2016ApJ...831..134V

According to the virial theorem, all gravitational systems in equilibrium sit on a plane in the three-dimensional parameter space defined by their mass, size, and second moment of the velocity tensor. While these quantities cannot be directly observed, there are suitable proxies: the luminosity L_k, half-light radius R_e, and dispersion σ_e. These proxies indeed lie on a very tight fundamental plane (FP). How do the black holes (BHs) in the centers of galaxies relate to the FP? Their masses are known to exhibit no strong correlation with total galaxy mass, but they do correlate weakly with bulge mass (when present), and extremely well with the velocity dispersion through the M_•∝σ_e^5.4 relation. These facts together imply that a tight plane must also exist defined by BH mass, total galaxy mass, and size. Here, I show that this is indeed the case using a heterogeneous set of 230 BHs. The sample includes BHs from zero to 10 billion solar masses and host galaxies ranging from low surface brightness dwarfs, through bulgeless disks, to brightest cluster galaxies. The resulting BH-size-luminosity relation M_•∝(L_k/R_e)^3.8 has the same amount of scatter as the M_*-σ relation and is aligned with the galaxy FP, such that it is just a reprojection of σ_e. The inferred BH-size-mass relation is M_•∝(M_*/R_e)^2.9. These relationships are universal and extend to galaxies without bulges. This implies that the BH is primarily correlated with its global velocity dispersion and not with the properties of the bulge. I show that the classical bulge-mass relation is a projection of the M_*-σ relation. When the velocity dispersion cannot be measured (at high z or low dispersions), the BH-size-mass relation should be used as a proxy for BH mass in favor of just galaxy or bulge mass.