Time dependence of the mass accretion rate in cluster cooling flows

We analyze two time-dependent cluster cooling flow models in spherical symmetry. The first assumes that the intracluster gas resides in a static external potential and includes the effects of optically thin radiative cooling and mass deposition. This corresponds to previous steady state cooling flow models calculated by White & Sarazin in 1987. Detailed agreement is found between steady state models and time-dependent models at fixed times in the simulations. The mass accretion rate (M)over dot is found either to increase or to remain nearly constant once flows reach a steady state. The time rate of change of (M)over dot is strongly sensitive to the value of the mass deposition parameter q but only mildly sensitive to the ratio beta of gravitational binding energy to gas temperature. We show that previous scaling arguments presented by Bertschinger in 1988 and by White in the same year are valid only for mature cooling flows with weak mass deposition (q less than or similar to 1). The second set of models includes the effects of a secularly deepening cluster potential and secondary infall of gas from the Hubble how. We find that such heating effects do not prevent the flows from reaching a steady state within an initial central cooling time.