I study non-radiative cooling of X-ray emitting gas via heat conduction along magnetic field lines inside magnetic flux loops in cooling flow clusters of galaxies. I find that such heat conduction can reduce the fraction of energy radiated in the X-ray band by a factor of 1.5-2. This non-radiative cooling joins two other proposed non-radiative cooling processes, which can be more efficient. These are mixing of cold and hot gas, and heat conduction initiated by magnetic fields reconnection between hot and cold gas. These processes when incorporated into the moderate cooling flow model lead to a general cooling flow model with the following ingredients. (1) Cooling flow does occur, but with a mass cooling rate about 10 times lower than in old versions of the cooling flow model. Namely, heating occurs such that the effective age of the cooling flow is much below the cluster age, but the heating can't prevent cooling altogether. (2) The cooling flow region is in a non-steady state evolution. (3) Non-radiative cooling of X-ray emitting gas can bring the model to a much better agreement with observations. (4) The general behavior of the cooling flow gas, and in particular the role played by magnetic fields, make the intracluster medium in cooling flow clusters similar in some aspects to the active solar corona.
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