WISDOM Project – XIX. Figures of merit for supermassive black hole mass measurements using molecular gas and/or megamaser kinematics

The mass (MBH) of a supermassive black hole (SMBH) can be measured using spatially-resolved kinematics of the region where the SMBH dominates gravitationally. The most reliable measurements are those that resolve the smallest physical scales around the SMBHs. We consider here three metrics to compare the physical scales probed by kinematic tracers dominated by rotation: the radius of the innermost detected kinematic tracer Rₘin normalised by respectively the SMBH's Schwarzschild radius (RSchw 2GMBH/c², where G is the gravitational constant and c the speed of light), sphere-of-influence (SOI) radius (RSOI GMBH/ₑ², where ₑ is the stellar velocity dispersion within the galaxy's effective radius) and equality radius the radius Rₑq at which the SMBH mass equals the enclosed stellar mass, MBH=M_* (Rₑq), where M_* (R) is the stellar mass enclosed within the radius R. All metrics lead to analogous simple relations between Rₘin and the highest circular velocity probed Vc. Adopting these metrics to compare the SMBH mass measurements using molecular gas kinematics to those using megamaser kinematics, we demonstrate that the best molecular gas measurements resolve material that is physically closer to the SMBHs in terms of RSchw but is slightly farther in terms of RSOI and Rₑq. However, molecular gas observations of nearby galaxies using the most extended configurations of the Atacama Large Millimeter/sub-millimeter Array can resolve the SOI comparably well and thus enable SMBH mass measurements as precise as the best megamaser measurements.