The M-$\sigma$ relation from the disruption of binaries from the galactic bulge

We present a novel explanation of the well known $M_{\bullet}-\sigma$ relation. In a triaxial potential binaries with chaotic orbits within a sphere that encompass $\sim100$ times the mass of the super-massive black-hole (SMBH) have a finite probability to be tidally disrupted by the SMBH. As a result one component loses energy and might itself break apart tidally and accreted onto the SMBH. More significantly, the other component, which gains energy, returns to the bulge and equilibrates its excess energy with the environment thereby changing the kinetic temperature, hence the velocity dispersion. We develop a mathematical model and find that its results are in agreement with the observed relation.