Towards an agnostic algorithm for sampling empirical structure models: The case of Uranus and Neptune
Stefano Wirth, Luca Morf, Ravit Helled
TL;DR
This work addresses the degeneracy in inferring planetary interior density profiles by proposing an agnostic, bias-minimizing sampling framework that abandons MCMC in favor of an optimization-based gradient-descent approach grounded in the Theory of Figures. Applied to Uranus and Neptune, the method yields ensembles of density and pressure profiles that reproduce observed mass, radius, and gravitational moments while revealing the full extent of solution-space diversity, especially in the deep interior. Key findings include median-density trends consistent with composition gradients, outer regions that are tightly constrained, and a systematic presence of density discontinuities around $r/R \\approx 0.65$ (Uranus) and $\\approx 0.70$ (Neptune), with the number and strength of discontinuities depending on the chosen criterion $c_D$. The approach offers a scalable, unbiased framework for exploring planetary interiors under limited data and can be extended to exoplanets, higher-order gravitational moments, and compositional analyses, advancing our ability to interpret gravitational-field data.
Abstract
We present an algorithm to efficiently sample the full space of planetary interior density profiles. Our approach uses as few assumptions as possible to pursue an agnostic algorithm. The algorithm avoids the common Markov Chain Monte Carlo method and instead uses an optimisation-based gradient-descent approach designed for computational efficiency. In this work, we use Uranus and Neptune as test cases and obtain empirical models that provide density and pressure profiles consistent with the observed physical properties (total mass, radius, and gravitational moments). We compare our findings to other work and find that while other studies are generally in line with our findings, they do not cover the entire space of solutions faithfully. Furthermore, we present guidance for modellers that construct Uranus or Neptune interior models with a fixed number of layers. We provide a statistical relation between the steepness classifying a density discontinuity and the resulting number of discontinuities to be expected. For example, if one classifies a discontinuity as a density gradient larger than 0.02 kg$\,$m$^{-4}$, then most solutions should have at most one such discontinuity. Finally, we find that discontinuities, if present, are concentrated around a planetary normalised radius of 0.65 for Uranus and 0.7 for Neptune. Our algorithm to efficiently and faithfully investigate the full space of possible interior density profiles can be used to study all planetary objects with gravitational field data.