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 \sigma^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-\sigma relations for black holes in massive bulges, to
include similar relations for nuclear clusters in lower-mass and disc galaxies.
