The interior of Uranus: Thermal profile, bulk composition and the distribution of rock, water and hydrogen and helium

TL;DR

This work tackles the poorly constrained interior of Uranus by deriving improved empirical density profiles and interpreting them with a novel random algorithm to infer internal temperature and composition. It combines a tenth-order Theory of Figures with an atmosphere-inclusive treatment to predict gravity moments up to $J_{14}$ and to explore two interior-model families, $U_{3, ext{comp}}$ and $U_{4, ext{comp}}$, that differ by the maximum materials per layer. The results show that most models are convective outside sharp boundaries, with a prevalent convective ionic $H_2O$ region that could sustain Uranus’ dynamo, and that including deep hydrogen–helium mixing shifts the interior toward rock-dominated solutions; central $H$–He mass fractions are tightly constrained near $0.1$ or less. The findings link high-order gravity measurements to composition-depth inferences and emphasize the need for future Uranus missions to refine $J_{6}$, $J_{8}$, and magnetic-field data to break degeneracies and inform planet formation and evolution scenarios.

Abstract

We present improved empirical density profiles of Uranus and interpret them in terms of their temperature and composition using a new random algorithm. The algorithm to determine the temperature and composition is agnostic with respect to the temperature gradient in non-isentropic regions and chooses randomly amongst all possible gradients that are stable against convection and correspond to an Equation of State compatible composition. Our empirical models are based on an efficient implementation of the Theory of Figures up to 10th order including a proper treatment of the atmosphere. The accuracy of 10th order ToF enables us to present accurate calculations of the gravitational moments of Uranus up to $J_{14}$: $J_{6} = ( 5.3078 \pm 0.3312)\cdot10^{-7}$, $J_{8} = (-1.1114 \pm 0.1391)\cdot10^{-8}$, $J_{10} = ( 2.8616 \pm 0.5466)\cdot10^{-10}$, $J_{12} = (-8.4684 \pm 2.0889)\cdot10^{-12}$ and $J_{14} = ( 2.7508 \pm 0.7944)\cdot10^{-13}$. We consider two interior models of Uranus that differ with respect to the maximal number of materials allowed per layer of Uranus (three vs. four composition components). The case with three materials does not allow Hydrogen and Helium in deeper parts of Uranus and results in a higher water abundance which leads to lower central temperatures. On the other hand, the models with four materials allow H-He to be mixed into the deeper interior and lead to rock-dominated solutions. We find that these four composition components models are less reliable due to the underlying empirical models incompatibility with realistic Brunt frequencies. Most of our models are found to be either purely convective with the exception of boundary layers, or only convective in the outermost region. Almost all of our models possess a region that is convective and consists of ionic H$_{2}$O which could explain the generation of Uranus' magnetic field.

Figures (19)

• Figure 1: Uranus' density $\rho$ as a function of the average normalised radius $r/R$. Other density profiles that have been published by previous studies are also presented for comparison Helled_2010Nettelmann_2013Vazan_2020. The panel on the right show the pressure as a function of density $P(\rho)$ in the outermost region. The white dots show the atmosphere model by Hueso_2020. The full $P(\rho)$ distribution is presented in appendix \ref{['app:data_summary']} in Figure \ref{['fig:P_of_rho']}.
• Figure 2: Relative difference of the Bessel gravitational moment solutions $J_2$ (top) and $J_8$ (bottom) compared to the ToF solutions. Results are shown for different numbers $N$ of used spheroids. We compare our implementation (This work) to the one used in Movshovitz_2022 (M&F 22) for different orders of the ToF. We also show the influence of using cubic spline interpolation with the parameter $n_x$ that stands for the number of points that were calculated without interpolation.
• Figure 3: Runtime in seconds of the calculations shown in Figure \ref{['fig:Bessel_convergence']} as a function of the number $N$ of used spheroids in the ToF. We refer to the legend and caption of Figure \ref{['fig:Bessel_convergence']} for further explanations.
• Figure 4: Our algorithm for inferring the temperature profile and composition of the empirical density (and pressure) profiles.
• Figure 5: Schematic drawing of the two possible composition models considered in this work. Top: U$_{3,\text{comp}}$ model that allows the presence of up to three elements simultaneously in a given layer. Bottom: U$_{4,\text{comp}}$ model that allows the presence of up to four elements simultaneously in a given layer. The ratios between different components are chosen freely and composition gradients are allowed but not necessary. The transition radii (white gaps) vary and are placed by the algorithm.