As shown above, the diffuse soft X-ray emission observed in and around galaxies clearly traces the global hot gas that is heated by the stellar and possibly AGN feedback. By comparing the X-ray observations directly with physical models and/or simulations of the feedback, we may further study its dynamics, which is so far hardly known. The following discussion is concentrated on the modeling of the feedback in galactic spheroids for their relative simplicity. The presence of substantial cool gas and star formation, as in typical disk galaxies, would certainly add complications.
5.1. Comparison of feedback models with X-ray observations
The missing feedback problem is particularly acute in so-called low Lx / LK spheroid-dominated galaxies. Empirically, the Ia SN rate is 0.019 SN yr-1) 10-0.42(B-K) [LK / (1010 L K)], where LK is the K-band luminosity of a galaxy . Adopting the color index B-K 4 for a typical spheroid, we estimate the total mechanical energy injection from Ia SNe (assuming 1051 ergs each) is LIa (1.3 × 1040 ergs s-1) [LK / (1010 L K)]. The mechanical energy release of an AGN can also be estimated empirically [54, 52] as LAGN = (1.1 × 1039 erg s-1) [LK / (1010 LK, )]2. Therefore, averaged over the time, Ia SNe are energetically more important than the AGN in a galactic spheroid with LK 1011 LK,. Furthermore, SN blastwaves provide a natural distributed heating mechanism for hot gas in galactic spheroids . Additional continuous heating is expected from converting the kinetic energy of stellar mass loss from randomly moving evolved stars to the thermal energy. On the other hand, the AGN feedback, likely occurring in bursts with certain preferential directions (e.g., in form of jets), can occasionally result in significant disturbances in global hot gas distributions, as reflected by the asymmetric X-ray morphologies observed in some elliptical galaxies (e.g., ). It is not yet clear as to what fraction of the AGN feedback energy is converted into the heating of the hot gas observed. In general, the lack of distributed cool gas in galactic spheroids makes it difficult to convert and release the thermal energy into radiation in wavelength bands other than the X-ray.
Not only the mechanical energy, the gas mass from the stars and Ia SNe, especially heavy elements, is missing as well. The mass injection rate is expected to be 0.026[LK / (1010 LK, )] M yr-1 with a mean iron abundance ZFe Z*,Fe + 4(MFe / 0.7 M), where Z*,Fe is the iron abundance of the stars while MFe is the iron mass yield per Ia SN (e.g., [53, 56, 57]). The above empirical estimates of the energy and mass feedback rates, which should be accurate within a factor of ~ 2, are typically a factor of ~ 102 greater than what are inferred from the diffuse X-ray emission. Naturally, one would expect that the missing feedback is gone with a wind (outflow), spheroid-wide or even galaxy-wide.
The notion that Ia SNe may drive galactic winds is not new (e.g., [57, 58, 59]). But could such winds explain the diffuse X-ray emission of galactic spheroids? It is easy to construct a 1-D steady-state supersonic wind model, assuming that the specific energy of the feedback (per mass) of a galaxy is large enough to overcome its gravitational bounding and that the CGM pressure (thermal or ram) is negligible (e.g., ). This supersonic wind model depends primarily on two feedback parameters: the integrated energy and mass input rates. However, the model in general fails miserably: it predicts a too low luminosity (by a factor of ~ 102), a too high temperature (a factor of a few), and a too steep radial intensity profile to be consistent with Chandra observations of low Lx / LB galactic spheroids, in particular those in M31 and M104 [37, 43]. Only few very low-mass and gas-poor spheroids, hence very faint in X-ray emission, still seem to be consistent with the 1-D wind model (e.g., ).
Figure 4. Sample snapshots of hydrodynamic simulations of the stellar feedback. (a) the 3-D simulated gas density distribution in an M31-like galactic spheroid. The slice is cut near the spheroid center and the units are in atoms cm-3, logarithmically. (b) Simulated 2-D large-scale gas density distribution in the r-z plane (in units of kpc). A nearly vertical magnetic field, similar to what is observed in the inter-cloud medium of the Galactic center, is included in the simulation to test its confinement effect on the spheroid wind; a reverse-shock is clearly visible. Also apparent are instabilities at the contact discontinuity between the shocked spheroid wind gas and the ejected materials from the initial starburst. The density is scaled in units of M kpc-3.
To compare with the X-ray emission, in fact one needs to account for 3-D effects. X-ray emission is proportional to the emission measure and is thus sensitive to the detailed structure of hot gas in a galactic spheroid. To realistically generate the inhomogeneity in the heating and chemical enrichment processes, Tang & Wang have developed a scheme to embed adaptively selected 1-D SNR seeds in 3-D spheroid-wide simulations of supersonic winds or subsonic outflows (e.g., [55, 61]). These 3-D simulations, reaching a resolution down to adaptively refined scales of a few pc (e.g., Fig. 4a), show several important 3-D effects :
All considered, subsonic outflows appear to be most consistent with the X-ray observations of diffuse hot gas in typical intermediate-mass galactic spheroids and elliptical galaxies.
5.2. The interplay between the feedback and galaxy evolution
Whether a galactic outflow is supersonic or subsonic depends not only on the specific energy of the ongoing feedback, but also on the properties of the CGM, which is a result of the past interplay between the feedback and the accretion of a galaxy or a group of galaxies from the intergalactic medium (IGM). Tang et al. have illustrated how this interplay may work, based on several 1-D hydrodynamic simulations in the context of galaxy formation and evolution . They approximate the feedback history as having two distinct phases: (1) an early starburst during the spheroid formation (e.g., as a result of rapid galaxy mergers) and (2) a subsequent long-lasting and slowly declining injection of mass and energy from evolved low-mass stars. An energetic outward blastwave is initiated by the starburst (including the quasar/AGN phase) and is sustained by the long-lasting stellar feedback. Even for a small galactic spheroid such as the one in the Milky Way, this blastwave may heat up the CGM on scales beyond the present virial radius, thus the gas accretion from the IGM into the galactic halo could be largely reduced (see also  for similar results from 3-D cosmological structure formation simulations). The long-lasting stellar feedback initially drives a galactic spheroid wind (Fig. 4b). As the mass and energy injection decreases with time, the feedback may evolve into a subsonic and quasi-stable outflow. This feedback/CGM interplay scenario provides a natural explanation to various observed phenomena: