The Impact of Elliptical Broad-Line Regions on Reverberation-Based Black Hole Mass Estimates

The virial factor $f$ is critical for accurate supermassive black hole (SMBH) mass measurements using reverberation mapping (RM) and the radius--luminosity ($R$--$L$) relation, yet its value remains highly uncertain. While traditional models assume axisymmetric broad-line region (BLR) geometries, growing evidence suggests that BLRs may possess more complex, asymmetric structures. We systematically investigate the impact of elliptical-disk BLR geometries on SMBH mass determinations through comprehensive numerical simulations. By computing emission-line profiles, emissivity-weighted time lags, and the corresponding virial factor $f$ over a wide range of eccentricities, orientations, and inclinations, we find that even in purely virialized systems, geometric effects alone can cause $f$ to vary by more than an order of magnitude and can mimic observational signatures typically attributed to radiation pressure. Additionally, local broadening introduces further systematic uncertainties in velocity width measurements, biasing $f$ by up to a factor of $\sim$3. Asymmetric BLR configurations also induce a scatter of $\sim$0.18 dex in the $R$--$L$ relation due to projection effects, comparable to the intrinsic scatter observed in RM studies. These results challenge the conventional attribution of RM uncertainties to non-virial motions or radiation pressure, and instead highlight the fundamental role of BLR geometry in SMBH mass measurements.