Do self-gravitating hard-sphere disk simulations form Darwin ellipsoids?

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 hs.$$h_s.$$ 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 (≳105$$( \gtrsim 10^5$$ 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.