Unification of the Fundamental Plane and Super-Massive Black Holes Masses

Abstract

According to the Virial Theorem, all gravitational systems in equilibrium sit on a plane in the 3D 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 Lk, half-light radius Re and dispersion sigma_e. These proxies indeed lie on a very tight Fundamental Plane (FP). How do the black holes 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 Mbh = sigma_e^5.4 relation. These facts together imply that a tight plane must also exist defined by black hole mass, total galaxy mass and size. Here I show that this is indeed the case using a heterogeneous set of 230 black holes. The sample includes BHs from zero to 10 billion solar masses and host galaxies ranging from low surface brightness dwarfs, through bulge-less disks, to brightest cluster galaxies. The resulting BH-size-luminosity Mbh=(Lk/Re)^3.8 has the same amount of scatter as the M-sigma relation and is aligned with the galaxy FP, such that it is just a re-projection of sigma. The inferred BH-size-mass relation is Mbh=(M_star/Re)^2.9. These relationships are universal and extend to galaxies without bulges. This implies that the black hole 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-sigma 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 black hole mass in favor of just galaxy or bulge mass.