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

## Summary (4 min read)

### 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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##### Citations

^{1}, University of California, Riverside

^{2}, Academia Sinica Institute of Astronomy and Astrophysics

^{3}, Southwest Research Institute

^{4}, Hazard Community and Technical College

^{5}, Australian Nuclear Science and Technology Organisation

^{6}, Kindai University

^{7}, Australian National University

^{8}, University of New South Wales

^{9}, University of Jena

^{10}, Swinburne University of Technology

^{11}

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##### Frequently Asked Questions (2)

###### Q2. What are the future works mentioned in the paper "Saturn’s probable interior: an exploration of saturn’s potential interior density structures" ?

In this paper, 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. Here the authors wish to summarize their findings for Saturn, and about planetary interior modeling in general, and to consider the strengths and weaknesses of their “ density first ” approach, versus traditional, composition-based modeling. The great variety of density profiles included in their sample may seem surprising and counterintuitive, but it is an unavoidable consequence of using an integrated quantity, in this case the external potential, to study the spatial distribution of local quantities, in this case the interior density and all properties of the planet that derive from it. 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.