Galactic Rotation Curves from Full-Disk Newtonian Modeling: The Lost and Found Framework
The approximately flat outer parts of spiral galaxy rotation curves are commonly interpreted as evidence for a discrepancy between the observed baryonic mass and the dynamical mass inferred from the measured orbital velocities. In many analyses, simplified mass estimates are often expressed through the relation $v^2(R)=GM(<R)/R$, which is exact only under spherical symmetry. Spiral galaxies, however, are flattened disk systems, for which mass exterior to the galactocentric radius under consideration can contribute non-negligibly to the gravitational field. We introduce the \textit{Lost and Found} (LF) model, a geometrically consistent Newtonian framework based on direct full-disk gravitational integration and a parametrized representation of the disk surface density. This approach is closely related to previous thin-disk treatments that compute the gravitational field from the full mass distribution, while providing a simplified parametrization suitable for systematic fitting across heterogeneous galaxy samples. We apply the LF model to a heterogeneous sample of disk galaxies spanning a broad range of masses and radial extents. The model reproduces the main observed features of the rotation curves, including the inner rise and the approximately flat outer behavior, while yielding systematically lower inferred masses compared to conventional dynamical mass estimates. Across the sample, the LF-inferred mass scales nearly linearly with the conventional dynamical mass, with a characteristic scaling factor $η_{\rm LF}\sim0.67$. These results suggest that part of the inferred mass discrepancy in disk galaxies may be associated with geometric assumptions in standard mass estimates, and highlight the importance of full-disk treatments when interpreting galactic rotation curves.