Journal ArticleDOI

# Saturn’s Probable Interior: An Exploration of Saturn’s Potential Interior Density Structures

11 Mar 2020-The Astrophysical Journal (IOP Publishing)-Vol. 891, Iss: 2, pp 109

AbstractThe gravity field of a giant planet is typically our best window into its interior structure and composition. Through comparison of a model planet's calculated gravitational potential with the observed potential, inferences can be made about interior quantities, including possible composition and the existence of a core. Necessarily, a host of assumptions go into such calculations, making every inference about a giant planet's structure strongly model dependent. In this work, we present a more general picture by setting Saturn's gravity field, as measured during the Cassini Grand Finale, as a likelihood function driving a Markov Chain Monte Carlo exploration of the possible interior density profiles. The result is a posterior distribution of the interior structure that is not tied to assumed composition, thermal state, or material equations of state. Constraints on interior structure derived in this Bayesian framework are necessarily less informative, but are also less biased and more general. These empirical and probabilistic constraints on the density structure are our main data product, which we archive for continued analysis. We find that the outer half of Saturn's radius is relatively well constrained, and we interpret our findings as suggesting a significant metal enrichment, in line with atmospheric abundances from remote sensing. As expected, the inner half of Saturn's radius is less well constrained by gravity, but we generally find solutions that include a significant density enhancement, which can be interpreted as a core, although this core is often lower in density and larger in radial extent than typically found by standard models. This is consistent with a dilute core and/or composition gradients.

Topics: Giant planet (54%), Saturn (54%), Gravitational potential (53%), Gravitational field (52%)

### 1.1. The Gravity Field as a Probe on the Interior

• There are a number of fundamental questions that the authors would like to understand about giant planets.
• Because the planets are fluid and rapidly rotating, they assume an oblate shape and their gravitational potential differs from that of a spherically symmetric body of the same mass.
• Initially, researchers had focused on finding a single, best-fit solution subject to a host of assumptions, chosen for computational convenience and not necessarily following reality.
• The main contribution of the present work is the introduction of a different approach to the task of inferring interior structure from gravity and the application of this approach to Saturn.

### 1.2.1. The Planets Have Three Layers

• Perhaps the most constraining assumption is the prototypical picture of three layers, each well mixed enough to be considered homogeneous.
• Investigators also adjust the abundance of heavy elements in the He-rich and He-poor layers, with little physical motivation other than it seems to facilitate finding an acceptable match to the gravity field.
• While a core–envelope structure is certainly a plausible one, and indeed rooted in well-studied planet formation theories, the assumption of compositionally homogeneous layers may well be a significantly limiting oversimplification.
• 2.2. The Interior Pressure–Temperature Profile Is Isentropic 2017; Militzer et al. 2019), but this interpolation is unlikely to capture fully the effects of composition gradients.

### 1.2.3. The Inferred Composition Relies on Equation of State Calculations

• As the field progresses, the equations of state (EOSs) used for modeling giant planet interiors become a better representation of reality.
• Nevertheless, the EOSs for all relevant materials and mixtures are not perfectly known.
• Simulations from first principles of hydrogen, helium, and their mixtures over the conditions relevant for giant planets have been carried out and partially validated against experimental data (Nettelmann et al. 2008; Militzer & Hubbard 2013).
• Today, there is good agreement between state-of-theart EOSs (Nettelmann 2017), but it should be kept in mind that DFT also suffers from approximations (Mazzola et al. 2018), and there remains an uncertainty of~2% in the hydrogen EOS, which increases significantly when it comes to predicting hydrogen–helium demixing (Morales et al. 2009).

### 1.2.4. Appreciating the Complexities

• The drawbacks of the assumptions discussed above have long been known and the reality that giant planets are surely more complicated than the traditional modeling framework allows for is generally accepted (e.g., Stevenson 1985).
• More recent investigations are attempting to allow for a more complex structure.
• Interior composition gradients due to remnants of formation (Leconte & Chabrier 2012; Helled & Stevenson 2017), core dredge-up (Militzer et al. 2016), convective mixing of primordial composition gradients (Vazan et al. 2016, 2018), and He sedimentation (Nettelmann et al. 2015; Mankovich et al. 2016) have been considered and were found to lead to different structures.
• Additionally, some investigators have begun using what may be referred to as “empirical” models.
• The authors explore systematically, in a Bayesian inference framework, the possible density profiles of Saturn.

### 2. Composition-independent Interior Density Calculation

• The premise of removing some assumptions and deriving composition-free interior density profiles (sometimes referred to as empirical models) is simple and in fact has been pursued in previous works (e.g., Marley et al.
• (We discuss similarities and differences with these works in Section 2.6 below.)the authors.the authors.
• All other properties of the planet will be inferences, rather than input parameters.
• Because there is unavoidable uncertainty associated with the measurement of the gravity field (and also with its theoretical calculation from interior models; see Section 2.4), this means that there must be a continuous distribution of possible density profiles that fit the gravity solution, and the authors must base their inferences on the entire distribution.

### 2.1. Overview

• This parameterization should be able to represent all the physically reasonable ( )r s curves without undue loss of generality, but this is not particularly difficult.
• To drive the sampling algorithm, the authors need a way to evaluate the relative likelihood of two model planets, and they do this by comparing how well they match Saturn’s observed mass and gravity field.
• The details of this calculation are given in Section 2.3.
• The authors need to first determine the planet’s equilibrium shape.
• There is no generally agreed-upon method of predicting the number of sampling steps required for convergence.

### 2.2. Parameterization of ( )r s

• These competing requirements are not easy to satisfy, and it may be that the best parameterization depends on the planet being studied as well as on the available sampling algorithms and computing resources.
• Figure 2 shows the density profiles of several Saturn models recently published by Mankovich et al. (2019, hereafter M19).
• This piecewise-continuous model should not be confused with the traditional three-layer one.
• Nevertheless, this demonstrates that, while the authors wish to be guided by physical models, their parameterization must not be overly constrained by them.
• The authors thus have 11 free parameters—three each for the three quadratic segments, plus the two “floating” transition radii.

### 2.3. Comparing Model and Observation

• MCMC sampling works by comparing, at every iteration, the likelihood of a proposed vector of parameter values, ( )yL , with that of the current vector of parameter values, ( )xL , and accepting or rejecting the proposed values with probability proportional to the relative likelihoods.
• The planetary properties that their models need to match are the gravity coefficients J and the planet’s mass MSat.
• In the work presented here, only J2, J4, and J6 were considered for the purpose of calculating the likelihood function.
• These phenomena are important in themselves and offer a promising avenue for studying further Saturn’s dynamic nature, but for the purpose of constraining the bulk interior structure, their net result is to increase the effective uncertainty of the loworder Js ascribed to solid-body rotation (Guillot et al. 2018).
• Unfortunately, for a sampling problem of this scope, this speedup is not enough to mitigate the speed disadvantage of CMS compared with ToF.

### 2.5. MCMC Sampling of Parameters

• There is a wide variety of MCMC sampling algorithms; all fundamentally seek to sample the posterior distribution by a sequence of random steps through parameter space.
• Luckily, this approach can benefit from parallelization with minimal overhead and is thus perfectly suitable to run on a large supercomputer.
• Simply put, the authors must decide when it is safe to stop the sampling run and use the obtained draws to calculate anything of interest, trusting that the sample distribution is similar enough to the underlying posterior.
• Recall that of the 11 parameters needed to define a ( )r s curve (Equation (3)), 2 are the normalized radii locating the points of possible density discontinuity; their values have a straightforward, physical meaning.
• Fixing values for the transition radii, one can sample the conditional joint posterior of the nine geometric parameters efficiently.the authors.

### 2.6. Relation to Previous Works

• While in previous sections the authors discussed the drawback of “standard” approaches, here it is worth discussing how their work compares to previously published alternative approaches.
• Saturn’s Density Profile and Inferred Properties.
• After obtaining an independent random sample from the posterior in parameter space, the authors examine the resulting distribution of density profiles.
• With this in mind perhaps the most precise statement to make is that at least half the density profiles in their sample show a discontinuity pronounced enough to be consistent with a heavy-element core transition.
• Perhaps the most useful aspect of the empirical-model approach is the possibility of finding unexpected solutions that can never arise where explicit composition modeling is used.

### 3.1. Inferences on Possible Composition

• Combining the two profiles to eliminate the radius variable results in a unique pressure–density relation, often called a barotrope.
• A possible approach is to compare the empirical barotropes obtained above to some reference barotrope and examine the “residual” density for possible constraints on composition.
• If a lower Y reference adiabat were chosen in the outer layers, larger density excess would be needed.
• But, again, the median is not the distribution.

### 4. Discussion and Conclusions

• The authors presented an empirical approach to using gravity data to explore the interior structures of fluid planets and applied it to Saturn using data from Cassiniʼs Grand Finale orbits.
• First, a point that was already made above but bears repeating: gravity data alone offer robust but loose constraints.
• As a result, the main finding the authors can report on, with respect to Saturn, is to confirm the well-known but often underappreciated suspicion that solutions to Saturn’s gravitational potential field exist that do not conform to a simple model of a few compositionally homogeneous and thermally adiabatic layers.
• The authors can contrast this with the seemingly more informative but less robust outcomes from traditional models.
• The trade-off for these precise, straightforward estimates is their unknown validity, being tied to very particular and often very simple a priori modeling framework for the planet.

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Recent Work
Title
Saturn’sProbable Interior: An Exploration of Saturn’s Potential Interior Density Structures
https://escholarship.org/uc/item/0vn7g57f
Journal
The Astrophysical Journal, 891(2)
ISSN
1538-4357
Authors
Movshovitz, Naor
Fortney, Jonathan J
Mankovich, Chris
et al.
Publication Date
2020-03-11
DOI
10.3847/1538-4357/ab71ff
Peer reviewed
University of California

Saturns Probable Interior: An Exploration of Saturns Potential Interior Density
Structures
Naor Movshovitz
1
, Jonathan J. Fortney
1
, Chris Mankovich
1,4
, Daniel Thorngren
2,5
, and Ravit Helled
3
1
Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA, USA; nmovshov@ucsc.edu
2
Department of Physics, University of California, Santa Cruz, CA, USA
3
Institute for Computational Science, Center for Theoretical Astrophysics & Cosmology, University of Zurich, Switzerland
4
Division of Geological and Planetary Sciences, Caltech, Pasadena, CA, US
5
Institute for research on exoplanets, Universite de Montreal, Montreal, Canada
Received 2019 July 23; revised 2020 January 25; accepted 2020 January 29; published 2020 March 11
Abstract
The gravity eld of a giant planet is typically our best window into its interior structure and composition. Through
comparison of a model planets calculated gravitational potential with the observed potential, inferences can be
made about interior quantities, including possible composition and the existence of a core. Necessarily, a host of
assumptions go into such calculations, making every inference about a giant planets structure strongly model
dependent. In this work, we present a more general picture by setting Saturn s gravity eld, as measured during the
Cassini Grand Finale, as a likelihood function driving a Markov Chain Monte Carlo exploration of the possible
interior density proles. The result is a posterior distribution of the interior structure that is not tied to assumed
composition, thermal state, or material equations of state. Constraints on interior structure derived in this Bayesian
framework are necessarily less informative, but are also less biased and more general. These empirical and
probabilistic constraints on the density structure are our main data product, which we archive for continued
analysis. We nd that the outer half of Saturns radius is relatively well constrained, and we interpret our ndings
as suggesting a signicant metal enrichment, in line with atmospheric abundances from remote sensing. As
expected, the inner half of Saturn s radius is less well constrained by gravity, but we generally nd solutions that
include a signicant density enhancement, which can be interpreted as a core, although this core is often lower in
density and larger in radial extent than typically found by standard models. This is consistent with a dilute core
Unied Astronomy Thesaurus concepts: Saturn (1426); Planetary interior (1248); Planetary structure (1256); Planet
formation (1241)
1. Introduction
1.1. The Gravity Field as a Probe on the Interior
There are a number of fundamental questions that we would
like to understand about giant planets. Do they have a heavy-
element core? If so, what is its mass? Is it distinct from
the overlying H/He envelope, or partially mixed into it? Is the
H/He envelope enriched in heavy elements compared to the
Sun? Is the envelope fully convective and well mixed?
Unfortunately, the vast mass of a giant planet is completely
hidden from view, so that we must use indirect methods to try
interiors of giant planets comes from interpreting their gravity
elds, as recently reviewed for Saturn by Fortney et al. (2018).
Because the planets are uid and rapidly rotating, they assume
an oblate shape and their gravitational potential differs from
that of a spherically symmetric body of the same mass. The
external gravitational potential
V
e
is a function of the colatitude
θ and distance r from the center of the planet, and is typically
written as an expansion in powers of
R
r
,
eq
where
R
e
q
is the
() ( ) ( ()
å
qq=- -
=
¥
Vr
GM
r
RrJP,1 cos.1
n
n
nne
1
eq
In Equation (1), P
n
are Legendre polynomials of degree n. The
coefcients J
n
(the Js) are measurable for many solar system
bodies by tting a multiparameter orbit model to Doppler
residuals of spacecraft on close approach. For uid planets in
hydrostatic equilibrium, where azimuthal and northsouth
symmetry hold, only even-degree coefcients are nonzero.
When an interior model for a planet is created, the J
n
values
are calculated as integrals of the interior density over the
planetary volume:
()() ( ) ()
ò
rqt=- ¢ ¢ ¢ ¢rMR J r P dcos . 2
n
n
n
n
eq
These model J
n
can then be compared to measured ones. As is
well known, the different Js sample the density at different
depths (with J
2
probing deepest) but with signicant overlap,
and with most of the weighting over the planets outer half in
radius. This point is illustrated in Figure 1.
The gravity eld is a nonunique feature of the interior mass
distribution. In other words, different mass distributions can
lead to identical gravity signals. This complicates the process of
making inferences about the interior structure based only on the
external gravity eld. In principle, one should explore a wide
range of possible interior structures, of possible
()
r
¢r
in
Equation (2), to see the full range of solutions that t the
gravity eld. Initially, researchers had focused on nding a
single, best-t solution subject to a host of assumptions, chosen
for computational convenience and not necessarily following
reality. More recently, there have been efforts to explore an
expanded range of interior structures, usually by making
alternative assumptions about the prototypical planet.
The Astrophysical Journal, 891:109 (19pp), 2020 March 10 https://doi.org/10.3847/1538-4357/ab71ff
1

The main contribution of the present work is the introduction
of a different approach to the task of inferring interior structure
from gravity and the application of this approach to Saturn. The
result is a suite of interior structure models of Saturn computed
with fewer assumptions and therefore showing a fuller range of
structures consistent with observation. We describe our method
in detail and compare it with previous work of similar spirit in
Sections 1.2.4 and 2.6.
1.2. Common Assumptions in Planetary Interior Models
There are typically at least three signicant assumptions or
choices that modelers make when constructing interior models
of giant planets, thereby implicitly constraining the possible
inferences from these models.
1.2.1. The Planets Have Three Layers
Perhaps the most constraining assumption is the prototypical
picture of three layers, each well mixed enough to be considered
homogeneous. For Jupiter and Saturn, these are a helium-poor
outer envelope, a helium-rich inner envelope, and a heavy-
element, usually constant-density core. Investigators also adjust
the abundance of heavy elements in the He-rich and He-poor
layers, with little physical motivation other than it seems to
facilitate nding an acceptable match to the gravity eld.
While a coreenvelope structure is certainly a plausible one,
and indeed rooted in well-studied planet formation theories, the
assumption of compositionally homogeneous layers may well
be a signicantly limiting oversimplication.
1.2.2. The Interior PressureTemperature Prole Is Isentropic
A typical assumption of interior modeling is that the
pressuretemperature prole is isentropic, lying on a single
(P, T) adiabat that is continued from a measured or inferred
temperature at 1bar. This second assumption is likely to be
true over some of the interior, but there are good reasons to
doubt that this holds throughout the interior.
Jupiter and Saturn have an atmospheric He depletion
compared to the Sun, and it has long been suggested that this
is due to He phase separation from liquid metallic hydrogen in
the deep interiors (Stevenson & Salpeter 1977; Fortney &
Hubbard 2003). There is likely a region with a He abundance
gradient starting between 1 and 2Mbar in both planets
(Nettelmann et al. 2015; Mankovich et al. 2016).Inmodels
that attempt to interpret the gravity eld, if such a layer is
included at all, it is by interpolation between the outer and inner
homogeneous layers (e.g., Wahl et al. 2017; Militzer et al. 2019),
but this interpolation is unlikely to capture fully the effects of
scale convection (Ledoux 1947), implying that heat is
transported via layered convection or radiation/conduction. This
leads to higher internal temperatures that in return allow higher
heavy-element enrichment at a given densitypressure. Indeed,
nonadiabatic structures have been recently suggested for all outer
planets in the solar system (e.g., Leconte & Chabrier 2012;
Vazan et al. 2016; Podolak et al. 2019).
1.2.3. The Inferred Composition Relies on Equation of State
Calculations
As the eld progresses, the equations of state (EOSs) used
for modeling giant planet interiors become a better representa-
tion of reality. Nevertheless, the EOSs for all relevant materials
and mixtures are not perfectly known. Simulations from rst
principles of hydrogen, helium, and their mixtures over the
conditions relevant for giant planets have been carried out and
partially validated against experimental data (Nettelmann et al.
2008; Militzer & Hubbard 2013). These EOSs for hydrogen
show good agreement with data up to
~1.5 Mbar
(e.g., Militzer
et al. 2016). However, the pressure at the bottom of Saturns
40Mbar, well beyond the realm of experiment. Recent
structure models used EOSs for hydrogen and helium based
on density functional theory (DFT) simulations (Nettelmann
et al. 2008; Miguel et al. 2016; Militzer et al. 2016). Until
recently, different EOSs led to different inferred compositions
for Jupiter due to different approaches to calculating the
entropy. Today, there is good agreement between state-of-the-
art EOSs (Nettelmann 2017), but it should be kept in mind that
DFT also suffers from approximations (Mazzola et al. 2018),
and there remains an uncertainty of
~2%
in the hydrogen EOS,
which increases signicantly when it comes to predicting
hydrogenhelium demixing (Morales et al. 2009).
The heavy elements must also be represented by an EOS
(typically for water or silicates), which introduces another
source of uncertainty. Therefore, the range of possible
compositions and internal structures from such interior models
Figure 1. Contribution functions of the gravitational harmonics J
2
(blue solid), J
4
(red dashed), and J
6
(yellow dotted) for a typical, three-layer Saturn model. The
contribution density (
()µJrdr
) is plotted in the left panel and the cumulative contribution in the right panel. The horizontal line intersects the curves at a depth
where the corresponding J reaches 90% of its nal value.
2
The Astrophysical Journal, 891:109 (19pp), 2020 March 10 Movshovitz et al.

cannot be taken to be the true range of allowed values, even if
the parameter space of possible EOSs, H/He/Z mixing ratios,
and outer/inner envelope transition pressures were thoroughly
explored.
1.2.4. Appreciating the Complexities
The drawbacks of the assumptions discussed above have
long been known and the reality that giant planets are surely
more complicated than the traditional modeling framework
allows for is generally accepted (e.g., Stevenson 1985). More
recent investigations are attempting to allow for a more
complex structure. Interior composition gradients due to
remnants of formation ( Leconte & Chabrier 2012; Helled &
Stevenson 2017), core dredge-up (Militzer et al. 2016),
convective mixing of primordial composition gradients (Vazan
et al. 2016, 2018), and He sedimentation (Nettelmann et al.
2015; Mankovich et al. 2016) have been considered and were
investigators have begun using what may be referred to as
empirical models. In this context, an empirical model is one
that is focused on the more direct connection between gravity
and density (e.g., Helled et al. 2009) or gravity and equilibrium
shape (Helled et al. 2015), without invoking the compositional
and thermodynamical origin of these structures.
The work we present here is in the spirit of empirical models.
We explore systematically, in a Bayesian inference framework,
the possible density proles of Saturn. We limit our assumptions
as much as possible, in order to nd the widest range of interior
structures with their probability distribution based on their
gravitational potential matching the observed eld.
2. Composition-independent Interior Density Calculation
The premise of removing some assumptions and deriving
composition-free interior density proles (sometimes referred
to as empirical models) is simple and in fact has been pursued
in previous works (e.g., Marley et al. 1995; Podolak et al. 2000;
Helled et al. 2009). (We discuss similarities and differences
with these works in Section 2.6 below.) The only information
that is needed to calculate a gravity eld is the density
everywhere inside the planet,
()
r
r
, and so this is the only
quantity we will directly vary. In fact, hydrostatic equilibrium
produces level surfacesclosed surfaces of constant density,
pressure, and potentialand therefore a one-dimensional
description of the mass distribution is sufcient: we can use
() ()
r
r=r s
, where s is the volumetric mean radius of the
unique level surface of density ρ.
All other properties of the planet will be inferences, rather
than input parameters. Because there is unavoidable uncertainty
associated with the measurement of the gravity eld (and also
with its theoretical calculation from interior models; see
Section 2.4), this means that there must be a continuous
distribution of possible density proles that t the gravity
solution, and we must base our inferences on the entire
distribution. In practice, because we can only ever consider a
nite number of solutions, this means that we must base our
inferences on a random sample from this unknown distribution
of allowed solutions.
In this section, we describe the process of obtaining this
random sample, as applied to Saturn. For the most part, the
same process would apply equally well to the other giant
planets. We mention in places modications that may be
needed if the same method is to be applied to Jupiter, Uranus,
or Neptune.
2.1. Overview
Formally, the distribution we are after is the posterior
probability
(∣ )r
Jp
, the probability that the planets interior
density follows
()
r
r= s
given that the gravity coefcients
were measured as
J
. This consists of several subtasks. First,
we must nd a suitable parameterization of
(
)r
s
. This
parameterization should be able to represent all the physically
reasonable
(
)r
s
curves without undue loss of generality, but
this is not particularly difcult. It is also necessary that the
range and behavior of the numeric values of all parameters are
such that they can be efciently sampled, e.g., with a Markov
Chain Monte Carlo (MCMC ) algorithm. This is easier said than
done, and the best parameterization may be different for
different planets.
For Saturn, we nd that a piecewise-quadratic function of
density as a function of normalized radius works best for the
bulk of the planet, with a quartic (degree 4) polynomial
required to represent the uppermost region (for
P 2GP
).
We describe this parameterization in detail in Section 2.2. Note
that this is one place where modications might be needed
before applying the same procedure to Jupiter or the ice giants.
To drive the sampling algorithm, we need a way to evaluate
the relative likelihood of two model planets, and we do this by
comparing how well they match Saturns observed mass and
gravity eld. The details of this calculation are given in
Section 2.3.
The likelihood calculation requires that we know the
equilibrium shape and gravity eld of a given density prole
to sufcient accuracy. Note that in Equation (2) the integrand is
known but the integration bounds are unknown. We need to
rst determine the planets equilibrium shape. The shape is
determined by a balance between the centrifugal acceleration of
the rotating planet and the gravitational acceleration. This is
therefore a circular problem, requiring an iterative calculation
to converge to a self-consistent solution.
We use an implementation of fourth-order Theory of Figures
(ToF) using the coefcients given in Nettelmann ( 2017) and
employ optimization techniques that allow us to solve the
hydrostatic equilibrium state to desired precision very quickly.
The details are given below in the Section 2.4.
The emphasis on speed is necessary, as the next subtask is to
employ a suitable MCMC algorithm to draw a large sample
of possible
r
. There is no generally agreed-upon method of
predicting the number of sampling steps required for
convergence.
6
By experimentation, we nd that our Saturn
parameterization requires tens of thousands of steps to become
independent of its seed state and has a long autocorrelation
time, requiring a large number of steps following convergence
to obtain the desired effective sample size. Producing a valid
sample required the computation of about 10 million model
planets in total. We give the details of our sampling method
and convergence tests in Section 2.5 and Appendix C.
The last step is calculating some derived physical quantities
of interest, based on the obtained
r
sample. Given the gravity
eld, the pressure on each level surface can be computed from
the hydrostatic equilibrium equation. And with knowledge of
the pressure and density at each level, we may begin to estimate
6
Or even of being sure that convergence was reached.
3
The Astrophysical Journal, 891:109 (19pp), 2020 March 10 Movshovitz et al.

other quantities of interest, e.g.,the helium fraction, the heavy-
element content, etc. These quantities are not determined
directly by the gravity eld but can be inferred, with additional
assumptions. We discuss the results of this analysis, as applied
to Saturn, in Section 3.
2.2. Parameterization of
(
)r
s
Our goal is to sample from a space of
(
)r
s
curves that is as
general as possible, making a minimum of assumptions about
(
)r
s
while still restricting the sample to physically meaningful
density proles and, importantly, keeping the number of free
parameters small, for sampling efciency. These competing
requirements are not easy to satisfy, and it may be that the best
parameterization depends on the planet being studied as well as
on the available sampling algorithms and computing resources.
When looking for a good parameterization of
(
)r
s
, we were
guided by previously published work on Saturns interior.
Traditionally derived models are less general than we would
like, but they are physically sound. Examining them exposes
the major features expected of a
(
)r
s
curve representing
Saturns interior. Figure 2 shows the density proles of several
hereafter M19). These models assume a three-layer structure
for Saturn along the lines of what was considered by
Nettelmann et al. (2013). They consist of a homogeneous
outer envelope with helium mass fraction
=
Y
Y
1
and water
mass fraction
=ZZ
1
, a homogeneous inner envelope with
=
Y
Y
2
and Z= Z
2
, and nally a central core with Z=1.
These models assume an additive-volume mixture of hydrogen,
helium, and water as described by the Saumon et al. (1995) and
French et al. (2009) EOSs, and are assumed to have adiabatic
temperature proles throughout the envelope with an iso-
thermal core.
The general feature is a monotonic and piecewise-smooth
function in three segments. This is not surprising, as these
models were all derived with the assumption of three layers of
homogeneous composition, commonly thought of as an upper
envelope, lower envelope, and core. While we do not wish to
make such a strong assumption, we nd it necessary to make
the much weaker assumption that
(
)r
s
is a monotonic,
piecewise-smooth function, with no more than (but possibly
fewer than!) two density discontinuities. Further, between
discontinuities, the density appears to follow very smooth
curves, suggesting that it may be well approximated by
m
for each segment, where
=
R
58, 232 km
m
is Saturns volumetric mean 1 bar radius
(Lindal et al. 1985). By experimentation, we nd no advantage
in using higher-order polynomials to approximate any of the
main segments.
This piecewise-continuous model should not be confused
with the traditional three-layer one. The assumption of density
being piecewise continuous is much less strict than that of
composition being piecewise constant, even if they lead to
visually similar plots. Nevertheless, it would be even better to
allow more discontinuities or, better yet, a variable number of
them. While this may seem like a relatively straightforward
generalization, it would in fact greatly increase the computa-
tional cost of sampling the parameter space. To understand
why, consider that each additional discontinuity in
(
)r
s
not
only introduces four additional parameters (the three para-
meters required to describe the quadratic plus the location of
the additional break point ), these parameters will also be highly
correlated with the rest. As it turns out, this correlation is
already evident with just two discontinuities. Informally, each
of the two density jumps can substitute for the other in the
large subset of models where only a single pronounced
discontinuity appears. This evident redundancy is by no
means proof that there cannot be more than two sharp density
jumps in Saturns interior. But it helps us accept, at least
temporarily, a compromise between maximum generality and
minimum CPU hours.
When we examine more closely the very top of the density
curves in Figure 2 we nd that the uppermost part of the
=
r
rR0.94
a m
same quadratic as the rest of the upper envelope. Instead, it is
more similar to a quartic polynomial. This is demonstrated
visually for one density prole in the inset of Figure 2 and in
more detail in Appendix A. In this low-pressure region, the
physical models are based on well-tested EOSs of H and He,
we would be well advised to constrain our proles to make use
of this information. In Appendix A we explain how we derive a
one-parameter family of quartic functions that keeps us
grounded to realistic density values in the region above r
a
,
while still allowing variation by letting the value of
()
r
r==sr
a
a
be sampled.
It is important to note that to date, all EOS-based models of
Saturn nd solutions consistent with the measured gravity eld
that predict a concentration of heavy elements in the envelope
of at most a few times the protosolar value (e.g., Helled &
Guillot 2013; Nettelmann et al. 2013; Militzer et al. 2019),
while Saturns atmospheric spectra indicate a higher value,
perhaps as high as 10 times the protosolar metallicity (Atreya
et al. 2016). In principle, atmospheric enrichment might not
represent the bulk composition of the outer envelope, as was
Figure 2. Three representative Saturn density proles from M19. These
proles were derived using the standard, three-layer assumption, and thus
represent only a subset of possible proles. On the other hand, they are known
to be in strict agreement (by construction) with theoretical EOSs throughout the
interior. The inset shows a zoomed-in view of the top part of the envelope.
The red solid line is the same curve as in the full-scale gure; the black dotted
line is a quadratic t, a good approximation of the upper envelope overall, and
the blue dashed line is a quartic t to the segment
>sR 0.9
4
m
, a much better
t there (Appendix A).
4
The Astrophysical Journal, 891:109 (19pp), 2020 March 10 Movshovitz et al.

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Journal ArticleDOI

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Journal ArticleDOI

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3,846 citations

Journal ArticleDOI
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