We present an investigation about the shape of the initial mass function (IMF) of early-type galaxies (ETGs), based on a joint lensing and dynamical analysis, and on stellar population synthesis models, for a sample of 55 lens ETGs identified by the Sloan Lens ACS (SLACS) Survey. We construct axisymmetric dynamical models based on the Jeans equations which allow for orbital anisotropy and include a dark matter halo. The models reproduce in detail the observed HST photometry and are constrained by the total projected mass within the Einstein radius and the stellar velocity dispersion (σ) within the SDSS fibers. Comparing the dynamically-derived stellar mass-to-light ratios (M * /L) dyn , obtained for an assumed halo slope ρ h ∝ r −1 , to the stellar population ones (M * /L) Salp , derived from full-spectrum fitting and assuming a Salpeter IMF, we infer the mass normalization of the IMF. Our results confirm the previous analysis by the SLACS team that the mass normalization of the IMF of high σ galaxies is consistent on average with a Salpeter slope. Our study allows for a fully consistent study of the trend between IMF and σ for both the SLACS and ATLAS 3D samples, which explore quite different σ ranges. The two samples are highly complementary, the first being essentially σ selected, and the latter volume-limited and nearly mass selected. We find that the two samples merge smoothly into a single trend of the form log α = (0.38 ± 0.04) × log(σ e /200 km s −1 ) + (−0.06 ± 0.01), where α = (M * /L) dyn /(M * /L) Salp and σ e is the luminosity averaged σ within one effective radius R e . This is consistent with a systematic variation of the IMF normalization from Kroupa to Salpeter in the interval σ e ≈ 90 − 270 km s −1 .

By means of high resolution 2D hydrodynamical simulations, we study the evolution of the hot ISM for a large set of early-type galaxy models, characterized by various degrees of flattening and internal rotation. The galaxies are described by state-of-the-art axisymmetric two-component models, tailored to reproduce real systems; the dark matter haloes follow the Navarro-Frenk-White or the Einasto profile. The gas is produced by the evolving stars, and heated by Type Ia SNe. We find that, in general, the rotation field of the ISM in rotating galaxies is very similar to that of the stars, with a consequent negligible heating contribution from thermalization of the ordered motions. The relative importance of flattening and rotation in determining the final X-ray luminosity L X and temperature T X of the hot haloes is a function of the galactic mass. Flattening and rotation in low mass galaxies favour the establishment of global winds, with the consequent reduction of L X . In medium-to-high mass galaxies, flattening and rotation are not sufficient to induce global winds, however, in the rotating models the nature of the gas flows is deeply affected by conservation of angular momentum, resulting in a reduction of both L X and T X .

We present a detailed diagnostic study of the observed temperatures of the hot X-ray coronae of early-type galaxies. Extending the investigation carried out in Pellegrini (2011) with spherical models, we focus on the dependence of the energy budget and temperature of the hot gas on the galaxy structure and internal stellar kinematics. By solving the Jeans equations we construct realistic axisymmetric three-component galaxy models (stars, dark matter halo, central black hole) with different degrees of flattening and rotational support. The kinematical fields are projected along different lines of sight, and the aperture velocity dispersion is computed within a fraction of the circularized effective radius. The model parameters are chosen so that the models resemble real ETGs and lie on the Faber-Jackson and Size-Luminosity relations. For these models we compute T * (the stellar heating contribution to the gas injection temperature) and T − g (the temperature equivalent of the energy required for the gas escape). In particular, different degrees of thermalisation of the ordered rotational field of the galaxy are considered. We find that T * and T − g can vary only mildly due to a pure change of shape. Galaxy rotation instead, when not thermalised, can lead to a large decrease of T * ; this effect can be larger in flatter galaxies that can be more rotationally supported. Recent temperature measurements T X , obtained with Chandra, are larger than, but close to, the T * values of the models, and show a possible trend for a lower T X in flatter and more rotationally supported galaxies; this trend can be explained by the lack of thermalisation of the whole stellar kinetic energy. Flat and rotating galaxies also show lower L X values, and then a lower gas content, but this is unlikely to be due to the small variation of T − g found here for them.

We present the analytical solution of the two-integrals Jeans equations for Miyamoto-Nagai discs embedded in Binney logarithmic dark matter haloes. The equations can be solved (both with standard methods and with the Residue Theorem) for arbitrary choices of the parameters, thus providing a very flexible two-component galaxy model, ranging from flattened discs to spherical systems. A particularly interesting case is obtained when the dark matter halo reduces to the Singular Isothermal Sphere. Azimuthal motions are separated in the ordered and velocity dispersion components by using the Satoh decomposition. The obtained formulae can be used in numerical simulations of galactic gas flows, for testing codes of stellar dynamics, and to study the dependence of the stellar velocity dispersion and of the asymmetric drift in the equatorial plane as a function of disc and halo flattenings. Here, we estimate the inflow radial velocities of the interstellar medium, expected by the mixing of the stellar mass losses of the lagging stars in the disc with a pre-existing gas in circular orbit.

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