The radial variation of hot gas density and temperature in bright elliptical galaxies noted in X-ray observations can be used to determine accurately the radial distribution of total galactic mass using the condition of hydrostatic equilibrium. However, we set out here to solve the inverse problem. Starting with the known distributions of total mass and of mass-losing stars in a specific large elliptical, NGC 4472, we attempt to solve the gasdynamical equations to recover the currently observed radial distribution of density and temperature in the hot interstellar gas. The galaxy is assumed to be initially gas free as a result of early type II supernova-driven galactic winds. In seeking this agreement, we consider a variety of assumptions for the evolution of the hot interstellar gas: mass dropout, variation of stellar mass loss with galactic radius, variation of the bolometric radiative cooling rate with galactic radius, and several supernova rates. After evolving for a Hubble time, none of these models accounts for several well-observed properties of the interstellar gas: gas temperatures that significantly exceed the mean stellar temperatures, positive temperature gradients within a few effective radii, shallow density gradients, and large interstellar gas masses. However, all of these discrepancies are lessened or disappear if large masses of hot gas are assumed to be present in the outer galactic potential at early times in galactic history. Most of the interstellar mass that contributes to cooling flows in galaxies like NGC 4472 has not come from mass lost by galactic stars since ~1 Gyr. The sustained inflow of hot gas from this circumgalactic environment may help to resolve other long-standing problems that have beset models of galactic cooling flows: the wide range of LX/LB for fixed LB; the variable and often low iron abundances observed in ellipticals, and the failure (so far) to observe pronounced rotational flattening in X-ray images of slowly rotating giant ellipticals. The characteristic hot gas temperature profile observed in many bright ellipticals has a maximum at about three effective radii (re). This can be understood as the mixing of gas ejected from stars with old circumgalactic gas flowing in from the halo. Since the circumgalactic gas is hotter than the stars, mass dropout is not needed to flatten the gas density gradient. Moreover, the agreement of the stellar mass in NGC 4472 (and NGC 4649) within re determined from stellar dynamics with the total mass determined by hot gas hydrostatic equilibrium is an additional argument against mass dropout.
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