Momentum-driven feedback and the M-\sigma relation in non-isothermal galaxies

We solve for the velocity fields of momentum-conserving supershells driven by feedback from supermassive black holes or nuclear star clusters (central massive objects: CMOs). We treat, for the first time, the case of CMOs embedded in gaseous protogalaxies with non-isothermal dark matter haloes having peaked circular-speed profiles. We find the CMO mass that is sufficient to drive any shell to escape any such halo. In the limit of large halo mass, relevant to real galaxies, this critical CMO mass depends only on the peak circular speed in the halo, scaling as M_crit \propto V_c,pk^4.