Energy- and momentum-conserving AGN feedback outflows

It is usually assumed that outflows from luminous AGN are either in the energy-conserving (non-radiative) or in the momentum-conserving (radiative) regime. We show that in a non-spherical geometry the effects of both regimes may manifest at the same time, and that it is the momentum of the outflow that sets the $M_{\rm BH}-\sigma$ relation. Considering an initially elliptical distribution of gas in the host galaxy, we show that a non-radiative outflow opens up a wide ``escape route'' over the paths of least resistance. Most of the outflow energy escapes in that direction. At the same time, in the directions of higher resistance, the ambient gas is affected mainly by the incident momentum from the outflow. Quenching SMBH growth requires quenching gas delivery along the paths of highest resistance, and therefore, it is the momentum of the outflow that limits the black hole growth. We present an analytical argument showing that such energy-conserving feedback bubbles driving leaky ambient shells will terminate SMBH growth once its mass reaches roughly the $M_\sigma$ mass derived earlier by King (2003) for momentum-conserving AGN outflows. Our simulations also have potentially important implications for observations of AGN jet feedback and starburst galaxy feedback. The collimation of the wide angle AGN outflow away from the symmetry plane, as found in our simulations, indicates that credit for work done by such outflows may sometimes be mistakenly given to AGN jets or star formation feedback since wide angle $v \sim 0.1 c$ outflows are harder to observe and the phase when they are present may be short.