THE EVOLUTION OF COOLING FLOWS IN CLUSTERS OF GALAXIES

Models are presented for the evolution of cooling flows in clusters of galaxies. Issues addressed include the time variation of the mass accretion rate and X-ray luminosity of the flows, the plausibility of forming stars, and possibly a central dominant galaxy, from the flows, and whether the flows must be in a steady state. The cluster potential is assumed to be dominated by a mildly evolving mass distribution (galaxies and dark matter), which acts as an effective heating source, Cosmological outer boundary conditions are imposed on the cluster gas. The gas cools by thermal bremsstrahlung and line radiation. In some models, heating by supernovae and by thermal conduction is included. All the models presume Ω = 1 and H0 = 50 km s-1 Mpc-1. Two classes of numerical models are considered. In the first class, the only initial external gravitational potential is that of the cluster. The flow proceeds in two stages. The first stage is an interplay between two gas accretion phases, one driven by radiative cooling and the other by gravitational heating arising from the deepening potential well of the cluster. In the limit of spherical symmetry, the phases each behave self-similarly inside the cluster core, and both are characterized by a growing gas density core which detaches from the original cluster core and moves inward. During this stage, the mass accretion rate and cluster X-ray luminosity increase with time. The cooling phase eventually dominates the evolution of the flow. Within an initial central cooling time, the flow undergoes an abrupt cooling transition to the second stage. The flow reaches a steady state within the cooling radius in this stage. By the end of the run, thermal instabilities have converted sufficient mass into stars to form a central dominant galaxy of ∼1012 M⊙. The star formation rate is sufficiently high, however, that the initial mass function must be heavily biased to low-mass stars (<1 M⊙) so as not to conflict with the observed colors of central dominant galaxies in existing cooling flows. In the second class of models, a central dominant galaxy is added to the cluster at the start of the evolution calculations. The evolution again proceeds in two stages. The presence of the galaxy breaks the self-similarity of the flow in the cluster center, but a vestige of self-similarity is retained in the density evolution at intermediate radii, and the gas settles toward the cluster center as in the first class. If the cluster potential is not allowed to evolve, the first stage is found to terminate in a rapid transition to steady state accretion within an initial cooling time. In the case of cluster evolution, however, the transition to the second stage is distinguished from that of the first class in that the approach to a steady state inflow is generally much slower, and a much smaller quantity of stars form. The evolution of the cluster prevents the inflowing gas from reaching a steady state within a cooling time. Instead, the gravitational heating due to cluster evolution allows the gas to achieve an inflow in which the cooling-induced decrease in local gas entropy occurs predominantly through an increase in gas density rather than a decrease in gas temperature. This situation arises because both the radiative cooling and the cluster evolution drive an inflow and thus an increasing gas density, while they oppose each other in their effect on the gas temperature. The gravitational heating maintains the gas at X-ray temperatures in the second stage even when the cooling time of the gas is ≲109 yr. In this scenario, very little star formation is required in order to produce gas density and temperature profiles in reasonable accord with those inferred for existing cooling flows. Although whether or not the flow undergoes a rapid transition to a steady state depends on the choice for initial gas density, the models demonstrate that for reasonable densities and temperatures, the approach to a steady state can be slowed by gravitational heating due to a deepening cluster potential well. This class of models may permit a reconciliation between the high accretion rates inferred from X-ray data in existing cooling flows and the severe constraints on the amount of accreted material imposed by optical, infrared, and radio measurements.