ABSTRACT
In several models of galaxy formation feedback occurs in cycles or mainly at high redshift. At times and in regions where feedback heating is ineffective, hot gas in the galaxy halo is expected to form a cooling flow, where the gas advects inward on a cooling timescale. Cooling flow solutions can thus be used as a benchmark for observations and simulations to constrain the timing and extent of feedback heating. Using analytic calculations and idealized 3D hydrodynamic simulations, we show that for a given halo mass and cooling function, steady-state cooling flows form a single-parameter family of solutions, while initially hydrostatic gaseous haloes converge on one of these solutions within a cooling time. The solution is thus fully determined once either the mass inflow rate ${\dot{M}}$ or the total halo gas mass are known. In the Milky Way halo, a cooling flow with ${\dot{M}}$ equal to the star formation rate predicts a ratio of the cooling time to the free-fall time of ∼10, similar to some feedback-regulated models. This solution also correctly predicts observed $\rm{O\,{\small VII}}$ and $\rm{O\,{\small VIII}}$ absorption columns, and the gas density profile implied by $\rm{O\,{\small VII}}$ and $\rm{O\,{\small VIII}}$ emission. These results suggest ongoing heating by feedback may be negligible in the inner Milky-Way halo. Extending similar solutions out to the cooling radius however underpredicts observed $\rm{O\,{\small VI}}$ columns around the Milky-Way and around other low-redshift star-forming galaxies. This can be reconciled with the successes of the cooling flow model with either a mechanism which preferentially heats the $\rm{O\,{\small VI}}$-bearing outer halo, or alternatively if $\rm{O\,{\small VI}}$ traces cool photoionized gas beyond the accretion shock. We also demonstrate that the entropy profiles of some of the most relaxed clusters are reasonably well described by a cooling flow solution.
Self-excited vibration of a circular plate subjected to frictional forces exerted in two regions on its outer circumference (A case of a circular plate subjected to ditributed frictional forces without the effect of internal resonances)
