Towards an agnostic algorithm for sampling empirical structure models

The case of Uranus and Neptune

¹ Department of Mathematics, Swiss Federal Institute of Technology Zurich (ETHZ), Rämistrasse 101, 8092 Zürich, Switzerland
² Department of Astrophysics, University of Zurich, Winterthurerstrasse 190, 8057 Zürich, Switzerland

^★★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 8 October 2025
Accepted: 18 December 2025

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 compared 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.