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

We solve for the velocity fields of momentum-conserving supershells driven from galaxy centres by steady winds from supermassive black holes or nuclear star clusters (central massive objects: CMOs). We look for the critical CMO mass that allows such a shell to escape from its host galaxy. In the case that the host galaxy dark matter halo is a singular isothermal sphere, we find that the critical CMO mass derived by King, which scales with the halo velocity dispersion as M_crit \propto σ^4, is necessary, but not by itself sufficient, to drive shells to large radii in the halo. Furthermore, a CMO mass at least 3 times the King value is required to drive the shell to the escape speed of the halo. In the case of CMOs embedded in protogalaxies with non-isothermal dark matter haloes, which we treat here for the first time, we find a critical CMO mass that \textit{is sufficient} to drive \textit{any} shell (under a steady wind) to escape \textit{any} galaxy with a peaked circular speed profile. In the limit of large halo mass, relevant to real galaxies, this critical CMO mass depends only on the value of the peak circular speed of the halo, scaling as M_crit \propto V_c,pk^4. Our results therefore relate to observational scalings between black hole mass and asymptotic circular speed in galaxy spheroids. They also suggest a natural way of extending analyses of M-σrelations for black holes in massive bulges, to include similar relations for nuclear clusters in lower-mass and disc galaxies.