Numerical Convergence of Physical Variables in Hydrodynamical Simulations of Cooling Clusters

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Abstract
Results from smoothed particle hydrodynamical (SPH) simulations of galaxy clusters are used to investigate the dependence of the final cluster X-ray properties on the numerical resolution and the assumed models for the physical gas processes. Two different spatially flat cosmological models have been considered: a low-density cold dark matter universe with a vacuum energy density ΩΛ = 0.7 (ΛCDM) and a cold+hot dark matter model (CHDM). For each of these models two different clusters have been extracted from a cosmological N-body simulation. A series of hydrodynamical simulations has then been performed for each of them using a TREESPH code. These simulations first include radiative cooling and then also conversion of cold gas particles into stars; because of supernova explosions these particles can release energy in the form of thermal energy to the surrounding intracluster gas. For a specific treatment for the thermal state of the gas, simulation runs have been performed with different numerical resolutions. This is in order to disentangle in the final results for the cluster profiles, the effects of the resolution from those due to the assumed model for the gas thermal evolution. The numerical resolution of the simulation is controlled by the number of gas particles Ng and the chosen value for the gas gravitational softening parameter εg. The latter is proportional to the minimum SPH smoothing length and therefore sets a maximum spatial resolution for the simulations. For the cooling runs, final X-ray luminosities have been found to be diverging according to LX ∝ 1/ε. The gas density profiles are also diverging at the cluster center. This is in agreement with previous findings. When cold gas particles are allowed to convert into stars, the divergences are removed. The final gas profiles show a well-defined core radius, and the temperature profiles are nearly flat. For the most massive test cluster in the ΛCDM model, these simulations show a prominent cooling flow in the cluster core. This cluster was analyzed in detail, running simulations with different star formation methods and increasing numerical resolution. A comparison between different runs shows that the results of simulations, based on star formation methods in which gas conversion into stars is controlled by an efficiency parameter c*, are sensitive to the numerical resolution of the simulation. In this respect star formation methods based instead on a local density threshold, as in Navarro & White, are shown to give more stable results. Final X-ray luminosities are found to be numerically stable, with uncertainties of a factor ~2. These simulations are also in good agreement with observational data when the final results are compared with the observed star formation rate and the luminosity-temperature relation from cooling flow clusters. Therefore I find that hydrodynamical simulations of cooling clusters can be used to give reliabe predictions of the cluster X-ray properties. For a given numerical resolution, the conversion of cool gas particles into stars as in Navarro & White should be preferred.
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