oscillating ring (red) at $-100\,\mathrm{km}$ from
resonance. Here the trajectories of haze particles start every phase interval
of $\frac{\pi}{3}$ during the up-and-down motion (so we are modeling 6
collisions per period). The ring has a continuous envelope of particles during
the period. Right panel: the envelope created in space by the trajectories of
particles post-collision using $\epsilon=0.5$. By looking at the lower left
insert panel one can see the slight variation of the vertical thickness
present in the haze profile; the thickness of the haze changes from $113$ to
$134$ m in the overall plot, and the variations in the zoom-in panel are only
of $4\,\mathrm{m}$ (less than a thick mark). A light ray has been plotted in
dashed gray to visualize the geometry of the occultations.

##  3 Transmission Model and Data Reduction

The transmission of light through the rings is modeled by the radiative
transfer equation simplified by the fact that the opacity $\kappa$ of the ring
particles is insensitive to wavelength. Moreover, we are working exclusively
in the $110-190\ \mathrm{nm}$ spectral bandpass of the High Speed Photometer
(HSP) within UVIS. Given the lower limit in the $100\,\mathrm{\mu m}$ range
for the particle size in the A-ring (Harbison et al., 2013), there is no
scattering; thus, we are firmly within the geometrical optics limit. As a
consequence of this, the intensity $I$ measured by UVIS is simply:

$I(r)=I_{0}e^{-\frac{\tau_{n}}{\mu}}+b$ (31)

where $I_{0}$ is the unocculted star signal, $b$ is background light entering
the detector that does not come from the star, $\tau_{n}$ is the normal
optical depth, and $\mu=\sin(B)$ where $B$ is the angle between the ring plane
and the light ray. Or, in terms of the transparency $T$

$T=\frac{I-b}{I_{0}}$ (32)

We will extract the observed optical depth by using:

$\ln{\frac{I(r)-b}{I_{0}}}=-\frac{\tau_{n}}{\mu}$ (33)

In order to isolate the star signal we determine the background signal ($b$)
and the unocculted star signal ($I_{0}$) following the methods of Colwell et
al. (2010). It is known that the detector becomes more sensitive over time
when the optical depth is close to $0$ (Colwell et al., 2010), which
introduces a systematic uncertainty on our measurements of $I_{0}$ that we are
not considering here. Since $I_{0}$ refers to the star, we focus on the
properties of the rings themselves by working with the transparency ($T$) and
the optical depth ($\tau$).

To the $217$ occultations of the BW region we apply a signal-to-noise filter,
or conversely a $\tau_{\mathrm{max}}$ filter (Colwell et al., 2010) of
$\tau_{\mathrm{max}}>1.5$ so that the variations in $\tau$ due to the BW are
distinguishable; this reduces the set of occultations to $130$ (see Appendix B
for the complete list). From this we select $60$ occultations with the
following criteria:

First, we maximize the range of geometries probed by our study without
oversampling any particular geometry. Six subsets are created where $B$ can be
low angle, $0^{\circ}$ to $30^{\circ}$, intermediate angle, $30^{\circ}$ to
$60^{\circ}$, or normal, $60^{\circ}$ to $90^{\circ}$. On the other hand,
$\phi$ can be radial ($|\cos{\phi}|>\frac{1}{2}$) or azimuthal
($|\cos{\phi}|<\frac{1}{2}$). We use $|\cos{\phi}|$ to partition the data
because it is the physically relevant quantity that appears in all the models
that will be tested. The smallest subset, being the low $B$ radial set, has a
total of $10$ occultations in the entire dataset, so all other subsets must
also contain $10$ occultations to not oversample any geometry, yielding $60$
occultations in total.

Secondly, in the subsets with more than $10$ occultations we select the
highest $\tau_{\mathrm{max}}$ occultation for each star, and if there are
fewer than $10$ unique stars we select the next highest $\tau_{\mathrm{max}}$
for each star until $10$ are reached. The final dataset is shown in Table 2.

Additional critical geometric parameters consist of the longitude of the
occultation in the rings relative to the moon Mimas, the vertical position of
Mimas, and the vertical velocity of Mimas. These parameters were determined
using the Navigation and Ancillary Information Facility (NAIF) SPICE toolkit
(Acton, 1996; Acton et al., 2018). These parameters generally depend on the
precise determination of Saturn’s pole, the ring plane radius being the most
uncertain parameter with errors of $<10\,\mathrm{km}$ for every occultation.
Errors in $B$ and $\phi$ are generally less than $1^{\circ}$. Knowing the
position and velocity of Mimas enables us to compute the phase of the wave
using equation 47a in SCL:

$\Phi_{\mathrm{theory}}=(4\Omega_{M}+\mu_{M})t-4\Theta-\frac{\pi}{4}$ (34)

where $\Omega_{M}$ and $\mu_{M}$ are the Keplerian and vertical frequencies of
Mimas and $\Theta$ is the azimuthal position of the occultation ($t=0$ at the
ascending node of Mimas). The gravitational moments used to determine the
frequencies were taken from Iess et al. (2019), while the other physical
parameters of Saturn and Mimas (mass, semi-major axis of Mimas, and
inclination of Mimas) were taken from Jet Propulsion Laboratory Horizons
database (Jacobson, 2010); these are presented in Table 3. We find a
theoretical value for the 5:3 vertical resonance location of
$r_{V}=131902\,\mathrm{km}$ and a wave amplitude of
$A_{V}=\frac{472\mathrm{m}}{\sqrt{\sigma_{100}}}$ where $\sigma_{100}$ is the
surface mass density in $100\,\mathrm{g/cm^{2}}$.

An alternative way of computing the phase is through observations. From the
position of the optical depth peaks and troughs in the data a phase can be
determined by fixing Gresh et al. (1986)’s best-fit parameters of the SCL
model—surface density $\sigma$ and viscosity $\nu$—while letting the phase
vary and applying a $\chi^{2}$ minimization scheme. This data-driven value for
the phase is reported in Table 2 as $\Phi_{\mathrm{obs}}$.

Occultation | $B\,\textrm{[deg]}$ | $\phi\,\textrm{[deg]}$ | $B_{\mathrm{eff}}\,\textrm{[deg]}$ | Lon. [deg] | $Z$ | $\textrm{sign}(\dot{Z})$ | $\Phi_{\mathrm{theory}}\textrm{[deg]}$ | $\Phi_{\mathrm{obs}}\textrm{[deg]}$ | $b$ | $I_{0}$ | $\tau_{\mathrm{max}}$
---|---|---|---|---|---|---|---|---|---|---|---
$B>60^{\circ}$ and $|\cos{\phi}|>\hskip 142.26378pt$Complete subset size = 10 occultations |
AlpCru(100)I | 68.2 | 154.5 | 70.1 | 213.6 | -5016.5 | 1 | 100 | - | 0.50 | 434.18 | 6.0
AlpCru(92)I | 68.2 | 169.6 | 68.5 | 114.7 | -4232.1 | 1 | 160 | - | 0.50 | 518.56 | 6.3
EpsCas(104)I | -70 | 187.8 | 70.1 | 152.8 | -2034.7 | -1 | 87 | - | 0.04 | 4.59 | 3.8
BetCru(98)I | 65.2 | 192.7 | 65.7 | 319 | 3954.6 | -1 | 248 | - | 0.23 | 274.05 | 7.1
BetCen(104)I | 66.7 | 209.3 | 69.4 | 321 | -1126.9 | -1 | 303 | - | 0.31 | 357.50 | 6.7
BetCen(105)I | 66.7 | 212.9 | 70.1 | 83.1 | 5023 | 1 | 62 | - | 0.15 | 313.44 | 5.2
BetCen(78)E | 66.7 | 45.9 | 73.3 | 206.5 | -5087.4 | -1 | 295 | - | 0.38 | 564.08 | 7.0
BetCen(77)E | 66.7 | 47.5 | 73.8 | 12.6 | 4961 | 1 | 161 | - | 0.34 | 593.24 | 7.4
BetCen(92)E | 66.7 | 55 | 76.2 | 75.5 | 4143.3 | -1 | 318 | - | 0.23 | 452.62 | 6.2
EpsCas(104)E | -70 | 122.3 | 79 | 123.6 | -4468.4 | 1 | 299 | - | 0.04 | 4.56 | 3.6
$B>60^{\circ}$ and $|\cos{\phi}|<\frac{1}{2}\hskip 142.26378pt$Complete subset size = 17 occultation |
BetCen(77)I | 66.7 | 270.4 | 89.8 | 257.3 | -1823.4 | 1 | 164 | - | 0.25 | 587.23 | 7.7
BetCen(85)I | 66.7 | 277.5 | 86.8 | 41.8 | -4526.9 | -1 | 210 | - | 0.71 | 1066.32 | 7.6
BetCen(89)I | 66.7 | 278 | 86.6 | 322.8 | -572.3 | 1 | 277 | - | 0.33 | 498.23 | 7.7
AlpCru(100)E | 68.2 | 93.9 | 88.5 | 188.8 | 807.8 | 1 | 288 | - | 0.50 | 428.76 | 5.8
BetCen(102)I | 66.7 | 249.1 | 81.3 | 0.9 | 4984.3 | -1 | 53 | - | 0.33 | 370.17 | 7.3
BetCen(104)E | 66.7 | 105.5 | 83.5 | 299.7 | -3402.1 | 1 | 154 | - | 0.29 | 342.97 | 8.6
BetCru(262)I | 65.2 | 257.3 | 84.2 | 18 | 5014.2 | -1 | 343 | - | 0.19 | 109.53 | 6.4
BetCru(253)I | 65.2 | 255.6 | 83.4 | 187.3 | -4002.7 | -1 | 157 | - | 0.20 | 106.17 | 6.5
GamCas(100)E | -66.3 | 72.5 | 82.5 | 93.5 | -3408.3 | 1 | 259 | - | 0.09 | 54.16 | 6.1
GamAra(37)I | 61 | 248.7 | 78.6 | 62.1 | 4684.1 | 1 | 133 | - | 0.12 | 53.50 | 4.2
$60^{\circ}>B>30^{\circ}$ and $|\cos{\phi}|>\frac{1}{2}\hskip 142.26378pt$Complete subset size = 35 occultations |
EpsCen(65)I | 59.6 | 226.6 | 68.1 | 343.1 | -1573.5 | -1 | 220 | 143 | 0.26 | 259.43 | 6.6
ZetCen(60)I | 53.6 | 228.1 | 63.8 | 258.7 | -4185.7 | -1 | 235 | 119 | 0.23 | 214.36 | 6.6
DelCen(98)I | 55.6 | 211.3 | 59.6 | 54.8 | 402.3 | 1 | 100 | 16 | 0.09 | 34.47 | 5.8
EpsLup(36)E | 51 | 44.9 | 60.2 | 135.3 | 2840.1 | -1 | 99 | 125 | 0.14 | 65.34 | 5.7
GamLup(32)E | 47.4 | 33.8 | 52.6 | 158.4 | 5097.4 | 1 | 310 | 333 | 0.15 | 145.17 | 5.4
LamSco(29)E | 41.7 | 148.3 | 46.3 | 222.1 | -4753.7 | -1 | 35 | 9 | 0.32 | 567.90 | 4.6
ZetPup(171)I | 38.6 | 202.1 | 40.8 | 120.2 | 2022.7 | -1 | 350 | 23 | 0.12 | 49.51 | 4.3
EtaLup(34)E | 44.5 | 357.1 | 44.5 | 90 | -3167.9 | -1 | 353 | 44 | 0.17 | 93.96 | 4.2
TheAra(40)E | 53.9 | 28.6 | 57.3 | 197.2 | -2036.3 | -1 | 269 | 218 | 0.09 | 24.98 | 4.1
KapCen(42)I | 48.5 | 168.5 | 49.1 | 237.3 | 228.4 | 1 | 88 | 62 | 0.24 | 82.26 | 4.1
$60^{\circ}>B>30^{\circ}$ and $|\cos{\phi}|<\frac{1}{2}\hskip 142.26378pt$Complete subset size = 37 occultations |
ZetCen(112)I | 53.6 | 239.7 | 69.6 | 112.5 | 4393.8 | -1 | 345 | 216 | 0.03 | 37.31 | 6.4
AlpAra(32)I | 54.4 | 277.9 | 84.4 | 258.6 | 3837.8 | -1 | 311 | 311 | 0.15 | 76.61 | 6.1
KapCen(35)E | 48.5 | 85.5 | 86.1 | 302.8 | -5074.8 | 1 | 279 | 305 | 0.12 | 92.03 | 5.8
TheCar(190)I | -43.3 | 252.4 | 72.2 | 219.3 | -4463.7 | 1 | 276 | 276 | 0.16 | 25.33 | 5.7
ZetCen(62)E | 53.6 | 70.1 | 75.9 | 8.9 | -4066 | 1 | 46 | 85 | 0.19 | 212.18 | 5.5
TheAra(41)E | 53.9 | 82.4 | 84.5 | 205.4 | 1720.2 | -1 | 193 | 193 | 0.09 | 23.52 | 5.5
DelCen(64)E | 55.6 | 110.6 | 76.4 | 194.3 | 4773.2 | 1 | 327 | 353 | 0.09 | 52.91 | 5.0
KapCen(36)I | 48.5 | 241.2 | 67 | 228.6 | -2480.8 | -1 | 329 | 200 | 0.24 | 88.26 | 5.0
DelPer(37)I | -54 | 264.5 | 86 | 145.7 | 4558.5 | -1 | 28 | 28 | 0.08 | 27.26 | 5.0
AlpLup(248)E | 53.9 | 111.4 | 75.1 | 73.2 | 4024.3 | -1 | 150 | 175 | 0.22 | 16.97 | 4.8
Occultation | $B\,\textrm{[deg]}$ | $\phi\,\textrm{[deg]}$ | $B_{\mathrm{eff}}\,\textrm{[deg]}$ |  Lon. [deg] | $Z$ | $\textrm{sign}(\dot{Z})$ | $\Phi_{\mathrm{theory}}\textrm{[deg]}$ | $\Phi_{\mathrm{obs}}\textrm{[deg]}$ | $b$ | $I_{0}$ | $\tau_{\mathrm{max}}$
---|---|---|---|---|---|---|---|---|---|---|---
$B<30^{\circ}$ and $|\cos{\phi}|>\frac{1}{2}\hskip 142.26378pt$Complete subset size = 11 occultations |
GamPeg(172)I | -20.3 | 36.7 | 24.7 | 283.4 | 4683.3 | 1 | 328 | 339 | 0.01 | 11.49 | 1.7
AlpCMa(274)E* | 13.5 | 40.4 | 17.5 | 188.9 | 2833.2 | -1 | 245 | 232 | 0.07 | 23.75 | 1.7
SigSgr(114)I | 29.1 | 330.2 | 32.6 | 262.3 | -492.7 | 1 | 160 | 133 | 0.04 | 33.50 | 2.7
AlpVir(34)E | 17.3 | 332.9 | 19.2 | 259.5 | 2584 | 1 | 207 | 220 | 0.50 | 979.08 | 1.7
GamPeg(32)I | -20.3 | 138.6 | 26.2 | 226.6 | 2857.3 | 1 | 342 | 342 | 0.53 | 149.51 | 1.8
AlpVir(8)I | 17.3 | 141.1 | 21.8 | 162.3 | 4488.6 | -1 | 144 | 221 | 5.00 | 1000.46 | 1.8
KapCMa(168)I | 29.3 | 175.9 | 29.4 | 268.8 | -3886.5 | 1 | 270 | 321 | 0.04 | 6.06 | 1.7
BetCMa(211)I# | 14.2 | 223.4 | 19.2 | 67.1 | -3637.3 | 1 | 1 | 232 | 0.05 | 43.52 | 1.8
EpsCMa(276)E | 26 | 50.6 | 37.5 | 217.7 | 4192 | -1 | 108 | 173 | 0.12 | 69.53 | 2.6
AlpVir(210)I# | 17.3 | 311.5 | 25.1 | 46 | -4015.8 | 1 | 259 | 104 | 0.04 | 138.22 | 2.5
$B<30^{\circ}$ and $|\cos{\phi}|<\frac{1}{2}\hskip 142.26378pt$Complete subset size = 20 occultations |
DelSco(236)I | 28.7 | 263.2 | 77.9 | 10.8 | -5022.7 | -1 | 171 | 168 | 0.65 | 22.98 | 2.8
AlpVir(232)E | 17.3 | 89.3 | 87.8 | 316.7 | -2068.6 | 1 | 284 | 284 | 0.46 | 125.75 | 2.3
AlpVir(34)I | 17.3 | 232.8 | 27.2 | 24 | 579.6 | 1 | 225 | 186 | 0.29 | 1021.26 | 2.2
SigSgr(244)I | 29.1 | 269.3 | 88.8 | 86.6 | -1341.1 | 1 | 313 | 313 | 0.04 | 16.60 | 2.1
GamPeg(36)E | -20.3 | 66.8 | 43.2 | 343.5 | 3206.1 | 1 | 59 | 34 | 0.45 | 141.95 | 1.8
GamPeg(211)E | -20.3 | 122.5 | 34.6 | 207.2 | 518.8 | -1 | 200 | 200 | 0.02 | 13.19 | 1.8
AlpVir(211)I | 17.3 | 267.2 | 81.2 | 103.3 | -2632.7 | 1 | 230 | 269 | 0.47 | 132.50 | 1.8
AlpVir(8)E | 17.3 | 91.3 | 85.9 | 197.2 | 3679.9 | -1 | 19 | 45 | 5.00 | 1053.60 | 1.7
KapCMa(168)E | 29.3 | 127.3 | 42.8 | 286.8 | -1613.5 | 1 | 229 | 284 | 0.04 | 6.19 | 1.7
AlpCMa(281)I* | 13.5 | 230.1 | 20.5 | 144.3 | 3652.3 | 1 | 143 | 195 | 0.06 | 27.91 | 1.7
Table 2: Dataset for this work. We divided the UVIS data set into 6 subsets comprising different geometries, each containing 10 occultations. The theoretical phases of the waves were computed using Eqn. (34) and the latest SPICE kernel for the longitude of the occultation relative to Mimas (the ‘Lon’ column). The observed phases were computed by fitting the phase of the wave in each occultation individually with fixed global parameters $(\beta,\nu,\sigma)$. $Z$ and $\mathrm{\dot{Z}}$ are the vertical position of Mimas with respect to the equator and the sign of its velocity respectively, both are required to compute the phase of the wave. “$\\#$” indicates that the difference between the theoretical and predicted phase is more than $90^{\circ}$. “$*$” indicates that the dispersion relation, or the predicted distances between peaks of optical depth, are inconsistent with the data for these occultations. | $GM\,[\mathrm{\frac{km^{3}}{s^{2}}}]$ | $a\,[\mathrm{km}]$ | $i\,[\mathrm{deg}]$
---|---|---|---
Mimas | $2.50349$ | $185539$ | $1.574$
Saturn | $37931206.2$ | - | -
Table 3: Physical parameters of Mimas and Saturn used in this work. Taken from
the JPL Horizons database (accessed in November 2019).

##  4 Ray-tracing model

The relationship between the slope of the wave and the extinction of starlight
in a UV occultation is non-linear, both in the SCL model and in our model. In
SCL the shape of the wave enters into the optical depth by determining the
limits of the optical depth path integral.

$\tau=\int^{l_{2}}_{l_{1}}n\bar{\sigma}_{\mathrm{ring}}dl$ (35)

where $n$ is the number density along the light ray,
$\bar{\sigma}_{\mathrm{ring}}$ is the average cross-section of absorbers over
the path of the light ray, and $l_{1}$ and $l_{2}$ are the ring’s entry and
exit points along the light ray.

Since we use SCL theory to trace the wave profile,
$\bar{\sigma}_{\mathrm{ring}}$, $n$, and the thickness of the ring $d$ are
uniform throughout the wave (Shu et al., 1983; Gresh et al., 1986). The
predicted haze is formed by relatively small particles (see Table 1) and hence
keeping $n$ uniform is compatible with the haze. A quick calculation shows
that, if the haze particles are an order of magnitude smaller than the ring’s,
they constitute $1\%$ of the mass at the peak of the haze’s optical depth.
This allows us to write the product $n\bar{\sigma}_{\mathrm{ring}}$ in terms
of the normal optical depth outside of the wave $\tau_{n0}$. Consider an
occultation along the wave where the light comes only from the direction
normal to the plane, then

$\tau_{n0}=\int^{d}_{0}n\bar{\sigma}_{\mathrm{ring}}dz=n\bar{\sigma}_{\mathrm{ring}}d$
(36)

$n\bar{\sigma}_{\mathrm{ring}}=\frac{\tau_{n0}}{d}$ (37)

So we can write

$\tau=\int^{l_{2}}_{l_{1}}\frac{\tau_{n0}}{d}dl$ (38)

In order to better match the Voyager I radio occultation, Gresh et al. (1986)
introduces a Gaussian enhancement to the optical depth without a physical
motivation for producing such an enhancement. We showed in section §2 that a
haze (Eqn. 29) is generated as a consequence of the rotation of self-gravity
wakes within the BW; adding this haze to Eqn. (35) we get:

$\tau=\int^{l_{2}}_{l_{1}}\frac{\tau_{n0}}{d}dl+\int^{l_{4}}_{l_{3}}\frac{\beta}{D}\left|\frac{dz}{dx}\right|_{\mathrm{max}}dl$
(39)

where $l_{3}$ and $l_{4}$ are the entry and exit points of the haze along the
light ray, $\beta$ is related to the number of particles released by the wakes
(see section §2) and $D$ is the vertical thickness of the haze.
$\left|\frac{dz}{dx}\right|_{\mathrm{max}}$ is the maximum slope over a period
at a given radial coordinate $x$. To simplify Eqn. (39) we make use of the
azimuthal and north-south symmetry of the rings which makes the integrals
depend only on the radial coordinate $x$. Then, if $B$ is the angle between
the rings and the light ray and $\phi$ is the angle between the radial
direction and the light ray (projected into the ring’s mean plane), we can
write:

$dl=\frac{dx}{\cos{B}\cos{\phi}}$ (40)

We know the cross-section $\bar{\sigma}_{\mathrm{ring}}$ and the column
density $n_{\mathrm{ring}}$ are not isotropic however (they depend on the
angle $\phi$), due to self-gravity wakes having a consistent average
orientation (Colwell et al., 2006). Thus, the normal optical depth obtained by
simply multiplying the oblique optical depth by $\sin{B}$ is not the true
normal optical depth defined in Eqn. (36). Nevertheless, it isn’t the goal of
our model to describe the dependence of $\bar{\sigma}_{\mathrm{ring}}$ or
$n_{\mathrm{ring}}$ on the geometry of the occultation, but only to explain
the variation of the optical depth in the wave with respect to the optical
depth outside the wave region. Thus, we define the background slanted optical
depth as:

$\tau_{0}=\frac{\tau_{n0}}{\sin{B}}=\frac{n\bar{\sigma}_{\mathrm{ring}}d}{\sin{B}}$
(41)

where $\tau_{0}$ is determined by averaging values for the optical depth
between $r=131600\,\mathrm{km}$ and $r=131700\,\mathrm{km}$, a region just
outside of the wave.

By substituting Eqns. (40) and (41) into Eqn. (39) we get the equation for the
predicted optical depth:

$\tau=\int^{r_{2}}_{r_{1}}\frac{\tau_{0}}{d}\sin{B}\frac{dx}{\cos{B}\cos{\phi}}+\int^{r_{3}}_{r_{4}}\frac{\beta}{D}\left|\frac{dz}{dx}\right|_{\mathrm{max}}\frac{dx}{\cos{B}\cos{\phi}}$
(42)

where the limits of integration $r_{1},r_{2}$ and $r_{3},r_{4}$ are the radial
coordinates of the entry and exit points of the light ray as it crosses the
ring and the haze respectively. These radial coordinates are computed by
numerically finding the intersection of the trajectories of incoming photons,

$z_{\mathrm{photon}}=\sin{B}\frac{x_{\mathrm{photon}}-x_{0}}{\cos{B}\cos{\phi}},$

with the curves $h(x)\pm\frac{d}{2}$ (for $r_{1}$ and $r_{2}$) and the curves
delimiting the upper and lower boundary of the haze depicted by the blue fill
in Figure 11 (for $r_{3}$ and $r_{4}$). Here $h(x)$ is the wave profile
derived in SCL (see Eqn. 5). To visualize the ray-tracing, consider the light
ray (dashed gray line) shown in Figure 11. The haze’s shape cannot be written
in closed form; therefore, in order to trace its upper and lower boundaries we
run the numerical procedure described in section §2.2 for each occultation.

Contrasting with Gresh et al. (1986)’s second integral in Eqn. (42), their
Gaussian enhancement adds three free parameters that vary independently:
width, amplitude, and position. On the other hand, Eqn. 42 adds one free-
parameter for the haze, $\beta=\Sigma\sigma_{\mathrm{haze}}$, which is related
to the number of particles released by the wake as they rotate and collide
with neighboring particles (see §2.2). All the spatial information of the
haze—shape, width, and position—is fixed by the wave profile $h$ (Eqn. 5)
which is also heavily constrained by the first integral in Eqn. (42). In Gresh
et al. (1986) the combinations of angles $\frac{\tan{B}}{\cos{\phi}}$ lead to
the definition of the angle
$\tan{B_{\mathrm{eff}}}=\frac{\tan{B}}{\cos{\phi}}$. Because this combination
does not occur in the second term in Eqn. (42), we prefer to leave the angle
$\phi$ explicit.

The shape of the wave itself is a function of the surface mass density
$\sigma$, the viscosity $\nu$, and the thickness $d$ which are free
parameters. New to our model is free-parameter $\beta$. The quantity to be
compared to the data is the transparency, which is the normalized stellar
counting rate after background subtraction:

$T_{\mathrm{model}}=e^{-\tau}=\exp{(-\int^{r_{2}}_{r_{1}}\frac{\tau_{0}}{d}\sin{B}\frac{dx}{\cos{B}\cos{\phi}}-\int^{r_{3}}_{r_{4}}\frac{\beta}{D}\left|{\frac{dz}{dx}}\right|_{\mathrm{max}}\frac{dx}{\cos{B}\cos{\phi}})}$
(43)

Note that while the thickness $d$ appears directly in the above expression it
does not figure as a relevant predictor in our model. As Gresh et al. (1986)
discovered, the effects of increasing $d$ on these ray-tracing simulations of
bending waves are equivalent to increasing the resolution of the simulation.
The optical depth profile becomes smoother in both cases because there are
more data points probing the slopes of the ring. Any difference between a
thickness of $10\ \mathrm{m}$ to $100\ \mathrm{m}$ in the rings would not make
a difference in a dataset that is fixed at a resolution of $400\,\mathrm{m}$.
Since the BW occultations at these resolutions can’t be used to determine the
thickness of the rings directly, we can fix the value of $d$ for all runs,
which we choose to be $15\,\mathrm{m}$. Similarly, the thickness of the haze,
which determines the limits of the second integral in Eqn. (43), is derived
from the dynamics in section §2 and also has values under the resolution of
our data; by the same argument we choose to fix $\epsilon$ (the coefficient of
restitution which determines the thickness of the haze) at two values, $0.5$
and $1$. We are then left with a total of three free parameters: the viscosity
$\nu$, the surface mass density $\sigma$, and the particles released by the
wakes per area times their cross-section $\beta$.

Note that by taking $\beta$ to be a constant for all occultations we assume
that the product $n_{\mathrm{haze}}\sigma_{\mathrm{haze}}$ does not depend on
the azimuthal viewing angle $\phi$. This becomes important in the next section
as we compare Eqn. (43) against the data.

##  5 Comparison with data

The ray-tracing code is used to compute the path integrals in Eqn. (43) for
$500$ light rays which are drawn at a radial separation of $400\,\mathrm{m}$
each, covering a radial distance of $200\,\mathrm{km}$ in the wave region,
hence matching the resolution of the data set (the exception to this is
EpsCas104E that has one null optical depth value within the BW: the
corresponding light ray is removed when simulating this occultation). We use a
reduced $\chi^{2}$ minimization method where we find the best parameter values
by comparing the model with the $60$ occultations shown in Table 2. The
reduced $\chi^{2}$ is given by:

$\chi^{2}_{R}=\frac{1}{D_{f}}\Sigma{\left(\frac{T-T_{\mathrm{model}}}{\Delta}\right)^{2}}$
(44)

where $\Delta=\frac{\sqrt{I-b}}{I_{o}}$ and $D_{f}$ is the degrees of freedom.
The uncertainty of $I_{0}$ and $b$ is negligible due to the small value of $b$
and the amount of data points used to compute them both. $T_{\mathrm{model}}$
is given by Eqns. (43), and $T$ is given by Eqn. (32). The wave profile
predicted by SCL theory has been modified to include a haze of particles whose
shape is computed using the particles’ velocity (Eqn. 30) and number density
(Eqn. 29) with $\beta$ as a free parameter. The total free parameters of our
model are $\beta$, the surface density $\sigma$, and the viscosity $\nu$,
while the thickness of the ring $d$ and the coefficient of restitution
$\epsilon$ have been fixed at values $d=15\,\mathrm{m}$ and $\epsilon=0.5$ and
$1$.

The best-fit parameters for the haze model and the SCL model (no haze), are in
Table 4. Figures 12 and 13 plot the best-fit optical depth profile for the
haze (black line) and SCL (dashed green line) models, and the observed UVIS
occultations (red line) for 7 occultations.

Figure 12: We compare an occultation of $\gamma$-Pegasus (rev 32I,
$B=20.3$,$B_{\mathrm{eff}}=26^{\circ}$ ) seen by Cassini (solid red) to our
model (solid black) with the parameters from the 3rd row in Table 4, and SCL
theory (dashed green) using the viscosity from Esposito et al. (1983)
($\nu=280\,\mathrm{cm^{2}/s}$) and our best-fitted $\sigma$ (our $\sigma$ is
used to ensure the peaks between both models better match the data). The error
bars in the data are displayed as a light red filled curve in the background
of the plot. The underprediction of the troughs in SCL is addressed in the new
model, as the haze layer increases the minimum optical depth of the
occultation, and the peaks fit better due to the shorter damping length. Note
that the predicted peaks are now broader in the new model due to the extended
height of the haze, which is something we also see in the data. The
discrepancies in the separation of the initial peaks of the model with respect
to the data persist. Figure 13: The haze model (3rd row of Table 4, solid
black lines), and the SCL model (dashed green; same parameters as in Figure
12) compared to occultations (red lines) of (a) $\beta$-Centauri (rev77I), (b)
$\kappa$-Centauri (rev35E) (c) $\alpha$-Virginis (rev232E), (d)
$\epsilon$-Cassiopeiae (rev104I) (e) $\eta$-Lupus (rev34E), and (f)
$\kappa$-Canis Majoris (rev168I). The error bars are displayed as a light red
filled curve in the background of the plot. In the top row we have azimuthal
occultations ($\cos{\phi}<1$), on the bottom row radial occultations
($\cos{\phi}>1$), at high, mid and low $B$ angles (from left to right). For
high $B_{\mathrm{eff}}$ occultations, the SCL theory predicts an unchanging
optical depth throughout the wave, but instead the data shows a symmetric rise
in the optical depth centered at $-80\,\mathrm{km}$ from resonance. Our model
reproduces both the position of the peak and the width of the Gaussian-like
enhancement. The amplitude of these peaks, however, does not vary with opening
angle $B$ as our model predicts: for azimuthal occultations the scaling with
$B$ of the optical depth is too rapid—the model underpredicts (a) and slightly
overpredicts (c) the peak of the enhancement—while for radial occultations the
scaling works well. The haze model also present a significant improvement when
matching the data of low $B_{\mathrm{eff}}$ occultations.

Table 4 shows that the models with the haze ($\beta\neq 0$) fit significantly
better than the SCL model for the dataset. The improvement for low
$B_{\mathrm{eff}}$ is represented in Figure 12. Most notably, Figure 12 shows
that the troughs of the optical depth pattern now coincide with the predicted
optical depth due to the Gaussian-like enhancement produced by the haze.
Moreover, the bases of the peaks are broader than those of SCL theory and
resemble the observed shape of the optical depth profile. The broadening of
the peaks’ bases is caused by the thickening of the haze at the minima and
maxima of the wave (shown in section §2.2). Note, however, that the position
of the peaks are unchanged by our modification, as we still use SCL’s
dispersion relation. Figure 12 also shows that the damping length due to the
viscosity of $\nu=576\,\mathrm{cm^{2}/s}$ closely matches the observed length
of the wave. This high viscosity increases the predicted value for the slope
of the wave which is $17^{\circ}$ for our best-fit model.

Table 4 shows that using the observed phase instead of the theoretical phase
improves the fit, but it does not change our values for $\beta$, $\nu$, and
$\sigma$ significantly. Likewise, changing the coefficient of restitution,
which changes the thickness of the haze (see Figure 11), does not alter the
agreement with the data.

Model | Phase | $\epsilon\,\mathrm{[-]}$ | $\chi^{2}_{R}\,\mathrm{[-]}$ | $\sigma\,\mathrm{[\frac{g}{cm^{2}}]}$ | $\nu\,\mathrm{[\frac{cm^{2}}{s}]}$ | $\beta\,[-]$
---|---|---|---|---|---|---
SCL | $\Phi_{\mathrm{theory}}$ | $-$ | $22.15$ | $36.4\pm 5$ | $3540\pm 40$ | $-$
Haze | $\Phi_{\mathrm{theory}}$ | $1$ | $5.56$ | $37.0\pm 0.4$ | $\ 623\pm 3$ | $1.48$
Haze | $\Phi_{\mathrm{obs}}$ | $1$ | $4.46$ | $36.7\pm 0.3$ | $\,\,576\pm 3$ | $1.39$
Haze | $\Phi_{\mathrm{obs}}$ | $0.5$ | $4.48$ | $36.6\pm 0.3$ | $\,\,576\pm 3$ | $1.37$
Table 4: Best fit results for the SCL and haze models. There is a significant
improvement in the fit of the haze model over the SCL model. Error bars show
the $1\sigma$ of the likelihood given by $e^{-\chi^{2}/2}$. The haze model has
3 free parameters, so all have $D_{f}=29819$ degrees of freedom. SCL has 2
free parameters and hence $D_{f}=29879$.

In Figure 13 we present 6 occultations, once for each of the data subsets in
Table 2. The occultations range from higher to lower $B$ from left to right in
the figure. Our model represents a significant improvement over SCL for both
radial (lower row) and azimuthal (upper tow) occultations. For higher
$B_{\mathrm{eff}}$ occultations, the SCL theory predicts little change in the
optical depth, while the haze model produces the shape of the optical depth
enhancement seen in the data. Nevertheless, we see in the top row that the
scaling with $B$ for the haze optical depth, which is simply $1/\sin{B}$ in
our model, fails to reproduce the amplitude of this optical depth enhancement
for some azimuthal occultations. While the amplitude fits well for
intermediate $B$ (b), for high and low $B$ angles our haze model underpredicts
(a) and overpredicts (c) the maximum of the smoothed optical depth by more
than $10\%$. This effect is not evident in the radial occultations, as the
troughs of the optical depth pattern match the data in (e) and (f), while (d)
is only slightly below the data’s amplitude value. This is a clear example of
a $\phi$-dependent optical depth (i.e. azimuthal brightness asymmetry). In
other words, the haze does not behave as an isotropic absorbing medium. A
potential explanation for this is discussed in the next section.

##  6 Discussion

Discrepancies between the optical depth predicted by the SCL model and those
measured by Cassini UVIS and Voyager I (Gresh et al., 1986), have been
partially addressed by our modifications to the theory (Figure 12). The
addition of haze particles released by self-gravity wakes changes the wave
profile, addressing problem (1) of the introduction (§1), and improving the
likelihood considerably (Table 4). Most of the explanatory power in the model
comes from the existence of the extra layer of particles.

However, Figure 13 shows that the observed haze optical depth presents
azimuthal brightness asymmetry ($\phi$-dependence). Given that the haze is
hypothesized to come from self-gravity wakes (which are responsible for the
$\phi$-dependence of the flat ring’s optical depth), the observed anisotropy
may be inherited from them. To improve our model to explain this anisotropy, a
more-in-depth analysis of the dynamics presented in section §2 is called for.
If the self-gravity wakes have a preferred direction inside the BW, and the
haze comes from said wakes, the haze will present an anisotropy; which, in
turn, can be compared against the observations as an additional test for the
dynamical model. Preliminarily, Figure 9 suggests that the azimuthal direction
may be a preferred orientation, but further analysis is needed to determine
the attractors in the rotation of wakes in the BW.

Figure 12 shows that the haze model’s fit of the observed wave damping length
yields a relatively high kinematic viscosity, partially addressing problem (3)
of the introduction (§1). The best-fit viscosity value more than doubles the
viscosity obtained from neighboring density waves (Tiscareno et al., 2007),
and it falls outside the upper error bars in Gresh et al. (1986) and Esposito
et al. (1983). We compare our value with the literature in Figure 14

Figure 14: Measurements of the viscosity of the rings made by various authors
around the bending wave region. The measurements done using the Mimas BW are
the ones at $131.9\,\mathrm{Mm}$. The values computed in this paper are the
filled triangles. This work’s values are considerably higher than the more
recurrent value of $\nu=280\,\mathrm{cm^{2}/s}$. Our measurement lies outside
the error bars of Gresh et al. (1986) and Esposito et al. (1983) (Tiscareno et
al. 2007 and Lissauer et al. 1984 didn’t include error bars). Esposito et al.
(1983), Gresh et al. (1986), Tiscareno et al. (2007) and Lissauer et al.
(1984) were using one occultation (or image) for their estimates; in this work
we use 60 occultations at different geometries.

Figure 14 shows that Gresh et al. (1986), Lissauer et al. (1984), and Esposito
et al. (1983) measured a lower viscosity than the best-fitted values of our
work. Esposito et al. (1983) used one occultation of $\delta$-Scorpii, while
Gresh et al. (1986) used one low-angle radio occultations of the Earth (at two
frequencies). Lissauer et al. (1984) used images of the shadows cast by the
elevated ring to estimate where the wave-train ended. In this work we use 60
occultations at better resolution (400 m vs. 2 km). The signal-to-noise is
also higher on average since we are looking at brighter sources, like
$\gamma$-Pegasus and $\beta$-Centauri.

We see two reasons why their viscosity is lower. The first one pertains to the
noise and the spatial resolution of the data: they didn’t fit all the cycles
because they couldn’t see them, so instead of fitting the complete wave-train
to find the viscosity they attempted to fit the damping length $\xi_{D}$,
which is the point at which the amplitude of the wave drops a factor of $e$.
This approach would work if the SCL theory were adequate for this BW, which we
see it isn’t in its entirety, as shown in the first row of Table 4 and Figures
12 and 13. This leads us to our second reason, the lack of a proper model for
the extra optical depth. Without the haze our value for the viscosity would
underpredict all of the peaks in 12. Moreover, in the case of Gresh et al.
(1986), the occultations were so low angle that the optical depth of the peak
was taken to be infinity (the slope of the wave and the viewing angle were
almost the same), which means they mostly fit the troughs and not the peaks
and we can see in Figure 12 that the underdamping in the SCL model’s fit is
more evident in the peaks than in the troughs. In summary, Gresh et al.
(1986), Lissauer et al. (1984) and Esposito et al. (1983) viscosity values are
lower than the ones suggested in this work because in our occultations the
late part of the wave-train can be clearly distinguished from the noise, and
their prediction for the damping length relies on the SCL theory.

Nevertheless, the value Gresh et al. (1986) computed for the region is still
higher than what Tiscareno et al. (2007) calculated from analyzing the 11:12
Prometheus density wave which used Cassini data and looks at weak bending
waves where SCL is adequate (although Tiscareno et al. 2007 points out that
the Prometheus 11:12 shows some non-linearity, it isn’t enough to expect his
results to be inaccurate, given their consistency with other estimates of the
viscosity in the region). To be consistent with Tiscareno et al. (2007) we
then have to consider that the BW itself increases the viscosity in a way that
a density wave does not.

To further address problem (3), we need to account for the back-reaction of
the wave to the wake’s rotation since we can expect that the inclusion of the
wake-wave interaction will affect the damping length of the theory. A way to
include this added dissipation is to consider the effects that the collisional
forces (Eqn. 16) have on the wakes’ vertical motion. In our current model the
collisions are taken to be north-south symmetric in the frame of the wake: the
same amount of particles are hitting the wake from above and from below, which
results in a net vertical force of zero. This is probably not true, and the
inelastic wake-wave collisions would cause a net damping of the vertical
motion of the wakes, and of the ring particles in general. Since we haven’t
included this effect, our model accounts for the extra dissipation by having a
high value in the kinematic viscosity of the SCL term, in which the viscous
effects of vertical collisions are modeled simply by $F_{\mathrm{visc}}/m=\nu
d\dot{z}/dx$ (where $m$ is the mass of the affected particle). However, even
our simple collision model suggests a more complex interaction that depends on
the slope, on the mass colliding with the self-gravity wake, and on the wake’s
size. The computation of this viscous force and the subsequent modification to
SCL’s wave profile are left to future work.

Angular momentum conservation between the rings and satellites indicates that
the viscosity responsible for the spreading of the rings, and that which damps
the waves, may not be related in a simple fashion (Tajeddine et al., 2017).
However, if the momentum transport in both cases is dominated by the motions
of self-gravity wakes a higher damping viscosity will be closely related to a
more efficient momentum transport in that region. Then, while the Mimas 5:3 BW
is relatively strong (the 2nd strongest BW in the rings), self-gravity wakes
allow the rings to transport the inflow of energy at resonance and form a
continuous wave. Despite this extra dissipation, the Mimas 5:3 BW still has a
slope of $\sim 15^{\circ}$, hence we can speculate that without the
enhancement in viscosity provided by the self-gravity wakes, the rings would
rift and create a gap similar to the ones seen in the Titan 0:-1 nodal BW
(Nicholson & Hedman, 2016) and in protoplanetary disks misaligned with their
star’s obliquity (Rowther et al., 2022). Without a theory of when (at what
slope and at what efficiency of viscous transport) warped disks break however,
this remains a speculation. An attempt at modeling the Titan 0:-1 nodal BW
using Cassini data may shed light on a general theory of rifted warps in
disks.

The more efficient angular momentum transport within the wave could also be
causing the diffusion of material into the neighboring regions. Consider the
values measured by Tiscareno et al. (2007) of the average surface mass density
of the Prometheus 12:11 wave, which lies about $100$ km closer to Saturn than
the Mimas 5:3 BW. The average measured surface density value is
$\sigma=46.1\,\mathrm{g/cm^{2}}$. This corresponds to an increase of $25\%$
with respect to our best-fit BW value (and a $24\%$ increase with respect to
Gresh et al. 1986’s best-fit value). The non-linear nature of this density
wave could be changing the local value of the surface density (Borderies et
al., 1986), but the Prometheus 12:11 is not strong enough for this effect to
change it more than $20\%$. So why does $\sigma$ increase in this region? A
possible explanation is that density waves, by their very nature of being
density-changing phenomena, are unreliable measures for $\sigma$, although
this wouldn’t explain why the measured values are overstimating the BW value.
Another more plausible explanation is that the higher viscosity in the BW
region is causing the migration of material. A local radial increase
(decrease) in viscosity has been shown to correspond to a local decrease
(increase) in surface density in protoplanetary disks (Lyra, W. et al., 2015).
If the change in viscosity is too abrupt, it can lead to an instability that
can excite the growth of vortices in the disk (Papaloizou & Pringle, 1985).
While this is not the case here, the increase in viscosity in the bending wave
region may be causing a decrease in surface density there, and a corresponding
increase in the nearby regions.

Comparing the values of $\Phi_{\mathrm{theory}}$ and $\Phi_{\mathrm{obs}}$ we
see a general agreement for the theoretical phase of the wave (Table 2). While
there are many waves with a phase difference of $\sim 20^{\circ}$, this is to
be expected considering that a radial offset of $10\,\mathrm{km}$, either in
the radial vector or the resonance position, and an error of $1^{\circ}$ in
the longitude of Mimas translates into a possible phase difference of
$25^{\circ}$ between theory and observation. Nevertheless, two stars show an
offset in the phase greater than $90^{\circ}$: AlpVir(210)I and BetCMa(276)E.
Note that the dispersion relation works well for these occultations once the
offset in the phase is corrected. That this only occurs in two out of $40$
occultations makes us hesitant to propose a mechanism driving such a big
offset. On the other hand, the offset could be due to a bigger systematic
error in the radial offset of these particular occultations.

While the used SCL dispersion relation does not improve the position of the
first peaks of the wave (problem (2) of the introduction §1) it does work
fairly well with most occultations (see Figure 13). Moreover, the broader
shape of the optical depth peaks partially alleviates some of the
discrepancies in the first peaks. Nevertheless, we find two low-angle
occultations in which the dispersion relation fails not only at the first
cycles but everywhere in the wave-train: AlpCMa(274)E and AlpCMa(281)I. A
possible cause of the dispersion relation difference can be a radial variation
in the surface mass density of the ring. By using the fitted value for the
isotropic model $\beta=1.39$, via equation
$\beta=\Sigma_{0}\bar{\sigma}_{\mathrm{haze}}$ (§2.2), assuming
$\bar{\sigma}_{\mathrm{haze}}=1\,\mathrm{cm^{2}}$ we estimate a maximum haze
surface mass density of $0.4\,\mathrm{g/cm^{2}}$ which is about $1\%$ the
best-fit for the ring’s surface mass density. The existence of the haze is
therefore consistent with SCL’s dispersion relation which assumes a constant
surface mass density throughout the wave and, by itself, is not evidence that
there is a significant change in the ring’s density.

Nevertheless, if the average cross-section of haze particles is bigger
($10\,\mathrm{cm}$, although we note this is contrary to what is suggested by
Table 1) and radially changing (there are bigger particles released when the
slope is bigger) there may still be a slow radial change of $5$ to $10\%$ in
the surface density which would affect the dispersion relation appreciably in
low $B_{\mathrm{eff}}$ angles occultations. Note that incorporating the wake-
wave interaction in the bending wave equation might also change the dispersion
relation of the wave, potentially improving the predicted wavelength near
resonance.

By gaining a better understanding of the Mimas 5:3 BW, we can help address
other problems in planetary rings science. First, consider the unexplained
high viscosity of the Mimas 5:3 density wave (Borderies et al., 1986). The
source of this viscosity may also be related to the rotation of self-gravity
wakes. Local increases in surface density may reduce the local shear-rate at
that location, causing wakes to rotate as they adjust their pitch-angle (see
Figure 7). While the rotation due to a changing shear-rate will be more gentle
than the one described in this work, it can contribute to the increase in
local viscosity seen in this density wave. Secondly, consider the puzzle set
by Tiscareno et al. (2013b) for the Iapetus -1:0 BW. This BW propagates
outwards through an abrupt increase in optical depth called ‘the inner A-ring
edge.’ Historically, researchers have interpreted this increase in optical
depth as an increase in the surface density $\sigma$. However, $\sigma$ also
affects the dispersion relation of waves, and no such abrupt change is seen in
the wavelength of the Iapetus -1:0 BW. If this edge corresponds to a point
where self-gravity wakes to start form, given how unstable self-gravity wakes
are to vertical perturbations within BWs, a haze of particles may contribute
to the slanted optical depth increase. Properly applying the rigid-bar and
haze models to these issues is left for future work.

##  7 Conclusion

We have shown that a necessary consequence of introducing rigid self-gravity
wakes to the linear bending wave theory is that bending waves generate an
extra layer of particles that is proportional to the amplitude of the slope.
Even if we consider the scenario of the wake’s partial disruption due to the
interaction with the bending wave, this disruption stills create the same
layer of haze particles, given that the relative vertical velocities between
the wakes and the ring are comparable to the characteristic vertical
velocities of the particles as the wave propagates.

Using a ray-tracing code, we tested this additional signal and found that the
extra layer of particles has considerable explanatory power for the Cassini
UVIS occultations of the BW compared to Shu, Cuzzi, & Lissauer (1983). Our
best model explains the most discrepant and surprising features of the Mimas
5:3 BW; namely, the enhancement of the signal for the cases of normal
occultations and a bigger than expected viscosity,
$\nu=576\,\mathrm{cm^{2}/s}$, more than two times bigger than the viscosity
computed from density waves. This shows that self-gravity wakes can be
especially effective in preventing the opening of gaps in a vertically
perturbed disk.

The rigid-bar model for self-gravity wakes proved to be successful not only in
the case of the bending wave but also in the flat ring, as it showed that
wakes tend to align at the same pitch-angle due to their mutual self-gravity.
Thus, this can explain why the average pitch-angle of the wakes is the same
throughout the A and B rings even if the ring properties, like surface mass
density and distance from Saturn, change considerably within them. The rigid-
bar model can be used as an analytical tool to investigate the motion of self-
gravity wakes in a perturbed environment, such as density waves.

While improvements have been made in explaining the Cassini UVIS dataset for
the Mimas 5:3 BW, there are still issues with the current bending wave theory
and we suggest a path to modify it in a way consistent with the results of
this paper. We find that the theoretical dispersion relation tends to fail in
the first cycles of the wave, and in some cases, it fails through the entire
wavetrain. The reason for this may be the back-reaction that the self-gravity
wakes’ motion has on the propagation of the wave. The inclusion of this effect
is suggested as a path to further improve the theory.

## Acknowledgements

This document is the results of the research project funded by NASA FINESST,
award No 80NSSC20K1379.

##  Appendix A Complete form of the torques on self-gravity wakes

Let $\mu_{G}=\frac{GM_{S}}{r^{3}}$ where $r$ is the radial distance from
Saturn, $M_{S}n$ is the mass of Saturn and $G$ the gravitational constant.

$\begin{array}[]{l}\tau_{\mathrm{tidal+wake};x^{\prime}}=4\big{(}\overbrace{\mu_{G}}^{\text{tidal
}}-\overbrace{\pi G\sigma k^{2}\cos^{2}\theta_{w}}^{\text{wake
}}\big{)}\frac{L^{3}}{24}WH\rho_{H}\left(\hat{z}^{\prime}\cdot\hat{x}\right)\left(\hat{y}^{\prime}\cdot\hat{x}\right)\\\
-2\big{(}\overbrace{\mu_{G}}^{\text{tidal }}+\overbrace{2\pi G\sigma
k^{2}\sin^{2}\theta_{w}}^{\text{wake
}}\big{)}\frac{L^{3}}{24}WH\rho_{H}\left(\hat{z}^{\prime}\cdot\hat{y}\right)(\hat{y^{\prime}}\cdot\hat{y})\\\
\overbrace{-4\pi G\sigma k\cos\theta_{w}\sin\theta_{w}}^{\text{wake
}}\frac{L^{3}}{24}WH\rho_{H}\left[(\hat{z^{\prime}}\cdot\hat{y})(\hat{y^{\prime}}\cdot\hat{x})+\left(\hat{z}^{\prime}\cdot\hat{x}\right)\left(\hat{y}^{\prime}\cdot\hat{y}\right)\right]\end{array}$
(A1)

$\begin{array}[]{l}\tau_{\mathrm{tidal+wake};y^{\prime}}=-4\big{(}\overbrace{\mu_{G}}^{\text{tidal
}}-\overbrace{\pi G\sigma k^{2}\cos^{2}\theta_{w}}^{\text{wake
}}\big{)}\frac{W^{3}}{24}LH{\rho_{H}}\left(\hat{z}^{\prime}\cdot\dot{x}\right)\left(\hat{x}^{\prime}\cdot\hat{x}\right)\\\
+2\big{(}\overbrace{\mu_{G}}^{\text{tidal }}+\overbrace{2\pi G\sigma
k^{2}\sin^{2}\theta_{w}}^{\text{wake
}}\big{)}\frac{W^{3}}{24}LH\rho_{H}\left(\hat{z}^{\prime}\cdot\dot{y}\right)\left(\hat{x}^{\prime}\cdot\hat{y}\right)\\\
+\overbrace{4\pi G\sigma k\cos\theta_{w}\sin\theta_{w}}^{\text{wake
}}\frac{W^{3}}{24}\rho_{H}LH\left[\left(\hat{z}^{\prime}\cdot\hat{y}\right)(\hat{x}\cdot\hat{x})+\left(\hat{z}^{\prime}\cdot\hat{x}\right)\left(\hat{x}^{\prime}\cdot
y^{\prime}\right)\right]\end{array}$ (A2)

$\begin{array}[]{l}\tau_{\mathrm{tidal+wake};z^{\prime}}=4\big{(}\overbrace{\mu_{G}}^{\text{tidal
}}-\overbrace{\pi G\sigma\cos^{2}\theta_{w}}^{\text{wake
}}\big{)}\frac{L^{3}}{24}\rho_{H}HW\left(\hat{x}^{\prime}\cdot\hat{x}\right)\left(\hat{y}^{\prime}\cdot\hat{x}\right)\\\
+2\big{(}\overbrace{\mu_{G}}^{\text{tidal }}+\overbrace{2\pi
G\sigma\sin^{2}\theta_{w}}^{\text{wake
}}\big{)}\frac{L^{3}}{24}\rho_{H}HW\left(\hat{x}^{\prime}\cdot\hat{y}\right)\left(\hat{y}^{\prime}\cdot\hat{y}\right)\\\
+\overbrace{4\pi G\sigma k\cos\theta_{w}\sin\theta_{w}}^{\text{wake
}}\frac{L^{3}}{24}\rho_{H}HW\left[(\hat{x}\cdot\hat{y})\left(\hat{y}^{\prime}\cdot\hat{x}\right)+(\hat{x}\cdot\hat{x})\left(\hat{y}^{\prime}\cdot\hat{y}\right)\right]\end{array}$
(A3)

Where $W$, $H$ and $L$ are the length of the principal axes of the wake.
Centered on the center of the wake, we set a coordinate systems along the
principal axes ($x^{\prime}$, $y^{\prime}$, $z^{\prime}$, respectively). $k$
is the wavenumber of the self-gravity wakes, which are treated as a plane-wave
field. $\rho_{H}$ is the density of the self-gravity wake and $\sigma$ is its
surface density. $\theta_{w}$ is the angle with respect to the azimuthal of
the long axis $L$ of the oriented wake field.

Consider the direction $\hat{\theta}$, which is the direction of the velocity
of a particle colliding using the Hill equations (see Figure 6). Consider
moreover, that the space density of colliding particles is given by
$\rho(z)=e^{-(\frac{z}{z_{0}})^{2}}$ where $z_{0}$ is the rings’ half-
thickness, and note that the $z$ coordinate can be written in terms of
$(x^{\prime},y^{\prime},z^{\prime})$. We can then write the torques due to
collisions of particles with a space density $\rho_{s}$, both due to the BW
shear (caused by the slope of the bending wave $\frac{dz}{dx}$) and due to the
Keplerian shear, as:

$\displaystyle\tau_{\mathrm{Kep+BWsh};x^{\prime}}=$ $\displaystyle
2*(1+\epsilon)\int^{b_{\mathrm{max}}}_{b_{\mathrm{min}}}\biggr{|}\overbrace{\vec{v}_{\theta}\cdot\hat{z}^{\prime}}^{\text{Keplerian
}}+(\overbrace{\frac{dz}{dx}(\hat{y}^{\prime}\cdot\hat{x})(\hat{z}^{\prime}\cdot\hat{z})}^{\text{BW
shear }}-\omega_{x^{\prime}})y^{\prime}(b)\biggr{|}$ (A4)
$\displaystyle+\biggr{[}\overbrace{\vec{v}_{\theta}\cdot\hat{z}^{\prime}}^{\text{Keplerian
}}+(\overbrace{\frac{dz}{dx}(\hat{y}^{\prime}\cdot\hat{x})(\hat{z}^{\prime}\cdot\hat{z})}^{\text{BW
shear
}}+\omega_{y^{\prime}})y^{\prime}(b)\biggr{]}y^{\prime}(b)\int^{W/2}_{-W/2}\rho_{s}(z(x^{\prime},y^{\prime}(b)))dbdx^{\prime}$

$\displaystyle\tau_{\mathrm{Kep+BWsh};y^{\prime}}=$
$\displaystyle-2*(1+\epsilon)\int^{b_{\mathrm{max}}}_{b_{\mathrm{min}}}\biggr{|}\overbrace{\vec{v}_{\theta}\cdot\hat{z}^{\prime}}^{\text{Keplerian
}}+(\overbrace{\frac{dz}{dx}(\hat{x}^{\prime}\cdot\hat{x})(\hat{z}^{\prime}\cdot\hat{z})}^{\text{BW
shear }}+\omega_{y^{\prime}})x^{\prime}(b)\biggr{|}$ (A5)
$\displaystyle+\biggr{[}\overbrace{\vec{v}_{\theta}\cdot\hat{z}^{\prime}}^{\text{Keplerian
}}+(\overbrace{\frac{dz}{dx}(\hat{x}^{\prime}\cdot\hat{x})(\hat{z}^{\prime}\cdot\hat{z})}^{\text{BW
shear
}}+\omega_{y^{\prime}})x^{\prime}(b)\biggr{]}x^{\prime}(b)\int^{L/2}_{-L/2}\rho_{s}(z(x^{\prime}(b),y^{\prime}))dbdy^{\prime}$

$\displaystyle\tau_{\mathrm{Kep+BWsh};z^{\prime}}=$
$\displaystyle-2*(1+\epsilon)\int^{b_{\mathrm{max}}}_{b_{\mathrm{min}}}\biggr{|}\overbrace{\vec{v}_{\theta}\cdot\hat{x}^{\prime}}^{\text{Keplerian
}}+(\overbrace{\frac{dz}{dx}(\hat{y}^{\prime}\cdot\hat{x})(\hat{z}^{\prime}\cdot\hat{z})}^{\text{BW
shear }}+\omega_{z^{\prime}})y^{\prime}(b)\biggr{|}$ (A6)
$\displaystyle+\biggr{[}\overbrace{\vec{v}_{\theta}\cdot\hat{z}^{\prime}}^{\text{Keplerian
}}+(\overbrace{\frac{dz}{dx}(\hat{y}^{\prime}\cdot\hat{x})(\hat{z}^{\prime}\cdot\hat{z})}^{\text{BW
shear}}+\omega_{z^{\prime}})y^{\prime}(b)\biggr{]}y^{\prime}(b)\int^{H/2}_{-H/2}\rho_{s}(z(y^{\prime}(b),z^{\prime}))dbdz^{\prime}$

Finally the torque caused by the radial gradient of the vertical acceleration
due to the bending wave:

$\vec{\tau}_{\mathrm{BWg},x^{\prime}}=-\frac{2}{3}\omega^{\prime
2}\frac{\partial z}{\partial
y^{\prime}}(\hat{z}\cdot\hat{z}^{\prime})(\hat{y}^{\prime}\times\hat{z}^{\prime})\rho_{H}HW\left(\frac{L}{2}\right)^{3}$
(A7)

$\vec{\tau}_{\mathrm{BWg},y^{\prime}}=-\frac{2}{3}\omega^{\prime
2}\frac{\partial z}{\partial
x^{\prime}}(\hat{z}\cdot\hat{z}^{\prime})(\hat{x}^{\prime}\times\hat{z}^{\prime})\rho_{H}HL\left(\frac{W}{2}\right)^{3}$
(A8)

$\vec{\tau}_{\mathrm{BWg},z^{\prime}}=-\frac{2}{3}\omega^{\prime
2}\frac{\partial z}{\partial
y^{\prime}}(\hat{z}\cdot\hat{x}^{\prime})(\hat{y}^{\prime}\times\hat{x}^{\prime})\rho_{H}HW\left(\frac{L}{2}\right)^{3}$
(A9)

##  Appendix B Table with the UVIS occultations of the Mimas 5:3 bending wave
and their geometry

The below table collects all UVIS occultations with $\tau_{\mathrm{max}}>1.5$
for the Mimas 5:3 bending wave region.

Occultation | $B\,\textrm{[deg]}$ | $\phi\,\textrm{[deg]}$ | $B_{\mathrm{eff}}\,\textrm{[deg]}$ | Lon. [deg] | $Z$ | $\textrm{sign}(\dot{Z})$ | $b$ | $I_{0}$ | $\tau_{\mathrm{max}}$ | $\Phi_{\mathrm{theory}}\textrm{[deg]}$
---|---|---|---|---|---|---|---|---|---|---
AlpAra(105)E | 54.4 | 77.1 | 80.9 | 125.7 | -4046 | -1 | 9950 | 16.94937 | 3.2 | 224
AlpAra(105)I | 54.4 | 355.8 | 54.5 | 314.9 | 4072 | -1 | 11381 | 18.1537 | 3.5 | 83
AlpAra(32)I | 54.4 | 277.9 | 84.4 | 258.6 | 3837.8 | -1 | 38303 | 75.4764 | 6.1 | 312
AlpAra(33)I | 54.4 | 278 | 84.3 | 12.4 | 2512 | 1 | 36362 | 84.57951 | 5.5 | 115
AlpAra(35)E | 54.4 | 116.1 | 72.6 | 343.7 | -4739.8 | -1 | 36982 | 75.47634 | 3.8 | 268
AlpAra(63)E | 54.4 | 106.8 | 78.3 | 221.1 | 3969.2 | -1 | 28626 | 38.80406 | 5.5 | 280
AlpAra(79)I | 54.4 | 6.5 | 54.6 | 207.1 | 688.9 | 1 | 24713 | 32.6135 | 3.6 | 34
AlpAra(85)E | 54.4 | 93.4 | 87.6 | 85.8 | 5119.7 | 1 | 24444 | 24.13635 | 3.5 | 61
AlpAra(85)I | 54.4 | 5.7 | 54.6 | 233.9 | 2256.6 | 1 | 24897 | 28.06419 | 3.5 | 306
AlpAra(86)E | 54.4 | 94.1 | 87 | 147.2 | 2714.7 | 1 | 21499 | 26.68124 | 3.5 | 118
AlpAra(86)I | 54.4 | 5 | 54.5 | 299.1 | -2625.6 | 1 | 22373 | 28.52165 | 3.5 | 348
AlpAra(90)E | 54.4 | 93.7 | 87.3 | 103 | 4970.2 | -1 | 20224 | 21.15087 | 3.6 | 7
AlpAra(90)I | 54.4 | 5.2 | 54.5 | 252.2 | 3600.4 | 1 | 20057 | 27.9169 | 3.5 | 251
AlpAra(96)E | 54.4 | 82.6 | 84.7 | 124.1 | 3156.1 | -1 | 19349 | 19.06265 | 3.5 | 141
AlpAra(96)I | 54.4 | 12.8 | 55.1 | 254.8 | 4942.7 | 1 | 19563 | 41.38505 | 3.6 | 271
AlpAra(98)E | 54.4 | 75.4 | 79.8 | 110.7 | 131.2 | -1 | 17371 | 56.24169 | 3.6 | 231
AlpAra(98)I | 54.4 | 8.4 | 54.7 | 250.9 | 4825.8 | -1 | 17202 | 65.79578 | 3.6 | 321
AlpCMa(274)E | 13.5 | 40.4 | 17.5 | 188.9 | 2833.2 | -1 | 23747 | 68.66385 | 1.7 | 246
AlpCMa(281)I | 13.5 | 230.1 | 20.5 | 144.3 | 3652.3 | 1 | 27911 | 57.7524 | 1.7 | 143
AlpCru(100)E | 68.2 | 93.9 | 88.5 | 188.8 | 807.8 | 1 | 428757 | 501.3685 | 5.8 | 289
AlpCru(100)I | 68.2 | 154.5 | 70.1 | 213.6 | -5016.5 | 1 | 434175 | 501.3685 | 6 | 102
AlpCru(92)I | 68.2 | 169.6 | 68.5 | 114.7 | -4232.1 | 1 | 518555 | 501.3384 | 6.3 | 160
AlpLup(248)E | 53.9 | 111.4 | 75.1 | 73.2 | 4024.3 | -1 | 16967 | 223.1837 | 4.8 | 150
AlpLyr(175)I | -35.2 | 233.8 | 50.1 | 318.5 | 1522.8 | -1 | 8139 | 21.61554 | 4 | 104
AlpLyr(202)E | -35.2 | 35 | 40.7 | 191.8 | -4393 | -1 | 8139 | 21.61554 | 3.7 | 147
AlpLyr(202)I | -35.2 | 236 | 51.6 | 90.2 | 3092.6 | -1 | 6485 | 19.1991 | 2.9 | 277
AlpLyr(206)I | -35.2 | 252.9 | 67.4 | 151.7 | -1264.5 | -1 | 6801 | 17.57583 | 3.5 | 82
AlpScoB(13)E | 32.2 | 117.8 | 53.4 | 255.1 | -831.6 | -1 | 3409 | 104.8678 | 1.8 | 204
AlpScoB(13)I | 32.2 | 195.6 | 33.1 | 216.3 | 2447 | -1 | 3499 | 123.7631 | 1.8 | 321
AlpScoB(29)I | 32.2 | 286.5 | 65.7 | 343.9 | -2274.3 | 1 | 3452 | 62.04907 | 2.3 | 173
AlpVir(116)I | 17.3 | 243.3 | 34.7 | 188.8 | 5001 | -1 | 161196 | 55.39598 | 1.7 | 22
AlpVir(124)E | 17.3 | 123.3 | 29.5 | 218.4 | -2033.2 | 1 | 153398 | 203.5833 | 2.2 | 138
AlpVir(134)I | 17.3 | 284.9 | 50.4 | 215.4 | -637.2 | 1 | 158112 | 311.6297 | 2.2 | 346
AlpVir(173)E | 17.3 | 93 | 80.4 | 329.3 | -666.8 | -1 | 124007 | 200.0026 | 1.7 | 265
AlpVir(173)I | 17.3 | 31.2 | 20 | 71.1 | 2787.3 | -1 | 124993 | 200.0026 | 1.7 | 358
AlpVir(210)I | 17.3 | 311.5 | 25.1 | 46 | -4015.8 | 1 | 138215 | 43.22061 | 2.5 | 259
AlpVir(211)I | 17.3 | 267.2 | 81.2 | 103.3 | -2632.7 | 1 | 132499 | 469.4783 | 1.8 | 231
AlpVir(232)E | 17.3 | 89.3 | 87.8 | 316.7 | -2068.6 | 1 | 125750 | 455.5824 | 2.3 | 284
AlpVir(30)I | 17.3 | 230.4 | 26 | 6.9 | -2187.3 | 1 | 532397 | 177.7105 | 2.5 | 262
AlpVir(34)E | 17.3 | 332.9 | 19.2 | 259.5 | 2584 | 1 | 489542 | 247.7773 | 1.7 | 207
AlpVir(34)I | 17.3 | 232.8 | 27.2 | 24 | 579.6 | 1 | 510631 | 145.8352 | 2.2 | 225
AlpVir(8)E | 17.3 | 91.3 | 85.9 | 197.2 | 3679.9 | -1 | 526802 | 2500.032 | 1.7 | 20
AlpVir(8)I | 17.3 | 141.1 | 21.8 | 162.3 | 4488.6 | -1 | 500231 | 2499.995 | 1.8 | 145
BetCen(102)I | 66.7 | 249.1 | 81.3 | 0.9 | 4984.3 | -1 | 370169 | 329.3598 | 7.3 | 55
BetCen(104)E | 66.7 | 105.5 | 83.5 | 299.7 | -3402.1 | 1 | 342974 | 288.9692 | 8.6 | 155
BetCen(104)I | 66.7 | 209.3 | 69.4 | 321 | -1126.9 | -1 | 357501 | 312.9495 | 6.7 | 304
BetCen(105)E | 66.7 | 100.5 | 85.5 | 47 | -3710.5 | -1 | 287316 | 145.6972 | 6.3 | 353
BetCen(105)I | 66.7 | 212.9 | 70.1 | 83.1 | 5023 | 1 | 313435 | 151.4026 | 5.2 | 61
BetCen(64)E | 66.7 | 101.7 | 85 | 282.5 | 1854.5 | 1 | 619944 | 469.1418 | 5.6 | 286
BetCen(75)I | 66.7 | 270.5 | 89.8 | 232.9 | -934 | 1 | 594336 | 465.1562 | 7.6 | 273
BetCen(77)E | 66.7 | 47.5 | 73.8 | 12.6 | 4961 | 1 | 593240 | 336.3192 | 7.4 | 160
BetCen(77)I | 66.7 | 270.4 | 89.8 | 257.3 | -1823.4 | 1 | 587228 | 252.7589 | 7.7 | 165
BetCen(78)E | 66.7 | 45.9 | 73.3 | 206.5 | -5087.4 | -1 | 564075 | 375.1392 | 7 | 293
BetCen(81)I | 66.7 | 275.6 | 87.6 | 147.8 | 4636 | -1 | 550954 | 281.519 | 7.1 | 19
BetCen(85)I | 66.7 | 277.5 | 86.8 | 41.8 | -4526.9 | -1 | 533158 | 354.402 | 7.6 | 210
BetCen(89)I | 66.7 | 278 | 86.6 | 322.8 | -572.3 | 1 | 498234 | 330.1721 | 7.7 | 277
BetCen(92)E | 66.7 | 55 | 76.2 | 75.5 | 4143.3 | -1 | 452618 | 230.9221 | 6.2 | 319
BetCen(96)I | 66.7 | 271.9 | 89.2 | 126.3 | -4796.6 | -1 | 444264 | 229.5003 | 6.3 | 239
BetCMa(211)I | 14.2 | 223.4 | 19.2 | 67.1 | -3637.3 | 1 | 43518 | 50.86847 | 1.8 | 1
BetCru(253)I | 65.2 | 255.6 | 83.4 | 187.3 | -4002.7 | -1 | 106166 | 202.3216 | 6.5 | 157
BetCru(262)I | 65.2 | 257.3 | 84.2 | 18 | 5014.2 | -1 | 109532 | 185.7413 | 6.4 | 345
BetCru(98)I | 65.2 | 192.7 | 65.7 | 319 | 3954.6 | -1 | 274051 | 232.4198 | 7.1 | 248
BetPer(116)E | -47.4 | 143 | 53.7 | 216.7 | 1129.6 | 1 | 2135 | 43.00056 | 2.7 | 1
BetPer(116)I | -47.4 | 165.1 | 48.4 | 208.1 | -79.3 | 1 | 2130 | 43.00056 | 2.8 | 22
BetPer(42)I | -47.4 | 229.7 | 59.2 | 122.9 | -4479.8 | -1 | 20003 | 43.61236 | 3 | 244
ChiCen(39)I | 47.6 | 175.5 | 47.6 | 101.6 | 4977.8 | 1 | 13121 | 59.66402 | 3.8 | 345
DelCen(183)E | 55.6 | 149.3 | 59.5 | 177.7 | 3370.9 | 1 | 11397 | 217.076 | 4.6 | 5
DelCen(185)E | 55.6 | 150 | 59.3 | 44.2 | -1532 | -1 | 10924 | 200.8173 | 4.5 | 336
DelCen(191)E | 55.6 | 138.4 | 62.9 | 335.4 | -1934.9 | 1 | 10932 | 211.6986 | 4.6 | 31
DelCen(194)I | 55.6 | 294.7 | 74 | 270.1 | -5045.4 | 1 | 11891 | 22.14481 | 4.1 | 54
DelCen(64)E | 55.6 | 110.6 | 76.4 | 194.3 | 4773.2 | 1 | 52909 | 90.11278 | 5 | 327
DelCen(64)I | 55.6 | 124.8 | 68.6 | 210.7 | 3089.3 | 1 | 53150 | 90.11546 | 4.9 | 229
DelCen(66)I | 55.6 | 135.4 | 64 | 124 | 3831.5 | -1 | 58632 | 45.05706 | 4.7 | 311
DelCen(98)I | 55.6 | 211.3 | 59.6 | 54.8 | 402.3 | 1 | 34467 | 90.11549 | 5.8 | 100
Occultation | $B\,\textrm{[deg]}$ | $\phi\,\textrm{[deg]}$ | $B_{\mathrm{eff}}\,\textrm{[deg]}$ | Lon. [deg] | $Z$ | $\textrm{sign}(\dot{Z})$ | $b$ | $I_{0}$ | $\tau_{\mathrm{max}}$ | $\Phi_{\mathrm{theory}}\textrm{[deg]}$
---|---|---|---|---|---|---|---|---|---|---
DelPer(36)E | -54 | 66.5 | 73.9 | 296 | 2858.5 | -1 | 13677 | 376.012 | 3.9 | 357
DelPer(37)I | -54 | 264.5 | 86 | 145.7 | 4558.5 | -1 | 13632 | 37.60231 | 5 | 29
DelPer(39)I | -54 | 264.6 | 86.1 | 45 | -4744 | -1 | 12741 | 38.59457 | 4.2 | 203
DelPer(41)I | -54 | 238.6 | 69.3 | 166.7 | -496 | -1 | 12185 | 63.56438 | 3.6 | 14
DelPer(60)I | -54 | 277.2 | 84.8 | 192.3 | 4814.5 | -1 | 11711 | 20.13449 | 3.6 | 16
DelSco(236)I | 28.7 | 263.2 | 77.9 | 10.8 | -5022.7 | -1 | 22981 | 647.6574 | 2.8 | 171
EpsCas(104)E | -70 | 122.3 | 79 | 123.6 | -4468.4 | 1 | 4557 | 37.14048 | 3.6 | 300
EpsCas(104)I | -70 | 187.8 | 70.1 | 152.8 | -2034.7 | -1 | 4586 | 37.14048 | 3.8 | 87
EpsCen(65)I | 59.6 | 226.6 | 68.1 | 343.1 | -1573.5 | -1 | 129715 | 127.6824 | 6.6 | 220
EpsCMa(173)I | 26 | 267 | 84 | 187.1 | 4165.8 | 1 | 40005 | 13.82051 | 2.8 | 341
EpsCMa(174)I | 26 | 267.1 | 84 | 47.5 | -3225.3 | -1 | 44805 | 15.57652 | 2.8 | 344
EpsCMa(276)E | 26 | 50.6 | 37.5 | 217.7 | 4192 | -1 | 69532 | 117.8264 | 2.6 | 109
EpsLup(36)E | 51 | 44.9 | 60.2 | 135.3 | 2840.1 | -1 | 32671 | 71.62883 | 5.7 | 100
EpsLup(37)E | 51 | 359.8 | 51 | 356.7 | -1132.5 | 1 | 31958 | 72.21977 | 3.7 | 135
EtaLup(34)E | 44.5 | 357.1 | 44.5 | 90 | -3167.9 | -1 | 46980 | 82.68831 | 4.2 | 353
EtaLup(34)I | 44.5 | 296.2 | 65.7 | 218.8 | 2575.6 | -1 | 47613 | 83.364 | 3.4 | 130
GamAra(37)I | 61 | 248.7 | 78.6 | 62.1 | 4684.1 | 1 | 26751 | 58.10375 | 4.2 | 133
GamCas(100)E | -66.3 | 72.5 | 82.5 | 93.5 | -3408.3 | 1 | 54160 | 93.44911 | 6.1 | 259
GamCol(173)E | -39.9 | 73.9 | 71.7 | 317.7 | 5020.4 | -1 | 1097 | 57.81785 | 1.9 | 226
GamCol(205)I | -42.6 | 260.4 | 79.7 | 269.1 | -2977.3 | -1 | 843 | 44.80865 | 2 | 354
GamGru(40)E | 35.1 | 193.1 | 35.8 | 187.8 | 4856.5 | -1 | 7292 | 122.9506 | 3.2 | 32
GamGru(41)E | 35.1 | 204.5 | 37.7 | 98.5 | -1794.7 | -1 | 7579 | 158.2192 | 2.6 | 122
GamGru(41)I | 35.1 | 282.7 | 72.6 | 98.2 | 4282.8 | -1 | 8154 | 166.3223 | 2.2 | 225
GamLup(30)E | 47.4 | 114.6 | 69 | 173.1 | -3263.2 | 1 | 77667 | 59.06165 | 4.6 | 303
GamLup(32)E | 47.4 | 33.8 | 52.6 | 158.4 | 5097.4 | 1 | 72586 | 72.70017 | 5.4 | 306
GamPeg(172)E | -20.3 | 74.1 | 53.5 | 209.7 | 5059.2 | -1 | 12683 | 12.42252 | 1.8 | 295
GamPeg(172)I | -20.3 | 36.7 | 24.7 | 283.4 | 4683.3 | 1 | 11491 | 12.42252 | 1.7 | 328
GamPeg(211)E | -20.3 | 122.5 | 34.6 | 207.2 | 518.8 | -1 | 13192 | 16.91014 | 1.8 | 200
GamPeg(32)I | -20.3 | 138.6 | 26.2 | 226.6 | 2857.3 | 1 | 74756 | 265.8262 | 1.8 | 343
GamPeg(36)E | -20.3 | 66.8 | 43.2 | 343.5 | 3206.1 | 1 | 70975 | 225.4201 | 1.8 | 60
KapCen(35)E | 48.5 | 85.5 | 86.1 | 302.8 | -5074.8 | 1 | 46015 | 57.75644 | 5.8 | 281
KapCen(36)I | 48.5 | 241.2 | 67 | 228.6 | -2480.8 | -1 | 44129 | 119.2304 | 5 | 330
KapCen(42)I | 48.5 | 168.5 | 49.1 | 237.3 | 228.4 | 1 | 41129 | 119.2288 | 4.1 | 88
KapCMa(168)E | 29.3 | 127.3 | 42.8 | 286.8 | -1613.5 | 1 | 6193 | 43.59726 | 1.7 | 229
KapCMa(168)I | 29.3 | 175.9 | 29.4 | 268.8 | -3886.5 | 1 | 6062 | 43.59726 | 1.7 | 270
KapSco(247)I | 43.4 | 273.4 | 86.4 | 326.7 | 4703.2 | -1 | 21356 | 409.719 | 3 | 21
LamSco(248)I | 41.7 | 283.9 | 75 | 261.3 | 585.4 | -1 | 58157 | 433.8854 | 3.7 | 343
LamSco(29)E | 41.7 | 148.3 | 46.3 | 222.1 | -4753.7 | -1 | 283952 | 159.058 | 4.6 | 35
LamSco(44)I | 41.7 | 235 | 57.2 | 62.7 | -841.5 | -1 | 252102 | 136.4883 | 6.1 | 254
MuCen(113)I | 48.7 | 239.1 | 65.8 | 24.1 | -3432.9 | -1 | 9633 | 23.36014 | 3.7 | 81
MuSco(43)E | 43.4 | 27 | 46.7 | 303.3 | 4005.9 | -1 | 86896 | 441.6879 | 4 | 130
PsiCen(38)I | 44.3 | 249.9 | 70.6 | 54.2 | 3180.3 | 1 | 1110 | 58.71554 | 2.2 | 137
SigSgr(11)I | 29.1 | 239.5 | 47.6 | 111 | -2828 | -1 | 118989 | 995.783 | 3.4 | 85
SigSgr(114)I | 29.1 | 330.2 | 32.6 | 262.3 | -492.7 | 1 | 33500 | 40.79908 | 2.7 | 160
SigSgr(244)E | 29.1 | 234.2 | 43.5 | 103.7 | 204.5 | 1 | 19555 | 40.79669 | 1.9 | 262
SigSgr(244)I | 29.1 | 269.3 | 88.8 | 86.6 | -1341.1 | 1 | 16595 | 40.79913 | 2.1 | 313
TheAra(40)E | 53.9 | 28.6 | 57.3 | 197.2 | -2036.3 | -1 | 12489 | 46.22136 | 4.1 | 270
TheAra(40)I | 53.9 | 356.2 | 53.9 | 317 | 4476.6 | -1 | 12490 | 46.22205 | 3.7 | 66
TheAra(41)E | 53.9 | 82.4 | 84.5 | 205.4 | 1720.2 | -1 | 11762 | 46.22204 | 5.5 | 194
TheCar(190)I | -43.3 | 252.4 | 72.2 | 219.3 | -4463.7 | 1 | 25334 | 159.9588 | 5.7 | 277
ZetCen(112)I | 53.6 | 239.7 | 69.6 | 112.5 | 4393.8 | -1 | 37307 | 34.73377 | 6.4 | 346
ZetCen(246)E | 53.6 | 79.3 | 82.2 | 254.4 | -4100.4 | 1 | 22256 | 232.1636 | 5.1 | 144
ZetCen(60)I | 53.6 | 228.1 | 63.8 | 258.7 | -4185.7 | -1 | 107181 | 113.7307 | 6.6 | 235