THE TEMPERATURE OF HOT GAS HALOS OF EARLY-TYPE GALAXIES

In ETGs, the hot gas comes from stellar mass losses produced by evolved stars, mainly during the red giant, asymptotic giant branch, and planetary nebula phases, and by SNe Ia that are the only ones observed in an old stellar population (e.g., Cappellaro et al. 1999). The first, more quiescent, type of losses originates from ejecta that initially have the velocity of the parent star, then individually interact with the mass lost from other stars or with the hot ISM and mix with it (Mathews 1990; Parriott & Bregman 2008).

For a galaxy of total stellar mass M_*, the evolution of the stellar mass-loss rate $\dot{M}_*(t)$ can be calculated using single-burst stellar population synthesis models (Maraston 2005) for Salpeter and Kroupa initial mass functions (IMFs), assuming, for example, solar abundance. Doing so at an age of 12 Gyr, a rate of $\dot{M}_* = B \times 10^{-11} \, L_B(L_{B,\odot })\;M_{\odot }$ yr⁻¹ is recovered, where L_B is the galactic B-band luminosity at an age of 12 Gyr, and B = 1.8 or B = 1.9 for the Salpeter or Kroupa IMF (see also Pellegrini 2011). This value is in reasonable agreement with the average derived for nine local ETGs from Infrared Space Observatory data (Athey et al. 2002) of $\dot{M}_*=7.8\times 10^{-12}\;L_B(L_{B,\odot })\;M_{\odot }$ yr⁻¹, an estimate based on individual observed values that vary by a factor of ∼10 and were attributed to different ages and metallicities.

The total mass-loss rate of a stellar population $\dot{M}$ is given by the sum $\dot{M}=\dot{M}_*+ \dot{M}_{\rm SN}$, where $\dot{M}_{\rm SN}$ is the rate of mass loss via SNe Ia events for the entire galaxy. $\dot{M}_{\rm SN}$ is given by $\dot{M}_{\rm SN} =M_{{\rm SN}}\, R_{\rm SN}$, where M_SN = 1.4 M_☉ is the mass ejected by one event and R_SN is the explosion rate. R_SN has been determined to be R_SN = 0.16(H₀/70)² × 10⁻¹² L_B(L_{B, ☉}) yr⁻¹ for local ETGs, where H₀ is the Hubble constant in units of km s⁻¹ Mpc⁻¹ (Cappellaro et al. 1999). More recent measurements of the observed rates of supernovae in the local universe (Li et al. 2011) give an SNe Ia rate in ETGs consistent with that of Cappellaro et al. (1999). For H₀ = 70 km s⁻¹ Mpc⁻¹, one obtains $\dot{M}_{\rm SN} =2.2 \times 10^{-13}$L_B(L_{B, ☉}) M_☉ yr⁻¹, which is ∼80 times smaller than the $\dot{M}_*$ derived above for an age of 12 Gyr; therefore, the main source of mass for the hot gas is provided by $\dot{M}_*$. A reasonable assumption is that the gas is shed by stars with a radial dependence that follows that of the stellar distribution, so that the density profile of the injected gas is ρ_gas(r)∝ρ_*(r), where ρ_*(r) is the stellar density profile. This assumption is adopted hereafter, and the characteristic temperatures presented below apply to a gas distribution following ρ_gas(r)∝ρ_*(r) (but see also Section 5).

The material lost by stars is ejected at a velocity of a few tens of km s⁻¹ and at a temperature of ≲ 10⁴ K (Parriott & Bregman 2008); it is subsequently heated to high, X-ray emitting temperatures by the thermalization of the stellar velocity dispersion as it collides with the mass lost from other stars or with ambient hot gas and is shocked. Another source of heating of stellar mass losses is provided by the thermalization of the kinetic energy of SNe Ia events. The internal energy given by these heating processes to the unit mass of injected gas is 3kT_inj/2μm_p (where k is the Boltzmann constant, m_p is the proton mass, and μm_p is the mean particle mass, with μ = 0.62 for solar abundance); T_inj is determined by the heating from thermalization of the motions of the gas-losing stars (T_star) and of the velocity of the SNe Ia ejecta (T_SN), and is written as (e.g., Gisler 1976; White & Chevalier 1983)

$$\begin{equation} T_{{\rm inj}}=T_{{\rm star}}+T_{{\rm SN}}={\dot{M}_* T_{*}+\dot{M}_{{\rm SN}} T_{{\rm ej}} \over \dot{M} }. \end{equation} \tag{ 1 }$$

Here, T_* is the equivalent temperature of the stellar motions (see below) and T_ej = 2μm_pE_SN/(3kM_SN) is the equivalent temperature of the kinetic energy E_SN of the SNe Ia ejecta, with E_SN = 10⁵¹ erg for one event (e.g., Larson 1974). T_ej can be calculated by assuming that a factor f of E_SN is turned into heat; f < 1, since radiative energy losses from expanding supernova remnants may be important, and values down to f = 0.1 have been adopted (Larson 1974; Chevalier 1974). A value of f = 0.85 should not be too wrong for the hot, diluted ISM of ETGs (e.g., Tang & Wang 2005). In this way, T_ej = (f/0.85)1.5 × 10⁹ K. By approximating $\dot{M} \simeq \dot{M}_*$ and using the estimates of Section 2.1 for $\dot{M}_{{\rm SN}}$ and $\dot{M}_*$ (for a Kroupa IMF) in the present epoch, one obtains T_SN ≃ 1.7(f/0.85) × 10⁷ K.

The injection temperature T_inj is then the sum of two parts: one (T_SN) is independent of the position within the galaxy where the gas is injected (i.e., independent of the radius in spherical symmetry) and is also constant from galaxy to galaxy (for a fixed IMF, age of the stellar population, and SNe Ia rate); however, for each ETG it can evolve with time if $\dot{M}_{{\rm SN}}$ and $\dot{M}_*$ evolve differently with time (Ciotti et al. 1991). The other part (T_star) is basically independent of time but has a radial dependence and changes with galaxy structure, i.e., with the total mass and its distribution. An average T_* is obtained by calculating the mass-weighted gas temperature gained by the thermalization of the stellar random motions 〈T_*〉:

$$\begin{equation} \langle T_{*}\rangle ={1\over k} {\mu m_p \over M_*} \int 4 \pi r^2 \rho _*(r) \sigma ^2 (r) \,dr, \end{equation} \tag{ 2 }$$

where σ(r) is the one-dimensional velocity dispersion of the stars. The integral term in Equation (2) is the same one that gives the kinetic energy associated with the stellar random motions [E_kin = 1.5∫4πr²ρ_*(r)σ²(r) dr] and enters the virial theorem for the stellar component; the mass-weighted temperature in Equation (2) is then often called "gas virial temperature." For a galaxy mass model made of stars and dark matter, characterized by ${\cal R}=M_h/M_*$, where M_h is the total dark mass and β = r_h/r_*, with r_h and r_* being the scale radii of the two mass distributions, 〈T_*〉 can be expressed using the central velocity dispersion σ_c as $\langle T_*\rangle =\mu m_p\, \sigma _c^2 \Omega ({\cal R},\beta)/k$ (e.g., Ciotti & Pellegrini 1992). The function Ω increases slightly for larger $\cal {R}$ and for lower β, that is, for a larger amount of gravitating mass or a higher mass concentration, but Ω is always <1, since σ(r) has in general a negative radial gradient (e.g., Section 4 and Figure 1). 〈T_*〉 is then proportional to σ²_c, and a simplified version of the virial temperature in Equation (2) that is often used is T_σ = μm_pσ²_c/k; T_σ, of course, overestimates the true 〈T_*〉.

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The mass-averaged injection temperature is finally given by

$$\begin{equation} \langle T_{{\rm inj}}\rangle = \langle T_{*}\rangle + 1.7(f/0.85)\times 10^7 \;{\rm K}, \end{equation} \tag{ 3 }$$

where, in general, the second term dominates, as is shown in Section 5 below.

In the case of mass losses flowing to the galactic center, the gas can be heated due to infall in the galactic potential and adiabatic compression; this process is sometimes referred to as "gravitational heating." The average change in gravitational energy per unit of gas mass flowing in through the galactic potential down to the galactic center is

$$\begin{equation} E_{\rm grav}^+ ={1\over M_*}\int ^{\infty }_{0}4\pi r^{2}\rho _*(r) [\phi (r) - \phi (0)] dr \end{equation} \tag{ 4 }$$

for galaxy mass distributions with a finite value of ϕ(0) (see also Ciotti et al. 1991). One can define the temperature equivalent to the energy in Equation (4) as 〈T⁺_grav〉 = 2μm_p E⁺_grav/3k. As 〈T_*〉, 〈T⁺_grav〉 is also ∝σ²_c and increases with larger ${\cal R}$ and smaller β, which, for inflowing gas, can be understood as a larger gas heating by compression during infall for a larger dark matter amount or for more concentrated dark matter.

Not all of E⁺_grav can be available for heating, though. If the inflow remains quasi-hydrostatic, then, according to the virial theorem, the energy radiated away is roughly one-half of the change in the gravitational potential energy, and the part available for the heating of the gas is the remaining half¹ (i.e., ∼0.5E⁺_grav). Actually, the energy available for heating will be much less than this. Inflows are caused by the radiative losses produced by the accumulation of stellar mass return, which makes the cooling time less than the galactic age; in the central regions, within a radius of ∼1 kpc, the cooling time can be as short as ≲ 10⁸ yr, even shorter than the infall time (e.g., Sarazin & White 1988; Pellegrini 2011). In these conditions, the gas departs from a slow inflow, becomes very dense and supersonic close to the center, and cools rapidly down to low temperatures, so that >0.5E⁺_grav is radiated away or goes into the kinetic energy of condensations (Sarazin & Ashe 1989). Furthermore, there is a possibility that not all the gas is hot when it reaches the galactic central regions if thermal instabilities develop and produce drop-outs from the flow; if gas cools and condenses out of the flow at large radii, then E⁺_grav can be much lower than in the definition above, and heating from infall in the gravitational potential is "lost" (Sarazin & Ashe 1989). In conclusion, without precise knowledge of how to compute E⁺_grav (which depends on the radius at which the injected gas drops below X-ray-emitting temperatures) and about what fraction of E⁺_grav is radiated or goes into the kinetic energy of the condensations, 〈T⁺_grav〉 remains a reference value. A more direct use can instead be made of the analogous temperature for escape 〈T⁻_grav〉, introduced in Section 3.

Because of the presence of a central MBH in ETGs, another potential source of gas heating could be provided by nuclear accretion. This subject has recently been studied intensely, through both observations and modeling, and it appears that the energy provided by accretion is of the order of that needed to cyclically offset the cooling of the inflowing gas in the central regions of gas-rich ETGs (e.g., Bîrzan et al. 2004; Million et al. 2010; Pellegrini et al. 2011). Therefore, the central MBH is believed to be a heat source that balances the radiative losses of the gas, acting mostly in the central cooling region. In gas-poor ETGs, the nuclear accretion energy is far lower due to the very small mass accretion rate, if present (Pellegrini et al. 2007), and the absorption of the energy output from accretion is likely not efficient. Given the role of the MBH outlined above, possible energy input from the MBH will not be considered as a source of global heating of gas.

In the left panel of Figure 2, the observed T's are compared with approximate estimates of the stellar temperature T_σ, T_inj, and the escape temperature 4T_σ (Section 3). The gas luminosity is also indicated with different colors; the L_X values are grouped in three ranges and have a roughly equal number of ETGs in each range. This grouping gives an indication of the gas flow status, based on previous works: a galactic wind leaving the galaxy at supersonic velocity has L_X < 10³⁸ erg s⁻¹ (e.g., Mathews & Baker 1971; Trinchieri et al. 2008); global subsonic outflows and partial winds can reach L_X ∼ 10⁴⁰ erg s⁻¹ (Ciotti et al. 1991; Pellegrini & Ciotti 1998); and a central inflow becomes increasingly more important in ETGs of increasingly large L_X. Magenta ETGs (10³⁸ erg s⁻¹ < L_X < 1.5 × 10³⁹ erg s⁻¹) should then host winds, subsonic outflows, and partial winds with a very small inflowing region of radius <100 pc; cyan ETGs (1.5 × 10³⁹ erg s⁻¹ < L_X < 1.2 × 10⁴⁰ erg s⁻¹) should host subsonic outflows and partial winds with an increasingly large inflowing region (of radius up to a few hundred pc); and black ETGs are hot gas-rich and mostly inflowing.³ The best fit found for X-ray bright ETGs is also shown in the left panel (O'Sullivan et al. 2003) and gives a good representation of the distribution of observed T's down to a range of low temperatures and gas contents never before explored. The slope of the fit (T∝σ^1.79_c) and that of the T_σ∝σ²_c relation are similar, with the fit being shallower; this could be because not all heating sources depend on σ²_c—see, e.g., the important SNe Ia contribution in T_inj, which produces a much flatter run of T_inj with σ_c (Figure 2). The fit was mostly based on gas-rich ETGs, whose T's show a trend with σ_c closer to that of T_σ (an aspect further addressed in Sections 5.2 and 6), while gas-poor ETGs depart most from it, since their T's change little for largely varying σ_c (as found by BKF; see Section 5.4).

The right panel of Figure 2 shows the stellar temperature 〈T_*〉, injection temperature 〈T_inj〉, escape temperature 〈T⁻_grav〉, and the characteristic temperature for the slowly outflowing gas 〈T^sub_esc〉, calculated for a set of representative galaxy mass models (Section 4). At any fixed σ_c and Sérsic index n, 〈T_*〉, 〈T⁻_grav〉, and 〈T^sub_esc〉 are larger at larger galaxy masses (${\cal R}$) and mass concentrations (smaller β).⁴ The dashed lines represent a reasonable upper limit to the values of each characteristic temperature, since they correspond to the most massive model ETGs with the most concentrated dark matter allowed for by recent studies (Section 4). The 〈T⁻_grav〉 curves lie below the simple approximation of the escape temperature given by 4T_σ; 〈T⁻_grav〉 = 4.8〈T_*〉 for ${\cal R}=3$, and ≃ 5.2〈T_*〉 for the three cases with ${\cal R}=5$.

As expected, all 〈T_*〉 curves lie below T_σ, which overestimates the kinetic energy associated with the stellar random motions (Section 2.2). Note that, according to the virial theorem, 〈T_*〉 is independent of orbital anisotropy, which just redistributes the stellar heating differently within a galaxy; the presence of ordered rotation in stellar motions instead requires a more careful consideration. For any fixed galaxy mass model, this rotation would leave the total stellar heating unchanged or lower it, depending on whether the entire stellar streaming motion or just a fraction of it is converted into heat (Ciotti & Pellegrini 1996). In the worst case that the stellar rotational motion is not thermalized at all, and the galaxy is a flat isotropic rotator, 〈T_*〉 in Figure 2 should be an overestimate of ∼30% of the temperature corresponding to stellar heating (Ciotti & Pellegrini 1996); the possible reduction of 〈T_*〉 should be lower than this, because the massive ETGs in Figure 2 are less flattened and more pressure-supported systems (e.g., Emsellem et al. 2011).

All observed T's are located above 〈T_*〉; thus, additional heating with respect to the thermalization of stellar kinetic energy is needed, as noticed previously using T_σ (e.g., Davis & White 1996; BKF). The gas can retain the memory of its injection temperature and have additional infall heating, as examined in Sections 5.2 and 5.3.

Finally, the values of 〈T_inj〉 for f = 0.85 are by far the largest temperatures in Figure 2, larger than 〈T⁻_grav〉 up to σ_c ∼ 250 km s⁻¹; therefore, SNe Ia should cause the escape of gas from all ETGs up to this σ_c, since the gas at every time is injected with an energy larger than required to leave the galaxy potential. This expectation is fulfilled by all ETGs with σ_c ≲ 200 km s⁻¹: their X-ray properties (a low L_X and T's on the order of 〈T^sub_esc〉) agree well with what is expected if outflows are important in them. This result has been suggested previously based on the low observed L_X; now, for the first time, it can be confirmed based on the observed T values. At σ_c > 200 km s⁻¹, instead, ETGs may have L_X far larger than expected for outflows (black symbols), and most ETGs where likely outflows are important (magenta or cyan symbols) have T's much lower than 〈T^sub_esc〉; these findings are discussed in Section 5.4.

I examine here the possibility that the additional heating with respect to the thermalization of the stellar kinetic energy is provided by infall heating and SNe Ia. Davis & White (1996) assumed that the hot gas is flowing in all ETGs and suggested that the observed temperatures are larger than T_σ because the luminous parts of ETGs are embedded in dark matter halos that are dynamically hotter than the stars; i.e., a form of "gravitational potential" way of heating the gas was invoked. This form can consist of an effect of the dark halo on stellar motions, which are then thermalized, or directly on the gas during infall (e.g., via E⁺_grav). The first possibility is excluded by the 〈T_*〉 curves in Figure 2 that are always lower than T_σ and that, through the Jeans equations, include the effect of a massive dark halo, chosen to be consistent with the current knowledge of the ETGs' structure. In the second possibility, heating from gas infall, E⁺_grav is potentially an important source of heating that increases with the amount and concentration of dark matter. This can be judged from Figure 2 after considering that 〈T⁻_grav〉 ∼ 5〈T_*〉 and that the temperature possibly attainable from infall was estimated to be T_infl ≲ 〈T⁻_grav〉 (see the end of Section 3). Note that T_infl (if it behaves like 〈T⁺_grav〉) could be ∝σ²_c, a trend close to that shown by the T's of gas-rich ETGs (BKF; see also Section 6).

Inflowing ETGs can also benefit from the SNe Ia energy input; for the unit mass of injected gas, this is written as $E_{{\rm SN}}^{{\rm tot}}=R_{{\rm SN}}E_{{\rm SN}}/ \dot{M}_*$. Both E⁺_grav and E^tot_SN contribute to the required additional thermal energy with respect to that gained from the stellar random motions, i.e., to ΔE_th = 3k(T − 〈T_*〉)/2μm_p. E⁺_grav and E^tot_SN in large part can be radiated in gas-rich ETGs, but they seem to far exceed the required ΔE_th. For example, for the highest L_X of Figure 2, ΔE_th ∼ (1–2) × 10⁴⁸ erg M⁻¹_☉ (i.e., ∼ 0.2–0.4 keV), when adopting an average galaxy mass model such as that represented by the thick black line in Figure 2. The energy spent in radiation can be estimated, in a stationary situation, as $L_X/\dot{M}_*$ (per unit of injected gas mass); using the L_X from BKF and deriving $\dot{M}_*$ as in Section 2.1 for the same distances in BKF and galactic B-magnitudes given by Hyperleda, for gas-rich ETGs one finds that $L_X/\dot{M}_*$ ranges between (0.5–3.4) × 10⁴⁸ erg M⁻¹_☉. The energy available is far larger than the sum of ΔE_th and $L_X/\dot{M}_*$: E^tot_SN = 7.3 × 10⁴⁸(f/0.85) erg M⁻¹_☉, and 0.5E⁺_grav ranges from 4 × 10⁴⁸ erg M⁻¹_☉ (σ_c ∼ 220 km s⁻¹) to 8 × 10⁴⁸ erg M⁻¹_☉ (σ_c ∼ 300 km s⁻¹) in the mass model with the thick black line in Figure 2. These results are detailed in Figure 3, where the values of ΔE_th and $L_X/\dot{M}_*$ for each galaxy are shown, together with various combinations of E⁺_grav and E^tot_SN. In conclusion, additional energy input for the gas, to account for the observed T's of gas-rich ETGs, seems to be available in a sufficient amount, even if f were to be <0.85.

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The gas is not mostly inflowing in all ETGs, while it is hotter than 〈T_*〉 in all of them. When in outflow, the radiative losses are far smaller, but energy is spent in extracting the gas from the galaxy and giving it a bulk velocity. I discuss here the possibility that heating from the SNe Ia energy input accounts for the observed ΔE_th of ETGs where outflows are likely important (those with low/medium L_X—the magenta and cyan symbols in Figure 2).

I assume that the SNe Ia energy is used for the uplift of gas and the kinetic energy with which gas escapes from the galaxy, and, neglecting radiative losses, that the remaining part can account for the observed ΔE_th. Then the energy balance per unit mass of injected gas is E^tot_SN = E_grav⁻ + E_out + ΔE_th, where ΔE_th is the same as in Section 5.2 and E_out = v²_out/2 is the mass-averaged kinetic energy of the escaping material per unit of gas mass. Figure 4 shows ΔE_th derived from this balance, with v_out = c_s, and c_s calculated for γ = 5/3 and kT = 0.3 keV, a temperature on the order of that observed for ETGs that are likely in outflow (Figure 2). Adopting v_out ∼ c_s (independent of σ_c) produces E_out at the upper end of those expected,⁵ and then the estimate of ΔE_th may be biased low. With v_out ∼ c_s, E_out is just ∼0.1(f/0.85)E^tot_SN. The energy needed for gas extraction, E⁻_grav, for the same galaxy mass model used in Section 5.2, varies instead from ∼1/3(f/0.85)E^tot_SN for σ_c = 150 km s⁻¹ to ∼E^tot_SN for σ_c = 250 km s⁻¹; this explains the strong dependence of the predicted ΔE_th on σ_c in Figure 4. It is clear from this figure that, for f = 0.85, SNe Ia can account for the required heating in all ETGs with low/medium L_X; for f = 0.35, however, the temperature increase would fall short of that required for all ETGs. In Figure 4, the energy losses due to radiation are also shown; their small size supports the hypothesis that in most cases they do not significantly affect the energy budget of the gas.

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Given the flat distribution of the observed points in Figure 4 and the steep behavior of the curves predicting ΔE_th, the value of f required to account for the observed ΔE_th increases with σ_c. In particular, the value f ∼ 0.85 that is required at high σ_c would produce an expected ΔE_th at low σ_c that is larger than observed. A possible solution could reside in the efficiency of the SNe Ia energy-mixing process. In massive, gas-rich ETGs, SNe Ia bubbles should disrupt and share their energy with the local gas within ∼3 × 10⁶ yr (Mathews 1990); for a Milky Way–size bulge in a global wind, however, three-dimensional hydrodynamical simulations of discrete heating from SNe Ia suggest a non-uniform thermalization of SNe Ia energy, with overheated gas from an SNe Ia explosion at the bulge center that is advected outward, carrying a large fraction of SNe Ia energy with it (Tang et al. 2009). For subsonic outflows, the mixing is expected to be more local and more complete (Lu & Wang 2011). The magenta and cyan ETGs in Figure 4 have gas densities and luminosities larger than those considered by Tang et al. (2009); however, if a discrete heating effect were still present at σ_c < 200 km s⁻¹, it could qualitatively explain a lower f for these galaxies. It is also possible that 〈T_*〉 has been overestimated (and then the observed ΔE_th underestimated) at low σ_c if these ETGs are less pressure-supported systems (e.g., Emsellem et al. 2011) and stellar rotational streaming is not entirely thermalized (Section 5.1). Another possible explanation could be that ETGs with σ_c > 200 km s⁻¹ are less outflow-dominated than those at lower σ_c (though this is not supported solely based on L_X, since magenta ETGs are found over the entire σ_c range in Figure 4), so that their E⁻_grav would be lower than assumed by the curves in Figure 4, and more SNe Ia energy would be available for heating. In this way, f could have a value <0.85 and possibly be similar for all ETGs.

Finally, a comparison of Figures 3 and 4 shows that the average ΔE_th is slightly larger for the X-ray brightest ETGs (for which it ranges between 0.1 and 0.5 keV) than for the X-ray faintest ones (0–0.3 keV); moreover, while the ΔE_th in Figure 3 can be explained even with f < 0.85, it is required that f ∼ 0.85 for the X-ray faintest ETGs with the largest σ_c in Figure 4. Both facts are the consequence of the large fraction of SNe Ia energy input that is used in gas extraction where outflows dominate, while all SNe Ia energy remains within the galaxies where inflow dominates.

In conclusion, even for ETGs with low/medium L_X, a fundamental X-ray property such as T can be accounted for by simple arguments, simply based on realistic galaxy mass models and reasonable SNe Ia heating capabilities. There may, however, be more energy available for the gas in ETGs with σ_c < 200 km s⁻¹ than can be accounted for by the present simple scenario.

The observed X-ray properties (low L_X and T ∼ 〈T^sub_esc〉) and the energy budget of the gas (e.g., 〈T⁻_grav〉 versus 〈T_inj〉) for ETGs with σ_c ≲ 200 km s⁻¹ are all consistent with the expectations for outflows; the large L_X and 〈T⁻_grav〉 larger than 〈T_inj〉 of ETGs with σ_c > 250 km s⁻¹ agree with gas that is mostly inflowing. For 200 < σ_c(km s⁻¹) < 250, however, ETGs show very different L_X and T, whose values seem unrelated to 〈T^sub_esc〉 and to the relative size of 〈T⁻_grav〉 and 〈T_inj〉 (Figure 2). For example, for f = 0.85, 〈T_inj〉 exceeds 〈T⁻_grav〉, but most T values lie well below 〈T^sub_esc〉, and even high L_X values (incompatible with outflows) are common. One primary explanation could be that f < 0.85; for example, for f = 0.35, 〈T_inj〉 becomes lower than 〈T⁻_grav〉 at σ_c ∼ 180 km s⁻¹ (Figure 2). Four ETGs with σ_c > 200 km s⁻¹ and very low L_X (Figure 2, magenta symbols), though, require that f be > 0.35 for them, and then that f vary from galaxy to galaxy, or that their gas be removed by other processes such as an active galactic nucleus outburst (e.g., Machacek et al. 2006; Ciotti et al. 2010), a merging, or an interaction (Read & Ponman 1998; Sansom et al. 2006; Brassington et al. 2007). An event similar to the latter two is unlikely in the recent past for three of these ETGs (NGC1023, NGC3115, NGC3379), which are very regular in their stellar morphological and kinematic properties, but is possible in the other ETG (NGC4621) that hosts a counter-rotating core (Wernli et al. 2002).

A second explanation could be that 〈T_inj〉 exceeds 〈T⁻_grav〉 only in the present epoch: while 〈T_*〉 and 〈T⁻_grav〉 are independent of time, T_SN may have been lower in the past (Equation (1)), to the point that 〈T_inj〉 may have been lower than 〈T⁻_grav〉 for more ETGs than in Figure 2 (which represents a snapshot of the present epoch). The gas then could have accumulated and radiative losses become important, even for the gas injected in later epochs. In fact, the population synthesis models of Section 2 predict that $\dot{M}_*$ was larger at early times (e.g., by ∼6 times at an age of 3 Gyr); to keep T_SN high in the past, then, from Equation (1), $\dot{M}_{{\rm SN}}$ must decrease with time t at a rate similar to or steeper than that of the stellar mass losses ($\dot{M}_*\propto t^{-1.3}$; Ciotti et al. 1991). Recent observational estimates indicate instead an SNe Ia rate that decays close to t⁻¹ (Maoz et al. 2011; Sharon et al. 2010); thus, T_SN and 〈T_inj〉 should be increasing with time, reaching the values of Figure 2 in the present epoch. Then a "cooling effect of the past" would explain a moderate or high L_X even where 〈T_inj〉 exceeds 〈T⁻_grav〉 in Figure 2.

Both explanations, f < 0.85 and/or a lower 〈T_inj〉 in the past, can account for the lack of a widespread presence of outflows for 200 < σ_c(km s⁻¹) < 250. Both, though, require a mechanism different from the SNe Ia energy input to cause degassing in some low L_X ETGs in the same σ_c range.

Another possibility is that partial winds become common for σ_c > 200 km s⁻¹: these ETGs host an inner inflow and an outer outflow (e.g., MacDonald & Bailey 1981), with variations in galactic structure causing different sizes of the inflowing regions and different L_X (Pellegrini & Ciotti 1998). This possibility holds even for f = 0.85 and for both kinds of time evolution of 〈T_inj〉; in fact, if the flow is decoupled, ETGs may host a central inflow even if 〈T_inj〉 is larger than 〈T⁻_grav〉. Similarly, the observed T's can be lower than 〈T^sub_esc〉 in ETGs where the outflow is only external; in this case, the observed T's may be lowered also by radiative losses in the central inflowing region.