  We calculate steady-state, one-dimensional hydrodynamic profiles of hot gas
in slowly accreting ("quiescent") galactic nuclei for a range of central black
hole masses $M_{\bullet}$, parametrized gas heating rates, and
observationally-motivated stellar density profiles. Mass is supplied to the
circumnuclear medium by stellar winds, while energy is injected primarily by
stellar winds, supernovae, and black hole feedback. Analytic estimates are
derived for the stagnation radius (where the radial velocity of the gas passes
through zero) and the large scale gas inflow rate, $\dot{M}$, as a function of
$M_{\bullet}$ and the gas heating efficiency, the latter being related to the
star-formation history. We assess the conditions under which radiative
instabilities develop in the hydrostatic region near the stagnation radius,
both in the case of a single burst of star formation and for the average star
formation history predicted by cosmological simulations. By combining a sample
of measured nuclear X-ray luminosities, $L_x$, of nearby quiescent galactic
nuclei with our results for $\dot{M}(M_{\bullet})$ we address whether the
nuclei are consistent with accreting in a steady-state, thermally-stable manner
for radiative efficiencies predicted for radiatively inefficiency accretion
flows. We find thermally-stable accretion cannot explain the short average
growth times of low mass black holes in the local Universe, which must instead
result from gas being fed in from large radii, due either to gas inflows or
thermal instabilities acting on larger, galactic scales. Our results have
implications for attempts to constrain the occupation fraction of SMBHs in low
mass galaxies using the mean $L_x-M_{\bullet}$ correlation, as well as the
predicted diversity of the circumnuclear densities encountered by relativistic
outflows from tidal disruption events.
