WISDOM Project -- XXIV. Cross-checking supermassive black hole mass estimates from ALMA CO gas kinematics and SINFONI stellar kinematics in the galaxy NGC 4751

Abstract: Supermassive black hole (SMBH) masses can be measured by observing the impacts of the SMBHs on dynamical tracers around them. We present high angular resolution ($0.19$ arcsec or $\approx24$ pc) Atacama Large Millimeter/submillimeter Array observations of the $^{12}$CO(3-2) line emission of the early-type galaxy NGC 4751, which reveal a highly-inclined regularly-rotating molecular gas disc with clear central Keplerian motions. Using a Hubble Space Telescope image to constrain the stellar mass distribution, we forward model the molecular gas kinematics and data cube in a Bayesian framework using the Kinematic Molecular Simulation code. Assuming a constant mass-to-light ratio ($M/L$), we infer a SMBH mass $M_\text{BH}=3.43^{+0.45}_{-0.44}\times10^9$ $\text{M}\odot$ and a F160W filter stellar $M/L$ $M/L\text{F160W}=(2.68\pm0.11)$ $\text{M}\odot/\text{L}{\odot,\text{F160W}}$ (all quoted uncertainties are at $3\sigma$ confidence). Assuming a linearly spatially-varying $M/L$, we infer $M_\text{BH}=2.79_{-0.57}^{+0.75}\times10^9$ $\text{M}\odot$ and $\left(M/L\text{F160W}\right)/\left(\text{M}\odot/\text{L}{\odot,\text{F160W}}\right)=3.07^{+0.27}{-0.35}-0.09^{+0.08}{-0.06}\,\left(R/\text{arcsec}\right)$, where $R$ is the galactocentric radius. We also present alternative SMBH mass estimates using the Jeans Anisotropic Modelling (JAM) method and SINFONI stellar kinematics. Assuming a cylindrically-aligned velocity ellipsoid (JAM$\text{cyl}$) we infer $M\text{BH}=(2.52\pm 0.36)\times10^9$ $\text{M}\odot$, while assuming a spherically-aligned velocity ellipsoid (JAM$\text{sph}$) we infer $M_\text{BH}=(3.24\pm0.87)\times10^9$ $\text{M}_\odot$. Our derived masses are all consistent with one another, but they are larger than (and inconsistent with) one previous stellar dynamical measurement using Schwarzschil's method and the same SINFONI kinematics.