https://doi.org/10.1140/epjs/s11734-025-01985-2
Regular Article
Do self-gravitating hard-sphere disk simulations form Darwin ellipsoids?
1
Observatório Nacional/MCTI, 20921-400, Rio de Janeiro, Brazil
2
Orbital Dynamics and Planetology Group (GDOP), São Paulo State University, 12516-410, Guaratinguetá, Brazil
a
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Received:
24
April
2025
Accepted:
20
September
2025
Published online:
3
October
2025
Abstract
The equilibrium shape of homogeneous fluid satellites results from a balance between self-gravity, planetary tides, and rotation, corresponding to Darwin ellipsoids—equilibrium figures associated with the Love numbers 
In this work, we simulate self-gravitating hard-sphere disks that produce a single Moon-mass satellite. The satellite’s shape is tracked over time to assess whether it resembles a Darwin ellipsoid, as predicted by theory. The Moon-like satellite forms near the planet’s Roche limit and grows by accreting both disk particles and recently coalesced proto-satellites. The energy dissipated during the merger impacts rapidly drives the satellite toward a spin-orbit equilibrium state and a triaxial ellipsoidal shape. Our results show that the numerical and analytical shapes exhibit discrepancies below 30%, indicating that this class of simulations is capable of reproducing the expected equilibrium shapes reasonably well, provided that the particle resolution of the disk is sufficiently high 
particles). The discrepancies decrease for satellites formed at larger distances from the planet, where weaker gravitational perturbations from both the planet and the remnant disk allow for greater structural homogenization.
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© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
