Revised entropy of the AQUA equation of state

¹ Institute of Space Research, German Aerospace Center (DLR), Rutherfordstr. 2, 12489 Berlin, Germany
² Institut für Physik, Universität Rostock, Albert-Einstein Str. 23–24, 18051 Rostock, Germany
³ Observatoire de la Côte d’Azur, Université Côte d’Azur, 96 Boulevard de l’observatoire, 06300 Nice, France

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

Received: 7 October 2025
Accepted: 11 March 2026

Abstract

Context. Water is a key constituent of the ice giants Uranus and Neptune, and potentially of Neptune-sized exoplanets. Understanding their present structure and evolution requires knowledge of the equation of state (EoS) and the entropy of the main constituents. Therefore, the EoS and entropy of water are crucial ingredients to ice giant models. The AQUA EoS was constructed to span a wide pressure-temperature (P–T) range by combining several existing water EoSs for different phases. However, one of the EoSs within AQUA contained a mistake in the equation used to compute the entropy.

Aims. We aim to provide a revised AQUA EoS that includes the entropy based on the revised equation. As the region of revised entropies covers the superionic regime, we also aim to improve AQUA in that phase by including further published absolute entropies and internal energies of superionic water derived from DFT-MD simulations and advanced post-processing.

Methods. We defined regions where entropy, density, and internal energy are interpolated whenever a continuous phase transition or no phase transition between the different EoS sources is expected to occur; otherwise, we allowed for jumps in entropy when a first-order phase transition is expected. We ensured a monotonic behaviour in entropy with pressure along isotherms and with temperature along isobars in the revised area of AQUA’s P–T space.

Results. We find that the entropies between the revised and original AQUA differ by up to ±10% in the regions relevant to the deep interiors of adiabatic Neptune-sized planets. The entropy in superionic water is found to be offset with respect to the neighbouring fluid phase by about −10%, which we take as evidence of a first-order phase transition with an associated latent heat release. Ice giant interiors are up to ~650 K colder and the latent heat could partially power the luminosity of Neptune.