On Magnetic Fields, Heating and Thermal Conduction in Halos, and the Suppression of Cooling Flows

We discuss the physics of thermal heat transport in halos of giant elliptical galaxies and clusters of galaxies and the consequences for cooling flows in such halos. We demonstrate that--contrary to widespread belief-- "tangled" magnetic fields do not reduce the effect of thermal conduction by an amount sufficient to allow one to ignore conduction in cooling flows. Thus, when thermal conduction is included, one can explain the observed density and temperature profiles with the rate of mass inflow reduced from the values commonly quoted in the literature by about a factor of 3-10 for M87. This effect, possibly in conjunction with very modest amounts of heating (possibly as a result of accretion onto, for example, a central black hole), can regulate the cooling flow in a thermally stable manner at mass accretion rates of less than 0.1 M_sun_ yr^-1^.