  Feedback heating from active galactic nuclei (AGNs) has been commonly invoked
to suppress cooling flows predicted in hot gas in elliptical galaxies, galaxy
groups and clusters. Previous studies have focused on if and how AGN feedback
heats the gas, but little paid attention to its triggering mechanism. Using
spherically symmetric simulations, we investigate how large-scale cooling flows
are accreted by central supermassive black holes (SMBHs) in eight well-observed
systems and find an interesting dichotomy. In massive clusters, the gas
develops a central cooling catastrophe within about the cooling time (typically
~ 100 - 300 Myr), resulting in a cold-mode accretion onto SMBHs. However, in
our four simulated systems on group and galaxy scales at a low metallicity
Z=0.3Z_{sun}, the gas quickly settles into a long-term state which has a cuspy
central temperature profile extending to several tens to about 100 pc. At the
more realistic solar metallicity, two groups (with R_{e} ~ 4 kpc) still host
the long-term hot-mode accretion. Both accretion modes naturally appear in our
idealized calculations where only cooling, gas inflow, and compressional
heating are considered. The long-term hot-mode accretion is maintained by the
quickly-established closeness between the timescales of these processes,
preferably in systems with low gas densities, low gas metallicities, and
importantly, compact central galaxies, which result in strong gravitational
acceleration and compressional heating at the intermediate radii. Our
calculations predict that central cuspy temperature profiles appear more often
in smaller systems than galaxy clusters, which instead often host significant
cold gas and star formation.
