# Ring Seismology of the Ice Giants Uranus and Neptune

Joseph A. A’Hearn Department of Physics, University of Idaho, Moscow, Idaho,
USA  Matthew M. Hedman Department of Physics, University of Idaho, Moscow,
Idaho, USA  Christopher R. Mankovich California Institute of Technology,
Pasadena, California, USA  Hima Aramona Reed College, Portland, Oregon, USA
Mark S. Marley Department of Planetary Sciences and Lunar and Planetary
Laboratory, University of Arizona, Tucson, Arizona, USA Joseph A’Hearn
<EMAIL_ADDRESS>
###### Abstract

We assess the prospect of using ring seismology to probe the interiors of the
ice giants Uranus and Neptune. We do this by calculating normal mode spectra
for different interior models of Uranus and Neptune using the stellar
oscillation code GYRE. These spectra provide predictions of where in these
planets’ ring systems the effects of interior oscillations might be detected.
We find that f-mode resonances with azimuthal order $m=2$ or $7\leq m\leq 19$
fall among the inner rings (6, 5, 4, $\alpha$, and $\beta$) of Uranus, while
f-mode resonances with $2\leq m\leq 12$ fall in the tenuous $\zeta$ ring
region. In addition, f-mode resonances with $m=2$ or $6\leq m\leq 13$ may give
azimuthal structure to Neptune’s tenuous Galle ring. We also find that g-mode
resonances may fall in the middle to outer rings of these planets. Although an
orbiter is most likely required to confirm the association between any waves
in the rings and planetary normal modes, the diversity of normal mode spectra
implies that identification of just one or two modes in the rings of Uranus or
Neptune would eliminate a variety of interior models, and thus aid in the
interpretation of Voyager observations and future spacecraft measurements.

##  1 Introduction

Three decades after the Voyager flybys of Uranus and Neptune, our knowledge of
their internal structure and composition is still quite limited (Helled &
Fortney, 2020). The ice giants Uranus and Neptune represent a distinct class
of planets with radii between those of terrestrial worlds like Earth and Venus
and gas giants like Jupiter and Saturn. The recent discovery that Neptune-mass
exoplanets are common (Suzuki et al., 2016) motivates the exploration of the
ice giants in our own solar system.

The deep interiors of the ice giants Uranus and Neptune are of special
interest because their internal structure and composition are distinct from
the gas giants Jupiter and Saturn. Their measured densities suggest that
Uranus and Neptune have substantial amounts of carbon, nitrogen, oxygen,
sulfur, and silicon, which form compounds like water, ammonia, methane, and
silicate minerals (Podolak et al., 2019). At the high pressures and
temperatures of planetary interiors, these “rocks” and “ices” display
interesting properties. Water, for example, can enter a superionic state where
the oxygen atoms become a lattice and the hydrogen nuclei are free to move,
which may play a role in explaining the non-axisymmetric non-dipolar magnetic
fields of Uranus and Neptune (Cavazzoni et al., 1999; Wilson et al., 2013).
Although numerical calculations and laboratory experiments are providing
better constraints on these exotic phases of these materials (Knudson et al.,
2012; Bethkenhagen et al., 2017; Millot et al., 2018, 2019; French &
Nettelmann, 2019), and methods to constrain interiors through high-precision
gravity measurements are being developed (Movshovitz & Fortney, 2022), there
remain many uncertainties regarding important parameters such as the mixing
properties of the various compounds. In addition, both the overall breakdown
and spatial distributions of hydrogen, helium, water, ammonia, methane, and
silicates within the planets are poorly constrained.

While the interiors of Uranus and Neptune remain largely hidden, their ring
systems are available for easier observation. Recent analyses of Saturn’s
rings have demonstrated that certain ring features are likely generated by
resonances with normal modes inside the planet (Hedman & Nicholson, 2013,
2014; French et al., 2016, 2019; Hedman et al., 2019; French et al., 2021;
Hedman et al., 2022), confirming predictions made by Marley & Porco (1993) and
providing new insights into that planet’s interior. Furthermore, Marley et al.
(1988) performed preliminary calculations and found that resonances with a few
Uranian normal modes could potentially fall close to some of the Uranian rings
(see Table 6 in the Appendices). This paper therefore seeks to build upon that
work and establish which planetary normal modes are most likely to fall close
to the rings of Uranus and Neptune and thus are the most promising candidates
for performing ring seismology at the ice giants.

In Section 2, we provide a brief overview of ring seismology and summarize the
current state of knowledge about the ring systems. In Section 3, we describe
the interior models that we use, and then we explain how we analyze the
gravitational potential and calculate the resonance locations. In Section 4,
we report the locations of these resonances and compare them with relevant
structures in the ring systems of the two planets. In Section 5, we discuss
which of the planetary normal mode resonances are most likely to be detectable
in the rings, with remarks on several individual rings of interest. Complete
tables of mode frequencies and associated resonance locations are provided in
the Appendices.

##  2 Background

Before describing our methods for computing normal mode resonance locations,
we first provide a brief overview of giant planet seismology in Subsection 2.1
and the relevant features in the rings of Uranus and Neptune in Subsection
2.2.

###  2.1 Overview of Ring Seismology

Planetary oscillations can be decomposed into a set of normal modes, each of
which oscillates at a frequency that depends on the planet’s profiles of
density, adiabatic sound speed, and rotation frequency. Familiar families of
oscillation modes include g-modes, whose restoring force is buoyancy, and
p-modes, whose restoring force is pressure. Three numbers characterize
planetary normal-mode oscillations: the number of radial nodes $n$, the
spherical harmonic degree $\ell$, and the azimuthal wavenumber $m$. In this
paper, we adopt the convention that $m>0$ correspond to prograde modes, i.e.
modes that propagate in the same direction as the planet’s rotation. While the
mode amplitude spectrum is generally unknown, for the simplest assumption of
energy equipartition, the strongest perturbations in the planet’s
gravitational field are produced by the fundamental modes ($n=0$; f-modes),
which can arise from a combination of predominantly gravity but also pressure
as the restoring force (Unno et al., 1979). Saturn’s f-modes, for example,
produce the most obvious features in its rings (Hedman & Nicholson, 2013,
2014; French et al., 2016, 2019, 2020; Hedman et al., 2019), as predicted by
Marley & Porco (1993). For this reason, we will primarily consider the f-modes
here.

Because the oscillation frequencies depend on the planet’s density profile,
measurements of these frequencies can probe the planet’s internal structure.
Efforts to detect oscillations with visible photometry, which were apparently
successful for Jupiter (Gaulme et al., 2011), have not yet been successful for
the ice giants (Rowe et al., 2017; Gaulme, 2017). Friedson (2020) found that
for reasonable amplitudes, detection of pressure or temperature variations due
to ice giant normal modes is not as promising as the prospect of detecting
their gravitational influence on an orbiting spacecraft.

Fortunately, we can potentially also detect planetary normal modes by treating
the ring material that orbits the planet as a seismograph. Any even $\ell-m$
mode is symmetric about the equator and would generate a Lindblad resonance in
ring material, which excites density waves; whereas any odd $\ell-m$ mode is
antisymmetric about the equator and would generate a vertical resonance, which
excites bending waves. The $\ell=m$ modes are the modes that are expected to
be the most easily observable at their Lindblad resonance locations because
the amplitude of the gravitational perturbation in the ring plane suffers no
destructive interference due to latitudinal variations in the phase of the
planetary oscillation. Because Saturn’s rings are the largest in our solar
system and the dataset on them was the most extensive, they were naturally the
first target for ring seismology. Voyager images and radio occultation
profiles of Saturn’s rings revealed spiral density waves and bending waves.
Some of these waves could be explained in terms of resonances with Saturn’s
moons, but other waves in the C ring were far from any known satellite
resonance (Rosen et al., 1991). Meanwhile, Marley (1991) and Marley & Porco
(1993), building upon ideas from Stevenson (1982), showed that certain normal
modes in Saturn’s interior could cause gravitational perturbations in the C
ring, and proposed potential correlations between some of the waves and
specific planetary f-modes. Once the Cassini mission arrived at Saturn, new
waves were detected (Baillié et al., 2011) and several features were confirmed
to be generated by resonances with planetary oscillations and asymmetries
(Hedman & Nicholson, 2013, 2014; French et al., 2016, 2019; Hedman et al.,
2019; French et al., 2021; Hedman et al., 2022). These studies yielded the
azimuthal wavenumber and the precise frequency for a set of planetary normal
modes that provide evidence for a stably stratified layer within the planet
(Fuller, 2014), an estimate for Saturn’s bulk rotation rate (Mankovich et al.,
2019), evidence for a diffuse core (Mankovich & Fuller, 2021), and constraints
on differential rotation (Dewberry et al., 2021).

###  2.2 The Rings and Inner Moons of Uranus and Neptune

Table 1 displays the semi-major axes and eccentricities of the thirteen
innermost Uranian moons and the seven innermost Neptunian moons. Table 2 shows
the semi-major axes and widths of the inner Uranian and Neptunian rings.

The Uranian ring system includes three broad rings ($\zeta$, $\nu$, and $\mu$)
and ten narrow rings (6, 5, 4, $\alpha$, $\beta$, $\eta$, $\gamma$, $\delta$,
$\lambda$, and $\epsilon$). The innermost Uranian moons, Cordelia and Ophelia,
each with diameters of $\sim 40$ km (Karkoschka, 2001), flank the $\lambda$
and $\epsilon$ rings and play a role in shepherding the $\epsilon$ ring
(French et al., 1991). The narrow rings except the $\lambda$ ring are
optically thick at visible wavelengths and are expected to be dominated by
centimeter- to meter-sized particles (Nicholson et al., 2018). Several small
moons of diameters 40-135 km (Ophelia, Bianca, Cressida, Desdemona, Juliet,
and Portia; Karkoschka 2001) are located between the $\epsilon$ and $\nu$
rings, and several more, of diameters 20-160 km (Rosalind, Cupid, Belinda,
Perdita, Puck, and Mab Karkoschka 2001; Showalter & Lissauer 2006), between
the $\nu$ and $\mu$ rings. Portia and Puck are the largest moons in these
regions, with diameters of 135 and 160 km, respectively. Beyond Mab are the
five largest Uranian moons: Miranda, in a class of its own with a diameter of
470 km; and then Ariel, Umbriel, Titania, and Oberon, with diameters of
1150-1580 km (Thomas, 1988).

moon |  $a$ (km) | $e$
---|---|---
moons of Uranus |  |
Cordelia | 49,752 | 0.00026
Ophelia | 53,763 | 0.00992
Bianca | 59,166 | 0.00092
Cressida | 61,767 | 0.00036
Desdemona | 62,658 | 0.00013
Juliet | 64,358 | 0.00066
Portia | 66,097 | 0.00005
Rosalind | 69,927 | 0.00011
Cupid | 74,392 | -
Belinda | 75,256 | 0.00007
Perdita | 76,417 | 0.00329
Puck | 86,004 | 0.00012
Mab | 97,736 | 0.00254
moons of Neptune |  |
Naiad | 48,228 | 0.00014
Thalassa | 50,075 | 0.00019
Despina | 52,526 | 0.00027
Galatea | 61,953 | 0.00020
Larissa | 73,548 | 0.00121
Hippocamp | 105,253 | 0.00001
Proteus | 117,647 | 0.00047
Table 1: Semi-major axes $a$ and eccentricities $e$ of the inner moons of Uranus, from Jacobson (1998) and Showalter & Lissauer (2006); and of Neptune, from Brozović et al. (2020).  ring |  $\bar{a}$ (km) |  $\bar{a}e$ (km)
---|---|---
narrow rings of Uranus |  |
6 | 41,838 | 43
5 | 42,235 | 80
4 | 42,572 | 45
$\alpha$ | 44,719 | 34
$\beta$ | 45,661 | 20
$\eta$ | 47,176 | -
$\gamma$ | 47,627 | 5
$\delta$ | 48,301 | -
$\lambda$ | 50,024 | -
$\epsilon$ | 51,150 | 406
broad rings of Uranus |  |  $W$ (km)
$\zeta$ (Voyager) | 38,300 | 2500
$\zeta$ (Keck) | 39,600 | 3500
$\nu$ | 67,300 | 3800
$\mu$ | 97,700 | 17,000
rings of Neptune |  |  $W$ (km)
Galle | 42,000 | 2,000
Le Verrier | 53,200 | 100
Lassell | 55,200 | 4,000
Arago | 57,200 | -
Galatea co-orbital | 61,953 | -
Adams | 62,933 | 15 (in arcs)
Table 2: Parameters of the rings of Uranus, from Nicholson et al. (2018); and
of Neptune, from de Pater et al. (2018). $\bar{a}$ is the mean semi-major
axis.

With such a quantity of moons exterior to the narrow rings, it is important to
distinguish planetary normal mode resonances from resonances with satellites.
A wave generated by a Lindblad resonance with an exterior moon propagates
outward, whereas a wave generated by a Lindblad resonance with an interior
moon or with most planetary normal modes propagates inward. Because we know
where the moons are, we can calculate where their Lindblad resonances fall.
Some of these line up nicely with some of the outer rings of Uranus, but none
of them line up well with the inner rings (Hedman & Chancia, 2021). For
example, the 6:5 resonance with Ophelia falls within the $\gamma$ ring (Porco
& Goldreich, 1987; Hedman & Chancia, 2021), even though its kinematics also
includes $m=0$ and $m=1$ normal modes (French et al., 1991). The $\epsilon$
ring is shepherded by Cordelia and Ophelia: its inner edge coincides with the
24:25 outer eccentric resonance with Cordelia, while the outer edge coincides
with the 14:13 inner eccentric resonance with Ophelia (Goldreich & Porco,
1987; French et al., 1991; French & Nicholson, 1995; Nicholson et al., 2018).

The $\eta$ ring’s kinematics are influenced by a 3:2 inner Lindblad resonance
with Cressida (Chancia et al., 2017). Finally, the $\delta$ ring’s kinematics
are well modeled by a single $m=2$ normal mode of the ring itself (French et
al., 1991). Nevertheless, no known satellite resonances are found close to the
6, 5, 4, $\alpha$, and $\beta$ rings.

Neptune has one large moon, Triton, which orbits with a high inclination and
in the retrograde direction. Proteus orbits Neptune about one third the
distance to Triton and is less than 1/500 the mass of Triton (Davies et al.,
1991; Stooke, 1994), but is Neptune’s next largest moon and can be considered
the outermost of the inner moons. The other inner moons, from outward in, are
the recently discovered 35-km diameter moon Hippocamp (Showalter et al.,
2019); then three moons of diameters of 150-200 km, namely, Larissa, Galatea,
and Despina; and finally Thalassa and Naiad, which have diameters of 60-80 km
(Karkoschka, 2003).

de Pater et al. (2018) recently reviewed the current state of our knowledge of
Neptune’s rings. Neptune has two broad faint rings, the Galle ring and the
Lassell ring, and four narrow rings: the Le Verrier ring, the Arago ring, an
unnamed ring that is co-orbital with Galatea, and the Adams ring (see Table
2). The optical depth of the Le Verrier ring is comparable to that of the
Adams ring outside of the arcs, while the optical depth of the Galle and
Lassell rings is two orders of magnitude lower (Porco et al., 1995). It is
still not entirely clear how any of these narrow rings are confined, and the
rings seem to lack the fine-scale structure that Saturn’s rings have.

##  3 Methods

In this Section, we first describe the interior models we use in Subsection
3.1. Next we explain how our frequency calculations account for rotation in
Subsection 3.2. Then we show how we identify the sources of gravitational
potential perturbations in Subsection 3.3. Last, we describe how we calculate
the resonance locations in Subsection 3.4.

###  3.1 Interior Models

Figure 1: Density (solid lines) and Brunt-Väisälä frequency (dotted) for
models of Uranus (top) and Neptune (bottom) as functions of the fractional
radius $r/R$. The scales of the $y$-axes are set equal for both the top and
bottom panels for easier comparison between Uranus and Neptune models.

The adiabatic normal mode spectrum for a star or planet depends on the profile
of mass density $\rho$, adiabatic sound speed $c$, and rotation rate $\Omega$.
The adiabatic sound speed $c$ is defined (Unno et al., 1979)

$c^{2}=\Gamma_{1}\frac{P_{0}}{\rho_{0}},$ (1)

where $P_{0}$ and $\rho_{0}$ are the pressure and density of the unperturbed
state, and the adiabatic exponent $\Gamma_{1}$ is

$\Gamma_{1}=\left(\frac{\partial\ln P}{\partial\ln\rho}\right)_{\rm{ad}}.$ (2)

Non-adiabatic regions are characterized by a non-zero Brunt-Väisälä frequency
$N$:

$N^{2}=-\frac{GM_{r}}{r}\left(\frac{d\ln\rho}{d\ln
r}-\frac{1}{\Gamma_{1}}\frac{d\ln P}{d\ln r}\right),$ (3)

where $G$ is the gravitational constant, $M_{r}$ is the mass interior to the
radius $r$. For clarity, $M$ without the subscript $r$ will refer to the total
mass of the planet. The Brunt-Väisälä frequency $N$ quantifies the angular
frequency with which a small parcel of gas oscillates radially with positive
or negative buoyancy under local pressure balance with its surrounding gas
(Unno et al., 1979).

Here we consider spherically symmetric models so that $\rho=\rho(r)$,
$c=c(r)$, and make the further assumption of rigid rotation so that
$\Omega(r)=\Omega={\rm constant}$. Although the cloud-level jet streams alone
guarantee some degree of differential rotation in Uranus and Neptune (Kaspi et
al., 2013), the uncertainty in their underlying bulk rotation rates means that
rigidly rotating models are sufficient for our purposes.

The system of equations required to solve for the frequencies of the normal
mode spectrum is found in classic works like Unno et al. (1979), and there is
now a publicly available and extensively validated asteroseismology software
package called GYRE that solves these equations (Townsend & Teitler, 2013).
This code has recently been used successfully for Saturn (Mankovich et al.,
2019; Markham et al., 2020).

Most models of the interiors of the ice giants make a number of simplifying
assumptions, such as a three-layer structure: a rock-rich core, a water-rich
envelope, and a hydrogen-rich atmosphere. Many models also assume an adiabatic
interior (see, for example, Scheibe et al. 2019). Thermal evolution models,
however, suggest that the classical assumption of an adiabatic interior is
inconsistent with the luminosities of Uranus and Neptune, and for this reason,
models are being explored that are not fully adiabatic, but instead have a
thermal boundary layer (see, for example, Scheibe et al. 2021; Stixrude et al.
2021). The gravitational harmonics determined by the Voyager flybys and
observations of the dynamics of rings and moons as well as theoretical
considerations based on laboratory experiments work together to constrain the
properties of the planets’ layers. Some fundamental aspects of ice giant
interiors, such as the ice-to-rock mass fraction, are poorly constrained
(Helled et al., 2020; Podolak et al., 2019). Even the rotation rates measured
by Voyager for Uranus (Desch et al., 1986; Warwick et al., 1986) and Neptune
(Warwick et al., 1989) have been called into question (Helled et al., 2010).
Given these uncertainties, it is worth considering a relatively broad range of
models.

Figure 1 shows profiles of density and Brunt-Väisälä frequency of Uranus and
Neptune interior models that we use in this work. What we have labeled the
adiabatic model is from Scheibe et al. (2019). Adiabatic oscillation
calculations in GYRE require the thermodynamic derivatives $\Gamma_{1}$ (Eq.
2) and

$\nabla_{\rm
ad}=\frac{1}{\Gamma_{1}}\left(\frac{\partial\textrm{ln}T}{\partial\textrm{ln}\rho}\right)_{\rm
ad},$ (4)

which we calculate numerically from the adiabatic model’s $\rho(P)$ and
$T(P)$. Because this model assumes a three-layer structure and features
density discontinuities owing to sudden composition changes, double mesh
points were inserted at the core-envelope and inner-outer envelope boundaries
($r/R\sim 0.07$ and $r/R\sim 0.75$, respectively, where $R$ is the planetary
radius) in order for GYRE to apply the appropriate jump conditions at those
locations.

We considered a Saturn model from Mankovich & Fuller (2021) and confirmed that
our mode calculation reproduced results consistent with theirs, up to the
expected error associated with our first-order treatment of rotation (see
below).

We have also constructed new models of Uranus and Neptune using the method
described in Mankovich & Fuller (2021). In brief, these models are calculated
using the fourth-order theory of figures (Nettelmann, 2017), assuming H-He-H2O
mixtures modeled using the MH13-SCvH equation of state for hydrogen and helium
(Militzer & Hubbard, 2013; Saumon et al., 1995; Miguel et al., 2016) and
Mazevet et al. (2019) for water, combined in an additive-volume approximation.
The Mazevet et al. (2019) equation of state for water transitions to that of
an ideal gas below $T=800$ K, following Scheibe et al. (2019). Rather than the
common practice of imposing discontinuous changes in composition, these models
impose a gradient in the water abundance between a homogeneous H-He dominated
outer envelope and a homogeneous H2O-dominated interior. Denoting the H, He
and H2O mass fractions by $X$, $Y$, and $Z$ so that $X+Y+Z=1$, we fix
$Y/(X+Y)=0.275$ throughout the model following the protosolar value estimated
by Asplund et al. (2009), and for $Z(r)$ assume a sigmoid function with four
free parameters specifying the inner and outer radii of the gradient region
and the $Z$ values at those boundaries (see Mankovich & Fuller 2021 for
details). One of these parameters is eliminated by the condition that the
model satisfy each planet’s equatorial radius.

Despite the stabilizing influence of the composition gradient, these models
assume for simplicity that temperature is adiabatically stratified, subject to
the boundary condition that $T=150$ K at $P=10$ bar, consistent with
atmosphere models (Fortney et al., 2011).

We present mode calculations for three Uranus models, one fitting $J_{2}$ and
$J_{4}$ exactly, and two that are offset in $J_{4}$ relative to the observed
value by approximately $\pm 1$ times the measurement uncertainty
$\sigma_{J_{4}}=1.30\times 10^{-6}$ (Jacobson, 2014). The models achieve this
by varying the width of the $Z$ gradient region and are accordingly labeled
“thin”, “medium” and “thick”, corresponding to $J_{4}$ offsets of $-0.89$, 0,
and $+0.79$ times $\sigma_{J_{4}}$, respectively. Another Uranus model we
label “shallow” has a $Z$ gradient region closer to the planet’s surface. For
Neptune we present a single model that fits $J_{2}$ and $J_{4}$. These models
are summarized in Table 3 and their interior structures shown in Figure 1,
where the stably stratified composition gradient regions are visible as
regions with positive Brunt-Väisälä frequency. Although we are primarily
focusing our discussion on f-modes, the models with stably stratified
composition regions also generate g-modes, which in principle can mix with the
f-modes, and can also be associated with their own resonances.

###  3.2 Normal Mode Frequency Calculation

Because the ratios of rotation rate to breakup frequency of Uranus and
Neptune,

$\frac{\Omega}{\sqrt{\frac{GM}{R^{3}}}}=\begin{cases}0.175\pm
0.004,&\text{Uranus}\\\ 0.155\pm 0.006,&\text{Neptune}\end{cases}$ (5)

(with uncertainties dominated by uncertainties in the bulk rotation rate; see
Desch et al. 1986; Warwick et al. 1986, 1989; Helled et al. 2010), are
significant, Coriolis accelerations are not negligible compared to the other
terms in the momentum equation. For this reason, the oscillation frequencies
given by GYRE are corrected to first order in the planet’s rotation rate
$\Omega$ (Ledoux, 1951; Unno et al., 1979). These corrections account for the
Doppler shift and the approximate intrinsic perturbation to mode frequencies
due to the Coriolis force.

GYRE evaluates an integral to provide the rotation splitting coefficient
$\beta$ that is involved in the Coriolis perturbation for solid-body rotation.
$\beta$ can also be calculated for a given mode with the equation (cf. Unno et
al. 1979)

$\beta=1-\frac{\int_{0}^{R}\left(2\xi_{r}\xi_{h}+\xi_{h}^{2}\right)\rho
r^{2}dr}{\int_{0}^{R}\left[\xi_{r}^{2}+\ell\left(\ell+1\right)\xi_{h}^{2}\right]\rho
r^{2}dr}$ (6)

where $\xi_{r}$ and $\xi_{h}$ are the radial and horizontal components of the
displacement eigenfunction (Unno et al., 1979), which is given by

$\displaystyle\mathbf{\xi}\left(r,\theta,\phi,t\right)=$
$\displaystyle\Biggl{[}\xi_{r}\left(r\right)\hat{r}+\xi_{h}\left(r\right)\left(\hat{\theta}\frac{\partial}{\partial\theta}+\hat{\phi}\frac{1}{\sin\theta}\frac{\partial}{\partial\phi}\right)\Biggr{]}$
(7) $\displaystyle\times Y_{\ell}^{m}\left(\theta,\phi\right)e^{i\sigma_{\ell
mn}t},$

We did verify that the $\beta$ values returned by GYRE are consistent with
this formula.

Because Uranus and Neptune rotate more slowly than Jupiter and Saturn, we do
not include second-order terms associated with oblateness and the centrifugal
force, which together tend to decrease the frequencies (Vorontsov & Zharkov,
1981). For this reason, our uncertainties in calculated mode frequencies, and
thus resonance locations, are one-sided.

We calculate the moment of inertia for each model, assuming spherical
symmetry, by integrating

$\frac{I}{MR^{2}}=\frac{8\pi}{3}\frac{\int\rho\left(r\right)r^{4}dr}{MR^{2}}$
(8)

Table 3 shows the parameters that were used for each model, as well as each
model’s calculated moment of inertia.

###  3.3 Sources of Gravitational Potential Perturbations

Following Marley & Porco (1993), we can write the total gravitational
potential as a sum of an unperturbed component and a perturbed component:

$\Phi=\Phi_{0}+\Phi^{\prime}\left(t\right),$ (9)

where full expressions for the unperturbed component $\Phi_{0}$ and for the
perturbed component $\Phi^{\prime}\left(t\right)$ can be found in Marley &
Porco (1993). The integrals for the perturbed gravitational harmonics are
taken over the Eulerian density perturbation

$\rho_{\ell mn}^{\prime}=\rho_{\ell
n}^{\prime}\left(r\right)Y_{\ell}^{m}\left(\theta,\phi\right)e^{-i\sigma_{\ell
mn}t}$ (10)

instead of over the unperturbed density $\rho$. In the above equation,
$\sigma_{\ell mn}$ is the oscillation frequency in the reference frame that
rotates with the planet, $r$ is the radius, $\theta$ is the colatitude, and
$\phi$ is the azimuthal angle. The spherical harmonics $Y_{\ell}^{m}$ are
defined in terms of the associated Legendre polynomials $P_{\ell}^{m}$.

The equations for the gravitational harmonics that appear in the equations for
$\Phi^{\prime}\left(t\right)$ can be reduced to

$MR^{\ell}J_{\ell
n}^{\prime}=-\left(\frac{4\pi}{2\ell+1}\right)^{1/2}e^{i\sigma_{\ell
0n}t}\int_{0}^{R}\rho_{\ell n}^{\prime}\left(r\right)r^{\ell+2}dr$ (11)

for $m=0$, and

$\displaystyle MR^{\ell}C_{\ell mn}^{\prime}=$
$\displaystyle\left(-1\right)^{\frac{m+|m|}{2}}\Biggl{[}\frac{2\ell+1}{4\pi}\left(\frac{\left(\ell-|m|\right)!}{\left(\ell+|m|\right)!}\right)\Biggr{]}^{1/2}$
(12) $\displaystyle\times e^{i\sigma_{\ell mn}t}\int_{0}^{R}\rho_{\ell
mn}^{\prime}\left(r\right)r^{\ell+2}dr$

for $m\neq 0$. $S_{\ell mn}^{\prime}$ can be made to vanish by an appropriate
choice of phase (Marley & Porco, 1993).

Figure 2: (Top) Radial displacement eigenfunctions for $\ell=2,6,10$, and
$14$, normalized to $\xi_{r}=1$ at $r=R$, from the medium Uranus model. The
Brunt-Väisälä frequency $N$, normalized to peak at 1, is shown with dotted
lines. (Middle) Eulerian density perturbation $\rho^{\prime}\left(r\right)$
for the same four modes. (Bottom) Integrand of Equations 11 and 12,
$\rho^{\prime}r^{\ell+2}$, relevant for gravitational potential perturbations,
for the same modes. The modes of lower oscillation degree $\ell$ are more
sensitive to inner layers of the planet, while the modes of higher oscillation
degree $\ell$ are more sensitive to outer layers of the planet.

The radial displacement eigenfunctions of $\ell=2,6,10,$ and $14$ from our
medium model are plotted in the top panel of Figure 2, normalized to
$\xi_{r}=1$ at $r=R$. We also show the normalized Brunt-Väisälä frequency $N$
as a dotted line to highlight its correlation with the warp in the radial
displacement eigenfunctions. The middle panel of Figure 2 shows the Eulerian
density perturbation of the same modes $\ell=2,6,10,$ and $14$. The bottom
panel of Figure 2 shows the integrand of Equations 11 and 12,
$\rho^{\prime}r^{\ell+2}$, for each of these modes. These plots illustrate how
the modes of lower spherical harmonic degree $\ell$ can probe deeper into the
planet, whereas the modes of higher spherical harmonic degree $\ell$ are more
sensitive to the outer layers and to the parts of the planet that are stably
stratified.

Only the spherical harmonic $Y_{\ell}^{0}$ contributes to the $J_{\ell
n}^{\prime}$ term due to the orthogonality of the spherical harmonics.
Likewise, for $m\neq 0$, the contribution to each $C_{\ell mn}^{\prime}$ term
is only from the $Y_{\ell}^{m}$ harmonic. Each oscillation mode thus
contributes to a single perturbed gravitational harmonic. The planet’s
rotation, however, can mix modes slightly. For Saturn, no more than 15% of the
rotationally corrected radial displacement eigenfunction could be attributed
to mode mixing (Marley & Porco, 1993); for Uranus and Neptune, which rotate
more slowly than Saturn, we can expect mode mixing to influence the radial
displacement eigenfunction to an even lesser extent.

###  3.4 Resonance Calculation

We calculate the pattern frequency $\Omega_{\rm{pat}}$ of each normal mode
seen from inertial space,

$\Omega_{\rm{pat}}=\frac{1}{m}\left(\sigma_{\ell n}^{0}+m\beta\Omega\right),$
(13)

where $\sigma_{\ell n}^{0}$ is the oscillation frequency in the non-rotating
limit for the mode specified by $\ell$ and $n$, and $\beta$ is the rotation
splitting coefficient defined in Equation 6. Our models for Uranus assume the
rotation rate provided in Helled et al. (2010), called the fast rotation rate,
corresponding to a period of 16.58 hours. The adiabatic model assumes the
classical, or slow, rotation rate, corresponding to a period of 17.24 hours
(Desch et al., 1986; Warwick et al., 1986). Our Neptune model likewise uses
the rotation rate from Helled et al. (2010), corresponding to a period of
17.46 hours, which in contrast is slower than the classical rotation rate
provided by Voyager 2 radio data, corresponding to a period of 16.11 hours
(Warwick et al., 1989). In general, resonance locations of a faster rotator
fall further inward than those of a slower rotator. Thus the rotation rate of
these planets is another parameter that ring seismology could help constrain,
similar to how Mankovich et al. (2019) calculated a seismological rotation
rate for Saturn.

model |  $R$ (km) |  $M$ ($\times 10^{25}$ kg) |  $\Omega$ ($\times 10^{-4}$ s-1) |  $J_{2}$ ($\times 10^{-4}$) |  $J_{4}$ ($\times 10^{-4}$) | $I/MR^{2}$
---|---|---|---|---|---|---
Uranus shallow | 25,559 | 8.68009 | 1.0527 | 35.1068 | -0.341705 | 0.2279
Uranus thin | 25,559 | 8.68009 | 1.0527 | 35.1068 | -0.330171 | 0.2204
Uranus medium | 25,559 | 8.68009 | 1.0527 | 35.1068 | -0.341705 | 0.2202
Uranus thick | 25,559 | 8.68009 | 1.0527 | 35.1069 | -0.351949 | 0.2200
Uranus adiabatic† | 25,559 | 8.68009 | 1.0124 | 35.107 | -0.342 | 0.2266
Neptune | 24,764 | 10.24092 | 0.9996 | 34.0843 | -0.334 | 0.2509
Table 3: Parameters of the planetary models. Equatorial radii $R$ are from
Archinal et al. (2018). $M$, $J_{2}$, and $J_{4}$ of Uranus are from Jacobson
(2014), while $M$, $J_{2}$, and $J_{4}$ of Neptune are from Jacobson (2009).
The classical Uranus spin rate $\Omega$ is from Voyager 2 radio data (Desch et
al., 1986; Warwick et al., 1986) and is used only in the adiabatic model,
while alternate spin rates $\Omega$ based on the shapes and gravitational
coefficients of the planets are from Helled et al. (2010) and are used in the
other models. The classical Neptune spin rate $\Omega$, not used in any of our
models, is $1.0834\times 10^{-4}$ s-1 (Warwick et al., 1989). The moments of
inertia are calculated using Equation 8.
† Scheibe et al. (2019)

Then we calculate the resonance location numerically. Lindblad resonances
occur at locations where the following relationship is satisfied (Goldreich &
Tremaine, 1979):

$m\left(n-\Omega_{\rm{pat}}\right)=\pm q\kappa,$ (14)

where the upper sign corresponds to an inner Lindblad resonance and the lower
sign corresponds to an outer Lindblad resonance, $q$ is a positive integer,
and $n$ and $\kappa$ are the resonance location’s mean motion and horizontal
epicyclic frequency, respectively. These are calculated according to the
second-order equations from Renner & Sicardy (2006).

Similarly, vertical resonances occur at locations where

$m\left(n-\Omega_{\rm{pat}}\right)=\pm b\mu,$ (15)

where the upper sign corresponds to an inner vertical resonance and the lower
sign corresponds to an outer vertical resonance, $b$ is a positive integer,
and $\mu$ is the resonance location’s vertical epicyclic frequency, which is
found using the equation from Shu et al. (1983):

$\mu^{2}+\kappa^{2}=2n^{2}$ (16)

We are focusing only on first-order resonances, which have $q=b=1$, though
higher-order resonances are possible (Marley, 2014).

Corotation resonances are potentially interesting to consider in the context
of Neptune’s Adams ring (see Section 5.4). Hence we compute corotation
resonance locations for Neptune following the same procedures as in A’Hearn et
al. (2021). That is, we find the radii where the mean motion $n$ matches the
pattern frequencies $\Omega_{\rm{pat}}$ associated with the $\ell=m$ modes we
have calculated from our Neptune model.

##  4 Results

Figures 3 and 4 show the calculated resonance locations with f-mode
oscillations for all our Uranus models. Figure 3 shows the Lindblad
resonances, while Figure 4 shows the vertical resonances. We show the
resonance locations up to $\ell=25$. For every mode, the resonance location of
the medium model falls in between those of the thin and thick models. The
resonance location of the thick model typically falls further outward (and the
resonance location of the thin model falls further inward), the $\ell=m=2$
mode being the sole exception. This general trend is set by the models’
different predictions for density in the outer envelope, which the higher $m$
modes are more sensitive to (cf. Figure 2).

The region of particular interest that we find spans from the $6$ ring to the
$\beta$ ring. Satellite resonances do not line up with these inner rings
(Hedman & Chancia, 2021), but several high-$m$ mode resonances from our models
do fall in this region. The predictions from our models as well as from Marley
et al. (1988) include resonance locations that fall in the $\zeta$ ring for
the modes with $\ell\leq 5$. The $\zeta$ ring, however, has low optical depth,
and would not be expected to sustain a wave.

Figure 3: Normal mode outer Lindblad resonance (OLR) location predictions from
all five Uranus models. The azimuthal order $m$ (equivalent to the number of
spiral arms) is the vertical axis, and distance from the center of Uranus is
the horizontal axis, shown in Uranus radii on the bottom and in km on the top.
OLRs can excite inward-propagating spiral density waves. The typical
uncertainty of these resonance locations is shown by one bar on the left.
Figure 4: Normal mode outer vertical resonance (OVR) location predictions from
all five Uranus models. The azimuthal order $m$ (equivalent to the number of
spiral arms) is the vertical axis, and distance from the center of Uranus is
the horizontal axis, shown in Uranus radii on the bottom and in km on the top.
OVRs can excite outward-propagating bending waves. The typical uncertainty of
these resonance locations is shown by one bar on the left.

Lindblad resonances that fall among the inner rings are listed in Table 4,
whereas vertical resonances are listed in Table 5. The resonance locations are
given in general ranges, based on all our models as well as the frequency
uncertainty from our calculations with first-order rotation corrections.
Predictions based on individual interior models are found in the Appendices.
We group the 6, 5, and 4 rings together, and the $\alpha$ and $\beta$ rings
together, in these tables, because the uncertainty associated with a mode’s
frequency or resonance location is comparable to the distance between these
narrow rings. Differential rotation confined to the outer layers of the
interior will have a greater effect on the higher-degree modes.

Among the 6, 5, and 4 rings, we expect that a Lindblad resonance with $m=2$ or
$7\leq m\leq 13$, or a vertical resonance with $9\leq m\leq 14$, could be from
a planetary normal mode. Among the $\alpha$ and $\beta$ rings, we expect that
a Lindblad resonance with $11\leq m\leq 18$, or a vertical resonance with
$13\leq m\leq 19$, could be from a planetary normal mode.

$\ell$ | $m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
---|---|---|---
6, 5, or 4 ring | 41,838; 42,235; 42,572
2 | 2 | 1995.2-2360.4 | 38571-43142
7 | 7 | 1585.4-1753.2 | 39244-41963
8 | 8 | 1517.2-1666.7 | 40166-42760
9 | 9 | 1460.4-1591.0 | 41088-43498
10 | 10 | 1412.3-1525.1 | 41981-44184
12 | 10 | 1506.0-1605.7 | 40565-42334
13 | 11 | 1450.1-1537.1 | 41532-43175
14 | 12 | 1402.8-1479.3 | 42409-43935
16 | 12 | 1466.5-1537.2 | 41338-42655
17 | 13 | 1419.4-1482.8 | 42176-43420
$\alpha$ or $\beta$ ring | 44,719; 45,661
11 | 11 | 1370.9-1467.8 | 42827-44822
12 | 12 | 1334.6-1418.1 | 43618-45417
13 | 13 | 1301.1-1375.0 | 44350-46012
14 | 14 | 1271.6-1337.3 | 45024-46562
15 | 15 | 1245.4-1304.3 | 45644-47070
16 | 14 | 1326.8-1387.5 | 43933-45262
17 | 15 | 1295.6-1350.5 | 44598-45850
18 | 16 | 1267.7-1318.0 | 45211-46397
19 | 15 | 1342.5-1395.1 | 43644-44776
20 | 16 | 1310.5-1359.2 | 44294-45382
21 | 17 | 1281.9-1327.2 | 44897-45949
22 | 18 | 1256.1-1298.6 | 45460-46480
Table 4: Predicted Lindblad resonance locations among the inner rings of Uranus. The ranges in frequencies and locations take into account the two most extreme models and include the error in the range given.  $\ell$ | $m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
---|---|---|---
6, 5, or 4 ring | 41,838; 42,235; 42,572
10 | 9 | 1517.1-1642.0 | 40245-42420
11 | 10 | 1460.6-1566.9 | 41241-43216
12 | 11 | 1412.5-1503.4 | 42159-43945
14 | 11 | 1486.3-1569.6 | 40966-42481
15 | 12 | 1435.2-1508.5 | 41868-43281
17 | 12 | 1496.8-1565.4 | 40850-42086
18 | 13 | 1446.7-1508.4 | 41704-42882
19 | 14 | 1403.1-1459.4 | 42486-43614
$\alpha$ or $\beta$ ring | 44,719; 45,661
14 | 13 | 1332.1-1402.8 | 43769-45302
15 | 14 | 1299.6-1362.7 | 44471-45896
16 | 15 | 1270.9-1327.7 | 45115-46446
18 | 15 | 1319.4-1373.0 | 44117-45303
19 | 16 | 1289.4-1338.8 | 44748-45881
20 | 17 | 1262.6-1308.4 | 45333-46422
21 | 16 | 1331.0-1379.2 | 43872-44922
22 | 17 | 1300.7-1345.8 | 44491-45511
23 | 18 | 1273.4-1315.8 | 45069-46063
24 | 19 | 1248.7-1288.8 | 45612-46582
Table 5: Predicted vertical resonance locations near the inner rings of
Uranus.

Although planetary normal mode resonances with $m>15$ may fall among the
$\eta$, $\gamma$, $\delta$, $\lambda$, and $\epsilon$ rings, we expect the
dynamics of these rings to be influenced more by the moons Cordelia and
Ophelia. The case has been made that while Cordelia is the outer shepherd of
the $\delta$ ring and the inner shepherd of the $\epsilon$ ring, Ophelia is
the outer shepherd of both the $\epsilon$ and $\gamma$ rings (Porco &
Goldreich, 1987). Recently, the $\eta$ ring was found to be influenced by
Cressida (Chancia et al., 2017). Nevertheless, at Saturn the $\ell=m=10$ mode
was found to be anomalously strong (Hedman et al., 2019), and with this
precedent the influence of planetary normal modes on the outer rings cannot be
ruled out.

Figure 5: Normal mode resonance location predictions for Neptune. The
azimuthal order $m$ is the vertical axis, and distance from the center of
Neptune is the horizontal axis, shown in Neptune radii on the bottom and in km
on the top. Both outer Lindblad resonances (OLR) and outer vertical resonances
(OVR) are shown. Several resonances fall in the Galle ring, and others may
perturb the inner moons Naiad and Thalassa.

At Neptune, modes with $\ell=m=16,...,21$ fall among the moons Naiad and
Thalassa. Naiad and Thalassa are themselves in a 73:69 resonance (Brozović et
al., 2020), made possible by Naiad’s high inclination (4.7∘). Should it be
found that this resonance does not fully account for their orbits,
perturbations from planetary normal modes may be part of the solution.
Furthermore, it is not known how Naiad reached its high inclination. The most
likely explanation is passage through a previous resonance with Despina
(Banfield & Murray, 1992). Another possibility, however, is that Naiad’s
inclination was excited instead by a vertical resonance with a Neptunian
normal mode.

The above calculations focused exclusively on the f-modes ($n=0$). It is also
interesting, however, to consider the $\ell=m$, $n=1$ g-modes for the non-
adiabatic models because these can also fall among the ring systems. In the
Uranian system (Figure 6), $\ell=m$, $n=1$ g-mode resonances are likely to
fall in the mid- to outer-ring system as well as the innermost moons. The
spread in resonance location predictions is greater for g-modes than for
f-modes because the g-mode spectrum is sensitive mainly to the Brunt-Väisälä
frequency $N(r)$. The thin-medium-thick triplet of models diverges more for
higher $\ell$, where the eigenfunctions have higher amplitudes in the g-mode
cavity.

Figure 6: Resonance location predictions for Uranian g-modes. The azimuthal
order $m$ is the vertical axis, and distance from the center of Uranus is the
horizontal axis, shown in Uranus radii on the bottom and in km on the top.
Only outer Lindblad resonances (OLR) with $\ell=m$, $n=1$ are shown.

In the Neptunian system (Figure 7), $\ell=m$, $n=1$ g-mode resonances are
likely to fall among the inner moons and middle to outer rings. There is even
a g-mode resonance that falls near the Adams ring, which we discuss in Section
5.4. G-modes at higher order ($n\geq 2$) are at lower frequency, and so their
outer Lindblad resonance locations would fall farther out. Nevertheless, their
gravity perturbations might be too small to matter because they are more
effectively confined deep down than the $n=1$ g-modes.

Figure 7: Resonance location predictions for Neptunian g-modes. The azimuthal
order $m$ is the vertical axis, and distance from the center of Neptune is the
horizontal axis, shown in Neptune radii on the bottom and in km on the top.
Outer Lindblad resonances (OLR) with $\ell=m$, $n=1$ are shown as filled-in
circles, while corotation resonances associated with these Lindblad resonances
are shown with open circles.

Note that the mean absolute model uncertainty, due to the approximate
treatment of rotation, in the resonance locations for all the f-modes we
calculated is $\delta r=610$ km for the Uranus models and $\delta r=450$ km
for the Neptune model. The maximum uncertainty in f-mode resonance locations
is $\delta r=805$ km for the Uranus models and $\delta r=557$ km for the
Neptune model. These both correspond to the $\ell=m=2$ f-mode. The uncertainty
generally decreases with increasing $m$.

To decrease the uncertainty, frequency-correction calculations may be carried
out to second-order using the perturbation theory techniques described in
Vorontsov & Zharkov (1981) and summarized in the Appendix of Marley (1990). We
leave the second-order calculations for possible future work. Further
knowledge of the planets’ interior structure, thermodynamic state, and
rotational state can also improve the precision of resonance location
predictions.

##  5 Discussion

We have found that planetary normal modes likely fall among the rings of
Uranus and Neptune. The next question is whether normal mode resonances would
be detectable in the Uranian and Neptunian rings. Potential signatures of such
resonances could include wave-like structures in the dense Uranian rings,
correlations with Uranus’s many narrow ring features, and longitudinal
structures in more dusty rings. Each of these methods is discussed in more
detail below.

Note that we do not attempt to compute the explicit amplitudes of the
gravitational perturbations associated with these perturbations. While Marley
& Porco (1993) estimated the strengths of the perturbations needed to produce
the observed waves in Saturn’s rings, more recent kronoseismology studies did
not find strong correlations between the anticipated and the observed
amplitudes of normal mode resonances (Hedman et al., 2019). Although it is
expected that torques decrease as the oscillation degree $\ell$ increases
because a smaller quantity of mass participates in the oscillation (Marley &
Porco, 1993), the intrinsic amplitude spectrum of normal modes in the planet
is unknown but the subject of much current research (Marley & Porco, 1993;
Markham & Stevenson, 2018; Wu & Lithwick, 2019; Markham et al., 2020). For
example, Markham & Stevenson (2018) examined mode excitation mechanisms in
giant planet interiors. They found that moist convective storms associated
with water condensation were not energetically feasible for Jupiter, but it
remains to be seen whether that mechanism could excite Uranian or Neptunian
modes to observable levels. Given all these theoretical uncertainties, we
leave the computation of amplitudes for future work.

###  5.1 Occultations of Narrow Dense Rings

One way to identify resonant perturbations would be the detection of density
waves similar to those seen in Saturn’s rings. This would best work in
Uranus’s narrow dense rings, where the search can be guided by a combination
of our location predictions and the range of the number of spiral arms $m$
corresponding to an approximation of the number of stellar occultation cuts
through the rings or images of the rings that would be necessary for
detection. Note that the resolution achievable with an occultation is limited
by the Fresnel scale: $\lambda_{F}\sim\sqrt{\lambda D}$. At $\lambda=0.2\mu$m,
the great distances $D$ of Uranus (19.2 AU) and Neptune (30 AU) from Earth
prevent ground-based or Earth-orbit-based stellar occultations from obtaining
better resolution than about 750 m and 950 m, respectively. Given that wave-
like variations in the dense rings have sub-kilometer wavelengths (Chancia &
Hedman, 2016; Horn et al., 1988), we expect resolutions of order 200 m is
needed to see any planet-generated waves. This resolution threshold, combined
with the number of cuts required to uniquely determine the number of arms,
which is of order the relevant $m$ (about 20 for the waves of interest here),
implies that the detection of planetary normal modes via wave identification
will have to await an orbiter of Uranus or Neptune. An orbiter can observe
occultations in the ultraviolet, visible, near-infrared, and radio parts of
the spectrum at a much closer distance, and it would be expected to produce
radial profiles of the rings at a range of longitudes with sufficient
resolution to determine the number of spiral arms in a wave. Observation of
the ring plane from different incidence angles will also allow bending waves
to be easily discerned from density waves.

While the overall shapes of the Uranian rings are dominated by factors that
can be attributed to free modes in the rings themselves, this does not exclude
the possibility that these rings can also preserve signals from planetary
normal modes. The Uranian $\alpha$ and $\beta$ rings are considered analogs of
the Saturnian Maxwell and Colombo ringlets (Porco, 1990; Chiang & Culter,
2003). These rings are considered in Chiang & Culter (2003), who present a
proof, relying on first steps taken by Borderies et al. (1983), that circular,
nodally locked rings are linearly stable to perturbations to their
inclinations and nodes. The Colombo ringlet, also named the Titan ringlet
because it is in a 1:0 apsidal resonance with Saturn’s moon Titan (Porco et
al., 1984; Nicholson et al., 2014), is located near several planetary normal
modes: $6\leq\ell\leq 15$, $5\leq m\leq 11$ (Mankovich et al., 2019). The
Maxwell ringlet inside the Maxwell Gap in Saturn’s C ring is perturbed by an
$\ell=m=2$ mode (French et al., 2016; Cuzzi et al., 2018), as predicted by
Fuller (2014). For this reason, we are also hopeful that either the $\alpha$
or $\beta$ rings, or both, are likewise perturbed by planetary normal modes.
The Uranian $\epsilon$ ring is also considered to have similar properties to
the Maxwell ringlet (French et al., 1991, 2016): sharp edges, a freely
precessing elliptical shape, and a linear width-radius relation. The mean
optical depth of the Maxwell ringlet is $\bar{\tau}=0.968$ (French et al.,
2016).

Horn et al. (1988) identified an inward-propagating density wave in the
$\delta$ ring, which they interpreted to be evidence of a moonlet interior to
the $\delta$ ring. Because no moonlet has yet been discovered interior to the
$\delta$ ring, we consider that such a density wave could be a candidate
planetary normal mode resonance. Horn et al. (1988) constrained the azimuthal
wavenumber $m$ of the resonance that generated this wave to be $48\leq m\leq
112$, based on three conditions: that the torque exceed a critical value
(Goldreich & Porco, 1987) for nonlinearity and the moonlet still remain
undetected in the Voyager Imaging search (Smith et al., 1986), that the
moonlet lie between the $\gamma$ and $\delta$ rings, and that the separation
in radius between first-order resonances be greater than half the width of the
$\delta$ ring (Horn et al., 1988). The range of possible $m$ values was thus
not constrained by an $m$-lobed pattern detected in the ring itself. If
instead the inward-propagating density wave in the $\delta$ ring is driven by
a planetary normal mode resonance, none of the conditions given by Horn et al.
(1988) would apply, though their measurement of the product of the wavelength
and the distance from the resonance $\lambda d=0.84\pm 0.07$ km (Horn et al.,
1988) makes it difficult for $m$ to be less than 10. From our resonance
location predictions, we can instead expect $17\leq m\leq 23$, should the
density wave in the $\delta$ ring be from an f-mode resonance.

In addition to density waves, Lindblad resonances with satellites can perturb
ring edges, generating non-circular shapes like those observed for the outer
edge of Saturn’s A and B rings (Goldreich & Tremaine, 1978; Nicholson et al.,
2014; El Moutamid et al., 2016; Tajeddine et al., 2017) and for Uranus’s
$\epsilon$ ring (French et al., 1991) and $\eta$ ring (Chancia et al., 2017).

###  5.2 Correlating Uranian Ring Features and Resonance Locations

Figure 8: Radial scan of Uranian rings compared to Lindblad resonance
predictions from the medium model. The cyan curve shows the mean ring normal
$I/F$ as a function of radius. While the named narrow rings are labeled, many
other narrow features can be seen in this scan. The tallest dash-dotted lines,
in red, show the locations of the $\ell=m$ mode resonances; the intermediate-
sized dash-dotted lines, in green, show the locations of the $\ell-m=2$ mode
resonances, and the shortest dash-dotted lines, in blue, show the locations of
the $\ell-m=4$ mode resonances. The integer above each dash-dotted line
corresponds to the azimuthal order $m$ of the mode.

Another potential way to identify these resonances would be to correlate the
locations of multiple narrow ring features with expected resonant locations.
For example, Figure 8 shows the brightness of the inner rings of Uranus as a
function of radius, from a high-phase image (C2685219; see Hedman & Chancia
2021) that showed many more narrow ring features than the named rings. These
features could potentially reflect additional locations where material is
confined by resonances. Vertical dash-dotted lines show Lindblad resonance
locations calculated from the medium model. Due to the uncertainty of our
calculations, lining up ring features with resonances in this way is to be
taken only as a demonstration of a way to correlate the models with ring
features. Nevertheless, the way that the $\ell=m=8,9,10,11,12$ modes line up
with either peaks or troughs in the ring brightness is suggestive of a role
they may play in perturbing ring material radially.

###  5.3 Longitudinal Variations in Diffuse Rings

The Uranian $\zeta$ ring and the Neptunian Galle ring have low optical depth.
Although many mode resonances fall within them, these rings are so tenuous
that we do not expect them to be capable of sustaining a wave. Longitudinal
structure can be driven by resonances with the planet’s magnetic field, as is
the case at magnetic Lindblad resonances in Saturn’s D Ring (Hedman et al.,
2009a; Chancia et al., 2019), and we encourage future work exploring potential
resonances between the Uranian and Neptunian magnetic fields and their ring
systems. Nonetheless, planetary normal modes, particularly the stronger
$\ell=m=2$ modes, should be kept in mind in studies of longitudinal structures
in diffuse rings.

###  5.4 Neptune’s Le Verrier and Adams Rings

Finally, we can consider whether planetary normal mode resonances could
influence the dynamics of Neptune’s narrow dusty rings in a detectable way.

While normal mode resonances could potentially play a role in confining the Le
Verrier ring (Brooks et al., 2021), testing this idea is challenging. For one,
the ring appears homogeneous in the Voyager images (Ferrari & Brahic, 1994),
which limits our ability to identify signatures of external perturbations.
Furthermore, this ring is also close to other resonances, including Thalassa’s
21:23 resonance (Gaslac Gallardo et al., 2020). In addition, the 2:1 resonance
with Neptune’s rotation frequency, whose value is known with less certainty,
falls in the Le Verrier ring region. Simply applying Kepler’s Third Law to
find the 2:1 resonance location, neglecting Neptune’s oblateness, yields

$a_{\rm{res}}=\left(\frac{GM}{n^{2}}\right)^{1/3}=\begin{cases}52,607\>\rm{km},&\text{fast}\\\
55,507\>\rm{km},&\text{slow}\end{cases}$ (17)

where the mean motion $n=2\Omega$, and “fast” and “slow” refer to estimates
for Neptune’s rotation rate given in Table 3. The Le Verrier ring is located
at 53,200 km (de Pater et al., 2018), in the middle of the resonance locations
according to the two different rotation rates (Warwick et al., 1989; Helled et
al., 2010).

It is also worth noting that $n=1$ g-mode resonances can occur near Neptune’s
Adams ring, because this could potentially mean that these resonances may be
relevant to understanding the arcs in that ring. The ring arcs in the Adams
ring comprise the most studied part of Neptune’s rings (see for example Porco
et al. 1995; de Pater et al. 2018). Ring arcs in the Saturnian system are
confined longitudinally by orbital resonances with moons (Spitale et al.,
2006; Hedman et al., 2007, 2009b, 2010; Cooper et al., 2008; A’Hearn et al.,
2019). Although several ideas have been proposed about the particular
resonance of Neptune’s ring arcs, consensus has not been reached. Initial
studies suspected that a 42:43 resonance with Galatea confined the ring arcs
(Goldreich et al., 1986; Porco, 1991; Namouni & Porco, 2002). Deviations from
the exact rate of different types of a corotation resonance with Galatea,
however, support other possible explanations (Renner et al., 2014).
Confinement due to shepherding by undetected satellites that are co-orbital
with the Adams ring arcs has been proposed (Lissauer, 1985; Salo & Hanninen,
1998; Renner et al., 2014), though more recent investigation claims to rule
out that co-orbital satellites could be the source of the dust (Giuliatti
Winter et al., 2020). Another recent idea is that the Adams ring arcs are in a
three-body resonance with Galatea and Larissa (Showalter et al., 2017).

One potential way to test whether planetary normal modes could be confining
material in the Adams ring is to take another look at the distribution and
extent of the arcs. A consequence of the initial 42:43 corotation resonance
theory of confinement was that each of the 42 corotation sites, also called
“pockets”, spans only $\sim 9^{\circ}$. This posed a problem for the
Fraternité arc, whose longitudinal extent is greater, and so it was assumed
that Fraternité occupies two corotation sites.

Should the ring arcs instead be the effect of a corotation resonance with an
oscillation degree $m<42$ , the corotation site would span a greater
longitudinal extent $\theta$, because

$\theta=\frac{360^{\circ}}{m},$ (18)

which for low enough $m$ could encompass the entirety of Fraternité in one
corotation site. Although Lindblad resonances do not confine material, each
Lindblad resonance can be associated with a corotation resonance, which can
confine material. The planetary corotation resonance from the $\ell=m=19$,
$n=1$ g-mode is calculated to fall in the range $62,844$-$63,230$ km, which
encompasses the Adams ring. The $m=19$ corotation site would span
approximately $19^{\circ}$, over twice as much as an $m=42$ corotation site,
and could thus encompass the entirety of the Fraternité arc. We hope to
explore this possibility further in a future work.

###  5.5 Conclusion

While none of the structures in the rings around Uranus and Neptune has yet
been firmly attributed to a planetary normal mode, the above considerations
indicate that ring seismology of the ice giants could be done when the
appropriate data becomes available. To the extent that different interior
models affect normal mode resonance locations, the detection of planet-driven
ring waves or perturbations to moon orbits would allow a variety of interior
models to be ruled out. These may provide evidence for or against a stably
stratified layer or a diffuse core. Stable stratification, which occurs in all
our interior models except the adiabatic model, allows the presence of
g-modes, which could possibly be detected if they fall in or near rings
(Fuller, 2014; Friedson, 2020). Should Uranus or Neptune additionally have a
diffuse core, for example, the f-mode and g-mode resonances would be modified
compared to our results here, particularly those corresponding to the $\ell=2$
modes most sensitive to $\rho(r)$ and $N(r)$ in the deep interior.

To evaluate these predictions, attempts can be made with Voyager and ground-
based observations. Nevertheless, we expect results from such observations to
be inconclusive, and an orbiter with a primary mission duration of at least a
year or two to measure pattern speeds to within around $0.1^{\circ}$/day would
likely be required to make the observations necessary to determine effects
from planetary normal mode resonances in the rings of Uranus and Neptune.

We are grateful to many individuals for useful comments and discussions,
including P. Nicholson, M. Tiscareno, M. Ćuk, L. Fletcher, M. Brozović, and K.
Volk. We thank two anonymous reviewers for their helpful comments that have
improved this paper. We also thank NASA for the support through the Solar
System Workings grants NNX15AH45G and 80NSSC21K1865.

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##  Appendix A Extended Tables of f-mode Frequencies and Resonance Locations

In this appendix, we present all the calculated f-mode frequencies and
resonance locations for all the models. Table 6 shows the initial estimates by
Marley et al. (1988). Our Lindblad resonance calculations are provided in
tables that contains two models each: the thick and medium models in Table 7,
the thin and adiabatic models in Table 8, and the shallow and Neptune models
in Table 9. Similarly, our vertical resonance calculations are then provided
in tables that contains two models each: the thick and medium models in Table
10, the thin and adiabatic models in Table 11, and the shallow and Neptune
models in Table 12. Error bars are one-sided because the frequencies and
resonance locations are only calculated to first order. Second-order
calculations would universally lower the frequencies, which would cause the
resonance locations to be more distant from the planet.

Table 6: Predicted Lindblad resonance locations among the inner rings of Uranus from Marley et al. (1988).  $\ell$ | $m$ | model |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
---|---|---|---|---
2 | 2 | 6 | 1683 | 48,300
|  | 5 | 1718 | 47,700
|  | 4 | 1767 | 46,800
3 | 3 | 6 | 2015 | 39,600
|  | 5 | 2025 | 39,500
|  | 4 | 2054 | 39,100
4 | 4 | 6 | 2005 | 38,100
|  | 5 | 2010 | 38,000
|  | 4 | 2030 | 37,800
5 | 5 | 6 | 1941 | 37,900
|  | 5 | 1941 | 37,900
|  | 4 | 1950 | 37,800
Table 7: Pattern frequencies and Lindblad resonance locations of different
models

Uranus Thick model
---
$\ell$ | $m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
$\ell=m$ |  |  |
2 | 2 | 2218.7 - 62.4 | 40196 + 772
3 | 3 | 2228.9 - 59.9 | 37054 + 678
4 | 4 | 2077.7 - 53.6 | 37198 + 653
5 | 5 | 1922.5 - 47.8 | 38123 + 644
6 | 6 | 1788.3 - 42.9 | 39263 + 641
7 | 7 | 1677.2 - 39.1 | 40418 + 639
8 | 8 | 1586.7 - 35.9 | 41503 + 638
9 | 9 | 1513.3 - 33.4 | 42481 + 636
10 | 10 | 1453.5 - 31.4 | 43345 + 634
11 | 11 | 1404.4 - 29.7 | 44106 + 632
12 | 12 | 1363.3 - 28.3 | 44780 + 629
13 | 13 | 1328.1 - 27.0 | 45386 + 626
14 | 14 | 1297.5 - 26.0 | 45939 + 623
15 | 15 | 1270.5 - 25.0 | 46451 + 620
16 | 16 | 1246.2 - 24.2 | 46928 + 617
17 | 17 | 1224.3 - 23.4 | 47377 + 614
18 | 18 | 1204.3 - 22.7 | 47802 + 611
19 | 19 | 1185.9 - 22.1 | 48205 + 608
20 | 20 | 1169.0 - 21.5 | 48589 + 605
21 | 21 | 1153.2 - 21.0 | 48955 + 602
22 | 22 | 1138.6 - 20.5 | 49305 + 600
23 | 23 | 1125.0 - 20.0 | 49640 + 597
24 | 24 | 1112.2 - 19.6 | 49962 + 594
25 | 25 | 1100.2 - 19.2 | 50271 + 592
$\ell-m=2$ |  |  |
4 | 2 | 3755.8 - 107.2 | 28306 + 551
5 | 3 | 2919.8 - 79.6 | 30955 + 575
6 | 4 | 2460.7 - 64.4 | 33234 + 592
7 | 5 | 2166.4 - 54.7 | 35208 + 604
8 | 6 | 1961.7 - 47.9 | 36916 + 613
9 | 7 | 1812.2 - 43.0 | 38388 + 618
10 | 8 | 1699.1 - 39.2 | 39654 + 621
11 | 9 | 1610.9 - 36.3 | 40749 + 623
12 | 10 | 1540.2 - 33.9 | 41705 + 623
13 | 11 | 1482.1 - 32.0 | 42553 + 622
14 | 12 | 1433.1 - 30.3 | 43314 + 621
15 | 13 | 1391.1 - 28.9 | 44007 + 619
16 | 14 | 1354.5 - 27.6 | 44644 + 617
17 | 15 | 1322.1 - 26.6 | 45234 + 615
18 | 16 | 1293.3 - 25.6 | 45784 + 613
19 | 17 | 1267.3 - 24.7 | 46299 + 611
20 | 18 | 1243.8 - 23.9 | 46784 + 609
21 | 19 | 1222.4 - 23.2 | 47242 + 606
22 | 20 | 1202.7 - 22.5 | 47675 + 604
23 | 21 | 1184.6 - 21.9 | 48087 + 602
24 | 22 | 1167.9 - 21.3 | 48478 + 599
25 | 23 | 1152.3 - 20.8 | 48852 + 597
$\ell-m=4$ |  |  |
7 | 3 | 3307.8 - 91.1 | 28487 + 535
8 | 4 | 2711.8 - 71.9 | 31153 + 562
9 | 5 | 2350.3 - 60.1 | 33349 + 581
10 | 6 | 2108.3 - 52.3 | 35187 + 593
11 | 7 | 1935.4 - 46.6 | 36743 + 602
12 | 8 | 1805.6 - 42.4 | 38080 + 607
13 | 9 | 1704.4 - 39.1 | 39246 + 610
14 | 10 | 1622.9 - 36.4 | 40278 + 612
15 | 11 | 1555.6 - 34.1 | 41203 + 613
16 | 12 | 1498.8 - 32.3 | 42041 + 614
17 | 13 | 1450.1 - 30.6 | 42807 + 613
18 | 14 | 1407.7 - 29.2 | 43512 + 612
19 | 15 | 1370.5 - 28.0 | 44165 + 611
20 | 16 | 1337.4 - 26.9 | 44772 + 610
21 | 17 | 1307.8 - 25.9 | 45340 + 608
22 | 18 | 1281.1 - 25.0 | 45873 + 607
23 | 19 | 1256.9 - 24.2 | 46375 + 605
24 | 20 | 1234.7 - 23.5 | 46849 + 603
25 | 21 | 1214.4 - 22.8 | 47298 + 601

Uranus Medium model
---
$\ell$ | $m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
$\ell=m$ |  |  |
2 | 2 | 2207.8 - 62.1 | 40328 + 774
3 | 3 | 2234.6 - 60.1 | 36991 + 678
4 | 4 | 2096.0 - 54.2 | 36981 + 651
5 | 5 | 1949.2 - 48.6 | 37774 + 641
6 | 6 | 1819.6 - 43.9 | 38812 + 636
7 | 7 | 1710.1 - 40.0 | 39899 + 634
8 | 8 | 1618.6 - 36.8 | 40956 + 632
9 | 9 | 1542.1 - 34.2 | 41951 + 630
10 | 10 | 1477.8 - 32.0 | 42870 + 629
11 | 11 | 1423.3 - 30.1 | 43714 + 627
12 | 12 | 1377.0 - 28.6 | 44481 + 626
13 | 13 | 1337.5 - 27.3 | 45174 + 624
14 | 14 | 1303.6 - 26.1 | 45797 + 621
15 | 15 | 1274.3 - 25.1 | 46358 + 619
16 | 16 | 1248.7 - 24.3 | 46867 + 616
17 | 17 | 1226.0 - 23.5 | 47335 + 614
18 | 18 | 1205.5 - 22.8 | 47771 + 611
19 | 19 | 1186.8 - 22.1 | 48180 + 608
20 | 20 | 1169.7 - 21.5 | 48568 + 605
21 | 21 | 1153.9 - 21.0 | 48937 + 602
22 | 22 | 1139.2 - 20.5 | 49289 + 600
23 | 23 | 1125.4 - 20.0 | 49626 + 597
24 | 24 | 1112.6 - 19.6 | 49949 + 594
25 | 25 | 1100.5 - 19.2 | 50259 + 592
$\ell-m=2$ |  |  |
4 | 2 | 3792.6 - 108.4 | 28123 + 548
5 | 3 | 2963.8 - 81.0 | 30649 + 571
6 | 4 | 2506.6 - 65.8 | 32828 + 587
7 | 5 | 2210.9 - 56.0 | 34735 + 598
8 | 6 | 2002.6 - 49.1 | 36413 + 606
9 | 7 | 1847.7 - 43.9 | 37895 + 612
10 | 8 | 1728.0 - 40.0 | 39211 + 616
11 | 9 | 1633.0 - 36.8 | 40381 + 618
12 | 10 | 1556.0 - 34.3 | 41424 + 619
13 | 11 | 1492.6 - 32.2 | 42352 + 620
14 | 12 | 1439.9 - 30.5 | 43178 + 619
15 | 13 | 1395.4 - 29.0 | 43917 + 618
16 | 14 | 1357.2 - 27.7 | 44584 + 617
17 | 15 | 1324.0 - 26.6 | 45192 + 615
18 | 16 | 1294.6 - 25.6 | 45753 + 613
19 | 17 | 1268.4 - 24.7 | 46275 + 611
20 | 18 | 1244.6 - 23.9 | 46764 + 609
21 | 19 | 1223.1 - 23.2 | 47224 + 606
22 | 20 | 1203.3 - 22.5 | 47660 + 604
23 | 21 | 1185.1 - 21.9 | 48073 + 602
24 | 22 | 1168.3 - 21.4 | 48466 + 599
25 | 23 | 1152.7 - 20.8 | 48840 + 597
$\ell-m=4$ |  |  |
7 | 3 | 3379.5 - 93.3 | 28084 + 528
8 | 4 | 2770.6 - 73.6 | 30711 + 555
9 | 5 | 2397.7 - 61.5 | 32909 + 574
10 | 6 | 2145.0 - 53.3 | 34785 + 587
11 | 7 | 1962.4 - 47.3 | 36406 + 597
12 | 8 | 1824.4 - 42.9 | 37819 + 603
13 | 9 | 1716.8 - 39.4 | 39058 + 608
14 | 10 | 1630.7 - 36.6 | 40150 + 611
15 | 11 | 1560.4 - 34.3 | 41117 + 612
16 | 12 | 1501.9 - 32.3 | 41983 + 613
17 | 13 | 1452.2 - 30.7 | 42766 + 613
18 | 14 | 1409.2 - 29.3 | 43481 + 612
19 | 15 | 1371.6 - 28.0 | 44140 + 611
20 | 16 | 1338.3 - 26.9 | 44752 + 610
21 | 17 | 1308.6 - 25.9 | 45323 + 608
22 | 18 | 1281.8 - 25.0 | 45858 + 607
23 | 19 | 1257.4 - 24.2 | 46361 + 605
24 | 20 | 1235.2 - 23.5 | 46836 + 603
25 | 21 | 1214.9 - 22.8 | 47286 + 601

Table 8: Pattern frequencies and Lindblad resonance locations of different
models

Uranus Thin model
---
$\ell$ | $m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
$\ell=m$ |  |  |
2 | 2 | 2191.7 - 61.6 | 40525 + 777
3 | 3 | 2235.0 - 60.1 | 36986 + 678
4 | 4 | 2108.1 - 54.6 | 36840 + 650
5 | 5 | 1968.8 - 49.3 | 37524 + 639
6 | 6 | 1843.2 - 44.7 | 38481 + 634
7 | 7 | 1735.2 - 40.8 | 39514 + 631
8 | 8 | 1643.7 - 37.6 | 40538 + 629
9 | 9 | 1566.4 - 34.9 | 41517 + 627
10 | 10 | 1500.6 - 32.6 | 42435 + 626
11 | 11 | 1444.4 - 30.7 | 43288 + 624
12 | 12 | 1395.9 - 29.1 | 44080 + 622
13 | 13 | 1353.6 - 27.6 | 44814 + 620
14 | 14 | 1316.7 - 26.4 | 45494 + 618
15 | 15 | 1284.2 - 25.3 | 46121 + 616
16 | 16 | 1255.6 - 24.4 | 46695 + 615
17 | 17 | 1230.6 - 23.6 | 47216 + 612
18 | 18 | 1208.5 - 22.8 | 47691 + 610
19 | 19 | 1188.9 - 22.2 | 48125 + 608
20 | 20 | 1171.2 - 21.6 | 48528 + 605
21 | 21 | 1155.0 - 21.0 | 48906 + 602
22 | 22 | 1140.0 - 20.5 | 49264 + 600
23 | 23 | 1126.2 - 20.0 | 49605 + 597
24 | 24 | 1113.2 - 19.6 | 49931 + 594
25 | 25 | 1101.1 - 19.2 | 50243 + 592
$\ell-m=2$ |  |  |
4 | 2 | 3818.5 - 109.2 | 27996 + 547
5 | 3 | 2997.4 - 82.1 | 30419 + 568
6 | 4 | 2542.4 - 67.0 | 32520 + 583
7 | 5 | 2246.1 - 57.1 | 34371 + 594
8 | 6 | 2035.9 - 50.1 | 36015 + 602
9 | 7 | 1878.4 - 44.8 | 37482 + 608
10 | 8 | 1756.0 - 40.8 | 38794 + 612
11 | 9 | 1658.0 - 37.5 | 39974 + 614
12 | 10 | 1577.8 - 34.9 | 41040 + 615
13 | 11 | 1511.0 - 32.7 | 42008 + 616
14 | 12 | 1454.5 - 30.8 | 42888 + 616
15 | 13 | 1406.3 - 29.2 | 43690 + 616
16 | 14 | 1364.8 - 27.9 | 44418 + 615
17 | 15 | 1329.0 - 26.7 | 45077 + 614
18 | 16 | 1297.9 - 25.7 | 45675 + 613
19 | 17 | 1270.6 - 24.8 | 46220 + 611
20 | 18 | 1246.2 - 24.0 | 46724 + 609
21 | 19 | 1224.3 - 23.2 | 47194 + 606
22 | 20 | 1204.3 - 22.6 | 47635 + 604
23 | 21 | 1185.9 - 22.0 | 48052 + 602
24 | 22 | 1169.0 - 21.4 | 48447 + 599
25 | 23 | 1153.3 - 20.9 | 48824 + 597
$\ell-m=4$ |  |  |
7 | 3 | 3438.3 - 95.2 | 27763 + 524
8 | 4 | 2820.2 - 75.1 | 30350 + 551
9 | 5 | 2440.1 - 62.8 | 32527 + 569
10 | 6 | 2181.5 - 54.4 | 34396 + 583
11 | 7 | 1993.7 - 48.2 | 36024 + 592
12 | 8 | 1850.8 - 43.6 | 37459 + 599
13 | 9 | 1738.3 - 39.9 | 38734 + 604
14 | 10 | 1647.5 - 37.0 | 39876 + 607
15 | 11 | 1572.8 - 34.6 | 40902 + 610
16 | 12 | 1510.4 - 32.5 | 41825 + 611
17 | 13 | 1457.8 - 30.8 | 42656 + 612
18 | 14 | 1412.9 - 29.4 | 43405 + 612
19 | 15 | 1374.1 - 28.1 | 44087 + 611
20 | 16 | 1340.1 - 27.0 | 44713 + 610
21 | 17 | 1309.9 - 26.0 | 45292 + 608
22 | 18 | 1282.8 - 25.1 | 45833 + 607
23 | 19 | 1258.3 - 24.3 | 46340 + 605
24 | 20 | 1236.0 - 23.5 | 46818 + 603
25 | 21 | 1215.5 - 22.8 | 47270 + 601

Uranus Adiabatic model
---
$\ell$ | $m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
$\ell=m$ |  |  |
2 | 2 | 2360.4 - 67.0 | 38571 + 747
3 | 3 | 2150.9 - 57.5 | 37943 + 692
4 | 4 | 1963.4 - 50.2 | 38627 + 672
5 | 5 | 1820.2 - 44.8 | 39537 + 662
6 | 6 | 1710.1 - 40.7 | 40451 + 655
7 | 7 | 1623.0 - 37.6 | 41314 + 649
8 | 8 | 1552.2 - 35.0 | 42115 + 645
9 | 9 | 1493.3 - 32.9 | 42858 + 640
10 | 10 | 1443.4 - 31.1 | 43547 + 637
11 | 11 | 1400.4 - 29.6 | 44189 + 633
12 | 12 | 1362.9 - 28.3 | 44787 + 630
13 | 13 | 1329.8 - 27.1 | 45349 + 626
14 | 14 | 1300.2 - 26.1 | 45876 + 623
15 | 15 | 1273.7 - 25.1 | 46373 + 620
16 | 16 | 1249.6 - 24.3 | 46843 + 617
17 | 17 | 1227.7 - 23.5 | 47289 + 614
18 | 18 | 1207.7 - 22.8 | 47712 + 611
19 | 19 | 1189.3 - 22.2 | 48115 + 608
20 | 20 | 1172.2 - 21.6 | 48499 + 605
21 | 21 | 1156.4 - 21.1 | 48866 + 602
22 | 22 | 1141.7 - 20.6 | 49217 + 600
23 | 23 | 1127.9 - 20.1 | 49553 + 597
24 | 24 | 1115.0 - 19.7 | 49876 + 594
25 | 25 | 1102.9 - 19.2 | 50187 + 592
$\ell-m=2$ |  |  |
4 | 2 | 3534.8 - 100.4 | 29473 + 571
5 | 3 | 2755.2 - 74.6 | 32175 + 594
6 | 4 | 2347.7 - 61.1 | 34291 + 607
7 | 5 | 2093.3 - 52.6 | 36022 + 615
8 | 6 | 1917.5 - 46.7 | 37481 + 620
9 | 7 | 1787.6 - 42.3 | 38739 + 623
10 | 8 | 1687.0 - 38.9 | 39843 + 624
11 | 9 | 1606.3 - 36.2 | 40826 + 624
12 | 10 | 1539.9 - 33.9 | 41711 + 623
13 | 11 | 1484.1 - 32.0 | 42515 + 622
14 | 12 | 1436.3 - 30.4 | 43251 + 621
15 | 13 | 1394.8 - 29.0 | 43930 + 619
16 | 14 | 1358.3 - 27.8 | 44559 + 617
17 | 15 | 1326.0 - 26.7 | 45145 + 615
18 | 16 | 1297.1 - 25.7 | 45694 + 613
19 | 17 | 1271.1 - 24.8 | 46209 + 611
20 | 18 | 1247.4 - 24.0 | 46694 + 609
21 | 19 | 1225.9 - 23.3 | 47153 + 606
22 | 20 | 1206.1 - 22.6 | 47587 + 604
23 | 21 | 1187.9 - 22.0 | 48000 + 602
24 | 22 | 1171.0 - 21.4 | 48392 + 599
25 | 23 | 1155.3 - 20.9 | 48767 + 597
$\ell-m=4$ |  |  |
6 | 2 | 4260.5 - 122.2 | 26026 + 509
7 | 3 | 3190.8 - 87.6 | 29179 + 546
8 | 4 | 2648.1 - 70.0 | 31649 + 570
9 | 5 | 2317.2 - 59.2 | 33665 + 585
10 | 6 | 2092.9 - 51.9 | 35359 + 596
11 | 7 | 1929.9 - 46.5 | 36813 + 603
12 | 8 | 1805.5 - 42.4 | 38083 + 607
13 | 9 | 1706.9 - 39.1 | 39208 + 610
14 | 10 | 1626.7 - 36.5 | 40215 + 612
15 | 11 | 1559.9 - 34.3 | 41126 + 613
16 | 12 | 1503.3 - 32.4 | 41957 + 613
17 | 13 | 1454.6 - 30.8 | 42719 + 613
18 | 14 | 1412.1 - 29.4 | 43422 + 612
19 | 15 | 1374.7 - 28.1 | 44074 + 611
20 | 16 | 1341.5 - 27.0 | 44682 + 610
21 | 17 | 1311.7 - 26.0 | 45251 + 608
22 | 18 | 1284.8 - 25.1 | 45785 + 607
23 | 19 | 1260.4 - 24.3 | 46288 + 605
24 | 20 | 1238.2 - 23.6 | 46763 + 603
25 | 21 | 1217.7 - 22.9 | 47212 + 601

Table 9: Pattern frequencies and Lindblad resonance locations of different
models

Uranus Shallow model
---
$\ell$ | $m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
$\ell=m$ |  |  |
2 | 2 | 2052.5 - 57.2 | 42337 + 805
3 | 3 | 2162.1 - 57.9 | 37812 + 690
4 | 4 | 2073.0 - 53.7 | 37254 + 657
5 | 5 | 1959.5 - 49.2 | 37642 + 643
6 | 6 | 1851.0 - 45.1 | 38373 + 636
7 | 7 | 1753.2 - 41.6 | 39244 + 633
8 | 8 | 1666.7 - 38.5 | 40166 + 631
9 | 9 | 1591.0 - 35.9 | 41088 + 629
10 | 10 | 1525.1 - 33.6 | 41981 + 628
11 | 11 | 1467.8 - 31.6 | 42827 + 626
12 | 12 | 1418.1 - 30.0 | 43618 + 625
13 | 13 | 1375.0 - 28.5 | 44350 + 623
14 | 14 | 1337.3 - 27.2 | 45024 + 621
15 | 15 | 1304.3 - 26.1 | 45644 + 618
16 | 16 | 1275.3 - 25.1 | 46214 + 616
17 | 17 | 1249.4 - 24.2 | 46741 + 613
18 | 18 | 1226.3 - 23.4 | 47229 + 611
19 | 19 | 1205.4 - 22.7 | 47685 + 608
20 | 20 | 1186.4 - 22.1 | 48112 + 605
21 | 21 | 1169.0 - 21.5 | 48514 + 603
22 | 22 | 1153.0 - 20.9 | 48895 + 600
23 | 23 | 1138.2 - 20.4 | 49256 + 598
24 | 24 | 1124.3 - 20.0 | 49601 + 595
25 | 25 | 1111.5 - 19.5 | 49930 + 593
$\ell-m=2$ |  |  |
4 | 2 | 3753.5 - 107.3 | 28318 + 553
5 | 3 | 2986.2 - 82.0 | 30495 + 570
6 | 4 | 2557.7 - 67.7 | 32390 + 584
7 | 5 | 2274.2 - 58.2 | 34088 + 594
8 | 6 | 2068.9 - 51.4 | 35631 + 602
9 | 7 | 1912.1 - 46.1 | 37041 + 607
10 | 8 | 1788.2 - 42.0 | 38327 + 612
11 | 9 | 1688.0 - 38.7 | 39500 + 614
12 | 10 | 1605.7 - 35.9 | 40565 + 616
13 | 11 | 1537.1 - 33.7 | 41532 + 617
14 | 12 | 1479.3 - 31.7 | 42409 + 617
15 | 13 | 1430.0 - 30.1 | 43206 + 616
16 | 14 | 1387.5 - 28.7 | 43933 + 615
17 | 15 | 1350.5 - 27.4 | 44598 + 614
18 | 16 | 1318.0 - 26.4 | 45211 + 612
19 | 17 | 1289.1 - 25.4 | 45778 + 611
20 | 18 | 1263.2 - 24.5 | 46306 + 609
21 | 19 | 1239.8 - 23.7 | 46799 + 606
22 | 20 | 1218.5 - 23.0 | 47263 + 604
23 | 21 | 1199.1 - 22.4 | 47701 + 602
24 | 22 | 1181.1 - 21.8 | 48115 + 600
25 | 23 | 1164.6 - 21.2 | 48508 + 598
$\ell-m=4$ |  |  |
7 | 3 | 3490.0 - 97.1 | 27489 + 521
8 | 4 | 2873.3 - 77.1 | 29975 + 547
9 | 5 | 2490.0 - 64.6 | 32091 + 566
10 | 6 | 2226.7 - 56.0 | 33930 + 580
11 | 7 | 2034.1 - 49.7 | 35547 + 591
12 | 8 | 1887.1 - 44.9 | 36978 + 598
13 | 9 | 1771.4 - 41.1 | 38252 + 603
14 | 10 | 1678.1 - 38.1 | 39391 + 607
15 | 11 | 1601.4 - 35.6 | 40414 + 609
16 | 12 | 1537.2 - 33.5 | 41338 + 610
17 | 13 | 1482.8 - 31.7 | 42176 + 611
18 | 14 | 1435.9 - 30.1 | 42942 + 611
19 | 15 | 1395.1 - 28.8 | 43645 + 610
20 | 16 | 1359.2 - 27.6 | 44294 + 609
21 | 17 | 1327.2 - 26.5 | 44897 + 608
22 | 18 | 1298.6 - 25.6 | 45460 + 606
23 | 19 | 1272.8 - 24.7 | 45988 + 605
24 | 20 | 1249.3 - 23.9 | 46484 + 603
25 | 21 | 1227.9 - 23.2 | 46953 + 601

Neptune model
---
$\ell$ | $m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
$\ell=m$ |  |  |
2 | 2 | 2292.5 - 45.4 | 41555 + 557
3 | 3 | 2251.9 - 42.6 | 38883 + 498
4 | 4 | 2088.3 - 38.1 | 39168 + 483
5 | 5 | 1940.9 - 34.3 | 40024 + 478
6 | 6 | 1821.9 - 31.3 | 40973 + 475
7 | 7 | 1726.4 - 28.9 | 41890 + 474
8 | 8 | 1648.6 - 27.0 | 42747 + 472
9 | 9 | 1583.8 - 25.4 | 43541 + 471
10 | 10 | 1529.0 - 24.1 | 44279 + 470
11 | 11 | 1481.7 - 22.9 | 44966 + 469
12 | 12 | 1440.4 - 21.9 | 45608 + 468
13 | 13 | 1404.0 - 21.0 | 46211 + 467
14 | 14 | 1371.5 - 20.2 | 46779 + 466
15 | 15 | 1342.2 - 19.5 | 47315 + 465
16 | 16 | 1315.7 - 18.9 | 47823 + 463
17 | 17 | 1291.5 - 18.3 | 48306 + 462
18 | 18 | 1269.4 - 17.8 | 48765 + 461
19 | 19 | 1249.0 - 17.3 | 49203 + 460
20 | 20 | 1230.1 - 16.9 | 49622 + 459
21 | 21 | 1212.6 - 16.5 | 50023 + 457
22 | 22 | 1196.3 - 16.1 | 50407 + 456
23 | 23 | 1181.0 - 15.7 | 50776 + 455
24 | 24 | 1166.7 - 15.4 | 51131 + 454
25 | 25 | 1153.3 - 15.1 | 51472 + 453
$\ell-m=2$ |  |  |
4 | 2 | 3803.1 - 76.1 | 29659 + 402
5 | 3 | 2969.9 - 57.1 | 32337 + 421
6 | 4 | 2526.1 - 46.9 | 34505 + 433
7 | 5 | 2247.0 - 40.5 | 36304 + 442
8 | 6 | 2053.6 - 36.0 | 37832 + 448
9 | 7 | 1910.6 - 32.6 | 39154 + 452
10 | 8 | 1799.8 - 30.1 | 40319 + 455
11 | 9 | 1711.0 - 28.0 | 41358 + 457
12 | 10 | 1637.8 - 26.3 | 42297 + 458
13 | 11 | 1576.2 - 24.8 | 43152 + 459
14 | 12 | 1523.4 - 23.6 | 43938 + 460
15 | 13 | 1477.6 - 22.5 | 44664 + 460
16 | 14 | 1437.4 - 21.6 | 45338 + 460
17 | 15 | 1401.6 - 20.8 | 45969 + 460
18 | 16 | 1369.6 - 20.0 | 46560 + 459
19 | 17 | 1340.8 - 19.4 | 47116 + 459
20 | 18 | 1314.6 - 18.7 | 47642 + 458
21 | 19 | 1290.6 - 18.2 | 48140 + 457
22 | 20 | 1268.7 - 17.7 | 48612 + 457
23 | 21 | 1248.4 - 17.2 | 49062 + 456
24 | 22 | 1229.7 - 16.8 | 49491 + 455
25 | 23 | 1212.2 - 16.4 | 49901 + 454
$\ell-m=4$ |  |  |
6 | 2 | 4638.9 - 93.8 | 25983 + 356
7 | 3 | 3461.9 - 67.4 | 29198 + 385
8 | 4 | 2863.8 - 54.0 | 31739 + 405
9 | 5 | 2498.8 - 45.7 | 33825 + 418
10 | 6 | 2251.3 - 40.1 | 35586 + 428
11 | 7 | 2071.3 - 36.0 | 37104 + 436
12 | 8 | 1933.8 - 32.9 | 38436 + 441
13 | 9 | 1824.9 - 30.4 | 39620 + 445
14 | 10 | 1736.2 - 28.3 | 40684 + 448
15 | 11 | 1662.3 - 26.6 | 41650 + 451
16 | 12 | 1599.6 - 25.2 | 42533 + 452
17 | 13 | 1545.6 - 24.0 | 43345 + 454
18 | 14 | 1498.5 - 22.9 | 44097 + 454
19 | 15 | 1457.0 - 21.9 | 44797 + 455
20 | 16 | 1420.1 - 21.1 | 45451 + 455
21 | 17 | 1387.0 - 20.3 | 46064 + 455
22 | 18 | 1357.2 - 19.6 | 46641 + 455
23 | 19 | 1330.0 - 19.0 | 47186 + 455
24 | 20 | 1305.2 - 18.4 | 47702 + 454
25 | 21 | 1282.4 - 17.9 | 48191 + 454

Table 10: Pattern frequencies and vertical resonance locations of different
models

Uranus Thick model
---
$\ell$ | $m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
$\ell-m=1$ |  |  |
3 | 2 | 3166.5 - 89.8 | 31763 + 613
4 | 3 | 2637.1 - 71.5 | 33162 + 611
5 | 4 | 2296.5 - 59.7 | 34825 + 615
6 | 5 | 2057.3 - 51.5 | 36463 + 621
7 | 6 | 1881.0 - 45.6 | 37981 + 625
8 | 7 | 1747.4 - 41.1 | 39344 + 628
9 | 8 | 1644.1 - 37.6 | 40545 + 629
10 | 9 | 1562.7 - 34.8 | 41593 + 629
11 | 10 | 1497.3 - 32.6 | 42507 + 628
12 | 11 | 1443.7 - 30.8 | 43311 + 627
13 | 12 | 1398.7 - 29.3 | 44029 + 625
14 | 13 | 1360.1 - 28.0 | 44680 + 623
15 | 14 | 1326.5 - 26.8 | 45276 + 620
16 | 15 | 1296.7 - 25.8 | 45828 + 618
17 | 16 | 1270.2 - 24.9 | 46344 + 615
18 | 17 | 1246.2 - 24.1 | 46827 + 612
19 | 18 | 1224.4 - 23.3 | 47284 + 610
20 | 19 | 1204.4 - 22.7 | 47715 + 607
21 | 20 | 1186.1 - 22.0 | 48125 + 605
22 | 21 | 1169.2 - 21.5 | 48514 + 602
23 | 22 | 1153.4 - 20.9 | 48886 + 599
24 | 23 | 1138.8 - 20.4 | 49241 + 597
25 | 24 | 1125.2 - 20.0 | 49581 + 594
$\ell-m=3$ |  |  |
5 | 2 | 4166.5 - 119.4 | 26475 + 515
6 | 3 | 3133.2 - 85.9 | 29574 + 551
7 | 4 | 2594.4 - 68.3 | 32113 + 575
8 | 5 | 2261.8 - 57.5 | 34235 + 591
9 | 6 | 2036.4 - 50.1 | 36027 + 602
10 | 7 | 1874.5 - 44.8 | 37549 + 609
11 | 8 | 1752.9 - 40.8 | 38853 + 614
12 | 9 | 1658.2 - 37.7 | 39983 + 616
13 | 10 | 1582.1 - 35.1 | 40977 + 617
14 | 11 | 1519.4 - 33.1 | 41863 + 618
15 | 12 | 1466.5 - 31.3 | 42664 + 617
16 | 13 | 1421.1 - 29.8 | 43394 + 616
17 | 14 | 1381.5 - 28.5 | 44067 + 615
18 | 15 | 1346.7 - 27.3 | 44689 + 613
19 | 16 | 1315.7 - 26.2 | 45269 + 612
20 | 17 | 1287.9 - 25.3 | 45812 + 610
21 | 18 | 1262.7 - 24.5 | 46322 + 608
22 | 19 | 1239.9 - 23.7 | 46803 + 606
23 | 20 | 1218.9 - 23.0 | 47257 + 603
24 | 21 | 1199.7 - 22.4 | 47688 + 601
25 | 22 | 1182.0 - 21.8 | 48097 + 599
$\ell-m=5$ |  |  |
8 | 3 | 3461.8 - 95.8 | 27679 + 520
9 | 4 | 2821.1 - 75.2 | 30374 + 550
10 | 5 | 2435.7 - 62.7 | 32590 + 570
11 | 6 | 2178.8 - 54.4 | 34444 + 584
12 | 7 | 1995.2 - 48.4 | 36022 + 594
13 | 8 | 1857.3 - 43.9 | 37386 + 600
14 | 9 | 1749.4 - 40.4 | 38583 + 605
15 | 10 | 1662.5 - 37.6 | 39648 + 608
16 | 11 | 1590.6 - 35.2 | 40606 + 609
17 | 12 | 1530.0 - 33.2 | 41476 + 610
18 | 13 | 1478.1 - 31.5 | 42271 + 610
19 | 14 | 1433.1 - 30.0 | 43004 + 610
20 | 15 | 1393.6 - 28.7 | 43683 + 609
21 | 16 | 1358.5 - 27.5 | 44314 + 608
22 | 17 | 1327.2 - 26.5 | 44904 + 607
23 | 18 | 1299.0 - 25.6 | 45457 + 606
24 | 19 | 1273.4 - 24.7 | 45977 + 604
25 | 20 | 1250.1 - 23.9 | 46469 + 602

Uranus Medium model
---
$\ell$ | $m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
$\ell-m=1$ |  |  |
3 | 2 | 3175.4 - 90.1 | 31704 + 612
4 | 3 | 2661.6 - 72.2 | 32959 + 608
5 | 4 | 2329.7 - 60.7 | 34494 + 611
6 | 5 | 2094.4 - 52.7 | 36031 + 615
7 | 6 | 1918.8 - 46.6 | 37482 + 619
8 | 7 | 1783.2 - 42.0 | 38818 + 621
9 | 8 | 1675.8 - 38.4 | 40033 + 623
10 | 9 | 1589.0 - 35.5 | 41133 + 624
11 | 10 | 1517.7 - 33.1 | 42127 + 624
12 | 11 | 1458.4 - 31.2 | 43021 + 623
13 | 12 | 1408.6 - 29.5 | 43823 + 622
14 | 13 | 1366.5 - 28.1 | 44541 + 621
15 | 14 | 1330.5 - 26.9 | 45184 + 619
16 | 15 | 1299.3 - 25.9 | 45767 + 617
17 | 16 | 1271.9 - 24.9 | 46301 + 615
18 | 17 | 1247.4 - 24.1 | 46796 + 612
19 | 18 | 1225.3 - 23.4 | 47259 + 610
20 | 19 | 1205.2 - 22.7 | 47695 + 607
21 | 20 | 1186.7 - 22.0 | 48107 + 605
22 | 21 | 1169.7 - 21.5 | 48499 + 602
23 | 22 | 1153.9 - 20.9 | 48872 + 599
24 | 23 | 1139.3 - 20.4 | 49228 + 597
25 | 24 | 1125.6 - 20.0 | 49569 + 594
$\ell-m=3$ |  |  |
5 | 2 | 4231.9 - 121.5 | 26202 + 511
6 | 3 | 3193.5 - 87.8 | 29202 + 545
7 | 4 | 2649.1 - 70.0 | 31670 + 568
8 | 5 | 2309.8 - 58.9 | 33760 + 584
9 | 6 | 2076.8 - 51.2 | 35559 + 596
10 | 7 | 1906.7 - 45.7 | 37125 + 604
11 | 8 | 1777.1 - 41.4 | 38500 + 609
12 | 9 | 1675.3 - 38.1 | 39711 + 613
13 | 10 | 1593.5 - 35.4 | 40782 + 615
14 | 11 | 1526.6 - 33.2 | 41731 + 616
15 | 12 | 1471.0 - 31.4 | 42576 + 616
16 | 13 | 1424.0 - 29.9 | 43335 + 616
17 | 14 | 1383.5 - 28.5 | 44025 + 615
18 | 15 | 1348.1 - 27.3 | 44658 + 613
19 | 16 | 1316.8 - 26.3 | 45245 + 611
20 | 17 | 1288.7 - 25.3 | 45792 + 610
21 | 18 | 1263.4 - 24.5 | 46305 + 608
22 | 19 | 1240.5 - 23.7 | 46787 + 606
23 | 20 | 1219.5 - 23.0 | 47243 + 603
24 | 21 | 1200.2 - 22.4 | 47675 + 601
25 | 22 | 1182.4 - 21.8 | 48085 + 599
$\ell-m=5$ |  |  |
8 | 3 | 3538.6 - 98.1 | 27279 + 514
9 | 4 | 2879.0 - 76.8 | 29967 + 543
10 | 5 | 2478.6 - 63.9 | 32214 + 564
11 | 6 | 2209.5 - 55.2 | 34125 + 579
12 | 7 | 2016.1 - 49.0 | 35773 + 590
13 | 8 | 1870.8 - 44.3 | 37205 + 598
14 | 9 | 1757.9 - 40.6 | 38459 + 603
15 | 10 | 1667.7 - 37.7 | 39564 + 607
16 | 11 | 1594.0 - 35.3 | 40549 + 609
17 | 12 | 1532.3 - 33.3 | 41435 + 610
18 | 13 | 1479.7 - 31.5 | 42241 + 610
19 | 14 | 1434.3 - 30.0 | 42980 + 610
20 | 15 | 1394.5 - 28.7 | 43663 + 609
21 | 16 | 1359.3 - 27.6 | 44297 + 608
22 | 17 | 1327.9 - 26.5 | 44888 + 607
23 | 18 | 1299.6 - 25.6 | 45443 + 606
24 | 19 | 1274.0 - 24.7 | 45965 + 604
25 | 20 | 1250.6 - 24.0 | 46457 + 602

Table 11: Pattern frequencies and vertical resonance locations of different
models

Uranus Thin model
---
$\ell$ | $m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
$\ell-m=1$ |  |  |
3 | 2 | 3176.8 - 90.2 | 31695 + 612
4 | 3 | 2678.2 - 72.8 | 32823 + 607
5 | 4 | 2354.5 - 61.6 | 34252 + 609
6 | 5 | 2122.9 - 53.6 | 35709 + 612
7 | 6 | 1948.1 - 47.6 | 37106 + 616
8 | 7 | 1811.8 - 42.9 | 38409 + 618
9 | 8 | 1702.9 - 39.2 | 39608 + 619
10 | 9 | 1614.1 - 36.2 | 40706 + 620
11 | 10 | 1540.5 - 33.8 | 41710 + 620
12 | 11 | 1478.6 - 31.7 | 42628 + 619
13 | 12 | 1425.8 - 29.9 | 43470 + 619
14 | 13 | 1380.3 - 28.4 | 44244 + 618
15 | 14 | 1340.8 - 27.2 | 44952 + 617
16 | 15 | 1306.6 - 26.0 | 45598 + 615
17 | 16 | 1276.7 - 25.1 | 46185 + 614
18 | 17 | 1250.6 - 24.2 | 46717 + 612
19 | 18 | 1227.5 - 23.4 | 47204 + 609
20 | 19 | 1206.7 - 22.7 | 47655 + 607
21 | 20 | 1187.9 - 22.1 | 48076 + 605
22 | 21 | 1170.6 - 21.5 | 48474 + 602
23 | 22 | 1154.7 - 21.0 | 48851 + 599
24 | 23 | 1139.9 - 20.5 | 49210 + 597
25 | 24 | 1126.1 - 20.0 | 49553 + 594
$\ell-m=3$ |  |  |
5 | 2 | 4283.1 - 123.2 | 25994 + 508
6 | 3 | 3241.6 - 89.3 | 28914 + 541
7 | 4 | 2693.2 - 71.4 | 31325 + 564
8 | 5 | 2349.6 - 60.1 | 33379 + 580
9 | 6 | 2112.4 - 52.3 | 35160 + 591
10 | 7 | 1938.3 - 46.6 | 36722 + 599
11 | 8 | 1804.9 - 42.2 | 38105 + 605
12 | 9 | 1699.2 - 38.7 | 39339 + 608
13 | 10 | 1613.3 - 35.9 | 40448 + 611
14 | 11 | 1542.3 - 33.6 | 41449 + 613
15 | 12 | 1482.6 - 31.7 | 42355 + 613
16 | 13 | 1432.0 - 30.0 | 43173 + 614
17 | 14 | 1388.8 - 28.6 | 43913 + 613
18 | 15 | 1351.6 - 27.4 | 44581 + 612
19 | 16 | 1319.1 - 26.3 | 45191 + 611
20 | 17 | 1290.4 - 25.4 | 45752 + 609
21 | 18 | 1264.7 - 24.5 | 46274 + 608
22 | 19 | 1241.5 - 23.8 | 46762 + 606
23 | 20 | 1220.3 - 23.0 | 47222 + 603
24 | 21 | 1200.9 - 22.4 | 47656 + 601
25 | 22 | 1183.0 - 21.8 | 48069 + 599
$\ell-m=5$ |  |  |
8 | 3 | 3604.4 - 100.2 | 26948 + 509
9 | 4 | 2931.5 - 78.5 | 29609 + 538
10 | 5 | 2521.9 - 65.2 | 31845 + 560
11 | 6 | 2245.4 - 56.3 | 33761 + 575
12 | 7 | 2045.8 - 49.8 | 35427 + 586
13 | 8 | 1894.6 - 44.9 | 36894 + 594
14 | 9 | 1776.2 - 41.1 | 38195 + 600
15 | 10 | 1681.0 - 38.0 | 39356 + 604
16 | 11 | 1603.0 - 35.5 | 40396 + 607
17 | 12 | 1538.3 - 33.4 | 41328 + 608
18 | 13 | 1483.7 - 31.6 | 42167 + 609
19 | 14 | 1437.0 - 30.1 | 42927 + 609
20 | 15 | 1396.4 - 28.8 | 43623 + 609
21 | 16 | 1360.7 - 27.6 | 44266 + 608
22 | 17 | 1329.0 - 26.6 | 44863 + 607
23 | 18 | 1300.5 - 25.6 | 45422 + 606
24 | 19 | 1274.7 - 24.8 | 45946 + 604
25 | 20 | 1251.3 - 24.0 | 46440 + 602

Uranus Adiabatic model
---
$\ell$ | $m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
$\ell-m=1$ |  |  |
3 | 2 | 3051.9 - 86.3 | 32550 + 626
4 | 3 | 2487.2 - 66.9 | 34478 + 631
5 | 4 | 2170.8 - 56.0 | 36153 + 634
6 | 5 | 1965.1 - 48.9 | 37591 + 635
7 | 6 | 1818.9 - 43.8 | 38839 + 635
8 | 7 | 1708.7 - 40.0 | 39935 + 634
9 | 8 | 1622.1 - 37.0 | 40911 + 633
10 | 9 | 1551.7 - 34.6 | 41789 + 631
11 | 10 | 1493.1 - 32.5 | 42587 + 629
12 | 11 | 1443.4 - 30.8 | 43318 + 627
13 | 12 | 1400.5 - 29.4 | 43992 + 625
14 | 13 | 1363.0 - 28.1 | 44616 + 622
15 | 14 | 1329.9 - 26.9 | 45199 + 620
16 | 15 | 1300.4 - 25.9 | 45743 + 617
17 | 16 | 1273.8 - 25.0 | 46255 + 615
18 | 17 | 1249.8 - 24.2 | 46737 + 612
19 | 18 | 1227.9 - 23.4 | 47193 + 610
20 | 19 | 1207.8 - 22.8 | 47625 + 607
21 | 20 | 1189.4 - 22.1 | 48035 + 605
22 | 21 | 1172.3 - 21.6 | 48426 + 602
23 | 22 | 1156.5 - 21.0 | 48799 + 600
24 | 23 | 1141.8 - 20.5 | 49155 + 597
25 | 24 | 1128.0 - 20.0 | 49496 + 595
$\ell-m=3$ |  |  |
5 | 2 | 3924.0 - 112.0 | 27549 + 534
6 | 3 | 2985.3 - 81.4 | 30539 + 566
7 | 4 | 2504.9 - 65.7 | 32871 + 586
8 | 5 | 2209.7 - 56.0 | 34769 + 599
9 | 6 | 2008.3 - 49.3 | 36362 + 607
10 | 7 | 1861.0 - 44.4 | 37730 + 612
11 | 8 | 1747.9 - 40.7 | 38926 + 615
12 | 9 | 1657.9 - 37.7 | 39987 + 617
13 | 10 | 1584.4 - 35.2 | 40938 + 617
14 | 11 | 1522.8 - 33.2 | 41800 + 617
15 | 12 | 1470.5 - 31.4 | 42587 + 617
16 | 13 | 1425.2 - 29.9 | 43310 + 616
17 | 14 | 1385.7 - 28.6 | 43978 + 615
18 | 15 | 1350.8 - 27.4 | 44599 + 613
19 | 16 | 1319.7 - 26.4 | 45179 + 611
20 | 17 | 1291.7 - 25.4 | 45722 + 610
21 | 18 | 1266.4 - 24.6 | 46232 + 608
22 | 19 | 1243.4 - 23.8 | 46714 + 606
23 | 20 | 1222.3 - 23.1 | 47170 + 604
24 | 21 | 1203.0 - 22.5 | 47601 + 601
25 | 22 | 1185.1 - 21.9 | 48012 + 599
$\ell-m=5$ |  |  |
8 | 3 | 3378.7 - 93.3 | 28129 + 528
9 | 4 | 2780.7 - 74.0 | 30667 + 555
10 | 5 | 2417.7 - 62.2 | 32751 + 573
11 | 6 | 2172.6 - 54.2 | 34509 + 585
12 | 7 | 1995.1 - 48.5 | 36023 + 594
13 | 8 | 1860.2 - 44.0 | 37347 + 600
14 | 9 | 1753.7 - 40.5 | 38521 + 604
15 | 10 | 1667.3 - 37.7 | 39572 + 607
16 | 11 | 1595.5 - 35.3 | 40522 + 609
17 | 12 | 1534.9 - 33.4 | 41388 + 610
18 | 13 | 1482.9 - 31.6 | 42182 + 610
19 | 14 | 1437.6 - 30.1 | 42914 + 610
20 | 15 | 1397.9 - 28.8 | 43592 + 609
21 | 16 | 1362.7 - 27.7 | 44224 + 608
22 | 17 | 1331.1 - 26.6 | 44815 + 607
23 | 18 | 1302.8 - 25.7 | 45369 + 606
24 | 19 | 1277.0 - 24.8 | 45891 + 604
25 | 20 | 1253.6 - 24.1 | 46383 + 602

Table 12: Pattern frequencies and vertical resonance locations of different
models

Uranus Shallow model
---
$\ell$ | $m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
$\ell-m=1$ |  |  |
3 | 2 | 3069.0 - 86.9 | 32429 + 625
4 | 3 | 2633.1 - 71.6 | 33195 + 613
5 | 4 | 2344.5 - 61.5 | 34349 + 612
6 | 5 | 2133.7 - 54.2 | 35589 + 614
7 | 6 | 1970.3 - 48.5 | 36828 + 616
8 | 7 | 1839.1 - 44.0 | 38029 + 618
9 | 8 | 1731.5 - 40.4 | 39172 + 620
10 | 9 | 1642.0 - 37.3 | 40245 + 621
11 | 10 | 1566.9 - 34.8 | 41241 + 621
12 | 11 | 1503.4 - 32.7 | 42159 + 621
13 | 12 | 1449.3 - 30.9 | 43000 + 620
14 | 13 | 1402.9 - 29.3 | 43769 + 619
15 | 14 | 1362.7 - 27.9 | 44471 + 618
16 | 15 | 1327.7 - 26.8 | 45115 + 616
17 | 16 | 1296.8 - 25.7 | 45707 + 614
18 | 17 | 1269.4 - 24.8 | 46254 + 612
19 | 18 | 1244.9 - 24.0 | 46762 + 610
20 | 19 | 1222.8 - 23.2 | 47237 + 607
21 | 20 | 1202.6 - 22.5 | 47683 + 605
22 | 21 | 1184.2 - 21.9 | 48103 + 602
23 | 22 | 1167.2 - 21.4 | 48501 + 600
24 | 23 | 1151.5 - 20.8 | 48879 + 598
25 | 24 | 1136.9 - 20.3 | 49239 + 595
$\ell-m=3$ |  |  |
5 | 2 | 4269.7 - 123.0 | 26049 + 510
6 | 3 | 3264.4 - 90.3 | 28780 + 541
7 | 4 | 2730.1 - 72.8 | 31043 + 563
8 | 5 | 2390.7 - 61.7 | 32997 + 578
9 | 6 | 2152.9 - 53.8 | 34719 + 590
10 | 7 | 1976.1 - 48.0 | 36253 + 598
11 | 8 | 1839.4 - 43.5 | 37627 + 604
12 | 9 | 1730.8 - 39.9 | 38859 + 608
13 | 10 | 1642.5 - 37.0 | 39967 + 611
14 | 11 | 1569.6 - 34.6 | 40966 + 613
15 | 12 | 1508.5 - 32.6 | 41868 + 613
16 | 13 | 1456.6 - 30.9 | 42686 + 613
17 | 14 | 1411.9 - 29.4 | 43433 + 613
18 | 15 | 1373.0 - 28.1 | 44117 + 612
19 | 16 | 1338.8 - 27.0 | 44748 + 611
20 | 17 | 1308.4 - 26.0 | 45333 + 609
21 | 18 | 1281.1 - 25.1 | 45879 + 607
22 | 19 | 1256.5 - 24.2 | 46390 + 606
23 | 20 | 1234.1 - 23.5 | 46870 + 604
24 | 21 | 1213.6 - 22.8 | 47323 + 602
25 | 22 | 1194.8 - 22.2 | 47753 + 600
$\ell-m=5$ |  |  |
8 | 3 | 3677.8 - 102.8 | 26590 + 505
9 | 4 | 2995.8 - 80.7 | 29186 + 535
10 | 5 | 2577.5 - 67.2 | 31387 + 556
11 | 6 | 2293.6 - 58.0 | 33288 + 572
12 | 7 | 2088.0 - 51.3 | 34949 + 584
13 | 8 | 1932.4 - 46.3 | 36412 + 592
14 | 9 | 1810.6 - 42.3 | 37711 + 598
15 | 10 | 1712.7 - 39.1 | 38869 + 602
16 | 11 | 1632.5 - 36.5 | 39910 + 605
17 | 12 | 1565.4 - 34.3 | 40850 + 607
18 | 13 | 1508.5 - 32.4 | 41704 + 608
19 | 14 | 1459.4 - 30.8 | 42486 + 608
20 | 15 | 1416.8 - 29.4 | 43206 + 608
21 | 16 | 1379.2 - 28.2 | 43871 + 607
22 | 17 | 1345.7 - 27.1 | 44491 + 606
23 | 18 | 1315.8 - 26.1 | 45069 + 605
24 | 19 | 1288.8 - 25.2 | 45612 + 604
25 | 20 | 1264.2 - 24.4 | 46123 + 602

Neptune model
---
$\ell$ | $m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
$\ell-m=1$ |  |  |
3 | 2 | 3211.4 - 63.9 | 33238 + 447
4 | 3 | 2659.9 - 50.8 | 34830 + 449
5 | 4 | 2326.8 - 42.8 | 36470 + 454
6 | 5 | 2103.6 - 37.5 | 37953 + 457
7 | 6 | 1943.3 - 33.7 | 39264 + 460
8 | 7 | 1822.2 - 30.8 | 40423 + 462
9 | 8 | 1726.8 - 28.6 | 41458 + 463
10 | 9 | 1649.4 - 26.7 | 42391 + 464
11 | 10 | 1584.9 - 25.2 | 43241 + 464
12 | 11 | 1530.1 - 23.9 | 44021 + 464
13 | 12 | 1482.9 - 22.8 | 44741 + 464
14 | 13 | 1441.6 - 21.8 | 45411 + 463
15 | 14 | 1405.1 - 20.9 | 46036 + 463
16 | 15 | 1372.5 - 20.2 | 46623 + 462
17 | 16 | 1343.1 - 19.5 | 47175 + 461
18 | 17 | 1316.6 - 18.9 | 47697 + 460
19 | 18 | 1292.3 - 18.3 | 48191 + 459
20 | 19 | 1270.1 - 17.8 | 48660 + 459
21 | 20 | 1249.7 - 17.3 | 49107 + 458
22 | 21 | 1230.8 - 16.8 | 49534 + 457
23 | 22 | 1213.2 - 16.4 | 49941 + 455
24 | 23 | 1196.8 - 16.0 | 50332 + 454
25 | 24 | 1181.5 - 15.7 | 50706 + 453
$\ell-m=3$ |  |  |
5 | 2 | 4256.1 - 85.7 | 27568 + 374
6 | 3 | 3230.4 - 62.5 | 30610 + 400
7 | 4 | 2702.6 - 50.6 | 33013 + 417
8 | 5 | 2377.7 - 43.2 | 34983 + 429
9 | 6 | 2155.7 - 38.1 | 36645 + 437
10 | 7 | 1993.3 - 34.4 | 38079 + 443
11 | 8 | 1868.6 - 31.5 | 39336 + 448
12 | 9 | 1769.4 - 29.2 | 40455 + 451
13 | 10 | 1688.1 - 27.3 | 41462 + 453
14 | 11 | 1620.2 - 25.8 | 42377 + 455
15 | 12 | 1562.3 - 24.4 | 43215 + 456
16 | 13 | 1512.2 - 23.3 | 43987 + 457
17 | 14 | 1468.5 - 22.3 | 44703 + 457
18 | 15 | 1429.8 - 21.4 | 45370 + 457
19 | 16 | 1395.3 - 20.6 | 45994 + 457
20 | 17 | 1364.2 - 19.8 | 46580 + 457
21 | 18 | 1336.2 - 19.2 | 47132 + 457
22 | 19 | 1310.6 - 18.6 | 47655 + 456
23 | 20 | 1287.2 - 18.1 | 48150 + 455
24 | 21 | 1265.6 - 17.6 | 48620 + 455
25 | 22 | 1245.7 - 17.1 | 49068 + 454
$\ell-m=5$ |  |  |
7 | 2 | 4980.5 - 101.1 | 24839 + 340
8 | 3 | 3673.9 - 71.9 | 28103 + 371
9 | 4 | 3013.5 - 57.1 | 30708 + 393
10 | 5 | 2612.4 - 48.1 | 32860 + 409
11 | 6 | 2341.5 - 42.0 | 34684 + 420
12 | 7 | 2145.3 - 37.6 | 36262 + 429
13 | 8 | 1995.9 - 34.2 | 37648 + 435
14 | 9 | 1878.1 - 31.5 | 38881 + 440
15 | 10 | 1782.3 - 29.3 | 39990 + 444
16 | 11 | 1702.8 - 27.5 | 40996 + 447
17 | 12 | 1635.6 - 26.0 | 41915 + 449
18 | 13 | 1577.8 - 24.7 | 42760 + 451
19 | 14 | 1527.6 - 23.5 | 43543 + 452
20 | 15 | 1483.5 - 22.5 | 44270 + 453
21 | 16 | 1444.3 - 21.6 | 44949 + 453
22 | 17 | 1409.2 - 20.8 | 45585 + 454
23 | 18 | 1377.6 - 20.1 | 46183 + 454
24 | 19 | 1348.9 - 19.4 | 46748 + 454
25 | 20 | 1322.8 - 18.8 | 47282 + 453

##  Appendix B Extended Tables of g-mode Frequencies and Resonance Locations

In this appendix, we present all the calculated g-mode frequencies and
resonance locations for all the models. Table 13 displays the $\ell=m$, $n=1$
Lindblad resonance locations for Uranus. Table 14 shows the $\ell=m$, $n=1$
Lindblad resonance locations and the corotation resonance locations associated
with them for Neptune.

Table 13: $\ell=m$, $n=1$, g-mode pattern frequencies and Lindblad resonance locations of non-adiabatic Uranus models | Thick | Medium | Thin | Shallow
---|---|---|---|---
$\ell$,$m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km) |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km) |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km) |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km)
1 | 1785.9 - 49.8 | 56259 +1070 | 1807.0 - 50.7 | 55820 +1069 | 1769.6 - 49.4 | 56605 +1079 | 1418.5 - 37.2 | 65597 +1172
2 | 1723.2 - 42.2 | 47569 + 793 | 1745.5 - 43.2 | 47163 + 794 | 1717.1 - 42.2 | 47682 + 798 | 1426.3 - 31.9 | 53959 + 819
3 | 1659.1 - 38.7 | 45108 + 715 | 1684.8 - 39.8 | 44647 + 716 | 1662.1 - 39.0 | 45053 + 719 | 1400.9 - 29.7 | 50490 + 726
4 | 1602.1 - 36.2 | 44230 + 679 | 1632.0 - 37.5 | 43688 + 681 | 1614.4 - 36.9 | 44005 + 683 | 1375.4 - 28.3 | 48962 + 684
5 | 1550.0 - 34.2 | 44004 + 659 | 1584.4 - 35.6 | 43365 + 662 | 1571.3 - 35.2 | 43605 + 663 | 1351.5 - 27.3 | 48211 + 659
6 | 1501.1 - 32.4 | 44118 + 646 | 1540.3 - 34.0 | 43367 + 649 | 1531.2 - 33.7 | 43539 + 651 | 1328.6 - 26.4 | 47857 + 644
7 | 1454.7 - 30.7 | 44438 + 636 | 1499.1 - 32.5 | 43556 + 640 | 1493.5 - 32.4 | 43665 + 642 | 1306.3 - 25.5 | 47740 + 632
8 | 1410.1 - 29.1 | 44897 + 628 | 1460.5 - 31.1 | 43859 + 634 | 1457.8 - 31.1 | 43912 + 635 | 1284.3 - 24.8 | 47780 + 624
9 | 1367.0 - 27.6 | 45457 + 622 | 1424.2 - 29.8 | 44233 + 628 | 1424.2 - 29.9 | 44234 + 630 | 1262.4 - 24.0 | 47932 + 617
10 | 1325.5 - 26.2 | 46091 + 616 | 1390.1 - 28.6 | 44653 + 623 | 1392.4 - 28.8 | 44604 + 626 | 1240.6 - 23.3 | 48168 + 611
11 | 1285.9 - 24.8 | 46774 + 611 | 1358.0 - 27.5 | 45103 + 618 | 1362.5 - 27.8 | 45004 + 622 | 1218.9 - 22.5 | 48469 + 606
12 | 1248.5 - 23.6 | 47482 + 606 | 1327.7 - 26.4 | 45576 + 614 | 1334.4 - 26.8 | 45422 + 618 | 1197.3 - 21.8 | 48823 + 601
13 | 1213.7 - 22.4 | 48194 + 602 | 1298.7 - 25.4 | 46068 + 610 | 1308.1 - 25.9 | 45848 + 614 | 1176.0 - 21.1 | 49218 + 597
14 | 1181.5 - 21.4 | 48898 + 598 | 1271.0 - 24.4 | 46577 + 606 | 1283.3 - 25.0 | 46278 + 610 | 1154.9 - 20.4 | 49644 + 592
15 | 1151.9 - 20.4 | 49585 + 593 | 1244.4 - 23.5 | 47096 + 602 | 1259.8 - 24.2 | 46712 + 606 | 1134.4 - 19.7 | 50093 + 588
16 | 1124.6 - 19.5 | 50250 + 589 | 1219.2 - 22.7 | 47619 + 599 | 1237.4 - 23.4 | 47152 + 603 | 1114.4 - 19.1 | 50557 + 584
17 | 1099.6 - 18.7 | 50891 + 584 | 1195.4 - 21.9 | 48138 + 596 | 1215.7 - 22.6 | 47601 + 599 | 1095.1 - 18.4 | 51030 + 580
18 | 1076.6 - 17.9 | 51508 + 579 | 1173.0 - 21.1 | 48646 + 593 | 1194.7 - 21.9 | 48057 + 596 | 1076.6 - 17.8 | 51506 + 576
19 | 1055.3 - 17.2 | 52102 + 574 | 1152.1 - 20.4 | 49142 + 590 | 1174.6 - 21.2 | 48515 + 593 | 1059.0 - 17.3 | 51980 + 572
20 | 1035.6 - 16.6 | 52672 + 570 | 1132.6 - 19.8 | 49622 + 587 | 1155.4 - 20.6 | 48968 + 590 | 1042.2 - 16.7 | 52449 + 568
21 | 1017.3 - 16.0 | 53220 + 565 | 1114.3 - 19.2 | 50088 + 583 | 1137.2 - 20.0 | 49413 + 587 | 1026.2 - 16.2 | 52911 + 564
22 | 1000.3 - 15.4 | 53748 + 560 | 1097.2 - 18.6 | 50538 + 580 | 1120.0 - 19.4 | 49848 + 584 | 1011.1 - 15.7 | 53363 + 560
23 | 984.4 - 14.9 | 54255 + 555 | 1081.1 - 18.1 | 50973 + 577 | 1103.9 - 18.9 | 50270 + 581 | 996.8 - 15.3 | 53805 + 556
24 | 969.6 - 14.4 | 54744 + 550 | 1065.9 - 17.6 | 51395 + 574 | 1088.6 - 18.4 | 50680 + 578 | 983.2 - 14.8 | 54236 + 552
25 | 955.7 - 14.0 | 55214 + 545 | 1051.7 - 17.2 | 51803 + 571 | 1074.2 - 17.9 | 51078 + 575 | 970.4 - 14.4 | 54654 + 548
Table 14: $\ell=m$, $n=1$, g-mode pattern frequencies and Lindblad resonance locations of our Neptune model $\ell$,$m$ |  $\Omega_{\rm{pat}}$ (deg/day) |  $r_{\rm{res}}$ (km) |  $r_{\rm{cor}}$ (km)
---|---|---|---
1 | 1808.1 - 34.7 | 58959 + 766 | 37170 + 481
2 | 1683.1 - 28.2 | 51058 + 578 | 38986 + 441
3 | 1566.6 - 24.6 | 49518 + 526 | 40893 + 433
4 | 1462.1 - 21.9 | 49668 + 502 | 42815 + 432
5 | 1369.0 - 19.7 | 50502 + 489 | 44732 + 432
6 | 1287.1 - 17.7 | 51646 + 480 | 46611 + 432
7 | 1215.6 - 16.1 | 52917 + 472 | 48417 + 431
8 | 1153.8 - 14.7 | 54217 + 465 | 50129 + 429
9 | 1100.5 - 13.5 | 55495 + 458 | 51736 + 426
10 | 1054.3 - 12.5 | 56722 + 451 | 53235 + 423
11 | 1014.1 - 11.6 | 57888 + 444 | 54630 + 418
12 | 979.0 - 10.8 | 58988 + 437 | 55927 + 414
13 | 948.2 - 10.1 | 60022 + 429 | 57132 + 408
14 | 920.9 - 9.5 | 60992 + 422 | 58253 + 402
15 | 896.7 - 8.9 | 61902 + 415 | 59299 + 396
16 | 875.0 - 8.5 | 62757 + 407 | 60274 + 390
17 | 855.5 - 8.0 | 63560 + 400 | 61186 + 385
18 | 837.9 - 7.6 | 64315 + 393 | 62041 + 379
19 | 821.9 - 7.3 | 65027 + 386 | 62843 + 373
20 | 807.3 - 6.9 | 65698 + 379 | 63597 + 367
21 | 793.9 - 6.6 | 66331 + 373 | 64308 + 361
22 | 781.7 - 6.4 | 66931 + 366 | 64978 + 355
23 | 770.4 - 6.1 | 67498 + 360 | 65612 + 349
24 | 759.9 - 5.9 | 68036 + 354 | 66211 + 344
25 | 750.2 - 5.7 | 68547 + 348 | 66779 + 339