As part of the mm-Wave Interferometric Survey of Dark Object Masses (WISDOM), we present a measurement of the mass of the supermassive black hole (SMBH) in the nearby early-type galaxy NGC 0383 (radio source 3C 031). This measurement is based on Atacama Large Millimeter/sub-milimeter Array (ALMA) cycle 4 and 5 observations of the 12CO(2-1) emission line with a spatial resolution of 58x32pc2 (0.18x0.1). This resolution, combined with a channel width of 10 km/s, allows us to well resolve the radius of the black hole sphere of influence (measured as R_SOI = 316pc = 0.98), where we detect a clear Keplerian increase of the rotation velocities. NGC 0383 has a kinematically-relaxed, smooth nuclear molecular gas disc with weak ring/spiral features. We forward-model the ALMA data cube with the Kinematic Molecular Simulation (KinMS) tool and a Bayesian Markov Chain Monte Carlo method to measure a SMBH mass of (4.2+/-0.7)x10^9 Msun, a F160W-band stellar mass-to-light ratio that varies from 2.8+/-0.6 Msun/Lsun in the centre to 2.4+/-0.3 Msun/Lsun at the outer edge of the disc and a molecular gas velocity dispersion of 8.3+/-2.1 km/s (all 3-sigma uncertainties). We also detect unresolved continuum emission across the full bandwidth, consistent with synchrotron emission from an active galactic nucleus. This work demonstrates that low-J CO emission can resolve gas very close to the SMBH (~140,000 Schwarzschild radii) and hence that the molecular gas method is highly complimentary to megamaser observations as it can probe the same emitting material.
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