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The energetics of flat and rotating early-type galaxies and their X-ray luminosity

The global energy budget of the hot gas flows in early-type galaxies is used to explain the recent observational results that S0 galaxies have lower mean X-ray luminosity $L_X$ per unit optical luminosity $L_B$ than do ellipticals. This could be explained with a higher heat input or with a lower gravitational energy (all per unit gas mass), at fixed $L_B$. The effect of the flattening of the mass distribution is investigated with galaxy models described by the Miyamoto-Nagai potential--density pair, to which a dark matter halo of various shapes is added. For these models the analytical expressions of the gravitational energy, and the stellar kinetic energy associated with various relative amounts of random motions and rotational streaming, are derived. It is found that rotation cannot produce a change in the flow phase of the hot gas, independently of the galaxy shape and the presence of dark matter. The effect of flattening instead can be substantial in reducing the binding energy of the hot gas. Thus S0s and possibly non spherical Es are less able to retain a significant halo of hot gas than rounder Es of the same $\lb$.