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

A recent analysis of the relationship between optical and X-ray properties for the Einstein sample of early-type galaxies has revealed that S0 galaxies have lower mean X-ray luminosity LX per unit optical luminosity LB than do ellipticals. In the same analysis, significant correlations are found between the X-ray properties and the axial ratios, such that the roundest systems have the highest LX/LB; this trend holds also for either Es or S0s alone. The systematic X-ray underluminosity of S0s with respect to Es could be explained with a higher heat input or with a lower gravitational energy (all per unit gas mass), at fixed LB. The gravitational energy could be lower because of their higher rotation rate, which decreases the effective potential, or their different mass distribution. These possibilities are examined here, by considering their role in the global energy budget of the hot gas flows in early-type galaxies. 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 two-component 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 given. 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.