Assessing indirect methods to determine black hole masses using NGC 4151

Accurately determining the black hole mass ($M_\mathrm{BH}$) in active galactic nuclei (AGN) is crucial to constraining their properties and to studying their evolution. While direct methods yield reliable measurements of $M_\mathrm{BH}$ in unobscured type 1 AGN, where the dynamics of stellar or gas components can be directly observed, only indirect methods can be applied to the vast majority of heavily absorbed type 2 AGN, which represent most of the AGN population. Since it is difficult to evaluate the accuracy and precision of these indirect methods, we utilize the nearby X-ray bright Seyfert galaxy NGC 4151, whose $M_\mathrm{BH}$ has been tightly constrained with several independent direct methods, as a laboratory to assess the reliability of three indirect methods that have been applied to obscured AGN. All three, the X-ray scaling method, the fundamental plane of black hole activity, and the M-$\sigma$ correlation, yield $M_\mathrm{BH}$ values consistent with those inferred from direct methods and can therefore be considered accurate. However, only the X-ray scaling method and the M-$\sigma$ correlation are precise because the substantial scatter in the fundamental plane of BH activity allows only for crude estimates. Of the four M-$\sigma$ correlations we used, only the one from Kormendy and Ho yields a value consistent with the dynamical estimates. This study suggests that the best approach to estimating the black hole mass in systems where direct dynamical methods cannot be applied is to utilize a combination of indirect methods, taking into account their different ranges of applicability.