Spatial surface gravity of small bodies from polyhedral shape models: slope, geopotential, and the resolution robustness of geopotential dispersion

Abstract

Three companion papers established that the area-weighted mean surface gravity ⟨g⟩ of a small body can be estimated from a polyhedral shape model, without an independently determined density (Papers I–III). A single scalar, however, cannot say where on a body the gravity is high or low, how steeply the surface is inclined to the local effective gravity, or where loose regolith is driven to accumulate. Here we extend the shape-based framework from the scalar mean to the spatially resolved surface fields: the effective gravity vector (self-gravitation plus rotation), the gravitational slope θ, and the surface geopotential U, evaluated facet by facet for seven asteroids from a polyhedral shape model together with an assumed uniform density and a specified spin state. The framework is validated against an independent per-facet reference. For (162173) Ryugu we compare with the gravitational products released by the Hayabusa2 team on the identical 49,152-facet shape model at the same density and rotation period. Agreement is essentially exact: mean slope 15.06° against 15.06°, RMS difference 0.48°, 99.96% of facets within 5° and 99.95% within 2°, Pearson r = 0.9989 in slope and 0.9898 in geopotential. For (101955) Bennu the global mean slope is 15.5° and the latitudinal profile rises from 10.5° near the equator to 19.9° at mid-latitude before falling toward the pole, reproducing the pattern reported from OSIRIS-REx data. The principal methodological result concerns stability under mesh refinement. Because slope is a local quantity, defined through individual facet normals, the area-weighted mean slope varies by 1.8% to 25.6% across meshes of the same body decimated from roughly 42,000 to 2,000 facets, whereas the normalized geopotential dispersion varies by only 0.9% to 5.9%, and is the more stable of the two on every body tested. The sharper distinction is stabilization: between the two finest meshes the geopotential dispersion is already near a plateau (changes of 0.00% to 0.60%), whereas the mean slope shows no comparable plateau within the tested resolution range, still rising by up to 8.2%. A slope value is therefore not reproducible unless the mesh resolution is quoted with it, whereas the geopotential dispersion exhibits practical convergence over the tested range. Comparing two independent shape models of Itokawa — the spacecraft-derived Gaskell model and a ground-based radar model — separates what survives a coarser shape from what does not. Bulk descriptors agree closely (area-weighted mean slope within 0.8%, geopotential dispersion within 7.9%), but the distribution tails do not: the radar model reaches only 41° with 1.7% of facets above 30°, against 149° and 4.8% for the Gaskell model. In this controlled case a ground-based shape therefore preserves the global gravity descriptors while failing to reproduce the distribution tails; one case does not establish the result generally, but it indicates where coarse models may and may not be relied upon. We verify the computed fields against two exact solutions: the closed-form potential of a homogeneous ellipsoid (Pearson r = 1.000000, mean relative error 0.18%) and the analytic slope field of a rotating sphere. The latter test is essential — a non-rotating sphere check cannot detect an error in the centrifugal term — and we recommend it as a standard verification for surface-gravity codes. Both tests are released as runnable scripts together with the analysis code.