# 4

J. J. Fortney, B. Militzer, C. R. Mankovich, R. Helled, S. M. Wahl, N.
Nettelmann, W. B. Hubbard, D. J. Stevenson, L. Iess, M. S. Marley, N.
Movshovitz

# Saturn’s Interior After the Cassini Grand Finale

J. J. Fortney, B. Militzer, C. R. Mankovich, R. Helled, S. M. Wahl, N.
Nettelmann, W. B. Hubbard, D. J. Stevenson, L. Iess, M. S. Marley, N.
Movshovitz

Abstract
We present a review of Saturn’s interior structure and thermal evolution, with
a particular focus on work in the past $\sim$5 years. Data from the _Cassini_
mission, including a precise determination of the gravity field from the Grand
Finale orbits, and the still ongoing identification of ring wave features in
Saturn’s C-ring tied to seismic modes in the planet, have led to dramatic
advances in our understanding of Saturn’s structure. Models that match the
gravity field suggest that differential rotation, as seen in the visible
atmosphere, extends down to at least a depth of 10,000 km (1/6th the planet’s
radius). At greater depths, a variety of different investigations all now
point to a deep Saturn rotation rate of 10 hours and 33 minutes. There is very
compelling evidence for a central heavy element concentration (“core”), that
in most recent models is 12-20 Earth masses. Ring seismology strongly suggests
that the core is not entirely compact, but is dilute (mixed in with the
overlying H/He), and has a substantial radial extent, perhaps out to around
one-half of the planet’s radius. A wide range of thermal evolution scenarios
can match the planet’s current luminosity, with progress on better quantifying
the helium rain scenario hampered by Saturn’s poorly known atmospheric helium
abundance. We discuss the relevance of magnetic field data on understanding
the planet’s current interior structure. We point towards additional future
work that combines seismology and gravity within a framework that includes
differential rotation, and the utility of a Saturn entry probe.

##  4.1 Introduction

Understanding the structure and interior composition of Saturn is a long-
standing problem in planetary sciences. Saturn is one of the solar system’s
two examples of hydrogen/helium (H/He) dominated planets, so along with
Jupiter it serves as a benchmark for our understanding of this class of
astrophysical objects. As such it serves as important point of comparison and
contrast with Jupiter, as Saturn formed in the same protoplanetary disk but
accreted less gas. The _Cassini_ Mission data obtained and analyzed over the
past $\sim$5 years, in particular data from the Grand Finale, have enabled a
revolution in our understanding of the planet, even in comparison to the
recent pre-Finale review chapter of Fortney et al. (2016). The purpose of this
contribution is to summarize and discuss this work, from a wide variety of
authors, while pointing to future work and observational needs.

There are many issues in planetary physics that are addressed by various kinds
of interior structure modeling. We would like to understand:

  * •

The mass of any central heavy element core, along with the current structure
of the core itself in relation to the overlying H/He envelope. This allows for
a better understanding of planet formation and the subsequent evolutionary
processes (Pollack et al., 1996; Wilson and Militzer, 2012; Helled and
Stevenson, 2017).

  * •

The distribution of He within the planet. It has long been suggested that H/He
phase separation could have taken place in Saturn leading to composition
gradients and He enrichment at depth (Stevenson and Salpeter, 1977; Hubbard et
al., 1999; Fortney and Hubbard, 2003; Mankovich and Fortney, 2020).

  * •

The enhancement and distribution of “metals” in the H/He envelope, compared to
their abundance in the Sun. The envelope enrichment can be compared to Jupiter
(Wong et al., 2004; Wahl et al., 2017) and spectroscopic investigation of
Saturn’s atmosphere to better understand Saturn’s formation (chapter by Atreya
et al.).

  * •

The temperature structure of the planet and any deviation from a simple
isentropic interior. Barriers to convection due to abundance gradients
(Leconte and Chabrier, 2012, 2013a; Mankovich and Fortney, 2020) can
dramatically alter the temperature structure and thermal evolution history
(Vazan et al., 2018).

  * •

The rotation rate of the planet and the depth of any differential rotation.
Rotation uncertainties impact the validity of models and understanding
differential rotation helps to understand the mechanics of convection within
the planet (chapter by Flasar et al.)

  * •

The location within Saturn (depth and extent) where the planet’s dynamo is
found. A better understanding of the generation of Saturn’s magnetic field
feeds into better knowledge of the planet’s interior structure (chapter by Cao
et al.)

There are several essential inputs for models of the interior structure of
giant planets. These include input physics, meaning the equation of state and
mixing properties of hydrogen, helium, and “heavy elements” like water and
other ices, and rocks. A recent review for hydrogen can be found in Helled et
al. (2020). A more general overview of the interior modeling of Saturn,
including a hierarchy of modeling approaches can be found in Fortney et al.
(2016). The other needed inputs are observational data that can shed light on
the structure and thermal evolution of the planet, and it is on these topics
where we will spend time reviewing recent important advances.

##  4.2 Observational data during Grand Finale orbits

###  4.2.1 Gravity field observations

During the terminal phase of the mission, started in April 2017, _Cassini_ was
inserted into a series of 22 highly eccentric orbits, with pericenter inside
the inner ring of Saturn, at altitudes between 2600 to 3900 km above the cloud
tops. From this privileged observation point Cassini carried out the first
detailed measurement of the planet’s gravity field.

Due to operational constraints and time sharing with other instruments, only
six out of the 22 Grand Finale orbits (GFO) could be used for gravity
measurements, which require that radio contact to ground is established via
the 4m high gain antenna. In these measurements, the gravity field is inferred
from the Doppler shift of a X band microwave signal sent from a ground antenna
at 7.2 GHz and coherently re-transmitted back to Earth at 8.4 GHz via an
onboard transponder. A detailed description of the experimental configuration
is described in Iess et al. (2019). As data from one of the passes were lost
due to a station misconfiguration, the actual gravity analysis is based on
only five pericenter passes.

The key gravity observable is the Doppler shift of the microwave carrier,
which is proportional to the (two-way) range rate of the spacecraft. _Cassini_
acts as a probe mass, freely falling in the gravity field of Saturn and the
other bodies of its system (most notably, the rings). The radio link allows to
observe the line-of-sight accelerations of the spacecraft with respect to a
fiducial point (the ground station), from which the gravitational forces
acting on the spacecraft can be derived. This well consolidated method to
derive planetary gravity fields was used in particular for Jupiter by the
spacecraft _Juno_ (Iess et al., 2018).

Two ingredients are crucial for a successful gravity determination, namely a
good dynamical model, accounting for all the forces acting on the spacecraft
(described in many cases by free parameters), and high quality Doppler data.
The gravity GFO were by design carried out at large solar elongation angles,
when charged particle noise from the interplanetary plasma is relatively low.
For this reason, the data quality was statistically equivalent in X and Ka
band, with a RMS Doppler noise of 0.020-0.088 mm s-1 at 30 s integration time,
depending on the local conditions at the antenna (mostly wind and water vapour
content along the line of sight). This noise is still much smaller (by a
factor of 40 and 200) than the Doppler signal produced by weakest the
harmonics (J3, J10) and Saturn’s ring.

Figure 4.1: Comparison of the gravity harmonics measured for Jupiter and Saturn with predictions from models assuming uniform rotation throughout the entire interior of both planets. The deviations are small for Jupiter while substantial discrepancies emerge for Saturn. This illustrates that the effects of differential rotation are much more important for Saturn, as discussed in Section 4.3 . Models are from Militzer et al. (2019).  Gravity | Cassini | Range of models with | Model with
---|---|---|---
harmonics | measurements | uniform rotation | differential rotation
$J_{2}$ | 16 290.573 $\pm$ 0.028 | 16 290.57 | 16 290.573
$J_{4}$ |  $-$935.314 $\pm$ 0.037 |  $-$935.31 |  $-$935.312
$J_{6}$ | 86.340 $\pm$ 0.087 | 80.74 $\ldots$ 81.76 | 86.343
$J_{8}$ |  $-$14.624 $\pm$ 0.205 |  $-$8.96 $\ldots$ $-$8.70 |  $-$14.616
$J_{10}$ | 4.672 $\pm$ 0.420 | 1.08 $\ldots$ 1.13 | 4.677
Table 4.1: The Cassini gravity measurements are compared with models that
assume Saturn’s interior to rotate uniformly or differentially. The
unexpectedly large magnitudes of $J_{6}$, $J_{8}$, and $J_{10}$ could not be
matched with uniform rotation models that have a planetary core mass and
envelope metallicity adjusted to exactly fit to the observed $J_{2}$ and
$J_{4}$ (Iess et al., 2019). This is also illustrated in Fig. 4.1.

The Doppler data were processed using the JPL MONTE orbit determination code
(Evans et al., 2016). The parameters of Saturn’s gravity field that were
estimated were Saturn’s GM, the zonal harmonic coefficients J2-J20 (shown in
Figure 4.1) the tesseral field of degree 2 (allowing for non-principal axis
rotation), and the mass of the rings. Although gravity measurements are
sensitive to the pole orientation, the pole position was adopted from the more
accurate ring occultation measurements of French et al. (2017). We refer to
Iess et al. (2019) for a discussion on the dynamical model and the orbital
fit.

While the purely zonal field absorbs most of the accelerations acting on
Cassini a surprising finding of the analysis was the impossibility to reduce
the Doppler residuals to the noise level in the combined analysis. Residual
accelerations of magnitude at the level of $4\times 10^{-7}$ m s-2 were found
within one hour from the epoch of closest approach. By comparison, Juno data
could be fit by a purely zonal field, although now, as more passes are
accumulated, tiny accelerations about a factor of 10-20 smaller than those
experienced by Cassini (2$\times$ $10^{-8}$ m s-2) also start showing up in
the analysis of Jupiter gravity (Durante et al., 2020).

The nature of these accelerations, a puzzling outcome of the Cassini data
analysis, is still unknown. They could be due to gravitational effects caused
by longitude-dependent wind dynamics, deep seated gravity anomalies, or normal
modes (Durante et al., 2017; Markham and Stevenson, 2018; Markham et al.,
2020, this list is not exhaustive) Although normal modes and a high degree
tesseral field are able to flatten the Doppler residuals, the preferred method
of analysis relied on the use of stochastic accelerations. Remarkably, the
estimated values of the zonal field and the B ring mass were nearly
insensitive to the method adopted in the analysis.

Understanding the source of these residual fields is one of the open questions
for both planets. Analysis of oscillations that show up on the surface of the
planets –and for Saturn in its rings–can provide unique constraints on
internal structure.

###  4.2.2 Ring seismological observations

Although _Cassini_ ’s journey ended almost four years before this chapter’s
writing, ongoing analysis of stellar occultation data gathered by _Cassini_
VIMS continues to expose ring waves that constrain Saturn’s interior. In their
initial study of the potential for Saturn f-modes to excite bending and
density waves in the C ring, Marley and Porco (1993) predicted locations for
about a dozen ring features. Hedman and Nicholson (2013) confirmed the f-mode
– ring wave connection by associating six waves with planetary oscillation
modes. Subsequently Hedman, Nicholson, French, and collaborators (Hedman and
Nicholson, 2014; French et al., 2016, 2019; Hedman et al., 2019; French et
al., 2021) have continued to identify ring waves using the entirety of the
_Cassini_ ring data set and now have identified more than thirty waves as
being clearly associated with Saturn modes. In particular these modes are all
_non-radial_ modes that propagate prograde in Saturn’s rotating reference
frame, and are conventionally assigned negative azimuthal wavenumbers $m<0$.
The modes span a range of angular degrees $\ell\geq 2$, azimuthal wavenumbers
$-\ell\leq m<0$, and latitudinal wavenumbers $\ell-|m|=\ell+m$. Fig. 4.2
provides a visual reference for these labels of angular structure. In addition
it is now clear that about 7 of these ring waves are influenced by
interactions between low degree f-modes and g-modes, providing probes of
static stability and composition gradients in Saturn’s interior, a first in
planetary seismology. The mechanism and utility of ring seismology for studies
of Saturn’s interior are reviewed in Mankovich (2020).

Each wave’s location (or pattern speed) and $m$ value together precisely
measure the oscillation mode frequency in the planet, which in in turn is
sensitive to Saturn’s rotation through a combination of the Doppler shift and
intrinsic rotation effects (e.g., the Coriolis force). Because the
wavefunctions of distinct planetary oscillation modes are concentrated in
different regions of the planetary interior, the full set of frequencies
provides an exceptional new window into the variation of density, buoyancy,
and rotation within Saturn’s interior. Fig. 4.3 summarizes the known
seismology constraints from the C ring, compared to the model of Mankovich et
al. (2019) who used the pattern frequencies of 14 waves to measure the solid
body rotation rate of Saturn as $P_{\rm Sat}=\rm
10h\,33m\,38s^{+1m\,52s}_{-1m\,19s}$.

The spate of subsequent ring wave detections by French et al. (2021) show
remarkable agreement with predictions from this 2019 model (Fig. 4.3) and
extend the catalogue of Saturn’s seismic constraints to angular degrees as
high as $\ell=19$ and latitudinal wavenumbers as high as $\ell+m=8$. With the
seismic constraints now spanning a swath of radial and angular structures
inside Saturn, it should be possible to further refine the seismic measurement
of Saturn’s bulk rotation and quantify departures from uniform rotation. The
full interpretation of this remarkable and growing dataset is still in its
early stages; recent findings from modeling efforts are summarized in §4.4
below.

Figure 4.2: The first few prograde spherical harmonics, a convenient basis for
describing the angular structure of Saturn’s nonradial modes of oscillation.
Angular degree $\ell$ increases downward, and azimuthal order $|m|$ increases
to the right. The latitudinal wavenumber $\ell+m$ increases away from the main
diagonal $\ell=-m$. Adapted from Mankovich (2020).  Figure 4.3: Observed wave
patterns in Saturn’s C ring (filled symbols) compared with expected resonance
locations for a theoretical spectrum of Saturn f-mode oscillations (open boxes
connected by colored lines). Modeled frequencies are shown for a three-layer
interior model from Mankovich et al. (2019) and are colored by the
corresponding normal mode’s latitudinal wavenumber $\ell+m$. Resonances
falling in the tenuous D ring close to Saturn or in the optically thick B ring
are unlikely to be observed.

###  4.2.3 He abundance

The determination of Saturn’s atmospheric helium abundance has long been a
pressing question, as it has important implications for the role of hydrogen-
helium phase separation in the planet’s deep interior (Stevenson and Salpeter,
1977; Fortney and Hubbard, 2003; Mankovich and Fortney, 2020). Since there has
not yet been a Saturn entry probe to measure atmospheric abundances in situ,
helium abundance determinations have been done via remote sensing, which is a
difficult data analysis problem. A review of the _Voyager_ -era remote sensing
issues, which typically combine radio occultation data and thermal infrared
spectroscopy, is provided in Conrath and Gautier (2000).

More recent determinations of Saturn’s atmospheric He abundance have come from
two papers. The first (Achterberg and Flasar, 2020) uses thermal infrared
spectroscopy from _Cassini_ ’s Composite Infrared Spectrometer (CIRS) of
wavelength regions that are sensitive to He-H2 collision-induced absorption
(CIA) features. They determine a He/H2 number ratio of 0.04–0.075, well below
the protosolar mixing ratio, suggesting significant amounts of helium have
rained down within the planet. Alternatively, Koskinen and Guerlet (2018) use
the _Cassini_ Ultraviolet Imaging Spectrograph (UVIS) (above the homopause)
and limb scans observed by CIRS (below the homopause) to create empirical
atmospheric structure models for Saturn (assuming an eddy diffusion
coefficient constrained by the vertical methane distribution), to assess the
atmosphere’s mean molecular weight with height. These authors find a volume
mixing ratio of $0.125\pm 0.025$, suggesting only modest He depletion. The
size of the error bar on both of these modern measurements, and more
importantly, the large difference between them, leads to significant
uncertainties in models of the planet’s current structure and evolution, and
stands in stark contrast with the precise determination of the He/H2 in
Jupiter’s atmosphere of $0.157\pm 0.003$. (von Zahn et al., 1998).

##  4.3 Interior structure models constrained by planet’s gravity field

###  4.3.1 Interior Model Background

Historically, the interior structure of Saturn was divided into distinct
layers as shown in Figure 4.4 (left). At the center is a compact core composed
predominately of rock and ice. The hydrogen-helium envelope is divided into an
outer region, where hydrogen forms molecules, and an inner region where
hydrogen is predicted to be a fluid atomic metal. The inner and outer envelope
is separated by a layer in which helium is immisicible and rains out of the
mixture (Stevenson and Salpeter, 1977). In some models (Püstow et al., 2016;
Mankovich and Fortney, 2020), the helium rain layer spans the entire inner
envelope resulting in a helium “ocean” forming outside the core.

The heavy element core could exhibit some degree of mixing, referred to as a
“dilute” or “fuzzy” core and characterized by a composition gradient extending
outwards into the inner envelope (Figure 4.4 right). Ab-initio materials
calculations have demonstrated that both rock and ice phases are soluble in
hydrogen at the conditions of Saturn’s core (González-Cataldo et al., 2014,
and references therein). However, gravitational stability of the heavy core
material could maintain a compact core against convective redistribution with
the overlying envelope (Moll et al., 2017).

In an attempt to be consistent with current (and still-evolving) usage, we
here characterize cores as either “compact” or “dilute/fuzzy.” Compact cores
consist of rocks, ice, or some combination thereof; their Z value (the heavy
elements mass fraction) is 1. The dilute/fuzzy core will have H/He “mixed in,”
with $Z<1$, and the fuzziness could be a portion of the core, or the entire
core, and this region may have a gradient in Z. Although there is not
necessarily a well-defined boundary between a dilute/fuzzy core and the
envelope, in some models its heavy element abundance is represented by a
constant-Z value (Ni 2020), while in others by a Gaussian-Z distribution (Wahl
et al., 2017), or another kind of gradient (Mankovich and Fuller, 2021). The
latter, discontinuous case is equivalent to what has been denoted inner
envelope metallicity (Nettelmann et al., 2013) or deeper envelope (Miguel et
al., 2016). For consistency with previous usage, in those cases (see Table
4.2, later) we tabulate as “core mass” only the mass of the compact core.
Otherwise, we count as core mass the mass of heavy elements in the
dilute/fuzzy region plus the mass of a central compact core, if any.

Figure 4.4: Sketches of Saturn’s possible internal structure. Left:
traditional layered structure. Right: Deep interiors with fuzzy core and
composition gradients. In this case Saturn’s deep interior is expected to be
non-adiabatic (adapted from Helled et al. (2020)).

Interior models of Saturn, with distinct layers similar to Figure 4.4 (left),
have previously been fit to pre-Grand Finale _Cassini_ data (Helled and
Guillot, 2013; Nettelmann et al., 2013). Updated modeling results using the
_Cassini_ Grand Finale gravity were presented in Iess et al. (2019) and
explored in greater detail in Militzer et al. (2019) and Ni (2020). For
further modeling details on typical model construction, see Fortney et al.
(2016).

Given our focus on _Cassini_ and space constraints of the chapter, we can only
touch on equation of state (EOS) issues. Recent tabulations of H/He EOS are
found in Militzer et al. (2016); Chabrier et al. (2019) and a recent review of
hydrogen is found in Helled et al. (2020). To be clear, uncertainties in the
equation of state directly translate into uncertainty of the interior
structure models. As an example of uncertainties still present in theory, we
note that predictions from density functional theory (DFT) (e.g., Militzer,
2013), more widely used in planetary EOS calculations, and Monte Carlo
calculations by Mazzola et al. (2018), yield 5% differences in the density of
pure hydrogen at 6000 K over a wide pressure range.

Due to progress with laboratory experiments, the entire pressure range in
Saturn’s and Jupiter’s interiors can now be probed with dynamic compression
techniques (Kritcher et al., 2020). While it has remained a challenge to
determine the density with an accuracy of 1%, the real difficulty lies in the
temperature measurements. In many shock wave experiments, temperature is not
measured but inferred from models.

Furthermore, the location of the hydrogen-helium immiscibility (“demixing”)
line in Fig. 4.5 is an additional source of uncertainty (see section 4.6.1 and
recent experimental work by Brygoo et al. (2021)). Some uncertainties are
introduced when different equations of state at low and high pressures are
combined into one model but most of them can be avoided if absolute entropies
(Militzer and Hubbard, 2013) are used that enable one to determine the
temperature for given entropy at different pressure points independently.

To conclude, one has to keep in mind that all the interior models constructed
using physical EOSs are affected by their uncertainties. As a result, the
inferred composition depends to some degree on the choice of the EOS.
Therefore, in order to advance our understanding of the giant planets progress
in experiment and computer simulations of relevant materials at high pressure
and temperature is required.

Since planets cool by convection, Saturn’s envelope is most often assumed to
have an adiabatic temperature profile. Under this assumption, the temperature
through the outer envelope is defined by an isentrope of the chosen equation
of state with entropy $S_{\rm mol}$ chosen to be consistent with the observed
temperature at 1 bar. Typically this is cited at $135\pm 5$K. In practice,
though, Militzer et al. (2019) use 143 K after correcting the 135 K value from
Lindal (1992) for a change in (very uncertain) helium abundance (Guillot,
1999). However, regions with static, stable compositional gradients, or
exhibiting double-diffusive convection (sometimes referred to as semi-
convection) can have super-adiabatic gradients (Leconte and Chabrier, 2013a;
Nettelmann et al., 2015).

The helium rain layer is one example of a region of possible non-adiabatic
behavior, while the core region is another. Different implementations of
helium rain in Saturn based on various phase diagrams have been adopted
(Fortney and Hubbard, 2003; Püstow et al., 2016; Militzer et al., 2019;
Mankovich and Fortney, 2020). Within a framework of a model that matches the
gravity field, any He abundance less the protosolar value in the outer regions
must be made up for with an enrichment at depth, which conserves total helium.

Figure 4.5: Phase diagram for a H-He mixture of a proto-solar composition as
predicted by numerical calculations together with representative _P-T_
profiles of Jupiter and Saturn. Green: Calculations from PBE with non-ideal
entropy of mixing (Morales et al., 2009). Orange: demixing region using the
vdW-DF and non-ideal entropy of mixing (Schöttler and Redmer, 2018). The red
line corresponding to the metallization of H as predicted by Mazzola et al.
(2018). Jupiter and Saturn isentropes are labeled. The figure is modified from
Helled et al. (2020).

###  4.3.2 Matching interior models to gravity measurements

The plausibility of a given interior structure model is judged by translating
the composition and temperature parameters for each layer into a density
distribution, from which the gravity field can be calculated and compared to
spacecraft measurements. It is not possible to uniquely infer the composition
and the distribution of the elements. As a result, the exact composition and
internal structure of Saturn remains unknown. However, more accurate gravity
data can be used to further constrain the interior. Prior to the _Cassini_
Grand Finale, the best measurements of Saturn’s gravity (Jacobson et al.,
2006) constrained only the first three even zonal harmonics $J_{2}$, $J_{4}$
and $J_{6}$. As summarized in Section 4.2.1 the gravity observations (Iess et
al., 2019) from _Cassini_ ’s Grand Finale orbits improved constraints on
$J_{2}$-$J_{6}$ and determined higher order moments of Saturn’s field, thus
placing much stricter constraints on the planet’s interior density
distribution.

The unprecedented precision of _Cassini_ Grand Finale gravity measurements, as
well as _Juno_ ’s measurements at Jupiter, necessitated the development of
more accurate methods for calculating self-consistently the shape and gravity
field (Hubbard, 2013; Wahl et al., 2017; Militzer et al., 2019; Nettelmann et
al., 2021).

A planet’s density profile influences the shape of the gravity field through
rotation and tides, which cause equipotential surfaces within the planet to
deviate from spherical symmetry. In the case of the rapidly rotating Jovian
planets, the non-spherical potential is dominated by the centrifugal force.
For a planet with a uniform rotation rate (i.e. no internal dynamics from
winds) both the density distribution and resulting gravity field are
axisymmetric and north-south symmetric, meaning that only even zonal
harmonics, $J_{2n}$, are non-zero. It is important to note, however, that even
for such a simplified case the relationship between $J_{n}$ and a particular
density distribution is inherently non-unique (Movshovitz and Fortney, 2022).

For a uniformly rotating planet in hydrostatic equilibrium, the magnitudes of
the even zonal harmonics decay as $|J_{2n}|\sim q_{\rm rot}^{n}$, where
$q_{\rm rot}$ is the ratio of the centrifugal and gravity accelerations at the
equator. As illustrated in Fig. 4.1 the gravity field of Jupiter measured by
the _Juno_ spacecraft (Folkner et al., 2017) is broadly consistent with this
relationship. By contrast, the observed Saturn field exhibits significant
divergence from the expected relationship for even moments $J_{6}$ and higher
(see Tab. 4.1). Iess et al. (2019) and Galanti et al. (2019) demonstrated that
Saturn’s observed gravity cannot be reproduced with models that assume uniform
rotation, and instead requires deep differential rotation (Hubbard, 1982).

This departure can be illustrated in terms of the simple thin-cord model shown
in Fig. 4.6. Here we have a Saturn model of mass $M_{S}$ in hydrostatic
equilibrium rotating with a single rotation period $P_{\rm Sat}$ (shaded
inset), The gravitational moments of this Saturn model are shown as red
pluses, falling to unobservable small values at $J_{12}$ and beyond. We
imagine this Saturn model’s equator to be encircled by a very thin cord of
total mass $\mu\ll M_{S}$, shown as a dashed line on the shaded inset. Using
Eq. (3) and Eq. (6) of Trubitsyn et al. (1978) , we have for the contribution
of this cord to the gravitational moment $J_{n}$,

$\Delta J_{n}=-\frac{\mu}{M_{S}}P_{n}(0),$

where $P_{n}(0)$ is the value of the n-th degree Legendre polynomial evaluated
at argument 0, the equator. The $\Delta J_{n}$ for the thin cord, assuming
$\mu/M_{S}=1.5\times 10^{-5}$, are plotted as open circles in Fig. 4.6. The
total observed $J_{n}$ are then the sum of the solid-body $J_{n}$ (pluses) and
the $\Delta J_{n}$, as shown in Fig. 4.6, qualitatively reproducing the
observed values.

The postulated thin cord has a gravitational signature similar to that of an
equatorial uplift induced by strong eastward zonal flow near the equator; we
note that the best-fit value for $\mu/M_{S}$ is orders of magnitude larger
than estimated values for Saturn’s total ring mass, so the rings cannot
possibly account for the progression in $J_{n}$.

Figure 4.6: Thin-cord model for Saturn’s high-order gravity harmonics. The red
pluses are $J_{n}$ calculated for a uniform rotating planet, the grey open
circles are the $J_{n}$ from thin cord, while orange squares are the combined
gravity signal. The grey filled circles are the _Cassini_ measurements
(plotted up to $J_{10}$) all larger than the (vertical) error bars, for
clarity.

###  4.3.3 Oblateness and rotation period

Images of the giant planets Jupiter and Saturn show a banded texture, which is
associated with winds of latitude-dependent velocities. In contrast, their
deep interiors are commonly thought to rotate rigidly. This rigid body
rotation rate is an important parameter for models of the giant planets as
Jupiter and Saturn in several aspects. It defines the reference system for the
observed winds and determines the centrifugal forces acting on the rotating
planet. The stronger such force, the more oblate the planetary shape becomes.
As shape and gravity field are intermingled, any inference of internal
structure through gravity field observations, such as acquired during the
Cassini Grand Finale, ultimately depends on the uncertainty in rotation rate.
Nevertheless, Saturn’s degree of flattening (oblateness) offers an independent
constraint on the rotation rate when combined with interior structure models
(Helled et al., 2009b).

In contrast to Jupiter, where magnetic field observations from space allowed
to infer Jupiter’s rotation period to high accuracy of less than 0.01 s, the
increasingly accurate Saturn magnetic field observations culminating in the
Cassini Grand Finale revealed an increasingly close spin axis—magnetic dipole
axis alignment, preventing determination of its rotation rate by magnetic
field observations. Nevertheless, an interplay of multiple Saturn (system)
observations and model predictions has, at about the time of the Cassini Grand
Finale, been converging toward a value around 10h 33m.

Several observables related to Saturn’s magnetosphere have periods, ranging
from the Voyager value of 10h 39m 22s inferred from radio emissions to 10h 47m
6s inferred from non-axisymmetric components of Cassini magnetic field data
(Giampieri et al., 2006). However, due to the time-variability of the magnetic
field data over rather short time-scales of days to decades, as revealed by
Cassini, none of these periods necessarily reflects the rotation of the deep
interior. Alternative methods predict faster rotation.

One method to predict a significantly faster rotation of 10h 32m is the
minimization method, where the dynamical height of the 100 mbar surface is
minimized by either using zonal wind or oblateness observations (Anderson and
Schubert, 2007). This method reproduces the system III rotation period of
Jupiter to within 1 minute (Helled et al., 2009b). Another method relies on
shape and gravity data. Using Pioneer and Voyager occultation data and Cassini
Grand Finale-like tight constraints on the low-order gravity field, the
rotation period was predicted to fall within 10h 31m and 10h 38m (Helled et
al., 2015).

By combining radio occultation oblateness measurements (Lindal et al., 1985),
Cassini grand finale gravity data on $J_{2}$–$J_{10}$, and interior models,
the rotation rate was constrained to within 1 minute around 10h 33m 34s
(Militzer et al., 2019).

Figure 4.7: Oblateness derived from CMS models with different rotation periods
compared to the radio occultation oblateness measurement by Lindal et al.
(1985), which determined an oblateness of 0.09822 $\pm$ 0.00018. Based on this
comparison, the derived a rotation period of 10:33:34 h $\pm$ 55 s (Militzer
et al., 2019) (red with shaded error) this is consistent with the rate of
10:32:45 h $\pm$ 46 s, inferred by Helled et al. (2015) and the one estimated
from ring seismology of 10:33:38 h${}^{+112\rm s}_{~{}-89\rm s}$ (Mankovich et
al., 2019) (green with shaded error).

Figure 4.7 compares Saturn interior models inferred by Militzer et al. (2019)
with differential rotation constructed for five different rotation periods
ranging from 10:30:00 to 10:47:06 h. For each period, it is possible to
construct interior models, including differential rotation on cylinders, that
match all even $J_{n}$. However, only a narrow range in rotation period is
consistent with the Lindal et al. (1985) radio occultation measurements. A
similar analysis performed on the identical radio occultation measurements
with solid-body rotation (Anderson and Schubert, 2007), suggests that the
contribution of differential rotation leads to a $\sim$50 s slower prediction
using this method.

The Militzer et al. (2019) estimate is in excellent agreement with the results
of Helled et al. (2015) who used gravity and shape data and calculated a
period of 10:32:45 h $\pm$ 46 s, and with the results based on the analysis of
ring seismology, with a period of 10:33:38 h${}^{+112\rm s}_{~{}-89\rm s}$
(Mankovich et al., 2019), which is further discussed in Section 4.4.2.

###  4.3.4 The influence of rotation period on other properties

The assumed rotation rate influences the inferred heavy element mass fraction
in the H/He envelope $Z$, with faster rotation allowing for higher atmospheric
$Z$-values but also a stronger possible heavy element gradients $Z(r)$ toward
the center (Nettelmann et al., 2013). Using _Cassini_ Grand Finale gravity
data, up to $4\times$ solar atmospheric $Z$ is found Militzer et al. (2019).
While this value is higher than that of comparable models for Jupiter, which
struggle to pass beyond $1\times$ solar, still falls short of the observed
methane enrichment of 9$\times$ solar, as was already discussed based on pre-
Finale data Fortney et al. (2016).

Beyond interior models and wind models, the rotation rate is an important
parameter; it also influences the moment of inertia and the Love numbers
$k_{nm}$, which quantify the planet’s tidal response to its satellites or
parent star. A lesson learned from Juno at Jupiter is that superposition of
dynamic effects changes its $k_{22}$ value by $\sim 5\%$ compared to the
static value (Durante et al., 2020). While Jupiter’s _observed_ $k_{22}$ value
is inferred from spacecraft tracking (Juno), for Saturn it is observations of
its moons’ positions acquired over decades. With _Cassini_ , that allowed for
a first estimate of Saturn’s $k_{22}$ value, yielding a value of $0.390\pm
0.024$ Lainey et al. (2017). Further spacecraft tracking of _Cassini_ and
astrometric/imaging data allowed Lainey et al. (2020) to refine ${\rm
Re}(k_{2})=0.382$ with a $3\sigma$ formal uncertainty of $\pm 0.017$ (Figure
4.8.)

Faster rotation favors lower static $k_{22}$ values from interior model
predictions. Notably, a rotation rate of 10h 34m or faster, as suggested by
the diverse independent approaches, brings the static model $k_{22}$ value
into agreement with the measured one, but the central value is not reached by
any static tidal response model (Wahl et al., 2017) unless the rotation period
would be further lowered to $\sim$ 10h 15m. However, given the uncertainty in
Saturn’s $k_{22}$ value, which is due to uncertainties in the tidal state of
Enceladus but also due to differences in numerical orbit predictions, it might
be too early to suggest that dynamic effects play a similarly small but non-
negligible role on Saturn as on Jupiter. Figure 4.8 illustrates the available
Saturn rotation periods and $k_{2}$-estimates.

Figure 4.8: Love number $k_{2}$ values and Saturn’s rotation period. Top:
observationally derived (open symbols) $k_{2}$ values of Lainey et al. (2017)
and Lainey et al. (2020) with $3\sigma$ formal uncertainty (dashed horizontal
lines) placed arbitrarily on x-axis; calculated static $k_{2}$ values of 3D
Saturn models (colored triangles: Wahl et al. (2017), colored diamonds: 2D
interior models of Nettelmann et al. (2013) fed into the 3D modeling tool of
Nettelmann (2019)), vertical yellow bar: $k_{2}$ of 2D models in Lainey et al.
(2020). Model $k_{2}$ values depend on assumed rotation period. Bottom:
predicted rotation rates from magnetic field (grey, blue), winds and shape
(orange), oblateness (red), and seismology-related ring features (dark red).

###  4.3.5 Inferred Depth of Differential Rotation

Large-scale zonal wind patterns in Saturn’s upper atmosphere have been
characterized and monitored using optical cloud tracking both preceding and
for the duration of the _Cassini_ mission (García-Melendo et al., 2011). The
most prominent feature of the observed wind profiles is a pronounced eastward
jet centered on the equator. Prior to the _Cassini_ Grand Finale gravity
measurements, the depth of these zonal flows was unconstrained, and their
effect was not considered in earlier modeling studies of Saturn’s interior
(Helled and Guillot, 2013; Nettelmann et al., 2013).

Although the possible contribution of deep winds to Saturn’s gravity is a
highly non-unique problem in principle, with the Grand Finale gravity
measurements it is possible to test a model in which the cloud-level velocity
profiles are mapped onto a hypothesized deep flow pattern. When the zonal-wind
velocity profile is extended to depths of many scale heights, it modifies the
axisymmetric gravitational field, meaning that the even $J_{2n}$ are modified
from the values expected for a uniformly rotating body in a predictable
fashion (Hubbard, 1982, 1999). Since the velocity profiles of the observed
winds have a north-south asymmetry, there must also be a small asymmetric
contribution to the gravity, manifesting itself as non-zero odd $J_{n}$
(Kaspi, 2013). Iess et al. (2019) reported measurements $J_{3}$ and $J_{5}$
that are thus interpreted as a consequence of differential rotation.

Iess et al. (2019) explores two different methods for incorporating
differential rotation into gravity models. The first treats interior dynamics
purely as uniform rotation on cylinders that extend all the way through the
planets equatorial plane. This simplified flow structure has the benefit that
it can be described using potential theory (Hubbard, 1982), and thus can be
integrated into the the CMS gravity simulation in a fully self consistent
manner (Wisdom and Hubbard, 2016). The contribution to the potential from
differential rotation modifies the shape of the equipotential surfaces, which
feeds back into the calculated gravitational field. The downside of this
method is that it cannot account for a decay of velocity with depth, which is
expected due to interactions with the magnetic field as hydrogen’s electrical
conductivity increases with increasing pressure (Cao and Stevenson, 2017).

Figure 4.9 shows a velocity profile for differential rotation on cylinders,
optimized to match the even $J_{2n}$ from the _Cassini_ Grand Finale, compared
with time and latitude averaged wind velocities from cloud tracking (García-
Melendo et al., 2011). The good agreement between the rotation on cylinders
model and the observed winds beyond $\sim$45,000 km from the rotation axis
suggest that the zonal winds extend to depths exceeding 10,000 km, and that
the dynamic contribution to the gravity is dominated by the equatorial jet.
Rotation on cylinders cannot adequately represent the high-latitude winds,
since at high latitudes the winds must be truncated, rather than extending all
the way through the center of the planet. Differential rotation on cylinders
is also north-south-symmetric by definition, so the odd $J_{n}$ cannot be
reproduced by this method.

Figure 4.9 is a corrected version of Fig. 5 of Militzer et al. (2019), where
the observed cloud-tracking wind speeds were incorrectly referenced to a frame
rotating with a period of 10:33:34 h rather than the System III period of
10:39:22.4 h. When this adjustment is made, lowering the ordinates of the
cloud-tracking data by about one percent, we obtain Fig. 4.9, with improved
agreement with the cloud-tracking data. As before the model reproduces the
eastward equatorial jet that rotates faster than the deep interior. But now,
the models also reproduce the westward jet that rotates 1% slower than the
deep interior. The CMS models in Militzer et al. (2019) cannot reproduce the
jets at higher latitudes because differential rotation on cylinders was
assumed and they would need to reach unphysically deep into the planet’s
interior to reproduce the high-latitude jets. If one instead terminates these
cylinders at finite depth, the jets at all latitudes can be reproduced
reasonably well (Iess et al., 2019) as is demonstrated with the following
approach.

Figure 4.9: Average of the rotation profiles in a suite of Saturn interior
models that match the observed even gravity harmonics. Differential rotation
must be several thousands of kilometers deep. Rotation on cylinders reproduces
the eastward equatorial jet that rotates about 3% faster than the deep
interior. The inset shows an illustration of the cylinders. The rotation
frequencies inferred by tracking the clouds in Saturn’s visible atmosphere
(García-Melendo et al., 2011) are shown for comparison.

The second method assumes the winds introduce a correction to the gravity
solution obtained from a uniformly rotating model. The thermal wind equation
(Kaspi, 2013; Galanti et al., 2017, 2019) or the gravitational thermal wind
equation (Kong et al., 2013) are employed to derive a local correction to
density from the assumed flow structure. This density change is integrated to
find corrections to the gravitational moments. This method allows for
consideration of more complicated wind fields at the expense of introducing
additional approximations, and neglecting the feedback of differential
rotation on shape of the planet’s internal density distribution. Iess et al.
(2019) includes calculations in which the observed $J_{n}$, including the odd
moments, are matched by optimizing a function describing decay of the surface
wind velocities with depth. For these models, wind depths of $\sim$9000 km
best match the _Cassini_ Grand Finale gravity, in rough agreement with the
prediction from rotation on cylinders.

For comparison, the depth of the winds in Jupiter was recently determined
thanks to the Juno mission (Iess et al., 2018; Kaspi et al., 2018; Guillot et
al., 2018). The depth was estimated to be 2000–3000 km, suggesting that 1% of
the outer part of Jupiter (by mass) is differentially rotating in patterns
similar to the observed atmospheric winds. This penetration depth is in
agreement with the one inferred using theoretical calculations of the ohmic
dissipation constraint associated with the metallization of hydrogen (e.g.,
Liu et al., 2008), but see also Christensen et al. (2020) for possible
complications.

It is expected that the winds can penetrate deeper regions in the case of
Saturn since hydrogen metallization occurs deeper due to its smaller mass and
resulting pressures (Cao and Stevenson, 2017). For Jupiter and Saturn the
depths of the winds correspond to about 95% and 80% of the total planetary
radius, respectively. A detailed discussion on that topic can be found in
Kaspi et al. (2020).

###  4.3.6 Results from recent Structure Models

Evaluation of interior parameters

Figure 4.10, adapted from Militzer et al. (2019), shows the interior model
parameters from ensembles of models assuming distinct layers, generated with
Monte Carlo sampling for two different rotation periods and core radii. Here a
core radius of $r_{c}$=0.188 corresponds to a compact core with an iron-
silicate (0.325:0.675) composition, while $r_{c}$=0.231 corresponds to a solar
iron-silicate-water ice (0.1625:0.3375:0.5) composition, assuming homogeneous
mixtures using the additive volume rule with their respective equations of
state. Panel (a) shows the distribution of heavy element mass between the
envelope and core. Militzer et al. (2019) infers the mass of Saturn’s core to
be $\sim$15–18 $M_{\oplus}$ with additional heavy elements (1.5–5
$M_{\oplus}$) distributed throughout the planet’s H/He envelope. This is in
good agreement with 4-layer Saturn models of Ni (2020) who obtains 12–18
$M_{\oplus}$ for the accumulated mass of heavy elements in a compact core and
an overlying diluted core layer.

Panels (b-f) of Figure 4.10 show the relationship between the compositional
parameters ($Y$ and $Z$) in the outer (mol) and inner (met) envelope, along
with the entropy $S$ as a proxy for envelope temperature, for the same
ensembles of interior models. Models with longer rotation periods are found to
be more restricted in parameter space, requiring higher interior entropies,
lower envelope metallicities and/or lesser degree of helium rainout. The
models with shorter rotation periods produce $Y_{\rm mol}$ that are compatible
with reanalyzed Voyager measurements of atmospheric helium,
$\sim$0.6–0.8$\times$solar (Conrath and Gautier, 2000), while models with
longer rotation rates require less depletion of helium in the outer envelope.
The _Cassini_ Grand Finale gravity permits models without enrichment of heavy
elements in the inner envelope ($Z_{\rm mol}\approx Z_{\rm met}$). By contrast
interior models of Jupiter fit to _Juno_ gravity data require either an
enrichment of $Z$ across the helium rain layer (Miguel et al., 2016) or an
extended ‘dilute’ core (Wahl et al., 2017), neither of which is required for
Saturn by gravity alone.

The core masses of $\sim$15–18 $M_{\oplus}$ found from Militzer et al. (2019)
models that use _Cassini_ Grand Finale gravity fall within the range of core
masses predicted from earlier modeling efforts, which ranged from 0-25
$M_{\oplus}$(Fortney et al., 2016). The significantly improved gravity field
certainly better constrains the interior, however, the derived core mass in a
structure model also depends sensitively on the modeling choices or
philosophy. For instance, previous models considered a wider range of
inhomogeneous $Z$ distributions, (Helled and Guillot, 2013) or explored
different equations of state (Nettelmann et al., 2012), compared to Militzer
et al. (2019).

Unknowns of the deep interior likely make the uncertainty in the core mass
larger than a few $M_{\oplus}$. For instance, if a larger fraction of Saturn’s
interior is stably stratified as indicated by the interpretation of ring
oscillations (Section 4), perhaps a relic of planet formation (Section 6), the
deep interior could be warmer and thus be more enriched that inferred in
models that only account for super-adiabaticity in the helium rain region
(Militzer et al., 2019; Ni, 2020), where super-adiabaticity may be low
(Mankovich and Fortney, 2020). The Militzer et al. (2019) models all assume a
compact core, an assumption that is in contrast with results from formation
models. As a result, the actual core size and the heavy element mass in
Saturn’s deep interior could be different if dilution of the core material by
the hydrogen helium envelope is considered (e.g., Wahl et al., 2017; Mankovich
and Fuller, 2021). Lastly, several Earth masses of He could have rained down
to the core, adding to the best-fit heavy element core mass (Mankovich and
Fortney, 2020).

In the H/He envelope, The 1–4$\times$solar uniform enrichment of all heavy
elements in the models of Militzer et al. (2019) is lower than would be
expected from observations atmospheric methane concentrations, consistent with
$\sim$9$\times$ solar enrichment of carbon (Fletcher et al., 2009). For
Saturn, atmospheric temperature and composition have never been measured in
situ. If the envelope is hotter than expected, it could trade off for higher
concentrations of heavy elements, without significantly affecting other model
predictions. Measurements of other heavy elements include both much lower (N/H
$\sim$3$\times$ solar) and higher S/H $\sim$13$\times$ solar) concentrations,
although these differences might reflect model dependence in determining
abundances, or from measuring regions of the atmosphere that are not well
mixed (Atreya et al., 2019).

For Jupiter, most elements measured by the Galileo Entry Probe showed an
enrichment in comparison to solar by about a factor of three (see chapter by
Atreya et al.). This may imply that Jupiter’s outer envelope is enriched in
comparison to a solar composition. On the other hand, structure models of
Jupiter that are based on state-of-the-art EOSs find solutions where Jupiter’s
atmospheric metallicity is about solar (Wahl et al., 2017), significantly
lower than the Galileo probe. For the water, current estimates based on
microwave radiometer (MWR) data of the equatorial region suggest that oxygen
is 1-2 solar (Li et al., 2020). This information would further constrain
structure models and in addition would provide critical information linked to
the origin and evolution of the planet (e.g., Helled and Lunine, 2014; Vazan
et al., 2018).

For both Jupiter and Saturn, if observed atmospheric element abundances are
taken as representative of the entire H/He envelope, or even lower bound for
the envelope $Z$, there appears to be a “heavy element crisis” in both Jupiter
and Saturn, as structure models yield lower values in both planets. The
implication of this situation is still being actively debated and models with
more complex metal enrichments as a function of depth are being explored
(Debras and Chabrier, 2019). As a result, linking the measured atmospheric
with internal structure models is challenging and it is unclear to what degree
the they can be used to further constrain the interior.

Figure 4.10:  Comparison of parameters from interior models with the preferred
rotation rate of 10:33:34 h from Militzer et al. (2019), compared to the
System III rotation rate. Core radii $r_{c}=0.188$ and $r_{c}=0.231$
correspond to rocky and rock-ice compositions respectively. (a) The
distribution of heavy element mass between the core and envelope, suggesting
around 20 $M_{\oplus}$ of heavy elements in total. (b) The variation of the
mass fraction elements heavier than hydrogen with entropy in the inner,
metallic envelope. Higher internal entropies leader to warmer, less dense
hydrogen, which can accommodate more heavy elements. (c) The variation of
helium mass fraction in the molecular and metallic envelopes. The Sun’s bulk
value is 0.275. (d) The variation of heavy element mass fraction in the
molecular and metallic envelopes. The value in the Sun is about 0.015. (e) The
tradeoff between heavy element and helium mass fractions in the molecular
envelope. As $Y$ decreases, $Z$ is able to increase. (f) The tradeoff between
heavy element and helium mass fractions in the metallic envelope. In panels
(b), (c), (d), (e), and (f) solar values are shown with a yellow star,
corresponding to an assumed end-member case with no partitioning of helium
through rain-out.

Constraints from models with deep subsurface differential rotation

The Saturn models of Militzer et al. (2019) described above assume that the
observed zonal flows can be extended downward to a certain depth. Militzer et
al. (2019) varied internal heavy element abundances, which mainly influence
$J_{2}$ and $J_{4}$, and the wind speeds of the cylindrical flows until good
agreement with the observed flows and all even moments was achieved. This
approach led to the finding of $J_{2}$ and $J_{4}$ being barely affected by
the winds, while the winds would modify $J_{6}$ by a large amount (compared to
Jupiter) of 6% (Iess et al., 2019). However, these Saturn models –like the
vast majority of planet models for gaseous planets– assume a spheroidal shape
derived under the assumption of an equipotential surface although the observed
winds are inconsistent with that assumption.

Ni (2020) adopted a broader approach by also allowing for winds that differ at
depth from the surface winds. Such winds are predicted if the assumption of
spheroidal shape is relaxed (Kong et al., 2019). Moreover, subsurface winds
are found to extend to greater depth, up to 20,000 km (Kong et al., 2019), and
thus have a non-negligible influence also on $J_{2}$ and $J_{4}$ (Ni, 2020).
Nevertheless, the resulting heavy element abundances from these models are
generally in good agreement with those of Militzer et al. (2019). Notably, Ni
(2020), as previously found by Saumon and Guillot (2004) and Nettelmann et al.
(2013), finds envelopes that are homogeneous in $Z$ over a wide range of
atmospheric He abundances and rotation rates. Despite this wide parameter
range, the obtained atmospheric metallicty falls within 0–8$\times$ solar,
which is lower than the observed C/H ratio in Saturn’s atmosphere, with a core
mass estimate of 12–18 $M_{\oplus}$.

Constraints from investigations that minimize modeling assumptions

Standard internal structure models of Saturn (and gaseous planets in general)
depend on model assumptions, in particular the assumed composition and the EoS
used by the modeler. An alternative and complementary approach to model
planetary interiors is to represent the density profile by a mathematical
function or other approaches. Then one searches for a density profile that
reproduces all the observations without assuming a specific composition and
number of layers (e.g., Helled et al., 2009a; Movshovitz et al., 2020). A
recent study by (Movshovitz et al., 2020) used polynomials to represent
Saturn’s interior. They used Markov-Chain Monte Carlo in order to map all the
possible interior density profiles of Saturn. Figure 4.11 shows the density
profiles that match Saturn’s gravity field and other physical properties.
While it is clear that it is not possible to uniquely determine Saturn’s
density profile, some key conclusions can be made. It was found that the outer
half of Saturn’s radius is relatively well constrained, as the gravity field
probes the outer half the planet’s mass distribution much better than the
inner half. The density structure of the outer layers was interpreted as a
significant metal enrichment which is consistent with atmospheric
measurements.

The deep interior of Saturn is harder to constrain, but the models indicate
the existence of a “core.” Note, however, that this inferred core has rather
low densities ($\sim$ 6 g cm-3) suggesting that Saturn’s deep interior
consists of a fuzzy core and/or composition gradients, although the
statistical significance of this finding is not high. Figure 4.11 serves as a
visual reminder that core properties are not well determined by gravity-field
data alone.

Figure 4.11: The posterior probability distribution of Saturn interior density
profiles as inferred from empirical models. The density profile is represented
by polynomials. The thick black line corresponds to the the sample-median of
density on each level surface. The dashed lines mark the the 16th and 84th
percentiles and the dotted lines mark the 2nd and 98th percentiles; between
the lines percentile value is indicated by color. A model from Militzer et al.
(2019) is shown in red, which also fits the gravity data, but is outside of
the most likely region from Movshovitz et al. (2020). The inset shows the
relatively uncertainty on the density profile as a function of radius, which
is less well constrained near the surface (where there is little mass) and in
the deep interior (where gravity data does not probe well).

##  4.4 Models constrained by ring seismology data

###  4.4.1 Evidence for stable stratification

Normal mode seismology is unique in its ability to probe interior composition
gradients directly. In a convective environment, the Brunt-Väisälä frequency
$N$ characterizing the buoyant response of the fluid is imaginary,
corresponding to convective instability: small, adiabatic, vertical fluid
displacements grow exponentially. The case of $N$ identically zero corresponds
to the limit of perfectly adiabatic convection, an idealization that describes
Saturn’s interior extremely well unless composition gradients or opacity
windows (Guillot et al., 1995) are present. If in an otherwise convective
environment one introduces a stabilizing (bottom-heavy) gradient in heavy
elements or helium, then $N$ takes real values corresponding to the
_oscillatory_ response of the fluid to adiabatic vertical displacements. These
oscillatory motions are internal gravity waves, which manifest rather
dramatically in the case of ring seismology at Saturn: some initially puzzling
constraints from _Cassini_ data (Hedman and Nicholson, 2013)—visible at
$m=-2,-3$ in Fig. 4.3—are now understood as the product of the prograde Saturn
f-modes anticipated by Marley (1990, 1991) and Marley and Porco (1993)
coupling to a spectrum of g-modes hosted in the deeper interior (Fuller,
2014). Fuller proposed that a stable stratification between Saturn’s core and
envelope gives rise to g-modes whose frequencies overlap with low-degree
$(\ell=2,3)$ f-modes. The result is a series of mixed modes that could explain
the unexpectedly dense spectrum of patterns observed in the rings at low
azimuthal orders ($m=-2,-3$). French et al. (2016) bolstered this hypothesis
by detecting an additional $m=-2$ density wave in the Maxwell ringlet,
confirming a prediction from Fuller (2014).

While a major conceptual step forward in light of the early _Cassini_ ring
wave detections, Fuller’s interior model was ultimately a simplistic
polytrope-based model only roughly constrained by Saturn’s $J_{2}$. These
basic ideas would face the test of new seismic constraints from subsequent
wave detections, new gravity constraints from the _Cassini_ Grand Finale, and
more systematic study using interior models based on physical equations of
state.

Among the multitude of more recent wave detections associated with Saturn
oscillations (French et al. 2019, Hedman et al. 2019, French et al. 2021),
only a single $m=-2$ wave called W76.44 in the inner C ring appears to be
associated with g-modes and thus Saturn’s composition stratification.
Mankovich and Fuller (2021) created new models based on the Militzer and
Hubbard (2013) ab initio hydrogen EoS, featuring a single stably stratified
region connecting an rock/ice-rich inner core to a hydrogen/helium-dominated
envelope, and found W76.44 to be associated with the lowest radial order
($n=1$) $\ell=2$, $m=-2$ g-mode of Saturn. This mode resembles a quasi-
interface mode trapped on the stratified core-envelope transition.
Consequently its frequency was observed to be remarkably sensitive to the
width of Saturn’s stably stratified core-envelope transition, a fact that
could not be exploited by Fuller (2014) when this pattern had not yet been
discovered.

Figure 4.12: Interior models inferred from ring seismology and _Cassini_ Grand
Finale gravity. Profiles colored by their likelihood are for a three-parameter
interior model; the best of these is superimposed in dashed black curves. The
broader distribution of grey profiles are for a seven-parameter interior
model. The top and middle panels show the heavy element mass fraction and mass
density as a function of mean radius; the bottom panel shows the Brunt-Väisälä
frequency normalized by Saturn’s dynamical frequency $\omega_{\rm
dyn}\equiv(GM_{\rm S}/R_{\rm S}^{3})^{1/2}$. Figure taken from Mankovich and
Fuller (2021).

Figure 4.12 displays the main results of the Mankovich and Fuller (2021)
models, obtained from joint fits to the $m=-2$ ring seismology and the zonal
harmonics $J_{2}$, $J_{4}$ and $J_{6}$ from the _Cassini_ Grand Finale. With
central densities of roughly $6\ {\rm g\ cm}^{-3}$ these models are very
similar to the most likely empirical models obtained by Movshovitz et al.
(2020) (see Fig. 4.11) but are considerably more tightly constrained, a
combination of the added seismic constraints and the stronger set of
assumptions involved in EoS-based models with particular parameterizations of
the composition structure.

The main finding of this joint retrieval is that in all cases the stably
stratified region is large, encompassing about 60% of Saturn’s interior by
radius (Figure 4.12). This dilute core model is consistent with the
composition gradient case sketched in Fig. 4.4. Such a broad transition proved
necessary in order to produce the modest typical stratification $N/\omega_{\rm
dyn}\sim 2$ implied by the g-mode frequencies observed in the rings, while
still maintaining the significant core-envelope density contrast required by
the even zonal gravity harmonics.

An implication of this model that warrants some skepticism is that the highly
electrically conductive inner regions are found to be stable to overturning
convection, at odds with the simplest view that Saturn produces its magnetic
field through a “standard” dynamo in a fully convective metallic hydrogen
interior. In reality, a stable stratification over the inner 60% of the
planet’s radius does not spell doom for its magnetic field generation. First,
a sizable fraction of the volume exterior to $\sim 0.6\,R_{\rm S}$ is strongly
electrically conductive (see Cao et al.) and so field generation by a dynamo
process in a shell outside the stratified region is possible. Second, the
seismology only requires that the inner 60% by radius be stably stratified in
a radius-averaged sense: the constraining modes are g-modes with radial
wavelengths on the order of $R_{\rm S}$, and hence are not greatly sensitive
to fine spatial structure. This means that the observations could also be
consistent with a more complicated composition profile, such as a convective
staircase (Wood et al., 2013) extending to $0.6\ R_{\rm S}$, provided that the
convective layers remain small compared to the g-mode radial wavelength
(Belyaev et al., 2015). It is thus possible for an atypical dynamo process to
take place in convective layers embedded within a metallic hydrogen interior
that is nonetheless stably stratified on average. As discussed in Section 4.7
and the chapter by Cao et al., the magnetic power spectrum observed by
_Cassini_ betrays a complicated structure that may require multiple dynamo
surfaces or multiple stable regions, and so we can say with certainty that the
structure presented in Fig. 4.12 is an oversimplification.

###  4.4.2 Constraints on rotation

The remainder of the new C ring wave detections are patterns with higher
azimuthal orders $|m|\geq 4$, where the f-modes are trapped closer to the
planetary surface and hence naturally decouple from the deep g-modes (Fuller,
2014; Mankovich and Fuller, 2021). As a result these new constraints say
little about the deep interior but are useful probes of Saturn’s rotation,
which is itself a major contributor to uncertainty in the interior structure
(see Sections 4.3.3 and 4.3.4). Fig. 4.3 compares the full set of ring waves
associated with Saturn oscillations to the prograde f-mode frequencies for a
simplistic interior model (three distinct layers, each homogeneous in its
composition) constrained by $J_{2}$ and $J_{4}$. Mankovich et al. (2019)
applied a family of such models and found that the overall fit to the f-mode
frequencies hinged strongly on the rotation rate assumed for Saturn’s
interior. Rotation enters the observable ring pattern frequencies via the
Doppler shift from the Saturn frame, and to a lesser extent through intrinsic
changes to mode frequencies due to the Coriolis force and higher order
rotation effects terms modifying the response of the fluid. Putting aside the
$m=-2,-3$ waves complicated by g-modes, Mankovich et al. used the remaining 14
f-mode frequencies that were known at the time to estimate Saturn’s bulk
rotation period as $P_{\rm Sat}=\rm 10h\,33m\,38s^{+1m\,52s}_{-1m\,19s}$.
While not a tighter constraint than all previous and contemporary estimates
(e.g., Read et al., 2009; Helled et al., 2015; Militzer et al., 2019), it is
remarkable that this independent and rather direct probe of Saturn’s rotation,
possible only in the era of _Cassini_ , delivered a consistent result.

Fig. 4.3 shows that the most recent ring detections (French et al., 2021)
agree extremely well with the Mankovich et al. (2019) predictions. However,
this broad-scale comparison conceals small but significant residuals: these
ring resonances are located to a relative precision of order $10^{-5}$ in
location or pattern speed, while the modeled pattern frequencies exhibit
relative departures from these at a level of $10^{-4}$ to $10^{-3}$. The
mismatch furthermore appears to depend systematically on the angular degree
$\ell$ of the f-mode in question (French et al., 2021), indicating that the
underlying model is not fully capturing Saturn’s true density or rotation
profile. Indeed all of the models discussed so far treat only rigid rotation,
a picture now clearly ruled out by Grand Finale gravity. The finer-scale
interpretation of the ring seismology constraints for models with realistic
flow profiles is an ongoing effort.

###  4.4.3 Synthesizing gravity science and ring seismology

No structure model that has been put forward for Saturn is satisfactory in all
respects. Based on existing modeling results, the _Cassini_ gravity alone does
little to constrain the details of the deep interior, beyond requiring
$\sim$15 $M_{\oplus}$ of heavy elements be concentrated near the center of the
planet. Thus, models with a deep stably stratified layer (Mankovich and
Fuller, 2021) are likely permitted, but are in no way required by the
_Cassini_ Grand Finale gravity alone. Conversely, works prioritizing good
agreement with Grand Finale gravity (Militzer et al., 2019; Ni, 2020) so far
assume layered structures that are not supported by ring seismology
constraints.

To summarize, the Saturn mode frequencies measured by Cassini ring science
present three pillars of support for a large-scale composition gradient
supporting g-modes:

  1. 1.

The fine ($\lesssim 1\%$) frequency splitting near the $m=-2$ and $m=-3$
f-modes requires a dense spectrum of “additional” oscillation modes with
frequencies similar to the f-mode frequencies $\omega_{f}/\omega_{\rm
dyn}\sim\sqrt{\ell}$.

  2. 2.

No non-f-mode detections have been made for $\ell\geq 4$ (see Fig. 4.3),
implying that the “additional” modes only extend up to frequencies
$\omega/\omega_{\rm dyn}\sim\sqrt{4}=2$, a signature of g-modes in a model
with $N_{\rm max}/\omega_{\rm dyn}\sim 2$.

  3. 3.

The additional $m=-2$ wave W76.44 is located some 10,000 km away from the
other $m=-2$ waves, a separation in line with the expected $\ell=2$ g-mode
period spacing for Saturn models with $N_{\rm max}/\omega_{\rm dyn}$ of order
a few.

Despite the zonal gravity harmonics not requiring a dilute core in Saturn,
these seismic constraints do appear to require one.

However, the published models oriented toward joint seismology/gravity fits
have their own limitations, chief of which is their strong assumption of a
single region over which $Z$ and $Y$ both vary with prescribed functional
forms; as we have seen, the reality is certainly more complicated. The
Mankovich and Fuller (2021) models are far from unique, and it will be
essential to study more general interior structures that decouple the heavy
element gradient from the helium gradient, and allow for more than one stable
region.

Some room for improvement of these seismology-forward models is that they have
so far not accounted for Saturn’s deep winds in a self-consistent way.
Although the winds make negligible contributions to the frequencies of the
_isolated_ low-degree modes that drove the main results of Mankovich and
Fuller (2021) regarding deep structure, the winds can play a strong role in
the fine low-degree frequency splittings and certaintly in the frequencies of
the more superficially confined high-degree f-modes Dewberry et al. (2021).
Finally the perturbative treatment of Saturn’s rotation adopted in most of the
seismology modeling efforts so far limits the accuracy of predicted mode
frequencies. This approach is just beginning to be supplanted by non-
perturbative techniques (Dewberry et al., 2021) that will be essential for
Saturn seismology in the longer term.

As is clear, even with the impressive gains in gravity and seismology data,
there is no single model that matches all available constraints. Given the
non-uniqueness of gravity data, and the still developing understanding of
seismology data, one model is not an achievable goal. Constraints on the total
heavy element mass within the planet, and the core mass, depend significantly
on the modeling framework chosen, including whether the interior profile is
close to adiabatic, whether the core is compact or dilute, and more generally
how helium and heavy elements are distributed within the planet. For
reference, in Table 4.2 we show modeling compilations of planetary parameters.
Recent models show a clear preference for a central heavy element
concentration of $\sim$12-24 $M_{\oplus}$, with relatively little heavy
elements in the H/He envelope. The interested reader must, however, delve into
the details of each of the papers to understand why these differences arise,
which we have done our best to briefly summarize.

Model reference | Total heavy | Core mass | Core | Core
---|---|---|---|---
| element mass (M⊕) | (M⊕) | radius ($R_{\rm s}$) | definition
Helled and Guillot (2013) |  5 $\dots$ 35 | 5 $\dots$ 20 |  $\sim$ 0.2 |
Nettelmann et al. (2013) | $17\ldots 27$ | $0\ldots 20$ | 0–0.2 | compact
Leconte and Chabrier (2012) | 26 $\dots$ 50 | 10 $\dots$ 21 | 0.2 $\dots$ 0.3 |
Vazan et al. (2016) (model $S_{3}$) |  $\sim$ 30 | 12 | 0.23 | dilute ($Z>0.5$)
Fig. 4.10 adopted from Militzer et al. (2019) | 18 $\dots$ 22 | 15 $\dots$ 18 | 0.19 $\dots$ 0.23 | compact
Movshovitz et al. (2020) | $24\ldots 32$ | – | – |
Ni (2020) | $16.5\ldots 19.5$ | $12\ldots 18$ | – | compact $+$ dilute
Mankovich and Fuller (2021) | $23\ldots 26$ | $20\ldots 24$ | $0.33\ldots 0.36$ | dilute ($Z>0.5$)
Table 4.2: Inferred total heavy-element mass, core masses, and core radii from
a variety of Saturn interior models. See the referenced papers for necessary
modeling details.

##  4.5 Clues from satellite migration

Measurements of Saturn’s tidal response have led to unique, if less readily
interpreted, constraints on interior structure. Using a combination of
astrometry and precision tracking of Titan’s orbit using _Cassini_ range-rate
data, Lainey et al. (2020) reported a rapid orbital expansion indicative of
very strong tidal dissipation inside Saturn. The radiometric data enabled a
precise determination of the imaginary part of Saturn’s Love number ${\rm
Im}(k_{2})=-(69\pm 57)\times 10^{-4}$ at Titan’s tidal frequency. The implied
quality factor $Q\sim 100$, some two orders of magnitude smaller than the
value expected from equilibrium tidal theory, lends overwhelming support to
the theory that Titan’s orbital evolution is dictated by resonance locking
with low-frequency oscillations inside Saturn (Fuller et al., 2016; Lainey et
al., 2017). For Enceladus, Tethys, Dione, and Rhea only weaker astrometric
constraints are available, but still their corresponding estimates of $Q$ and
orbit evolution timescales similarly favor the resonance locking hypothesis.
In this picture the moons’ orbits expand on Saturn’s evolution timescale,
tracking the spectrum of Saturn oscillation frequencies that in turn change
with Saturn’s evolving structure.

These planetary oscillations could take the form of inertial waves intrinsic
to any Saturn model (including conventional layered models; see Ogilvie and
Lin, 2004; Wu, 2005) or, if Saturn does host a stable stratification, mixed
inertial/gravity waves (Fuller et al., 2016). The basic mechanism thus does
not strictly require static stability $N>0$, but having one generally
increases the likelihood of resonant capture, since in that case mixed
inertial/gravity waves would offer up a denser and broader spectrum of
resonances capable of ensnaring satellite orbits (Fuller et al., 2016; André
et al., 2019).

##  4.6 Results from Thermal Evolution Models

Models of the thermal evolution of giant planets aim to match a few particular
quantities, typically the planet’s $T_{\rm eff}$ (or $T_{\rm int}$) and radius
at the planet’s known mass and age. The planet’s current energy balance is an
important aspect. The total thermal luminosity yields $T_{\rm eff}$, but since
$T_{\mathrm{eff}}^{4}=T_{\mathrm{eq}}^{4}+T_{\mathrm{int}}^{4}$, and since
$T_{\rm eq}$ depends on the planet’s measured Bond albedo, the _intrinsic_
component of thermal flux (rather than re-radiated solar flux) is the
important quantity.

A thermal evolution model aims to understand the secular cooling of the planet
either from some initial formation model, or, more commonly, for an arbitrary
hot initial state, not tied to any particular formation mechanism. The
radiative atmosphere serves as the bottleneck for cooling of the interior
(Hubbard et al., 1999; Fortney et al., 2011). Potential problems with
arbitrary initial conditions are discussed in Marley et al. (2007) in the
context of formation energy lost in the formation process. Other processes
linked to the planetary formation that can affect the thermal evolution
include the physics of the accretion shock (e.g., Cumming et al., 2018) and
the existence of composition gradients (e.g., Vazan et al., 2016).

###  4.6.1 Models with H/He phase separation

Mankovich and Fortney (2020) coupled the thermal evolution of both Saturn and
Jupiter to a variant of the _ab initio_ He/H phase diagram of Schöttler and
Redmer (2018). Since Jupiter and Saturn must obey the same physics, the
evolution of both planets were treated as a coupled problem for the first
time. The phase diagram of Schöttler and Redmer (2018) was allowed to “float”
in temperature-pressure space, in terms of onset temperature for phase
separation, and models needed to match $T_{\rm eff}$ and radius for each
planet and Jupiter’s accurately measured $Y_{atmos}$. Given the complexity of
the phase space a Monte Carlo approach was employed, marginalizing over
temperature shifts of the phase diagram, Saturn’s uncertain Bond albedo, and
any effects due to superadiabatic temperature gradients in regions with a He
composition gradient (Stevenson and Salpeter, 1977; Fortney and Hubbard, 2003;
Nettelmann et al., 2015; Mankovich et al., 2016).

A narrow range of fully consistent solutions was indeed found. These solutions
require a modest temperature shift of 500 K hotter He phase separation onset,
little to no superadiabacity in either Jupiter or Saturn, and an upward
revision Saturn’s Bond albedo to be $\sim$0.5, in line with a published upward
revision to Jupiter’s value from _Cassini_ fly-by data (Li et al., 2018). Of
particular note in these models is that He phase separation proceeds
differently in Saturn than in Jupiter (see Fig. 4.13). In Jupiter, the He
phase separation region takes up very little mass within the planet, and the
vast reservoir of hot liquid metallic hydrogen allows He-rich droplets to re-
dissolve, slightly enriching the metallic region in He overall.

However, in Saturn He phase separation begins far earlier in the planet’s
evolution, and the separation region affects a far larger region of the planet
in mass-shell-space. Essentially all of the liquid metallic hydrogen region
reaches He immiscibility due to Saturn’s much lower pressure at the bottom of
the H/He envelope compared to Jupiter. This quickly leads to He-rich droplets
“pooling up” atop the core, which in the Mankovich and Fortney (2020) models
was treated as a distinct water/rock core. These models suggest that He phase
separation is not merely a perturbation on the planet’s overall structure (as
in Jupiter) but radically alters the mass distribution and cooling history of
Saturn. It is therefore clear that in order to better understand the evolution
and internal structure of Saturn improved knowledge of the H/He phase diagram
is required, and most importantly, a measurement of the He abundance in
Saturn’s atmosphere.

Figure 4.13: Profiles of the helium mass fraction within Saturn (left) and
Jupiter (right) as a function of mass shell and age in evolutionary models
(Mankovich and Fortney, 2020) coupled to a modified version of the Schöttler
and Redmer (2018) H/He phase diagram designed to reproduce the Galileo $Y_{\rm
atm}$ measurement in Jupiter (von Zahn et al. 1998; blue line). The top x-axis
is in Earth masses, $M_{\rm E}$. Note the very different scales for the two
planets.

###  4.6.2 Uncertainty in the initial conditions and relation to formation

Another type of Saturn evolution model includes composition gradients in the
deep interior (e.g., Leconte and Chabrier, 2013b; Vazan et al., 2016). Such
interiors are non-adiabatic which could be a result of a primordial
composition gradient due to the formation process itself (Lozovsky et al.,
2017; Helled and Stevenson, 2017; Valletta and Helled, 2020) and/or because of
the erosion of a central core and immiscibility effects (e.g., of helium in
metallic hydrogen). Composition gradients can inhibit convection and influence
the heat transport inside Saturn. If the composition gradient is steep and the
luminosity is not very large, overturning convection is suppressed and mixing
and homogenization are prevented. In that case the heat transport would be
dominated by conduction/radiation or a less efficient type of convection
(layered/fingering convection).

If giant planets indeed form with primordial composition gradients, their
long-term evolution and current-state internal structure are affected. Using a
planet evolution code that accounts for the effect of the heavy-elements on
the planet self-consistently, also including helium rain (see Vazan et al.
2016 for details) it was found that the innermost regions of Saturn are stable
against large-scale convection. As a result, the heat cannot be transported
efficiently in these regions, while the outer envelope is convective
throughout the entire evolution. The increasing temperature gradient between
the innermost non-adiabatic and non-convective region and the outer convective
region leads to a small convective penetration inward during the long-term
evolution, leading to a moderate heavy-element enrichment of the outer regions
as time progresses. An example for such non-adiabatic models is shown in
Figure 4.14, which shows an example of a Saturn model with primordial
composition gradients and its inferred current-state internal structure. Of
note in this model that metal enrichment in the outer half of the H/He
envelope is not representative of the entire envelope.

Figure 4.14: An example of a not fully-adiabatic model of Saturn. Left: The
primordial (dashed-black) and current-state (solid-red) heavy-element
distribution in Saturn. Right: The density (blue) and temperature (red)
profiles for the current-state internal structure. (adapted from Vazan et al.
(2016).)

To summarize, Saturn’s interior might be well represented by distinct layers,
but instead, is expected to have more complex structure which includes
composition gradients. These two different possible structures are sketched in
Fig. 4.4.

##  4.7 Current understanding of structure in light of magnetic field data

In principle, Saturn’s magnetic field provides information about the internal
structure that is not accessible by any other means at present because the
dynamo process responsible for the field implies aspects of the convection and
layering that are not directly obtainable from gravity or even seismology.
Importantly, the magnetic field provides insight to the dynamics (i.e.
velocity field) of the conducting interior. This section is a brief summary;
please see the extensive review by Cao et al. elsewhere in the volume.

Many of the basic questions about the field are lacking a good explanation at
present, including the following:

  1. 1.

Why is the field so small? The dipole field is twenty times smaller than
Jupiter’s.

  2. 2.

Why is the field (and its higher harmonics) so spin-axisymmetric?

  3. 3.

Why is the field spectrum so different from other planets?

  4. 4.

How is the existence of this field reconciled with claimed structures inferred
from gravity and ring seismology?

There are a number of complications that make addressing these questions
difficult. In the “classical” view of the Saturnian field there is a layer,
possibly quite thick, of metallic hydrogen. The expected convective motions
can easily provide the magnetic Reynolds number $\sim$100 needed for field
regeneration (Stevenson, 2003). In this simple view, Saturn should not differ
that much from Earth or the “classical” view of Jupiter. Importantly, there is
no static stability in this picture since vertical motions of substantial
radial extent are needed to sustain a dynamo (Busse, 1975).

Neither Jupiter nor Saturn are this simple, and the differences between the
two planets might lie in these complications. First, there is adequate
electrical conductivity in dense, hot molecular hydrogen for a dynamo effect
or at least for magnetohydrodynamic (MHD) effects. This is an old idea
(Smoluchowski, 1975), and is supported by the Juno results (Bolton et al.,
2017). It permits dynamo generation out to larger radii than previously
supposed, perhaps relevant to explaining the power observed in higher
harmonics and perhaps permitting dynamo action in more than one layer rather
than in a single thick uniform convecting shell.

Furthermore, helium differentiation promotes the presence of a layer that
inhibits large vertical motions. Jupiter has strong evidence for helium rain
from Galileo probe results, while Saturn has possibly contradictory evidence
even though Saturn should exhibit the larger effect because of its presumed
lower internal entropy. Saturn’s heat flow permits but does not demand a
helium rain-out explanation. Still, a layer of helium rain-out is expected in
Saturn and is likely more substantial than in Jupiter, favoring a more complex
dynamo activity, with field generation both above and below a layer that may
inhibit large scale vertical motions.

Cassini gravity (like Juno gravity) clearly detected deep-seated differential
rotation. The visible winds on Saturn extend to a much deeper level (but
perhaps to a similar pressure and temperature) as those on Jupiter. It is not
known whether this leads to a different MHD or dynamo effect for the two
planets.

Lastly, some aspect of their formation has led to compositional gradients in
the heavy elements with consequences for convective structure. The inference
from ring seismology (Mankovich and Fuller, 2021) and satellite evolution
(Lainey et al., 2020) of static stability (g-modes) and a dilute core provides
additional support for a more complicated layered structure or layered dynamo
within Saturn. Since molecular hydrogen may support dynamo action for
temperature exceeding a few thousand degrees, especially if differential
rotation is substantial and aids field generation, there may be no difficulty
in creating compositional gradients out to 0.6 of Saturn’s radius, as
suggested by Mankovich and Fuller (2021).

Returning then to the questions posed:

  1. 1.

The difference in field strength between Jupiter and Saturn is not yet
explained but hints at the failure of simple scaling laws, perhaps because
structure matters and it is not sufficient to characterize a planet by mass
density, rotation, heat flow, conductivity and size(Stevenson, 2003).

  2. 2.

The spin-axisymmetry is certainly promoted by the presence of differential
rotation (Stevenson, 1980) but it’s not yet clear whether this is a sufficient
explanation; it could be that the dynamo operates differently under the
conditions needed to filter the non-symmetric terms. Moreover, there is no
fundamental difficulty with a dynamo that is externally symmetric and the non-
axisymmetry could be present in regions or at length scales that are not
currently observable. Either the helium rain region or the overlying
differential rotation in molecular hydrogen may be involved.

  3. 3.

The non-power-law character to the power spectrum might suggest a layered
dynamo with no single “radius of dynamo generation.”

  4. 4.

More complicated dynamo models are suggested for both Jupiter and Saturn,
meaning that a single shell picture could be insufficient.

##  4.8 Conclusions and Future Work

Often, Saturn is viewed simply as a smaller version of Jupiter. While both
planets have many similarities, they also have fundamental differences and
each planet is clearly unique. The _Cassini_ Grand Finale orbits provided an
unprecedented data set on Saturn’s gravitational field that has yielded strong
new constraints on the planet’s interior structure. Studies of ring features
linked to normal mode oscillations in the planet’s interior are still yielding
new results that will continue to provide novel, complementary constraints on
the interior structure.

At some point, no more new ring features will be found, and our “complete”
_Cassini_ data set will be the precise gravity field and interior normal mode
constraints. At that point no new additional data may be forthcoming for quite
some time. New advances on those longer time horizons will likely have to come
from theory. The EOS of hydrogen, and helium, and their mixtures, from ab
initio calculations, vetted by high-pressure experiments, will progress.
Experimental constraints on the H/He phase diagram can give us greater
confidence in the effects of phase separation on the planet.

Measurements of the elemental abundances in giant planet atmospheres is
critical to further constraining their formation, evolution, and internal
structure. While it is still unclear whether such a measurements represent the
planetary bulk, the enrichment of specific elements and the measurement of the
noble gases can be used to put limits on enrichment mechanisms tied to
planetary origin (e.g., Mousis et al., 2016, and references therein). As a
result, it is desirable to send a entry probe into Saturn’s atmosphere,
comparable to Jupiter’s Galileo probe, and measure the abundance of noble
gases (in particular helium and neon) and other key elements and isotopes.
Perhaps the most important measurement for structure models would be the
helium abundance, as a definitive, precise measurements in Saturn, to go along
with Jupiter, strongly constrains the role of He phase separation in the
current structure of Saturn and the role of He sedimentation in the cooling
history of the planet (Mankovich and Fortney, 2020).

A complete picture of the planet today, that melds seismology and gravity
field data, including differential rotation, and the role of He phase
separation, is still a work in progress. It is significant that seismology has
had such an important role is understanding the deepest regions of the planet,
where the gravity field tells us less than we would hope. Clearly, the
_Cassini_ revolution for Saturn’s interior is still yet unfinished, but we
should be energized by the significant advances that _Cassini_ has delivered.

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